id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
0809.0386
|
# MULTIPLICATIVE DIOPHANTINE EXPONENTS OF HYPERPLANES
Yuqing Zhang Brandeis University, Waltham MA 02454-9110 yqzhang@brandeis.edu
###### Abstract.
We study multiplicative Diophantine approximation property of vectors and
compute Diophantine exponents of hyperplanes via dynamics.
## 1\. Introduction
For a vector ${\bf y}=(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}$ its Diophantine
exponent is defined by
$\omega({\bf y})=\textrm{sup}\\{v|\textrm{ }\exists\infty\textrm{ many
}{\bf{q}}=(q_{1},q_{2},\ldots,q_{n})\in{\mathbb{Z}}^{n}\textrm{ with }$
$|{\bf{q}}{\bf
y}+p|=|q_{1}y_{1}+q_{2}y_{2}+\ldots+q_{n}y_{n}+p|<\|{\bf{q}}\|^{-v}\textrm{
for some }p\in{\mathbb{Z}}\\}$ (1.1)
In this standard definition the size of the vector ${\bf{q}}$ is measured by
its supreme norm and all its entries with smaller absolute values than the
norm are simply ignored. A more elaborate way of measuring ${\bf{q}}$ is by
$\prod_{\times}({\bf{q}})=\prod_{i=1\ q_{i}\neq 0}^{n}|q_{i}|$
And we have the multiplicative version of Diophantine exponent for ${\bf y}$:
$\omega^{\times}({\bf y})=\textrm{sup}\\{v|\textrm{ }\exists\infty\textrm{
many }{\bf{q}}\in{\mathbb{Z}}^{n}\textrm{ with }|{\bf{q}}{\bf
y}+p|<{\prod_{\times}({\bf{q}})}^{-v/n}\textrm{ for some }p\in{\mathbb{Z}}\\}$
(1.2)
Besides, ${\bf y}$ can be viewed as a column vector and be equipped with the
$\sigma$ version of Diophantine exponent:
$\sigma({\bf y})=\textrm{sup}\Big{\\{}v|\textrm{ }\exists\infty\textrm{ many
}q\in{\mathbb{Z}}\textrm{ with }\left\|\begin{array}[]{c}qy_{1}+p_{1}\\\
\vdots\\\ qy_{n}+p_{n}\\\ \end{array}\right\|<|q|^{-v}\textrm{ for some }{\bf
p}\in{\mathbb{Z}}^{n}\ \Big{\\}}$
We further define the multiplicative Diophantine exponent
$\omega^{\times}(\mu)$ of a Borel measure $\mu$ to be the $\mu$-essential
supreme of the $\omega^{\times}$ function, that is,
$\omega^{\times}(\mu)=\textrm{sup}\\{v|\textrm{ }\mu\\{{\bf y}|\textrm{
}\omega^{\times}({\bf y})>v\\}>0\\}$ (1.3)
And for a smooth submanifold M of ${\mathbb{R}}^{n}$ with measure class of its
Riemannian volume denoted by $\mu$, we set
$\omega^{\times}(M)=\omega^{\times}(\mu)$. $\omega(\mu)$ and $\omega(M)$ are
defined in an analogous way.
From these definitions a couple of inequalities are derived:
$\omega^{\times}({\bf y})\geqq\omega({\bf y})$ (1.4)
$\omega^{\times}(M)\geqq\omega(M)$ (1.5)
Suppose $L$ is an affine subspace of ${\mathbb{R}}^{n}$. $\omega(L)$ has been
studied to great lengths, both by elementary methods as in [J] and by
quantitative nondivergence by D. Kleinbock in [K1][K3].
By comparison the multiplicative exponent of $L$ is inherently more
complicated. Here advanced mathematical tools appear to be much more
desirable, if not indispensable.In this paper we apply nondivergence which has
been developed and strengthened in [K1][K2][K3] to find out multiplicative
exponents of hyperplanes and their nondegenerate submanifolds.
One of the major theorems we are to establish is:
###### Theorem 1.1.
Let $L$ be a hyperplane of ${\mathbb{R}}^{n}$ parameterized by
$(x_{1},\ldots,x_{n-1},a_{1}x_{1}+\ldots+a_{s-1}x_{s-1}+b)$ with $a_{i}\neq
0,i=1,\ldots,s-1$,$s\leqslant n$ and $M$ a non-degenerate submanifold in $L$.
We have
$\textrm{
}\omega^{\times}(L)=\omega^{\times}(M)=max\big{\\{}n,\dfrac{n}{s}\sigma(a_{1},\ldots,a_{s-1},b)\big{\\}}$
## 2\. Quantitative nondivergence and proof of Theorem 1.1
###### Definition 2.1.
$W^{\times}_{v}=\\{{\bf y}\in R^{n}\textrm{ }|\omega^{\times}({\bf y})\geqq
v\\}$
###### Definition 2.2.
$\Omega_{n+1}=\operatorname{SL}(n+1,{\mathbb{R}})\diagup\operatorname{SL}(n+1,{\mathbb{Z}})$
$\Omega_{n+1}$ is non-compact, and
$\Omega_{n+1}=\bigcup_{\epsilon>0}K_{\epsilon}$ (2.1)
where $K_{\epsilon}=\\{\Lambda\in\Omega_{n+1}|\textrm{
}\|v\|\geq\epsilon\textrm{ for all nonzero }v\in\Lambda\\}$. Each
$K_{\epsilon}$ is compact.
We associate ${\bf y}$ with a $(n+1)\times(n+1)$ matrix
$u_{\bf y}=\left(\begin{array}[]{cc}1&{\bf y}\\\ 0&I_{n}\end{array}\right)$
And for $\textbf{t}=(t_{1},\ldots,t_{n})$ with $t_{i}\geqslant 0$, set
$t=\sum_{i=1}^{n}t_{i}$,
$g_{\textbf{t}}=diag(e^{t},e^{-t_{1}},\ldots,e^{-t_{n}})$
The following lemma will enable us to study multiplicative diophantine
approximation via $g_{\textbf{t}}$ action. It is essentially the same as Lemma
2.1 of [K2] and a proof can be found there.
###### Lemma 2.3.
Suppose we are given a positive integer $k$ and a set $E$ of $(x,{\bf
z})\in{\mathbb{R}}^{n+1}$ which is discrete and homogeneous with respect to
positive integers. $z_{i}\geqslant 1$ for $i_{1},\ldots,i_{k}$ and $z_{i}=0$
for the rest $i$. Take $v>n$ and $c_{k}=\dfrac{v-n}{kv+n}$,then the following
are equivalent:
1. (1)
$\exists(x,{\bf z})\in E$ with arbitrarily large $\|{\bf z}\|$ such that
$|x|\leq{\prod_{\times}({\bf z})}^{-v/n}$
2. (2)
$\exists$ an unbounded set of $\textbf{t}\in{\mathbb{R}}_{+}^{n}$ such that
for some $(x,{\bf z})\in E\backslash\\{0\\}$ one has
max$(e^{t}|x|,e^{-t_{i}}|z_{i}|)\leqq e^{-c_{k}t}$
Accordingly $Z^{n+1}$ is decomposed as
$\bigcup_{k=0}^{n}{\mathbb{Z}}^{n+1}_{k}$, where
${\mathbb{Z}}^{n+1}_{k}=\big{\\{}(p,q_{1},q_{2},\ldots,q_{n})\in{\mathbb{Z}}^{n+1}\big{|}\textrm{
exactly k entries of }(q_{1},q_{2},\ldots,q_{n})\textrm{ nonzero}\big{\\}}$
(2.2)
Take $v>n$, ${\bf y}\in{\mathbb{R}}^{n}$ and $E=\\{(|{\bf{q}}{\bf
y}+p|,{\bf{q}})\textrm{ }|\textrm{
}p\in{\mathbb{R}},{\bf{q}}\in{\mathbb{R}}^{n}\\}$ we see
(2.3.1) is equivalent to $\omega^{\times}({\bf y})\geqq v$ and (2.3.2)
equivalent to
$g_{\textbf{t}}u_{\bf y}{\mathbb{Z}}^{n+1}_{k}\textrm{ contains at least one
nonzero vector with norm }\leq e^{-c_{k}t}$ $\textrm{ for an unbounded set of
}\textbf{t}\in{\mathbb{R}}_{+}^{n+1}$ (2.3)
For any fixed $v>n$, $c_{1}>c_{2}>\ldots>c_{n}$.
$c_{k}=\dfrac{v-n}{kv+n}\Leftrightarrow v=\dfrac{n+nc_{k}}{1-kc_{k}}$
If $\gamma_{k}({\bf y})=$ sup {$c_{k}|\textrm{ }(2.3)\textrm{ holds }\\}$then
by preceding lemma
$\omega^{\times}({\bf y})=max_{\begin{subarray}{c}1\leqslant k\leqslant
n\end{subarray}}\dfrac{n+n\gamma_{k}({\bf y})}{1-k\gamma_{k}({\bf y})}$ (2.4)
Suppose $\lambda$ is a measure on ${\mathbb{R}}^{n}$, and $v\geq n$ , by
definition $\omega^{\times}(\lambda)\leq v$ iff $\lambda(W^{\times}_{u})=0$
for any $u>v$.By Borel-Cantelli Lemma, a sufficent condition for
$\omega^{\times}(\lambda)\leq v$ is
$\sum_{t=1}^{\infty}\lambda(\\{{\bf y}|g_{\textbf{t}}u_{\bf
y}{\mathbb{Z}}^{n+1}_{k}\textrm{ contains at least one nonzero vector with
norm}\leq e^{-d_{k}t}\\})<\infty$ $\forall d_{k}>c_{k},\quad 1\leq k\leq n$
(2.5)
(2.6) provides one way of determining the upper bounds of
$\omega^{\times}(\lambda)$.To make it more explicit,quantitative nondivergence
is needed.
###### Lemma 2.4.
Let $k$, $N$ $\in{\mathbb{N}}$ and $C,D,\alpha,\rho>0$ and suppose we are
given an $N$-Besicovitch metric space $X$, a ball $B=B(x_{0},r_{0})\subset X$,
a measure $\mu$ which is $D$-Federer on $\tilde{B}=B(x_{0},3^{k}r_{0})$ and a
map $h$: $\tilde{B}\rightarrow\operatorname{GL}_{k}({\mathbb{R}})$. Assume the
following two conditions hold:
1. (1)
$\forall\quad\Gamma\subset{\mathbb{Z}}^{k}$, the function
$x\rightarrow\|h(x)\Gamma\|$ is $(C,\alpha)$-good on $\tilde{B}$ with respect
to $\mu$;
2. (2)
$\forall\quad\Gamma\subset{\mathbb{Z}}^{k}$,
$\|h(\cdot)\Gamma\|_{\mu,B}\geq\rho^{rk(\Gamma)}$
Then for any positive $\epsilon\leq\rho$ one has
$\mu(\\{x\in B|\textrm{ }h(x){\mathbb{Z}}^{k}\notin K_{\epsilon}\\})\leq
kC(ND^{2})^{k}(\dfrac{\epsilon}{\rho})^{\alpha}\mu(B)$ (2.6)
Lemma 2.4 implies the following proposition:
###### Proposition 2.5.
Let $X$ be a Besicovitch metric space, $B=B(x,r)\subset X$, $\mu$ a measure
which is D-Federer on $\tilde{B}=B(x,3^{n+1}r)$ for some $D>0$ and $f$ a
continuous map from $\tilde{B}$ to ${\mathbb{R}}^{n}$. Take $v>n$,
$c_{k}=\dfrac{v-n}{kv+n},k=1,\ldots,n$ and assume that
1. (1)
$\exists c,\alpha>0$ such that all the functions
$x\rightarrow\|g_{t}u_{f(x)}\Gamma\|$, $\Gamma\subset{\mathbb{Z}}^{n+1}$ are
$(c,\alpha)$\- good on $\tilde{B}$ with respect to $\mu$
2. (2)
for any $d_{k}>c_{k}$, $\exists T=T(d_{k})>0$ such that for any $t\geqq T$ and
any $\Gamma\subset{\mathbb{Z}}^{n+1}$ one has
$\|g_{\textbf{t}}u_{f(\cdot)}\Gamma\|_{\mu,B}\geq e^{-rk(\Gamma)d_{k}t}$
Then $\omega^{\times}(f_{*}(\mu|_{B}))\leq v$.
###### Proof.
We set $\lambda=f_{*}(\mu|B)$. (2.4.1) is the same as (2.5.1). (2.5.2) implies
(2.4.2) for any $t>T(\dfrac{c_{k}+d_{k}}{2})$. Hence by lemma 2.4,
$\lambda(\\{{\bf y}|g_{\textbf{t}}u_{\bf y}{\mathbb{Z}}^{n+1}_{k}\textrm{
contains at least one nonzero vector with norm}\leq
e^{-d_{k}t}\\})\leq\lambda(\\{{\bf y}|\textrm{ }g_{t}u_{\bf
y}{\mathbb{Z}}^{n+1}\notin K_{e^{-d_{k}t}}\\})=\mu(\\{x\in B|\textrm{
}h(x){\mathbb{Z}}^{n+1}\notin K_{e^{-d_{k}t}}\\}$ $\leq const\cdot
e^{-\alpha\frac{d_{k}-c_{k}}{2}t}$ for all but finitely many
$t\in{\mathbb{N}}$. Therefore (2.5) follows. ∎
Suppose ${\mathbb{R}}^{n+1}$ has standard basis $e_{1},\ldots,e_{n+1}$ and we
extend its Euclidiean structure to $\bigwedge^{j}({\mathbb{R}}^{n+1})$, then
for all index sets $I\subseteq\\{1,2,\ldots,{n+1}\\}$,
$\\{e_{I}\big{|}e_{I}=e_{i_{1}}\wedge\ldots\wedge e_{i_{j}},\sharp I=j\\}$
form an orthogonal basis of $\bigwedge^{j}({\mathbb{R}}^{n+1})$. We identify
$\Gamma$, subgroup of ${\mathbb{Z}}^{n+1}$ of rank $j$ with ${\bf
w}\in\bigwedge^{j}({\mathbb{R}}^{n+1})$ and reproduce the calculations done in
[K3].
$g_{t}u_{\bf y}{\bf
w}=\sum_{\begin{subarray}{c}I\end{subarray}}e^{-\sum_{\begin{subarray}{c}i\in
I\end{subarray}}t_{i}}\langle e_{I},{\bf w}\rangle
e_{I}+\sum_{\begin{subarray}{c}J\end{subarray}}e^{t-\sum_{\begin{subarray}{c}i\in
J\end{subarray}}t_{i}}(\sum_{i=0}^{n}\langle e_{i}\wedge e_{J},{\bf w}\rangle
y_{i})e_{0}\wedge e_{J}$
$\widetilde{f}=(1,f_{1},\ldots,f_{n})=(1,g_{1},\ldots,g_{s})R$
for $(1,g_{1},\ldots,g_{s})$ linearly independent and $R$ a $(s+1)\times(n+1)$
matrix, set
$C_{J}({\bf w})=\langle e_{i}\wedge e_{J},{\bf w}\rangle y_{i})e_{J},\sharp
J=j-1$
Up to some constant
$\big{\|}g_{t}u_{f}{\bf w}\big{\|}=max\Big{(}e^{-\sum_{\begin{subarray}{c}i\in
I\end{subarray}}t_{i}}\big{\|}\langle e_{I},{\bf w}\rangle\big{\|},\quad
e^{t-\sum_{\begin{subarray}{c}i\in J\end{subarray}}t_{i}}\big{\|}RC_{J}({\bf
w})\big{\|}\Big{)}$ (2.7)
(2.5.2) can be rewritten as
$\forall d_{k}>c_{k},\exists T=T(d_{k})>0\textrm{ such that for any }t\geqq
T\textrm{ and any }\Gamma\in{\mathbb{Z}}^{n+1}\textrm{ of rank j one has}$
$max\Big{(}e^{-\sum_{\begin{subarray}{c}i\in
I\end{subarray}}t_{i}}\big{\|}\langle e_{I},{\bf w}\rangle\big{\|},\quad
e^{t-\sum_{\begin{subarray}{c}i\in J\end{subarray}}t_{i}}\big{\|}RC_{J}({\bf
w})\big{\|}\Big{)}\geqq e^{-jd_{k}t},1\leqq j\leqq n$ (2.8)
Note that $j$ and $k$ are two independent variables: $j$ denotes rank of
subgroup of ${\mathbb{Z}}^{n+1}$ and $k$ the number of nonzero entries in the
vectors used for approximation.
For hyperplane $L$ parameterized in Theorem 1.1,
$R=\left(I_{n}\begin{array}[]{c}b\\\ a_{1}\\\ \vdots\\\ a_{s-1}\\\ 0\\\
\vdots\\\ 0\\\ \end{array}\right)$ (2.9)
$a_{i}\neq 0,1\leq i\leq s-1$
$f=(1,x_{1},\ldots,x_{n-1},a_{1}x_{1}+\ldots a_{s-1}x_{s-1}+b),a_{i}\neq
0,1\leqslant i\leqslant s-1$ (2.10)
Thanks to Lemma 4.6 of [K2] we know that $\|RC_{J}({\bf w})\|\geq 1$ for
$j>1$, so (2.8) is automatically fulfilled for such subgroups of $Z^{n+1}$ .We
only need to check for subgroups of rank 1, or vectors, for a negation of
(2.8).
${\bf w}=(p_{0},p_{1},\ldots,p_{n})\in{\mathbb{Z}}^{n+1}$ $\big{\|}RC_{J}({\bf
w})\big{\|}=\left\|\begin{array}[]{c}p_{0}+bp_{n}\\\ p_{1}+a_{1}p_{n}\\\
p_{2}+a_{2}p_{n}\\\ \vdots\\\ p_{s-1}+a_{s-1}p_{n}\\\ p_{s}\\\ \vdots\\\
p_{n-1}\\\ \end{array}\right\|$
To avoid $\|RC_{J}({\bf w})\|\geq 1$,$p_{s},p_{s+1},\ldots,p_{n-1}$ must all
be zero. Since $a_{i}\neq 0,1\leqslant i\leqslant s-1$,$p_{1},\ldots,p_{s-1}$
and $p_{n}$ must all be nonzero. By previous notation ${\bf
w}=(p_{0},p_{1},\ldots,p_{n})\in{\mathbb{Z}}_{s}^{n+1}$. And
$\big{\|}RC_{J}({\bf w})\big{\|}=\left\|\begin{array}[]{c}p_{0}+bp_{n}\\\
p_{1}+a_{1}p_{n}\\\ p_{2}+a_{2}p_{n}\\\ \vdots\\\
p_{s-1}+a_{s-1}p_{n}\end{array}\right\|$
The above observations coupled with (2.8) supply a handy tool for establishing
upper bounds of multiplicative exponents of hyperplanes:
###### Proposition 2.6.
Let $X$ be a Besicovitch metric space, $B=B(x,r)\subset X$, $\mu$ a measure
which is D-Federer on $\tilde{B}=B(x,3^{n+1}r)$ for some $D>0$ and $f$ a
continuous map from $\tilde{B}$ to ${\mathbb{R}}^{n}$ defined in (2.10). Take
$v>n$, $c_{s}=\dfrac{v-n}{sv+n}$ and assume that
1. (1)
$\exists c,\alpha>0$ such that all the functions
$x\rightarrow\|g_{t}u_{f(x)}\Gamma\|$, $\Gamma\subset{\mathbb{Z}}^{n+1}$ are
$(c,\alpha)$\- good on $\tilde{B}$ with respect to $\mu$
2. (2)
for any $d_{s}>c_{s}$, $\exists T=T(d_{s})>0$ such that for any $t\geqq T$ and
any ${\bf w}\in{\mathbb{Z}}^{n+1}_{s}$ one has
$max\Big{(}e^{-t_{i}}\big{\|}\langle e_{i},{\bf w}\rangle\big{\|},\quad
e^{t}\big{\|}RC_{J}({\bf w})\big{\|}\Big{)}\geqq e^{-d_{s}t}$
Then $\omega^{\times}(f_{*}(\mu|_{B}))\leq v$.
###### Lemma 2.7.
Let $\mu$ be a measure on a set $B$, take $v>n$ and $c_{s}=\frac{v-n}{sv+n}$
and let $f$ as in (2.10) be a map such that $\mathrm{(2.6.2)}$ does not hold.
Then $f(B\bigcap supp\textrm{ }\mu)\subset W_{u}^{\times}$ for some $u>v$.
###### Proof.
The assumption of the lemma says that $\exists$ an unbounded sequence
$\mathbf{t}_{i}$ and a sequence of ${\bf w}\in Z^{n+1}_{s}$ such that $\forall
x\in B\bigcap supp\textrm{ }\mu\quad$
$\|g_{\mathbf{ti}}u_{f(x)}{\bf w}\|<e^{-d_{s}t_{i}}$.
Hence $g_{\mathbf{ti}}u_{f(x)}{\mathbb{Z}}^{n+1}_{s}$ has at least one nonzero
vector with norm $<e^{-d_{s}t_{i}}$ $\Rightarrow\gamma_{s}(f(x))\geqq d_{s}$.
By (2.4) the lemma holds. ∎
###### Theorem 2.1.
Let $\mu$ be a Federer measure on a Besicovitch metric space $X$,$L$ an affine
subspace of ${\mathbb{R}}^{n}$ as in Theorem 1.1. Let $f:X\rightarrow L$ be a
continuous map which is $\mu-$good and $\mu-$nonplanar in L described in
(2.10). Then the following are equivalent for $v>n$
1. (1)
$\\{x\in$ supp $\mu|f(x)\notin W^{\times}_{u}\\}$ is nonempty for any $u>v$
2. (2)
$w^{\times}(f_{*}\mu)\leqslant v$
3. (3)
(2.6.2) holds for $R$ of (2.9)
###### Proof.
Suppose (2.8.2) holds then $\\{x\in$ supp $\mu|f(x)\notin W^{\times}_{u}\\}$
has full measure for any $u>v$ hence (2.8.1) holds
If (2.8.3) holds then (2.6.2) implies (2.5.2) for (2.10) map by previous
discussion concerning subgroups of various ranks. According to Lemma 1.1 of
[K3] (2.5.1) is met. Proposition(2.5) is therefore applicable to establish
(2.8.2).
If (2.8.3) does not hold , no ball $B$ intersecting supp $\mu$ satisfies
(2.6.2). By Lemma 2.7 $f(B\bigcap$ supp $\mu)\subset W_{u}^{\times}$ for some
$u>v$. This contradicts (2.8.1).
∎
The above theorem shows that if $\exists y\in L$ with
$\omega^{\times}(y)\geqslant v$ then the set $\\{y\in
L|\omega^{\times}(y)\geqslant v\\}$ has full measure. And we have the theorem
for nondegenerate submanifold:
###### Theorem 2.2.
Let $M$ be a nondegenerate submanifold of $L$ as in Theorem 1.1
$\omega^{\times}(L)=\omega^{\times}(M)=inf\\{\omega^{\times}({\bf y})|{\bf
y}\in L\\}=inf\\{\omega^{\times}({\bf y})|{\bf y}\in M\\}$
Furthermore from Theorem 2.8 we derive
$\omega^{\times}(L)=max\big{\\{}n,sup\\{v\big{|}(2.6.2)\textrm{ }does\textrm{
}not\textrm{ }hold\textrm{ }for\textrm{ }R\\}\big{\\}}$ (2.11)
###### Theorem 2.3.
Let $L$ be a hyperplane of ${\mathbb{R}}^{n}$ parameterized by
$(x_{1},\ldots,x_{n-1},a_{1}x_{1}+\ldots+a_{s-1}x_{s-1}+b)$ with $a_{i}\neq
0,i=1,\ldots,s-1$
$\textrm{
}\omega^{\times}(L)=max\big{\\{}n,\frac{n}{s}\sigma(a_{1},\ldots,a_{s-1},b)\big{\\}}$
###### Proof.
(2.6.2) $\Leftrightarrow\forall d_{s}>c_{s}$,$\exists T>0$ such that $\forall
t>T,$
$\forall{\bf w}=(p_{0},p_{1},\ldots,p_{n})\in{\mathbb{Z}}^{n+1}_{s}$ with
$p_{i}\neq 0$ for $1\leqslant i\leqslant s-1$ and $i=n$,
$max\Big{(}e^{-t_{i}}|p_{i}|,e^{t}\|RC_{J}({\bf w})\|\Big{)}\geqslant
e^{-d_{s}t}$. Lemma 2.3 is applicable with
$k=s,v=\dfrac{n+nc_{s}}{1-sc_{s}},|x|=\|RC_{J}({\bf
w})\|,z_{i}=p_{i},E={(|x|,{\bf z})}\subset{\mathbb{R}}^{n+1}$
(2.6.2) is equivalent to $\exists N>0\textrm{ such that }\forall(x,{\bf z})\in
E\textrm{ with }\|{\bf z}\|>N$
$\left\|\begin{array}[]{c}p_{1}+a_{1}p_{n}\\\ p_{2}+a_{2}p_{n}\\\ \vdots\\\
p_{s-1}+a_{s-1}p_{n}\\\
p_{0}+bp_{n}\end{array}\right\|\geqslant(p_{1}p_{2}\ldots
p_{s-1}p_{n})^{-v/n}$ (2.12)
By assuming $\|p_{i}+a_{i}p_{n}\|\leqslant 1$ for $1\leqslant i\leqslant s-1$,
we know up to some constant (2.12) is the same as
$\left\|\begin{array}[]{c}p_{1}+a_{1}p_{n}\\\ p_{2}+a_{2}p_{n}\\\ \vdots\\\
p_{s-1}+a_{s-1}p_{n}\\\ p_{0}+bp_{n}\end{array}\right\|\geqslant
p_{n}^{-sv/n}$
Therefore by (2.11)
$\omega^{\times}(L)=max\big{\\{}n,\frac{n}{s}\sigma(a_{1},\ldots,a_{s-1},b)\big{\\}}$
∎
Theorem 1.1 is obtained by combining Theorem 2.9 and Theorem 2.10.
## 3\. A special class of hyperplanes and elementary approach
We prove a special case of Theorem 1.1 via elementary methods here, namely,
for a hyperplane $L\subset{\mathbb{R}}^{n}$ parameterized as
$(x_{1},\ldots,x_{n-1},b)$ $\textrm{
}\omega^{\times}(L)=max\\{n,n\sigma(b)\\}$.
First, in the definition of $\omega^{\times}({\bf y})$ ${\bf
y}=(x_{1},\ldots,x_{n-1},b)$, if we set ${\bf{q}}=(0,0,\ldots,0,q)$ we see
that $\omega^{\times}({\bf y})\geqslant n\sigma(b)$, the coefficient $n$
arising from the denominator on the right hand side of the equality of (1.2).
Therefore $\omega^{\times}(L)\geqslant max\\{n,n\sigma(b)$.
Second, $\prod_{\times}({\bf{q}})=\prod_{i=1\ q_{i}\neq
0}^{n}|q_{i}|\leqslant\|{\bf{q}}\|^{n}$
so $\omega^{\times}({\bf y})\leqslant n\omega({\bf y})$,and
$\omega^{\times}(L)\leqslant n\omega(L)=n\sigma(b)$.
Together we have $\omega^{\times}(L)=max\\{n,n\sigma(b)\\}$.
Acknowledgement. The author is grateful to Professor Kleinbock for helpful
discussions.
## References
* [J] V. Jarnik, Eine Bemerkung zum Übertragimgssatz, Büalgar. Akad. Nauk. Izv. Mat. Inst. (3) (1959), 169–175.
* [C] J. Cassels, An introduction to Diophantine Approximation, Cambridge Tracts in Math. Vol. 45, Cambridge Univ. Press,Cambridge,1957.
* [K1] D. Kleinbock, Extremal subspaces and their submanifolds, Geom. Funct. Anal. 13 (2003) 437–466.
* [K2] D. Kleinbock, An extension of quantitative nondivergence and application to diophantine exponents, Preprint (2005).
* [KM] D. Kleinbock and G.A. Margulis, Flows on homogenous spaces and Diophantine approximation on manifolds, Ann. Math. 148 (1998) 339–360.
* [M] K. Mahler, Über das Mass der Menge aller S-ZAHLEN, Math. Ann. 106(1932), 131–139.
* [S] W. Schmid, Diophantine Approximation, Springer, Berlin, 1980.
* [Sp1] V. Sprindžuk, More on Mahler’s conjecture (in Russian), Doklady. Akad. Nauk. SSSR 155 (1964), 54–56.
* [Sp2] V. Sprindžuk, Achievements and problems in Diophantine approximation theory, Russian . Math. Surveys. 35 (1980), 1–80.
|
arxiv-papers
| 2008-09-02T10:07:37
|
2024-09-04T02:48:57.607356
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuqing Zhang",
"submitter": "Yuqing Zhang",
"url": "https://arxiv.org/abs/0809.0386"
}
|
0809.0419
|
11institutetext: Max-Planck Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany 22institutetext: School of Mathematics and Statistics,
Newcastle University, Newcastle upon Tyne, NE1 7RU, U.K.
# High-resolution radio continuum survey of M33
III. Magnetic fields
F. S. Tabatabaei 11 M. Krause 11 A. Fletcher 22 and R. Beck 11
###### Abstract
Aims. We study the magnetic field structure, strength, and energy density in
the Scd galaxy M33.
Methods. Using the linearly polarized intensity and polarization angle data at
3.6, 6.2 and 20 cm, we determine variations of Faraday rotation and
depolarization across M33. We fit a 3-D model of the regular magnetic field to
the observed azimuthal distribution of polarization angles. We also analyze
the spatial variation of depolarization across the galaxy.
Results. Faraday rotation, measured between 3.6 and 6.2 cm at an angular
resolution of 3$\arcmin$ (0.7 kpc), shows more variation in the south than in
the north of the galaxy. About 10% of the nonthermal emission from M33 at 3.6
cm is polarized. High degrees of polarization of the synchrotron emission
($>20\%$) and strong regular magnetic fields in the sky plane ($\simeq
6.6\,\mu$G) exist in-between two northern spiral arms. We estimate the average
total and regular magnetic field strengths in M33 as $\simeq$ 6.4 and 2.5
$\mu$G, respectively. Under the assumption that the disk of M33 is flat, the
regular magnetic field consists of horizontal and vertical components: however
the inferred vertical field may be partly due to a galactic warp. The
horizontal field is represented by an axisymmetric ($m=0$) mode from 1 to 3
kpc radius and a superposition of axisymmetric and bisymmetric ($m=0+1$) modes
from 3 to 5 kpc radius.
Conclusions. An excess of differential Faraday rotation in the southern half
together with strong Faraday dispersion in the southern spiral arms seem to be
responsible for the north-south asymmetry in the observed wavelength dependent
depolarization. The presence of an axisymmetric $m=0$ mode of the regular
magnetic field in each ring suggests that a galactic dynamo is operating in
M33. The pitch angles of the spiral regular magnetic field are generally
smaller than the pitch angles of the optical spiral arms but are twice as big
as simple estimates based on the mean-field dynamo theory and M33’s rotation
curve. Generation of interstellar magnetic fields from turbulent gas motions
in M33 is indicated by the equipartition of turbulent and magnetic energy
densities.
###### Key Words.:
galaxies: individual: M33 – radio continuum: galaxies – galaxies: magnetic
field – galaxies: ISM
††offprints: F. Tabatabaei
tabataba@mpifr-bonn.mpg.de
## 1 Introduction
Magnetic fields in galaxies can be traced by radio polarization measurements.
The linearly polarized intensity gives information about the magnetic field
strength and orientation in the plane of the sky, Faraday rotation
measurements enable us to determine the magnetic field strength and direction
along the line of sight and depolarizing effects can be sensitive to both
components.
M33, the nearest Scd galaxy at a distance of 840 kpc, with its large angular
size [high-frequency radio continuum emission was detected in an area of
$35\arcmin\times 40\arcmin$, 39] and medium inclination of 56∘, allows
determination of the magnetic field components both parallel and perpendicular
to the line of sight equally well. The first RM study of M33, based on
polarization observations at 11.1 and 21.1 cm [2], suggested a bisymmetric
regular magnetic field structure in the disk of M33. [8] confirmed the
presence of this bisymmetric field using two further polarization maps at 6.3
and 17.4 cm. However, these results may be affected by weak polarized
intensity and the consequent high uncertainty in RM in the southern half of
M33, due to the low-resolution (7.7$\arcmin$ or 1.8 kpc) and low-sensitivity
of the observations.
Our recent observations of this galaxy provided high-sensitivity and high-
resolution ($3\arcmin\simeq 0.7\,{\rm kpc}$) maps of total power and linearly
polarized intensity at 3.6, 6.2, and 20 cm presented by [39]. These data are
ideal to study the rotation measure (RM), magnetic fields (structure and
strength), and depolarization effects in detail.
[39] found a north-south asymmetry in the polarization distribution that is
wavelength-dependent, indicating a possible north-south asymmetry in Faraday
depolarization. Investigation of this possibility requires a knowledge of the
distribution of RM, turbulent magnetic field and thermal electron density in
the galaxy. Furthermore, depolarization is best quantified using the
nonthermal degree of polarization rather than the fraction of the total radio
emission that is polarized. [38] developed a new method to separate the
thermal and nonthermal components of the radio continuum emission from M33
which yielded maps of the nonthermal synchrotron emission and the synchrotron
spectral index variations across the galaxy [37]. The nonthermal maps are used
in this paper to determine the nonthermal degree of polarization at different
wavelengths, and hence to study how the radio continuum emission from
different parts of M33 is depolarized.
By combining an analysis of multi-wavelength polarization angles with modeling
of the wavelength dependent depolarization, [20] and [6] derived the 3-D
regular magnetic field structures in M31 and M51, respectively. The high
sensitivity of our new observations allows a similar study for M33.
We first determine the nonthermal degree of polarization using the new
nonthermal maps (Sect. 2). Then we calculate a map of the RM intrinsic to M33
with a spatial resolution of 3$\arcmin$ or 0.7 kpc and probe its mean value in
rings in the galactic plane in Sect. 3. The regular magnetic field structure
is derived in Sect. 4 and the estimated strengths are presented. We derive a
map for the observed depolarization and discuss the possible physical causes
of depolarization sources in Sect. 5. Furthermore, we discuss the resulting
vertical fields and pitch angles in Sect. 6. The estimated energy density of
the magnetic field is compared to the thermal and turbulent energy densities
of the interstellar gas.
Table 1: Integrated nonthermal flux densities and average nonthermal degree of polarization at 180$\arcsec$. $\lambda$ | Snth | SPI | $\bar{{\rm P}}_{\rm nth}$
---|---|---|---
(cm) | (mJy) | (mJy) | $\%$
3.6 | 370 $\pm$ 60 | 38 $\pm$ 4 | 10.3 $\pm$ 2.0
6.2 | 696 $\pm$ 110 | 79 $\pm$ 5 | 11.3 $\pm$ 1.9
20 | 1740 $\pm$ 65 | 115 $\pm$ 10 | 6.6 $\pm$ 0.6
## 2 Nonthermal degree of polarization
The degree to which synchrotron emission is polarized reflects the degree of
coherent structure in the magnetic field within one beam-area; a purely
regular magnetic field will produce about 75%111If a nonthermal spectral index
of $\alpha_{n}=\,1$ is used. fractional polarization of synchrotron emission.
The quantity of interest is the degree of polarization of the synchrotron
emission or ‘nonthermal degree of polarization’, P${}_{\rm nth}={\rm
PI}/I_{\rm nth}$, where PI is the intensity of the linearly polarized emission
and $I_{\rm nth}$ is the intensity of the nonthermal emission.
Since we observe the total intensity $I$, which consists of both nonthermal
$I_{\rm nth}$ _and_ thermal $I_{\rm th}$ intensities at cm wavelengths, Pnth
cannot be calculated straightforwardly. To date, Pnth has been estimated by
assuming either a fixed ratio $I_{\mathrm{th}}/I$ or $I_{\mathrm{nth}}$ has
been derived assuming a fixed spectral index of the synchrotron emission. Here
we use a new, more robust method for determining the distribution of
$I_{\mathrm{nth}}$ by correcting H$\alpha$ maps for dust extinction, using
multi-frequency infra-red maps at the same resolution, and thus independently
estimating $I_{\mathrm{th}}$ [38]. Using the PI maps of Tabatabaei et al. [39]
and the nonthermal maps obtained by Tabatabaei et al. [38], we derived maps of
Pnth at different wavelengths.
Figure 1 shows Pnth at 3.6 cm. High nonthermal degrees of polarization
(P${}_{\rm nth}>$ 30%) are found in several patches of M33, with the most
extended region of high Pnth in the northern part of the magnetic filament
identified by Tabatabaei et al. [39], inside the second contour at DEC
$>\,30^{\circ}\,54\arcmin$ in Fig. 1. The high Pnth also exist at 6.2 cm in
these regions.
Integrating the polarized and nonthermal intensity maps in the galactic plane
out to galactocentric radius of R $\leq$ 7.5 kpc, we obtained the flux
densities of the nonthermal ${\rm S}_{\rm nth}$ and linearly polarized ${\rm
S}_{\rm PI}$ emission along with the average nonthermal degree of polarization
$\bar{{\rm P}}_{\rm nth}={\rm S}_{\rm PI}/{\rm S}_{\rm nth}$. Table 1 gives
${\rm S}_{\rm nth}$, ${\rm S}_{\rm PI}$, and $\bar{{\rm P}}_{\rm nth}$ at
different wavelengths, all at the same angular resolution of 180$\arcsec$. At
3.6 and 6.2 cm, $\bar{{\rm P}}_{\rm nth}$ is the same, demonstrating that
Faraday depolarization effects are not significant at these wavelengths: the
weaker SPI at 3.6 cm is due to lower synchrotron emissivity, as expected from
the power-law behavior of synchrotron emission with respect to frequency.
However, Faraday depolarization effects are possibly important at 20 cm
reducing $\bar{{\rm P}}_{\rm nth}$.
Figure 1: Nonthermal degree of polarization at 3.6 cm, with an angular
resolution of 2${\arcmin}$ (the beam area is shown in the left-hand corner).
Overlayed are contours of the linearly polarized intensity at 3.6 cm with
levels of 0.1 and 0.3 mJy/beam.
## 3 Rotation measures
When linearly polarized radio waves propagate in a magneto-ionic medium, their
polarization vector is systematically rotated. The amount of Faraday rotation
depends on the wavelength ($\lambda$) of the radio emission, the strength of
the magnetic field along the line of sight (that is the regular field in the
line of sight $\,B_{{\rm reg}\parallel}$), and the number density of thermal
electrons ($n_{e}$) along the line of sight ($l$):
$\displaystyle\frac{\Delta\phi}{\rm rad}$ $\displaystyle=$ $\displaystyle
0.81\left(\frac{\lambda}{\rm m}\right)^{2}\int_{0}^{\frac{L}{\rm
pc}}\left(\frac{\,B_{{\rm reg}\parallel}}{\mu{\rm
G}}\right)\left(\frac{n_{e}}{{\rm cm}^{-3}}\right){\rm d}\left(\frac{l}{\rm
pc}\right),$ (1) $\displaystyle=$ $\displaystyle 0.81\lambda^{2}\mathcal{R}$
where $L$ is the path length through the magneto-ionic medium. Hence, the
measured polarization angle ($\phi=\frac{1}{2}\,{\rm arctan}\frac{\rm U}{\rm
Q}$) differs from the intrinsic polarization angle ($\phi_{i}$) as
$\phi-\phi_{i}=\Delta\phi\equiv\lambda^{2}\,\mathcal{R}.$ (2)
When Faraday depolarization is small (Faraday-thin condition), $\mathcal{R}$
does not depend on wavelength [35] and an estimate for $\mathcal{R}$ can be
obtained from measurements of the polarization angles at two wavelengths:
$\frac{\,\mathrm{RM}}{\rm rad\,m^{-2}}=\frac{(\phi_{1}/{\rm
rad})\,-\,(\phi_{2}/{\rm rad})}{\left(\frac{\lambda_{1}}{\rm
m}\right)^{2}\,-\,\left(\frac{\lambda_{2}}{\rm m}\right)^{2}}.$ (3)
In this definition, the unknown intrinsic polarization angle of the source (or
sources along the line of sight) cancels. Positive RM indicates that
$\,B_{{\rm reg}\parallel}$ points towards us.
Part of the measured RM is due to the interstellar medium of M33 (intrinsic RM
or $\,\mathrm{RM_{i}}$), the rest is due to the Galactic foreground medium
($\,\mathrm{RM_{fg}}$), $\,\mathrm{RM}=\,\mathrm{RM_{i}}+\,\mathrm{RM_{fg}}$.
The foreground rotation measure in the direction of M33 is mainly caused by
the extended Galactic magnetic bubble identified as region $A$ by Simard-
Normandin & Kronberg [34]. Assuming that the intrinsic contributions of the
extragalactic sources 3C41, 3C42, and 3C48 (in a $5^{\circ}\times 5^{\circ}$
region around M33) themselves cancel out and the intragalactic contribution is
negligible, [7] and [36] found a foreground rotation measure of $-57\pm 10$
rad m-2 for those sources. For the polarized sources in the $2^{\circ}\times
2^{\circ}$ M33 field, [8] found a foreground RM of $-55\pm 10$ rad m-2. About
the same value was obtained by [21]. In the following we use
$\,\mathrm{RM_{fg}}=-55\,{\rm rad\,m^{-2}}$.
Using the polarization data of Tabatabaei et al. [39], we first obtained the
distribution of RM between 3.6 and 20 cm across M33 (Fig. 2, left panel),
showing a smooth distribution of RM in the northern half of the galaxy.
However, stronger and more abrupt fluctuations in RM occur in the southern
half of the galaxy, which are not due to the $\pm\,n\,73$ rad m-2 ambiguity in
RM between these wavelengths ($\pm
n\pi/\mid\lambda_{1}^{2}-\lambda_{2}^{2}\mid$). Weak polarized emission in the
southern half at 20 cm [presented in 39] can be linked to these RM variations.
Between 3.6 and 6.2 cm, RM varies less than between 3.6 and 20 cm in the south
of the galaxy (Fig. 2, right panel). This indicates that the relation between
$\Delta\phi$ and $\lambda^{-2}$ in Eq. (2) is not linear over the interval
between 3.6 and 20 cm due to Faraday depolarization at 20 cm in the south of
M33 and so RM measured at these wavelengths is not a good estimator for
$\mathcal{R}$.
Figure 2: Left: observed rotation measure map of M33 ($\,{\rm rad\,m^{-2}}$)
between 3.6 and 20 cm with contours of 3.6 cm polarized intensity. Contour
levels are 0.1, 0.2, 0.4, 0.6, 0.8 mJy/beam. Right: observed rotation measure
between 3.6 and 6.2 cm with contours of 6.2 cm polarized intensity. Contour
levels are 0.3, 0.4, 0.8, 1.2, 1.6 mJy/beam. The angular resolution in both
maps is 3${\arcmin}$ (the beam area is shown in the left-hand corners). The
straight line in the left panel shows the minor axis of M33.
Figure 3: Left: rotation measure map of M33 ($\,{\rm rad\,m^{-2}}$) between
3.6 and 6.2 cm after correction for the foreground
$\,\mathrm{RM_{fg}}=-55\,{\rm rad\,m^{-2}}$, with an angular resolution of
3${\arcmin}$ (the beam area is shown in the left-hand corner). Overlayed are
contours of 6.2 cm polarized intensity. Contour levels are 0.3, 0.4, 0.8, 1.2,
1.6, 3.2, 6.4 mJy/beam. Also shown are rings at 1, 3, and 5 kpc radii as used
in Sect. 4. Right: the distribution of the estimated error in RM is shown in
the right panel.
Figure 3 shows $\,\mathrm{RM_{i}}$ between 3.6 and 6.2 cm which varies in a
range including both positive and negative values. Comparing
$\,\mathrm{RM_{i}}$ with the overlayed contours of PI222Note that PI is
related to the magnetic field in the plane of the sky that is a combination of
both a mean field (or coherent regular field) and anisotropic (compressed or
sheared) random fields. $\,\mathrm{RM_{i}}$ is related to only coherent
regular field along the line of sight., $\,\mathrm{RM_{i}}$ seems to vary more
smoothly in regions of high PI. The apparent agreement between the ordered
magnetic field in the plane of the sky and the regular magnetic field in the
line of sight is clearest in the north and along the minor axis and is best
visible in the magnetic filament between the arms IV and V in the north-west
of M33 [Fig. 4, see also 39] where $\,\mathrm{RM_{i}}$ shows small variation
within the PI contours. This indicates that the ordered magnetic field in this
region is mainly regular. Sign variations of RMi are more frequent (arising
locally) in the southern half, where there is no correlation with PI, than in
the northern half of the galaxy. This indicates that the regular magnetic
field is more affected by local phenomena [like starforming activity, e.g. see
39] in the south than in the north of the galaxy. The local RMi variations
between large positive and negative values may represent loop-like magnetic
field structures going up from and down to the plane (e.g. Parker loops). This
is particularly seen in the central part of the galaxy besides regions in the
southern arm II S (Fig. 4).
Figure 3 also shows that the magnetic field is directed towards us on the
western minor axis (at azimuth $\theta\approx 110\degr$ and $\theta\approx
290\degr$), but has an opposite direction on the eastern side. The large RMi
values in regions with small electron density, e.g. on the eastern and western
minor axis and in a clumpy distribution in the central south near the major
axis with 30∘ 25$\arcmin<$DEC$<$30∘ 30$\arcmin$ [38], indicates a strong
magnetic field along the line of sight and/or large path length through the
magneto-ionic medium. Considerable Faraday rotation measured on the eastern
and western minor axis hints to deviation from a purely toroidal structure for
the large-scale magnetic field [22, 4]. Furthermore, along with the magnetic
field parallel to the disk, the presence of a vertical field component to the
galactic disk is indicated in kpc-scale regions of large RMi but small $n_{e}$
values. The existence of the vertical magnetic field in M33 (that is strong
near the major axis) is shown in Sect. 4.
## 4 The magnetic field
### 4.1 The regular magnetic field structure
Figure 4: Left: Optical image (B-band, taken from the STScI Digitized Sky
Survey) of M33 with overlayed vectors of intrinsic magnetic field in the sky-
plane with 3$\arcmin$ angular resolution. The length gives the polarized
intensity at 3.6 cm, where 1$\arcmin$ is equivalent to 0.37 mJy/beam. Right: A
sketch of the optical arms [31] overlayed on the linearly polarized intensity
with an angular resolution of 2$\arcmin$ (contours and grey scale) at 3.6 cm
[see 39].
The $\lambda 3.6\,{\rm cm}$ polarization angles were corrected for Faraday
rotation and rotated by $90\degr$ to obtain the intrinsic orientation of the
regular magnetic field component in the plane of the sky ($\,B_{{\rm
reg}\perp}$). Figure 4 shows the derived $\,B_{{\rm reg}\perp}$ superimposed
on an optical image of M33. The orientation of $\,B_{{\rm reg}\perp}$ shows a
spiral magnetic field pattern with a similar pitch angle to the optical arms
in the north and south, but larger pitch angles in the east and west of the
galaxy.
The apparent continuity of the regular magnetic field straight through the
center of M33 is remarkable. The Faraday rotation map (Fig. 3) shows that the
field is oppositely directed on the east and west sides of the center.
Unfortunately the current data lack the resolution to investigate the field
properties in this region further.
The behavior of $\,B_{{\rm reg}\perp}$ along the minor axis ($\theta\approx
110\degr$, $\theta\approx 290\degr$) is also informative. $\,B_{{\rm
reg}\perp}$ on the minor axis clearly has a strong radial component (Fig. 4).
If we assume that $\,B_{\rm reg}$ lies solely in the disc, so that $\,B_{{\rm
reg}\parallel}$ is produced by the galaxy’s inclination (i.e. $\,B_{z}=0$),
then the change of sign in RM along the minor axis (Fig. 3) indicates that the
direction of the radial component of $\@vec{\,B_{\rm reg}}$ is towards center
on both sides of the minor axis. This means that the dominant azimuthal mode
of $\,B_{\rm reg}$ cannot be the bisymmetric $m=1$ mode suggested by earlier
studies [2, 8]: we expect the dominant mode to be even.
In order to identify the 3-D structure of the _regular_ magnetic field
$\,B_{\rm reg}$ we fit a parameterized model of $\,B_{\rm reg}$ to the
observed polarization angles at different wavelengths using the method
successfully employed by Berkhuijsen et al. [6] and Fletcher et al. [20] to
determine the magnetic field structures of M51 and M31 respectively.
At each wavelength the maps in the Stokes parameters $Q$ and $U$ are averaged
in sectors of $10\degr$ opening angle and $2\,{\rm kpc}$ radial width in the
range $1\leq r\leq 5$ kpc. The size of the sectors is chosen so that the area
of the smallest sector is roughly equivalent to one beam-area. We also take
care that the standard deviation of the polarization angle in each sector is
greater than the noise (if the sector sizes are too small then fluctuations in
angle due to noise can be larger than the standard deviation). Then the
average $Q$ and $U$ intensities are combined to give the average polarization
angle $\psi=0.5\arctan{U/Q}$ and the average polarized emission intensity
$\mathrm{PI}\sqrt{(Q^{2}+U^{2})}$ in each sector.
The observed polarization angles are related to the underlying properties of
the regular magnetic field in M33 by333Note that we choose different symbols
for the average polarization angle $\phi$ and the average polarization angle
in sectors $\psi$.
$\psi=\psi_{\mathrm{0}}(\,B_{{\rm
reg}\perp})+\lambda^{2}\mathrm{RM_{i}}(\,B_{{\rm
reg}\parallel})+\lambda^{2}\,\mathrm{RM_{fg}},$ (4)
where $\psi_{\mathrm{0}}$ is the intrinsic angle of polarized emission,
$\mathrm{RM_{i}}$ is the Faraday rotation experienced by a photon as it passes
through the magneto-ionic medium of M33 and $\,\mathrm{RM_{fg}}$ is foreground
Faraday rotation due to the Milky Way. $\,B_{{\rm reg}\perp}$ is the component
of the regular magnetic field of M33 that lies in the sky-plane and $\,B_{{\rm
reg}\parallel}$ the component directed along the line-of-sight.
The cylindrical components of the regular field in the disk of M33
$\@vec{\,B_{\rm reg}}=(\,B_{r},\,B_{\theta},\,B_{z})$ can be represented in
terms of the Fourier series in the azimuthal angle $\theta$:
$\displaystyle\,B_{r}$ $\displaystyle=$ $\displaystyle B_{0}\sin
p_{0}+B_{1}\sin p_{1}\cos(\theta-\beta_{1})$ $\displaystyle+\,B_{2}\sin
p_{2}\cos 2(\theta-\beta_{2}),$ $\displaystyle\,B_{\theta}$ $\displaystyle=$
$\displaystyle B_{0}\cos p_{0}+B_{1}\cos p_{1}\cos(\theta-\beta_{1})$
$\displaystyle+\,B_{2}\cos p_{2}\cos 2(\theta-\beta_{2}),$
$\displaystyle\,B_{z}$ $\displaystyle=$ $\displaystyle
B_{z0}+B_{z1}\cos(\theta-\beta_{z1})+B_{z2}\cos 2(\theta-\beta_{z2}),$
where $B_{m}$ and $B_{zm}$ are the amplitude of the mode with azimuthal wave
number $m$ in the horizontal and vertical fields, $p_{m}$ is the constant
pitch angle of the $m$’th horizontal Fourier mode (i.e. the angle between the
field and the local circumference) and $\beta_{m}$ and $\beta_{zm}$ are the
azimuths where the non-axisymmetric modes are maximum. The amplitudes of the
Fourier modes are obtained in terms of the variables $B_{m}$ in units of
$\,{\rm rad\,m^{-2}}$ : in order to obtain amplitudes in Gauss independent
information is required about the average thermal electron density and disc
scale-height. Useful equations describing how $\,B_{{\rm reg}\perp}$ and
$\,B_{{\rm reg}\parallel}$ are related to the field components of Eq. (4.1)
can be found in Appendix A of Berkhuijsen et al. [6].
Using Eqs. (4) and (4.1), and taking into account the inclination ($56\degr$)
and major-axis orientation ($23\degr$) of M33, we fit the three-dimensional
$\@vec{\,B_{\rm reg}}$ to all of the observed polarization angles in a ring by
minimizing the residual
$S=\sum_{\lambda,n}\left[\frac{\psi_{n}-\psi(\theta_{n})}{\sigma_{n}}\right]^{2},$
(6)
where $\psi_{n}$ is the observed angle of polarization, $\psi(\theta_{n})$ the
modelled angle in the sector centred on azimuth $\theta_{n}$ and $\sigma_{n}$
are the observational errors. The Fisher test is used to ensure that the fits
at each wavelength are statistically, equally good [see Appendix B in 6].
Errors in the fit parameters are estimated from the region of parameter space
where $S\leq\chi^{2}$ at the $2\sigma$ level.
### 4.2 Results of fitting
Figure 5: Polarized intensity (PI) at 6 cm in the ring 1–3 kpc, observed
(points with error bars) and expected from the modelled regular magnetic field
(lines) with PI$\propto\,B_{{\rm reg}\perp}^{2}$ given by the different fits
shown in Table. LABEL:tab:fit. (a) using a fitted regular magnetic field
containing the modes $m=0+z0+z1$. (b) as (a) but for the fit containing the
modes $m=0+1+z1$. (c) as (a) but for the fit containing the modes $m=0+2$.
Figure 6: As figure 5 but for the ring 3–5 kpc using a fitted regular magnetic
field containing the modes $m=0+1+z1$ (top) and $m=0+2$ (bottom).
We applied the method described in Sect. 4 to the polarization maps at 3.6 and
6.2 cm. A preliminary examination of the data at all three wavelengths showed
that the 20 cm polarization angles do not have the same
$\psi\propto\lambda^{2}$ Faraday rotation dependence as the angles at 3.6 and
6.2 cm. This is most probably because the 20 cm signal is strongly depolarized
by Faraday effects (see Sect. 5) and so only photons from an upper layer of
the emitting region are detected in polarization. If the depolarization is
constant at a given radius, a ‘depth’ parameter can be used to take account of
this effect [6, 20]. However, in the case of M33 depolarization is strongly
asymmetric (Fig. 11) and this method does not lead to consistent results: we
therefore work only with the polarization angles at 3.6 and 6.2 cm to model
the regular magnetic field.
We fixed the foreground Faraday rotation, $\,\mathrm{RM_{fg}}$ in Eq. (4), to
$-55\,{\rm rad\,m^{-2}}$ (see Sect. 3). For each ring we found more than one
statistically good fit to the observed polarization angles, three different
fits in the ring 1–3 kpc and two fits in the ring 3–5 kpc, the fitted
parameters are shown in Table LABEL:tab:fit. All of the fits require the
presence of more than one azimuthal Fourier mode and have two common
characteristics: the presence of an $m=0$ mode that has a significant
amplitude; the pitch angle of the $m=0$ mode is in the range $40\degr\lesssim
p\lesssim 50\degr$. The reason why several equally good fits are found is the
rather weak large-scale intrinsic rotation measure signal, i.e. the amplitude
of systematic rotation measure variations is rather low compared to local
fluctuations. This is a sign that the regular magnetic field of M33 is not as
well-ordered and strong, relative to the small-scale field, as that of, for
example, M31 [20].
The model regular magnetic field given by Eq. 4.1 is fitted to the observed
polarization _angles_ , in order to obtain the results shown in Table
LABEL:tab:fit. Since we have not made use of the observed polarized
_intensity_ we can try to use this to select the best regular magnetic field
model for each of the two rings from the fits given in Table LABEL:tab:fit. We
compare the predicted azimuthal pattern of polarized intensity from the model
fields with the observed polarized intensity (PI) at 6.2 (Faraday
depolarization effects are negligible at 6.2 cm and emissivity is higher than
at 3.6 cm due to the spectral index, thus giving a stronger signal). The model
described by Eq. 4.1 is not designed to reproduce the observed PI so we cannot
make meaningful statistical assessments about the relative merits of the
different fits in a given ring. But we can judge whether or not the fits are
better or worse than each other in explaining the location of the main maxima
and minima in the observed pattern of PI and so try to select a preferred
model field for each ring.
Figures 5 and 6 shows the square of the predicted plane-of-sky regular
magnetic field $\,B_{{\rm reg}\perp}^{2}$ for each of the fits given in Table
LABEL:tab:fit and the observed 6.2 cm polarized intensity, both normalized to
avoid having to use a prescription for the poorly known synchrotron
emissivity. In the case of energy equipartition between cosmic rays and
magnetic fields the polarized intensity would be proportional to a higher
power of $\,B_{{\rm reg}\perp}$ than $2$ and maxima would be more pronounced.
In the ring 1–3 kpc, the model field with the components of $m=0+z0+z1$ 444
$z0$ and $z1$ are the first and secound Fourier modes of the vertical field.
(Fig. 5a) reproduces the broad features of the observed polarized emission
better than the models using the modes $m=0+1+z1$ (Fig. 5b) and $m=0+2$ (Fig.
5c). The match to the observed PI is far from perfect in Fig. 5a but this is
the only model field that can account for the strong excess of PI at 6 cm in
the northern half of the disc at these radii. In the ring 3–5 kpc the fit
using $m=0+1+z1$ (Fig. 6, top) is better at reproducing the general pattern of
polarized emission at 6 cm than the other statistically good fit using $m=0+2$
(Fig. 6, bottom). Again, the match to observations is not perfect, but the
model with $m=0+2$ would produce a strong maximum in PI at $\theta\sim
90\degr$ that is not observed.
To summarize: we select the statistically good fit using the modes $m=0+z0+z1$
in the ring 1–3 kpc and that using $m=0+1+z1$ in the ring 3–5 kpc as being the
best descriptions of the regular magnetic field in M33 (the parameters of
these two preferred fits are given in columns 3 and 6 of Table LABEL:tab:fit).
Our reason is that the $\,B_{{\rm reg}\perp}^{2}$ produced by these models
produces a much closer match to the observed pattern of PI at 6.2 cm than
other statistically good models. Fig. 7 shows the regular magnetic field in a
face-on view of the galaxy (thus the vertical field components cannot be
seen).
Table 2: Parameters of the fitted models and their $2\sigma$ errors. $RM_{\mathrm{fg}}$ is the Faraday rotation measure arising in the Milky Way, $B_{m}$ and $p_{m}$ are the amplitude and pitch angle of the mode with wave number $m$, and $\beta_{m}$ is the azimuth where a mode with azimuthal wave number $m$ is maximum. The minimum value of the residual $S$ and the value of $\chi^{2}$ are shown for each model in the bottom lines. The combination of azimuthal modes of $m=0+z0+z1$ and $m=0+1+z1$ can best re-produce the observed polarized intensity in the 1-3 and 3-5 kpc rings, respectively (Figs 5 and 6). | Units | Radial range (kpc)
---|---|---
| | 1–3 | | 3–5
RMfg | $\rm{rad\,m^{-2}}$ | $-55$ $\pm 45$ | $-55$ ${}^{+6}_{-9}$ | $-55$ ${}^{+30}_{-59}$ | | $-55$ ${}^{+30}_{-60}$ | $-55$ $\pm 19$
$B_{0}$ | $\rm{rad\,m^{-2}}$ | $-30$ ${}^{+11}_{-20}$ | $-69$ $\pm 4$ | $-14$ $\pm 2$ | | $-13$ $\pm 3$ | $-103$ $\pm 9$
$p_{0}$ | deg | $48$ $\pm 12$ | $51$ $\pm 2$ | $42$ $\pm 4$ | | $42$ ${}^{+1}_{-7}$ | $41$ $\pm 2$
$B_{1}$ | $\rm{rad\,m^{-2}}$ | | | $-12$ $\pm 3$ | | $-9$ $\pm 2$ |
$p_{1}$ | deg | | | $28$ ${}^{+1}_{-9}$ | | $14$ ${}^{+11}_{-7}$ |
$\beta_{1}$ | deg | | | $-56$ ${}^{+12}_{-1}$ | | $-67$ ${}^{+22}_{-39}$ |
$B_{2}$ | $\rm{rad\,m^{-2}}$ | | $-41$ $\pm 3$ | | | | $-67$ $\pm 6$
$p_{2}$ | deg | | $-87$ $\pm 6$ | | | | $-103$ $\pm 8$
$\beta_{2}$ | deg | | $-12$ $\pm 3$ | | | | $-22$ $\pm 4$
$B_{z0}$ | $\rm{rad\,m^{-2}}$ | $-14$ $\pm 15$ | | | | |
$B_{z1}$ | $\rm{rad\,m^{-2}}$ | $-52$ ${}^{+22}_{-35}$ | | $-16$ $\pm 5$ | | $-14$ $\pm 4$ |
$\beta_{z1}$ | deg | $32$ ${}^{+21}_{-16}$ | | $24$ ${}^{+1}_{-10}$ | | $9$ $\pm 9$ |
$m$ | | $0+z0+z1$ | $0+2$ | $0+1+z1$ | | $0+1+z1$ | $0+2$
$S$ | | $56$ | $83$ | $57$ | | $57$ | $82$
$\chi^{2}$ | | $85$ | $85$ | $84$ | | $84$ | $85$
Figure 7: The regular magnetic field described by our favoured fitted model
(see Sect. 4.2 for details). The galaxy has been deprojected into a face-on
view, so only the disk-plane components of the magnetic field are shown, the
vertical components are not visible. The dashed line shows the major axis of
M33, with north to the top. The ring boundaries are at 1, 3 and 5 kpc.
### 4.3 The equipartition magnetic field strengths
The strengths of the total magnetic field $\,B_{\mathrm{tot}}$ and its regular
component $\,B_{\rm reg}$ can be found from the total synchrotron intensity
and its degree of linear polarization Pnth. Assuming equipartition between the
energy densities of the magnetic field and cosmic rays
($\varepsilon_{CR}=\varepsilon_{\,B_{\mathrm{tot}}}=\,B_{\mathrm{tot}}^{2}/8\pi$),
$\displaystyle\,B_{\mathrm{tot}}=\big{[}\frac{4\pi(2\alpha_{n}+1)\,{\mathrm{K}}^{\prime}\,I_{n}\,E_{\mathrm{p}}^{1-2\alpha_{n}}\,(\frac{\nu}{2c_{1}})^{\alpha_{n}}}{(2\alpha_{n}-1)\,c_{2}(\alpha_{n})\,L\,c_{3}}\big{]}^{\frac{1}{\alpha_{n}+3}}$
(7)
[5], where ${\mathrm{K}}^{\prime}={\mathrm{K}}+1$ with ${\mathrm{K}}$ the
ratio between the number densities of cosmic ray protons and electrons,
$I_{n}$ is the nonthermal intensity, $L$ the pathlength through the
synchrotron emitting medium, and $\alpha_{n}$ the mean synchrotron spectral
index. $E_{\mathrm{p}}=938.28$ MeV $=1.50\times 10^{-3}$ erg is the proton
rest energy and
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\left.3e/(4\pi{m_{\mathrm{e}}}^{3}c^{5})=\frac{6.26428\cdot
10^{18}}{{\rm erg}^{2}.\,{\rm s.\,G}},\right.$ $\displaystyle
c_{2}(\alpha_{n})$ $\displaystyle=$ $\displaystyle{1\over
4}c_{3}\,\left(\alpha_{n}+5/3\right)/(\alpha_{n}+1)\,\Gamma[(3\alpha_{n}+1)/6]$
$\displaystyle\times\,\Gamma[(3\alpha_{n}+5)/6].$
For a region where the field is completely regular and has a constant
inclination $i$ with respect to the sky plane ($i=0^{o}$ is the face-on view),
$c_{3}=[{\cos}\,(i)]^{(\alpha_{n}+1)}$. If the field is completely turbulent
and has an isotropic angle distribution in three dimensions,
$c_{3}=(2/3)^{(\alpha_{n}+1)/2}$. If the synchrotron intensity is averaged
over a large volume, $[{\cos}\,(i)]^{(\alpha_{n}+1)}$ has to be replaced by
its average over all occurring values of $i$.
The strength of the regular magnetic field in the plane of the sky can be
estimated from the observed nonthermal degree of polarization [32]:
$\displaystyle{\rm P}_{\rm nth}$ $\displaystyle=$
$\displaystyle\left.\left(\frac{3\gamma\,+\,3}{3\gamma\,+\,7}\right)\,\times\right.$
$\displaystyle\left.\,\,\left[1+\frac{(1-q)\,\pi^{1/2}\,\Gamma[(\gamma+5)/4]}{2q\Gamma[(\gamma+7)/4]F(i)}\right]^{-1},\right.$
$\displaystyle F(i)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\,\int_{0}^{2\pi}\,\left(1-{\sin^{2}{i}}\,\sin^{2}{\theta}\right)^{(\gamma+1)/4}\,{\rm
d}\theta,$
with $\,B_{\rm reg}/\,B_{\rm tur}=q^{2/(1+\gamma)}$, $\gamma=2\alpha_{n}+1$,
and $\theta$ the azimuthal angle ($\,B_{\rm tur}$ is the turbulent magnetic
field). This formula assumes that the regular magnetic field has a single
orientation, is parallel to the disk and, taken over the galaxy as a whole,
has no further preferential orientation with respect to any fixed direction in
space.
The determined average values of $I_{n}$, $\alpha_{n}$, and Pnth with the
assumed values of ${\mathrm{K}}\,(\simeq$ 100) and $L\,(\simeq\,1\,{\rm
kpc}/{\rm cos}\,i$) lead to $\,B_{\mathrm{tot}}=\,6.4\,\pm\,0.5\,\mu$G and
$\,B_{\rm reg}=\,2.5\,\pm\,1.0\,\mu$G for the disk of M33 ($R<$7.5 kpc). The
strongest regular magnetic field is found in between the northern arms IV N
and V N (in the magnetic filament) with $\,B_{\rm reg}\simeq 6.6\,\mu$G where
$\,B_{\mathrm{tot}}\simeq 8.3\,\mu$G.
The regular magnetic field strength estimated from the mean rotation measure
in sectors and assuming $n_{e}\simeq 0.03$ cm-3 (see Sect. 5) is
$1.4\,\scriptsize{{}^{+0.9}_{-0.5}}$ $\mu$G and $0.6\pm 0.1\,\mu$G in the ring
1–3 kpc and 3–5 kpc, respectively. In the ring 1–3 kpc, the regular field
strength is consistent with that estimated from the equipartition assumption,
while it is much smaller in the ring 3–5 kpc. The more frequent RM variations
in both amplitude and sign in the second ring (see Fig. 3) indicate a line-of-
sight field component with many reversals and a small mean RM within each
sector, whereas the field component in the sky plane causes significant
polarized emission (which is insensitive to the field reversals).
The equipartition total magnetic field strength of $6\,\pm\,0.5\mu$G is
slightly higher than the total field obtained by [8], $4\,\pm\,1\mu$G,
assuming the minimum total energy requirement of the disk-like synchrotron
source. This is not surprising because a difference of $\simeq$20% is expected
between the minimum and equipartition magnetic field strengths for field
strengths of about $5\,\mu$G [5]. Furthermore, the mean nonthermal fraction
from the standard thermal/nonthermal separation method is lower than that from
the new method [38], resulting in a weaker equipartition magnetic field.
As the polarized intensity (PI/0.75, corrected for the maximum fractional
polarization in a completely regular field) is related to the regular magnetic
field strength, and the nonthermal intensity ($I_{n}$) to the total magnetic
field strength in the plane of the sky, $I_{n}\,-\,$(PI/0.75) gives the
nonthermal emission due to the turbulent magnetic field $\,B_{\rm tur}$. Using
this intensity with Eq. (7) and assuming a completely turbulent field yields
the distribution of $\,B_{\rm tur}$ across the galaxy. Figure 8 shows strong
$\,B_{\rm tur}$ ($>7\,\mu$G) in the central region of the galaxy, the arm I S,
and parts of the northern arm I N.
Figure 8: Distribution of the turbulent magnetic field strength, $\,B_{\rm
tur}$ ($\mu$G), in M33 with an angular resolution of 3${\arcmin}$ (the beam
area is shown in the left-hand corner).
Using the mean synchrotron flux density, synchrotron spectral index, and
degree of polarization in rings, we also derive the average field strengths in
rings. Figure 9 shows some fluctuations but no systematic increase or decrease
of these strengths with galactocentric radius. The small bump at $4.5<R<5.5$
kpc is due to the M33’s magnetic filament.
Figure 9: Variation of the mean total, regular, and turbulent magnetic field
strengths in rings of 500 pc width with galactocentric radius.
## 5 Depolarization
The depolarization observed at a certain wavelength is defined as the ratio of
the nonthermal degree of linear polarization Pnth and the theoretical maximum
value $p_{0}$ (75% for $\alpha_{n}=\,1$). Generally, depolarization may be
caused by instrumental effects as the bandwidth and beamwidth of the
observations or by the wavelength-dependent Faraday depolarization. Bandwidth
depolarization occurs when the polarization angles vary across the frequency
band, reducing the observed amount of polarized emission. It is given by
$sinc\,(2{\rm RM}\,{\lambda}^{2}\,\delta\nu/\nu)$, where $\delta\nu$ is the
bandwidth of the observations [e.g. 28]. In our study, the wavelengths,
bandwidths and RM values lead to a negligible bandwidth depolarization.
Beamwidth depolarization occurs when polarization vectors of different
orientation are unresolved in the telescope beam. In order to compensate this
effect, the ratio of the nonthermal degree of polarization at two wavelengths
is used at a same angular resolution,
${\rm DP}_{\lambda_{2}/\lambda_{1}}={{\rm P}_{\rm nth}^{\lambda_{2}}\over{\rm
P}_{\rm nth}^{\lambda_{1}}},$ (9)
where, $\lambda_{2}>\lambda_{1}$. The observed depolarization ${\rm
DP}_{\lambda_{2}/\lambda_{1}}$, that is only wavelength _dependent_ , is
called the Faraday depolarization.
We derived the depolarization DP20/3.6 using the maps of nonthermal and
polarized intensity at 20 and 3.6 cm at the same angular resolution of
180$\arcsec$ (Fig. 11, top panel). The southern half of the galaxy is highly
depolarized compared to the northern half. While DP20/3.6 changes between 0.0
and 0.5 in the south, it varies between 0.3 and 1.0 in the north. Considerable
depolarization is found at the positions of the prominent HII regions NGC604
(RA = 1h 34m 32.9s, DEC = 30∘ 47$\arcmin$ 19.6$\arcsec$), NGC595 (RA = 1h 33m
32.4s, DEC = 30∘ 41$\arcmin$ 50$\arcsec$) and IC133 (RA = 1h 33m 15.3s, DEC =
30∘ 53$\arcmin$ 19.7$\arcsec$) as can be expected due to their high densities
of thermal electrons. The strongest depolarization in the inner galaxy occurs
in the main southern arm I S. No depolarization (DP $\simeq$ 1) is seen on the
eastern end of the minor axis and some northern regions.
There are several mechanisms that can lead to wavelength-dependent Faraday
depolarization [9, 35]. _Differential Faraday rotation_ occurs when
synchrotron emission originates in a magneto-ionic medium containing a regular
magnetic field. The polarization plane of the radiation produced at different
depths within the source is rotated over different angles by the Faraday
effect and this results in a decrease in the measured degree of polarization.
_Faraday dispersion_ is depolarization due to fluctuations in the rotation
measure within a beam, caused by the turbulent magnetic field and distribution
of thermal electrons along the line of sight. When this dispersion is
intrinsic to the source, it is called internal Faraday dispersion. In case of
a dispersion in an external screen it is called external Faraday dispersion.
This depolarization effect may be responsible for the north-south asymmetry in
the polarized emission from M33, if an asymmetry in distribution of the
foreground magneto-ionic medium exists. However, as M33 cannot be resolved in
the available foreground surveys, like RM [21] and H$\alpha$ (Wisconsin
H$\alpha$ mapper) surveys, we do not discuss this depolarization further.
Finally _rotation measure gradients_ on the scale of the beam or larger due to
systematic variation in the regular magnetic field can also lead to
depolarization. The regular field in M33 is not strong enough nor is the
inclination of the galaxy high enough for this effect to be significant [in
contrast the highly regular field of the strongly inclined galaxy M31 does
produce strong RM gradients, 20].
The depolarization due to internal Faraday dispersion [given by 35] is
$\displaystyle{\rm DP}_{r}$ $\displaystyle=$
$\displaystyle\left.{1-e^{-2\sigma_{\rm RM}^{2}\,\lambda^{4}}\over
2\sigma_{\rm RM}^{2}\,\lambda^{4}},\right.$ $\displaystyle\sigma_{\rm RM}$
$\displaystyle=$ $\displaystyle 0.81\langle n_{e}\rangle\,\,B_{\rm
tur}\,\sqrt{L\,d\,/\,f},$ (10)
where the dispersion in rotation measure is $\sigma_{\mathrm{RM}}$, with $L$
the pathlength through the ionized medium, $f$ the filling factor of the
Faraday-rotating gas along the line of sight [$\simeq$ 0.5, 3], and $d$ the
turbulent scale [$\simeq$ 50 pc, 26]. Using the H$\alpha$ emission measure
($EM=\int{n_{e}^{2}.\,dl}$) and a clumping factor $f_{\rm c}=\,\langle
n_{e}\rangle^{2}/\langle n_{e}^{2}\rangle$ describing the variations of the
electron density, $\langle n_{e}\rangle$ can be determined by $\langle
n_{e}\rangle=\sqrt{{f_{\rm c}\,EM/L}}$. For the local interstellar medium,
[24] found $f_{\rm c}\simeq\,0.05$. Assuming a thickness of $\simeq 1$ kpc for
the thermal electrons in the disk of the galaxy [the Galactic value, 14] and
correcting for the inclination of M33, $L\simeq 1800$ pc. Then the extinction
corrected H$\alpha$ ($EM$) map of M33 [38] generates a distribution of
$\langle n_{e}\rangle$ across the galaxy with a mean value of $\simeq$ 0.05
cm-3 and a most probable value of $\simeq$ 0.03 cm-3 (Fig. 10), that is in
agreement with the estimated values in our galaxy [14] and other nearby
galaxies [23, 18]. Note that a more realistic approach would consider
different filling factors and electron densities for the thin and thick disk
of the galaxy. However, because the only information we have is a
superposition of these components along the line of sight, we are not able to
distinguish the role of each component. The resulting $\langle n_{e}\rangle$
and $\,B_{\rm tur}$ obtained in Sect.4.3 (Fig. 8) enable us to estimate DPr at
3.6 and 20 cm. The left-bottom panel in Fig. 11 shows the ratio of DPr at 20
and 3.6 cm.
Figure 10: Histogram of the thermal electron density $\langle n_{e}\rangle$
(cm-3) distribution across M33, derived from an extinction corrected H$\alpha$
[38]. The mean and standard deviation (stddev) of the distribution are given
also.
Figure 11: Top: observed depolarization ${\rm DP}_{20/3}$ between 3.6 and 20
cm (Eq. 9). Bottom-left: estimated depolarization ${\rm DP}_{20/3}$ due to
dispersion in Faraday rotation (Eq. 10), and bottom-right: estimated
depolarization ${\rm DP}_{20/3}$ due to differential Faraday rotation (Eq.
11). The straight line shows the minor axis of M33.
The other Faraday depolarization effect that is strong in M33, differential
Faraday rotation, is given by [9] and [35] as,
${\rm DP}_{u}={\rm sinc}\,\,(2\,{\rm RM}_{i}\,\lambda^{2}),$ (11)
where for simplicity we assume that the disk of M33 can be represented as a
uniform slab. Using the RMi map in Fig. 3, we estimated DPu between 20 and 3.6
cm across the galaxy (Fig. 11, bottom-right). As small variations in RMi
produce large changes in the sinc function in Eq. (11), the resulting ${\rm
DP}_{u}$ is not smoothly distributed among neighboring pixels.
Qualitatively, both kinds of Faraday depolarization contribute to the observed
depolarization in M33. The global phenomenon, the north-south asymmetry, is
visible in both DPu as weaker depolarization (DP${}_{20/3}\sim 1$) in the
north and also in DPr as stronger depolarization (DP${}_{20/3}\sim 0$) in the
south part of the central region. However, locally e.g. at the positions of
HII complexes and the southern spiral arms, DPr could explain the observed
depolarization. The contributions of DPu and DPr vary region by region. A more
quantitative comparison requires a combination of DPu and DPr across the
galaxy, but this needs a detailed modelling of depolarization along with
distribution of the filling factors $f$ and $f_{c}$, the pathlength $L$, and
the turbulent scale $d$ across the galaxy.
In the south of M33, a strong turbulent condition was already indicated from
the HI line-widths being larger than in the north [17], which could be
connected to the high starformation activities in the southern arms
particularly in the main arm I S [39]. Hence, we conclude that the highly
turbulent southern M33 along with a magneto-ionic medium containing vertical
regular magnetic fields, reduce the degree of polarization of the integral
emission from the southern half and cause the wavelength-dependent north-south
asymmetry in polarization (or depolarization).
## 6 Discussion
### 6.1 Vertical magnetic fields
The model regular magnetic field described in Table LABEL:tab:fit and shown in
Fig. 7 has a vertical component in each ring. In the inner ring, the
combination of the modes $m=z0$ and stronger $m=z1$ produces a sinusoidal
vertical field that is strongest near the major axis: pointing away from us at
$\theta\simeq 30\degr$ and towards us at $\theta\simeq 210\degr$. In the outer
ring the vertical field is also strongest near the major axis, at
$\theta\simeq 10\degr$ (directed away) and $\theta\simeq 190\degr$ (directed
towards).
The presence of a vertical component to the regular magnetic field was already
indicated locally by our rotation measure maps (Sect. 3). However, the large
scale vertical field required by our fits requires a global origin. The strong
$\,B_{z}$ along the major axis together with the large line of sights through
the magneto-ionic medium on the eastern and western minor axis (Sect. 3)
suggest that warp may play a role. In other words, the ‘vertical’ field that
we identify may be due to the severe warp in M33 [29, 27, 31, 13]. The inner
HI disk investigated by [29], shows a warp beginning at a radius of $\simeq$ 5
kpc with a change in the inclination angle of $40\degr$ at 8 kpc. The warp of
the optical plane begins as close to the center as the first arm system at 2
kpc (the center of our inner ring), with a change in the arm inclination of
$>15\degr$ at 3 kpc and $25\degr$ at 5 kpc [31]. The model in Sect. 4.2
assumes a constant inclination of $i=56\degr$ but in a strongly warped disc
$i$ varies with radius and azimuth. In this case, even if $\,B_{\rm reg}$ only
has components in the _warped_ disk plane $B_{\mathrm{d}}$, as for e.g. M51
[6], M31 [20], and NGC6946 [3], there will be an _apparent_ vertical component
$\hat{\,B_{z}}$ (as well as an apparent disk parallel component
$\hat{B_{\mathrm{d}}}$) with respect to the _average_ disk-plane.
The ratio $\hat{\,B_{z}}/\hat{B}_{\mathrm{d}}=\tan i_{w}$ where $i_{w}$ is the
warp inclination. So for $w_{i}\simeq 15\degr$ in the inner ring
$\hat{\,B_{z}}/\hat{B}_{\mathrm{d}}\simeq 0.3$ and in the outer ring
$w_{i}\simeq 25\degr$ gives $\hat{\,B_{z}}/\hat{B}_{\mathrm{d}}\simeq 0.5$.
Our model field in Table LABEL:tab:fit has $\,B_{z}/B_{\mathrm{d}}\leq 2\pm 1$
in the ring $1$–$3$kpc and $\,B_{z}/B_{\mathrm{d}}\leq 1.0\pm 0.4$ in the ring
$3$–$5$kpc. This indicates that, in the outer ring, the vertical field could
be mainly due to the warp. However, a real vertical field of a broadly
comparable strength to the disk field can exist in the inner ring.
### 6.2 Magnetic and spiral-arm pitch angles
The pitch angles of the horizontal component of the regular magnetic field are
high: $\,p_{\mathrm{B}}=48\degr$ in the ring 1–3 kpc and
$\,p_{\mathrm{B}}=42\degr$ in the ring 3–5 kpc. These magnetic field pitch
angles are however lower than the pitch angles of the optical arm segments
identified by Sandage & Humphreys [31], which are typically
$\,p_{\mathrm{a}}=60\degr$–$70\degr$. A combination of shear from the
differential rotation, producing an azimuthal magnetic field with
$\,p_{\mathrm{B}}=0\degr$, and compression in spiral arm segments, amplifying
the component of the field parallel to the arms $\,p_{\mathrm{B}}=65\degr$,
may be responsible for the observed $\,p_{\mathrm{B}}\simeq 40\degr$. However,
this type of alternate stretching and squeezing of the field could not produce
the $m=0$ azimuthal mode that is found in both rings, unless the pre-galactic
field was of this configuration. The presence of a significant $m=0$ azimuthal
mode of $\,B_{\rm reg}$ can be explained if a large-scale galactic dynamo is
operating in M33: the axisymmetric mode has the fastest growth rate in disk
dynamo models [e.g. 4]. This does not mean that a dynamo is the origin of
_all_ of the regular magnetic field structure in M33. In particular it would
be a strange coincidence if the large $\,p_{\mathrm{B}}$, higher than the
typical $\,p_{\mathrm{B}}$ in other disc galaxies by a factor of $\sim 2$, is
not connected to the open spiral arms with $\,p_{\mathrm{a}}\simeq 65\degr$.
A rough estimate of the magnetic field pitch angles expected due to a simple
mean-field dynamo can be obtained by considering the ratio of the alpha-effect
— parameterizing cyclonic turbulence generating radial field $\,B_{r}$ from
azimuthal $\,B_{\theta}$ — to the omega-effect — describing differential
rotation shearing radial field into azimuthal. This can be written as [33]
$\tan{\,p_{\mathrm{B}}}=\frac{\,B_{r}}{\,B_{\theta}}\simeq\frac{1}{2}\sqrt{\frac{\pi\alpha}{hG}},$
(12)
where $\alpha$ is a typical velocity of the helical turbulence, $h$ is the
scale height of the dynamo active layer and
$G=R\,\mathrm{d}\Omega/\mathrm{d}R$ gives the shear rate due to the angular
velocity $\Omega$. Using the HI rotation curve derived by [12], $\alpha\sim
1\,{\rm km\,s^{-1}}$ as a typical value [30], and an HI scale height of 250pc
at $R$=2kpc increasing steadily to 650pc at $R$=5kpc [1] we obtain approximate
pitch angles of $\,p_{\mathrm{B}}\simeq 20\degr$ and $\,p_{\mathrm{B}}\simeq
15\degr$ for the rings 1-3 and 3-5 kpc, respectively. These are only about 1/2
to 1/3 of the fitted pitch angles of the m=0 modes. Models specific to M33,
which allow for dynamo action as well as the large scale gas-dynamics of the
galaxy, are required to understand the origin of the large $\,p_{\mathrm{B}}$
as well as the vertical component of the regular magnetic field.
### 6.3 Energy densities in the ISM
Figure 12: Energy densities and their variations with galactocentric radius in
M33.
The energy densities of the equipartition magnetic fields in the disk ($B_{\rm
t}^{2}/8\pi$ and $B_{u}^{2}/8\pi$ for the total and regular magnetic fields,
respectively) are shown in Fig. 12. The thermal energy density of the warm
ionized gas, $\frac{3}{2}\langle n_{e}\rangle kT_{e}$, is estimated from the
H$\alpha$ map assuming $T_{e}\simeq 10^{4}$ K (see also Sect. 5). Assuming the
pressure equilibrium between the warm and hot ionized gas with $T_{e}\simeq
10^{6}$ K and an electron density of $\simeq 0.01\langle n_{e}\rangle$ [e.g.
19], the energy density of the hot ionized gas is about the same order of
magnitude as the warm ionized gas energy density. For the neutral gas, we
derive the energy density of $\frac{3}{2}\langle n\rangle kT$ using the
average surface density of total (molecular + atomic) gas given by [11] and an
average temperature of $T\simeq 50$ K [40]. The warm neutral gas with a
typical temperature of $\simeq 6000$ K has roughly the same thermal energy
density as the cold neutral gas, due to a $\simeq 100$ times smaller density
[e.g. 19]. Assuming a constant scale height of the disk of 100pc [as used for
NGC6946 3], we obtain a gas density of $\langle n\rangle\simeq$ 6 cm-3 at R=1
kpc (that is about 8 times smaller than the corresponding value in NGC6946) to
$\simeq$ 2 cm-3 at R=5 kpc (about 3.5 times smaller than that in
NGC6946)555Hence, the radial profile of the energy density of the neutral gas
is much flatter in M33 than in NGC6946 [see Fig. 5 in 3]. The total thermal
energy density shown in Fig. 12 includes the contribution of the warm, hot
ionized and the cold, warm neutral gas. Figure. 12 also shows the kinetic
energy density of the turbulent motion of the neutral gas estimated using a
turbulent velocity of 10 km s-1 [13, 25].
Generally, the energy densities of all components are about the same order of
magnitude as in the Milky Way [15], but one order of magnitude smaller than in
NGC6946 [3]. The small thermal energy density compared to the total magnetic
field energy density shows that the M33’s ISM is a low $\beta$ plasma [$\beta$
is defined as the thermal to magnetic energy density ratio, similar results
were found in the Milky Way and NGC6946 15, 3]. On the other hand, the small
thermal energy density compared to that of the turbulent motions indicates
that turbulence in the diffuse ISM is supersonic [as in NGC6946 3]. This is in
agreement with 3-D MHD models for the ISM [16].
The energy densities of the total magnetic field and turbulent gas motions are
about the same. This hints to the generation of interstellar magnetic fields
from turbulent gas motions for R$<$8 kpc in M33. For a comparison at larger
radii, detection of the magnetic fields is required using deep surveys of
Faraday rotation of polarized background sources. This seems to be promising
as the scale length of the total magnetic field [$\simeq$24 kpc, 38] is much
larger than the present detection limit of the radio emission.
The energy density of the regular field is about 3–7 times smaller than that
of the total field. Furtheremore, it shows more variations with radius than
the total field strength, with a maximum increase at $4.5<R<5.5$ kpc.
### 6.4 Comparison with M31
Comparing magnetic fields in our neighbors, M33 and M31, is instructive
specially because a similar method was used to find out the 3-D magnetic field
structure in both galaxies. It is interesting to see that the total magnetic
field strength is about the same in M33 and M31 ($\simeq 7\,\mu$G), but the
regular field strength in M33 is about half of that in M31 ($\simeq 4\,\mu$G).
Furthermore, in contrast to M33, M31 has a disk plane parallel regular field
without a vertical component. In other words, the large-scale magnetic field
is well-ordered and strong relative to the small-scale field in M31. The
regular magnetic field is fitted by a dominant m=0 mode in M31 [20] that is
much stronger than that in M33.
The larger pitch angles of the horizontal magnetic field in M33 than in M31 is
not due to a smaller shear in M33. From their rotation curves [10, 12], the
shear rate at the relevant radii is larger in M33 ($>10$ km s-1 kpc-1 at
$1<R<5$ kpc) than in M31 ($\simeq$ 6 km s-1 kpc-1 at $8<R<12$ kpc).
The fact that M33 has a higher star formation efficiency than its 10-times
more massive neighbor, M31, may be a clue to the origin of their differences.
Strong starformation activities in the inner part of M33 could cause vertical
distribution of magneto-ionic matter and hence the vertical magnetic field.
Furthermore, stronger turbulence in the interstellar medium can be generally
caused by high starformation rate increasing the dynamo alpha-effect and hence
providing large pitch angles of the horizontal magnetic field.
## 7 Summary
The distributions of linearly polarized intensity and polarization angle at
3.6, 6.2, and 20 cm along with the maps of nonthermal intensity and nonthermal
spectral index [obtained from the new separation method, 38] yielded high-
resolution distributions of RM, nonthermal degree of polarization, and Faraday
depolarization in M33. Furthermore, we derived the 3-D structure of the
regular magnetic field by fitting the observed azimuthal distributions of the
polarization angle within two rings of 2 kpc width in the radial range 1 to 5
kpc. The main results and conclusions are as follows:
* 1\.
The average nonthermal degree of polarization is P${}_{\rm nth}\simeq$ 10% (at
3.6 cm) for $R<7.5$ kpc and $>\,20\%$ in parts of the magnetic filament. Due
to Faraday depolarization Pnth decreases to $\simeq$ 6% at 20 cm.
* 2\.
The intrinsic Faraday rotation shows larger small-scale variations and weaker
correlation with PI in the south than in the north of M33. The higher
starformation activity in the southern arms could increase the turbulent
velocities of interstellar clouds and disturb the regular field configuration.
On the other hand, a good correlation between RMi and PI in the magnetic
filament in the north-west of M33 shows that here the magnetic field is mainly
regular.
* 3\.
The average equipartition strengths of the total and regular magnetic fields
are $\,B_{\mathrm{tot}}\simeq 6.4\,\mu$G and $\,B_{\rm reg}\simeq 2.5\,\mu$G
for $R<7.5$ kpc. The regular magnetic field strength is higher within the ring
at $4.5<R<5.5$ kpc, which contains the magnetic filament that has a maximum
regular field of $\,B_{\rm reg}\simeq 6.6\,\mu$G. Strong turbulent magnetic
fields ($\,B_{\rm tur}>7\,\mu$G) occur in the extended central region and the
arms I S and part of II S.
* 4\.
The 3-D structure of the regular magnetic field can be explained by a
combination of azimuthal modes of $m=0+z0+z1$ in the ring 1–3 kpc and
$m=0+1+z1$ in the ring 3–5 kpc. The horizontal magnetic field component
follows an arm-like pattern with pitch angles smaller than those of the
optical arm segments, indicating that large-scale gas-dynamical effects such
as compression and shear are not solely responsible for the spiral magnetic
lines. The significant axisymmetric mode (m=0) in both rings indicates that
galactic dynamo action is present in M33.
* 5\.
The presence of vertical magnetic fields, shown by the best-fit model of the
3-D field structure ($z1$) and indicated by the Faraday rotation distribution
across the galaxy, is possibly due to both global (e.g. M33’s warp or
interaction with M31) and local (e.g. starformation activities, Parker loops)
phenomena. The warp can better explain the origin of the vertical field in the
outer ring (3–5 kpc).
* 6\.
In the southern half of M33, an excess of differential Faraday rotation
together with strong Faraday dispersion seem to be responsible for the north-
south asymmetry in the observed depolarization (which is wavelength
dependent).
* 7\.
The energy densities of the magnetic field and turbulence are about the same,
confirming the theory of generation of interstellar magnetic fields from
turbulent gas motions. Furthermore, it seems that the ISM in M33 can be
characterized by a low $\beta$ plasma and dominated by a supersonic
turbulence, as the energy densities of the magnetic field and turbulence are
both higher than the thermal energy density.
###### Acknowledgements.
We are grateful to E. M. Berkhuijsen, U. Klein and E. Krügel for valuable and
stimulating comments. FT was supported through a stipend from the Max Planck
Institute for Radio Astronomy (MPIfR). AF thanks the Leverhulme Trust for
financial support under research grant F/00 125/N.
## References
* Baldwin [1981] Baldwin, J. E. 1981, in Structure and Evolution of Normal Galaxies, ed. S. M. Fall & D. Lynden-Bell, 137–147
* Beck [1979] Beck, R. 1979, Ph.D. Thesis, Rheinische Friedrich-Wilhelms-Universitaet, Bonn.
* Beck [2007] Beck, R. 2007, A&A, 470, 539
* Beck et al. [1996] Beck, R., Brandenburg, A., Moss, D., Shukurov, A., & Sokoloff, D. 1996, ARA&A, 34, 155
* Beck & Krause [2005] Beck, R. & Krause, M. 2005, Astronomische Nachrichten, 326, 414
* Berkhuijsen et al. [1997] Berkhuijsen, E. M., Horellou, C., Krause, M., et al. 1997, A&A, 318, 700
* Broten et al. [1988] Broten, N. W., MacLeod, J. M., & Vallee, J. P. 1988, Ap&SS, 141, 303
* Buczilowski & Beck [1991] Buczilowski , U. R. & Beck , R. 1991, A&A, 241, 47
* Burn [1966] Burn, B. J. 1966, MNRAS, 133, 67
* Carignan et al. [2006] Carignan, C., Chemin, L., Huchtmeier, W. K., & Lockman, F. J. 2006, ApJ, 641, L109
* Corbelli [2003] Corbelli, E. 2003, MNRAS, 342, 199
* Corbelli & Salucci [2007] Corbelli, E. & Salucci, P. 2007, MNRAS, 374, 1051
* Corbelli & Schneider [1997] Corbelli, E. & Schneider, S. E. 1997, ApJ, 479, 244
* Cordes & Lazio [2002] Cordes, J. M. & Lazio, T. J. W. 2002, ArXiv Astrophysics e-prints
* Cox [2005] Cox, D. P. 2005, ARA&A, 43, 337
* de Avillez & Breitschwerdt [2007] de Avillez, M. A. & Breitschwerdt, D. 2007, ApJ, 665, L35
* Deul & van der Hulst [1987] Deul, E. R. & van der Hulst, J. M. 1987, A&AS, 67, 509
* Dumke et al. [2000] Dumke, M., Krause, M., & Wielebinski, R. 2000, A&A, 355, 512
* Ferrière [2001] Ferrière, K. M. 2001, Reviews of Modern Physics, 73, 1031
* Fletcher et al. [2004] Fletcher, A., Berkhuijsen, E. M., Beck, R., & Shukurov, A. 2004, A&A, 414, 53
* Johnston-Hollitt et al. [2004] Johnston-Hollitt, M., Hollitt, C. P., & Ekers, R. D. 2004, in The Magnetized Interstellar Medium, ed. B. Uyaniker, W. Reich, & R. Wielebinski, 13–18
* Krause [1990] Krause, M. 1990, in IAU Symposium, Vol. 140, Galactic and Intergalactic Magnetic Fields, ed. R. Beck, R. Wielebinski, & P. P. Kronberg, 187–196
* Krause et al. [1989] Krause, M., Hummel, E., & Beck, R. 1989, A&A, 217, 4
* Manchester & Mebold [1977] Manchester, R. N. & Mebold, U. 1977, A&A, 59, 401
* Milosavljević [2004] Milosavljević, M. 2004, ApJ, 605, L13
* Ohno & Shibata [1993] Ohno, H. & Shibata, S. 1993, MNRAS, 262, 953
* Reakes & Newton [1978] Reakes, M. L. & Newton, K. 1978, MNRAS, 185, 277
* Reich [2006] Reich, W. 2006, ArXiv Astrophysics e-prints
* Rogstad et al. [1976] Rogstad, D. H., Wright, M. C. H., & Lockhart, I. A. 1976, ApJ, 204, 703
* Ruzmaikin et al. [1988] Ruzmaikin, A. A., Sokolov, D. D., & Shukurov, A. M., eds. 1988, Astrophysics and Space Science Library, Vol. 133, Magnetic fields of galaxies, Chapter VI.4
* Sandage & Humphreys [1980] Sandage, A. & Humphreys, R. M. 1980, ApJ, 236, L1
* Segalovitz et al. [1976] Segalovitz, A., Shane, W. W., & de Bruyn, A. G. 1976, Nature, 264, 222
* Shukurov [2004] Shukurov, A. 2004, ArXiv Astrophysics e-prints
* Simard-Normandin & Kronberg [1980] Simard-Normandin, M. & Kronberg, P. P. 1980, ApJ, 242, 74
* Sokoloff et al. [1998] Sokoloff, D. D., Bykov, A. A., Shukurov, A., et al. 1998, MNRAS, 299, 189
* Tabara & Inoue [1980] Tabara, H. & Inoue, M. 1980, A&AS, 39, 379
* Tabatabaei et al. [2007a] Tabatabaei, F. S., Beck, R., Krause, M., Krügel, E., & Berkhuijsen, E. M. 2007a, Astronomische Nachrichten, 328, 636
* Tabatabaei et al. [2007b] Tabatabaei, F. S., Beck, R., Krügel, E., et al. 2007b, A&A, 475, 133
* Tabatabaei et al. [2007c] Tabatabaei, F. S., Krause, M., & Beck, R. 2007c, A&A, 472, 785
* Wilson et al. [1997] Wilson, C. D., Walker, C. E., & Thornley, M. D. 1997, ApJ, 483, 210
|
arxiv-papers
| 2008-09-02T12:18:49
|
2024-09-04T02:48:57.612568
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. S. Tabatabaei, M. Krause, A. Fletcher, and R. Beck",
"submitter": "Fatemeh Sadat Tabatabaei Asl",
"url": "https://arxiv.org/abs/0809.0419"
}
|
0809.0555
|
# Topological Black Holes in Brans-Dicke-Maxwell Theory
A. Sheykhi1,2111sheykhi@mail.uk.ac.ir and H. Alavirad1 1Department of Physics,
Shahid Bahonar University, P.O. Box 76175, Kerman, Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
###### Abstract
We derive a new analytic solution of $(n+1)$-dimensional $(n\geq 4)$ Brans-
Dikce-Maxwell theory in the presence of a potential for the scalar field, by
applying a conformal transformation to the dilaton gravity theory. These
solutions describe topological charged black holes with unusual asymptotics.
We obtain the conserved and thermodynamic quantities through the use of the
Euclidean action method. We also study the thermodynamics of the solutions and
verify that the conserved and thermodynamic quantities of the solutions
satisfy the first law of black hole thermodynamics.
Keywords: Brans-Dicke; dilaton; black holes.
## I Introduction
Among the alternative theories of general relativity, perhaps, the most well-
known theory is the scalar-tensor theory which was pioneered several decades
ago by Brans and Dicke BD , who sought to incorporate Mach’s principle into
gravity. Compared to Einstein’s general relativity, Brans-Dicke (BD) theory
describes the gravitation in terms of the metric as well as scalar fields and
accommodates both Mach’s principle and Dirac’s large number hypothesis as new
ingredients. Although BD theory has passed all the possible observational
tests Will , however, the singularity problem remains yet in this theory. In
recent years this theory got a new impetus as it arises naturally as the low
energy limit of many theories of quantum gravity such as the supersymmetric
string theory or the Kaluza-Klein theory. Besides, recent observations show
that at the present epoch, our Universe expands with acceleration instead of
deceleration along the scheme of standard Friedmann models and since general
relativity could not describe such Universe correctly, cosmologists have
attended to alternative theories of gravity such as BD theory. Due to highly
nonlinear character of BD theory, a desirable pre-requisite for studying
strong field situation is to have knowledge of exact explicit solutions of the
field equations. And as black holes are very important both in classical and
quantum gravity, many authors have investigated various aspects of them in BD
theory Sen ; Sen2 ; Sen3 . It turned out that the dynamic scalar field in the
BD theory plays an important role in the process of collapse and critical
phenomenon. The first four-dimensional black hole solutions of BD theory was
obtained by Brans in four classes Brans . It has been shown that among these
four classes of the static spherically symmetric solutions of the vacuum BD
theory of gravity only two are really independent, and only one of them is
permitted for all values of $\omega$. It has been proved that in four
dimensions, the stationary and vacuum BD solution is just the Kerr solution
with constant scalar field everywhere Hawking . It has been shown that the
charged black hole solution in four-dimensional Brans-Dicke-Maxwell (BDM)
theory is just the Reissner-Nordstrom solution with a constant scalar field,
however, in higher dimensions, one obtains the black hole solutions with a
nontrivial scalar field Cai1 . This is because the stress energy tensor of
Maxwell field is not traceless in the higher dimensions and the action of
Maxwell field is not invariant under conformal transformations. Accordingly,
the Maxwell field can be regarded as the source of the scalar field in the BD
theory Cai1 . Other studies on black hole solutions in BD theory have been
carried out in Kim ; Deh1 ; Gao ; Kim2 .
On another front, it has been realized that in four dimensions the topology of
the event horizon of an asymptotically flat stationary black hole is uniquely
determined to be the two-sphere $S^{2}$ Haw1 ; Haw2 . The “topological
censorship theorem” of Friedmann, Schleich and Witt indicates the
impossibility of non spherical horizons FSW1 ; FSW2 . However, when the
asymptotic flatness of spacetime is violated, there is no fundamental reason
to forbid the existence of static or stationary black holes with nontrivial
topologies. It was shown that for asymptotically AdS spacetime, in the four-
dimensional Einstein-Maxwell theory, there exist black hole solutions whose
event horizons may have zero or negative constant curvature and their
topologies are no longer the two-sphere $S^{2}$. The properties of these black
holes are quite different from those of black holes with usual spherical
topology horizon, due to the different topological structures of the event
horizons. Besides, the black hole thermodynamics is drastically affected by
the topology of the event horizon. It was argued that the Hawking-Page phase
transition Haw3 for the Schwarzschild-AdS black hole does not occur for
locally AdS black holes whose horizons have vanishing or negative constant
curvature, and they are thermally stable Birm . The studies on the topological
black holes have been carried out extensively in many aspects Lem ; Cai2 ;
Vanzo ; Bril1 ; Cai3 ; Cai4 ; Shey1 ; shey2 ; Cri ; Cri2 ; Cri3 ; MHD ;
jplemos ; huang ; rgcai0 ; rbmann ; Ban ; rgcai1 ; rgcai2 . In this paper, we
would like to study the topological black hole solutions in
$(n+1)$-dimensional BDM theory for an arbitrary value of $\omega$ and
investigate their properties.
In the next section, we review the basic equations and the conformal
transformation between the action of the dilaton gravity theory and the BD
theory. In section III, we construct charged topological black hole solutions
in BDM theory and investigate their properties. In section IV, we study the
thermodynamical properties of the solutions and calculate the conserved
quantities through the use of the Euclidean action method. The last section is
devoted to summary and discussion.
## II Basic equations and Conformal Transformations
The action of the $(n+1)$-dimensional Brans-Dicke-Maxwell theory with one
scalar field $\Phi$ and a self-interacting potential $V(\Phi)$ can be written
as
$I_{G}=-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n+1}x\sqrt{-g}\left(\Phi{R}-\frac{\omega}{\Phi}(\nabla\Phi)^{2}-V(\Phi)-F_{\mu\nu}F^{\mu\nu}\right)-\frac{1}{8\pi}\int{d^{n}x\sqrt{h}\Phi
K},$ (1)
where ${R}$ is the scalar curvature, $V(\Phi)$ is a potential for the scalar
field $\Phi$, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, and $A_{\mu}$ is the electromagnetic potential.
The factor $\omega$ is the coupling constant. The last term in Eq. (1) is the
Gibbons-Hawking surface term. It is required for the variational principle to
be well-defined. The factor $K$ represents the trace of the extrinsic
curvature for the boundary and $h$ is the induced metric on the boundary. The
equations of motion can be obtained by varying the action (1) with respect to
the gravitational field $g_{\mu\nu}$, the scalar field $\Phi$ and the gauge
field $A_{\mu}$ which yields the following field equations
$\displaystyle
G_{\mu\nu}=\frac{\omega}{\Phi^{2}}\left(\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(\nabla\Phi)^{2}\right)-\frac{V(\Phi)}{2\Phi}g_{\mu\nu}+\frac{1}{\Phi}\left(\nabla_{\mu}\nabla_{\nu}\Phi-
g_{\mu\nu}\nabla^{2}\Phi\right)$
$\displaystyle+\frac{2}{\Phi}\left(F_{\mu\lambda}F_{\nu}^{\
\lambda}-\frac{1}{4}F_{\rho\sigma}F^{\rho\sigma}g_{\mu\nu}\right),$ (2)
$\displaystyle\nabla^{2}\Phi=-\frac{n-3}{2(n-1)\omega+2n}F^{2}+\frac{1}{2(n-1)\omega+2n}\left((n-1)\Phi\frac{dV(\Phi)}{d\Phi}-(n+1)V(\Phi)\right),$
(3) $\displaystyle\nabla_{\mu}F^{\mu\nu}=0,$ (4)
where $G_{\mu\nu}$ and $\nabla$ are, respectively, the Einstein tensor and
covariant differentiation in the spacetime metric $g_{\mu\nu}$. It is apparent
that the right hand side of Eq. (2) includes the second derivatives of the
scalar field, so it is hard to solve the field equations (2)-(4) directly. We
can remove this difficulty by a conformal transformation. Indeed, the BDM
theory (1) can be transformed into the Einstein-Maxwell theory with a
minimally coupled scalar field via the conformal transformation
$\displaystyle\bar{g}_{\mu\nu}=\Phi^{\frac{2}{n-1}}g_{\mu\nu},$
$\displaystyle\bar{\Phi}=\frac{n-3}{4\alpha}\ln\Phi,$ (5)
where
$\alpha=\frac{n-3}{\sqrt{4(n-1)\omega+4n}}.$ (6)
Using this conformal transformation, the action (1) transforms to
$\bar{I}_{G}=-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n+1}x\sqrt{-\bar{g}}\left({\bar{R}}-\frac{4}{n-1}(\bar{\nabla}\
\bar{\Phi})^{2}-\bar{V}(\bar{\Phi})-e^{-\frac{4\alpha\bar{\Phi}}{n-1}}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}\right),$
(7)
where ${\bar{R}}$ and $\bar{\nabla}$ are, respectively, the Ricci scalar and
covariant differentiation in the spacetime metric $\bar{g}_{\mu\nu}$, and
$\bar{V}(\bar{\Phi})$ is
$\bar{V}(\bar{\Phi})=\Phi^{-\frac{n+1}{n-1}}V(\Phi).$ (8)
This action is just the action of the $(n+1)$-dimensional Einstein-Maxwell-
dilaton gravity, where $\bar{\Phi}$ is the dilaton field and
$\bar{V}(\bar{\Phi})$ is a potential for $\bar{\Phi}$. $\alpha$ is an
arbitrary constant governing the strength of the coupling between the dilaton
and the Maxwell field. Varying the action (7), we can obtain equations of
motion
$\displaystyle\bar{{R}}_{\mu\nu}=\frac{4}{n-1}\left(\bar{\nabla}_{\mu}\bar{\Phi}\bar{\nabla}_{\nu}\bar{\Phi}+\frac{1}{4}\bar{V}(\bar{\Phi})\bar{g}_{\mu\nu}\right)+2e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\left(\bar{F}_{\mu\lambda}\bar{F}_{\nu}^{\
\lambda}-\frac{1}{2(n-1)}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma}\bar{g}_{\mu\nu}\right),$
(9)
$\displaystyle\bar{\nabla}^{2}\bar{\Phi}=\frac{n-1}{8}\frac{\partial\bar{V}}{\partial\bar{\Phi}}-\frac{\alpha}{2}e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma},$
(10)
$\displaystyle\bar{\nabla}_{\mu}\left(e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\bar{F}^{\mu\nu}\right)=0.$
(11)
Comparing Eqs. (2)-(4) with Eqs. (9)-(11), we find that if
$\left(\bar{g}_{\mu\nu},\bar{F}_{\mu\nu},\bar{\Phi}\right)$ is the solution of
Eqs. (2)-(4) with potential $\bar{V}(\bar{\Phi})$, then
$\left[{g}_{\mu\nu},{F}_{\mu\nu},{\Phi}\right]=\left[\exp\left({\frac{-8\alpha\bar{\Phi}}{(n-1)(n-3)}}\right)\bar{g}_{\mu\nu},\bar{F}_{\mu\nu},\exp\left({\frac{4\alpha\bar{\Phi}}{n-3}}\right)\right],$
(12)
is the solution of Eqs. (9)-(11) with potential $V(\Phi)$.
## III Topological black holes in BDM theory
The solutions of the field equations (9)-(11) for various metrics have been
constructed by many authors (see e.g. CHM ; Cai22 ; Clem ; DF ; DF2 ; SDR ;
SDR2 ). Here we would like to obtain the topological black hole solutions of
the field equations (2)-(4) in BDM theory, by applying the conformal
transformations (12) on the corresponding solutions in the dilaton gravity
theory. The $(n+1)$-dimensional topological black hole solution of the field
equations (9)-(11) has been obtained by one of us in shey2 for two Liouville-
type dilaton potentials
$\bar{V}(\bar{\Phi})=2\Lambda_{0}e^{2\zeta_{0}\bar{\Phi}}+2\Lambda
e^{2\zeta\bar{\Phi}},$ (13)
where $\Lambda_{0}$, $\Lambda$, $\zeta_{0}$ and $\zeta$ are constants. In
shey2 the spacetime metric was written in the form
$d\bar{s}^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}{R^{2}(r)}h_{ij}dx^{i}dx^{j},$
(14)
where $f(r)$ and $R(r)$ are functions of $r$ which should be determined, and
$h_{ij}$ is a function of coordinate $x_{i}$ which spanned an
$(n-1)-$dimensional hypersurface with constant scalar curvature $(n-1)(n-2)k$.
Here $k$ is a constant which characterized the hypersurface. Without loss of
generality, one can take $k=0,1,-1$, such that the black hole horizon or
cosmological horizon in (14) can be a zero (flat), positive (elliptic) or
negative (hyperbolic) constant curvature hypersurface. The Maxwell equations
can be integrated immediately to give
$\bar{F}_{tr}=\frac{qe^{\frac{4\alpha\bar{\Phi}}{n-1}}}{(rR)^{n-1}},$ (15)
where $q$, an integration constant, is related to the electric charge of black
hole. Defining the electric charge via
$Q=\frac{1}{4\pi}\int\exp\left[{-4\alpha\bar{\Phi}/(n-1)}\right]\text{
}^{*}\bar{F}d{\Omega},$ we get
${Q}=\frac{q\Omega_{n-1}}{4\pi},$ (16)
where $\Omega_{n-1}$ represents the volume of constant curvature hypersurface
described by $h_{ij}dx^{i}dx^{j}$. Notice that $Q$ is invariant under the
conformal transformation (12). Using the ansatz
$R(r)=e^{\frac{2\alpha\bar{\Phi}}{n-1}},$ (17)
one can show that the system of equations (9)-(10) have solutions of the form
shey2
$\displaystyle
f(r)=-\frac{k(n-2)(\alpha^{2}+1)^{2}b^{-2\gamma}r^{2\gamma}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}-\frac{m}{r^{(n-1)(1-\gamma)-1}}+\frac{2q^{2}(\alpha^{2}+1)^{2}b^{-2(n-2)\gamma}}{(n-1)(\alpha^{2}+n-2)}r^{2(n-2)(\gamma-1)}$
(20) $\displaystyle+$
$\displaystyle\frac{2\Lambda(\alpha^{2}+1)^{2}b^{2\gamma}}{(n-1)(\alpha^{2}-n)}r^{2(1-\gamma)},$
$\displaystyle R(r)=\left(\frac{b}{r}\right)^{\gamma},$
$\displaystyle\bar{\Phi}=\frac{(n-1)\alpha}{2(1+\alpha^{2})}\ln\left(\frac{b}{r}\right),$
where $b$ is an arbitrary constant and $\gamma=\alpha^{2}/(\alpha^{2}+1)$. In
the above expression, $m$ appears as an integration constant and is related to
the mass of the black hole. In order to fully satisfy the system of equations,
we must have shey2
$\zeta_{0}=\frac{2}{\alpha(n-1)},\hskip
22.76228pt\zeta=\frac{2\alpha}{n-1},\hskip
22.76228pt\Lambda_{0}=\frac{k(n-1)(n-2)\alpha^{2}}{2b^{2}(\alpha^{2}-1)}.$
(21)
Notice that here $\Lambda$ is a free parameter which plays the role of the
cosmological constant. Using the conformal transformation (12), the
$(n+1)$-dimensional topological black hole solutions of BDM theory can be
obtained as
$ds^{2}=-U(r)dt^{2}+\frac{dr^{2}}{V(r)}+r^{2}{H^{2}(r)}h_{ij}dx^{i}dx^{j},$
(22)
where $U(r)$, $V(r)$, $H(r)$ and $\Phi(r)$ are
$\displaystyle
U(r)=-\frac{k(n-2)(\alpha^{2}+1)^{2}b^{-2\gamma(\frac{n-1}{n-3})}r^{2\gamma(\frac{n-1}{n-3})}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}-\frac{mb^{(\frac{-4\gamma}{n-3})}}{r^{n-2}}r^{\gamma\left(n-1+\frac{4}{n-3}\right)}$
$\displaystyle+\frac{2q^{2}(\alpha^{2}+1)^{2}b^{-2\gamma\left(n-2+\frac{2}{n-3}\right)}}{(n-1)\left(\alpha^{2}+n-2\right)r^{2[(n-2)(1-\gamma)-\frac{2\gamma}{n-3}]}}+\frac{2\Lambda(\alpha^{2}+1)^{2}b^{2\gamma(\frac{n-5}{n-3})}}{(n-1)(\alpha^{2}-n)}r^{2(1-\frac{\gamma(n-5)}{n-3})},$
(23) $\displaystyle
V(r)=-\frac{k(n-2)(\alpha^{2}+1)^{2}b^{-2\gamma(\frac{n-5}{n-3})}r^{2\gamma(\frac{n-5}{n-3})}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}-\frac{mb^{(\frac{4\gamma}{n-3})}}{r^{n-2}}r^{\gamma(n-1-\frac{4}{n-3})}$
$\displaystyle+\frac{2q^{2}(\alpha^{2}+1)^{2}b^{-2\gamma\left(n-2-\frac{2}{n-3}\right)}}{(n-1)(\alpha^{2}+n-2)r^{2[(n-2)(1-\gamma)+\frac{2\gamma}{n-3}]}}+\frac{2\Lambda(\alpha^{2}+1)^{2}b^{2\gamma(\frac{n-1}{n-3})}}{(n-1)(\alpha^{2}-n)}r^{2(1-\gamma(\frac{n-1}{n-3}))},$
(24) $\displaystyle H(r)=\left(\frac{b}{r}\right)^{\frac{(n-5)\gamma}{n-3}},$
(25)
$\displaystyle\Phi(r)=\left(\frac{b}{r}\right)^{\frac{2(n-1)\gamma}{n-3}}.$
(26)
Applying the conformal transformation, the electromagnetic field and the
scalar potential become
$\displaystyle F_{tr}$ $\displaystyle=$
$\displaystyle\frac{qb^{(3-n)\gamma}}{r^{(n-3)(1-\gamma)+2}},$ (27)
$\displaystyle V(\Phi)$ $\displaystyle=$ $\displaystyle
2\Lambda_{0}\Phi^{\frac{n(\alpha^{2}+1)+\alpha^{2}-3}{\alpha^{2}(n-1)}}+2\Lambda\Phi^{2}.$
(28)
It is worth noting that in the case $k\neq 0$, these solutions are ill-defined
for the string case where $\alpha=1$ (this is corresponding to
$\omega=(n-9)/4$). It is also notable to mention that the electric field
$F_{tr}$ and the scalar field $\Phi(r)$ go to zero as $r\rightarrow\infty$.
When $\omega\rightarrow\infty$ ($\alpha=0=\gamma$), these solutions reduce to
$\displaystyle
U(r)=V(r)=k-\frac{m}{r^{n-2}}+\frac{2q^{2}}{(n-1)(n-2)r^{2(n-2)}}-\frac{2\Lambda}{n(n-1)}r^{2},$
(29)
which describes an $(n+1)$-dimensional asymptotically (anti)-de Sitter
topological black holes with a positive, zero or negative constant curvature
hypersurface (see e.g. Bril1 ; Cai3 ). It is easy to show that the Kretschmann
scalar $R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ diverge at $r=0$ and
therefore there is an essential singularity at $r=0$. As one can see from Eqs.
(23)-(24), the solutions are also ill-defined for $\alpha=\sqrt{n}$ with
$\Lambda\neq 0$ (corresponding to $\omega=-3(n+3)/4n$). The cases with
$\alpha<\sqrt{n}$ and $\alpha>\sqrt{n}$ should be considered separately. In
the first case where $\alpha<\sqrt{n}$, there exist a cosmological horizon for
$\Lambda>0$, while there is no cosmological horizons if $\Lambda<0$. Indeed,
in the latter case ($\alpha<\sqrt{n}$ and $\Lambda<0$) the spacetimes exhibit
a variety of possible casual structures depending on the values of the metric
parameters $\alpha$, $m$, $q$ and $k$ shey2 . Therefore, our solutions can
represent topological black hole, with inner and outer event horizons, an
extreme topological black hole, or a naked singularity provided the parameters
of the solutions are chosen suitably. In the second case where
$\alpha>\sqrt{n}$, the spacetime has a cosmological horizon for $\Lambda<0$
despite the value of curvature constant $k$, while for $\Lambda>0$ we have
cosmological horizon in the case $k=1$ and naked singularity for $k=0,-1$.
## IV Thermodynamics of topological BD black hole
We now turn to the investigation of the thermodynamics of charged topological
BD black holes we have just found. The Hawking temperature of the topological
black hole on the outer horizon $r_{+}$ can be calculated using the relation
$T_{+}=\frac{\kappa}{2\pi}=\frac{U^{\text{
}^{\prime}}(r_{+})}{4\pi\sqrt{U/V}},$ (30)
where $\kappa$ is the surface gravity. Then, one can easily show that
$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle-\frac{(\alpha^{2}+1)}{2\pi(n-1)}\left(\frac{k(n-2)(n-1)b^{-2\gamma}}{2(\alpha^{2}-1)}r_{+}^{2\gamma-1}+\Lambda
b^{2\gamma}r_{+}^{1-2\gamma}+q^{2}b^{-2(n-2)\gamma}r_{+}^{(2n-3)(\gamma-1)-\gamma}\right)$
(31) $\displaystyle=$
$\displaystyle-\frac{k(n-2)(\alpha^{2}+1)b^{-2\gamma}}{2\pi(\alpha^{2}+n-2)}r_{+}^{2\gamma-1}+\frac{(n-\alpha^{2})m}{4\pi(\alpha^{2}+1)}{r_{+}}^{(n-1)(\gamma-1)}$
$\displaystyle-\frac{q^{2}(\alpha^{2}+1)b^{-2(n-2)\gamma}}{\pi(\alpha^{2}+n-2)}{r_{+}}^{(2n-3)(\gamma-1)-\gamma}.$
If we compare Eq. (31) with the temperature obtained in the dilaton gravity
theory shey2 , we find that the temperature is invariant under the conformal
transformation (12). This is due to the fact that the conformal parameter is
regular at the horizon. Equation (31) also shows that when $k=0$, the
temperature is negative for two cases (_i_) $\alpha>\sqrt{n}$ despite the sign
of $\Lambda$, and (_ii_) positive $\Lambda$ despite the value of $\alpha$. As
we argued above in these two cases we encounter cosmological horizons.
Physically it is not easy to accept the negative temperature, the temperature
on the cosmological horizon should be defined as $T=|\kappa|/2\pi$ so that it
becomes a positive since on the cosmological horizon the surface gravity is
negative.
The ADM (Arnowitt-Deser-Misner) mass $M$, entropy $S$ and electric potential
$U$ of the topological black hole can be calculated through the use of the
Euclidean action method CaiSu . In this approach, first the electric potential
and the temperature are fixed on a boundary with a fixed radius $r_{+}$. The
Euclidean action has two parts; bulk and surface. The first step to make the
Euclidean action is to substitute $t$ with $i\tau$. This makes the metric
positive definite:
$ds^{2}=U(r)d\tau^{2}+\frac{1}{V(r)}dr^{2}+r^{2}H^{2}(r)h_{ij}dx^{i}dx^{j}.$
(32)
There is a conical singularity at the horizon $r=r_{+}$ in the Euclidean
metric CaiSu . To eliminate it, the Euclidian time $\tau$ is made periodic
with period $\beta$, where $\beta$ is the inverse of Hawking temperature. Now
we obtain the Euclidean action of $(n+1)$-dimensional Brans-Dicke-Maxwell
theory. The Euclidean action can be calculated analytically and continuously
changing of action (1) to Euclidean time $\tau$, i.e.,
$I_{GE}=-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n+1}x\sqrt{g}\left(\Phi{R}-\frac{\omega}{\Phi}(\nabla\Phi)^{2}-V(\Phi)-F_{\mu\nu}F^{\mu\nu}\right)-\frac{1}{8\pi}\int{d^{n}x\sqrt{h}\Phi(K-K_{0})},$
(33)
where $K_{0}$ is the trace of the extrinsic curvature on the metric $h$ for
our metric background with $q=0$ and $m=0$, which must be added so that it can
normalize the Euclidean action to zero in this spacetime Brown . Using the
metric (32), we find
$\displaystyle R$ $\displaystyle=$
$\displaystyle-g^{-1/2}(g^{1/2}U^{\prime}V/U)^{\prime}-2G^{0}_{0},$ (34)
$\displaystyle K$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{V}\left[rHU^{\prime}+2(n-1)\left(UH+rUH^{\prime}\right)\right]}{2rHU},$
(35)
where $G^{0}_{0}$ is the (00) component of the Einstein tensor. Inserting
$U(r)$ and $V(r)$ from (23) and (24) with $q=0$ and $m=0$ in $K$ we obtain the
extrinsic curvature for the metric background
$\displaystyle K_{0}$ $\displaystyle=$
$\displaystyle\left({\frac{b}{r}}\right)^{{\frac{2\gamma}{n-3}}}\left(-{\frac{k\left(n-2\right)\left({\alpha}^{2}+1\right)^{2}{b}^{-2\,\gamma}{r}^{2\,\gamma}}{\left({\alpha}^{2}-1\right)\left({\alpha}^{2}+n-2\right)}}+{\frac{2\Lambda\,\left({\alpha}^{2}+1\right)^{2}{b}^{2\,\gamma}{r}^{2-2\,\gamma}}{\left(n-1\right)\left({\alpha}^{2}-n\right)}}\right)^{1/2}$
(36)
$\displaystyle\times\left[2{b}^{2\,\gamma}\left(\alpha^{2}-1\right)\left(n\gamma-n+3-5\,\gamma\right)\left({\alpha}^{2}+n-2\right)n\Lambda\,{r}^{2-2\,\gamma}\right.$
$\displaystyle\left.+\left(n-2\right){r}^{2\,\gamma}{b}^{-2\,\gamma}\left(n-{\alpha}^{2}\right)\left(n-1\right)^{2}k\left(\gamma\,n-n-6\,\gamma+3\right)\right]\left(n-3\right)^{-1}r^{-1}$
$\displaystyle\times\left[2{b}^{2\,\gamma}\Lambda\,\left(\alpha^{2}-1\right)\left({\alpha}^{2}+n-2\right){r}^{2-2\,\gamma}+{r}^{2\,\gamma}{b}^{-2\,\gamma}k\left(n-1\right)\left(n-2\right)\left(n-{\alpha}^{2}\right)\right]^{-1}$
Substituting Eqs. (34)-(36) in action (33) and using Eqs. (23)-(28), after a
long calculation, we obtain the Euclidean action as
$\displaystyle
I_{GE}=\beta\frac{\Omega_{n-1}}{16\pi}\left(\frac{b^{(n-1)\gamma}(n-1)m}{(\alpha^{2}+1)}\right)-\frac{\Omega_{n-1}}{4}\left(b^{(n-1)\gamma}r_{+}^{(n-1)(1-\gamma)}\right)-\beta\frac{\Omega_{n-1}q^{2}}{8\pi\Upsilon
r_{+}^{\Upsilon}},$ (37)
where $\Upsilon=(n-3)(1-\gamma)+1$. According to Ref. Brown ; Brown2 ; Brown3
; Brown4 , the thermodynamical potential can be given by $I_{GE}$, we get
$I_{GE}=\beta M-S-\beta Uq,$ (38)
where $M$ is the ADM mass, $S$ and $U$ are, respectively, the entropy and the
electric potential. Comparing Eq. (37) with Eq. (38), we find
${M}=\frac{b^{(n-1)\gamma}(n-1)\Omega_{n-1}}{16\pi(\alpha^{2}+1)}m,$ (39)
$S=\frac{b^{(n-1)\gamma}r_{+}^{(n-1)(1-\gamma)}}{4}\Omega_{n-1},$ (40)
$U=\frac{qb^{(3-n)\gamma}\Omega_{n-1}}{\Upsilon r_{+}^{\Upsilon}}.$ (41)
Comparing the conserved and thermodynamic quantities calculated in this
section with those obtained in shey2 , we find that they are invariant under
the conformal transformation (12). It is worth emphasizing that in BD theory,
where we have the additional gravitational scalar degree of freedom, the
entropy of the black hole does not follow the area law kang . This is due to
the fact that the black hole entropy comes from the boundary term in the
Euclidean action formalism. Nevertheless, the entropy remains unchanged under
the conformal transformations. Finally, we consider the first law of
thermodynamics for the topological black hole. It is a matter of calculation
to show that the the conserved and thermodynamic quantities obtained above
satisfy the first law of black hole thermodynamics
$dM=TdS+Ud{Q}.$ (42)
## V Summary and discussion
To conclude, in $(n+1)$-dimensions, when the $(n-1)$-sphere of black hole
event horizn is replaced by an $(n-1)$-dimensional hypersurface with positive,
zero or negative constant curvature, the black hole is called as a topological
black hole. The construction and analysis of these exotic black holes in anti-
de Sitter (AdS) space is a subject of much recent interest. This interest is
motivated by the correspondence between the gravitating fields in an AdS
spacetime and conformal field theory on the boundary of the AdS spacetime. In
this paper, we further generalized these exotic solutions by constructing a
class of $(n+1)$-dimensional $(n\geq 4)$ topological black holes in BDM theory
in the presence of a potential for the scalar field. In contrast to the
topological black holes in the Einstein-Maxwell theory, which are
asymptotically AdS, the topological black holes we found here, are neither
asymptotically flat nor (A)dS. Indeed, the scalar potential plays a crucial
role in the existence of these black holes, as the negative cosmological
constant does in the Einstein-Maxwell theory. When $k=\pm 1$, these solutions
do not exist for the string case where $\alpha=1$ (corresponding to
$\omega=(n-9)/4$). Besides they are ill-defined for $\alpha=\sqrt{n}$ with
$\Lambda\neq 0$ (corresponding to $\omega=-3(n+3)/4n$). We obtained the
conserved and thermodynamic quantities through the use of the Euclidean action
method, and verified that the conserved and thermodynamic quantities of the
solutions satisfy the first law of black hole thermodynamics. We found that
the entropy does not satisfy the area law. We also found that the conserved
and thermodynamic quantities are invariant under the conformal transformation.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran.
## References
* (1) C. Brans and R. H. Dicke, Phys. Rev. $\mathbb{124}$, 925 (1961).
* (2) C. M. Will, Theory and Experiment in Gravitational Physics, (Cambridge University Press, Cambridge, 1993).
* (3) A. Sen, Phys. Rev. Lett. $\mathbb{69}$, 1006 (1992).
* (4) G. W. Gibbons and K. Maeda, Ann. Phys. (N.Y.) $\mathbb{167}$, 201 (1986).
* (5) V. Frolov, A. Zelinkov and U. Bleyer, Ann. Phys. (Leipzig) $\mathbb{44}$, 371 (1987).
* (6) C. H. Brans, Phys. Rev. $\mathbb{125}$, 2194 (1962).
* (7) S. W. Hawking, Commun. Math. Phys. $\mathbb{25}$, 167 (1972).
* (8) R. G. Cai and Y. S. Myung, Phys. Rev. D. $\mathbb{56}$, 3466 (1997).
* (9) H. Kim, Phys. Rev. D, $\mathbb{60}$, 024001 (1999).
* (10) M. H. Dehghani, J. Pakravan, and S. H. Hendi, Phys. Rev. D 74, 104014 (2006).
* (11) C. J. Gao and S. N. Zhang, gr-qc/0604083.
* (12) H. Kim, Nuovo Cim. B 112 (1997) 329.
* (13) S. W. Hawking and G. F. Ellis. The large scale structure of spacetime (Cambridge University Press, Cambridge, England, 1973).
* (14) S. W. Hawking. Commun. Math. Phys. 25, 152 (1972).
* (15) J. L. Friedman, K. Schleich and D. M. Witt, Phys. Rev. Lett. 71, 1486 (1993).
* (16) J. L. Friedman, K. Schleich, and D. M. Witt, Phys. Rev. Lett. 75, 1872 (1995).
* (17) S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983).
* (18) D. Birmingham, Class. Quant. Gravit. 16, 1197 (1999).
* (19) J. P. S. Lemos, Phys. Lett. B 353, 46 (1995).
* (20) R. G. Cai and Y. Z. Zhang, Phys. Rev. D 54, 4891 (1996).
* (21) L. Vanzo, Phys. Rev. D 56, 6475 (1997).
* (22) D. R. Brill, J. Louko, and P. Pelda n, Phys. Rev. D 56, 3600 (1997).
* (23) R. G. Cai and K. S. Soh, Phys. Rev. D 59 (1999) 044013.
* (24) R. G. Cai, J.Y. Ji, and K. S. Soh, Phys. Rev. D 57(1998) 6547 .
* (25) A. Sheykhi, Phys. Letts. B 662 (2008) 7.
* (26) A. Sheykhi, Phys. Rev. D 76(2007) 124025 .
* (27) J. Criso stomo, R. Troncoso, and J. Zanelli, Phys. Rev. D 62, 084013 (2000).
* (28) R. Aros, R. Troncoso, and J. Zanelli, Phys. Rev. D 63, 084015 (2001).
* (29) R. G. Cai, Phys. Rev. D 65, 084014 (2002).
* (30) M. H. Dehghani, Phys. Rev. D 70, 064019 (2004).
* (31) J. P. Lemos, Class. Quant. Gravit. 12 (1995) 1081.
* (32) C. G. Huang and C. B. Liang, Phys. Lett. A 201(1995) 27 .
* (33) R. G. Cai, Nucl. Phys. B 524 (1998)639.
* (34) R. B. Mann, Class. Quant. Gravit. 14, L109 (1997).
* (35) M. Banados, A. Gomberoff, and C. Mart nez, Class. Quant. Gravit. 15 (1998) 3575.
* (36) R. G. Cai, Phys. Rev. D 65(2002) 084014.
* (37) R. G. Cai, Phys. Lett. B 582(2004) 237 .
* (38) K. C. K. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B447 (1995) 441 .
* (39) R. G. Cai and Y. Z. Zhang, Phys. Rev D 64 (2001) 104015.
* (40) G. Clement, D. Gal’tsov and C. Leygnac, Phys. Rev. D 67(2003) 024012.
* (41) M. H Dehghani and N. Farhangkhah, Phys. Rev. D 71 (2005) 044008.
* (42) M. H Dehghani, Phys. Rev. D 71 (2005) 064010.
* (43) A. Sheykhi, M. H. Dehghani, N. Riazi, Phys. Rev. D 75 (2007) 044020.
* (44) A. Sheykhi, M. H. Dehghani, N. Riazi and J. Pakravan Phys. Rev. D 74 (2006) 084016.
* (45) R. G. Cai, R. K. Su, and P. K. N. Yu, Phys. Lett. A 195, (1994) 307.
* (46) J.D. Brown, E.A. Martinez and J.W. York Jr., Phys. Rev. Lett. 66 (1991) 2281.
* (47) H.W. Braden, J.D. Brown, B.F. Whiting and J.W. York Jr., Phys. Rev. D 42 (1990) 3376.
* (48) B.F. Whiting and J.W. York Jr., Phys. Rev. Lett. 61 ( 1988 ) 1336.
* (49) J.W. York Jr., Phys. Rev. D 33 ( 1986 ) 2092.
* (50) G. Kang, Phys. Rev. D 54, (1996) 7483.
|
arxiv-papers
| 2008-09-03T07:13:29
|
2024-09-04T02:48:57.622036
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi and H. Alavirad",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0809.0555"
}
|
0809.0648
|
# Semi-inclusive processes
at low and high transverse momentum
Alessandro Bacchetta1 Daniel Boer2 Markus Diehl3 and Piet J. Mulders2
1Jefferson Lab, 12000 Jefferson Ave, Newport News, VA 23606, USA
2Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands
3Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany
###### Abstract
This talk reports on recent work where we studied the connection between the
description of semi-inclusive DIS at high transverse momentum (based on
collinear factorization) and low transverse momentum (based on transverse-
momentum-dependent factorization). We used power counting to determine the
leading behavior of the structure functions at intermediate transverse
momentum in the two descriptions. When the power behaviors are different, two
distinct mechanisms are present and there can be no matching between them.
When the power behavior is the same, the two descriptions must match. An
explicit calculation however shows that for some observables this is not the
case, suggesting that the transverse-momentum-dependent-factorization
description beyond leading twist is incomplete.
## 1 Introduction
The cross section for polarized semi-inclusive DIS can be written in terms of
18 structure functions [2]. Each of them depends, among other variables, on
the square of the transverse momentum of the outgoing hadron,
$P_{h\perp}^{2}$, with respect to the virtual photon direction. For
theoretical considerations, it is often preferable to consider the transverse
momentum $q_{T}^{2}\approx P_{h\perp}^{2}/z^{2}$. The problem involves three
scales, namely the scale of nonperturbative QCD dynamics, which we represent
by the nucleon mass $M$, the transverse momentum $q_{T}$, and the photon
virtuality $Q$, which we require to be large compared with $M$.
At high $q_{T}$ ($q_{T}\gg M$) the structure functions can be described using
collinear factorization, i.e., in terms of collinear distribution and
fragmentation functions together with perturbative radiation. At low-$q_{T}$
($q_{T}\ll Q$) the structure functions can be described using Collins–Soper
TMD factorization [3, 4], i.e., in terms transverse-momentum-dependent (TMD)
parton distribution and fragmentation functions. The low- and high-$q_{T}$
domains overlap for $M\ll q_{T}\ll Q$ (intermediate $q_{T}$), where both
descriptions can hence be applied.
Studying the relation between the high-$q_{T}$ and low-$q_{T}$ regimes is
important both from the theoretical and phenomenological point of view. We
observe that in some cases the calculations in the two regimes have to give
the same result at intermediate $q_{T}$, i.e., they have to match. If this
does not occur, we can make the important conclusion that there is some flaw
in the formalism. We observe in other cases that the two calculations describe
different mechanisms and therefore do not have to match. Both of them have to
be taken into consideration independently in the overlap region.
## 2 Matches and mismatches: general discussion
To assess whether the high-$q_{T}$ and low-$q_{T}$ calculations have to match
or not in the intermediate-$q_{T}$ region it is sufficient to study the power
behavior of the structure functions in the two regimes.
It is important to realize that the power expansions are done in two different
ways in the two descriptions. At low $q_{T}$, first we expand in
$(q_{T}/Q)^{n-2}$ and neglect terms with $n$ bigger than a certain value (so
far, analyses have been carried out only up to $n=3$, i.e., twist-3). To study
the behavior at intermediate $q_{T}$ we further expand in $(M/q_{T})^{k}$.
Conversely, at high $q_{T}$ we first expand in $(M/q_{T})^{n}$ (also in this
case, analyses are available up to $n=3$, i.e., twist-3). To study the
intermediate-$q_{T}$ region, we further expand in $(q_{T}/Q)^{k-2}$.
We can encounter two different situations. For simplicity, we will refer to
them as type-I and type-II observables.
### 2.1 Type-I observables
Consider, e.g., a structure functions described by two contributions
$F={A}\,\frac{M^{2}}{M^{2}+q_{T}^{2}}+{B}\,\frac{q_{T}^{2}}{Q^{2}}\frac{M^{2}}{M^{2}+q_{T}^{2}}.$
(1)
At low transverse momentum, term $B$ is neglected from the very beginning
because it is of order $(q_{T}/Q)^{2}$ (twist-4). The remaining term is
$F^{\text{twist-2}}_{\rm
low}={A}\,\frac{M^{2}}{M^{2}+q_{T}^{2}}=A\,\frac{M^{2}}{q_{T}^{2}}+{\cal
O}\left(\frac{M^{4}}{q_{T}^{4}}\right),$ (2)
where the second step identifies the leading term at $q_{T}\gg M$.
At high transverse momentum, both terms $A$ and $B$ are twist-2 and are taken
into consideration. However, the second term is neglected if a further
expansion in $q_{T}/Q$ is performed, to study the regime of intermediate
transverse momentum, i.e.,
$F^{\text{twist-2}}_{\rm
high}=A\,\frac{M^{2}}{q_{T}^{2}}+B\,\frac{M^{2}}{Q^{2}}=A\,\frac{M^{2}}{q_{T}^{2}}+{\cal
O}\left(\frac{q_{T}^{2}}{Q^{2}}\right).$ (3)
Therefore, the leading terms in the two expansions are the same. In this case,
the calculations at high and low transverse momentum must yield exactly the
same result at intermediate transverse momentum [5, 6]. If a mismatch occurs,
it means that one of the calculations is incorrect or incomplete.
### 2.2 Type-II observables: expected mismatches
Consider the example of a structure functions composed by two terms
$F={A}\,\frac{M^{4}}{M^{4}+q_{T}^{4}}+{B}\,\frac{q_{T}^{2}}{Q^{2}}\frac{M^{2}}{M^{2}+q_{T}^{2}}.$
(4)
At low transverse momentum, term $B$ is neglected from the very beginning
because it is of order $(q_{T}/Q)^{2}$ (twist-4). What is left is
$F^{\text{twist-2}}_{\rm
low}={A}\,\frac{M^{4}}{M^{4}+q_{T}^{4}}=A\,\frac{M^{4}}{q_{T}^{4}}+{\cal
O}\left(\frac{M^{8}}{q_{T}^{8}}\right),$ (5)
where in the second step we expanded in $M/q_{T}$.
At high transverse momentum, the term $A$ is now twist-4 and it is usually
neglected. Only the second term is kept and gives
$F^{\text{twist-2}}_{\rm
high}=B\,\frac{q_{T}^{2}}{Q^{2}}\,\frac{M^{2}}{q_{T}^{2}}$ (6)
In this case, if the calculations at high and low transverse momentum are
performed at their respective leading twist, they correspond to two different
contributions to the cross section and will not lead to the same result at
intermediate transverse momentum. In order to “match”, the calculations would
have to be carried out in both regimes up to the sub-subleading order. We
could call this situation an “expected mismatch”, since it is simply due to
the difference between the two expansions.
## 3 Matches and mismatches: semi-inclusive DIS case
In Tab. 3 we list the power behavior of the structure functions at
intermediate transverse momentum, as obtained from the limits of the
low-$q_{T}$ and high-$q_{T}$ calculation. For details of the calculation, we
refer to Ref. Bacchetta:2008xw. In the names of the structure functions, the
first and second subscript respectively specifies the polarization of the beam
and the target. When present, the third subscript refers to the polarization
of the photon.
Behavior of SIDIS structure functions in the region $M\ll q_{T}\ll Q$. Empty
fields indicate that no calculation is available (in this case, twist $4$
indicates observables that are zero when calculated up to twist-three
accuracy). Yes/no in parentheses: expected answers based on analogy, rather
than actual calculation. low-$q_{T}$ calculation high-$q_{T}$ calculation
power exact observable twist power twist power match match $F_{UU,T}$ 2
$1/q_{T}^{2}$ 2 $1/q_{T}^{2}$ yes yes $F_{UU,L}$ 4 2 $1/Q^{2}$
$F^{\cos\phi_{h}}_{UU}$ 3 $1/(Q\mskip 1.5muq_{T})$ 2 $1/(Q\mskip 1.5muq_{T})$
yes no $F^{\cos 2\phi_{h}}_{UU}$ 2 $1/q_{T}^{4}$ 2 $1/Q^{2}$ no
$F^{\sin\phi_{h}}_{LU}$ 3 $1/(Q\mskip 1.5muq_{T})$ 2 $1/(Q\mskip 1.5muq_{T})$
yes (no) $F^{\sin\phi_{h}}_{UL}$ 3 $1/(Q\mskip 1.5muq_{T})$ (yes) (no)
$F^{\sin 2\phi_{h}}_{UL}$ 2 $1/q_{T}^{4}$ (no) $F_{LL}$ 2 $1/q_{T}^{2}$ 2
$1/q_{T}^{2}$ yes yes $F^{\cos\phi_{h}}_{LL}$ 3 $1/(Q\mskip 1.5muq_{T})$ 2
$1/(Q\mskip 1.5muq_{T})$ yes no $F^{\sin(\phi_{h}-\phi_{S})}_{UT,T}$ 2
$1/q_{T}^{3}$ 3 $1/q_{T}^{3}$ yes yes $F^{\sin(\phi_{h}-\phi_{S})}_{UT,L}$ 4 3
$1/(Q^{2}\mskip 1.5muq_{T})$ $F^{\sin(\phi_{h}+\phi_{S})}_{UT}$ 2
$1/q_{T}^{3}$ 3 $1/q_{T}^{3}$ yes (yes) $F^{\sin(3\phi_{h}-\phi_{S})}_{UT}$ 2
$1/q_{T}^{3}$ 3 $1/(Q^{2}\mskip 1.5muq_{T})$ no $F^{\sin\phi_{S}}_{UT}$ 3
$1/(Q\mskip 1.5muq_{T}^{2})$ 3 $1/(Q\mskip 1.5muq_{T}^{2})$ yes (no)
$F^{\sin(2\phi_{h}-\phi_{S})}_{UT}$ 3 $1/(Q\mskip 1.5muq_{T}^{2})$ 3
$1/(Q\mskip 1.5muq_{T}^{2})$ yes (no) $F^{\cos(\phi_{h}-\phi_{S})}_{LT}$ 2
$1/q_{T}^{3}$ (yes) (yes) $F^{\cos\phi_{S}}_{LT}$ 3 $1/(Q\mskip
1.5muq_{T}^{2})$ (yes) (no) $F^{\cos(2\phi_{h}-\phi_{S})}_{LT}$ 3 $1/(Q\mskip
1.5muq_{T}^{2})$ (yes) (no)
In summary, the calculation at high $q_{T}$ is done using standard collinear
factorization, as done in, e.g., Ref. Mendez:1978zx,Koike:2006fn and in Ref.
Eguchi:2006mc for the subleading-twist sector. To obtain the power behavior at
intermediate $q_{T}$, we need to perform an expansion in $q_{T}/Q$. The
calculation has no fundamental difficulties and allows us to fill in the third
column of Tab. 3. The blank entries correspond to the structure functions that
have not yet been computed in the high-transverse-momentum regime.
The calculation at low $q_{T}$ is done using TMD factorization [3, 4]. The
behavior of the TMD functions at intermediate transverse momentum can be
calculated perturbatively by considering diagrams as the ones depicted in Fig.
1.
Figure 1: Example diagrams for the calculation of the high-$p_{T}$ behavior
TMD parton distribution functions. $\Phi_{A}^{\alpha}$ represent the quark-
gluon-quark correlator. The dashed lines represent the final-state cut.
We calculated the power behavior of all twist-2 and twist-3 TMD functions,
which allowed us to fill in the second column of Tab. 3. Two structure
functions cannot be calculated as they require twist-4 contributions, which
are beyond the current limits of the TMD factorization framework.
The fourth column of Tab. 3 is obtained by comparing the second and third
column. The structure functions with a “yes” are type-I observables, those
with a “no” are type-II. The values in parentheses are expectations based on
analogy with similar structure functions, since the high-$q_{T}$ calculations
are not available.
Beside studying the power behavior, we also calculated the explicit form of
some of the TMD functions at intermediate transverse momentum, namely the ones
requiring only the evaluation of diagrams analogous to that of Fig. 1(a). A
calculation of the Sivers function, requiring the evaluation of diagrams like
that of Fig. 1(b), was already performed in Refs. Ji:2006ub,Koike:2007dg.
The explicit calculations allows us to check if for type-I observables the
explicit expressions obtained from high and low transverse momentum exactly
match or not. The results are listed in the fifth column of Tab. 3. The
entries in parentheses are conjectures based on analogy rather than actual
calculation.
### 3.1 Type-I structure functions
For type-I structure functions (“yes” in the column “power match”), we know
from power counting that the two calculations describe the same physics and
should therefore exactly match. In these cases, the high-$q_{T}$ calculation
corresponds to the perturbative tail of the low-$q_{T}$ effect. The two
mechanisms need not be distinguished. Using resummation it should be possible
to construct expressions for these observables that are valid at any $q_{T}$,
as was done for the Drell-Yan analog of $F_{UU,T}$ in Ref. Collins:1984kg.
Only five of these structure functions have been calculated explicitly:
$F_{UU,T}$, $F_{LL}$ and $F^{\sin(\phi_{h}-\phi_{S})}_{UT,T}$ (Sivers
structure function) present an exact matching [6, 11], while in our work we
showed that $F^{\cos\phi_{h}}_{UU}$ and $F^{\cos\phi_{h}}_{LL}$ do not match.
In analogy to these results, we expect that also
$F^{\sin(\phi_{h}+\phi_{S})}_{UT}$ (Collins structure function) and
$F^{\cos(\phi_{h}-\phi_{S})}_{LT}$ will match exactly, while problems will
occur with all the others, since they are twist-3 in the low-$q_{T}$ regime,
and the TMD factorization formalism is probably complete only at twist 2.
The structure function $F^{\sin(\phi_{h}-\phi_{S})}_{UT,T}$, related to the
Sivers function, is an example of a match between high- and low-$q_{T}$. Some
of the consequences of the calculation are:
* •
the leading (twist-3, in this case) contribution of the high-$q_{T}$
calculation corresponds to the tail of the Sivers function at intermediate
$q_{T}$, it is not a competing effect and should not be summed to the Sivers
function;
* •
it is conceivable to construct an expression that extends the high-$q_{T}$
calculation to $q_{T}\approx M$, through a smooth merging into the Sivers
function;
* •
since the structure function falls as $1/q_{T}^{3}$, it is safe to use
$q_{T}$-weighted asymmetries to extract the Sivers function.
As an example of a mismatch we consider the structure function
$F_{UU}^{\cos\phi_{h}}$, related to the Cahn effect. We show in this case the
main steps of the calculation to explain the nature of the problem. In the
low-$q_{T}$ formalism, the expression for this observable is [2]
$\displaystyle F_{UU}^{\cos\phi_{h}}=\frac{2M}{Q}\,\mathcal{C}\biggl{[}$
$\displaystyle-\frac{\hat{\bm{h}}{\mskip-1.5mu}\cdot{\mskip-1.5mu}\bm{k}_{T}}{M_{h}}\biggl{(}xh\,H_{1}^{\perp}+\frac{M_{h}}{M}\,f_{1}\frac{\tilde{D}^{\perp}}{z}$
$\displaystyle-\frac{\hat{\bm{h}}{\mskip-1.5mu}\cdot{\mskip-1.5mu}\bm{p}_{T}}{M}\biggl{(}xf^{\perp}D_{1}+\frac{M_{h}}{M}\,h_{1}^{\perp}\frac{\tilde{H}}{z}\biggr{)}\biggr{]},$
(7)
where the convolution means
$\displaystyle\mathcal{C}\bigl{[}wfD\bigr{]}=\sum_{a}x\mskip 1.5mue_{a}^{2}$
$\displaystyle\int
d^{2}\bm{p}_{T}\,d^{2}\bm{k}_{T}\,d^{2}\bm{l}_{T}\,\delta^{(2)}\bigl{(}\bm{p}_{T}-\bm{k}_{T}+\bm{l}_{T}+\bm{q}_{T}\bigr{)}$
$\displaystyle\times
w(\bm{p}_{T},\bm{k}_{T})\,f^{a}(x,p_{T}^{2})\,D^{a}(z,k_{T}^{2})\,U(l_{T}^{2})\,.$
(8)
The term $U$ denotes the so-called soft factor. It is obtained in the
factorization proof for twist-two observables. Here we assume we can use it
also for twist-three observables. The terms with $h_{1}^{\perp}$ and
$H_{1}^{\perp}$ fall off as $1/p_{T}^{3}$ or $1/k_{T}^{3}$ and are power
suppressed compared to the terms with $f^{\perp}$ and $\tilde{D}^{\perp}$ when
$q_{T}\gg M$. For intermediate $q_{T}$ we therefore have
$F_{UU}^{\cos\phi_{h}}=-\frac{2q_{T}}{Q}\sum_{a}x\mskip
1.5mue_{a}^{2}\,\biggl{[}xf^{\perp
a}(x,q_{T}^{2})\,\frac{D_{1}^{a}(z)}{z^{2}}-f_{1}^{a}(x)\,\frac{\tilde{D}^{\perp
a}(z,q_{T}^{2})}{z}\biggr{]}$ (9)
at leading power. In this case there is no leading contribution from the soft
factor taken at large transverse momentum. The tail of the functions at
$q_{T}\gg M$ can be calculated perturbatively and yields
$\displaystyle xf^{\perp q}(x,p_{T}^{2})$
$\displaystyle=\frac{\alpha_{s}}{2\pi^{2}}\,\frac{1}{2\bm{p}_{T}^{2}}\,\biggl{[}\mskip
1.5mu\frac{L(\eta^{-1})}{2}\,f_{1}^{q}(x)+\bigl{(}P^{\prime}_{qq}\otimes
f_{1}^{q}+P^{\prime}_{qg}\otimes f_{1}^{g}\bigr{)}(x)\biggr{]}\,,$
$\displaystyle\frac{\tilde{D}^{\perp q}(z,k_{T}^{2})}{z}$
$\displaystyle=-\frac{\alpha_{s}}{2\pi^{2}}\,\frac{1}{2z^{2}\mskip 1.5mu{\bm
k}{}_{T}^{2}}\,\biggl{[}\mskip
1.5mu\frac{L(\eta_{\smash{h}}^{-1})}{2}\,D_{1}^{q}(z)-2C_{F}D_{1}^{q}(z)$
$\displaystyle\hskip 85.35826pt+\bigl{(}D_{1}^{q}\otimes
P^{\prime}_{qq}+D_{1}^{g}\otimes P^{\prime}_{gq}\bigr{)}(z)\biggr{]}\,,$ (10)
where $L(y)=2C_{F}\ln y-3C_{F}\,,$ and $P_{qq}^{\prime}$, $P_{gq}^{\prime}$,
$P_{qg}^{\prime}$ are kernels specific to the functions under consideration.
The parameters $\eta$ and $\eta_{h}$ are related to the choice of a
nonlightlike gauge (Wilson line) in the calculation of the functions and
fulfill the relation $\sqrt{\eta\eta_{h}}=q_{T}^{2}/Q^{2}$. Putting these
ingredients together we arrive at
$\displaystyle F_{UU}^{\cos\phi_{h}}$ $\displaystyle=-\frac{1}{Q\mskip
1.5muq_{T}}\,\frac{\alpha_{s}}{2\pi^{2}z^{2}}\sum_{a}x\mskip
1.5mue_{a}^{2}\,\biggl{[}f_{1}^{a}(x)\,D_{1}^{a}(z)\,L\biggl{(}\frac{Q^{2}}{q_{T}^{2}}\biggr{)}$
$\displaystyle\qquad+f_{1}^{a}(x)\,\bigl{(}D_{1}^{a}\otimes
P_{qq}^{\prime}+D_{1}^{a}\otimes P_{gq}^{\prime}\bigr{)}(z)$
$\displaystyle\qquad+\bigl{(}P_{qq}^{\prime}\otimes
f_{1}^{a}+P_{qg}^{\prime}\otimes
f_{1}^{g}\bigr{)}(x)\,D_{1}^{a}(z)-2C_{F}\mskip
1.5muf_{1}^{a}(x)\,D_{1}^{a}(z)\biggr{]}\,.$ (11)
This expression differs from the one obtained at high $q_{T}$ by the last term
$2C_{F}f_{1}^{a}(x)\,D_{1}^{a}(z)$. At this point we are forced to conclude
that the description of twist-3 structure functions is incomplete in the TMD-
factorization formalism. However, it is interesting to note that adding a term
$f_{1}^{a}(x)\,D_{1}^{a}(z)z^{-2}U(q_{T}^{2})/2$ within brackets in Eq. (9)
would be sufficient to cure this problem. It is however not clear how such an
expression would be obtained from a factorized formula.
### 3.2 Type-II structure functions
For type-II structure functions (“no” in the column “power match”) the
low-$q_{T}$ and high-$q_{T}$ calculations at leading order pick up two
different components of the full structure function. They therefore describe
two different mechanisms and do not match.
An example of a type-II observable is the structure function $F^{\cos
2\phi_{h}}_{UU}$, related at low $q_{T}$ to the Boer–Mulders function [12].
Some studies of this structure functions have recently appeared [13, 14].
However, some considerations have to be kept in mind:
* •
the leading contribution from the high-$q_{T}$ calculation (often referred to
as a pQCD or radiative correction) is a competing effect that has to be taken
into account;
* •
it is at present not possible to construct an expression that extends the
high-$q_{T}$ calculation to $q_{T}\approx M$, since this requires a smooth
merging into unknown twist-4 contributions in TMD factorization;
* •
Using $q_{T}$-weighted asymmetries to extract the Boer–Mulders function is not
a good idea, since the high-$q_{T}$ mechanism dominates the observable;
* •
a solution to the above problems could be to consider observables that are
least sensitive to the effect of radiative corrections, for instance by
considering specific combinations of structure functions.
We stress that the above considerations apply not only to semi-inclusive DIS,
but also to Drell–Yan and $e^{+}e^{-}$ annihilation [15]. Drell–Yan data have
been already used to extract [16] the Boer–Mulders function, without taking
into account radiative corrections, while the extraction [17] of the Collins
function from $e^{+}e^{-}$ relies on the cancellation of radiative effects
through the construction of suitable experimental observables [18, 15].
## References
* [1]
* [2] A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders and M. Schlegel, JHEP 02, 093 (2007), hep-ph/0611265.
* [3] J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981).
* [4] X. Ji, J.-P. Ma and F. Yuan, Phys. Rev. D71, 034005 (2005), hep-ph/0404183.
* [5] J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B250, 199 (1985).
* [6] X. Ji, J.-W. Qiu, W. Vogelsang and F. Yuan, Phys. Rev. Lett. 97, 082002 (2006), hep-ph/0602239.
* [7] A. Bacchetta, D. Boer, M. Diehl and P. J. Mulders, JHEP 08, 023 (2008), arXiv:0803.0227 [hep-ph].
* [8] A. Mendez, Nucl. Phys. B145, 199 (1978).
* [9] Y. Koike, J. Nagashima and W. Vogelsang, Nucl. Phys. B744, 59 (2006), arXiv:hep-ph/0602188.
* [10] H. Eguchi, Y. Koike and K. Tanaka, Nucl. Phys. B763, 198 (2007), arXiv:hep-ph/0610314.
* [11] Y. Koike, W. Vogelsang and F. Yuan, Phys. Lett. B659, 878 (2008), arXiv:0711.0636 [hep-ph].
* [12] D. Boer and P. J. Mulders, Phys. Rev. D57, 5780 (1998), hep-ph/9711485.
* [13] V. Barone, A. Prokudin and B.-Q. Ma (2008), arXiv:0804.3024 [hep-ph].
* [14] B. Zhang, Z. Lu, B.-Q. Ma and I. Schmidt (2008), arXiv:0807.0503 [hep-ph].
* [15] D. Boer (2008), arXiv:0804.2408 [hep-ph].
* [16] B. Zhang, Z. Lu, B.-Q. Ma and I. Schmidt, Phys. Rev. D77, 054011 (2008), arXiv:0803.1692 [hep-ph].
* [17] M. Anselmino et al., Phys. Rev. D75, 054032 (2007), hep-ph/0701006.
* [18] K. Abe et al., Phys. Rev. Lett. 96, 232002 (2006), hep-ex/0507063.
* [19]
|
arxiv-papers
| 2008-09-03T15:35:00
|
2024-09-04T02:48:57.627632
|
{
"license": "Public Domain",
"authors": "Alessandro Bacchetta (JLab), Daniel Boer (VU Amsterdam), Markus Diehl\n (DESY), Piet J. Mulders (VU Amsterdam)",
"submitter": "Alessandro Bacchetta",
"url": "https://arxiv.org/abs/0809.0648"
}
|
0809.0708
|
# Relativistisk rapsodi
Frank G. Borg
###### Abstract.
F religgande ess behandlar den Speciella Relativitetsteorin med historiska
kommentarer, samt diverse f rs k att modifiera teorin. Trots mer n hundra r p
nacken finns det inget som tyder p att Relativitetsteorin kommer att ers ttas
inom n ra framtid med n gon b ttre lokal teori f r rumtiden.
Jyv skyl Universitet, Karleby Universitetscenter Chydenius, POB 567,
67101-Karleby, Finland. Email: borgbros@netti.fi
###### Contents
1. I Den speciella relativitetsteorin
1. 1 Inledning
2. 2 Symmetri I
3. 3 Egentid och gauge-teori
4. 4 Einstein vs Poincar
5. 5 Konstant acceleration
6. 6 Geometri och fysik
7. 7 Symmetri II. No-interaction-theorem (NIT)
8. 8 Symmetri III. Begreppet massa
9. 9 Tid och relativitet
2. II Kvantrum och SR-modifikationer
1. 10 Dubbel speciell relativitet (DSR)
2. 11 Kvantrum
3. 12 Avslutning
## Part I Den speciella relativitetsteorin
### 1\. Inledning
Den speciella (SR) och allm nna relativitetsteorin111I b rjan anv nde Einstein
enbart beteckningen ’relativitetsprincip’. Planck inf rde 1906 beteckningen
’Relativtheorie’ medan A. H. Bucherer lanserade ’Relativit tstheorie’ som b
rjade nyttjas av Einstein 1907. Felix Klein f reslog 1910 utan framg ng namnet
’Invariantentheorie’. Efter 1915 skiljde Einstein mellan den ’speciella’ och
den ’allm nna relativitetsteorin’. Se J. J. Stachel, Einstein’s miraculous
years: Five papers that changed the face of physics (Princeton U. P., 1998) s.
102. (ART) utg r tv av den mod rna fysikens grundpelare j msides med
kvantmekaniken. Relativitetsteorins st llning r - f r att f lja den inslagna
metaforiska linjen - till den grad grundmurad att man kanske s llan gnar den n
gon st rre uppm rksamhet. F rvisso finns det en och annan som t ex hoppas
kunna p visa att Lorentz-symmetrin inte g ller till hundra procent222D.
Mattingly, ”Modern tests of Lorentz invariance” gr-qc/0502097; M. Pospelov &
M. Romalis, ”Lorentz invariance on trial,” Physics Today 40-46 (July 2004); C.
M. Will, ”Special relativity: A centennary perspective,” gr-qc/0504085. Will
anm rker att ”On the 100th anniversary of special relativity, we see that the
theory has been so thoroughly integrated into the fabric of modern physics
that its validity is rarely challenged, except by cranks and crackpots. It is
ironic then, that during the past several years, a vigorous theoretical and
experimental effort has been launched, on an international scale, to find
violations of special relativity. The motivation for this effort is not a
desire to repudiate Einstein, but to look for evidence of new physics ’beyond’
Einstein, such as apparent violations of Lorentz invariance that might result
from models of quantum gravity. So far, special relativity has passed all
these new high-precision tests, but the possibility of detecting a signature
of quantum gravity, stringiness, or extra dimensions will keep this effort
alive for some time to come”. Vi terkommer till fr gan om ’bortom’ Einstein-
fysiken i senare avsnitt. H r kan man inflika att inom mod rn fysik r det ju
en inbyggd reflex att associera symmetrier med brutna symmetrier varf r inte
heller Lorentz kan undg misstankar om ’symmetribrott’.. SR utg r en s
integrerad del av de ”kovarianta” och eleganta f ltteoriformalismerna att den
kanske sj lv n stan hamnat ur sikte333Detta r ett f rh llande som B. Greene
tar upp i f rordet till en ny utg va av A. Einstein, The meaning of relativity
(Princeton U. P., 2005, f rsta utg van 1922 baserad p f rel sningar vid
Princeton universitetet 1921, sista uppl. utkom postumt 1956). ”As a
professional physicist, it is easy to become inured to relativity. Whereas the
equations of relativity were once startling statements fashioned within the
language of mathematics, physicists how now written relativity into the very
mathematical grammar of fundamental physics. Within this framework, properly
formulated mathematical equations automatically take full account of
relativity, and so by mastering a few mathematical rules one becomes
technically fluent in Einstein’s discoveries. Nevertheless, even though
relativity has been systematized mathematically, the vast majority of
physicists would say that they still don’t ’feel relativity in their bones’
(…) but I can also attest to the undiminished feeling of awe I experience each
time I pay sufficient attention to the details hidden within mathematics
streamlined for relativistic economy, and come face to face with the true
meaning of relativity. Space and time form the very arena of reality. The
seismic shift in this arena caused by relativity is nothing short of an
upheavel in our basic conception of reality.” (s. vii - viii.) (B.G. har
slarvat en aning med f rordet och lyckas stava namnen Riemann, Friedman och
Schwarzschild fel samt kallar den senare f r en ryss.) Einsteins framst llning
r elegant och koncis och i litter r klass med Diracs ber mda The principles of
quantum mechanics (Oxford U. P. 1930).. Emellertid, den revolution som
Einstein startade i fysiken r l ngt fr n ver. Rummets och tidens problematik
kommer att fascinera m nniskan s l nge hon r kapabel att t nka. I denna ess
som kretsar kring SR kommer vi att anv nda metrikkonventionen $(+---)$ f r
Minkowski-metriken $g_{\mu\nu}$.
### 2\. Symmetri I
Relativitetsteorin444A. Einstein, ”Zur Elektrodynamik bewegter K rper,” Ann.
d. Phys. 17, 549-560 (1905). formulerades (principiellt) som ett svar p
diskrepansen som hade uppkommit mellan Newtons mekanik och Maxwells
elektrodynamik. Klassisk mekanik omfattade ’Galileis relativitetsprincip’ som
f r tv inertiala referensystem i relativ translatorisk r relse postulerar
transformationen (1D-fallet)
$\displaystyle x^{\star}$ $\displaystyle=$ $\displaystyle x-vt$ (1)
$\displaystyle t^{\star}$ $\displaystyle=$ $\displaystyle t$ (2)
Detta r vad det ’praktiska f rnuftet’ mer eller mindre f reskriver. Fig.1 ger
en geometrisk beskrivning av transformationen (1). M ttlinjen (Eichkurve) EE⋆
som tr ffar tidsaxlarna i punkterna E och E⋆ r inf rd f r att vi skall f
$t=t^{\star}$. N mligen, f r att erh lla de fysikaliska koordinatv rdena ($t$,
$t^{\star}$) p axlarna $t$ och $t^{\star}$ skall vi dela med avst ndena OE
respektive OE⋆: $t$ = OT:OE; $t^{\star}$ = OT⋆:OE⋆.
Figure 1. Galilei-transformationen
Einstein ins g att Galilei-transformationen (1, 2) r inkonsistent med kravet p
att ekvationen f r ljusets utbredning (vi s tter h refter ljushastigheten i
vakuum $c$ = 1)
$t^{2}-x^{2}=0$ (3)
r invariant f r inertialsystem och att den kr ver en symmetrisk behandling av
tiden och rummet i motsats till (1, 2) och Fig.1. I Fig.2 visas en
symmetriserad transformation f r $x$ och $t$.
Figure 2. Lorentz-transformationen
Den streckade diagonalen r en del av ljuskonen (3) och parabeln r en del av
den nya m ttlinjen
$t^{2}-x^{2}=1.$ (4)
Den konventionella presentationen av den nya symmetriska transformationen
brukar skrivas
$\displaystyle x^{\star}$ $\displaystyle=$ $\displaystyle\gamma(v)(x-vt)$ (5)
$\displaystyle t^{\star}$ $\displaystyle=$ $\displaystyle\gamma(v)(t-vx)$ (6)
$\displaystyle\gamma(v)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-v^{2}}}.$ (7)
Uttrycket f r gamma-faktorn f ljer fr n invarianskravet och leder till Ekv.(4)
f r m ttlinjen. Man kan ocks best mma $\gamma$-faktorn genom att transformera
tillbaka till ($x$, $t$)-systemet som anv nder samma ekvationer (5)-(6) men
med hastigheten $-v$. Transformationsformlerna f ljer ocks genom att observera
att (4) kan parametriseras med hj lp av de hyperboliska funktionerna,
$\displaystyle t$ $\displaystyle=$ $\displaystyle\cosh(\phi_{o})$ (8)
$\displaystyle x$ $\displaystyle=$ $\displaystyle\sinh(\phi_{0})$ (9)
En transformation (5, 6) motsvaras d av en ’rotation’
$\displaystyle\phi_{0}\rightarrow\phi_{0}-\phi\quad\Rightarrow$ $\displaystyle
x^{\star}=\sinh(\phi_{0}-\phi)=\cosh(\phi)\left(\sinh(\phi_{0})-\tanh(\phi)\cosh(\phi_{0})\right)$
$\displaystyle=\gamma(x-vt).$
Fr n uttrycket $v=\tanh(\phi)$ f r hastigheten kan man enkelt h rleda den
relativistiska formen f r additionsteoremet av hastigheter
$u\oplus v=\frac{u+v}{1+uv}.$ (10)
Den geometriska tolkningen ovan g r tillbaka p Hermann Minkwoski och har
populariserats bl a av Max Born och Rolf Nevanlinna.555Max Born, Die Relativit
tstheorie Einsteins (3. uppl. Springer, 1922); Rolf Nevanlinna,
Suhteellisuusteorian periaatteet (WSOY, 1963). En annan elegant geometrisk h
rledning av transformationsformlerna baseras p den s.k. Bondi-faktorn, se H.
Bondi, Relativity and common sense (Dover 1980) - anv nds t ex av M. Ludvigsen
i General relativity. A geometric approach (Cambridge U.P., 1999), kap. 3.
Minkowskis geometrisering (1908) var ett avg rande steg som sedan gjorde det m
jligt f r att Einstein att utveckla den allm nna relativitetsteorin.
### 3\. Egentid och gauge-teori
I Minkowski-rum ber knas ett objekts ’egentid’ $\tau$ genom ekvationen
$\tau=\int_{\Gamma}\sqrt{1-v^{2}}\,dt,$ (11)
d r $\Gamma$ betecknar dess bana i rumtiden. Givet en bana s r motsvarande f
rlopp av egentid en invariant storhet. Egentiden mellan tv punkter A och B
beror av vilken bana man f ljer vilket leder till den sk ’tvilling-paradoxen’.
Betr ffande ’tvilling-paradoxen’ (eller ’klockparadoxen’, ’Uhren-paradoxon’)
har jag faktiskt f r m nga r sedan tr ffat p en matematiker som vid
kaffebordet ih rdigt argumenterade f r att dess ’l sning’ kr ver ART och ett
beaktande av gravitationen.666R. Dugas har i History of mechanics (Dover 1988)
ett kort avsnitt som heter ”Pseudo-paradoxes in special relativity” (495-7) d
r han betr ffande tvilling-paradoxen anm rker att ”this paradox is so a
pseudo-paradox: it is not in the field of validity of the Lorentz-Einstein
theory” utan ”outside the scope of special relativity”. Man kan undra vad som
gett upphov till en tradition som ansett att accelererande r relse ligger
utanf r SR? Att inertiala referenssystem har en s rst llning i SR betyder ju
inte att SR inte kan hantera accelererande referens-system. Ist llet f r en
anomali kan ’tvilling-paradoxen’ anv ndas just f r att framh va en fundamental
egenhet hos SR. Det kan anm rkas att A. Einstein ocks i ett skede 1918 s kte
efter en sorts dynamisk f rklaring till ’tvilling-paradoxen’ baserad p ART.
’Tvilling-paradoxen’ formulerad i termer av m nskliga resen rer g r f. .
tillbaka p en uppsats av Langevin 1911, se diskussionen i A. I. Miller, Albert
Einstein’s special theory of relativity. Emergence (1905) and early
interpretations (1905-1911) (Springer, 1998), s. 242-248. Som synes r
’paradoxen’ dock en direkt f ljd av v gberoendet av integralen (11). Man kan h
r uppfatta en analogi till m tt-teorier (gauge theories). Faktiskt r det m
jligt att anv nda just egentiden och dess v gberoende som ett s tt att belysa
relativitetsteorins inneb rd i analogi med dubbelspaltexperimentet i
kvantmekaniken. H rifr n kan man d direkt g vidare till den allm nna
relativitetsteorin (ART) och f rklara att den modifierar SR genom att mass-
energi ocks p verkar egentiden. Sj lva begreppet egentid r t mligen enkelt att
definiera popul rt som den tid som en ideal medf ljande klocka skulle ange.
Fr n (11) ser vi att maximal egen-tid mellan tv punkter A och B i rumtid
motsvarar en bana med l ngsammast m jliga f rlopp ($v$ = 0). Om vi t nker oss
att en s dan bana motsvarar en fri kropps r relse s ligger det n ra till hands
att s ka dylika r relser som l sningar till en minimering av den negativa
egentiden. Mer exakt s anv nds inom SR formen
$S=-m\int\sqrt{1-v^{2}}\,dt$ (12)
f r verkan d r $m$ betecknar objektets (vilo)massa. Ekv.(12) representerar den
enklaste formen av en relativistisk invariant icke-trivial verkan. Genom
verkan (12) kommer vi n rmare en koppling mellan relativitetsteori och m tt-
teori eftersom verkan f rekommer i kvantmekaniken genom fas-faktorn
$\exp(iS/\hbar)$ (denna fas-faktor lyftes fram av Dirac777P. A. M. Dirac, ”The
Lagrangian in quantum mechanics,” Physikalische Zeitschrift der Sowjetunion,
Band 3, Heft 1 (1933); se ocks ”On the analogy between classical and quantum
mechanics,” Rev. Mod. Phys. 17 (2 & 3), 195-9 (1945). Feynmans
doktorsavhandling godk ndes 1942 och ges ut av L. M. Brown tillsammans med
Diracs artikel i Feynman’s thesis - a new approach to quantum mechanics (World
Scientific, 2005). och utvecklades senare av R. P. Feynman till v
gintegralformalismen).
### 4\. Einstein vs Poincar
Som bekant framlade Henri Poincar Reltativitetsprincipen f re Einstein och
visade bl a att transformationen (5)-(6), som han kallade f r888Den tidigaste
formen av (5) - (6) f rmodas vara Woldemar Voigts framst llning fr n 1887 som
behandlade transformationerna som matematiskt knep vid l sningen av v
gekvationerna. ’Lorentz transformation’, bildar en matematisk grupp. nd r det
Einstein som av de flesta r knas som Relativtetsteorins upphovsman.999Enligt
en modern myt, som nyligen t ex propagerats av en PSB-’dokument r’ och som
visats bl a i Sverige (magasinet ”Vetenskapens v rld”) och Finland, skulle
Mileva Maric varit en dold medf rfattare till Einsteins SR-artikel 1905. John
Stachel ser sig n dsakad att i 2005 rs upplaga av Einstein’s miraculous years
(Princeton U. P., 2005) punktera denna historia som f tts genom en feltolkning
av ett citat av Abraham Joffe. P ”Vetenskapens v rlds” webbsida hittar man bl
a p st endet att ”Vetenskapens v rld lyfter ocks fram en rysk version av
uppsatsen, f rvarad i ett arkiv i Moskva, d r man kan l sa dubbelnamnet
Einstein-Maric.” Stachels unders kningar visar att dokumentet i fr ga r en
kopia av en kopia av en passage ur en rysk popul rvetenskaplig bok fr n 1962
som feltolkar ett citat av Joffe som ingenstans skrivit att n gon artikel av
Einstein skulle ha undertecknats med ’Einstein-Marty’. Det finns emellertid en
passage d r Joffe i en artikel 1955 ber ttar om de revolutionerande artiklarna
fr n 1905 vars f rfattare, ”an unknown person at that time, was a bureaucrat
at the Patent Office in Bern, Einstein-Marity (Marity - the maiden name of his
first wife, which by Swiss custom is added to husbands family name.)” Denna
passage (d r man utel mnar parentesen) utg r hela grundstenen f r
mytfabrikationen kring Mileva Maric som medf rfattare till SR! Upphovskvinnan
till myten r den serbiska f rfattarinnan, Desanka Trubohovic-Gjuric, som
skrivit en biografi ver Mileva Maric versatt till tyska 1983. En skillnad
mellan Einstein och Poincar har sagts vara att Poincar var en ’fixare’ som n
jde sig med att f rs ka ’lappa’ ihop Lorentzs elektronteori. Enligt
vetenskapshistorikern Peter Galison101010P. L. Galison & D. G. Burnett,
’Einstein, Poincar & modernity: a conversation” (2003),
http://www.aip.org/history/einstein/essay-einsteins-time.htm. R. Torretti sin
sida anser att Poincar s ’misslyckande’ till stor del best mdes av hans
filosofiska syn: ”I am therefore inclined to attribute Poincar ’s failure to
another aspect of his philosophy, namely, his conventionalism”. (R. Torretti,
Relativity and geometry (Dover, 1996) s. 87.) Einstein utvecklade med tiden en
t mligen sofistikerad syn p filosofi och epistemologi men med den brasklappen
att samtidigt som forskaren beh ver filosofisk-epistemologiska insikter har
denne inte r d med att binda sig vid n gon slutgiltig position en r i sista
ndan r det verkligheten som best mmer och den kan vi aldrig fullst ndigt k
nna. Vi kan inte i f rv g veta vad vi har att v nta oss. Principer och ’ismer’
skall inom forskningen vara v gledande men inte spika fast m let. Ph. Frank
verraskades d Einstein ’ vergav’ den sorts positivism som synbart verkar ha
varit utm rkande f r skapandet av SR, medan M. Born t ex h ll fast vid en
deduktiv uppfattning av vetenskapen vilken Einstein inte heller omfattade. r
1919, d Einstein i ett slag blev v rldsber md f r den stora allm nheten efter
solf rm rkelseexpeditionen vars m tningar kom att st da hans teori, skrev han
ett intressant inl gg f r Berliner Tageblatt (25.12.1919) vilket diskuteras
och terges av A. Adam i ”Farewell to certitude: Einstein’s novelty on
induction and deduction, fallibism” (J. for General Philosophy of Science 31,
19-37, 2000). I denna korta skrivelse framl gger Einstein fallibism-id n in
nuce, som senare skulle bli Karl Poppers varum rke (Logik der Forschung
(Springer, 1935)). Adam spekulerar i huruvida Popper (som d var 17 r och h ll
p att intressera sig f r Einsteins teori) l st artikeln och inspirerats direkt
av denna, en sak som dock kanske g r varken fr n eller till. Popper har ju
ofta betonat att hans filosofi inspirerats av Einsteins relativitetsteori och
att denna aktualiserat fr gan om kunskapens gr nser i och med att den 300 r
gamla tanketraditionen (Newton) raserades; en tradition (som f rst s inte
sammanfaller med Newtons egna tankar) vilken betraktade mekaniken n stan som
lika ’s ker’ som matematiken. P ett st lle verkar A. Adam dock hugga i sten i
sin uppsats. N mligen, d Einstein i Tageblatt skriver att en ’skarpsinnig
forskare’ ofta hamnar att samtidigt nyttja motstridiga teorier menar Adam att
Einstein haft m jligen Maxwell eller Poincar i tankarna, medan Einstein sj lv
”did not hold contradictory theories”. Snarare r det v l s att Einsteins annus
mirabilis var en str lande uppvisning i att just balansera med motsatser, hur
han i en artikel lanserar hypotesen om ljuskvanta medan han i SR-artikeln
terigen st der sig p Maxwells f ltteori. Den ’skarpsinniga forskaren’ Einstein
fr mst hade i tankarna f r man f rmoda var honom sj lv och med all r tt! Medan
Einstein sannolikt antog att motsatserna skulle kunna f renas i en mer fullst
ndig teori kan Niels Bohrs komplementraritetsfilosofi (Atomteori och
naturbeskrivning (Aldus, 1967)) ses som ett uttryck f r antagandet att
motsatserna karakt riserar en intrinsisk dubbelhet hos naturen i relation till
det epistemiska subjektet. Ett central medvetet ’motstridig’ metod hos Bohr r
t ex att analysera m tningar av kvantsystem i termer av ’klassiska’ m
tinstrument som ju ’egentligen’ ocks borde beskrivas kvantmekaniskt.
representerade Poincar en teknologisk syn p vetenskapen och en sorts
”reparative reason” medan Einstein s kte efter en verenst mmelse mellan teori
och verklighet. Thibault Damour111111T. Damour, ”Poincar , relativity,
billiards and symmetry” (hep-th/0501168). radar upp en imponerande r cka uppt
ckter av Poincar som kan r knas till Relativitetsteorin men nd r hans slutsats
att Poincar inte kan s gas ha skapat Relativitetsteorin s som en ny
fundamental ram f r fysiken. Poincar t ex fortsatte efter 1905 att h lla fast
vid Lorentzs distinktion mellan sann tid (’le temps vrai’) och skenbar tid
(’le temps apparent’). Enligt Poincar s syns tt skulle det s ledes inte
uppkomma n gon verklig tidsskillnad i ’tvilling-paradoxen’. Det f refaller som
om Poincar t nkte sig att problemen i Lorentzs teori handlade om att fixa
dynamiken och att han inte uppfattade saken som att Relativistetsteorin l ste
problemen genom att omgestalta kinematiken.121212Man kan se ett snarlikt f rh
llningss tt i Hjalmar Tallquists 80-sidiga l nga uppsats ”De fysikaliska f
rklaringarna af aberrationen” (Festskrift Tillegnad Anders Donner (Helsingfors
1915)). I uppsatsens avslutning tas faktiskt Relativistetsteorin upp men
Tallquists omst ndiga genomg ng av de ’fysikaliska’ (dynamiska) f rklaringarna
till aberrationen visar att han vid denna tid inte helt vertygats om v rdet av
Einsteins kinematiska l sning. Liksom Poincar och m nga andra hade Tallquist
en sorts nostalgiskt f rh llande till etern. I Naturvetenskapliga uppsatser
(Schildts, 1924) gnar Tallquist ett kapitel t ”V rldseterns historia” d r han
avslutningsvis utbrister, ”finnes det en v rldseter eller ej, s m ste det
uppriktiga svaret lyda, att det kan man ej veta”. Tallquists elev Gunnar
Nordstr m blev tidigt medveten om Relatitivtetseorins revolution ra betydelse
samt ins g genast att Minkowskis ”dj rft matematiskt betraktelses tt kastade
ett fullkomligt nytt ljus” ver teorin, f r att citera ur hans stilrena versikt
”Rum och tid enligt Einstein och Minkowski” ( fversigt af Finska Vetenskaps-
Societetens F rhandlingar. LII. A, N.o 4, 1909-1910). Emellertid, den
dynamiska tolkningstraditionen (Lorentz) lever vidare. Se t ex H. R. Brown &
O. Pooley, ”Minkowski space-time: a glorious non-entity,” physics/0403088, som
karakt riserar den dynamiska/konstruktiva tolkningens k rnpunkt som s : ”What
is definitive of this position is the idea that constructive explanation of
’kinematic’ phenomena involves investigation of the details of the dynamics of
the complex bodies that exemplify the kinematics” (s. 11). Filosofen Harvey
Brown har ocks p g ng boken Dynamical relativity. Space time from a dynamical
perspective (Oxford U. P., 2005). A. Einstein betonade sj lv att SR var
ofullst ndig s tillvida att entiteter som ’klockor’ och ’m tstavar’ inte
visats motsvara l sningar till kovarianta ekvationer. I ett brev till A.
Sommerfeld 14.1.1908 (citeras i Brown & Pooley, s. 5) j mf rde Einstein
relationen mellan SR och en hypotetiskt mer fullst ndig teori som den mellan
termodynamik och statistisk mekanik, en j mf relse som senare anv ndes av J.
Bell. Andra k nda namn som efterlyst dynamisk f rklaring till ’Lorentz-
Fitzgerald-kontraktionen’ r W. Pauli och A. S. Eddington. Antar man att SR
bygger p en approximativ symmetri leds man ocks till fr gan om hur den skall f
rklaras i termer av en ’djupare’ struktur. D rtill understryker Damour att
Poincar i sina pre-Einstein-1905 artiklar fortfarande laborerar med effekter
av ordningen $O(v)$ och verkar vara tveksam att acceptera att Lorentz-
transformationerna h ller exakt.131313S. Katzir, ”Poincar ’s relativistic
physics. Its origins and nature,” Physics in Perspective 7, 268-292 (2005), g
r delvis emot Damours slutsatser. Enligt Kazir tog Poincar
Relativitetsprincipen p fullt allvar och bem dade sig om att visa att
elektrodynamik och gravitationsteori kunde formuleras i samklang med denna. F
r Poincar utgjorde s ledes Relativitetsprincipen en sorts gr nsvillkor, medan
den f r Einstein utgjorde en deduktiv utg ngspunkt. Kazir betonar ocks att
Poincar uppfattade Lorentzs ’lokala tid’ (’le temps apparent’) som den tid som
m ts av klockor, medan Lorentzs ’sanna tid’ enbart r en konstruktion. Mitt
intryck r dock att Damours argumentation v ger tyngre betr ffande inskr
nkningarna hos Poincar s Relativitetsprincip. M nga fysiker och filosofer f
ljde l ngt efter 1905 fortfarande de Lorentz-Poincar ska linjerna. Eino Kaila
skrev nnu p 20-talet om ’etern’ som han likst llde med sorts elektromagnetisk
bakgrundstr lning som erbjuder en global referensram, ’substraatti’. ”Niinpian
kuin t m my nnet n, ei Einsteinin radikaalista relativismia ajan ja avaruuden
suhteen en voidaan pit pystyss ; ei voidaan silloin en Einsteinin tavoin esim.
v itt , ett jokaisella taivaankappaleella on oma yksityinen aikansa, jonka ei
tarvitse olla sama kuin muiden. Niin pitk lle kuin ’eetteri’ ulottuu on my
skin aika yksi ja sama.” (”Filosofisia huomautuksia relativiteettiteoriaan,”
Aika 14, 269-285 (1920). F rkortad version i Valitut teokset 1 (Otava, 1990).)
Uppenbarligen verkar det som Kaila d rmed skulle exempelvis ha avvisat
’tvillingparadoxen’. Senare i verket ”Einstein-Minkowskin invarianssiteoria”
(Ajatus 21, 5-121 (1958)) omfattar Kaila relativitetsteorin i hela sin
fysikaliska vidd. En aning ironiskt r att Kaila noterar med tillfredsst llelse
att den ldre Einstein s.a.s. vergivit den Mach-inspirerade ”fenomenologiska
fysiken”. Kailas egen filosofiska st ndpunkt verkade ju ursprungligen att ha
lett honom vilse betr ffande relativitetsteorins inneb rd medan Einstein trots
sina ’f rvillelser’ tr ffade r tt. Men liksom t ex Ph. Frank uppfattade Kaila
kanske inte nyanserna i Einsteins t nkande och metoder utan f rs kte pressa in
honom i tr nga filosofiska fack. F r en versikt av Einstein som filosof som
betonar kontiuniteten i hans t nkande se D. A. Howard, ”Albert Einstein,
philosophy of science,” Standford encyclopedia of philosophy (2004),
http://plato.stanford.edu/entries/einstein-philscience. Howard anm rker att de
som uppfattade 1905 rs Einstein som Machisk fullblodspositivist gl mmer att
han med sina unders kningar kring Brownsk r relse samma r s kte bevis p
atomernas existens, i direkt motsats till Machs d varande uppfattning. Poincar
dog 1912 utan att n gonsin verkat ha omfattat Relativitetsteorins djupg ende
betydelse s som vi f rst r (?) den idag. nd har Damour fog f r p st endet att
det hade, ifall ett nobelpris ha utdelats f r Relatitivtetseorin f re 1912,
varit r ttvist att dela det mellan Einstein, Lorentz och Poincar .141414Tv av
Poincar s centrala artiklar om relativitetsteori, ”Sur la dynamique de l’
lectron,” C. R. Acad. Sci., 140, 1504-1508 (1905); ”Sur la dynamique de l’
lectron,” Rend. Circ. Math. Palermo 21, 29-175, (1906), kan h mtas fr n sajten
”Historic papers”,
http://home.tiscali.nl/physis/HistoricPaper/HistoricPapers.html. P arkivet
hittas ocks Einsteins centrala artiklar, samt artiklar av Minkowski, Laue,
m.fl. Tv m nader f re sin d d skrev Poincar ess n ”L’espace et le temps” (ing
r i Derni res pens es (Flammarion, 1913)) som avslutas med de ofta citerade
orden: ”Ajourd’hui certains physiciens veulent adopter une convention
nouvelle. Ce n’est pas qu’ils soient contraints; ils jugent cette convention
nouvelle plus commode, voil tout; et ceux qui ne sont pas de cet avis pouvent
l gimitimement conserver l’ancienne pour ne pas troubler leurs vieilles
habitudes. Je crois, entre nous, que c’est ce qu’ils feront encore longtemps.”
### 5\. Konstant acceleration
Ett viktigt specialfall inom den relativistiska kinematiken r konstant
acceleration. Att ett objekt erfar konstant acceleration $a$ (i
$x$-riktningen) i ett Minkowski-rum inneb r att den under ett
egentidsintervall $d\tau$ erfar en hastighets kning $a\,d\tau$ j mf rt med ett
medf ljande referenssytem. Anv nder man additionsteoremet (10) f r vi f r
hastighets kningen i ’laboratoriereferenssystemet’
$dv=\frac{v+a\,d\tau}{1+va\,d\tau}-v\approx(1-v^{2})a\,d\tau=(1-v^{2})^{3/2}a\,dt.$
(13)
Detta kan skrivas som
$\frac{d}{dt}\left(\frac{dx}{d\tau}\right)=a.$ (14)
Ekvationen (14) kan tolkas som ekvationen f r ett objekt som p verkas av en
konstant kraft $F=ma$, varf r den relativistiska generaliseringen av Newtons
lag blir151515M. Planck torde ha varit den f rste som skrev ned denna form av
relativistisk generalisering av Newtons ekvation i fallet med Lorentz-kraften
(”Das Prinzip der Relativit t und die Grundgleichungen der Mechanik,”
Verhandlungen der Deutschen Physikalischen Gesellschaft 4, 136-141, 1906). I
artikeln 1905 hade Einstein nnu inte utvecklat en relativistisk mekanik.
(vilomassan betecknas med $m$)
$m\frac{du}{dt}=\frac{d}{dt}\left(\frac{du}{d\tau}\right)=F$ (15)
med anv ndning av den relativistiska definitionen av hastighet, $u=dx/d\tau$.
Kraften $F$ i (3) r den kraft som erfars av objektet visavis ett tempor rt
medf ljande inertialt referenssytem. Inf r vi den relativistiska impulsen
$p=mu$ kan (15) skrivas p den v lbekanta formen $dp/dt=F$. Eftersom vi har
identiskt
$\left(\frac{dt}{d\tau}\right)^{2}-\left(\frac{dx}{d\tau}\right)^{2}=1$ (16)
kan vi nyttja parametriseringen
$\displaystyle\frac{dt}{d\tau}$ $\displaystyle=$
$\displaystyle\cosh(\phi(\tau))$ (17) $\displaystyle\frac{dx}{d\tau}$
$\displaystyle=$ $\displaystyle\sinh(\phi(\tau))$ (18)
som insatt i (15) leder till $d\phi/d\tau=a$; dvs, den konstanta
accelerationen har som l sning den hyperboliska banan
$\displaystyle x(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\left(\cosh(a\tau)-1\right)+x_{0}$ (19)
$\displaystyle t(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\sinh(a\tau)$ (20)
som ligger p en hyperbel $\left(x-x_{0}+1/a\right)^{2}-t^{2}=1/a^{2}$.
Definierar vi energin $E$ sedvanligt som ’kraften g nger v gen’ erh ller vi f
r den hyperboliska banan
$E(t)-E(0)=\int_{0}^{t}F\,dx=m\left(\cosh(a\tau)-1\right)=m\left(\frac{1}{\sqrt{1-v^{2}}}-1\right)$
(21)
vilket ger det relativistiska uttrycket f r kinetisk energi som funktion av
hastigheten $v$. Det relativistiska energibegreppet kan belysas genom att
betrakta ett raketexempel som f ljer f reg ende h rledning men i (13) skall vi
nyttja impulskonservering ist llet. N mligen, antag att raketens massa vid
tidpunken $t$ r $m$ och att dess propulsionssystem slungar ut en massa $\Delta
m$ med hastigheten $w$ s att raketens massa (sett ur raketens referenssystem)
ndras med $dm$ under ett tidsintervall $dt$, d f r vi f r hastighets ndringen
$dV$,
$m\,dV+w\,dm=0\Rightarrow dV=-w\frac{dm}{m},$ (22)
som genom motsvarande resonemang som f r (13) (med $a\,d\tau=dV$) ger i
’laboratoriereferenssystemet’
$dv=-\left(1-v^{2}\right)w\frac{dm}{m}\Rightarrow
v=\frac{(m_{0}/m)^{2w}-1}{(m_{0}/m)^{2w}+1}$ (23)
f r hastigheten $v$, d r $m_{0}$ betecknar raketens initialmassa d $v=0$ (f r
$w\ll 1$ n rmar sig den sista delen i (23) K. Tsiolkovskys (1837-1935)
klassiska raketekvation). En po ng h r r att raketens mass- ndring $dm$ inte r
lika med den massa $\Delta m$ som (accelereras i raketens motor och) slungas
ut utan dessa r relaterade genom $dm={\Delta m}/\sqrt{1-v^{2}}$ vilket f ljer
av energikonserveringen och (21). Einstein presenterade mass-energi formeln n
gon m nad161616A. Einstein, ”Ist die Tr gheit eines K rpers von seinem
Energieinhalt abh ngig?,” Ann. d. Phys. 18, 639-641 (1905). Einsteins h
rledning av (21) var begr nsad till fallet med en elektrisk partikel i ett
konstant elektriskt f lt. Som sagt, 1905 hade han inte nnu tagit steget fullt
ut till en relativistisk mekanik. efter relativitetsartikeln fast n formeln
(21) (f r en elektron som accelererar i ett konstant elektriskt f lt) redan
ing r i relativitetsartikeln. Einstein var inte n jd med ’beviset’ f r mass-
energi formeln och s kte i flera r efter ett generellt bevis; hans sista
publicerade bevisvariant r fr n 1946171717M. Jammer ger en intressant versikt
av olika ansatser f r att grunda ’$E=mc^{2}$’ i Concepts of mass in
contemporary physics & philosophy (Princeton U. P., 1999), kap. 3. Ovan
understr ks att $m$ betyder h r ’vilomassan’ av den anledningen att i
litteraturen figurerar ett begrepp som ’relativistisk massa’ som har vissa
historiska anor men som r t mligen pass inom mod rna framst llningar av SR.
Jammer diskuterar ocks debatten kring relativistisk massa. En f rsk kritisk
versikt om dess anv ndning ges av G. Oas, physics/0504110. . Numera v gar man
kanske s ga att gruppteorin har klarlagt massans roll i galileisk och
relativistisk dynamik (se nedan).
### 6\. Geometri och fysik
Fallet med konstant acceleration r ocks av intresse eftersom det leder till
det icke-triviala problem att f rs ka konstruera ett global referens- och
koordinatsystem f r det accelererande objektet. Efter ekvivalensprincipens inf
rande (Einstein 1907) s motsvarar detta uppgiften att konstruera ett globalt
referenssystem i fallet med ett konstant homogent gravitationsf lt. N r man
s.a.s. tittar i historiens bakspegel kan det f refalla som om utvecklingen av
ART kunde ha tagit en helt annan fart ifall man f rst noggrannt studerat
exemplet med konstant acceleration ur differentialgeometrisk och topologisk
synvinkel. Detta r f rst s ett anakronistiskt syns tt eftersom ven dessa
grenar av matematiken var delvis i sin linda. ven inom ART tog det t ex l nge
innan man f rstod att skilja mellan kta singulariteter och s dana som enbart
berodde p ’ol mpligt’ val av koordinatsystem (s som i fallet Schwarzschild-
radien)181818Naturen av Schwarzschild koordinat-singulariteten klarlades
definitivt f rst av M. D. Kruskal i ”Maximal extension of Schwarzschild
metric,” Phys. Rev. 119, 1743-1745 (1960), ven om t ex en l sning fanns redan
implicit i ett arbete av A. S. Eddington fr n 1924. F r en historisk notis om
fallet se W. Rindler, Essential relativity (2. ed. Springer, 1977), s. 150.
Som ett till gg kan man erinra sig att inom m tt-teorier kan vi ha v xelverkan
fast n kr kningen f rsvinner verallt utom i vissa singul ra punkter. Aharonov-
Bohm-effekten (Phys. Rev. 115, 485, 1959) r ett ber mt exempelfall.
Gravitationell effekt i rumtid med $R_{\mu\nu\kappa\lambda}=0$ m ste d remot
rimligtvis vara utesluten.. Einstein har sj lv p pekat att ett av de st rsta
hindren n r han efter SR f rs kte utveckla ART var att han l nge f rbis g
skillnaden mellan koordinater som mer eller mindre godtyckliga matematiska
konventioner och den intrinsiska geometrisk-fysikaliska inneb rden.191919J.
Earman & C. Glymour har f ljt upp Einsteins vingliga v g i ”Lost in the
tensors: Einstein’s struggles with the covariance principles 1912-1916,” Stud.
Hist. Phil. Sci. 9 (4), 251-278 (1978). Einstein v gleddes i b rjan av
’kovarians-principen’ enligt vilken fysikens ’lagar’ skall kunna formuleras p
en koordinat-oberoende form. Emellertid, d Einstein i samarbete med Marcel
Grossman unders kte m jligheten av en f ltekvation av formen
$R_{\mu\nu}=\kappa T_{\mu\nu}$ (24)
som f rknipper geometrin (Riemann-tensorn $R_{\mu\nu\kappa\lambda}$) med
materien (energi-impuls-tensorn $T_{\mu\nu}$ ), s var de till en b rjan f ngna
av ett gravt matematiskt missf rst nd. N mligen, de antog att $R_{\mu\nu}=0$
implicerar att $R_{\mu\nu\kappa\lambda}=0$ (ett f rh llande som endast g ller
i tv dimensioner); dvs, att Riemann-tensorn f rsvinner identiskt. H rav skulle
det f lja att ingen v xelverkan kan f rekomma eftersom $T_{\mu\nu}=0$ i rummet
mellan materien (i vakuum) skulle implicera att d rst des r
$R_{\mu\nu\kappa\lambda}=0$. Einstein konstruerade f ljaktligen ett sinnrikt
argument202020Argumentet presenterades som en ’Bemerkung’ till A. Einstein &
M. Grossmann, ”Entwurf einer verallgemeinerten Relativit tstheorie und einer
Theorie der Gravitation,” Zeitschrift f r Mathematik und Physik 62, 225-261
(1913). H largumentet diskuteras av R. Torretti (1996, s. 162 - 168). kallat f
r ’h largumentet’ f r att bevisa att fysiken inte kan vara generellt
kovariant! ven om Einstein korrigerade sitt misstag (sj lva argumentet r
korrekt men fr gan r vilka slutsatser man skall dra) r der det fortfarande
delade meningar om vilken sorts fysikalisk inneb rd man kan eventuellt tillm
ta kravet p diffeomorfism-invarians.212121Se t ex J. Barbour, ”Dynamics of
pure shape, relativity and the problem of time” gr-qc/0309089 samt uppsatser i
Physics meets philosophy at the Planck scale (C. Callender & N. Huggett, eds.,
Cambridge U. P. 2001). Att Einstein fick problem med differentialgeometrin r
knappast att f rv na. Vid den aktuella tiden var teorin f r m ngfalder som
sagt inte nnu kodifierad och h ll p att formuleras av kanske fr mst H. Weyl,
E. Cartan och O. Veblen. Weyls Die Idee der Riemannschen Fl che (1913) r en av
milstolparna i preciseringen av m ngfaldsbegreppet. Med H. Whitneys
”Differentiable manifolds” (Annals of Mathematics 37, 645-680, 1936) r den mod
rna syntesen ett faktum. Problemet ansluter sig till fr gan vad som menas med
en punkt i rumtiden: givet tv rumtids-m ngfalder $M_{1}$ och $M_{2}$, kan vi
identifiera punkter i $M_{1}$ med punkter i $M_{2}$? Einsteins l sning var ett
tidigt exempel p insikten om att fysikaliskt meningsfulla storheter m ste vara
’gauge-invarianta’ (i detta fall f r gruppen av diffeomorfismer).222222C.
Rovelli formulerar Einsteins slutsats som: ”Contrary to Newton and to
Minkowski, there are no spacetime points where particles and fields live.
There are no spacetime points at all. The Newtonian notions of space and time
have disappeared”. (C. Rovelli: Quantum gravity (Cambridge U. P., 2004) s.
74.) Rumtidspunkterna har ingen objektiv realitet, endast ’koincidenser’
mellan partiklar kan tillskrivas objektiv verklighet. H nvisning till
koincidenser leder emellertid sm ningom till nya sv righeter genom
kvantmekaniken och kvantf ltteorin; n mligen, begreppet lokalisering r
problematisk.232323H. Bacry, Localizability and space in quantum physics (LNP
308, Springer 1988).
### 7\. Symmetri II. No-interaction-theorem (NIT)
Vad inneb r det att en teori r ’relativistiskt invariant’? Dagens fysiker har
det i ryggm rgen att Lagrange-funktionerna skall vara relativistiskt
invarianta. Kompletterar man Lorentz-transformationerna med translationer och
rotationer erh ller vi Poincar -gruppen $\mathcal{P}$. En m rklig konsekvens
som uppdagades under 60-talet r att invarians under Poincar -gruppen f r
$N$-partikelsystem utesluter v xelverkan; dvs, det enda fallet kompatibelt med
’manifest kovarians’ r en samling fria partiklar. Detta resultat g r under
namnet No-Interaction-Theorem (NIT) som kanske s llan tas upp i l rob
cker.242424Teoremet presenterades i originalversion av D. G. Currie, T. F.
Jordan & E. C. G. Sudarshan i ”Relativistic invariance and Hamiltonian
theories of interacting particles,” Rev. Mod. Phys. 35, 350-75 (1963), som ett
par r senare utstr cktes till godtyckligt antal partiklar av H. Leytweyler
(Nuovo Cimento 37, 556-67, 1965). En grundlig genomg ng av den kanoniska
versionen av teoremet och beviset ges i l roboken E. C. G. Sudarshan & N.
Mukunda, Classical dynamics - A modern perspective (Wiley, 1974). Ett bevis
som ist llet baseras p Lagrange-formulering presenterades av G. Marmo, N.
Mukunda & E. C. G. Sudarshan i ”Relativistic particle dynamics. Lagrangian
proof of the no-interaction-theorem,” Phys. Rev. D30, 2110-6 (1984). En
elegant rendering av denna version terfinns i l roboken G. Esposito, G. Marmo
& E. C. G. Sudarshan, From classical to quantum mechanics (Cambridge U. P.,
2004), kap. 16. F r Galilei-gruppen d remot kan man l tt konstruera icke-
trivial v xelverkan p basen av translationsinvarianta och rotationssymmetriska
interaktionspotentialer. Men detta lyckas inte i det relativistiska fallet
ifall vi kr ver att partiklarnas banor $(x(t),t)$ i Minkowski-rummet skall i
likhet med koordinat-systemen ven Lorentz-transformeras (’manifest
koviarians’, ’word-line condition’). Detta f ljer av att boost-generatorerna
$K_{i}$ (som genererar Lorentz-transformationer) kommer att bero av dynamiken,
de r inte rent kinematiska. N mligen, eftersom Poincar -gruppen f reskriver
kommuteringsrelationerna f r boost-generatorerna s s tter de d rmed ocks
villkor f r dynamiken. Detta villkor visar sig vara s str ng att det endast
kan uppfyllas f r fria, icke-v xelverkande partiklar. I Hamilton-dynamisk
formulering252525Elementa f r Hamilton-mekanik, Poisson-klamrar mm terfinns i
uppsatsen F. Borg, ”Symplektiska strukturer i fysiken” (2000),
http://www.netti.fi/~borgbros/artiklar/sympl.pdf. utmynnar kravet p manifest
kovarians i villkoret
$\\{K^{j},x^{k}\\}=x^{j}\\{H,x^{k}\\}$ (25)
d r $H$ betecknar Hamilton-funktionen. En nyckel-relation som tillsammans med
(25) leder till NIT r $\\{K^{j},H\\}=P^{j}$ d r $P^{j}$ betecknar
generatorerna f r translationerna. F r fria partiklar kan (25) enkelt
uppfyllas med $K^{j}=x^{j}H$. En annan intressant omst ndighet r att Poisson-
klammer mellan tv boost-generatorer ger en rotation (rotationsgeneratorer
betecknas med $J$),
$\\{K^{i},K^{j}\\}=-\epsilon_{ijk}J^{k}$ (26)
Detta leder till den s.k. Thomas-precessionen f r t ex elektroner som
’kretsar’ kring en atomk rna.262626L. H. Thomas, ”The motion of the spinning
electron,” Nature 117 (2945), 514 (1926) (”Letters to the Editor”). H.
Goldstein, Classical mechanics (Addison-Wesley, 1989), 2. uppl., har
kommenterat effekten: ”The spatial rotation resulting from the successive
application of two parallel axes Lorentz transformations has been declared
every as bit as paradoxical as the more frequently discussed apparent
violations of common sense, such as the ’twin paradox”’ (s. 287).
Hur r det d m jligt att vi trots NIT kan ha ’relativistisk’ v xelverkan? Den
konventionella omv gen r att inf ra f lt,272727S som Peres noterar leder denna
utv g i sin tur till nya problem …. ”the infinite number of dynamical field
variables gives rise to new difficulties: divergent sums over states, far
worse than those appearing when there is a finite number of continuous
variables”. (A. Peres, Quantum theory - concepts and methods (Kluwer, 1995) s.
256.) av vilka det elektromagnetiska (EM) f ltet $A_{\mu}$ (vektorpotentialen)
r det prim ra exemplet (Faraday, Maxwell), som f rmedlar v xelverkan; d.v.s.,
partiklar v xelverkar med f lt ist llet f r direkt med varandra genom
potentialfunktioner. Emellertid finns det en intressant knorr n r det g ller
elektrodynamiken. N mligen, det klassiska EM-f ltet har inga egna
frihetsgrader. Ekvationen (vi f ruts tter bivillkoret
$\partial_{\mu}A^{\mu}=0$, s.k. ’Lorentz-gauge’)
$\partial_{\mu}\partial^{\mu}A_{\nu}=4\pi J_{\nu}$ (27)
som f rknippar f ltet ($A$) med materien (str mmen $J$) kan omv ndas enligt
$A_{\nu}(x)=\int G(x-y)J_{\nu}(y)\,dy$ (28)
genom en Greens funktion $G$ varigenom f ltet $A$ kan elimineras ur teorin som
formuleras direkt i term av partikeldata. Wheeler och Feynman p b rjade en
dylik omformulering av elektrodynamiken (’absorber theory’) under mitten av
1940-talet282828J. A. Wheeler & R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945);
Rev. Mod. Phys. 21, 425 (1949). F reg ngare var H. Tetrode och A. D. Fokker.
Det var f rs ket att kvantisera Wheeler-Feynman-teorin som i Feynmans
dr:avhandling ledde fram till v gintegralmetoden, se L. M. Brown (2005).
vilken sedermera har utvecklats till ’action-at-a-distance electrodynamics’
med avst ndsv xelverkan. A. O. Barut har nyttjat den f r en omformulering
(’self-field’ teorin) av kvant-elektrodynamiken liksom F. Hoyle och J. V.
Narlikar vilka s sent som 1996 gav ut en l robok i mnet.292929F. Hoyle & J. V.
Narlikar, Lectures on cosmology and action at a distance electrodynamics
(World Scientific, 1996). Intresset f r avst ndsv xelverkan beror bl.a. p att
den proberar fysikens teoretiska grunder samt att den ger nya synvinklar p
renormaliseringsproblemet i kvantelektrodynamiken och sj lvenergiproblemet i
klassisk elektrodynamik.303030M. Frisch diskuterar inkonsistensproblematiken i
klassisk elektrodynamik i Inconsistency, asymmetry, and non-locality. A
philosophical investigation of classical electrodynamics (Oxford U. P., 2005).
Maxwells ekvationer kombinerade med punkt-partiklar och Lorentz-kraften leder
till problem, s som divergerande sj lv-energi. Frischs tes r att det r om jigt
att kombinera Maxwell, Lorentz och punktpartiklar i en konsistent modell.
Frischs po ng r emellertid att fysiker trots allt kommer v l till r tta med en
inkonsistent teori genom att p f ra extra villkor och kan d rmed nyttja den
med h g precision. F r en matematisk studie av bl.a. Abraham- och Lorentz-
modellen se H. Spohn, Dynamics of charged particles and their radiation field
(Cambridge U. P., 2004), math-ph/9908024. (Sj lva formuleringen kan kanska
verka ’ful’ eftersom exempelvis gauge-teori-aspekten verkar att hamna i
skymundan.) Problemkomplexet anknyter ocks till Bohms program f r en
ontologisk tolkning av kvantmekaniken.313131D. Bohm & B. J. Hiley, The
undivided universe (Routledge, 1993). Relativistisk invarians diskuteras i
kap. 12. H rvidlag tycks det r da en f rvirring betr ffande huruvida det finns
en relativistisk Bohm-formulering eller ej. Om man som Bohm inf r en partikel-
ontologi f r kvantmekaniken s r det klart att vi erh ller en icke-lokal teori
(t ex EPR-situationen)323232EPR efter Einstein, Podolsky & Rosen vars
beskrivning av ’entaglement-situationen’ i kvantmekaniken r en klassiker
(Phys. Rev. 47, 777-780, 1935). EPR spin-varianten lanserades av D. Bohm i
Quantum theory (Prentice-Hall, 1951). vilken ocks var en av Bohms po nger.
NIT-resultatet antyder ocks att det vore verraskande ifall formuleringen
skulle leda till en manifest kovariant teori. D remot r formuleringen
relativistisk kovariant i den meningen att observablerna statistiskt sett
uppfyller kovarianskravet. S ledes, relativistisk kovarians r d rmed faktiskt
mer relativ n absolut i Bohms formulering och teorin r ppen f r situationer
med ett ’ verskridande’ av relativitetsteorin och kvantmekaniken d r s.a.s.
icke-kovarians kan bli observerbar. I kvantf ltteori m ter man problemet med
partiklar som ’skapas’ och ’f rintas’ vilket verkar vara sv rt att f rena med
en fundamental partikel-ontologi.333333D. D rr et al. (Phys. Rev. Lett. 93, 9,
090402, 2004) har dock byggt vidare p J. S. Bells arbeten (Phys. Rep. 137, 49,
1986) och formulerat kvantf ltteori med Bohmska partikel-banor d r partiklar
skapas och f rintas. H. Nikolic har i sin tur vidareutvecklat en kovariant
kvantisering, utg ende fr n s.k. De Donder-Weyl formalism, som han menar att
naturligt bygger p deterministiska partikelbanor (hep-th/0407228). I
standardkvantmekaniken (J. v. Neumann)343434J. v. Neumann, Mathematische
Grundlagen der Quantenmechanik (Springer, 1932). f rblir andra sidan ’v
gkollapsen’ ett prek rt problem om man vill beskriva den med en kovariant
dynamik.
### 8\. Symmetri III. Begreppet massa
Idag klassifierar vi partiklar i term av massa, spin och helicitet som bygger
p representationer av Poincar -gruppen. Den definitiva framst llningen av
detta klassifikationssystem r E. P. Wigners artikel fr n 1939353535E. P.
Wigner, ”On unitary representations of the inhomogenous Lorentz group,” Annals
of Mathematics, 40, 1, 149-204 (1939). Med ’inhomogenous Lorentz group’ menas
samma som Poincar -gruppen. om vilken Sternberg har anm rkt: ”It is difficult
to overestimate the importance of this paper, which will certainly stand as
one of the great intellectual achievements of our century. It has not only
provided a framework for physical research for elementary particles, but has
also had profound influence on the development of modern mathematics, in
particular the theory of group representations.”363636S. Sternberg, Group
theory and physics (Cambridge U. P., 1994), s. 149. Wigners arbete var
kulmineringen p en serie arbeten bl.a. av Majorana, Proca, Klein, Murray,
Weyl, von Neumann och Dirac. Wigner ger speciellt erk nnande t Dirac: ”The
subject of this paper was suggested to me as early as 1928 by P. A. M. Dirac
who realized even at that date the connection of representations with quantum
mechanical equations. I am greatly indebted to him also for many fruitful
conservations about this subject, especially during the years 1934/35, the
outgrowth of which the present papers is ” (Wigner 1939, s. 156.).
Dirac373737P. A. M. Dirac, ”Forms of relativistic dynamics,” Rev. Mod. Phys.
21, 3, 392-399 (1949). Det kan f. . n mnas att det fanns en personlig l nk
mellan E. P. Wigner och Dirac genom att Dirac gifte sig med Wigners syster
Margit 1937. har ocks senare betonat att relativistisk dynamik (’klassisk’
eller kvantum) handlar om representationer av Poincar -gruppen; problemet g
ller att finna nya dynamiska system vars observabler satisfierar Poincar
-algebrans kommuteringsrelationer. Classical dynamics (1974)383838Boken bygger
f. . delvis p Mukundas f rel sningsanteckningar fr n V. Bargmanns kurs vid
Princeton universitetet 1964-5. av Sudarshan och Mukunda r ett prominent
exempel p en till mpning av detta program b de f r relativistisk och galileisk
dynamik393939I sammanhanget r det v rt att n mna J. -M. Souriau som varit en
pionj r i att f rena symmetribegreppet med dynamik genom en generell
symplektisk formulering av fysiken (Structure des syst mes dynamiques (Dunod,
1970); eng. vers., Birkh user 1997). Souriau har ocks bidragit till teorin f r
geometrisk kvantisering (Souriau-Kostant teorin). och f ljande utl ggning r
inspirerad av detta.
En kanonisk (symplektisk) realisering av Galilei-gruppen $\mathcal{G}$ (uppsp
nd av rotationer, rums- och tidstranslationer, samt Galilei-transformationer
av typ Ekv. (1)-(2)) inneb r att Lie-gruppen $\mathcal{G}$ r realiserad s som
kanoniska transformationer p ett fasrum $M$. Varje element $g$ i $\mathcal{G}$
motsvaras av en kanonisk transformation $\Phi_{g}:M\rightarrow M$ s dan att
$\Phi_{g}\circ\Phi_{h}=\Phi_{gh}$; d.v.s. den definierar en representation av
Galilei-gruppen $\mathcal{G}$ p gruppen av kanoniska transformationer,
Kan$(M)$, som vi h r kan definiera som gruppen av diffeomorfismer ver $M$
vilka l mnar Poisson-klammern (PK) invariant. Denna representation inducerar
vidare en representation av Lie-algebran $L(\mathcal{G})$ p Lie-algebran av
$\mbox{Kan}(M)$ vilken sammanfaller med m ngden av Hamiltonska vektorf lt p
$M$; d.v.s., vektorf lt $X_{F}$ som alstras av funktioner
$F:M\rightarrow\mathbf{R}$ genom $X_{F}f=\\{f,F\\}$ f r en godtycklig reell
funktion $f$ p $M$ ( $Xf$ betecknar derivatan av $f$ i riktningen $X$:
$Xf=df(X)=X^{i}\partial_{i}f$). Givna tv element $a$, $b$ i $L(\mathcal{G})$
som motsvaras av generatorerna $F_{a}$ och $F_{b}$, d har vid det generella
sambandet404040Tilldelningen $a\rightarrow F_{a}$ r vad man efter Souriau
kallar f r en momentum mapping fr n $L(\mathcal{G})$ till Kan$(M)$ d r
$\mathcal{G}$ betecknar systemets symmetrigrupp. Momentum-avbildningen
generaliserar E. Noethers resultat om sambandet mellan symmetrier och
invarianter och g r tillbaka p S. Lie. (F r en historisk notis se J. E.
Marsden & T. S. Ratiu, Introduction to mechanics and symmetry (2. ed.
Springer, 1999) s. 369-70.)
$\\{F_{a},F_{b}\\}=F_{[a,b]}+C(a,b)$ (29)
d r $F_{[a,b]}$ betecknar en generator motsvarande Lie-produkten $[a,b]$ av
$a$ och $b$ i $L(\mathcal{G})$. $C(a,b)$ r en bilinj r funktion, en sorts
’integrationskonstant’ och ett exempel p en kohomologi. Genom omdefiniering av
generatorer (genom att exempelvis addera konstanter) kan man f rs ka eliminera
antalet ’integrationskonstanter’. Det intressanta r att f r Galilei-gruppen
$\mathcal{G}$ (i motsats till fallet f r Poincar -gruppen) terst r dock ett
element som inte g r att eliminera, n mligen $M$ i
$\\{P_{i},G_{j}\\}=-\delta_{ij}M.$ (30)
H r st r $P_{i}$ f r generatorn av translationer l ngs $x_{i}$-axeln, och
$G_{i}$ f r generatorn av Galilei-transformationen i dito riktning. Gruppen
har allts en icke-trivial kohomologi parametriserad av $M$ som kallas ’massa’
(Souriau 1970).414141V. Bargmann var antagligen den f rste att studera (som
ett exempel) representationer av Galilei-gruppen i ”On unitary ray
representations of continuous groups,” Annals of Mathematics 59, 1-46 (1954).
Han visade bl.a. att Schr dinger-ekvationen motsvarar en Galilei-
representation (f r en enkel demonstration se Borg (2000)). En f ljd av att
massan r ett neutralt element (= kommuterar med alla andra element) i Galilei-
representationen r att massan r ’superselekterad’ (begrepp inf rt av Wick,
Wightman och Wigner, Phys. Rev. 88, 101-105, 1952); d.v.s., ett tillst nd kan
inte vara en superposition av tillst nd med olika massor. (I Wigner-Bargmann
formuleringen manifesteras kohomologin i odeterminerade fasfaktorer f r de
unit ra operatorer som representerar symmetrigruppen.) Enligt Primas betyder
detta att ”the Galilei group gives the final explanation of the concept of the
conservation of matter introduced into chemistry by Antoine Laurent de
Lavoisier (1743-1794) and the law of definite proportions due to Joseph Louis
Proust (1754-1826) and John Dalton (1766-1844).” (H. Primas, Chemistry,
quantum mechanics and reductionism (2. ed., Springer 1983) s. 73.) Primas
uppger ocks att begreppet Galilei-transformation mellan referens-system inf
rdes av Ph. Frank i ”Die Stellung des Relativit tsprinzips im System der
Mechanik und Elektrodynamik,” Sitzungsberichte d. math.-naturwiss. Kl. Akad.
Wiss. Wien 118, 373-446 (1909). Begreppet ’referenssystem’ i sin tur
lanserades av Ludwig Lange 1885. L. Ricci har ett roligt inl gg i Nature (vol.
434, s. 717, 7 april 2005) under rubriken ”Dante’s insight into galilean
invariance. The poet’s vividly imagined flight unwittingly captures a physical
law of motion”. Det handlar om Dante Alighieris Divina Commedia (ca 1310) och
raderna 115-117 d r Dante beskriver en flygf rd p monstret Geryons rygg.
Enligt Ricci visar beskrivningen att Dante intuitivt f rstod att de
fysikaliska betingelserna r likadana som i ett vilande referenssystem, f rutom
vindens p verkan. … ”Dante intuitively grasped the concept of invariance but,
unlike Galileo, he did not pursue this idea any further. Still, it seems he
was well ahead of his time with regard to the views about laws of nature held
in the Middle Ages”. Generatorn av tidstranslationer betecknas med $H$
(Hamilton-funktion) och dess enda icke-triviala PK r
$\\{G_{i},H\\}=P_{i}$ (31)
P basen av PK-relationerna kan man visa att424242Som element av m ngden av
reella funktioner p fasrummet r det naturligt att konstruera dylika polynom. F
r en abstrakt Lie-algebra kan man ocks konstruera polynom genom att inf ra
tensor-produkten p algebran (ett vektor-rum) och genom att identifiera
$X\otimes Y-Y\otimes X$ med $[X,Y]$. Detta leder till s.k. ’enveloping
algebras’, se t ex A. O. Barut & R. Raczka, Theory of group representations
and applications (World Scientific, 1986) s. 249-251.
$C=\mathbf{P}^{2}-2MH$ (32)
r en Casimir-invariant (kommuterar med alla element). Ekv (32) kan skrivas p
en mer bekant form
$H=\frac{\mathbf{P}^{2}}{2M}+\mbox{konst.}$ (33)
som r den icke-relativistiska Hamilton-funktionen f r en fri partikel med
’massan’ $M$. Motsvarande unders kning kan g ras f r Poincar -gruppen
$\mathcal{P}$. Om man anv nder 4-vektorbeteckningen och s tter $P^{0}=H$ (och
anv nder metrikkonventionen $(+---)$) s kan den f rsta Casimir-invarianten
skrivas
$C_{1}=P_{\mu}P^{\mu}=M^{2}$ (34)
som omv nt implicerar en Hamilton-funktion
$H=\sqrt{\mathbf{P}^{2}+M^{2}}.$ (35)
J mf r434343P. Feyerabend har f. . gjort sig tolk f r tesen att SR och
klassisk mekanik r ’inkompatibla teorier’ och att massbegreppet i den ena
teorin inte kan j mf ras med massbegreppet i den andra teorin (se Jammer 1999,
s. 57). Emellertid, om man utg r fr n att b da teorierna m ste h nf ra sig
till en och samma verklighet (de r s.a.s. representativa modeller) blir det
rimligt att j mf ra begreppen i dem. Matematiskt g r det ocks att studera
deformationer eller kontraktioner av Lie-algebran $L(\mathcal{G})$ p
$L(\mathcal{P})$ som visar hur vi en viss mening kan ’interpolera’ mellan
teorierna (V. Guillemin & S. Sternberg, Symplectic techniques in physics
(Cambridge U. P., 1984) s. 114-116; Barut & Raczka (1986), s. 44-46). Inom
vetenskapsfilosofin har man kanske ibland haft sv rt att f rst fysikernas
pragmatiska s tt att laborera med approximationer och verg ngar fr n en teori
till en annan beroende p ’validitetsomr det’. Men endel fysiker bidrar kanh
nda sj lva ibland till missf rst nden genom att besk ftigt beskriva fysiken
som en matematiskt exakt vetenskap eller som en ’teori om allting’. vi den
relativistiska versionen (35) med (33) genom att utveckla den f rra i serie av
$\mathbf{P}^{2}$ d $\mathbf{P}^{2}\ll M^{2}$ ser vi att f r relativistisk
symmetri r den additiva konstanten i (33) inte l ngre godtycklig utan ges av
$M$ (’$Mc^{2}$’). S ledes, relativistisk symmetri leder till att
’grundenerginiv n’ r fixerad till v rdet $E_{0}=Mc^{2}$. M rk ven att medan
massan r ett element i Galilei-algebran, r den enbart en konstant i det
relativistiska fallet.
Jammer (1999) terv nder upprepade g nger till ’mysteriet’ varf r ljusets
hastighet $c$ figurerar i mass-energi formeln; bygger $E_{0}=Mc^{2}$ via n gon
sorts omv g p elektrodynamiken? Dock, SR handlar om en symmetrisering mellan
rummet och tiden, varf r det m ste finnas en konstant ’$c$’ som f rknippar
$x$\- och $t$-dimensionen och som g r att vi kan j mf ra $\Delta x$ med
$\Delta t$. Det r elektrodynamiken som (’anakronistiskt’) rver denna konstant
genom att teorin formuleras s som kompatibel med relativistisk kinematik.
Galilei-symmetrin som har den enkla m ttlinjen $t$ = konstant (se avsnitt 2)
undviker en sammanblandning av rum och tid, medan relativistisk symmetri har m
ttlinjen (4) vilken fodrar en konversionsfaktor mellan rum och tid. Mer
korrekt vore kanske att s ga att det p abstrakt niv inte finns n gon
uppdelning i rum- och tidsdimension, det r vi som inf r $c$ som g r denna
uppdelning (i t ex meter vs sekund). Aristoteles som ryggade tillbaka inf r
att dividera kvantiteter med olika enheter (s som l ngd/tid) kanske skulle ha
k nt sig mera hemmastadd i abstrakt relativistisk dynamik utan $c$.
### 9\. Tid och relativitet
SR har sj lvfallet djupt p verkat senare filosoferanden kring ’tidens v sen’.
Den har inte l st n gra av tidens filosofiska problem, snarare f r ndrat
perspektiven och gett upphov till nya fr gor. M nga har i SR funnit st d f r
en statisk uppfattning (’tenseless time’) kontra en dynamisk uppfattning
(’tiden som fl dar’, ’tensed time’) av tiden. Sett ur Minkowski-rum perspektiv
r allting som h nt och h nder redan fixerat i rumtiden. Einstein sj lv verkar
ha varit ben gen att tolka tiden statiskt, d r upplevelsen av tiden som fl dar
r en sorts ’illusion’.444444Ett av de klassiska Einstein-citaten r fr n ett
brev till Michele Bessos familj efter denna n ra v ns bortg ng 1955: ”Nun ist
er mir auch mit dem Abschied von dieser sonderbaren Welt ein weinig
vorausgegangen. Dies bedeutet nichts. F r uns glaubige Physiker hat die
Scheidung zwischen Vergangenheit, Gegenwart und Zukunft nur die Bedeutung
einer wenn auch hartn ckigen Illusion”. Sj lvfallet r inte SR n dv ndig f r
att uppfatta verkligheten som fixerad i rumtiden, men efter SR och Minkowski g
r det inte l ngre att undvika att uppfatta verkligheten i term av rumtiden. M
nga filosofer och anh ngare av statisk tid har inspirerats av SR, som t ex
Michael Lockwood.454545M. Lockwood, The labyrinth of time. Introducing the
universe (Oxford U. P., 2005). Lockwood gjorde sig bem rkt som
vetenskapsfilosof genom Mind, brain and the quantum (Oxford U. P., 1989).
Lockwood har tidigare tillsammans med David Deutsch utrett de logiska m
jligheterna f r tidsresor.464646D. Deutsch, M. Lockwood, ”The quantum physics
of time travel,” Scientific American 270, 50-56 (1994). Lockwood (2005) g r
igenom hela galleriet med tidens pil, entropi, statisk kontra dynamisk tid,
kvantgravitation, tidresor, osv, (Wheeler-Feymann teorin avf rdas f. . som ”a
dead duck”) men kanske de intressantaste anm rkningarna hittas mot slutet av
boken. I fysiken (dess matematiska modeller) representeras ’tiden’ ofta (
tminstone lokalt) som en parameter $t$, men det som intresserar oss h r r hur
den subjektiva tiden, upplevelsen av tiden som ett fl de, h nger ihop med den
fysiska tiden $t$. Lockwood terv nder h rvidlag till Everetts m nga-v rldars
tolkning av kvantmekaniken.474747H. Everett, ”’Relative state’ formulation of
quantum mechanics,” Rev. Mod. Phys. 29, 454-462 (1957); B. S. DeWitt, N.
Graham, The many-worlds interpretation of quantum mechanics (Princeton U. P.,
1973). Medvetandet f rknippas med en speciell medvetande-bas
$|\alpha_{i}\rangle$ i en Everett-dekomposition av universums v gfunktion
$|\Phi\rangle$,
$|\Phi\rangle=\sum|\alpha_{i}\rangle\otimes|\beta_{i}\rangle.$ (36)
Nu-upplevelsen (space-time-actuality) s gs uppst d $|\alpha_{i}\rangle$
motsvarar en dekoherensbas, vilket hos Everett antagligen motsvarar de s.k.
minnes-tillst nden (memory states), som har klassisk pr gel. P
neurofysiologisk niv h nvisar Lockwood till teorierna om ’thalamocortical
oscillations’ och en form av ’cortical phase locking’.484848Lockwood h nvisar
bl.a. till R. Llinas, U. Ribary, ”Coherent 40-Hz oscillation characterizes
dream state in humans,” Proceedings of the National Academy of Sciences USA,
2078-2081 (1993). S ledes, tidsmedvetandet kan uppdelas i ’specious presents’
(subjektiva ’tidskvantum’, en sorts minsta tidsupplevelse som uppfattas som en
helhet, t ex observationen av ett stj rnfall p himlen) som betingas av att
speciella neuroner i thalamus oskillerar i fas med varandra (en sorts koherent
tillst nd). N r dessa koherenta tillst nd bryter samman och ers tts av nya,
stegar s.a.s. medvetandet fram i tiden (en tanke/f rnimmelse ers tts av en
annan tanke/f rnimmelse).
## Part II Kvantrum och SR-modifikationer
### 10\. Dubbel speciell relativitet (DSR)
P senare tid har det lanserats modifikationer av SR under den samlande
beteckningen ”double special relativity” (DSR)494949G. Amelino-Camelia,
”Double special relativity,” gr-qc/0207049 v1; J. Magueijo & L. Smolin,
”Lorentz invariance with an invariant energy scale,” hep-th/0112090;
”Generalized Lorentz invariance with an invariant energy scale,” gr-
qc/0207085; ”Gravity’s rainbow,” gr-qc/0305055. Varf r stanna vid ’dubbel-
relativitet’? J. Kowalski-Glikman & L. Smolin har inf rt en tredje invariant
motsvarande den ’kosmologiska konstanten’ $\Lambda$ i ”Triple special
relativity” (hep-th/0406276). DSR kan ocks l sas som ’deformed special
relativity’ som introducerades av F. Cardone & R. Mignani 1998 och st r f r en
teori d r metriken $g_{\mu\nu}(E)$ beror p energin $E$ ist llet f r att vara
konstant som i SR (Cardone & Mignani, Energy and Geometry: an introduction to
deformed special relativity (World Scientific, 2004); ”Energy-dependent metric
for gravitation and the breakdown of local Lorentz invariance,” Annales de la
Fondation Louis de Broglie 27 no 3, 423-442, 2002). (Energi-beroende metrik
diskuteras ocks av Magueijo & Smolin, i ovan citerade gr-qc/0305055, men utan
h nvisningar till Cardone & Mignani.) Ytterligare en form av relativitetsteori
lanserades av L. Nottali 1993 som han kallat ’scale-relativity’ baserad p en
sorts id om fraktal rumtid men dess logik ppnade sig inte f r undertecknad
trots ett f rs k till l sning av hans detaljerade versikt ”Scale-relativity
and quantization of the universe. I. Theoretical framework” (Astronomy and
Astrophysics, 327, 867-889, 1997). d r man uppfattar ljushastigheten $c$ som
invariant men tillf r ocks ytterligare en l ngd-invariant, alternativt mass-
eller impuls-invariant. Motiveringen r att Planck-l ngden
$L_{P}=\sqrt{G\hbar/c^{3}}$ ($\approx 10^{-35}$ m, eller Planck-massan
$M_{P}=\sqrt{\hbar c/G}\approx 10^{-8}\,\mbox{kg}\,\approx 10^{19}$ proton-
massor), som karakt riserar en hypotetisk kvantgravitationsregim, borde vara
densamma f r alla observat rer och inte p verkas av Lorentz-Fitzgerald
kontraktionen: ”all observers agree that there is an invariant energy scale,
which we take to be the Planck scale $E_{P}$” (Magueijo & Smolin, 2001). Denna
motivering som s dan verkar ganska tunn; snarare handlar helt enkelt om
antagandet att ’Planck-niv ’ effekter fenomenologiskt f rv ntas yttra sig i en
modifikation av t ex Lorentz-transformationerna. En ansats bygger p en ndring
av den relativistiska ’dispersionsrelationen’ (Magueijo & Smolin 2002),
$E^{2}f_{1}(E,\lambda)^{2}-\mathbf{p}^{2}f_{2}(E,\lambda)^{2}=m^{2}.$ (37)
$\lambda$ betecknar en karakt ristisk l ngdparameter som antas vara av
storleksordningen $L_{P}$, $\lambda\approx L_{P}$. F r $f_{1}\neq f_{2}$ erh
ller vi s.k. ’variable speed of light’-teorier (VSL) d r ’hastigheten’
$v=dE/dp$ f r fotoner ($m$ = 0) kommer att bero av energin (eller frekvensen)
vilket r av ett visst intresse inom astrofysiken; n mligen, en VSL-effekt
skulle kunna sk njas som en frekvensberoende tidsf rdr jning hos fotoner fr n
gammak llor.505050F r en kartl ggning av m jliga observationer av DSR-effekter
se T. Jacobson et al., Phys. Rev. D67, 124011 (2003). J. Magueijo har v ckt
endel uppm rksamhet med sin bok Faster than light - the story of a scientific
speculation (Perseus Books, 2003) som inneh ller en f rgstark skildring av
hans f rs k att lansera den variabla ljushastighetsteorin och hur han upplevt
sig bli motarbetad. ”Though we get some glimpses here of theorists grappling
with an elusive idea, too much of the story comes off as puerile”, anser G.
Johnson i en recension i New York Times (Feb. 9, 2003) med h nvisning till
Magueijos angrepp p ’etablissemanget’ . Magueijos bidrag ”A genuinly evolving
universe” i Wheeler-Festschriften J. D. Barrow et al., Science and the
ultimate realtiy - Quantum theory, cosmology and complexity (Cambridge U. P.,
2004) s. 528-549, avsl jar en n stan manisk aversion mot tanken p universella
of r nderliga lagar, h rav tesen om ’muterande naturlagar’ … ”I am very fond
of this view of the nature of physical law, primarily because I dislike the
alternative current of thought: that there is ’a law’, and that we shall know
it; that we are close to the end of theoretical physics; that we may dream of
a ’final theory’. Such mystical views are too ’lawyer-minded’ for my taste”
(ibidem s. 530). I bakgrunden verkar ocks vara en sorts klaustrofobisk k nsla
av att en v rld med t ex konstant ljushastighet skulle i praktiken om jligg ra
interstell ra rymdresor och m ten med intressanta utomjordingar. Sant, f r oss
som v xt upp med ET och rymdbarerna i Star Wars och Liftarens guide till
galaxen s r det onekligen en trist tanke att vi f r all framtid vore liksom
inst ngda under en liten kupa i solsystemet. Magueijo spekulerar att man kunde
resa med verljushastighet (lokalt r dock $v<c$) l ngs kosmiska str ngar
(ibidem s. 546). Fysikern och f rfattaren Alastair Reynolds har med sin
Relevation Space-rymdsaga (fyra b cker sedan 2000) dock visat att rymdepik r m
jlig trots subluminal transport ifall man har m jlighet till l mplig
kryoteknik, eller man kan hindra ldrandet. I ett enkelt exempel p DSR
modifieras Lorentz-transformationer $\Lambda$ verkande p momentum-rummet
enligt
$\displaystyle U^{-1}(p_{0})\,\Lambda\,U(p_{0})$ (38) $\displaystyle
U(p_{0})=\exp(\lambda p_{0}D)$ (39) med dilatationsoperatorn $\displaystyle
D=p_{\mu}\frac{\partial}{\partial p_{\mu}}$ (40)
Transformationen $U$ h nger ihop med funktionerna $f_{1}$ och $f_{2}$ i (37)
genom
$U(p_{0},\mathbf{p})=\left(f_{1}(E,\lambda)p_{0},f_{2}(E,\lambda)\mathbf{p}\right).$
(41)
F rutsatt att $|\lambda p_{0}|<1$ (garanterar att $U(p_{0})$ konvergerar) har
vi
$U(p_{0})p_{\mu}=\frac{p_{\mu}}{1-\lambda p_{0}}.$ (42)
F ljaktligen f r vi f r en ’boost’-transformation l ngs $z$-axeln i
$(p,E)$-rummet
$\displaystyle\tilde{p}_{0}$ $\displaystyle=$
$\displaystyle\frac{\gamma(p_{0}-vp_{z})}{1-\lambda(1-\gamma)p_{0}-\lambda\gamma
vp_{z}}$ (43) $\displaystyle\tilde{p}_{z}$ $\displaystyle=$
$\displaystyle\frac{\gamma(p_{z}-vp_{0})}{1-\lambda(1-\gamma)p_{0}-\lambda\gamma
vp_{z}}.$ (44)
Magueijo och Smolin kallar (43)-(43) f r Fock-Lorentz transformationer
eftersom de (i rumtids-version) redan lanserades av V. Fock515151V. Fock, The
theory of space-time and gravitation (Pergamon Press, 1964). i ett f rs k att
modifiera Lorentz-transformationer f r kosmologiska avst nd. DSR-teorier
bygger p en modifiering av Lorentz-boost operatorerna som i f reg ende fall
motsvarar (i energi-momentum rummet)
$\displaystyle\tilde{K}^{i}$ $\displaystyle=$ $\displaystyle
U^{-1}(p_{0})K^{i}U(p_{0})=K^{i}+\lambda p^{i}D$ (45) $\displaystyle K^{i}$
$\displaystyle=$ $\displaystyle p^{i}\frac{\partial}{\partial
p_{0}}-p_{0}\frac{\partial}{\partial p_{i}}.$ (46)
Enda kommutator som ndras i Poincar -algebran r
$\displaystyle[\tilde{K}^{i},P^{j}]$ $\displaystyle=$ $\displaystyle
P^{0}\delta_{ij}+\lambda P^{i}P^{j}\,,$ (47)
$\displaystyle[\tilde{K}^{i},P^{0}]$ $\displaystyle=$ $\displaystyle
P^{i}+\lambda P^{i}P^{0}\,,$ (48)
som reduceras till de traditionella formerna d $\lambda=0$. Ekv (45)-(46) r
ett enkelt exempel p en s.k. deformation av Poincar -algebran som de senaste
tv decennierna studerats under namnet av $\kappa$-Poincar algebran.525252J.
Lukierski, A. Nowicki & H. Ruegg, ”New quantum Poincar algebra and
$\kappa$-deformed field theory,” Phys. Lett. B293, 344 (1992). Polska forskare
f refaller ha varit speciellt aktiva p detta omr de. F r en versikt
(Ph.D-avhandling) av ”$\kappa$-teorier” se A. Agostini, ”Fields and symmetries
in $\kappa$-Minkowski non-commutative space time,” hep-th/0312305. En r cka av
olika deformationer kan alstras via ekv (45) genom att modifiera $U$. Kraven p
att Lie-algebra Jacobi-likheten m ste satisfieras, samt att rotationsinvarians
m ste respekteras, begr nsar v sentligen m jliga deformationer till
$\kappa$-varianterna. Olika till tna deformationer kan visas motsvara olika
val av koordinatsystem p ett de Sitter-rum (J. Kowalski-Glikman & S. Nowak,
hep-th/0204245; J. Kowalski-Glikman, hep-th/0207279; F. Girelli & E. R.
Levine, gr-qc/0412079). F r transformationerna (38)-(39) erh ller vi f ljande
invariant
$m^{2}=\frac{p^{2}}{\left(1-\lambda p_{0}\right)^{2}}\quad(p^{2}\equiv
p_{\mu}p^{\mu}),$ (49)
som definierar ’massan’ $m$. Identifierar vi ’energin’ enligt $E=p^{0}$ f ljer
relationen
$E=\frac{\gamma m}{1+\lambda\gamma m},$ (50)
f r kinetisk energi f r en partikel genom att transformera m.h.a. (43) mellan
partikelns referenssystem ($p_{z}=0$) och ’laboratoriet’. Ekv (50) inneb r att
energin begr nsas genom $E<1/\lambda\approx E_{P}$ (Planck-energin). En id r
att detta inf r en energi cut-off som i princip kan eliminera divergenser i
kvantf lteorier. En annan slutsats r att den modifierade Poincar -symmetrin g
ller endast f r den ’sub-Planckiska’ regimen. Detta kan verka mots gelsefullt
eftersom makroskopiska kroppar antas best av partiklar med $E<E_{P}$ men nd f
rmodas satisfiera de konventionella SR-transformationerna. Problemet med att
massan begr nsas av Planck-massan brukar h nvisas till som ’the soccer-ball
problem’ (Amelino-Camelia). En tanke r att f r sub-Planck partiklar m ste vi
ocks modifiera summeringsreglerna f r impuls och energi f r vilka Magueijo och
Smolin (2002) f resl r en icke-linj r summering (betecknad $\oplus$)
$\frac{p_{i}^{(tot)}}{1-\frac{\lambda}{N}p_{0}^{(tot)}}=\sum_{k=1}^{N}\frac{p_{i}^{(k)}}{1-\lambda
p_{0}^{(k)}}\equiv p_{i}^{(1)}\oplus\dots\oplus p_{i}^{(N)},$ (51)
d r $p_{i}^{(tot)}$ betecknar energi-impulsen f r en kropp sammansatt av $N$
partiklar. I transformations-formlerna (43)-(44) f r $p_{i}^{(tot)}$ ers tts
$\lambda$ med $\lambda/N$ vilka d rf r n rmar sig Lorentz-transformationerna f
r stora $N$. Dessa egendomliga summerings- och transformationsregler som beror
p hur man uppdelar en kropp i icke-sammansatta ’partiklar’ antyder p
principiella problem med denna typs DSR.535353R. Alosio et al., ”A note on
DSR-like approach to space-time,” Phys. Lett. B610, 101-106 (2005). H r
argumenteras att ifall man f rs ker inf ra en invariant l ngd i rumtiden i
likhet med fallet f r momentum-rummet ovan hamnar vi att ”abandon the group
structure of the translations and leave[s] a space-time structure where points
with relative distances smaller or equal to the invariant scale cannot be
unambigiously defined”. F. Girelli & E. R. Levine gr-qc/0412079 har gett en
allm nnare teoretisk bakgrund f r Magueijo-Smolin momentum-additionen (51).
Enligt denna r (51) att j mf ra med relativistisk addition av hastigheter (10)
som har att g ra med en komposition av tv relativa referenssystem. H.
Bacry545454H. Bacry, ”The foundations of the Poincar group and the validity of
general relativity,” Reports on Mathematical Physics 53 (3), 443-473 (2003).
Kan n mnas att Bacry (enligt egen utsago) tydligen ofta diskuterat fysik med
A. Connes som r k nd som en pionj r inom icke-kommutativ geometri. har utg tt
fr n $\kappa$-Poincar -algebra med relationen
$P_{0}=2\kappa\sinh\left(\frac{\tilde{P}_{0}}{2\kappa}\right).$ (52)
N r parametern $\kappa\rightarrow\infty$ terf r vi $P_{0}=\tilde{P}_{0}$.
$P_{0}$ uppfattas som en klassisk additiv energi-parameter medan
$\tilde{P}_{0}$ st r f r den fysikaliska energin. S ledes, en galaxs massa
$M$, som best r av $N$ stj rnor var och en med massan $m$, best ms enligt
Bacry av relationen
$\sinh\left(\frac{M}{2\kappa}\right)=N\sinh\left(\frac{m}{2\kappa},\right).$
(53)
varav f ljer att $M<Nm$; d.v.s., ’summan’ r mindre n delarna. Genom att anta
att $\kappa\approx 10^{11}$ solmassor (motsvarar en typisk galax) anser Bacry
att man kan f rklara och avf rda ”the dark matter affair” ven om tankeg ngen
inte verkar helt solklar.555555Begreppet ’m rk materia’ h rr r sig fr n
observationen att ifall man r knar ihop all materia som kan observeras i
galaxer m.h.a. av optiska teleskop och radioteleskop s tyder stj rnornas
snabba r relser i galaxer p att det m ste finnas ett betydande verskott av
materia som inte ’syns’ f r att kunna f rklara r relserna (genom den m rka
materiens gravitationella p verkan). Yttre delar av galaxer roterar snabbare n
vad den skenbara massan enligt Keplers lagar skulle f reskriva. Hypotesen om m
rk materie g r tillbaka p F. Zwicky och dennes observationer under 1930-talet
av galaxerna i Comaklustern. Einstein diskuterar metoden f r att best mma
andelen m rk materia i The meaning of relativity, appendix 1. Enligt
uppskattningar best r 90-95% av universum av m rk materia och energi (’m rk
energi’ f rmodas f rklara universums accelererande expansion). VIRGOHI21 som
uppt cktes i februari 2005 med radioteleskop (tack vare ’v te-signalen’ p v gl
ngden 21 cm) tros vara en galax helt best ende av m rk materia. V tgasens
dynamik implicerar n rvaron av den ’osynliga’ galaxen. Existensen av ett s dan
m rk galax inneb r rimligtvis en d dsst t f r de flesta ’bortf rklarings-
teorierna’. (F r en popul r versikt av ’m rk materia’ se
http://en.wikipedia.org/wiki/Dark_matter.) ”[T]he issue of dark matter
arguable constitutes the most astrophysically interesting aspect of modern
cosmology” (J. A. Peacock, Cosmological physics (Cambridge U. P., 1999) s.
368), men huruvida denna fr ga har n got att direkt g ra med
relativtetsteorins fundament, eller kan ’bortf rklaras’ med revisioner av
denna, r ovisst. I sig r existensen av m rk materia inget att f rv na sig ver
… ”It is not surprising that most of the matter in the universe should be
dark: there is no reason why everything should shine” … (M. Rees, Perspectives
in astrophysical cosmology (Cambridge U. P., 1995) s. 33). M rk materia som
bara v xelverkar via gravitationen (s.k. WIMP) vore sj lvfallet aningen
kufisk. Ett annat aktuellt fenomen som ocks satt fart p m nga relativisters
fantasi r den s.k. Pioneer-anomalin (se t ex J. G. Harnett & F. J. Oliveira,
”Anomalous acceleration of the Pioneer spacecraft matches the acceleration
predicted by Carmelian cosmology,” gr-qc/0504107. Analys av bandata fr n
rymdsonderna Pioneer-10 och 11 antyder p att de uts tts f r en of rklarlig
extra radiell acceleration mot solen av storleksordningen $(8.74\pm
1.33)\times 10^{-10}$ m/s2 (Anderson et al, Phys. Rev. Lett. 81, 2858-2861,
1998). (Igen hittas en t mligen bra popul r versikt p Wikipedia,
http://en.wikipedia.org/wiki/Pioneer_anomaly.) Man f rmodar att ’anomalin’
blir k nnbar p ca 20 AU:s avst nd fr n solen (AU = ’astronomical unit’ =
medelavst ndet mellan solen och jorden). En s dan effekt skulle fodra en f
rklaring p super-Planckisk niv ist llet f r p sub-Planckisk niv . Ett m rkligt
samband som lett till en del spekulerande (exv Hartnett & Oliveira) r att den
anomala accelerationen r av samma storleksordning som $Hc\approx 6.9\times
10^{-10}$ m/s2 d r $H$ betecknar Hubble-konstanten ($\approx 2.3\times
10^{-18}$ s-1). I Bacrys version blir deformationerna nu ist llet endast m
rkbara i kosmologiska sammanhang f r galaktiska system medan i Magueijo-
Smolin-teorin effekterna g llde den sub-Planckska regimen. Det kan verka aning
l ngs kt att modifiera SR f r kosmologiska skalor eftersom gravitationen
dominerar p denna niv och vi borde verg till ART.
Ett centralt problem med deformerade Poincar -algebran r att de formulerats f
r momentum-rummet. Det finns ingen automatisk koppling till Minkowski
rumtiden. Detta h nger ihop med att det saknas enhetliga principer f r
fysikaliska tolkningar av de algebraiska schematan. S ledes, n r vi tidigare h
nvisade till ’hastigheten’ $v=dE/dp$ i samband med (37) s hade vi ingen grund
f r att identifiera detta uttryck med hastighet i den ’vanliga’ meningen
$v=dx/dt$. Den kanske mest systematiska ansatsen p att f rbinda momentumrummet
och rumtiden baseras p teorin f r kvantgrupper (icke-kommutativa Hopf-
algebran) som uppr ttar en sorts dualitet mellan $\kappa$-Poincar -algebra och
$\kappa$-Minkowski rumtid.565656F r en definition av dessa termer se t ex
Agostinis artikel hep-th/0312305. 90-talets kanske mesta kvantgruppsguru har
varit Shahn Majid som gett en pedagogisk versikt och introduktion i
Foundations of quantum group theory (Cambridge U. P., 1995, pb ed. 2000). Hans
millenium-artikel ”Quantum groups and non-commutative geometry,” J. of
Mathematical Physics 41 (6), 3892-3942 (2000), uttrycker stora f rhoppningar p
att kvantgrupper och icke-kommutativ geometri r den r tta v gen till fysikens
innersta hemligheter. ”In fact, there are fundamental reasons why one needs
noncommutative geometry for any theory that pretends to be a fundamental one.
Since gravity and quantum theory both work extremely well in their separate
domains, this comment refers mainly to a theory that might hope to unify the
two. As a matter of fact I believe that, through noncommutative geometry, this
’Holy Grail’ of theoretical physics may now be in sight.” Liksom Penroses
twistorer (se nedan) s inst ller sig fr gan huruvida den eleganta matematiken
enbart h ller samma gammla vin i nya l glar eller ifall den faktiskt kan
inspirera till helt ny fysik. En kanske mer ’fysikalisk’ koppling har gjorts
av Magueijo, Smolin, m. fl. N mligen, utg ngspunkten r att teorin skall terge
planv gsl sningen $\exp(ip_{\mu}x^{\mu}/\hbar)$ f r fria partiklar och beh lla
uttrycket f r den invarianta fasen $p_{\mu}x^{\mu}/\hbar$. Detta implicerar
via (37) en sorts dual energiberoende rumtidsmetrik av formen575757D. Kimberly
et al., gr-qc/0303067; J. Magueijo & L. Smolin, gr-qc/0305055.
$ds^{2}=\frac{dt^{2}}{f_{1}(E,\lambda)^{2}}-\frac{d\mathbf{x}^{2}}{f_{2}(E,\lambda)^{2}},$
(54)
som motsvarar en dual avbildning till (41) given genom
$U(x)=\left(\frac{t}{f_{1}(E,\lambda)},\frac{\mathbf{x}}{f_{2}(E,\lambda)}\right).$
(55)
En konsekvens r att den relativistiska ’egentiden’ $\tau$ (11) nu kommer att
bero p partikelns energi,
$\Delta\tau=\sqrt{1-v^{2}}\,\Delta
t\,\frac{f_{1}({E}_{0},\lambda)}{f_{1}(\tilde{E}_{0},\lambda)}=\sqrt{1-v^{2}}\,\Delta
t\left(1+(\gamma-1)\lambda E_{0}\right)^{-1},$ (56)
d r $\tilde{E}_{0}$ betecknar den transformerade (enligt (43)-(44)) vilo-
energin $E_{0}$. ’Tvilling-paradoxen’ i denna version leder till att
tidsdifferensen mellan tvillingarna ocks kommer att bero p deras massa och
uppenbarligen g r den tyngre tvillingens klocka en aning l ngsammare. En
principiell f ljd av (56) r att tv partiklar med olika energier enligt (54)
kan vara associerade med skilda metriker vid en och samma punkt $(t,x)$
(Magueijo & Smolin har av denna anledning kallat (54) f r en ’regnb gsmetrik’,
’rainbow metric’). ”Thus, quanta of different energies see different classical
geometries” (M&S, gr-qc/0305055). Detta har t ex tolkats som s att varje
partikel definierar ett eget momentum-beroende referenssystem som inneb r att
vid j mf relse mellan tv referenssystem m ste man f rutom den relativa
hastigheten (Lorentz) ocks ta h nsyn till partiklarnas momentum. Dylika verl
ggningar har f ranlett Magueijo och Smolin att h vda att fr gan om den ’r tta’
duala rumtiden till momentum-rummet ”is the wrong question to ask and that,
instead, there is no single classical spacetime geometry when effects of order
$L_{P}$ are taken into account. Instead, we propose that classical spacetime
is to leading order in $L_{P}$ represented by a one parameter family of
metrics, parametrized by the ratio $E/E_{P}$ (…) This may seem nonsensical if
one regards the geometry of spacetime as primary. However, in the currently
well studied theories of quantum gravity, including loop quantum gravity and
string theory, the classical geometry of spacetime is not primary. Rather, it
emerges as a low energy coarse grained description of a very different quantum
geometry” (ibidem). Detta ’emerges as …’, borde man inflika, r tillsvidare n
rmast en from f rhoppning. Smolin, som r en av de fr msta proponenterna f r
’loop-gravity’, brukar understryka vikten av teorins bakgrundsoberoende;
d.v.s., den f r inte f ruts tta n gon a priori metrik, och hans mer generella
slutsats h rav r att teorin inte l ngre kan leva p n gon m ngfald … ”what is
left is only algebra, representation theories of algebras, and
combinatorics”.585858L. Smolin, ”Quantum theories of gravitation” i J. D.
Barrow et al., 2004, s. 523. Einstein avlsutar ocks The meaning of relativity
(1956 uppl.) med anm rkningen att kvantfenomen ”does not seem to be in
accordance with a continuum theory, and must lead to an attempt to find purely
algebraic theory for the description of reality. But nobody knows how to
obtain the basis for such a theory.” Som s dan verkar detta vara en n got
anemisk id och igen saknar man en v gledande fysikalisk tanke eller princip.
Liksom Wheelers dictum ”it all comes from higgledy-piggledy” f refaller det
tminstone inte tillsvidare ha transmuterats till n gon genomgripande
fysikteori.
### 11\. Kvantrum
Variationerna p temat deformerade Poincar -algebran r intressanta ur den
synvinkeln att ifall en konsistent kvantgravitation-teori r m jlig s kan DSR-
teorier och liknande ge vinkar om diverse sorts approximationer till den
’fullst ndiga teorin’. En modell som i detta sammanhang verkar ofta teruppt
ckas av forskare r det veterligen f rsta f rslaget till en kvantiserad
rumtidsteori som lades fram av Hartland S. Snyder,595959H. S. Snyder,
”Quantized space-time,” Phys. Rev. 71 (1), 38-41 (1947). Trots efterforskning
p internet hittade jag inga n rmare biografiska detaljer om Hartland Sweet
Snyder (1913-1962). Claus Montonen lyckades gr va fram en kort minnesruna i
Physics Today (July 1962). Uppenbarligen dog Snyder i sviterna av en hj
rtattack den 22 maj 1962. I Kip S. Thorns Black holes & time warps (W. W.
Norton & Company 1994, s. 212) hittar jag ett kort beskrivning av Snyder som
uppenbarligen r baserad p Thornes intervju med Robert Serber, en annan av
Oppenheimers elever. ”Snyder was different from Oppenheimer’s other students.
The others came from middle-class families; Snyder was working class. Berkeley
rumor had it that he was a truck driver in Utah before turning physicist. As
Robert Serber recalls: ’Hartland pooh-poohed a lot of things that were
standard for Oppie’s students, like appreciating Bach and Mozart and going to
string quartets and liking fine food and liberal politics (…) Of all Oppie’s
students Hartland was the most independent (…) Hartland had more talent for
difficult mathematics than the rest of us’, recalls Serber. ’He was very good
at improving the cruder calculations that the rest of us did’.” F rutom svarta
h l f refaller Snyder ha varit k nd i samband med uppfinnandet av ’strong
focusing’ tekniken r 1952 f r acceleratorer. Snyders kvantrumtidsteori f ljdes
genast upp av C. N. Yang i Phys. Rev. 72, 874 (1947) som inf r
translationsinvarians. Snyder fick f rmodligen uppslaget till teorin fr n sin
l rare Oppenheimer som i sin tur tippsats av W. Pauli som f tt id n fr n W.
Heisenberg; n mligen, Heisenberg hade f r Pauli framkastat tanken om att inf
ra en os kerhetsrelation f r koordinater. Se F. A. Schaposnik, ”Three lectures
on noncommutative field theories” (hep-th/0408132). Pauli kommenterar Snyder
(1947) i ett brev till Bohr (28.1.1947): ”On the other hand I am looking as
critical as you on this idea of so called universal length. If this length -
let us call it $l_{0}$ \- is understood to be of geometrical nature, such
theories or models will always lead to strange consequences for large momenta
of the order $h/l_{0}$ in a field of purely classical experiments where the
quantum of action should not play any role. Recently we discussed here at Z
rich a mathematically ingeniuous proposal of Snyder, which, however, seems to
be a failure for reasons of physics of the type just mentioned.” (O. Stoyanov
har satt ut brevet p webben, se www.math.uiuc.edu/~stoyanov/math428-M1/,
kopierad fr n Wolfgang Pauli, Scientific Correspondence vol. III, s. 414, ed.
Karl von Meyenn, Springer-Verlag, 1985.) Heisenberg intresserade sig ocks f r
en sorts gitterteori (lattice theory, Gitterwelt) f r rumtiden under
1930-talet. Kanh nda p.g.a. den negativa responsen fr n Bohr s publicerade han
aldrig n got om teorin ven om hans id er blev k nda och diskuterade bland
fysikerna (se B. Carazza & H. Kragh, ”Heisenberg’s laittice world: The 1930
theory sketch,” Am. J. Phys. 63, 7, 595-605, 1995). Bohr ans g (i ett brev
till Mott 18.10.1929) att hypotesen om en minsta l ngdenhet i naturen stred
mot relativitetsteorin: ”To my view all such limitations would interfere with
the beauty and consistency of the theory [of relativity] to far great extent”.
N mligen, gittermodellen utgick fr n proton Compton-l ngden som minsta l
ngdenhet och denna r f rst s ingen relativistisk invariant. En tid senare
(1.4.1930) tillade han dock: ”Although I still think that my arguments were
correct, I have revised my attitude towards the matter and think now that very
general arguments can be given in favour of such a limitation of space
determinations, and that this very point is of essential importance as regards
obtaining consistency in the apparent chaos of relativity quantum mechanics.”
(Citaten fr n Carazza & Kragh 1995.) Trots en viss enstusiasm vergav Bohr id n
om den granul ra etermodellen bl.a. eftersom Heisenbergs modell ledde till
brott mot laddnings-konservering. Pauli verkar redan fr n b rjan ha ogillat
tanken om minsta l ngdenheter och varnade att ”those who are making holes in
continuous space should mind where they step” (Pauli under en kongress i
Odessa 1930, citerad i Carazza & Kragh 1995). F r en versikt av olika f rs k
att generalisera/modifiera rumtidsbegreppet se N. A. M. Monk, ”Conceptions of
space-time: problems and possible solutions,” Stud. Hist. Phil. Mod. Phys. 28
(1), 1-34 (1997). Monk har samarbetat med Basil Hiley, Bohms n ra medarbetare
under 80-talet, som det senaste rtiondet f rs kt utveckla en sorts algebraisk
teori f r fysiken och intresserat sig f r ’icke-kommutativ kvantgeometri’ (se
Hileys bidrag i Quo vadis quantum mechanics? A. Elitzur, S. Dolev & N.
Kolenda, eds. (Springer, 2005) s. 299-324). C. F. v. Weizs cker, en elev till
Heisenberg, har under en l ng era arbetat p en sorts kvantlogiskt fundament
(’Urtheorie’) f r fysiken i ett f rs k att s.a.s. f rena Einstein, Bohr och
Heisenberg (se Aufabau der Physik (Carl Hanser Verlag, 1985)). En intressant
omst ndighet r att Weizs cker (Aufbau, kap.9, ”Spezielle Relativit tstheorie”)
kommer fram till gruppen $SO(2,4)$ som ocks centralt figurerar i R. Penroses
twistorteori. Penrose har i sin tur i ver 40 r spunnit p med sina twistorer
men i sitt epos The road to reality (Jonathan Cape, 2004) hamnar han tillst
att teorin tillsvidare fr mst har gett en alternativ formulering av k nda
resultat inom relativistisk fysik. Han betonar att twistor-teorin inte r en
fysikalisk teori, men Penroses f rhoppning r dock att den skall bana v g f r
ny fysik i likhet med Hamilton-formalismen i klassisk fysik. ”Hamiltonian
theory did not introduce physical changes, but it provided a different outlook
on classical physics that later proved to be just what was required for the
new quantum theory”… (ibid. s. 1004). elev till J. R. Oppenheimer (Snyder och
Oppenheimer var f. . de f rsta att presentera ett scenario f r hur ett svart h
l bildas genom en implosion av en tryckfri v tska i ett gemensamt arbete i
Phys. Rev. 56, 455-459, 1939). Under 30- och 40-talet inspirerades arbeten
kring rumtids-kvantisering av tanken att man p detta s tta skulle kunna tg rda
divergensproblemen i kvantf ltteorierna. ”We hope that the introduction of
such a unit of length [parametern $a$ nedan] will remove many of the
divergence troubles in present field theory”, anm rker Snyder i inledningen
till sin artikel. I och med att renormaliseringstekniken utvecklades (Feynman,
Dyson, Tomonaga, Schwinger, Bethe, m fl) bortf ll denna motivering och arbeten
p rumdtidskvantiseringar stannande mer eller mindre av. Divergensproblemen
inom kvantgravitation och utveckligen av icke-kommutativ geometri har
emellertid terupplivat intresset f r dylika teorier (vore intressant att f lja
upp hur citationskurvan f r Snyders artikel utvecklats med
tiden).606060Snyders papper varit mer eller mindre bortgl md tills nyligen d
den teruppt ckts i samband med utvecklingen av icke-kommutativ geometri,
kvantgrupper och DSR. Man skulle kanske tycka att kvantrumtid hade intresserat
dem som studerat relationer mellan kvantteori och gravitationen. ven om t ex
C. W. Misner, K. S. Thorne & J. A. Wheeler i Gravitation (Freeman, 1973)
spekulerar om pregeometri, ’space-time foam’, etc, n mns inte Snyder. Tv f
rska b cker i kvantgravitation, C. Rovelli (2004) och C. Kiefer (Quantum
gravity (Oxford U. P., 2004)), n mner varkendera kvantrumtid eller Snyder. F
rsta g ngen jag sj lv st tte p en h nvisning till Snyder (1947) var i von
Weizs ckers Aufbau (1985). En av de f b cker verhuvudtaget som varit dedikerad
t mnet rumdtid och kvantmekanik torde vara D. I. Blokhintsev, Space and time
in the microworld (Reidel, 1973). En t mligen direkt procedur f r att erh lla
en icke-kommutativ geometri r att tolka koordinaterna $x^{i}$ som
rotationsoperatorer i ett utvidgat rum. Detta r vad Snyders g r genom att
uppfatta koordinaterna som hermitska operatorer, vars egenv rden motsvarar de
m tbara storheterna (”possible results of measurements”), genom att utg fr n
den reella kvadratiska formen
$-\eta^{2}=\eta_{0}^{2}-\eta_{1}^{2}-\eta_{2}^{2}-\eta_{3}^{2}-\eta_{4}^{2}\,,$
(57)
som f r konstant $\eta$ beskriver ett 4-dimensionellt de Sitter-rum.
Koordinaterna $x^{\mu}$ tolkas d som differentialoperatorer av formen (anv
ndande metriken $h=\mbox{diag}(1,-1,-1,-1,-1)$ och
$\eta_{\alpha}=h_{\alpha\beta}\eta^{\beta}$)
$x^{\mu}=ia\left(\eta^{\mu}\frac{\partial}{\partial\eta_{4}}-\eta^{4}\frac{\partial}{\partial\eta_{\mu}}\right)\quad(\mu=0,1,2,3),$
(58)
med $a$ som ”the natural unit of length”. F r dessa operatorer f r vi
$\displaystyle[x^{\mu},x^{\nu}]$ $\displaystyle=$
$\displaystyle\frac{ia^{2}}{\hbar}L^{\mu\nu}\quad\mbox{med}$ (59)
$\displaystyle L^{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle
i\hbar\left(\eta^{\mu}\frac{\partial}{\partial\eta_{\nu}}-\eta^{\nu}\frac{\partial}{\partial\eta_{\mu}}\right).$
(60)
F r $\mu$ = 1, 2, 3 har $x^{\mu}$ s ledes formen av en rotationsoperator med
egenfunktioner av formen $e^{im\phi}$ som d rf r implicerar egenv rdena $ma$,
f r $m\in\mathbf{Z}$; d.v.s., multipler av l ngdenheten $a$. F r tidsoperatorn
$x^{0}$ d remot har vi en ’boost’-typ operator som inte leder till n gon
motsvarande ’kvantisering’ av tiden. Att ’tiden’ h r tack vare pseudometriken
ist llet f r ett kontinuerligt spektrum kan m h nda antyda en po ng med att
’tiden’ skiljs ut med sitt speciella f rtecken i metriken. Kvantiseringen av
rumtiden verkar s ledes implicera en asymmetri, diskret vs kontinuerlig,
mellan rum och tid som inte direkt terfinns i SR. F r impulsvariablerna valde
Snyder (ett av m nga matematiskt rimliga alternativ) formen
$p^{\mu}=\frac{\hbar}{a}\frac{\eta^{\mu}}{\eta^{4}},$ (61)
vilket r f renligt med den konventionella relationen
$L^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}$ och ekv (60). Fr n (61) f ljer t ex
kommuteringsrelationen
$[x^{1},p_{1}]=i\hbar\left\\{1+\left(\frac{a}{\hbar}\right)^{2}p_{1}^{2}\right\\}.$
(62)
Ekv (62) implicerar i sin tur en os kerhetsrelation av typen (som numera g r
under beteckningen ’generalized uncertainty relation/principle’, GU)
$\Delta x\Delta
p\geq\frac{\hbar}{2}\left\\{1+\left(\frac{a}{\hbar}\right)^{2}{\langle
p^{2}\rangle}\right\\},$ (63)
som bl a aktualiserats inom str ngteorin.616161M. Maggiore, ”Quantum groups,
gravity, and the generalized uncertainty principle,” Phys. Rev. D 49, 10,
5182-7, 1994; L. J. Garay, ”Quantum gravity and minimum length,” Int. J. Mod.
Phys. A10, 145, 1995 (gr-qc/9403008). (Mer exakt borde vi f r (63) ha tagit
kvadratroten av medelv rdet av kvadraten av h gra ledet i (62),
$\sqrt{\langle(\dots)^{2}\rangle}$.) Ekv (63) inneb r vidare att
$\Delta x\approx\frac{\hbar}{2}\left\\{\frac{1}{\Delta
p}+\left(\frac{a}{\hbar}\right)^{2}\Delta p\right\\}.$ (64)
Minimerar vi h gra delen i (64) visavis $\Delta p$ f r vi olikheten $\Delta
x>a$ (minimiv rdet motsvarar $\Delta p=\hbar/a$) vilket ter visar att $a$
karakt riserar en sorts minsta m tbara l ngdenhet hos rummet.
I en uppf ljande artikel626262H. S. Snyder, ”The electromagnetic field in
quantized space-time,” Phys. Rev. 72, 1, 68-71, 1947. formulerade Snyder
Maxwells ekvationer i denna modell f r kvantrumtid och antydde att Diracs,
Procas och Kleins ekvationer kunde behandlas analogt, men detta projekt f
refaller ha avbrutits. (Snyders sista vetenskapliga publikation i Physical
Review r fr n r 1955.) Sju r senare publicerar Hellund och Tanaka en
artikel636363E. J. Hellund & K. Tanaka, ”Quantized space-time,” Phys. Rev. 94,
1, 192-195, 1954. H&T-modellen r en sorts reciprok variant av Snyder-modellen.
baserad p en variation av Snyders modell d r de bl a uppm rksammar relationen
(63) och unders ker Dirac-ekvationen och m jligheten av l sningar som ger
’time-localized states’. De senaste ren kan utvecklingen s gs ha fortsatt i
form av f lt-teori p icke-kommutativa rum.646464F r versikter se Schaposnik
(hep-th/0408132); A. Tureanu, ”Some aspects of quantum field and gauge
theories on noncommutative space-time,” Helsingfors Universitet 2004,
Ph.D.-avhandling, e-version (pdf):
http://ethesis.helsinki.fi/julkaisut/mat/fysik/vk/tureanu/. Ett enkelt s tt
att inf ra icke-kommutativitet r att f r klassiska kommutativa koordinater
$x^{i}$ definiera koordinatoperatorer
$\hat{x}^{i}=x^{i}-\frac{1}{2\hbar}\Theta^{ij}p_{j},$ (65)
d r $p_{j}=-i\hbar\partial/\partial x^{j}$ och $\Theta$ r en konstant
antisymmetrisk matris. Av (65) f ljer att
$[\hat{x}^{i},\hat{x}^{j}]=\Theta^{ij}.$ (66)
Calmet (2005)656565X. Calmet, ”Space-time symmetries of noncommutative
spaces,” Phys. Rev. D 71, 085012, 2005. argumenterar i omv nd riktning; givet
(66) s kan vi inf ra kommuterande koordinater $x^{i}$ via (65).
### 12\. Avslutning
I detta skede, mer n 100 r efter publiceringen av SR, kan vi konstatera att
Relativitetsteorin r mer bef st n n gonsin. Spekulationer om avvikelser fr n
SR har f rblivit blott spekulationer. Som antytts terst r dock m nga
fundamentala problem inom fysiken som kan ha konsekvenser f r SR. P det
teoretiska omr det kan emellertid sk njas en sorts
utmattning.666666Matematikern-fysikern Peter Woit har i sin Not even wrong.
The failure of string theory and the continuing challenge to unify the laws of
physics (Jonathan Cape, 2006) g tt till attack mot str ng-teorin eftersom den
inte har lett till en enda verifierbar f ruts gelse och menar att den mer
liknar religion n vetenskap. De mer begr nsade ansatserna till kvant-
gravitation har ej heller lett till n got genombrott. Lee Smolin som jobbat
med loop-kvantisering verkar i The trouble with physics: The rise of string
theory, the fall of a science, and what comes next (Houghton Mifflin, 2006)
ocks ganska desperat och efterlyser en ny Einstein. Smolins bok pryds f
rresten av en p rmbild visande ett par skor med hopbundna skosn ren … ’Str
ng-teorins’ 20- riga kenvandring har inte verkat f ra oss n rmar det f rlovade
landet som skulle bl a klara upp kvant-gravitationen. Men liksom tidigare
under historiens g ng kan den f rl sande insikten komma n rsomhelst fr n ett
ov ntat h ll.
|
arxiv-papers
| 2008-09-03T20:57:48
|
2024-09-04T02:48:57.633763
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Frank G. Borg",
"submitter": "Frank G. Borg",
"url": "https://arxiv.org/abs/0809.0708"
}
|
0809.0823
|
# Strangeness production in STAR
###### Abstract
We present a summary of strangeness enhancement results comparing data from
Cu+Cu and Au+Au collisions at $\sqrt{S_{NN}}=200GeV$ measured by the STAR
experiment. Relative yields in central Cu+Cu data seem to be higher than the
equivalent sized peripheral Au+Au collision. In addition, strange particle
production from these two systems is compared in terms of a statistical model,
applying a Grand-Canonical ensemble and also applying a canonical correlation
volume for the strange particles. Thermal fit results from the Grand-Canonical
formalism shows little dependence on the system size but, when considering a
strange canonical ensemble, strangeness enhancement shows a strong dependency
on the correlation volume.
## 1 Introduction
Strangeness production is expected to be enhanced in heavy ion collision as
one of the possible signatures of the Quark-Gluon Plasma [1]. An increase of
the strange quark density should result in an increase of the strange hyperon
production thus, the measured integrated yield per participant should be
sensitive to the density of quarks in the system. Experimentally, strangeness
enhancement has been observed in A+A collisions with respect to p+p collisions
when comparing the normalized strange particle yields scaled by factors such
as the number of participants in the collision, or the number of wounded
nucleons [2, 3]. Thermal models with a canonical approach show that this
observed relative strangeness enhancement can also be explained by considering
a suppression due to limitations on the phase space available for strange
particles production in small systems [4, 5]. These models have been quite
successful in describing the level of enhancement and the observed hierarchy
with the number of strange valence quarks, but up to now, no model has been
able to describe simultaneously all features of the measured strangeness
enhancement. In this paper, we present the latest results from the analysis of
the STAR experiment with respect to the strangeness enhancement including data
from Au+Au and Cu+Cu collisions at RHIC maximum energy. In addition, we apply
a statistical thermal model fit to study the system size dependence of thermal
parameters.
## 2 Strangeness Enhancement
Fig. 1.: Strangeness enhancement plots, particle yields from Au+Au and Cu+Cu
data normalized by $<N_{part}>$ relative to yields from p+p as a function of
$<N_{part}>$. Left plot shows the relative enhancement of $\Lambda$, $\Xi$ and
$\Omega+\bar{\Omega}$, and the right plot shows the same data for
$\bar{\Lambda}$, $\bar{\Xi}$ and $\Omega+\bar{\Omega}$.
Figure 1 left side shows the system size dependence of the strangeness
enhancement ratio for $\Lambda$, $\Xi$ and $\Omega+\bar{\Omega}$ measured in
STAR for Au+Au and Cu+Cu at collision energy of 200 GeV per nucleon pairs.
Figure 1 right shows the same plot for $\bar{\Lambda}$, $\bar{\Xi}$ and
$\Omega+\bar{\Omega}$. Particle yields were measured for different event
centrality classes and normalized by the equivalent mean number of participant
nucleons $<N_{part}>$ and then divided by the equivalent ratio measured in p+p
collisions at the same energy. The $\Lambda$ and $\bar{\Lambda}$ yields were
corrected for feed-down from $\Xi$ decays. In both Au+Au and Cu+Cu data, a
large strangeness enhancement is observed even in the most peripheral
centrality bin. The data also shows a strong dependency with the system size
and do not seem to saturate as expected from a Grand-Canonical Thermal model.
Strangeness enhancement is higher for strange particles than for anti-
particles, which can be a result of the non zero net baryon density. The
striking result is that the strangeness enhancement observed in central Cu+Cu
data is higher than the peripheral Au+Au collision with equivalent
$<N_{part}>$. This is an indication that the production mechanism of these
strange particles does not scale with the pure geometrical parameterization of
the system size. Normalized proton yields were included in the plots of figure
1 as a reference to show that the relative discrepancy between the two systems
is unique to the strangeness sector. The measured proton yields in STAR are
inclusive [7], and in this plot these proton yields were subtracted by the
$\Lambda$ yield factorized by the decay branching ratio to correct for the
feed-down into the proton yield. The normalized proton yields do not show any
dependence with the system size and also no difference between Au+Au and Cu+Cu
data.
Fig. 2.: Energy dependence of the relative strangeness enhancement for
$\Lambda$, $\bar{\Lambda}$, $\Xi$, $\bar{\Xi}$ and $\Omega+\bar{\Omega}$,
observed at central collisions of Au+Au and Pb+Pb, compiling RHIC and SPS
results. SPS results shown here are from experiment NA57 [3].
Figure 2 shows the collision energy dependence of the strangeness enhancement
factor for the most central events for $\Lambda$, $\bar{\Lambda}$, $\Xi$,
$\bar{\Xi}$, $\Omega+\bar{\Omega}$. SPS data for the low energy points were
extracted from Pb+Pb data of experiment NA57 [3]. The observed strangeness
enhancement is decreasing with collision energy for the $\Lambda$ and $\Xi$,
as predicted by Grand-Canonical statistical model, but, an opposite trend is
observed for the anti-particles that yields higher enhancement in RHIC data
compared to SPS data. Also clear in this plot is the strange quark content
hierarchy observed in the enhancement as predicted by statistical thermal
models.
## 3 Statistical thermal model results
Data was analyzed using the THERMUS code [6] and considering particle ratios
that includes $p$ ($\bar{p}$), $\pi^{\pm}$, $K^{\pm}$, $\Lambda$
$(\bar{\Lambda})$, $\Xi^{\pm}$, $\Omega^{\pm}$, and $\phi$. Protons were
corrected for the feed-down of the $\Lambda$ decay that resulted in a
reduction of the inclusive proton yield of approximately $30\%$. Since STAR
does not yet have $\Sigma$ measurements, the feed-down contribution from this
particle was estimated using the ratio of $\Sigma/\Lambda=0.35$ [8]. Final
reduction of the inclusive proton yields was on the order of $35\%$ to $45\%$.
To estimate the error due to this feed-down correction, the relative $\Sigma$
yield to $\Lambda$ was varied over $50\%$. The overall effect on the final
thermal parameters was less than $10\%$.
Fig. 3.: Thermal model fit chemical freeze-out temperature $T_{ch}$, and
strangeness chemical potential $\mu_{S}$, as a function of $<N_{part}>$ for
Au+Au and Cu+Cu data. For comparison, the same fit was applied to p+p data and
included in this plot.
Figure 3 shows the chemical freeze-out temperature $T_{ch}$ and the
strangeness chemical potential $\mu_{S}$ as a function of the system size,
$<N_{part}>$. $T_{ch}$ is around 155 MeV and seems to show no dependence on
the system size and also no sensitivity to the colliding systems, yielding the
same results for Au+Au and Cu+Cu data. The baryon chemical potential $\mu_{B}$
and strangeness chemical potential $\mu_{S}$ also showed no difference between
the fits to Au+Au and Cu+Cu data. To study the validity of the statistical
thermal fit considering a Grand Canonical formulation, we have applied the
same fit to the p+p data. The chemical freeze-out temperature that results
from this fit is slightly lower, around 150 MeV.
Fig. 4.: Strangeness under-saturation parameter of the thermal model from fits
to particle ratios of Au+Au and Cu+Cu data for the different system sizes. The
result from a fit to the p+p data is also included.
Figure 4 shows the dependence on the system size of the strangeness under-
saturation factor $\gamma_{S}$ obtained from the statistical thermal fit with
a Grand-Canonical ensemble for the Au+Au and Cu+Cu data. Cu+Cu data shows the
same results and behavior of the peripheral Au+Au data. The $\gamma_{S}$
reaches unity only above $<N_{part}>$ greater than approximately 100. Most of
the Cu+Cu data is below that limit. The $\gamma_{S}$ factor can be interpreted
as a measure of the validity of the Grand-Canonical formalism to describe the
data in the strange sector. Under this assumption, from the results shown in
figure 4, it is clear that only the most central bins of Cu+Cu collision can
be well described with this approach. $\gamma_{S}$ from the fit to the p+p
data is also shown in figure 4, and shows a much lower value, around 0.6,
indicating that the strange particles ratios in p+p collisions cannot be well
described with this model.
Fig. 5.: Plot shows the strange particle to pion ratio as predicted by the
thermal model considering a Grand-Canonical ensemble with a strange canonical
system, as a function of the radius of the strangeness canonical size.
As an alternative to the Grand Canonical formulation (GC) where the
conservation of the quantum numbers are ensured on average, the THERMUS code
allows for a strangeness canonical (SC) ensemble where only the quantum
numbers of the strange particles are required to conserve exactly. In this
formalism, a correlation volume of strange particle production is defined as
the sub-volume where the strangeness chemical equilibrium is restricted. We
applied the fit to the Au+Au 200 GeV data using this approach and obtained a
strangeness correlation volume radius of approximately $5fm$, for the most
central event centrality class. This value is much higher than the value
presented for a similar analysis on SPS data [3] that was around $1fm$. To
understand the effect of this correlation volume on the final strange particle
yields, we studied the variation of the strange particle yields relative to
the pions for different correlation volumes. Figure 5 shows the results of the
model prediction in solid lines and the symbols represent the experimental
values of the most central Au+Au 200 GeV data. It is interesting to note that
the $\phi/\pi$ ratio does not depend on the correlation volume. The value of
the correlation volume obtained from the best fit to the data indicates that
the system is already in the region where the strange particle production is
saturated, and thus, consistent with a system where the strangeness is already
equilibrated.
## 4 Conclusions
Strange particle production is enhanced in heavy ion collisions compared to
p+p. This enhancement increases with the system size and does not seem to
saturate as expected by models considering Grand Canonical Statistical models.
When using $<N_{part}>$ as a scaling factor, Cu+Cu central collision show
higher strangeness enhancement than peripheral Au+Au collision. Application of
thermal model fit to Au+Au and Cu+Cu data seem to show almost no difference in
the thermal parameters. In addition, the thermal fit parameters considering a
Grand-Canonical approach seem to show no dependency on the system size except
for the $\gamma_{S}$ parameter. However, when considering a strangeness
Canonical formalism in the statistical thermal model, strangeness production
shows strong dependence on the correlation volume. Fits to the data indicate
that the correlation volume is large, around $5fm$, above the saturation
point.
## Acknowledgments
We wish to thank RHIC and STAR group and the U.S. DOE Office of Science for
the support in the participation of this conference.
## References
* [1] J. Rafelski and B. Mueller, Phys. Rev. Lett., 48 (1982) 1066.
* [2] B.I. Abelev et. al. (STAR Collaboration), Phys. Rev. Lett., 77, 044908 (2008).
* [3] F. Antinori et al. (NA57 Collaboration), J. Phys. G, 32 (2006) 427-441.
* [4] A. Tounsi, A. Mischke and K.Redlich, Nucl. Phys. A715, (2003) 565.
* [5] I. Kraus, et al, Phys. Rev. C76, 064903 (2007).
* [6] S. Wheaton and J. Cleymans, J. Phys. G. 31, (2005) S1069.
* [7] A. Iordanova for the STAR Collaboration, Particle Production at RHIC, These proceedings.
* [8] B.I. Abelev et al. (STAR Collaboration) Phys.Rev.Lett. 97 (2006) 152301.
|
arxiv-papers
| 2008-09-04T14:23:02
|
2024-09-04T02:48:57.643771
|
{
"license": "Public Domain",
"authors": "J. Takahashi and R. Derradi de Souza",
"submitter": "Jun Takahashi",
"url": "https://arxiv.org/abs/0809.0823"
}
|
0809.1031
|
# On the role of the nonlocal Hartree-Fock exchange in ab initio quantum
transport:
H2 in Pt nanocontacts revisited
Y. García
Instituto de Ciencia de Materiales,
Universidad de Valencia, E-46071 Valencia, Spain
and
J.C. Sancho-García
Departamento de Química-Física,
Universidad de Alicante, E-03080 Alicante, Spain E-mail: Yamila.Garcia@uv.es
###### Abstract
We propose a practical way to overcome the ubiquitous problem of the
overestimation of the zero-bias and zero-temperature conductance, which is
associated to the use of local approximations to the exchange-correlation
functional in Density-Functional Theory when applied to quantum transport.
This is done through partial substitution of the local exchange term in the
functional by the nonlocal Hartree-Fock exchange. As a non-trivial example of
this effect we revisit the smallest molecular bridge studied so far: a H2
molecule placed in between Pt nanocontacts. When applied to this system the
value of the conductance diminishes as compared to the local-exchange-only
value, which is in close agreement with results predicted from Time-Dependent
Current-Density-Functional Theory. Our results issue a warning message on
recent claims of perfect transparency of a H2 molecule in Pt nanocontacts.
It is widely admitted that a first-principles methodology based on Density-
Functional Theory (DFT) is the only compromise between accuracy and
computational resources that can help pave the way towards molecular-
engineered nanoscale devices[1]. The DFT manageability is based, for the most
part, in the use of the Kohn-Sham scheme (DFT-KS)[2, 3]. In addition, the
central track doing a practical DFT-KS scheme lies on the definition of an
approach to the true exchange-correlation (xc) functional[4]. Initially, the
manageability of DFT-KS comes upon the introduction of the Local-Density
Approximation (LDA) to the xc functional[3]. (Henceforth, extensions beyond
LDA within local exchange models, like generalized gradient approximations
(GGA)[5], and Meta-GGA[6], are referred as semi-local methods and labeled as
LDA methods for the sake of simplicity).
This ab-initio based method combined with Landauer’s formalism is nowadays
routinely applied to compute the conductance, $G$, of nanoscaled systems, i.e
metallic nanocontacts and single-molecule junctions[7, 8, 9]. However, the use
of DFT-KS electronic structure within LDA approximations, would present two
problems in this regard: (i) an obvious one related to the use of LDA and
related approximations to approach the true xc-potential, $v_{xc}$[4, 9]; and
(ii) a much more subtle one which could be associated with the absence of
dynamical corrections in the DFT-KS schemes[10, 11].
Regarding the first point, the reliability of LDA results relies naturally on
the existence of fairly homogeneous, well-behaved electronic densities. While
this is the case for metallic nanocontacts, molecular bridges, definitely, do
not satisfy this premise[10]. For weakly-coupled molecules conducting in the
Coulomb blockade regime, where charge localization is strong, failures of LDA
are traced back to the self-interaction and derivative discontinuity problems,
which are inherent to local or semi-local approximations[13, 12, 14, 9].
Interestingly, this problem is not present in more sophisticated exact
exchange (EXX) nonlocal approaches, as recently emphasized by others[15].
Concerning the second point, the use of Time-Dependent Density-Functional
Theory (TDDFT)[16] has been shown to give the exact total current if the exact
$v_{xc}$ is known[10], and it reduces in the zero-frequency limit to the
standard static treatment only when the adiabatic approximation is
invoked[10]. Furthermore, as initially suggested by Vignale and Kohn[11], the
apparent impossibility to approach the exact $v_{xc}$ through successive local
approximations could be circumvented by a new form of functional where the
local current density plays a central role or, equivalently, a nonlocal
density-based theory. Recent work[10] based on Time-Dependent Current-Density-
Functional Theory (TDCDFT) has shown that dynamical effects manifest
themselves in the appearance of a dynamical potential, $v_{\rm xc}^{\rm dyn}$,
that always opposes the electrostatic potential and lowers the conductance
obtained within LDA methods, even in linear response. The origin of this
potential can be traced back to the nonlocal response to an electric field
which is captured by TDCDFT[11, 10]. It is difficult to assess to what extent
the resulting dynamical potential accounts for the lack of nonlocality of the
starting LDA approximation or if it represents a true intrinsic dynamical
effect which is present even for the exact DFT-KS potential[9] (probably
both).
The role played by the dynamical term in increasing the resistance is
compatible with another seemingly unrelated fact: The polarizability of
molecular systems (e.g., hydrogen[17] or polymeric[18] chains) is severely
overestimated by LDA, while Hartree-Fock (HF), TDCDFT, and exact many-body
calculations –all of them nonlocal schemes– yield similar values. We note here
that a recent work has shown that the conductance can be expressed in terms of
the polarizability[19].
In the light of the above discussion, one might ask: to what extent can
nonlocal approaches to the $v_{xc}$ (out of the realm of standard static DFT-
KS theory) mimic the missing dynamical corrections? In this work we explore
the influence of a nonlocal potential -such as HF exchange- on the conductance
of a molecular bridge (see Figure 1). To this end we choose to study electron
transport in a H2 molecule bridging Pt nanocontacts[20, 21, 22] (see inset in
Figure 2), using nonlocal functionals customized for this problem. We show
that the HF-like exchange has a strong influence on the conductance of this
bridge, rapidly decreasing as the percentage of HF-like increases in the
hybrid functional. By adjusting this percentage, with the aid of accurate
quantum chemistry calculations in clusters, we conclude that previously
reported calculations based on LDA might have systematically overestimated the
value of the conductance in this system. Finally, for this benchmark system,
we compare the numerical results derived from our method, the nonlocal
functionals for an effective DFT-KS formalism, with those obtained within
TDCDFT[10]. The conductance values are quite similar, which further supports
the conclusions reached here. The average value of the conductance for such
system turns out to be $\approx 0.2\times 2e^{2}/h=0.2\times G_{0}$, in stark
contrast to LDA calculations yielding $\approx 1.0\times G_{0}$ and in support
of a previous work by some of us[21].
Our theoretical approach have been thus developed in accordance with the
scheme that follows. The central track doing an effective DFT-KS scheme lies
on the definition of the $v_{xc}$[1, 23, 4]. Rooted on the adiabatic
connection theorem, which formally justified the design of nonlocal
functionals (usually referred as hybrid functionals[24]), and based on the
experience gained after original Becke’s proposal[25], we adopt the following
expression for an approach to the $v_{xc}$:
$v_{xc}=\alpha v_{x}^{HF}+\left(1-\alpha\right)v_{x}^{local}+v_{c}^{local},$
(1)
where $v_{x}^{HF}$ is the HF-like potential contribution, which accounts for
nonlocal effects, the $v_{x,c}^{local}$ contributions are given by any of the
available exchange (x) or correlation (c) local functionals and, finally,
$\alpha$ is a fitting parameter. With this parameterized potential, we will
solve the mean-field equations derived from DFT-KS formalism for a finite
system, the cluster. This finite cluster is selected in such a way that the
boundary conditions determined by the problem are fulfilled, as discussed in
the next paragraphs. Finally, the electronic states derived from the DFT-KS
are deployed in conjunction with Landauer formalism for zero temperature
quantum transport characterization[26]. Concerning the software employed for
the numerical calculations, we use the code ALACANT (ALicante Ab initio
Computation Applied to NanoTransport)[27] which is interfaced to
GAUSSIAN03[28].
We first compute the LDA conductance ($\alpha=0$) of the system shown in the
inset of Figure 2 where the H2 molecule lies along the Pt nanocontacts
axis[29]. In agreement with previous works[20, 30, 22], we obtain a single
fully conducting channel at the Fermi level which yields $G\approx 1.0\times
G_{0}=2e^{2}/h$. This result supports the implementation for the computation
of the current[31].
To probe in a systematic way the influence of the nonlocal exchange on the
electronic structure and concomitant change in the conductance, the
coefficient $\alpha$ is now varied. Strictly speaking $\alpha$ should not be a
constant, but a function of spatial coordinates, $\alpha=\alpha(\vec{r})$. It
is expected that HF-like exchange is more needed to describe electronic
density regions governed by just one electron where the self-interaction error
is more notorious[32]. This is particularly the case for the H2 molecule. The
value of $\alpha$, on the other hand, should decrease as we move into the bulk
electrodes where LDA or GGA is expected to perform better. As a compromise
between both situations we take $\alpha$ to be constant in the nanocontacts
region close to the H2 molecule where the coordination is low, i.e., the
bridge Pt-H2-Pt (see inset in Figure 3). The rest of the system is described
by the self-energy corresponding to a parameterized Bethe lattice. Figure 1
shows the effect of the HF-like exchange on the conductance. For values close
to 50% the conductance has almost dropped to zero.
The drop in conductance as the amount of HF-like exchange increases at the
bridge can also be understood from a purely microscopic point of view. This
construction is based on the approach to the exact non-correlated density-
based scheme for determine the electronic structure[23], and its combination
with the accepted Landauer formalism for determine the electronic
conductance[7]. Therefore, under these two conditions, the electronic
structure limited to the HOMO-LUMO gap, $\Delta_{HOMO-LUMO}$, will play a key
role in the final outcome of the electronic conductance[33]. Figure 3 shows
the effect of the HF-like contribution on the HOMO-LUMO gap ($\Delta_{HOMO-
LUMO}$) for the isolated Pt-H2-Pt bridge. This effect is well known in
molecules and bulk semiconductors where LDA approximations usually
underestimate the real value of the charge gap[34, 35, 36]. The one electron
Green’s function ($g(\epsilon)$) of the isolated Pt-H2-Pt bridge has poles at
the position of the molecular orbitals energies. Once the isolated system is
coupled to the rest of the electrodes these poles shift and broaden in a way
dictated by the self-energies $\Sigma_{\rm L}$ and $\Sigma_{\rm R}$
representing the electrodes:
$g^{(\pm)}(\epsilon)=[(\epsilon\pm\,i\delta){I}-{H}-\Sigma_{\rm L}-\Sigma_{\rm
R}]^{-1}.$ (2)
The Fermi level is likely to lie in the gap between the HOMO and LUMO states
of the Pt-H2-Pt cluster since no significant charge transfer is expected
between Pt atoms. As can be seen in Figure 3, the LDA gap is almost zero for
this bridge. From the opening of the gap as $\alpha$ increases one could
easily have anticipated a strong decrease in conductance since this depends on
the Green’s functions through the well-known expression[26]:
$G=\frac{2e^{2}}{h}Tr[\hskip 5.69054pt{\Gamma_{L}}(\epsilon)\hskip
5.69054pt{g^{+}}(\epsilon)\hskip 5.69054pt{\Gamma_{R}}(\epsilon)\hskip
5.69054pt{g^{-}}(\epsilon)].$ (3)
Finally, to estimate $\alpha$ we follow a well-established methodology in
semiconductors and insulators. The value is chosen so that $\Delta_{HOMO-
LUMO}$ matches the charge gap, $GAP$. Accordingly, it is defined as
$GAP=2E_{0}(N)-E_{0}(N+1)-E_{0}(N-1)$, where $N$ represents the number of
electrons, and $E_{0}$ is the total energy for the selected ground states in a
Kohn-Sham system of independent electrons. Matching between $\Delta_{HOMO-
LUMO}$ and $GAP$ will be enough to guarantee accuracy for mean-field Green’s
functions[37, 38], and therefore for the conductance value, expressions 2 and
3. The choice of the finite region where to perform the search of $\alpha$ is
constrained by three conditions: (i) it must contain the zone which is
restricting the current flow, in our case the H2 molecule; (ii) the HOMO and
LUMO must not be localized on the electrodes; and (iii) screening effects from
the contacts should be -at least- partially included. These three conditions
reduce the range of possibilities to the cluster shown in Figure 3, and limit
the selection of system (previously referred as the extended molecule[9, 39]).
One way to estimate to what extent screening is included in such small cluster
is to look for a quantity such as the internal mode frequency of the H2
molecule, $w_{H_{2}}$, and check how much it changes with increasing the size
of the cluster. We find that larger clusters do not appreciably change that
quantity, which is given by $w_{H_{2}}\approx w_{H_{2}}^{0}/2$ where
$w_{H_{2}}^{0}$ is the internal mode of an isolated H2. This change in the
frequency with respect to the free molecule suggests an intermediate regime of
coupling.
To find out ground state energies and therefore the right $GAP$, we begin by
considering the mean-field HF method. To approach the exact many-body solution
to the $GAP$, we invoke hierarchical correlated ab initio methods such as
Möller-Plesset perturbation (MP) or Coupled-Cluster (CC) theories. The
corresponding results are presented in Figure 4. The $GAP$ for this system,
$\sim 5$ eV, critically differs from the $\Delta_{HOMO-LUMO}$, described from
the electronic structure in pure DFT-LDA calculations $\sim 0$ eV, see Figure
3. Therefore, and considering the scheme we propose to correct this pitfall,
the $\Delta_{HOMO-LUMO}$ for the system is extracted from Figure 4. Comparing
those results with the whole information contained in curve of Figure 3, we
estimate that the percentage of HF exchange in Eq. 1 should be $\sim 60$
percent, which gives a very low value for the conductance (see Figure 1). It
is the main result we should draw from this paper.
Note that there is an increasing concern about the influence of exact exchange
for conduction phenomena at the molecular level[41, 40]. Although a rule of
thumb for the needed admixture of exact exchange is provided here, we indeed
believe that it will be difficult to unambiguously give the optimum weight of
exact exchange for an uniformly good description of all systems. Instead of
this, we would like to remark how the results clearly show that inclusion of
exact exchange is necessary for accurate descriptions. We should point that
the same amount of HF exchange is obtained (with error band of 10 percent), if
the contour condition to find out its value is limited to the matching between
the HOMO eigenvalue and the negative of the ionization potential,
$\epsilon_{HOMO}=-IP$[42]. Concerning future work and despite being
technically challenging, it would be definitively interesting to investigate
the performance on these benchmark systems of a new generation of functionals
describing $v_{xc}$. For instance, we should mention orbital-based
functionals[43], self-interaction corrected functionals[44, 45], the hybrid
functionals based on a screened Coulomb potential[46, 47], and the last
generation of $v_{xc}$ density-functionals belonging to the M06 suite of
methods[48, 49]. Furthermore, more studies for closely related molecules and
nanocontacts will be highly welcomed too.
If we were able to follow unambiguously the same procedure for larger systems
the value for $\alpha$ would probably be reduced, although we do not expect it
to go to zero. Unable to get rid of this uncertainty we compare our results to
those obtained with TDCDFT following the work by Sai et al.[10]. A dynamical
counteracting potential appears whenever there is a change in the macroscopic
density which modifies the local-density conductance:
$G=\frac{G_{\rm local}}{1+G_{\rm local}R^{\rm dyn}}$ (4)
From the average density profile along the z-direction (see Figure 5) we
compute the corrections to the local conductance. The electronic density is
fixed in the electrodes to the bulk value, considering for the number of free-
electrons as the mean experimental atomic-point-contact transmission. In case
of Pt-nanocontacts, this number implies a range between one- and two-free-
electrons per Pt atom[20]. The rest of the parameters involved in the
calculation of $R^{\rm dyn}$ were estimated in agreement with the method
proposed by Sai et al.[10]. Our calculations brings the conductance down to
$\approx 0.2\times G_{0}$. (Details are included in Ref. [50]). Nevertheless,
this procedure is not free from ambiguities either, but it points towards
similar conclusions as the ones drawn above.
In summary, we have presented a route to approach the DFT-KS electronic
transport calculations based on the controlled addition of EXX contribution to
$v_{xc}$. The selection of the fraction of HF-like needed is based on first-
principles procedures. Our results agree with the ones obtained with the
method developed by Sai et al[10] within TDCDFT, and supports the
functionality of DFT methods based on nonlocal models for $v_{xc}$. Its
agreement opens up possibilities to formally justify a connection between
nonlocal DFT and TDCDFT in the zero-frequency limit. Furthermore, more work is
needed on this respect since the exact amount of nonlocality might be
difficult to estimate given: (i) the difficulties for the definition of the
region where it should operate; and (ii) the correct implementation of this
behavior. However, we can at least define an upper limit for this nonlocal HF
correction, and give arguments in favour of its non-zero value, as DFT-LDA
methods considered. Within this context, the numerical calculation of the
conductance for a H2 molecule in Pt nanocontacts have been corrected, lowering
in one order of magnitude the standard DFT-LDA calculation. Note, however,
that the statements and the conclusions that will be made are not limited to
calculations of conductance in this benchmark system. It is widely accepted
that nonlocal contributions to $v_{xc}$ do play a significant role to get
beyond DFT-LDA results either in molecular systems or bulk materials. For
instance, we could mention the polarizability of polymeric chains[18], the
magnetic coupling and the band structure in systems such as antiferromagnetic
insulators[51, 52] and magnetic oxides[53]. Finally, our theory is that the
overestimation of the conductance is not intrinsic to the DFT-KS approach.
This wrong behavior could be greatly improved by a controlled design of an
exchange-correlation potential which incorporates nonlocal contributions.
We gratefully acknowledge M. DiVentra’s detailed revision of this manuscript.
Y. García is indebted to Professor Francesc Illas for recommending the use of
quantum chemistry methods, and also acknowledges valuable discussions with
J.J. Palacios and E.Louis (Alicante, Spain) along last years. J.C.S.G. also
thanks the “Ministerio de Educación y Ciencia” (CTQ2007-66461/BQU) of Spain
for a research contract under the “Ramón y Cajal” program and the “Generalitat
Valenciana” for further economic support.
## References
* [1] W. Kohn, Reviews of Modern Physics 71, 50 (1999).
* [2] P. Hohenberg and W. Kohn, Phys. Rev. Lett. 136, B864 (1964).
* [3] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
* [4] R. K. Nesbet, Modern Physics Letters B 18, 73 (2004).
* [5] J. P. Perdew, J. Chevary, S. Vosko, K. A.Jackson, M. R. Pederson, D. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).
* [6] V. N. Staroverov, G. E. Scuseria, J. Tao, and J. P. Perdew, J. Chem. Phys. 119, 12129 (2003).
* [7] N. Agraït, A. L. Yeyati, and J. M. van Ruitenbeek, Physics Reports 377, 81 (2003), and references therein.
* [8] J. J. Palacios, A. J. Pérez-Jiménez, E. Louis, E. SanFabián, J. A. Vergés, and Y. García, in _Computational Chemistry: Reviews of Current Trends_ , edited by J. Leszczynski (World Scientific, Singapore-New Jersey-London-Hong Kong, 2005), vol. 9.
* [9] M. Koentopp, C. Chang, K. Burke, and R. Car, J. Phys.: Condens. Matter 20, 083203 (2008).
* [10] N. Sai, M. Zwolak, G. Vignale, and M. D. Ventra, Phys. Rev. Lett. 94, 186810 (2005).
* [11] G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996).
* [12] F. Evers, F. Weigend, and M. Koentopp, Phys. Rev. B 69, 2354411 (2004).
* [13] J. J. Palacios, Phys. Rev. B 72, 125424 (2005).
* [14] C. Toher, A. Filippetti, S. Sanvito, and K. Burke, Phys. Rev. Lett 95, 146402 (2005).
* [15] S.-H. Ken, H. U. Baranger, and W. Yang, J. Chem. Phys. 126, 201102 (2007).
* [16] K. Burke, J. Werschnik, and E. K. U. Gross, J. Chem.Phys. 123, 062206 (2005).
* [17] S. Kummel, L. Kronik, and J. P. Perdew, Phys. Rev. Lett. 93, 213002 (2004).
* [18] M. van Faassen, Ph.D. thesis, Rijksuniversiteit Groningen (2005).
* [19] P. Bokes and R. W. Godby, Phys. Rev. B 69, 245420 (2004).
* [20] R. H. M. Smit, Y. Noat, C. Untiedt, N. D. Lang, M. C. van Hemert, and J. M. van Ruitenbeek, Nature (London) 419, 906 (2002).
* [21] Y. García, J. J. Palacios, E. SanFabián, J. A. Vergés, A. J. Pérez-Jiménez, and E. Louis, Phys. Rev. B 69, 041402(R) (2004).
* [22] K. S. Thygesen and K. W. Jacobsen, Phys. Rev. Lett. 94, 036807 (2005).
* [23] R. M. Martin, _Electronic Structure. Basic Theory and Practical Methods_ (Cambridge University Press, 2004).
* [24] C. Adamo and V. Barone, Chem. Phys. Lett. 274, 242 (1997).
* [25] A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
* [26] S. Datta, _Electronic transport in mesoscopic systems_ (Cambridge University Press, Cambridge, 1995).
* [27] E. Louis, J. A. Vergés, J. J. Palacios, A. J. Pérez-Jiménez, and E. SanFabián, Phys. Rev. B 67, 155321 (2003).
* [28] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, et al., Gaussian 03, Revision B.01, Gaussian, Inc., Pittsburgh PA, 2003.
* [29] The molecular structure has been optimized using a DFT-LDA approach with localized atomic-orbital basis sets for Pt (lan2dz) and H (cc-pVTZ). Subsequently, an enhanced molecular bridge Pt-H2-Pt (${}^{1}\Sigma_{g}$) can be defined with a Pt-H (H-H) distance of 3.2 (1.7) bohr. Dipersions from these values due to different selections of exchange-correlation functionals and basis sets are expected not to be relevant for the conductance calculations further performed. These results agree fairly well with previous works[20, 21, 22].
* [30] F. Pauly, J. Heurich, J. C. Cuevas, W. Wenzel, and G. Shon, Nanotechnology 14 (8), R29 (2003).
* [31] J. J. Palacios, A. J. Pérez-Jiménez, E. Louis, E. SanFabián, and J. A. Vergés, Phys. Rev. B 66, 035322 (2002).
* [32] J. Jaramillo, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys 118, 1068 (2003).
* [33] A. W. Ghosh and S. Datta, J. Comp. El. 1, 515 (2002).
* [34] R. L. Martin and F. Illas, Phys. Rev. Lett. 79, 1539 (1997).
* [35] J. Muscat, A. Wander, and N. Harrison, Chem. Phys. Lett. 342, 397 (2001).
* [36] M. Nolan, S. OCallagban, G. Fagas, J. C.Greer, and T. Frauenheim, Nanoletters 7, 34 (2007).
* [37] F. Bickelhaupt and E. Baerends, Rev. Comput. Chem. 15, 1 (2000).
* [38] W.Schattke, M. V. Hove, F. G. de Abajo, R. D. Muino, and N. Mannella, _Solid-State Photoemission and related Methods: Theory and Experiment_ (Wiley-VCH Verlag GmbH and Co.KGaA, 2003).
* [39] A. Nitzan and M. A. Ratner, Science 300, 1384 (2003).
* [40] W. Weimer, W. Hieringer, F. D. Sala, and A. Gorling, Chem. Phys. 309, 77 (2005).
* [41] M. Zhuang, P. Rocheleau, and M. Ernzerhof, J. Chem. Phys. 122, 154705 (2005).
* [42] E. Livshits and R. Baer, Phys. Chem. Chem. Phys. 9, 2932 (2007).
* [43] A. Gorling, J. Chem. Phys. 123, 062203 (2005).
* [44] P. Mori-Sánchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 125, 201102 (2006).
* [45] C. D. Pemmaraju, T. Archer, D. Sánchez-Portal, and S. Sanvito, Phys. Rev. B 75, 045101 (2007).
* [46] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
* [47] J. Heyd, J. E. Peralta, G. E. Scuseria, and R. L. Martin, J. Chem. Phys. 123, 174101 (2005).
* [48] Y. Zhao and D. G. Truhlar, Acc. Chem. Res. 41, 157 (2008).
* [49] R. Valero, R. Costa, I. D. P. R. Moreira, D. G. Truhlar, and F. Illas, J. Chem. Phys. 128, 114103 (2008).
* [50] Y. García, Ph.D. thesis, University of Alicante (2007).
* [51] R. L. Martin and F. Illas, Phys. Rev. Lett. 79, 1539 (1997).
* [52] J. K. Perry, J. Tahir-Kheli, and W. A. Goddard, Phys. Rev. B 63, 144510 (2001).
* [53] R. Grau-Crespo, F. Cora, A. A. Sokol, N. H. de Leeuw, and C. R. A. Catlow, Phys. Rev. B 73, 035116 (2006).
* •
Figure 1. Reduction of the conductance due to the progressive substitution of
the local xc-potential by the nonlocal HF-like contribution. The inset shows
the conductance as function of energy calculated in a nonlocal DFT
approximation for $\alpha=0.6$.
* •
Figure 2. Transmission as a function of energy calculated in a local DFT
approximation for the Pt-H2-Pt bridge shown in the inset.
* •
Figure 3. Enhancement of the HOMO-LUMO gap ($\Delta_{HOMO-LUMO}$) due to the
substitution of local xc potential by the nonlocal HF contribution. The inset
shows the Pt-H2-Pt system bridging the Platinum electrodes.
* •
Figure 4. Ground state energy calculations of the charge gap, $GAP$, for the
system shown in the inset. $GAP=2E_{0}(N)-E_{0}(N+1)-E_{0}(N-1)$, where N
represents the number of electrons, and $E_{0}$ is the total energy for the
selected ground states. Total energy calculations were done using wave-
function based methods: Hartree-Fock (HF), Möller-Plesset perturbation theory
at the second, third and fourth order (MP2, MP3, MP4, respectively) and
Coupled-Cluster theory with single, doubles and single, doubles, and
perturbatively estimated triples [CCSD, CCSD(T), respectively] methods.
* •
Figure 5. Electron density profile after planar average (dashed line) and
three-dimensional average of the microscopic density.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
|
arxiv-papers
| 2008-09-05T13:59:18
|
2024-09-04T02:48:57.653000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Garcia and JC. Sancho-Garcia",
"submitter": "Yamila Garccia-Martinez",
"url": "https://arxiv.org/abs/0809.1031"
}
|
0809.1099
|
# On different ways to quantize Soft-Collinear Effective Theory
Christian W. Bauer Ernest Orlando Lawrence Berkeley National Laboratory,
University of California, Berkeley, CA 94720 Oscar Catà Ernest Orlando
Lawrence Berkeley National Laboratory, University of California, Berkeley, CA
94720 Grigory Ovanesyan Ernest Orlando Lawrence Berkeley National
Laboratory, University of California, Berkeley, CA 94720
###### Abstract
Collinear fields in soft collinear effective theory (SCET) can be made
invariant under collinear gauge transformations by multiplying them with
collinear Wilson lines. We discuss how we can quantize SCET directly in terms
of these gauge invariant fields, allowing to directly calculate $S$ matrix
elements using the gauge invariant collinear fields. We also show how for each
collinear direction SCET can be written in terms of fields whose interactions
are given by the usual QCD Lagrangian, and how external operators coupling
these different directions can be constructed.
Soft collinear effective theory (SCET) Bauer:2000ew ; Bauer:2000yr ;
Bauer:2001ct ; Bauer:2001yt is by now a rather mature effective field theory
with wide applications in $B$ physics and collider physics. SCET describes QCD
in the kinematic regime where the energy of particles is far in excess of
their (invariant) mass. Short distance physics is contained in Wilson
coefficients which are determined order by order in perturbation theory. Long
distance physics on the other hand is described by separate collinear fields
for each light-like direction, together with Wilson lines $Y_{n}$ describing
the usoft physics interactions between the different collinear directions.
Since there are no direct interactions between collinear fields in different
directions, gauge invariance requires the presence of Wilson lines to render
collinear fermions and gauge bosons gauge invariant.
There are several equivalent versions of SCET used in the literature. The
original formulation of SCET Bauer:2000ew ; Bauer:2000yr described the
interactions between the gauge dependent collinear quark $\xi_{n}$ and gluon
$A_{n}$ fields, with the leading order Lagrangian given by111We here omit any
reference to gauge-fixing and potential ghost terms. We will address the
quantization of gauge fields later on.
$\displaystyle{\cal{L}}_{I}^{n}(\xi_{n},A_{n})$ $\displaystyle=$
$\displaystyle{\bar{\xi}}_{n}\left[in\\!\cdot\\!D_{n}+iD\\!\\!\\!\\!/_{n}^{\perp}\frac{1}{i{{\bar{n}}}\\!\cdot\\!D_{n}}iD\\!\\!\\!\\!/_{n}^{\perp}\right]\frac{{\bar{n}}\\!\\!\\!/}{2}\xi_{n}$
(1) $\displaystyle-\frac{1}{2}{\mathrm{Tr}}\,F^{n}_{\mu\nu}F_{n}^{\mu\nu}\,,$
with the standard definition of the covariant derivative and the field
strength tensor
$iD_{n}^{\mu}=i\partial^{\mu}_{n}+g_{s}A_{n}^{\mu}\,,\qquad
F_{n}^{\mu\nu}=\frac{i}{g_{s}}[D_{n}^{\mu},D_{n}^{\nu}]\,,\\\ $ (2)
where the partial derivative $\partial_{n}$ is given in terms of the label
operator introduced in Bauer:2001ct
$i\partial_{n}^{\mu}={\bar{n}}\\!\cdot\\!{\cal P}\frac{n^{\mu}}{2}+{\cal
P}_{\perp}^{\mu}+in\\!\cdot\\!\partial\frac{{\bar{n}}^{\mu}}{2}\,.$ (3)
In order to construct gauge invariant operators containing collinear fermions,
these fermions are required to appear in the gauge invariant combination
$\chi_{n}=W_{n}^{\dagger}\xi_{n}\,,\\\ $ (4)
where $W_{n}$ is the collinear Wilson line Bauer:2001ct
$W_{n}={\rm P}\exp\left[-ig_{s}\\!\int_{0}^{\infty}\\!{\rm
d}s\,\,{\bar{n}}\\!\cdot\\!A_{n}({\bar{n}}s+x)\right]\,.$ (5)
Using a simple field redefinition, one can easily obtain the collinear
Lagrangian in terms of these gauge invariant combinations
$\displaystyle{\cal{L}}_{II}^{n}(\chi_{n},A_{n})$ $\displaystyle=$
$\displaystyle{\bar{\chi}}_{n}W^{\dagger}_{n}\left[in\\!\cdot\\!D_{n}+iD\\!\\!\\!\\!/_{n}^{\perp}\frac{1}{i{{\bar{n}}}\\!\cdot\\!D_{n}}iD\\!\\!\\!\\!/_{n}^{\perp}\right]\frac{{\bar{n}}\\!\\!\\!/}{2}W_{n}\chi_{n}$
(6) $\displaystyle-\frac{1}{2}{\mathrm{Tr}}\,F^{n}_{\mu\nu}F_{n}^{\mu\nu}\,.$
Since the fields $\chi_{n}$ are gauge invariant, the combination
$W_{n}^{\dagger}D_{n}^{\mu}W_{n}$ has to be gauge invariant as well. Thus, we
can define Arnesen
${\cal{D}}_{n}^{\mu}=W_{n}^{\dagger}D_{n}^{\mu}W_{n}\,.$ (7)
The gauge invariant derivative operator ${\cal{D}}_{n}^{\mu}$ can be written
in terms of the partial derivative and a gauge invariant gluon field
${\cal{B}}_{n}^{\mu}$
$i{\cal{D}}_{n}^{\mu}=i\partial_{n}^{\mu}+g_{s}{\cal{B}}_{n}^{\mu}\,,$ (8)
where
${\cal{B}}_{n}^{\mu}=\left[\frac{1}{{\bar{n}}\\!\cdot\\!\partial}\left[i{\bar{n}}\\!\cdot\\!{\cal
D}_{n},i{\cal
D}_{n}^{\mu}\right]\right]=\frac{1}{g_{s}}\left[W_{n}^{\dagger}iD_{n}^{\mu}W_{n}\right]\,,$
(9)
and the derivatives only act within the square brackets. In terms of these
fields, the Lagrangian reads
$\displaystyle{\cal{L}}_{III}^{n}(\chi_{n},{\cal B}_{n})$ $\displaystyle=$
$\displaystyle{\bar{\chi}}_{n}\left[in\\!\cdot\\!{\cal D}_{n}+i{\cal
D}\\!\\!\\!\\!/_{n}^{\perp}\frac{1}{i{\bar{n}\\!\cdot\\!\partial}}i{\cal
D}\\!\\!\\!\\!/_{n}^{\perp}\right]\frac{{\bar{n}}\\!\\!\\!/}{2}\chi_{n}$ (10)
$\displaystyle-\frac{1}{2}{\mathrm{Tr}}\,{\cal{F}}^{n}_{\mu\nu}{\cal{F}}_{n}^{\mu\nu}\,,$
where we have defined
${\cal
F}_{n}^{\mu\nu}=\frac{i}{g_{s}}[{\cal{D}}_{n}^{\mu},{\cal{D}}_{n}^{\nu}]\,.$
(11)
A sample of Feyman rules for the three different formulations of SCET is shown
in Fig. 1.
$\displaystyle V^{(1)}_{{\cal L}_{I}}$ $\displaystyle=$
$=igT^{A}\\!\left[n^{\mu}+\frac{\gamma^{\mu}_{\perp}p\\!\\!\\!/_{\perp}}{\bar{n}\cdot
p}+\frac{p\\!\\!\\!/^{\prime}_{\perp}\gamma^{\mu}_{\perp}}{\bar{n}\cdot
p^{\prime}}-\frac{p\\!\\!\\!/_{\perp}^{\prime}p\\!\\!\\!/_{\perp}}{\bar{n}\cdot
p^{\prime}\bar{n}\cdot p}\bar{n}^{\mu}\right]\\!\frac{{\bar{n}}\\!\\!\\!/}{2}$
$\displaystyle V^{(2)}_{{\cal L}_{I}}$ $\displaystyle=$
$=ig^{2}\\!\left[\frac{T^{A}T^{B}}{\bar{n}\cdot(p-q)}\gamma_{\perp}^{\mu}\gamma_{\perp}^{\nu}+\frac{T^{B}T^{A}}{\bar{n}\cdot(q+p^{\prime})}\gamma_{\perp}^{\nu}\gamma_{\perp}^{\mu}\right]\\!\frac{{\bar{n}}\\!\\!\\!/}{2}+(\ldots)$
$\displaystyle V^{(1)}_{{\cal L}_{II}}$ $\displaystyle=$ $=V^{(1)}_{{\cal
L}_{I}}+igT^{A}\\!\left[\frac{1}{\bar{n}\cdot(p-p^{\prime})}\left(\frac{p^{2}}{\bar{n}\cdot
p}-\frac{p^{\prime 2}}{\bar{n}\cdot
p^{\prime}}\right)\bar{n}^{\mu}\right]\\!\frac{{\bar{n}}\\!\\!\\!/}{2}$
$\displaystyle V^{(2)}_{{\cal L}_{II}}$ $\displaystyle=$ $=V^{(2)}_{{\cal
L}_{I}}+(\ldots)$ $\displaystyle V^{(1)}_{{\cal L}_{III}}$ $\displaystyle=$
$=igT^{A}\\!\left[n^{\mu}+\frac{\gamma^{\mu}_{\perp}p\\!\\!\\!/_{\perp}}{\bar{n}\cdot
p}+\frac{p\\!\\!\\!/^{\prime}_{\perp}\gamma^{\mu}_{\perp}}{\bar{n}\cdot
p^{\prime}}\right]\\!\frac{{\bar{n}}\\!\\!\\!/}{2}$ $\displaystyle
V^{(2)}_{{\cal L}_{III}}$ $\displaystyle=$
$=ig^{2}\\!\left[\frac{T^{A}T^{B}}{\bar{n}\cdot(p-q)}\gamma_{\perp}^{\mu}\gamma_{\perp}^{\nu}+\frac{T^{B}T^{A}}{\bar{n}\cdot(q+p^{\prime})}\gamma_{\perp}^{\nu}\gamma_{\perp}^{\mu}\right]\\!\frac{{\bar{n}}\\!\\!\\!/}{2}$
$\displaystyle\Delta_{{\cal L}_{III}}$ $\displaystyle=$
$=-i\frac{\delta^{AB}}{k^{2}+i\epsilon}\left(g_{\mu\nu}-\frac{{\bar{n}}_{\mu}k_{\nu}+{\bar{n}}_{\nu}k_{\mu}}{{\bar{n}}\cdot
k}\right)$ Figure 1: A subset of Feynman rules for the three different
formulations of SCET. The (…) denote terms which do not contribute to the
tadpole diagram of Fig. 2b) in Feynman gauge.
It is well known that the dynamics of SCET with a single collinear direction
is identical to full QCD. This is of course expected, since one can perform a
simple Lorentz boost along the direction $n$ to make all momentum components
of the collinear field similar in magnitude. Since this eliminates any large
ratio of scales, the interactions have to be those of full QCD. This implies
that for example the wave function renormalization in SCET is equivalent to
that of full QCD, as was first shown in Bauer:2000ew ; Bauer:2000yr . This
equivalence has been used in the literature in order to simplify perturbative
calculations in SCET (vid., for instance, Becher:2006qw ).
It is the purpose of this paper to study the relationship between different
formulations of SCET. We work out the relationship between SCET using gauge
dependent and gauge invariant degrees of freedom, as well as the relationship
between full QCD and collinear fields in a single direction further. One of
the features of the original formulation of SCET is that collinear gluons are
coupled to the quark fields in a non-linear way. This means that there are an
infinite number of vertices consisting of quark-antiquark and an arbitrary
number of collinear gluons, whose Feynman rules get increasingly complicated.
This makes the theory particularly unfriendly for computations beyond the one-
loop order.
We will show how to quantize SCET directly in terms of the gauge invariant
degrees of freedom, and write the theory as a path integral over these gauge
invariant fields. We will also discuss how to re-express the theory using only
the interactions of full QCD. This first gives a precise field theoretical
understanding of the well known property of SCET that the dynamics in a given
collinear direction are equivalent to that of full QCD. Our formulation using
directly the generating functional will extend this result to include
interactions between different collinear directions through local operators.
One can hope that these results will simplify the perturbative calculation of
matching coefficients in the future, since much of the SCET calculations are
now identical to the corresponding QCD results.
It will prove instructive, however, to first illustrate this equivalence
between different formulations of SCET using a simple one-loop calculation.
Consider the two point correlator of two gauge invariant fermion fields
$\langle 0|T\chi_{n}(x)\bar{\chi}_{n}(y)|0\rangle=\langle
0|TW^{\dagger}_{n}(x)\xi_{n}(x)\bar{\xi}_{n}(y)W_{n}(y)|0\rangle\,.$ (12)
The Fourier transform of this correlator is what is known in the literature as
the jet function, and plays a crucial role in any process containing external
collinear particles. In the original formulation of SCET in terms of $\xi_{n}$
and $A_{n}$ fields there are four diagrams contributing at one loop, which are
shown in Fig. 2. The first two diagrams are entirely built out of interactions
contained in the Lagrangian of the theory, while in the last two diagrams one
of the gluon couplings comes from the Wilson lines $W_{n}$ or
$W_{n}^{\dagger}$. Using the Feynman rules given in Fig. 1, one can easily
obtain the result
$\displaystyle
D_{I,a}=g_{s}^{2}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}C_{F}(2-D)\int\frac{d^{D}k}{(2\pi)^{D}}\left[\frac{1}{2}\frac{1}{(k^{2}+i\epsilon)((k+p)^{2}+i\epsilon)}-\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}\frac{1}{(k^{2}+i\epsilon){\bar{n}}\\!\cdot\\!(k+p)}\right]\,,$
(13) $\displaystyle
D_{I,b}=g_{s}^{2}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\left(\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}\right)^{2}C_{F}(2-D)\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{(k^{2}+i\epsilon){\bar{n}}\\!\cdot\\!(k+p)}\,,$
(14) $\displaystyle
D_{I,c}=D_{I,d}=g_{s}^{2}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}C_{F}(n\\!\cdot\\!{\bar{n}})\int\frac{d^{D}k}{(2\pi)^{D}}\frac{{\bar{n}}\\!\cdot\\!(k+p)}{(k^{2}+i\epsilon)((k+p)^{2}+i\epsilon){\bar{n}}\\!\cdot\\!k}\,.$
(15)
Note that the tadpole diagram is canceled exactly against the second term in
the first diagram. Performing the remaining integrals and summing the diagrams
one obtains the well known result Manohar:2003vb
$\displaystyle
D_{I}=i\frac{\alpha_{s}C_{F}}{4\pi}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}\left(\frac{\mu^{2}}{-p^{2}}\right)^{\epsilon}\left[\frac{4}{\epsilon^{2}}+\frac{3}{\epsilon}+7-\frac{\pi^{2}}{3}\right]\,.$
(16)
Figure 2: Diagrams contributing to the gauge invariant jet function at one
loop.
We can repeat this calculation using the formulation of SCET in terms of
$\chi_{n}$ and $A_{n}$ fields. This removes the last two diagrams of Fig. 2,
since there are no Wilson lines in the definition of the correlator when
written in terms of $\chi_{n}$ fields. However, the extra Wilson lines in the
collinear Lagrangian change the Feynman rules in the way shown in Fig. 1.
While this does not change the result for the second diagram, the first
diagram is now
$\displaystyle D_{II,a}$ $\displaystyle=$ $\displaystyle
g_{s}^{2}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}C_{F}\int\frac{d^{D}k}{(2\pi)^{D}}\Bigg{[}(2-D)\left(\frac{1}{2}\frac{1}{(k^{2}+i\epsilon)((k+p)^{2}+i\epsilon)}-\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}\frac{1}{(k^{2}+i\epsilon){\bar{n}}\\!\cdot\\!(k+p)}\right)$
(17)
$\displaystyle+2n\\!\cdot\\!{\bar{n}}\frac{{\bar{n}}\\!\cdot\\!(k+p)}{(k^{2}+i\epsilon)((k+p)^{2}+i\epsilon){\bar{n}}\\!\cdot\\!k}-(n\\!\cdot\\!{\bar{n}})^{2}\frac{{\bar{n}}\\!\cdot\\!p}{p^{2}}\frac{1}{(k^{2}+i\epsilon){\bar{n}}\\!\cdot\\!k}\Bigg{]}\,.$
The first two terms reproduce the result for $D_{I,a}$, and the third
reproduces $D_{I,c}+D_{I,d}$. Finally, the fourth term in $D_{II,a}$ vanishes,
since it is odd as $k\to-k$. Thus, the sum of all diagrams is identical in
both versions of the theory.
Showing that we can reproduce this result using the fully gauge invariant
$\chi_{n}$ and ${\cal B}_{n}$ fields is a little more tricky. This is because
now there are no Wilson lines whatsoever, neither in the definition of the
correlator nor in the Lagrangian of Eq. (10). Thus, it is not immediately
obvious how the contributions from diagrams $D_{c}$ and $D_{d}$ are reproduced
in this case. However, care has to be taken when deriving the gluon
propagator. The ${\cal B}_{n}$ field is by construction explicitly invariant
under collinear gauge transformations, and the usual procedure of adding an
arbitrary gauge-fixing term to the Lagrangian is not valid. However, from the
definition of ${\cal B}_{n}^{\mu}$ in Eq. (9) one easily verifies the
constraint ${\bar{n}}\\!\cdot\\!{\cal B}_{n}=0$ (see Eqs. (29) and (30)
below). Thus, the propagator of a ${\cal B}_{n}$ field has to satisfy the
condition ${\bar{n}}_{\mu}\Delta_{\cal B}^{\mu\nu}=0$. As we will discuss in
more detail later, the propagator takes the form Bassetto:1984dq ;
Veliev:2001gp
$(\Delta_{{\cal
B}})_{\mu\nu}^{ab}(k)=\frac{-i\delta^{ab}}{k^{2}+i\epsilon}\left(g_{\mu\nu}-\frac{{\bar{n}}_{\mu}k_{\nu}+{\bar{n}}_{\nu}k_{\mu}}{{\bar{n}}\\!\cdot\\!k}\right)\,.$
(18)
Using this propagator for the gauge invariant gluon field one can easily
verify that
$D_{III,a}=D_{II,a}\,,\quad D_{III,b}=D_{II,b}\,.$ (19)
Thus, the three formulations of SCET give identical results to one another for
the two point correlator of two gauge invariant collinear fermion fields.
In order to generalize this discussion to any matrix element, we quantize the
theory directly in terms of the various fields. This is achieved by using the
path integral formulation, working directly with the generating functional of
the theory
$Z[J]=\int\\!{\cal{D}}{\bar{\xi}}_{n}{\cal{D}}\xi_{n}{\cal{D}}A_{n}^{\mu}\,{\mathrm{exp}}\left[i\int\\!d^{4}x\,{{\cal{S}}_{I}}(\xi_{n},A_{n}^{\mu},J_{n})\right]\,,$
(20)
where we have defined
$\displaystyle{\cal{S}}_{I}$ $\displaystyle=$
$\displaystyle\sum_{n}\big{[}{\cal{L}}^{n}_{I}+{\bar{J}}_{n}^{\xi}\xi_{n}+{\bar{\xi}}_{n}J_{n}^{\xi}+{\bar{J}}_{n}^{\chi}W^{\dagger}_{n}\xi_{n}+{\bar{\xi}}_{n}W_{n}J_{n}^{\chi}$
(21) $\displaystyle+J_{n\mu}^{A}A_{n}^{\mu}+J_{n\mu}^{\cal B}{\cal
B}_{n}^{\mu}(A_{n})$ $\displaystyle+\sum_{k}{J}_{k}{\cal
O}_{k}\left(W_{n}^{\dagger}\xi_{n},{\cal B}_{n}^{\mu}(A_{n})\right)\,.$
A few comments are in order to understand our notation. First, the integration
in Eq. (20) is over all fields with different directions $n$. Second, the
subscripts $I,II,III$ indicate which version of SCET we are using, with
Lagrangians given in Eqs. (1), (6) and (10) above. Third, we have added
separate currents for the gauge invariant fields
$\chi_{n}=W_{n}^{\dagger}\xi_{n}$ and ${\cal B}_{n}^{\mu}={\cal
B}_{n}^{\mu}(A_{n})=\frac{1}{g_{s}}[W_{n}^{\dagger}iD_{n}^{\mu}W_{n}]$, as
well as for the gauge dependent fields $\xi_{n}$ and $A_{n}$. This allows us
to calculate correlators with gauge invariant fields, such as the jet
function, as well as those with gauge dependent fields, as is often done in
matching calculations to QCD. Finally, we have indicated currents $J_{k}$ for
any local operator in SCET. Such operators are typically written in terms of
the gauge invariant fields, and an example would be the production current for
two collinear fields in opposite directions, ${\cal
O}_{2}=\bar{\chi}_{n}\Gamma\chi_{{\bar{n}}}$.
In order to obtain the generating functional with the Lagrangian written in
terms of $\chi_{n}$ fields, we make the field redefinition given in Eq. (4),
which just amounts to a change in the integration variable in the generating
functional. Since $W_{n}^{\dagger}W_{n}=1$, one can easily show that the
integration measure is the same when written in terms of the $\chi_{n}$ fields
${\cal D}\xi_{n}{\cal D}\bar{\xi}_{n}{\cal D}A_{n}^{\mu}={\cal D}\chi_{n}{\cal
D}\bar{\chi}_{n}{\cal D}A_{n}^{\mu}\,.$ (22)
Thus, the generating functional can be written as
$Z[J]=\int\\!{\cal{D}}{\bar{\chi}}_{n}{\cal{D}}\chi_{n}{\cal{D}}A_{n}^{\mu}\,{\mathrm{exp}}\left[i\\!\int\\!d^{4}x\,{\cal{S}}_{II}(\chi_{n},A_{n}^{\mu},J_{n})\right]\,,$
(23)
with
$\displaystyle{\cal{S}}_{II}$ $\displaystyle=$
$\displaystyle\sum_{n}\big{[}{\cal{L}}^{n}_{II}+{\bar{J}}_{n}^{\xi}W_{n}\chi_{n}+{\bar{\chi}}_{n}W_{n}^{\dagger}J_{n}^{\xi}+{\bar{J}}_{n}^{\chi}\chi_{n}+{\bar{\chi}}_{n}J_{n}^{\chi}$
(24) $\displaystyle+J_{n\mu}^{A}A_{n}^{\mu}+J_{n\mu}^{\cal B}{\cal
B}_{n}^{\mu}(A_{n})\big{]}$ $\displaystyle+\sum_{k}{J}_{k}{\cal
O}_{k}(\chi_{n},{\cal B}_{n}(A_{n}))\,.$
In other words, any matrix element written in terms of $\xi_{n}$ and $A_{n}$
fields is identical to the matrix element written in terms of $\chi_{n}$ and
$A_{n}$ fields, as long as the interactions between the fields are given by
the Lagrangian ${\cal L}_{II}$ instead of ${\cal L}_{I}$.
Next, we discuss the relation between the gauge dependent gluon field $A_{n}$
and the gauge invariant field ${\cal B}_{n}$. The Yang Mills action is given
by
$Z_{\rm YM}=\int\\!{\cal D}A_{n}^{\mu}\,e^{iS_{\rm YM}[A_{n}]}\,,$ (25)
where
$S_{\rm YM}[A]=-\frac{1}{2}\int\\!d^{4}x\,\sum_{n}{\rm
Tr}\,F^{n}_{\mu\nu}F_{n}^{\mu\nu}\,.$ (26)
Recall that the relation between these two fields is given by
${\cal{B}}_{n}^{\mu}=\frac{1}{g_{s}}\left[W_{n}^{\dagger}iD_{n}^{\mu}W_{n}\right]\,,$
(27)
where $\partial_{n}^{\mu}$ acts only within the square brakets. Since the
Wilson lines $W_{n}$ are unitary, the Yang-Mills action can be written in
terms of the ${\cal B}_{n}^{\mu}$ fields as
$S_{\rm YM}[{\cal B}]=-\frac{1}{2}\int\\!d^{4}x\,\sum_{n}{\rm Tr}\,{\cal
F}^{n}_{\mu\nu}{\cal F}_{n}^{\mu\nu}\,,$ (28)
where ${\cal F}_{n}^{\mu\nu}$ is given in Eq. (11). However, in order to write
the generating functional in terms of the fields ${\cal B}_{n}$ requires
changing the integration measure as well, and that is where additional care
has to be taken. From the definition of the ${\cal B}_{n}$ field we can
immediately see that
${\bar{n}}\\!\cdot\\!{\cal B}_{n}=0\,,$ (29)
which follows from the well known relation of Wilson lines
${\bar{n}}\\!\cdot\\!D_{n}\,W_{n}=W_{n}\,{\bar{n}}\\!\cdot\\!\partial_{n}\,.$
(30)
Thus, while there are four components of the $A_{n}^{\mu}$ field, there are
only three components for the ${\cal B}_{n}^{\mu}$ field, making the Jacobian
for the change in the integration measure singular.
Of course, the fact that the ${\cal B}_{n}^{\mu}$ field has less independent
components than the $A_{n}^{\mu}$ field is not unexpected, given that the
former is gauge independent, while the latter contains all the gauge
redundancy. The only way one can obtain a meaningful definition of a Jacobian
factor is by removing the gauge redundancy and thus considering only three of
the four components of the $A_{n}^{\mu}$ field. This can be achieved using the
usual Faddeev-Popov procedure, by inserting a representation of unity into the
path integral (25) in the following form
$1=\int\\!{\cal D}\alpha(x)\,\delta[G(A_{n}^{\alpha})]\,\det\left(\frac{\delta
G(A_{n}^{\alpha})}{\delta\alpha}\right)\,,$ (31)
where $G(A_{n}^{\alpha})$ is some gauge-fixing function linear in the gauge
field. Here $\alpha(x)$ defines a specific gauge transformation and
$A_{n}^{\alpha}$ denotes the (infinitesimally) gauge transformed field
$(A^{\mu}_{n})^{\alpha}=A^{\mu}_{n}+\frac{1}{g_{s}}D^{\mu}_{n}\alpha\,.$ (32)
Note that for infinitesimal gauge transformations (from which all finite
transformations can be constructed), the determinant of $\delta
G/\delta\alpha$ is in general a function of $(A^{\mu}_{n})^{\alpha}$ but
independent of $\alpha$.
Following the standard treatment, the gauge invariance of both the action and
the integration measure allows one to write
$Z_{\rm YM}=\int\\!{\cal D}\alpha\int\\!{\cal
D}A_{n}^{\mu}\,\delta[G(A_{n})]\,E_{G}[A_{n}]\,,$ (33)
where we have defined
$E_{G}[A_{n}]=\det\left(\frac{\delta
G(A_{n}^{\alpha})}{\delta\alpha}\right)\\!\\![A_{n}]\,e^{iS_{\rm
YM}[A_{n}]}\,.$ (34)
One should remember that the determinant ${\rm det}(\delta G/\delta\alpha)$ is
independent of $\alpha$ and therefore the integral over the gauge freedom is
just a global factor that can be safely ignored. The important feature of this
way of writing the path integral is that the integration measure ${\cal
D}A_{n}^{\mu}\,\delta[G(A_{n})]$ contains only three components of the
$A_{n}^{\mu}$ field, and can thus be related to the integration measure of the
${\cal B}_{n}^{\mu}$ field. This allows us to formally write
${\cal D}A_{n}^{\mu}\,\delta[G(A_{n})]=J_{G}[{\cal B}_{n}]\,{\cal D}{\cal
B}_{n}^{\mu}\,\delta[{\bar{n}}\\!\cdot\\!{\cal B}_{n}]\,,$ (35)
where the Jacobian factor for the change of the integration measure
$J_{G}[{\cal B}_{n}]$ depends on the choice of the gauge-fixing condition $G$.
Combining these results together we find
$Z_{\rm YM}=\int\\!{\cal D}{\cal
B}_{n}^{\mu}\,\,\delta[{\bar{n}}\\!\cdot\\!{\cal B}_{n}]\,J_{G}[{\cal
B}_{n}]\,E_{G}[A_{n}({\cal B}_{n})]\,.$ (36)
Everything in this generating functional is known, except for the explicit
form of the Jacobian $J_{G}[{\cal B}_{n}]$ and the determinant inside
$E_{G}[A_{n}({\cal B}_{n})]$. Due to the non-linear nature of Eq. (27), their
expressions for a general gauge-fixing condition $G$ are very difficult to
derive. However, from the Faddeev-Popov procedure it is obvious that the
generating functional is identical for all choices of the gauge-fixing
condition $G$, since it was introduced as an arbitrary function in Eq. (31).
Thus, any choice of $G(A_{n})$ will do, and the easiest choice is light-cone
gauge, which uses
$G(A_{n})\equiv G_{\rm LC}(A_{n})={\bar{n}}\\!\cdot\\!A_{n}\,.$ (37)
In this case, we have ${\bar{n}}\\!\cdot\\!A_{n}=0$, which immediately implies
$W_{n}=1$, making the relation between the $A_{n}^{\mu}$ and ${\cal
B}_{n}^{\mu}$ fields trivial:
${\cal B}_{n}^{\mu}=A_{n}^{\mu}\,.$ (38)
Thus, in this particular gauge we find
$J_{G_{\rm LC}}[{\cal B}_{n}]=1\,,\qquad E_{G_{\rm LC}}[{\cal
B}_{n}]={\mathrm{det}}({\bar{n}}\\!\cdot\\!\partial)\,e^{iS_{\rm YM}[{\cal
B}_{n}]}\,,$ (39)
and we obtain the final form of the generating fuctional in terms of ${\cal
B}^{\mu}_{n}$ fields as
$Z_{\rm YM}=\int\\!{\cal D}{\cal
B}_{n}^{\mu}\,\delta[{\bar{n}}\\!\cdot\\!{\cal
B}_{n}]\,{\mathrm{det}}({\bar{n}}\\!\cdot\\!\partial)\,e^{iS_{\rm YM}[{\cal
B}_{n}]}\,.$ (40)
In other words, the Yang-Mills action in terms of the gauge invariant gluon
field ${\cal B}_{n}^{\mu}$ is identical to the one in terms of the field
$A_{n}^{\mu}$ in the light-cone gauge. Therefore, all Feynman rules for the
${\cal{B}}_{n}^{\mu}$ fields are identical to Feynman rules for the
$A_{n}^{\mu}$ fields in the light-cone gauge. In particular, this justifies
Eq. (18) as the right form of the gluon propagator for the ${\cal
B}_{n}^{\mu}$ fields. Incidentally, notice also that the determinant in Eq.
(40) is independent of the gauge field and therefore can be ignored, meaning
that the formulation with ${\cal B}_{n}^{\mu}$ fields is ghost-free. This
obviously complies with the well known fact that the light-cone gauge is
unitary and ghost fields decouple (see, for instance, tHooft ).
Having worked out how one can quantize SCET directly in terms of the gauge
invariant degrees of freedom, we next ask whether it is possible to write the
generating functional of SCET in terms of fields, whose interactions are given
by the interactions of full QCD. As we will show, this is indeed possible if
we restrict ourselves to leading order in the power counting, but requires
separate fields for each different collinear direction. We will also show how
to construct external operators coupling these different fields to one
another, such that any leading order correlation function in SCET can be
reproduced using only fields whose coupling to other fields is described by
the Lagrangian of full QCD. We do want to emphasize that this by no means
implies that SCET as an effective theory is useless. The power of SCET comes
from understanding the interactions between fields in different directions,
and while we can reproduce any leading order operator using fields that
resemble full QCD, we can neither easily implement power corrections, nor can
we derive the form of the leading order operators without the construction of
SCET. However, we can use this equivalence to calculate matrix elements in
SCET using the familiar Feynman rules of QCD, which will in general simplify
the required calculations at higher orders in perturbation theory.
We start by making the Ansatz
$Z[J]=\int\\!{\cal{D}}{\bar{\psi_{n}}}{\cal{D}}\psi_{n}{\cal{D}}A_{n}^{\mu}{\mathrm{exp}}\left[i\int\\!d^{4}x\,S_{\rm
QCD}(\psi_{n},A_{n},J)\right],$ (41)
where $S_{\rm QCD}$ is defined by
$\displaystyle S_{\rm QCD}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sum_{n}\big{[}{\cal{L}}^{\rm
QCD}_{n}+{\bar{J}}^{\xi}_{n}{\cal{M}}^{\xi}_{n}\psi_{n}+{\bar{\psi}}_{n}{\bar{{\cal{M}}}}^{\xi}_{n}J^{\xi}_{n}+{\bar{J}}^{\chi}_{n}{\cal{M}}^{\chi}_{n}\psi_{n}$
(42)
$\displaystyle+{\bar{\psi}}_{n}{\bar{{\cal{M}}}}^{\chi}_{n}J^{\chi}_{n}+J_{n\mu}^{A}A_{n}^{\mu}+J_{n\mu}^{\cal
B}{\cal B}_{n}^{\mu}(A_{n})\big{]}$
$\displaystyle+\sum_{k}J_{k}{\cal{Q}}_{k}(\psi_{n},A_{n})\,,$
with
${\cal L}_{n}^{\rm QCD}=\bar{\psi}_{n}\,iD\\!\\!\\!\\!/\,\psi_{n}\,.$ (43)
The set of operators ${\cal Q}_{k}$ couple $k$ fields in different directions
$n_{1},...,n_{k}$. Our goal is to find expressions for ${\cal M}_{n}$ and
${\cal Q}_{k}$, such that the generating functional in Eq. (41) is equivalent
to the generating functional of SCET.
Let’s begin by setting all currents in the action to zero, leaving only the
Lagrangian ${\cal L}_{n}^{\rm QCD}$. One can write
$\psi_{n}(x)=\left(P_{n}+P_{\bar{n}}\right)\psi_{n}(x)\,,$ (44)
with the projection operators $P_{n}$ and $P_{\bar{n}}$ defined by
$P_{n}=\frac{n\\!\\!\\!/{\bar{n}}\\!\\!\\!/}{4}\,,\qquad
P_{\bar{n}}=\frac{{\bar{n}}\\!\\!\\!/n\\!\\!\\!/}{4}\,,$ (45)
and define
$\xi_{n}\equiv P_{n}\psi_{n}\,,\qquad\phi_{n}\equiv P_{\bar{n}}\psi_{n}\,.$
(46)
This allows us to write
$\displaystyle Z[J=0]$ $\displaystyle=$
$\displaystyle\int{\cal{D}}{\bar{\xi}_{n}}{\cal{D}}\xi_{n}{\cal{D}}{\bar{\phi}_{n}}{\cal{D}}\phi_{n}{\cal{D}}A_{n}^{\mu}$
$\displaystyle\times{\mathrm{exp}}\left[\sum_{n}i\\!\int\\!d^{4}x\,(\bar{\xi}_{n}+\bar{\phi}_{n})iD\\!\\!\\!\\!/\,(\xi_{n}+\phi_{n})\right]\,.$
Using the well-known formula for Gaussian integration,
$\displaystyle\int\\!{\cal{D}}\phi{\cal{D}}{\bar{\phi}}\,\,{\mathrm{exp}}\left[i\int\\!d^{4}x({\bar{\phi}}M\phi+{\bar{J}}\phi+{\bar{\phi}}J)\right]$
$\displaystyle\qquad={\mathrm{det}}(-iM)\,{\mathrm{exp}}\left[-i\int\\!d^{4}x{\bar{J}}\frac{1}{M}J\right]\,,$
(48)
it is straightforward to perform the integrals over $\phi_{n}$ explicitly. We
find
$Z[J=0]=\int\\!{\cal{D}}{\bar{\xi}_{n}}{\cal{D}}\xi_{n}{\cal{D}}A_{n}^{\mu}{\mathrm{exp}}\left[i\\!\int\\!d^{4}x\,\sum_{n}{\cal
L}_{n}^{\rm SCET}\right]\,,$ (49)
where
${\cal L}_{n}^{\rm
SCET}={\bar{\xi}}_{n}\left[in\\!\cdot\\!D+iD\\!\\!\\!\\!/_{\perp}\frac{1}{i{\bar{n}}\\!\cdot\\!D}iD\\!\\!\\!\\!/_{\perp}\right]\frac{{\bar{n}}\\!\\!\\!/}{2}\xi_{n}\,.$
(50)
Note that in getting to Eqs. (49) and (50) no expansion has been made, only
integration of modes in the generating functional. Also note that in Eq.(49)
we have omitted the determinant factor in Eq. (48). Indeed it is easy to show
that
$\displaystyle\det\left(\frac{n\\!\\!\\!/}{2}\bar{n}\\!\cdot\\!D\right)=\int\\!{\cal{D}}\eta_{n}{\cal{D}}{\bar{\eta}_{n}}\,\,{\mathrm{exp}}\left[-\\!\int\\!d^{4}x\,{\bar{\eta}_{n}}\left(\frac{n\\!\\!\\!/}{2}\bar{n}\\!\cdot\\!D\right)\eta_{n}\right]$
$\displaystyle=\int\\!{\cal{D}}\eta_{n}^{\prime}{\cal{D}}{\bar{\eta}}_{n}^{\prime}\,\,{\mathrm{exp}}\left[-\\!\int\\!d^{4}x\,{\bar{\eta}}_{n}^{\prime}\left(\frac{n\\!\\!\\!/}{2}W^{\dagger}_{n}\bar{n}\\!\cdot\\!DW_{n}\right)\eta_{n}^{\prime}\right]$
$\displaystyle=\det\left(\frac{n\\!\\!\\!/}{2}\bar{n}\\!\cdot\\!\partial\right),$
(51)
where we have defined $\eta_{n}^{\prime}=W_{n}^{\dagger}\eta_{n}$. Thus the
determinant is just an overall constant and can be ignored.
We can now move on and consider the addition of current terms in the action.
Keeping the currents ${\bar{J}}_{n}{\cal M}_{n}$ and ${\bar{\cal M}}_{n}J_{n}$
for the fields $\psi_{n}$ and $\bar{\psi}_{n}$, but still neglecting the
currents $J_{k}$ for the local operators ${\cal Q}_{k}$, and again performing
the integrals over $\phi_{n}$ and $\bar{\phi}_{n}$ gives
$\displaystyle
Z[J_{k}=0]=\int\\!{\cal{D}}{\bar{\xi}_{n}}{\cal{D}}\xi_{n}{\cal{D}}A_{n}^{\mu}\,{\mathrm{exp}}\left[i\\!\int\\!d^{4}x\,S^{\rm
SCET}(J_{k}=0)\right]\,,$
with
$\displaystyle S^{\rm SCET}(J_{k}=0)$ $\displaystyle=$
$\displaystyle\sum_{n}{\cal
L}_{I}^{n}+{\bar{J}}^{\xi}_{n}{\cal{M}}^{\xi}_{n}{\cal{R}}_{n}\xi_{n}+{\bar{\xi}}_{n}{\bar{\cal{R}}}_{n}{\bar{\cal{M}}}^{\xi}_{n}J^{\xi}_{n}+{\bar{J}}^{\chi}_{n}{\cal{M}}^{\chi}_{n}{\cal{R}}_{n}\xi_{n}+{\bar{\xi}}_{n}{\bar{\cal{R}}}_{n}{\bar{\cal{M}}}^{\chi}_{n}J^{\chi}_{n}+J_{n\mu}^{A}A_{n}^{\mu}+J_{n\mu}^{\cal
B}{\cal B}_{n}^{\mu}(A_{n})$ (53) $\displaystyle-\left(\bar{J}_{n}^{\xi}{\cal
M}_{n}^{\xi}+\bar{J}_{n}^{\chi}{\cal
M}_{n}^{\chi}\right)\frac{1}{i{\bar{n}}\\!\cdot\\!D}\frac{{\bar{n}}\\!\\!\\!/}{2}\left(\bar{\cal
M}_{n}^{\xi}J_{n}^{\xi}+\bar{\cal M}_{n}^{\chi}J_{n}^{\chi}\right)\,.$
Here we have defined
${\cal{R}}_{n}=\left[1+\frac{1}{i{\bar{n}}\\!\cdot\\!D}iD\\!\\!\\!\\!/_{\perp}\frac{{\bar{n}}\\!\\!\\!/}{2}\right]\,.$
(54)
In order for this action to be equal to the action of SCET given in Eq. (21)
(still with $J_{k}=0$), requires
$\displaystyle{\cal{M}}^{\xi}_{n}{\cal{R}}_{n}\xi_{n}\equiv\xi_{n}\,,\qquad{\cal{M}}^{\chi}_{n}{\cal{R}}_{n}\chi_{n}\equiv
W_{n}^{\dagger}\xi_{n}\,,$ (55)
in addition to having the second line in Eq. (53), corresponding to contact
terms arising when taking two derivatives of the generating functional with
respect to the currents $J_{n}^{\xi/\chi}$, vanish. There are two possible
solutions for each of the ${\cal{M}}^{\xi}_{n}$ and ${\cal{M}}^{\chi}_{n}$ to
satisfy Eq. (55), namely
$\displaystyle{\cal{M}}^{\xi}_{n}$ $\displaystyle={\cal{R}}_{n}^{-1}$
$\displaystyle{\rm or}\qquad{\cal{M}}^{\xi}_{n}$ $\displaystyle=P_{n}\,,$
$\displaystyle{\cal{M}}^{\chi}_{n}$
$\displaystyle=W_{n}^{\dagger}{\cal{R}}_{n}^{-1}$ $\displaystyle{\rm
or}\qquad{\cal{M}}^{\chi}_{n}$ $\displaystyle=W_{n}^{\dagger}P_{n}\,.$ (56)
While both of these solutions for ${\cal M}_{n}$ give the same answer, the
second choice is in practice much easier to use. This is because choosing
${\cal{M}}^{\xi}_{n}={\cal R}_{n}^{-1}$ in Eq. (42) adds couplings between
fermions and gluons to the current terms, complicating perturbative
calculations significantly. Furthermore, for the second solution the second
line in Eq. (53) vanishes as desired. Therefore, for
${\cal{M}}^{\xi}_{n}=P_{n}$ and ${\cal{M}}^{\chi}_{n}=W^{\dagger}_{n}P_{n}$ we
obtain for $J_{k}=0$ the desired result $S^{\rm SCET}=S_{I}$, where $S_{I}$ is
defined in Eq. (21).
Finally, we add the currents for the local operators ${\cal Q}_{k}$ back to
the action. Since these operators couple fields with different $n$’s to one
another, integrating out the $\phi_{n}$ fields is very complicated. However,
there is a simple choice for the operators ${\cal Q}_{k}$ that will directly
reproduce the form $\sum_{k}J_{k}{\cal O}_{k}$ present in the final answer,
Eq. (21). This is achieved by taking
${\cal Q}_{k}(\psi_{n},A_{n})={\cal O}_{k}(W_{n}^{\dagger}P_{n}\psi_{n},{\cal
B}_{n}(A_{n}))\,,$ (57)
with $P_{n}$ defined in Eq. (45). Since $P_{n}\psi_{n}=\xi_{n}$, this choice
eliminates any dependence on $\phi_{n}$ in ${\cal Q}_{k}$. Thus, the integrals
over $\phi_{n}$ can be performed as before and we therefore find
${\cal Q}_{k}(\psi_{n},A_{n})={\cal O}_{k}(W_{n}^{\dagger}\xi_{n},{\cal
B}_{n}(A_{n}))\,.$ (58)
In conclusion, the generating functional in terms of QCD fields
$Z[J]=\int{\cal{D}}{\bar{\psi_{n}}}{\cal{D}}\psi_{n}{\cal{D}}A_{n}^{\mu}{\mathrm{exp}}\left[i\int\\!d^{4}x\,S_{\rm
QCD}(\psi_{n},A_{n},J)\right]\,,$ (59)
with $S_{\rm QCD}$ defined by
$\displaystyle S_{\rm QCD}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sum_{n}\big{[}{\cal{L}}^{\rm
QCD}_{n}+{\bar{J}}^{\xi}_{n}P_{n}\psi_{n}+{\bar{\psi}}_{n}P_{\bar{n}}J^{\xi}_{n}+{\bar{J}}^{\chi}_{n}W_{n}^{\dagger}P_{n}\psi_{n}$
(60)
$\displaystyle+{\bar{\psi}}_{n}P_{\bar{n}}W_{n}J^{\chi}_{n}+J_{n\mu}^{A}A_{n}^{\mu}+J_{n\mu}^{\cal
B}{\cal B}_{n}^{\mu}(A_{n})\big{]}$
$\displaystyle+\sum_{k}J_{k}{\cal{O}}_{k}(W^{\dagger}_{n}P_{n}\psi_{n},{\cal
B}_{n}(A_{n}))\,,$
is identical to the generating functional defined in Eqs. (20) and (21) in
terms of SCET fields. This proves that the collinear sector of SCET is
equivalent to a theory containing multiple copies of QCD, where the only
interactions between them are contained in the local operators ${\cal O}_{k}$.
So far we have only considered the collinear sector of SCET, but of course it
is well known that usoft degrees of freedom are required in order to reproduce
the long distance dynamics of QCD. On the other hand, it is also well known
that at leading order in the effective theory the interactions between usoft
and collinear particles can be removed to all orders in perturbation theory by
using the field redefinition Bauer:2001yt
$\xi_{n}\to Y_{n}\xi_{n}\,,$ (61)
where
$Y_{n}={\rm P}\exp\left[ig\\!\int_{0}^{\infty}\\!{\rm
d}s\,\,n\\!\cdot\\!A(ns+x)\right]\,.$ (62)
Thus, we can include the interactions with the usoft gluons by making a
similar field redefinition on the fields $\psi_{n}$. This implies that the
action given in Eq. (42), but now with
${\cal L}_{n}^{\rm
QCD}=\bar{\psi}_{n}\,\left(iD\\!\\!\\!\\!/+gn\\!\cdot\\!A_{us}\frac{{\bar{n}}\\!\\!\\!/}{2}\right)\,\psi_{n}\,$
(63)
reproduces both the collinear and usoft interactions of the collinear fields.
In conclusion, we have shown how SCET can be quantized either in terms of
gauge dependent or gauge invariant fields. In practice, most calculations in
the literature are performed using the gauge dependent degrees of freedom,
whereas the external operators have to depend on the gauge invariant fields.
Using our results, one can perform the calculations directly in terms of the
gauge invariant fields, reducing the number of Feynman diagrams significantly.
We have then moved on to show how the collinear sector of SCET is equivalent
to a theory constructed out of multiple decoupled copies of full QCD, in the
sense that each copy describes the interactions of fields in a given direction
and the different copies do not interact with one another. We have also shown
in detail how to construct the local operators describing precisely the
interactions between the different copies of QCD, such that any SCET
correlator at leading order can be reproduced.
###### Acknowledgements.
We would like to thank Iain Stewart for helpful discussions. This work was
supported in part by the Director, Office of Science, Office of High Energy
Physics of the U.S. Department of Energy under the Contract DE-AC02-05CH11231.
CWB acknowledges support from an LDRD grant from LBNL. OC would like to thank
the Fulbright Program and the Spanish Ministry of Education and Science for
financial support under grant number FU2005-0791 and also LBNL and UCB-BCTP
for their hospitality this past year.
## Appendix: Diagrammatic proof of the equivalence of QCD and SCET with one
collinear direction
In the main body of this paper we have shown that any collinear SCET diagram
can be obtained using a gener- ating functional in which the interactions
between the fields are equivalent to full QCD, but the external cur- rents are
modified to contain projection operators. This relation was first discussed in
Bauer:2000yr and used in Becher:2006qw to calculate jet functions in SCET.
In this appendix we want to prove this identity diagrammatically for the
correlator containing two collinear fermions and $N$ collinear gluons.
We will accomplish this by working out in both theories the Feynman diagrams
for $N$ gluons coupled to a fermion line, from which the correlator can be
constructed. Using this result we will then show that both of these
calculations lead to equivalent answers. Note that there are $N!$ possible
color structures, and for each of them the QCD result has to equal the SCET
result. We begin by showing this equivalence for the color structure
$T^{a_{1}}\,T^{a_{2}}\,\ldots\,T^{a_{N}}$, and then discuss how the result can
be modified to include the other color structures as well.
Define $Q^{(N)}$ and $S^{(N)}$ to be the QCD and SCET cor- relators for this
color structure in momentum space, multiplied by a factor of $p_{i}^{2}$ for
each internal propagator and with the factor $g_{s}^{N}$ removed. This gives
$\displaystyle
Q^{(N)}=P_{n}p\mspace{-10.0mu}/\mspace{3.0mu}_{0}\gamma^{\mu_{1}}p\mspace{-10.0mu}/\mspace{3.0mu}_{1}\dots\gamma^{\mu_{N}}p\mspace{-10.0mu}/\mspace{3.0mu}_{N}P_{\bar{n}}\,,$
(64) $\displaystyle S^{(N)}=\sum_{k=1}^{N}S^{(N-k)}L_{k}\,.$ (65)
The first equation follows simply from the QCD Feynman rules, while the SCET
equation is a recurrence formula, that takes into account all the
possibilities of having $k$ out of the $N$ gluons being emitted from a single
vertex. $L_{k}$ is therefore the Feynman rule for $k$-gluon emissions from a
single vertex, multiplied by a factor of $\prod_{i}p_{i}^{2}/g_{s}$ to account
for the removal of the factors $p_{i}^{2}$ and $g_{s}$, as discussed above:
$L_{k}\frac{{\bar{n}}\\!\\!\\!/}{2}=i(\bar{n}\\!\cdot\\!p_{N})\frac{p_{N-k+1}^{2}\cdots
p_{N-1}^{2}}{(-g)^{k}}V_{k}\,,$ (66)
with
$\displaystyle
V_{k}=\frac{i(-g)^{k}\bar{n}^{\mu_{N-k+2}}\dots\bar{n}^{\mu_{N-1}}}{\bar{n}\\!\cdot\\!p_{N-k+1}\dots\bar{n}\\!\cdot\\!p_{N-1}}\frac{\bar{n}\mspace{-10.0mu}/\mspace{3.0mu}}{2}\times$
(67)
$\displaystyle\times\Big{(}\gamma^{\mu_{N-k+1}}_{\perp}\gamma^{\mu_{N}}_{\perp}-\bar{n}^{\mu_{N}}\gamma^{\mu_{N-k+1}}_{\perp}\frac{p\mspace{-10.0mu}/\mspace{3.0mu}_{N}^{\perp}}{\bar{n}\\!\cdot\\!p_{N}}+$
$\displaystyle+\frac{p\mspace{-10.0mu}/\mspace{3.0mu}_{N-k}^{\perp}p\mspace{-10.0mu}/\mspace{3.0mu}_{N}^{\perp}}{\bar{n}\\!\cdot\\!p_{N-k}\bar{n}\\!\cdot\\!p_{N}}\bar{n}^{\mu_{N-k+1}}\bar{n}^{\mu_{N}}-\bar{n}^{\mu_{N-k+1}}\frac{p\mspace{-10.0mu}/\mspace{3.0mu}_{N-k}^{\perp}\gamma^{\mu_{N}}_{\perp}}{\bar{n}\\!\cdot\\!p_{N-k}}\Big{)}\,.$
We will show the equivalence $Q^{(N)}=S^{(N)}$ by induction. For $N=0$ it is
straightforward:
$\displaystyle Q^{(0)}$
$\displaystyle=P_{n}p\mspace{-10.0mu}/\mspace{3.0mu}_{0}P_{\bar{n}}=\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\bar{n}\\!\cdot\\!p_{0},$
(68) $\displaystyle S^{(0)}$
$\displaystyle=\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\bar{n}p_{0}=Q^{(0)}\,.$
(69)
Next, we assume that the statement $Q=S$ holds for $0,1,\dots N-1$ to show
that this leads to $Q^{(N)}=S^{(N)}$. This implies
$Q^{(N)}=\sum_{k=1}^{N}Q^{(N-k)}L_{k}\,.$ (70)
To prove Eq. (70) we rewrite the general QCD correlator $Q^{(N)}$ by pushing
the projection operator $P_{n}$ in Eq. (64) through the
$p\mspace{-10.0mu}/\mspace{3.0mu}_{n}$ and $\gamma^{\mu_{n}}$, to obtain
$Q^{(N)}=\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\sum_{m=0}^{N}\sum_{l=1}^{C_{2N+1}^{2m}}\chi^{2m,2N+1}_{i_{1}\dots
i_{2m}}\,,$ (71)
where
$\displaystyle\chi^{2m,2N+1}_{i_{1}\dots
i_{2m}}={(-1)}^{i_{1}+\dots+i_{2m}-(1+\dots+2m)}\perp_{i_{1}}\dots\perp_{i_{2m}}$
$\displaystyle\quad\quad\quad\times(\bar{n}_{j_{1}}n_{j_{2}}\bar{n}_{j_{3}}\dots
n_{j_{2N-2m}}\bar{n}_{j_{2N+1-2m}})\,.$ (72)
Here $C_{k}^{l}$ denotes the binomial coefficient for $l$ choose $k$, and we
have used a shorthand notation in which $\bar{n}_{j}$ corresponds to
$\bar{n}\\!\cdot\\!p$ for even $j$ and to $\bar{n}^{\mu}$ for odd $j$ and
accordingly $\perp_{j}$ corresponds to
$p\mspace{-10.0mu}/\mspace{3.0mu}_{\perp}$ for even $j$, while
$\gamma_{\perp}^{\mu}$ for odd $j$.
We would like to comment on how we obtained this result. Expanding each
$\gamma$ matrix on the right hand side of the Eq. (64) according to
$\gamma^{\alpha}=\bar{n}^{\alpha}\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}+n^{\alpha}\frac{\bar{n}\mspace{-10.0mu}/\mspace{3.0mu}}{2}+\gamma^{\alpha}_{\perp}$
will result in terms with fixed number $0\leq N_{\perp}\leq 2N+1$ of
$\gamma_{\perp}$’s, together with $(2N+1-N_{\perp})$ of
$n\mspace{-10.0mu}/\mspace{3.0mu}$ or
${\bar{n}}\mspace{-10.0mu}/\mspace{3.0mu}$. Since the
$n\mspace{-10.0mu}/\mspace{3.0mu}$ and
${\bar{n}}\mspace{-10.0mu}/\mspace{3.0mu}$ terms have to alternate, and the
projection operator forces the first and last term to be
$n\mspace{-10.0mu}/\mspace{3.0mu}$, $N_{\perp}$ has to be an even number.
As a next step, we work out the sum on the right hand side of Eq. (70). Note
that the term $L_{k}$ contains factors of $p_{i}^{2}$ in the numerators, while
there are no such terms on the left hand side of Eq. (70). However, both
$Q^{(N-k)}$ and $L_{k}$ contain terms with
$p\mspace{-10.0mu}/\mspace{3.0mu}_{\perp}$, which can lead to
$p_{\perp}^{2}=p^{2}-n\\!\cdot\\!p\,{\bar{n}}\\!\cdot\\!p$. After a
straightforward, but lengthy calculation, one can show that
$\sum_{k=1}^{N}Q^{(N-k)}L_{k}=\frac{n\mspace{-10.0mu}/\mspace{3.0mu}}{2}\sum_{m=0}^{N}\sum_{l=1}^{C_{2N+1}^{2m}}\chi^{2m,2N+1}_{i_{1}\dots
i_{2m}}\,.$ (73)
Thus, both sides of Eq. (70) are equal and we have thus shown that
$Q^{(N)}=S^{(N)}$ for all values of $N$.
So far we have only dealt with the term with color structure $T^{a_{1}}\dots
T^{a_{N}}$. Keeping the general color structure allows us to write
$\displaystyle Q^{(N)}\rightarrow\sum_{l=1}^{N!}Q^{(N)}_{i_{1}\dots
i_{N}}T^{a_{i_{1}}}\dots T^{a_{i_{N}}}\,,$ (74) $\displaystyle
S^{(N)}\rightarrow\sum_{l=1}^{N!}S^{(N)}_{i_{1}\dots i_{N}}T^{a_{i_{1}}}\dots
T^{a_{i_{N}}}\,.$ (75)
What we have shown so far is that
$Q^{(N)}_{1,2,\dots,N}=S^{(N)}_{1,2,\dots,N}$. However, it is clear that the
proof goes through for any color permutation, with obvious replacements to
account for the different orderings of the gluons. Finally, notice that triple
or quartic gluon vertices do not change the result, since they are the same in
QCD and SCET. This completes the proof.
## References
* (1) C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D 63, 014006 (2000) [arXiv:hep-ph/0005275].
* (2) C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [arXiv:hep-ph/0011336].
* (3) C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134 (2001) [arXiv:hep-ph/0107001].
* (4) C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 65, 054022 (2002) [arXiv:hep-ph/0109045].
* (5) C. M. Arnesen, J. Kundu and I. W. Stewart, Phys. Rev. D 72, 114002 (2005) [arXiv:hep-ph/0508214].
* (6) T. Becher and M. Neubert, Phys. Lett. B 637, 251 (2006) [arXiv:hep-ph/0603140].
* (7) A. V. Manohar, Phys. Rev. D 68, 114019 (2003) [arXiv:hep-ph/0309176].
* (8) A. Bassetto, M. Dalbosco, I. Lazzizzera and R. Soldati, Phys. Rev. D 31, 2012 (1985).
* (9) E. V. Veliev, Phys. Lett. B 498, 199 (2001).
* (10) G. ’t Hooft, Nucl. Phys. B 75, 461 (1974), L. H. Ryder, Cambridge, Uk: Univ. Pr. ( 1985) 443p.
|
arxiv-papers
| 2008-09-07T14:33:27
|
2024-09-04T02:48:57.658799
|
{
"license": "Public Domain",
"authors": "Christian W. Bauer, Oscar Cata and Grigory Ovanesyan",
"submitter": "Christian Bauer",
"url": "https://arxiv.org/abs/0809.1099"
}
|
0809.1130
|
# Charged rotating dilaton black strings in AdS spaces
Ahmad Sheykhi 111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
We derive a class of charged rotating dilaton black string solutions in the
background of anti-de-Sitter spaces with an appropriate combination of three
Liouville-type dilaton potentials. We also present the suitable counterterm
which removes the divergences of the action in the presence of dilaton
potential. The solutions are analyzed and their thermodynamics is discussed by
using the counterterm method.
There has been considerable attention in the past years in dilaton gravity. It
is of great importance to investigate the effect of the dilaton field on the
properties of the solutions. It was found that the dilaton field changes the
causal structure of the black hole and leads to the curvature singularities at
finite radii. In the absence of dilaton potential, exact solutions of charged
dilaton black holes have been constructed by many authors CDB1 ; CDB2 . These
black holes are all asymptotically flat. The presence of Liouville-type
dilaton potential, which is regarded as the generalization of the cosmological
constant, changes the asymptotic behavior of the solutions to be neither
asymtotically flat nor (anti)-de Sitter [(A)dS]. Indeed, it has been shown
that with the exception of a pure cosmological constant, no dilaton de-Sitter
or anti-de-Sitter black hole solution exists with the presence of only one
Liouville-type dilaton potential MW . In the presence of one or two Liouville-
type potential, black hole spacetimes which are neither asymptotically flat
nor (A)dS have been explored by many authors (see e.g. CHM ; Cai ; Clem ;
Shey0 ). Although these kind of solutions may shed some light on the possible
extensions of AdS/CFT correspondence, they are physically less interesting due
to their unusual asymptotic behavior.
On the other side, the construction and analysis of black hole solutions in
(A)dS space is a subject of much recent interest. This is primarily due to
their relevance for the AdS/CFT correspondence. It was argued that the
thermodynamics of black holes in AdS spaces can be identified with that of a
certain dual CFT in the high temperature limit Witt . Having the AdS/CFT
correspondence idea at hand, one can gain some insights into thermodynamic
properties and phase structures of strong ’t Hooft coupling CFTs by studying
thermodynamics of AdS black holes. Recently, the dilaton potential leading to
(anti)-de Sitter-like solutions of dilaton gravity has been found Gao1 (see
also Gao2 ). It was shown that the cosmological constant is coupled to the
dilaton in a very nontrivial way. With the combination of three Liouville-type
dilaton potentials, a class of static dilaton black hole solutions in (A)dS
spaces has been obtained by using a coordinates transformation which recast
the solution in the schwarzschild coordinates system Gao1 . The purpose of the
present Letter is to construct a class of charged rotating dilaton black
string solutions in the background of AdS spacetime. We will also present the
suitable counterterm which removes the divergences of the action. Finally, we
analyze the solutions and calculate their conserved and thermodynamic
quantities by using the counterterm method inspired by AdS/CFT correspondence.
Our starting point is the four-dimensional Einstein-Maxwell-dilaton action
$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(R\text{
}-2\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi)-e^{-2\alpha\Phi}F_{\mu\nu}F^{\mu\nu}\right)$
(1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\Theta(\gamma),$
where ${R}$ is the Ricci scalar curvature, $\Phi$ is the dilaton field, and
$V(\Phi)$ is a potential for $\Phi$. $\alpha$ is a constant determining the
strength of coupling of the scalar and electromagnetic field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and $A_{\mu}$ is the electromagnetic potential.
The last term in Eq. (1) is the Gibbons-Hawking surface term. It is required
for the variational principle to be well-defined. The factor $\Theta$
represents the trace of the extrinsic curvature for the boundary
${\partial\mathcal{M}}$ and $\gamma$ is the induced metric on the boundary.
While $\alpha=0$ corresponds to the usual Einstein-Maxwell-scalar theory,
$\alpha=1$ indicates the dilaton-electromagnetic coupling that appears in the
low energy string action in Einstein’s frame. In this Letter, we examine
action (1) with three Liouville-type dilaton potentials Gao1
$V(\Phi)=\frac{2\Lambda}{3(\alpha^{2}+1)^{2}}\left[{\alpha}^{2}\left(3\,{\alpha}^{2}-1\right){e^{-2\Phi/\alpha}}+\left(3-{\alpha}^{2}\right){e^{2\,\alpha\,\Phi}}+8\,{\alpha}^{2}{e^{\Phi(\alpha-1/\alpha)}}\right],$
(2)
where $\Lambda$ is the cosmological constant. It is clear the cosmological
constant is coupled to the dilaton in a very nontrivial way. This type of
dilaton potential can be obtained when a higher dimensional theory is
compactified to four dimensions, including various supergravity models Gid .
The equations of motion can be obtained by varying the action (1) with respect
to the gravitational field $g_{\mu\nu}$, the dilaton field $\Phi$, and the
gauge field $A_{\mu}$ which yields the following field equations
${R}_{\mu\nu}=2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)+2e^{-2\alpha\Phi}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{4}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right),$ (3)
$\nabla^{2}\Phi=\frac{1}{4}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{-2\alpha\Phi}F_{\lambda\eta}F^{\lambda\eta},$
(4) $\partial_{\mu}\left(\sqrt{-g}e^{-2\alpha\Phi}F^{\mu\nu}\right)=0.$ (5)
Our aim here is to construct charged rotating black string solutions of the
field equations (3)-(5) and investigate their properties. The metric of four-
dimensional rotating solution with cylindrical or toroidal horizons can be
written as Lem
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-f(r)\left(\Xi dt-
ad\phi\right)^{2}+r^{2}R^{2}(r)\left(\frac{a}{l^{2}}dt-\Xi
d\phi\right)^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{l^{2}}R^{2}(r)dz^{2},$
$\displaystyle\Xi^{2}$ $\displaystyle=$ $\displaystyle 1+\frac{a^{2}}{l^{2}},$
(6)
where $a$ is the rotation parameter. The functions $f(r)$ and $R(r)$ should be
determined and $l$ has the dimension of length which is related to the
constant $\Lambda$ by the relation $l^{2}=-3/\Lambda$. The two dimensional
space, $t$=constant and $r$ =constant, can be (i) the flat torus model $T^{2}$
with topology $S^{1}\times S^{1}$, and $0\leq\phi<2\pi$, $0\leq z<2\pi l$,
(ii) the standard cylindrical model with topology $R\times S^{1}$, and
$0\leq\phi<2\pi$, $-\infty<z<\infty$, and (iii) the infinite plane $R^{2}$
with $-\infty<\phi<\infty$ and $-\infty<z<\infty$. We will focus upon (i) and
(ii). The Maxwell equation (5) can be integrated immediately to give
$\displaystyle F_{tr}$ $\displaystyle=$ $\displaystyle\frac{q\Xi
e^{2\alpha\Phi}}{r^{2}R^{2}(r)},$ $\displaystyle F_{\phi r}$ $\displaystyle=$
$\displaystyle-\frac{a}{\Xi}F_{tr},$ (7)
where $q$, an integration constant, is related to the electric charge of black
string. Inserting the Maxwell fields (Charged rotating dilaton black strings
in AdS spaces) and the metric (6) in the field equations (3) and (4), we can
write these equation for $a=0$ as
$\displaystyle
2r^{4}R^{3}R^{\prime}f^{\prime}+r^{4}R^{4}f^{\prime\prime}+2r^{3}R^{4}f^{\prime}+r^{4}R^{4}V(\Phi)-2q^{2}e^{2\alpha\Phi}=0,$
(8) $\displaystyle
2r^{3}R^{4}f^{\prime}+r^{4}R^{4}f^{\prime\prime}+8r^{3}R^{3}fR^{\prime}+4r^{4}R^{3}fR^{\prime\prime}+2r^{4}R^{3}R^{\prime}f^{\prime}$
$\displaystyle+4r^{4}R^{4}f\Phi^{\prime
2}+r^{4}R^{4}V\left(\Phi\right)-2q^{2}e^{2\alpha\Phi}=0,$ (9) $\displaystyle
8r^{3}R^{3}fR^{\prime}+2r^{3}R^{4}f^{\prime}+2r^{4}R^{3}f^{\prime}R^{\prime}+2r^{2}R^{4}f+2r^{4}R^{2}fR^{\prime
2}$
$\displaystyle+2r^{4}R^{3}fR^{\prime\prime}+r^{4}R^{4}V\left(\Phi\right)+2q^{2}e^{2\alpha\Phi}=0,$
(10) $\displaystyle
r^{4}R^{4}{\Phi^{\prime}}{f^{\prime}}+r^{4}R^{4}{\Phi^{\prime\prime}}{f}+2r^{3}R^{4}\Phi^{\prime}f+2r^{4}R^{3}R^{\prime}{\Phi^{\prime}}{f}-r^{4}R^{4}\frac{\partial{V}}{4\partial{\Phi}}-\alpha
q^{2}e^{2\alpha\Phi}=0,$ (11)
where the “prime” denotes differentiation with respect to $r$. Subtracting Eq.
(9) from Eq. (8) gives
$\displaystyle 2R^{\prime}+rR^{\prime\prime}+rR\Phi^{\prime 2}=0.$ (12)
Then we make the ansatz
$R(r)=e^{\alpha\Phi}.$ (13)
Substituting this ansatz in Eq. (12), it reduces to
$\displaystyle
r\alpha\Phi^{\prime\prime}+2\alpha\Phi^{\prime}+r(1+{\alpha}^{2})\Phi^{\prime
2}=0,$ (14)
which has a solution of the form
$\Phi(r)=\frac{\alpha}{\alpha^{2}+1}\ln(1-\frac{b}{r}),$ (15)
where $b$ is an integration constants. Inserting (15), the ansatz (13), and
the dilaton potential (2) into the field equations (8)-(11), one can show that
these equations have the following solution
$f(r)=-\frac{c}{r}\left(1-{\frac{b}{r}}\right)^{{\frac{1-{\alpha}^{2}}{1+{\alpha}^{2}}}}-\frac{\Lambda}{3}\,{r}^{2}\left(1-{\frac{b}{r}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}},$
(16)
where $c$ is an integration constant. The above solutions will fully satisfy
the system of equations (8)-(11) provided we have $q^{2}(1+\alpha^{2})=bc$.
One can also check that these solutions satisfy equations (3)-(5) in the
rotating case where $a\neq 0$. It is apparent that this spacetime is
asymptotically (anti)-de-Sitter. The Kretschmann scalar
$R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ and the Ricci scalar $R$
diverge at $r=0$ and therefore there is an essential singularity located at
$r=0$. The explicit form of the Kretschmann scalar is complicated and we do
not present it here, however, the Ricci scalar curvature of the metric has a
simpler form and can be written as
$R=-\frac{2\,{q}^{2}{\alpha}^{2}b\left(1-{\frac{b}{r}}\right)^{{\frac{1-\alpha^{2}}{{1+\alpha}^{2}}}}}{\left(r-b\right)^{2}\left({\alpha}^{2}+1\right){r}^{3}}+\frac{2\Lambda\,\left(1-{\frac{b}{r}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}}\left(2\left(r-b\right)^{2}+4\,{\alpha}^{2}{r}(r-b)+2\,{\alpha}^{4}{r}^{2}-{\alpha}^{2}{b}^{2}\right)}{\left({\alpha}^{2}+1\right)^{2}\left(r-b\right)^{2}}.$
(17)
One can easily check that for the arbitrary values of the dilaton coupling
constant, $R\rightarrow 4\Lambda$ as $r\rightarrow\infty$. We also find out
that for arbitrary $\alpha$ the Kretschmann scalar approach $8\Lambda^{2}/3$
as $r\rightarrow\infty$. This confirm our above discussion that our solution
is asymptotically anti-de-Sitter. In the absence of a nontrivial dilaton
($\alpha=0$), the solution reduces to the asymptotically (anti)-de-Sitter
charged rotating black string Lem . However, for $\alpha\neq 0$ the solution
is qualitatively different. As one can see from Eq. (17), the surface $r=b$ is
a curvature singularity except for the case $\alpha=0$ when it is a
nonsingular inner horizon. This is consistent with the idea that the inner
horizon is unstable in the Einstein-Maxwell theory. Therefore, our solutions
describe black strings in the case $b<r_{+}$, where $r_{+}$ is the outer
horizon of the black string (the root of Eq. $f(r)=0$). The above discussions
will become more clear if we look on the figures 1 and 2, where we have
plotted the function $f(r)$ versus $r$ for different values of the dilaton
coupling $\alpha$ and the charge parameter $q$. In the case $r=b$, it is clear
from (Charged rotating dilaton black strings in AdS spaces) and (13) that for
$\alpha\neq 0$ the area of the event horizon goes to zero (since
$R(r)\rightarrow 0$ in this case).
In summary, compared to the charged black holes/strings in (A)dS universe, the
dilaton version has some remarkable properties. In the first place, there may
be three horizons in the Reissner Nordstr m de Sitter spacetime, i.e., black
hole event horizon, black hole Cauchy horizon and cosmic event horizon.
However, the charged dilaton black holes/strings in (A)dS spaces has at most
two horizons. Here the inner Cauchy horizon disappears. This is due to the
fact that the inner horizon is unstable, as pointed by Garfinkle CDB2 .
Figure 1: The function $f(r)$ versus $r$ for $\alpha=1$, $b=2$ and
$\Lambda=-1$. $q=0.5$ (bold line), $q=1$ (continuous line) and $q=1.5$ (dashed
line).
Figure 2: The function $f(r)$ versus $r$ for $b=1$, $q=2$ and $\Lambda=-1$.
$\alpha=0$ (bold line), $\alpha=1$ (dashed line).
Next, we calculate the conserved quantities of the solutions. For
asymptotically (anti)-de-Sitter solutions, the way that one can calculate
these quantities and obtain finite values for them is through the use of the
counterterm method inspired by (A)dS/CFT correspondence Mal . In this paper we
deal with the spacetimes with zero curvature boundary, $R_{abcd}(\gamma)=0$,
and therefore the counterterm for the stress energy tensor should be
proportional to $\gamma^{ab}$. We find the suitable counterterm which removes
the divergences in the form
$I_{ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right).$
(18)
One may note that in the absence of a dilaton field where we have
$V(\Phi)=2\Lambda=-6/l^{2}$, the above counterterm has the same form as in the
case of asymptotically (A)dS solutions with zero-curvature boundary. Having
the total finite action $I=I_{G}+I_{\mathrm{ct}}$ at hand, one can use the
quasilocal definition to construct a divergence free stress-energy tensor BY .
Thus the finite stress-energy tensor in four dimensional Einstein-dilaton
gravity with three Liouville-type dilaton potentials (2) can be written as
$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta\gamma^{ab}+\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right)\gamma^{ab}\right],$
(19)
The first two terms in Eq. (19) are the variation of the action (1) with
respect to $\gamma_{ab}$, and the last two terms are the variation of the
boundary counterterm (18) with respect to $\gamma_{ab}$. To compute the
conserved charges of the spacetime, one should choose a spacelike surface
$\mathcal{B}$ in $\partial\mathcal{M}$ with metric $\sigma_{ij}$, and write
the boundary metric in ADM (Arnowitt-Deser-Misner) form:
$\gamma_{ab}dx^{a}dx^{a}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$
where the coordinates $\varphi^{i}$ are the angular variables parameterizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (19) can be written as
$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(20)
where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. For boundaries with timelike
($\xi=\partial/\partial t$) and rotational
($\varsigma=\partial/\partial\varphi$) Killing vector fields, one obtains the
quasilocal mass and angular momentum
$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b},$ (21)
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}.$
(22)
These quantities are, respectively, the conserved mass and angular momenta of
the system enclosed by the boundary $\mathcal{B}$. Note that they will both
depend on the location of the boundary $\mathcal{B}$ in the spacetime,
although each is independent of the particular choice of foliation
$\mathcal{B}$ within the surface $\partial\mathcal{M}$. The mass and angular
momentum per unit length of the string when the boundary $\mathcal{B}$ goes to
infinity can be calculated through the use of Eqs. (21) and (22). We find
$\displaystyle{M}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}(\alpha^{2}-1)b^{3}}{24\pi
l^{3}(\alpha^{2}+1)^{3}}+\frac{(3\Xi^{2}-1)c}{16\pi l},$ (23)
$\displaystyle{J}$ $\displaystyle=$ $\displaystyle\frac{3\Xi
c\sqrt{\Xi^{2}-1}}{16\pi}.$ (24)
For $a=0$ ($\Xi=1$), the angular momentum per unit volume vanishes, and
therefore $a$ is the rotational parameters of the spacetime. The entropy of
the dilaton black string still obeys the so called area law of the entropy
which states that the entropy of the black hole is a quarter of the event
horizon area Beck . This near universal law applies to almost all kinds of
black holes, including dilaton black holes, in Einstein gravity hunt . It is a
matter of calculation to show that the entropy per unit length of the string
is
${S}=\frac{r_{+}^{2}\Xi(1-\frac{b}{r_{+}})^{\frac{2\alpha^{2}}{\alpha^{2}+1}}}{4l}.$
(25)
By analytic continuation of the metric we can obtain the temperature and
angular velocity of the horizon. The analytical continuation of the Lorentzian
metric by $t\rightarrow i\tau$ and $a\rightarrow ia$ yields the Euclidean
section, whose regularity at $r=r_{+}$ requires that we should identify
$\tau\sim\tau+\beta_{+}$ and $\phi\sim\phi+i\beta_{+}\Omega_{+}$ where
$\beta_{+}$ and $\Omega_{+}$ are the inverse Hawking temperature and the
angular velocity of the horizon. We find
$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{f^{\text{
}^{\prime}}(r_{+})}{4\pi\Xi}=\frac{\left(1-{\frac{b}{r_{+}}}\right)^{{\frac{{\alpha}^{2}-1}{{\alpha}^{2}+1}}}}{2\pi
r_{+}^{2}\Xi({\alpha}^{2}+1)}\left(-\frac{\Lambda}{3}r_{+}^{2}\left({\alpha}^{2}r_{+}+r_{+}-b\right)+\frac{c\left({\alpha}^{2}r_{+}+r_{+}-2\,b\right)}{2r_{+}}\left(1-{\frac{b}{r_{+}}}\right)^{{\frac{1-3\,{\alpha}^{2}}{{\alpha}^{2}+1}}}\right)$
$\displaystyle\Omega_{+}$ $\displaystyle=$ $\displaystyle\frac{a}{\Xi l^{2}}.$
(26)
The next quantity we are going to calculate is the electric charge of the
string. To determine the electric field we should consider the projections of
the electromagnetic field tensor on special hypersurface. The normal vectors
to such hypersurface are
$u^{0}=\frac{1}{N},\text{ \ }u^{r}=0,\text{ \ }u^{i}=-\frac{V^{i}}{N},$ (27)
where $N$ and $V^{i}$ are the lapse function and shift vector. Then the
electric field is $E^{\mu}=g^{\mu\rho}e^{-2\alpha\phi}F_{\rho\nu}u^{\nu}$, and
the electric charge per unit length of the string can be found by calculating
the flux of the electric field at infinity, yielding
${Q}=\frac{\Xi q}{4\pi l}.$ (28)
The electric potential $U$, measured at infinity with respect to the horizon,
is defined by Cvetic
$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(29)
where $\chi=\partial_{t}+\Omega\partial_{\phi},$ is the null generator of the
event horizon. One can easily show that the vector potential $A_{\mu}$
corresponding to the electromagnetic tensor (Charged rotating dilaton black
strings in AdS spaces) can be written as
$A_{\mu}=-\frac{q}{r}\left(\Xi\delta_{\mu}^{t}-a\delta_{\mu}^{\phi}\right).$
(30)
Substituting (30) in (29) we obtain the electric potential as
$U=\frac{q}{\Xi r_{+}}.$ (31)
Having the conserved and thermodynamic quantities of the rotating dilaton
black string at hand, we are in a position to check the first law of black
hole thermodynamics. Numerical calculations show that the conserved quantities
calculated above satisfy the first law of thermodynamics
$dM=TdS+\Omega d{J}+Ud{Q}.$ (32)
In conclusion, we constructed a new class of charged rotating solutions in
four-dimensional Einstein-Maxwell-dilaton gravity with cylindrical or toroidal
horizons in the presence of dilaton potentials and investigate their
properties. These solutions are asymptotically anti-de Sitter. The
cosmological constant couples the dilaton field in a nontrivial way. The
dilaton potential with respect to the cosmological constant includes three
Liouville-type potentials. This is consistent with the arguments in MW that
no (anti)-de-Sitter version of dilaton black holes exist with only one
Liouville-type dilaton potential. We found the suitable counterterm which
removes the divergences of the action in the presence of three Liouville-type
dilaton potential. We also computed the conserved and thermodynamic quantities
of the solutions by using the conterterm method and verified numerically that
these quantities satisfy the first law of black hole thermodynamics.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran.
## References
* (1) G. W. Gibbons and K. Maeda, Nucl. Phys. B298, 741 (1988);
T. Koikawa and M. Yoshimura, Phys. Lett. B189, 29 (1987);
D. Brill and J. Horowitz, ibid. B262, 437 (1991).
* (2) D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D 43, 3140 (1991);
R. Gregory and J. A. Harvey, ibid. 47, 2411 (1993);
M. Rakhmanov, ibid. 50, 5155 (1994);
G. T. Horowitz and A. Strominger, Nucl. Phys. B 360 (1991) 197.
* (3) S. J. Poletti, D. L. Wiltshire, Phys. Rev. D 50 (1994) 7260 ;
S. J. Poletti, J. Twamley and D. L. Wiltshire, Phys. Rev. D 51 (1995) 5720;
S. Mignemi and D. L. Wiltshire, Phys. Rev. D 46 (1992) 1475.
* (4) K. C. K. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B447, 441 (1995).
* (5) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev D 57, 6547 (1998);
R. G. Cai and Y. Z. Zhang, ibid. 64, 104015 (2001).
* (6) G. Clement, D. Gal’tsov and C. Leygnac, Phys. Rev. D 67, 024012 (2003);
G. Clement and C. Leygnac, ibid. 70, 084018 (2004).
* (7) A. Sheykhi, M. H. Dehghani, N. Riazi, Phys. Rev. D 75, 044020 (2007);
A. Sheykhi, N. Riazi, Phys. Rev. D 75, 024021 (2007);
A. Sheykhi, Phys. Rev. D 76, 124025 (2007);
A. Sheykhi, Phys. Lett. B 662 (2008) 7.
* (8) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).
* (9) C. J. Gao, S. N. Zhang, Phys. Rev. D 70 (2004) 124019\.
* (10) C. J. Gao, S. N. Zhang, Phys. Lett. B 605 (2005) 185 ;
C. J. Gao, S. N. Zhang, Phys. Lett. B 612 (2005) 127.
* (11) S. B. Giddings, Phys. Rev. D 68 (2003) 026006;
E. Radu, D. H. Tchrakian, Class. Quant. Gravit. 22, 879 (2005).
* (12) J. P. S. Lemos, Class. Quantum Gravit. 12, 1081 (1995);
J. P. S. Lemos, Phys. Lett. B 353, 46 (1995).
* (13) J. Maldacena, Adv. Theor. Math. Phys., 2, 231 (1998);
E. Witten, Adv. Theor. Math. Phys., 2, 253 (1998);
O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rep.
323,183 (2000).
* (14) J. D. Brown and J. W. York, Phys. Rev. D 47, 1407 (1993).
* (15) J. D. Beckenstein, Phys. Rev. D 7, 2333 (1973);
G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
* (16) C. J. Hunter, Phys. Rev. D 59, 024009 (1998);
S. W. Hawking, C. J Hunter and D. N. Page, ibid. 59, 044033 (1999).
* (17) M. Cvetic and S. S. Gubser, J. High Energy Phys. 04, 024 (1999);
M. M. Caldarelli, G. Cognola and D. Klemm, Class. Quant. Gravit. 17, 399
(2000).
|
arxiv-papers
| 2008-09-06T04:28:26
|
2024-09-04T02:48:57.664641
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0809.1130"
}
|
0809.1131
|
# Higher dimensional slowly rotating dilaton black holes in AdS spacetime
A. Sheykhia,b111sheykhi@mail.uk.ac.ir and M. Allahverdizadeh a aDepartment of
Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran
bResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
###### Abstract
In this paper, with an appropriate combination of three Liouville-type dilaton
potentials, we obtain the higher dimensional charged slowly rotating dilaton
black hole solution for asymptotically anti-de Sitter spacetime. The angular
momentum and the gyromagnetic ratio of such a black hole are determined for
the arbitrary values of the dilaton coupling constant. It is shown that the
dilaton field modifies the gyromagnetic ratio of the rotating dilaton black
holes.
###### pacs:
04.70.Bw, 04.20.Ha, 04.50.+h
## I Introduction
Over the past years there has been a growing interest for studying the
rotating black hole solutions in the background of anti-de Sitter (AdS)
spacetimes. This interest is motivated by the correspondence between the
gravitating fields in an AdS spacetime and conformal field theory on the
boundary of the AdS spacetime Witt1 . It was argued that the thermodynamics of
black holes in AdS spaces can be identified with that of a certain dual
conformal field theory (CFT) in the high temperature limit Witt2 . In the
AdS/CFT correspondence, the rotating black holes in AdS space are dual to
certain CFTs in a rotating space Haw , while charged ones are dual to CFTs
with chemical potential Cham . The most general higher dimensional uncharged
rotating black holes in AdS space have been recently found Haw ; Gib . As far
as we know, rotating black holes for the Maxwell field minimally coupled to
Einstein gravity in higher dimensions, do not exist in a closed form and one
has to rely on perturbative or numerical methods to construct them in the
background of asymptotically flat kunz1 ; Aliev2 and AdS kunz2 spacetimes.
There has also been recent interest in constructing the analogous charged
rotating solutions in the framework of gauged supergravity in various
dimensions Cvetic0 ; Cvetic1 .
On another front, scalar coupled black hole solutions with different
asymptotic spacetime structure is a subject of interest for a long time. There
has been a renewed interest in such studies ever since new black hole
solutions have been found in the context of string theory. The low energy
effective action of string theory contains two massless scalars namely dilaton
and axion. The dilaton field couples in a nontrivial way to other fields such
as gauge fields and results into interesting solutions for the background
spacetime CDB1 ; CDB2 . These scalar coupled black hole solutions CDB1 ; CDB2
, however, are all asymptotically flat. It was argued that with the exception
of a pure cosmological constant, no dilaton-de Sitter or anti-de Sitter black
hole solution exists with the presence of only one Liouville-type dilaton
potential MW . In the presence of one or two Liouville-type potentials, black
hole spacetimes which are neither asymptotically flat nor (A)dS have been
explored by many authors (see e.g. CHM ; Cai ; Clem ; Shey0 ). Recently, the
“cosmological constant term” in the dilaton gravity has been found by Gao and
Zhang Gao1 ; Gao2 . With an appropriate combination of three Liouville-type
dilaton potentials, they obtained the static dilaton black hole solutions
which are asymptotically (A)dS in four and higher dimensions. The motivations
for studying such dilaton black holes with nonvanishing cosmological constant
originate from supergravity theory. Gauged supergravity theories in various
dimensions are obtained with negative cosmological constant in a
supersymmetric theory. In such a scenario AdS spacetime constitutes the vacuum
state and the black hole solution in such a spacetime becomes an important
area to study Witt1 .
In the light of all mentioned above, it becomes obvious that further study on
the rotating black hole solutions in a spacetime with nonzero cosmological
constant in the presence of dilaton-electromagnetic coupling is of great
importance. The properties of charged rotating dilaton black holes, for an
arbitrary dilaton coupling constant, in the small angular momentum limit in
four Hor1 ; Sheykhi2 and higher dimensions have been studied Sheykhi3 ; ShAll
. Recently, in the presence of a Liouville-type dilaton potential, one of us
has constructed a class of charged slowly rotating dilaton black hole
solutions in arbitrary dimensions Sheykhi4 . Unfortunately, these solutions
Sheykhi4 are neither asymptotically flat nor (A)dS. Besides, they are ill-
defined for the string case where $\alpha=1$. More recently, a class of slowly
rotating charged dilaton black hole solutions in four-dimensional anti-de
Sitter spacetime has been found Ghosh . Until now, higher dimensional charged
rotating dilaton black hole solutions for an arbitrary dilaton-electromagnetic
coupling constant in the background of anti-de Sitter spacetime have not been
constructed. Our aim in this paper is to construct a higher dimensional
charged rotating dilaton black hole solution for asymptotically AdS spacetime
in the small angular momentum limit with an appropriate combination of three
Liouville-type dilaton potentials. We then determine the angular momentum and
the gyromagnetic ratio of such a black hole for the arbitrary values of the
dilaton coupling constant. We will restrict ourselves to the rotation in one
plane, so our black hole has only one angular momentum parameter.
## II Field equations and solutions
We consider the $n$-dimensional $(n\geq 4)$ theory in which gravity is coupled
to the dilaton and Maxwell field with an action
$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n}x\sqrt{-g}\left(R\text{
}-\frac{4}{n-2}\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi)-e^{-{4\alpha\Phi}/({n-2})}F_{\mu\nu}F^{\mu\nu}\right)$
(1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n-1}x\sqrt{-h}\Theta(h),$
where ${R}$ is the scalar curvature, $\Phi$ is the dilaton field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, and $A_{\mu}$ is the electromagnetic potential.
$\alpha$ is an arbitrary constant governing the strength of the coupling
between the dilaton and the Maxwell field. The last term in Eq. (1) is the
Gibbons-Hawking surface term. It is required for the variational principle to
be well defined. The factor $\Theta$ represents the trace of the extrinsic
curvature for the boundary ${\partial\mathcal{M}}$ and $h$ is the induced
metric on the boundary. While $\alpha=0$ corresponds to the usual Einstein-
Maxwell-scalar theory, $\alpha=1$ indicates the dilaton-electromagnetic
coupling that appears in the low energy string action in Einstein’s frame. For
an arbitrary value of $\alpha$ in AdS space the form of the dilaton potential
in arbitrary dimensions is chosen as Gao2
$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle\frac{\Lambda}{3(n-3+\alpha^{2})^{2}}\left[-\alpha^{2}(n-2)\left(n^{2}-n\alpha^{2}-6n+\alpha^{2}+9\right)e^{[{-4(n-3)\Phi}/{(n-2)\alpha}]}\right.$
(2)
$\displaystyle\left.+(n-2)(n-3)^{2}(n-1-\alpha^{2})e^{{4\alpha\Phi}/({n-2})}+4\alpha^{2}(n-3)(n-2)^{2}e^{[{-2\Phi(n-3-\alpha^{2})}/{(n-2)\alpha}]}\right].$
Here $\Lambda$ is the cosmological constant. It is clear the cosmological
constant is coupled to the dilaton in a very nontrivial way. This type of
dilaton potential can be obtained when a higher dimensional theory is
compactified to four dimensions, including various supergravity models Gid .
In the absence of the dilaton field the action (1) reduces to the action of
Einstein-Maxwell gravity with cosmological constant. Varying the action (1)
with respect to the gravitational field $g_{\mu\nu}$, the dilaton field $\Phi$
and the gauge field $A_{\mu}$, yields
$R_{\mu\nu}=\frac{4}{n-2}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{4}g_{\mu\nu}V(\Phi)\right)+2e^{{-4\alpha\Phi}/({n-2})}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{2(n-2)}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right),$ (3)
$\nabla^{2}\Phi=\frac{n-2}{8}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{{-4\alpha\Phi}/({n-2})}F_{\lambda\eta}F^{\lambda\eta},$
(4)
$\partial_{\mu}{\left(\sqrt{-g}e^{{-4\alpha\Phi}/({n-2})}F^{\mu\nu}\right)}=0.$
(5)
We would like to find $n$-dimensional rotating solutions of the above field
equations. For small rotation, we can solve Eqs. (3)-(5) to first order in the
angular momentum parameter $a$. Inspection of the $n$-dimensional Kerr
solutions shows that the only term in the metric that changes to the first
order of the angular momentum parameter $a$ is $g_{t\phi}$. Similarly, the
dilaton field does not change to $O(a)$ and $A_{\phi}$ is the only component
of the vector potential that changes. Therefore, for infinitesimal angular
momentum we assume the metric being of the following form
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-U(r)dt^{2}+{dr^{2}\over
W(r)}-2af(r)\sin^{2}{\theta}dtd{\phi}$ (6)
$\displaystyle+r^{2}R^{2}(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\cos^{2}\theta d\Omega_{n-4}^{2}\right),$
where $d\Omega^{2}_{n-4}$ denotes the metric of a unit $(n-4)$\- sphere. The
functions $U(r)$, $W(r)$, $R(r)$ and $f(r)$ should be determined. In the
particular case $a=0$, this metric reduces to the static and spherically
symmetric cases. For small $a$, we can expect to have solutions with $U(r)$
and $W(r)$ still functions of $r$ alone. The $t$ component of the Maxwell
equations can be integrated immediately to give
$F_{tr}=\sqrt{\frac{U(r)}{W(r)}}\frac{Qe^{{4\alpha\Phi}/({n-2})}}{\left(rR\right)^{n-2}},$
(7)
where $Q$, an integration constant, is the electric charge of the black hole.
In general, in the presence of rotation, there is also a vector potential in
the form
$A_{\phi}=-aQC(r)\sin^{2}\theta.$ (8)
The asymptotically (A)dS static ($a=0$) black hole solution of the above field
equations was found in Gao2 . Here we are looking for the asymptotically (A)dS
solution in the case $a\neq 0$. Our strategy for obtaining the solution is the
perturbative method suggested by Horne and Horowitz Hor1 . Inserting the
metric (6), the Maxwell fields (7) and (8) into the field equations (3)-(5),
one can show that the static part of the metric leads to the following
solutions Gao2 :
$\displaystyle U(r)$ $\displaystyle=$
$\displaystyle\left[1-\left(\frac{r_{+}}{r}\right)^{n-3}\right]\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{1-\gamma\left(n-3\right)}-\frac{1}{3}\Lambda
r^{2}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma},$ (9)
$\displaystyle W(r)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}\left[1-\left(\frac{r_{+}}{r}\right)^{n-3}\right]\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{1-\gamma\left(n-3\right)}-\frac{1}{3}\Lambda
r^{2}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma}\Bigg{\\}}$
(10)
$\displaystyle\times\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma(n-4)},$
$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle\frac{n-2}{4}\sqrt{\gamma(2+3\gamma-n\gamma)}\ln\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right],$
(11) $\displaystyle R(r)$ $\displaystyle=$
$\displaystyle\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma/2},$
(12)
while we obtain the following solution for the rotating part of the metric
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{2\Lambda
r^{2}}{(n-1)(n-2)}\left[1-\left({\frac{r_{-}}{r}}\right)^{n-3}\right]^{\gamma}+\left(n-3\right)\left(\frac{r_{+}}{r}\right)^{n-3}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{({n-3-\alpha^{2}})/({n-3+\alpha^{2}})}$
(13)
$\displaystyle+\frac{(\alpha^{2}-n+1)(n-3)^{2}}{\alpha^{2}+n-3}r_{-}^{n-3}r^{2}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma}\times\int\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma(2-n)}\frac{dr}{r^{n}},$
$\displaystyle C(r)=\frac{1}{r^{n-3}}.$ (14)
We can also perform the integration and express the solution in terms of the
hypergeometric function
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{2\Lambda
r^{2}}{(n-1)(n-2)}\left[1-\left({\frac{r_{-}}{r}}\right)^{n-3}\right]^{\gamma}+\left(n-3\right)\left(\frac{r_{+}}{r}\right)^{n-3}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{(n-3-\alpha^{2})/(n-3+\alpha^{2})}$
(15)
$\displaystyle+\frac{(\alpha^{2}-n+1)(n-3)^{2}}{(1-n)(\alpha^{2}+n-3)}(\frac{r_{-}}{r})^{n-3}\left[1-\left(\frac{r_{-}}{r}\right)^{n-3}\right]^{\gamma}$
$\displaystyle\times_{2}F_{1}\left(\left[(n-2)\gamma,\frac{n-1}{n-3}\right],\left[\frac{2n-4}{n-3}\right],\left({\frac{r_{-}}{r}}\right)^{n-3}\right).$
Here $r_{+}$ and $r_{-}$ are, respectively, the event horizon and Cauchy
horizon of the black hole, and the constant $\gamma$ is
$\gamma=\frac{2\alpha^{2}}{(n-3)(n-3+\alpha^{2})}.$ (16)
The charge $Q$ is related to $r_{+}$ and $r_{-}$ by
$Q^{2}=\frac{(n-2)(n-3)^{2}}{2(n-3+\alpha^{2})}r_{+}^{n-3}r_{-}^{n-3},$ (17)
and the physical mass of the black hole is obtained as follows Fang
${M}=\frac{\Omega_{n-2}}{16\pi}\left[(n-2)r^{n-3}_{+}+\frac{n-2-p(n-4)}{p+1}r^{n-3}_{-}\right],$
(18)
where we have ignored the term of the order of $a^{2}$ in the mass expression
for the anti-de Sitter dilatonic black hole Ghosh ; Aliev3 . Here
$\Omega_{n-2}$ denotes the area of the unit $(n-2)$-sphere and the constant
$p$ is
${p}=\frac{(2-n)\gamma}{(n-2)\gamma-2}.$ (19)
It is apparent that the metric corresponding to (9)-(15) is asymptotically
(A)dS. For $\Lambda=0$, the above solutions recover our previous results for
asymptotically flat rotating dilaton black holes ShAll . In the special case
$n=4$, the static part of our solution reduces to
$U(r)=W(r)=\left(1-{\frac{r_{+}}{r}}\right)\left(1-{\frac{r_{-}}{r}}\right)^{{({1-\alpha}^{2})}/({1+{\alpha}^{2})}}-\frac{1}{3}\Lambda
r^{2}\left(1-{\frac{r_{-}}{r}}\right)^{{2\alpha^{2}}/({1+\alpha^{2}})},$ (20)
$\Phi\left(r\right)=\frac{\alpha}{{\alpha}^{2}+1}\ln\left(1-{\frac{r_{-}}{r}}\right),$
(21)
$R\left(r\right)=\left(1-{\frac{r_{-}}{r}}\right)^{\alpha^{2}/(1+{\alpha}^{2})},$
(22)
while the rotating part reduces to
$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\left(1-\frac{r_{-}}{r}\right)^{{(1-\alpha^{2})}/({1+\alpha^{2}})}\left(1+\frac{(1+\alpha^{2})^{2}r^{2}}{(1-\alpha^{2})(1-3\alpha^{2})r^{2}_{-}}+\frac{(1+\alpha^{2})r}{(1-\alpha^{2})r_{-}}-\frac{r_{+}}{r}\right)$
(23)
$\displaystyle+\frac{r^{2}(1+\alpha^{2})^{2}}{(1-\alpha^{2})(1-3\alpha^{2})r^{2}_{-}}\left(1-\frac{r_{-}}{r}\right)^{{2\alpha^{2}}/({1+\alpha^{2}})}+\frac{1}{3}\Lambda
r^{2}\left(1-{\frac{r_{-}}{r}}\right)^{{2\alpha^{2}}/({1+\alpha^{2}})},$
which is the four-dimensional asymptotically (A)dS charged slowly rotating
dilaton black hole solution presented in Ghosh . One may also note that in the
absence of a nontrivial dilaton ($\alpha=0=\gamma$), our solutions reduce to
$U\left(r\right)=W(r)=\left[1-\left({\frac{r_{+}}{r}}\right)^{n-3}\right]\left[1-\left({\frac{r_{-}}{r}}\right)^{n-3}\right]-\frac{1}{3}\Lambda
r^{2},$ (24)
$f\left(r\right)=(n-3)\left[\frac{r^{n-3}_{-}+r^{n-3}_{+}}{r^{n-3}}-\left(\frac{r_{+}r_{-}}{r^{2}}\right)^{n-3}\right]+{\frac{2\Lambda\,{r}^{2}}{\left(n-1\right)\left(n-2\right)}},$
(25)
which describe the $n$-dimensional charged Kerr-(A)dS black hole in the limit
of slow rotation.
Next, we calculate the angular momentum and the gyromagnetic ratio of these
rotating dilaton black holes which appear in the limit of slow rotation
parameter. The angular momentum of the dilaton black hole can be calculated
through the use of the quasilocal formalism of Brown and York BY . According
to the quasilocal formalism, the quantities can be constructed from the
information that exists on the boundary of a gravitating system alone. Such
quasilocal quantities will represent information about the spacetime contained
within the system boundary, just like the Gauss’s law. In our case the finite
stress-energy tensor can be written as
$T^{ab}=\frac{1}{8\pi}\left(\Theta^{ab}-\Theta h^{ab}\right),$ (26)
which is obtained by variation of the action (1) with respect to the boundary
metric $h_{ab}$. To compute the angular momentum of the spacetime, one should
choose a spacelike surface $\mathcal{B}$ in $\partial\mathcal{M}$ with metric
$\sigma_{ij}$, and write the boundary metric in ADM form
$\gamma_{ab}dx^{a}dx^{a}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$
where the coordinates $\varphi^{i}$ are the angular variables parametrizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions, respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (26) can be written as
$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{n-2}\varphi\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(27)
where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. For boundaries with rotational
($\varsigma=\partial/\partial\varphi$) Killing vector field, we can write the
corresponding quasilocal angular momentum as follows
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{n-2}\varphi\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b},$
(28)
provided the surface $\mathcal{B}$ contains the orbits of $\varsigma$.
Finally, the angular momentum of the black holes can be calculated by using
Eq. (28). We find
${J}=\frac{a\Omega_{n-2}}{8\pi}\left(r^{n-3}_{+}+\frac{(n-3)(n-1-\alpha^{2})r^{n-3}_{-}}{(n-3+\alpha^{2})(n-1)}\right).$
(29)
For $a=0$, the angular momentum vanishes, and therefore $a$ is the rotational
parameter of the dilaton black hole. For $n=4$, the angular momentum reduces
to
${J}=\frac{a}{2}\left(r_{+}+\frac{3-\alpha^{2}}{3(1+\alpha^{2})}r_{-}\right),$
(30)
which restores the angular momentum of the four-dimensional Horne and Horowitz
solution Hor1 . Finally, we calculate the gyromagnetic ratio of this rotating
dilaton black hole. As we know, the gyromagnetic ratio is an important
characteristic of the Kerr-Newman-AdS black hole. Indeed, one of the
remarkable facts about a Kerr-Newman black hole in asymptotically flat
spacetime is that it can be assigned a gyromagnetic ratio $g=2$, just as an
electron in the Dirac theory. It should be noted that, unlike four dimensions,
the value of the gyromagnetic ratio is not universal in higher dimensions
Aliev3 . Besides, scalar fields such as the dilaton, modify the value of the
gyromagnetic ratio of the black hole and consequently it does not possess the
gyromagnetic ratio $g=2$ of the Kerr-Newman black hole Hor1 . Here, we wish to
calculate the value of the gyromagnetic ratio when the dilatonic black hole
has an asymptotic AdS behavior. The magnetic dipole moment for this
asymptotically AdS slowly rotating dilaton black hole can be defined as
${\mu}=Qa.$ (31)
The gyromagnetic ratio is defined as a constant of proportionality in the
equation for the magnetic dipole moment
${\mu}=g\frac{QJ}{2M}.$ (32)
Substituting $M$ and $J$ from Eqs. (18) and (29), the gyromagnetic ratio $g$
can be obtained as
$g=\frac{(n-1)(n-2)[(n-3+\alpha^{2})r^{n-3}_{+}+(n-3-\alpha^{2})r^{n-3}_{-}]}{(n-1)(n-3+\alpha^{2})r^{n-3}_{+}+(n-3)(n-1-\alpha^{2})r^{n-3}_{-}}.$
(33)
One can see that in the linear approximation in the rotation parameter $a$,
the above expression for $g$ turns out to be same as that found in ShAll for
the asymptotically flat slowly rotating dilaton black hole. This means that
the dilaton potential (cosmological constant term) does not change the
gyromagnetic ratio of the rotating (A)dS dilaton black holes, as discussed in
Ghosh . However, the dilaton field modifies the value of the gyromagnetic
ratio $g$ through the coupling parameter $\alpha$ which measures the strength
of the dilaton-electromagnetic coupling. This is in agreement with the
arguments in Hor1 . We have shown the behavior of the gyromagnetic ratio $g$
of the dilatonic black hole versus $\alpha$ in Fig. 1. From this figure we
find out that the gyromagnetic ratio decreases with increasing $\alpha$ in any
dimension. In the absence of a nontrivial dilaton $(\alpha=0=\gamma)$, the
gyromagnetic ratio reduces to
${g}=n-2,$ (34)
which is the gyromagnetic ratio of the $n$-dimensional Kerr-Newman black hole
with a single angular momentum in the limit of slow rotation Aliev2 . When
$n=4$, it reduces to
${g}=2-\frac{4\alpha^{2}r_{-}}{(3-\alpha^{2})r_{-}+3(1+\alpha^{2})r_{+}},$
(35)
which is the gyromagnetic ratio of the four-dimensional slowly rotating
dilaton black hole Hor1 ; Ghosh .
Figure 1: The behavior of the gyromagnetic ratio $g$ versus $\alpha$ in
various dimensions for $r_{-}=1$, $r_{+}=2$. $n=4$ (bold line), $n=5$
(continuous line), and $n=6$ (dashed line).
## III Summary and Conclusion
It is well known that in the presence of one Liouville-type dilaton potential,
no de Sitter or anti-de Sitter dilaton black hole exists even in the absence
of rotation MW . In this paper, with an appropriate combination of three
Liouville-type dilaton potential proposed in Gao2 , we showed that such
potential leads to higher dimensional slowly rotating charged dilaton black
holes solutions in an anti-de Sitter spacetime. The presence of such an AdS
dilatonic charged rotating black hole is inevitably associated with an
accompanying scalar field with appropriate Liouville-type potential. Their
study therefore may lead to a better understanding of the origin of the dark
matter in the universe. We started from the nonrotating charged dilaton black
hole solutions in anti-de Sitter spacetime Gao2 and then successfully
obtained the solution for the rotating charged dilaton black hole in higher
dimensions by introducing a small angular momentum and solving the equations
of motion up to the linear order of the angular momentum parameter. We
discarded any terms involving $a^{2}$ or higher powers in $a$ where $a$ is the
rotation parameter. For small rotation, the only term in the metric which
changes is $g_{t\phi}$. The vector potential is chosen to have a nonradial
component $A_{\phi}=-aQC(r)\sin^{2}{\theta}$ to represent the magnetic field
due to the rotation of the black hole. As expected, our solution $f(r)$
reduces to the Ghosh and SenGupta solution for $n=4$, while in the absence of
the dilaton field $(\alpha=0=\gamma)$, it reduces to the $n$-dimensional
slowly rotating Kerr-Newman-AdS black hole. We calculated the angular momentum
$J$ and the gyromagnetic ratio $g$ which appear up to the linear order of the
angular momentum parameter $a$. Interestingly enough, we found that the
dilaton field modifies the value of the gyromagnetic ratio $g$ through the
coupling parameter $\alpha$ which measures the strength of the dilaton-
electromagnetic coupling. This is in agreement with the arguments in Hor1 .
Finally, we would like to mention that in this paper we only considered the
higher dimensional charged slowly rotating black hole solutions with a single
rotation parameter in the background of AdS spacetime. In general, in more
than three spatial dimensions, black holes can rotate in different orthogonal
planes, so the general solution has several angular momentum parameters.
Indeed, an $n$-dimensional black hole can have $N=[(n-1)/2]$ independent
rotation parameters, associated with $N$ orthogonal planes of rotation where
$[x]$ denotes the integer part of $x$. The generalization of the present work
to the case with more than one rotation parameter and arbitrary dilaton
coupling constant is now under investigation and will be addressed elsewhere.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran.
## References
* (1) E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998);
J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
* (2) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).
* (3) S. W. Hawking, C. J. Hunter and M. Taylor, Phys. Rev. D 59, 064005 (1999).
* (4) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D 60, 064018 (1999);
M. Cvetic and S. S. Gubser, JHEP 9904, 024 (1999);
R. G. Cai and K. S. Soh, Mod. Phys. Lett. A 14, 1895 (1999).
* (5) G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope, Phys. Rev. Lett. 93, 171102 (2004);
G. W. Gibbons, H. Lu, D. N. Page and C. N. Pope, J. Geom. Phys. 53, 49 (2005).
* (6) J. Kunz, F. Navarro-Lerida and J. Viebahn, Phys. Lett. B 639, 362(2006).
* (7) A. N. Aliev, Phys. Rev. D 74, 024011 (2006).
* (8) J. Kunz, F. Navarro-Lerida and E. Radu, Phys. Lett. B 649, (2007) 463;
A. N. Aliev, Phys. Rev. D 75, 084041 (2007);
H. C. Kim, R. G. Cai, Phys. Rev. D 77, 024045 (2008);
Y. Brihaye and T. Delsate, arXiv:0806.1583.
* (9) M. Cvetic, H. Lu and C. N. Pope, Phys. Lett. B 598, 273 (2004);
M. Cvetic, H. Lu and C. N. Pope, Phys. Rev. D 70, 081502 (2004).
* (10) M. Cvetic and D. Youm, Nucl. Phys. B 476, 118 (1996);
Z. W. Chong, M. Cvetic, H. Lu, and C. N. Pope, Phys. Rev. D 72, 041901 (2005).
* (11) G. W. Gibbons and K. Maeda, Nucl. Phys. B 298, 741 (1988);
T. Koikawa and M. Yoshimura, Phys. Lett. B 189, 29 (1987);
D. Brill and J. Horowitz, Phys. Lett. B 262, 437 (1991).
* (12) D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D 43, 3140 (1991);
R. Gregory and J. A. Harvey, ibid. 47, 2411 (1993);
G. T. Horowitz and A. Strominger, Nucl. Phys. B 360, 197 (1991).
* (13) S. J. Poletti, D. L. Wiltshire, Phys. Rev. D 50, 7260 (1994) ;
S. J. Poletti, J. Twamley and D. L. Wiltshire, Phys. Rev. D 51, 5720 (1995).
* (14) K. C. K. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B 447, 441 (1995).
* (15) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev D 57, 6547 (1998).
* (16) G. Clement, D. Gal’tsov and C. Leygnac, Phys. Rev. D 67, 024012 (2003).
* (17) A. Sheykhi, M. H. Dehghani, N. Riazi, Phys. Rev. D 75, 044020 (2007);
A. Sheykhi, N. Riazi, Phys. Rev. D 75, 024021 (2007);
A. Sheykhi, Phys. Rev. D 76, 124025 (2007);
A. Sheykhi, Phys. Lett. B 662, 7 (2008).
* (18) C. J. Gao, S. N. Zhang, Phys. Rev. D 70, 124019 (2004).
* (19) C. J. Gao, S. N. Zhang, Phys. Lett. B 605, 185 (2005).
* (20) J. H. Horne and G. T. Horowitz, Phys. Rev. D 46, 1340 (1992).
* (21) A. Sheykhi and N. Riazi, Int. J. Theor. Phys. 45, 2453 (2006).
* (22) A. Sheykhi and N. Riazi, Int. J. Mod. Phys. A, Vol. 22, No. 26, 4849 (2007).
* (23) A. Sheykhi, et al., Phys. Lett. B 666, 82 (2008).
* (24) A. Sheykhi, Phys. Rev. D 77, 104022 (2008).
* (25) T. Ghosh and S. SenGupta, Phys. Rev. D 76, 087504 (2007).
* (26) S. B. Giddings, Phys. Rev. D 68 (2003) 026006;
E. Radu, D. H. Tchrakian, Classical Quantum Gravity 22, 879 (2005).
* (27) H. Z. Fang, Nucl. Phys. B 767, 130 (2007).
* (28) A. N. Aliev, Phys. Rev. D 75, 084041 (2007); Classical Quantum Gravity 24, 4669 (2007).
* (29) J. Brown and J. York, Phys. Rev. D 47, 1407 (1993).
|
arxiv-papers
| 2008-09-06T04:35:17
|
2024-09-04T02:48:57.668734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi and M. Allahverdizadeh",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0809.1131"
}
|
0809.1139
|
# Fractality feature in oil price fluctuations
M. Momeni momeni@shahroodut.ac.ir Faculty of Physics, Shahrood University of
Technology, Shahrood, Iran I. Kourakis Queen’s University Belfast, Center
for Plasma Physics, BT7 1 NN Northern Ireland, UK K. Talebi Faculty of
mining engineering , Shahrood University of Technology, Shahrood, Iran
###### Abstract
The scaling properties of oil price fluctuations are described as a non-
stationary stochastic process realized by a time series of finite length. An
original model is used to extract the scaling exponent of the fluctuation
functions within a non-stationary process formulation. It is shown that, when
returns are measured over intervals less than 10 days, the Probability Density
Functions (PDFs) exhibit self-similarity and monoscaling, in contrast to the
multifractal behavior of the PDFs at macro-scales (typically larger than one
month). We find that the time evolution of the distributions are well fitted
by a Lévy distribution law at micro-scales. The relevance of a Lévy
distribution is made plausible by a simple model of nonlinear transfer.
###### pacs:
02.50.-r,02.50.Ey,64.60.al
## I introduction
Many natural or man-made phenomena, such as turbulence flows, fluctuations in
finance (stock market), seismic recording, internet traffic, climate change,
etc., are characterized by randomness or stochasticity Fris ; bisk ; Kolm1 ;
noroz ; mant ; mant1 ; bouch ; abra ; farah ; ken ; keil ; vere ; voig ; bufe
; saic ; kosc ; barb ; frae . Analysis of non-stationary stochastic processes
referring to quantities which fluctuate widely and are uncertain has been a
problem of fundamental interest for a long time. Over the past two decades,
oil price has increased very sharply, rising from $25 per barrel in January
1986 to a peak of close to $ 122 per barrel in the last week of July 2008. The
effects of oil price fluctuations on the world economy are undeniable and
particularly evident from the international reports. Oil price data as a time
series exhibit complex patterns and seemingly appear to be a chaotic system.
Indeed, the behavior of oil price fluctuations can be efficiently modelled by
the Ising model which was proposed for stock-price fluctuations pler and by
the the cascade model developed based on fractal concepts which was used for
hydrodynamics and magnetohydrodynamic turbulence ghas ; muzy ; mom . Here, we
employ the latter technique to characterize the statistical properties of the
oil-price time series, which sensitively distinguish between self-similarity
and multi-fractality in a time series. The model is based on two-point
increments of oil price, yielding a comprehensive and scale-dependent
characterization of the statistical properties of the system via an associated
Probability Density Function(PDF). It is necessary to stress that the data
series is represented by a finite number of records which do not constitute a
stationary process. The effect of non-stationarity on the detrended
fluctuation analysis has been investigated in Ref. hu . To eliminate the
effect of sinusoidal trend, we apply the Fourier Detrended Fluctuation
Analysis (F-DFA). After the elimination of the trend we use the Multifractal
Detrended Fluctuation Analysis (MF-DFA) to analysis the data set. The MF-DFA
methods are the modified version of detrended fluctuation analysis (DFA) to
detect multifractal properties of time series. The detrended fluctuation
analysis (DFA) method introduced by Peng et al. peng has became a widely-used
technique for the determination of (multi)fractal scaling properties and the
detection of long-range correlations in noisy, non-stationary time series hu ;
peng . It has successfully been applied to diverse fields such as DNA
sequences buld ; buld1 , heart rate dynamics ken ; ken1 , neuron spiking bles
, human gait haus , long-time weather records kosc ; ivan , cloud structure
ivan1 , geology mala , ethnology alad , economical time series mant ; liu ,
solid state physics kant , sunspot time series sadegh , and cosmic microwave
background radiation keil ; vere .
We propose a method for generating a stationary process analysis out of a non-
stationary process. The fact is that stationary stochastic systems often show
scaling in a statistical sense, coincident with non-Gaussian leptokurtic
(heavy- tailed) statistics. Importantly, identification of the associated
scaling exponents implies the ability to interpret and estimate the behavior
of the fluctuations as well as the detection of long-range correlations. A
self-similar Brownian walk with Gaussian PDFs, which has scaling exponent
$1/2$, is a good example of the process where shows uncorrelation on all
temporal scales. We try to determine the scaling properties of the PDFs that
are leptokurtic at micro-scales. The scaling exponents can be determined
through the scaling behavior of the moments, usually characterized by
computing structure functions. It is said that the fluctuations are self-
similar (monofractal) if scaling exponents of the moments exhibit a linear
power-law dependence. In contrast a nonlinear dependence infers to
multifractal scaling, which is caused by intermittent small-scale structures
of oil price fluctuations. A similar feature has been found in physical
systems for example, in velocity and magnetic fields of the solar wind kian ;
kian1 ; gold as well as in magnetohydrodynamic turbulence studied via direct
numerical simulations mom . Finally, distributed price changes are
characterized by a stable Lévy distribution in the central part of the
distribution.
The paper is structured as follows. In Section II we describe our data set.
The MF-DFA method is briefly presented in Section III and shown that the
scaling exponent determined via the MF-DFA method are identical to those
obtained by the standard multifractal formalism based on PDF analysis. In
Section IV we employ a recently developed technique kian ; kian1 ; mom that
sensitively distinguishes between self-similarity and multifractality in times
series. By analyzing the temporal evolution of price dynamics, we demonstrate
the strongly the non-Gaussian behavior of the returns of oil price and scale-
dependent behavior of the PDFs. Also we explain the Hurst exponents analysis.
The micro-scale PDFs resemble leptokurtic Lévy distribution which will be
discussed in Section V. Finally, in Section VI we will summarize all results
discussed throughout this paper.
## II The data
Over a twelve-year period, on average, the price of oil has increased from $25
per barrel in January 1986 to a peak of close to $ 122 per barrel in the last
week of July 2008 . Oil price as recorded in international markets sit offer
us an almost unique possibility to gain information on the stochastic dynamics
state in a very large scale range, say 1 day up to 200 days. Our database
consists of about 5695 daily price values which seem to provide a set of data
points which will be sufficient to obtain the scaling properties the system.
Fig.1 presents the daily fluctuations in oil price $p(t)$ in the period
1986-2008. It is evident from the figure that the fluctuations do not
constitute a stationary process, for instance one can show that the variance
of the signal in some window does not remain stable upon increasing the window
size. Let us introduce the increments (or returns) $\delta p(t,\tau)$ defined
by, $\delta p(t,\tau)=p(t+\tau)-p(t)$. The resulting series for $\delta
p(\tau)$ is shown in the inset graph of Fig.1.
Figure 1: Semi-log plot of oil price series over the period 1986-2008. Inset:
daily return of the oil price index.
It is straightforward to show, by measuring the average and the variance of
$\delta p(t,\tau)$ in a moving window, that $\delta p(t,\tau)$ is stationary.
Upon initiating the analysis of the distribution of oil price returns, the
mean, standard deviation, skewness, and kurtosis of the return series are
calculated (see Table 1). It is easy to show that the skewness of a Gaussian
function is zero so that the negative value of skewness, $\lambda=-0.606749$,
is a hallmark of departure of the PDFs from the Gaussian distribution (such as
a leptokurtic distribution), which confirms the existence of intermittency in
the fluctuations. On the other hand the large value of kurtosis,
$\kappa=9.55225$, with respect to Gaussian kurtosis $(\kappa=3)$, show that
the tails of the return distribution are fatter than the Gaussian ones.
Table 1: Mean, standard deviation, skewness, and kurtosis of the oil price returns. Mean | Standard Deviation | Skewness | Kurtosis
---|---|---|---
0.0124274 | 0.729173 | -0.606749 | 9.55225
## III The MF-DFA analysis
The MF-DFA method is a modified version of detrended fluctuation analysis to
detect multifractal properties of a time series. Omitting unnecessary details,
a brief summary of the method for calculating MF-DFA based on fractal concepts
can be formulated in five steps. We take the price series $p_{k}$ with the
size of $N$ and follow the steps as follow:
$\bullet$ step 1: Determine the ” profile”
$Y(i)=\sum_{k=1}^{i}[p_{k}-\langle p\rangle],\hskip 28.45274pti=1,....,N\,$
(1)
where $\langle p\rangle$ is the mean of the series. Subtraction of the mean
$\langle p\rangle$ is not compulsory, since it would be eliminated by the
detrending later in the third step.
$\bullet$ step 2: Divide the profile $Y(i)$ into $N_{s}\equiv int(N/s)$
nonoverlapping segments of equal lengths $s$. Since the length $N$ of the
series is often not a multiple of the considered time scale $s$, a short part
at the end of the profile may remain. In order not to disregard this part of
the series , the same procedure is repeated starting from the opposite end.
$\bullet$ step 3: Calculate the local trend for each of the $2N_{s}$ segments
by a least-square fit of the series. Then determine the variance
$F^{2}(s,\nu)\equiv\frac{1}{s}\sum_{i=1}^{s}\\{Y[(\nu-1)s+i]-y_{\nu}(i)\\}^{2},\
$ (2)
for each segment $\nu$, $\nu=1,...,N_{s}$ and
$F^{2}(s,\nu)\equiv\frac{1}{s}\sum_{i=1}^{s}\\{Y[N-(\nu-
N_{s})s+i]-y_{\nu}(i)\\}^{2},\ $ (3)
for $\nu=N_{s}+1,...,2N_{s}$. Here we use the linear fitting polynomial
$y_{\nu}(i)$ in segment $\nu$.
$\bullet$ step 4: Average over all segments to obtain the $q$-th order
fluctuation function, defined as:
$F_{q}(s)\equiv\biggl{\\{}\frac{1}{2N_{s}}\sum_{\nu=1}^{2N_{s}}[F^{2}(s,\nu)]^{q/2}\biggr{\\}}^{1/q},\
$ (4)
where, in general, the index variable $q$ can take any real value except zero.
We repeat steps 2, 3 and 4 on several timescales $s$. It is apparent that
$F_{q}(s)$ will increase with increasing $s$.
$\bullet$ step 5: Determine the scaling behavior of the fluctuation functions
by analyzing log-log plots of $F_{q}(s)$ versus $s$.
Figure 2: The fluctuation function $F_{2}(s)$ as a function of box size $s$
for the returns of the oil index.
A power law relation between $F_{q}(s)$ and $s$ indicates the presence of
scaling: $F_{q}(s)\sim s^{\alpha(q)}$. In general, the exponent $\alpha(q)$
may depend on $q$. For stationary time series such as fGn (fractional Gaussian
noise), $Y(i)$ in Eq. (1), will be a fBm (fractional Brownian motion) signal,
so, $0<\alpha(q=2)<1.0$ sadegh1 . As a result, the fluctuation function
$F_{2}(s)$ shows a scaling behavior, $\alpha(2)$, which is identical to the
well-known Hurst exponent $H$. The Hurst exponent is called the scaling
exponent or correlation exponent, and represents the correlation properties of
the signal. If $H=0.5$, there is no correlation and the signal is an
uncorrelated signal, if $H<0.5$, the signal is anticorrelated, if $H>0.5$,
there is positive correlation in the signal. We obtain the following estimate
for the Hurst exponent, $H=0.52\pm 0.02$ as we can see in Fig. (2). Since
$H>0.5$ it is concluded that oil price returns show persistence, i.e, a
certain correlation among consecutive increments. For $s\sim 150$ the
empirical data deviate from the initial scaling behavior, as we can see in
Fig. (2). This indicates that oil price tends to loose its memory after a
period of the order of 200 days or less.
## IV Statistical self-similarity
A set of time series $\delta p(t,\tau)$ is obtained for each value of
nonoverlapping time lag $\tau$. The return of the stochastic variable $\delta
p(t,\tau)$ is said to be self-similar with parameter $\alpha$ $(\alpha\geq)$,
if for any $\lambda$
$\delta p(\tau)\stackrel{{\scriptstyle EL}}{{=}}\lambda^{\alpha}\delta
p(\lambda\tau).\ $ (5)
The relation (5) is interpreted as an equality in law (EL), that is the two
sides of the equation have the same statistical properties. For the associated
cumulative probability distribution, it follows that
$\wp(\delta p(\tau)\leq\rho)=\wp(\lambda^{\alpha}\delta
p(\lambda\tau)\leq\rho),\ $ (6)
for any real $\rho$. This implies for the probability density $P$
$P[\delta p(\tau)]=\lambda^{-\alpha}P_{s}[\lambda^{-\alpha}\delta p_{s}],\ $
(7)
introducing the master PDF $P_{s}$ with $\delta p_{s}=\delta p(\lambda\tau)$.
According to Eq. (7), there is a family of PDFs that can be collapsed to a
single curve $P_{s}$, if $\alpha$ is independent of $\tau$. This is known as
monoscaling in contrast to multifractal scaling observed, e.g., for oil price
returns at macro-scales.
Figure 3: The PDFs of the normalized oil price fluctuations for four
different scales, $(a)$: $\tau=1$, $(b)$: $\tau=20$, $(c)$: $\tau=60$, and
$(d)$: $\tau=200$. For comparison, the Gaussian distribution with the same
variance is depicted(solid line)
To characterize quantitatively the observed stochastic process, we measure the
PDF $P(\delta p)$ of the price fluctuations for $\tau$ ranging from 1 to 200
days. The number of data in each set decreases from the maximum value of 5695
$(\tau=1)$ to the minimum value of 5495 $(\tau=200)$. In Fig.(3) we show the
four selected PDFs (normalized with the variance $<\delta p(\tau)^{2}>^{1/2}$)
for $\tau=1$, $\tau=20$, $\tau=60$, and $\tau=200$ from the top ( left side)
to the bottom (right side) respectively. The distributions lose their
leptokurtic shape, as $\tau$ increases. Due to the lack in correlation among
distant fluctuations, the associated distributions become approximately
Gaussian at macro-scale. The scaling behavior of the distribution at coarser
time scales has two different regimes. At micro-scales (typically shorter than
10 days), correlations between successive price changes are dominant. This may
be due to several reasons, such as oil pipe line damage, weather changes, or
local variations in the internal (US) oil availability. Interestingly, the
PDFs at the micro-scales have the same leptokurtic shape, exhibit monoscaling,
and do not change fundamentally, and resemble closely Lévy distributions, see
Fig. (4). On the other hand, at macro-scale (typically larger than one month)
permanent crisis in the Middle East and north African govern the price drift
and corresponding to a Gaussian regime. This is coincide with a multifractal
feature, as we can see in Fig. (5). However the micro-and macro time scales
regimes can be led to a linear and nonlinear scaling-dependence respectively.
Figure 4: The PDFs of the oil price fluctuations on micro-scales. We can see
that the shape of the PDFs do not change fundamentally as a sequent of the
monofractality. Figure 5: The PDFs of the oil price fluctuations on macro
scales. The PDFs become Gaussian as the time scales is increased. This is
coincide with multifractality feature.
First, let us consider the scaling as defined by the structure functions. The
generalized structure function of order $n$ are simply defined as
$S^{n}(\tau;\pm\infty)=\langle|\delta
p|^{n}\rangle=\int_{-\infty}^{+\infty}|\delta p|^{n}P(\delta p,\tau)d(\delta
p).\ $ (8)
The analysis which follows is also valid for the moments; however, structure
functions are typically calculated for a data series. The arguments do not
apply to structure functions of odd order, which not only may have negative
coefficients, but could in fact even change the sign of the scaling range. The
proof will, however, remain valid for odd orders if the structure functions
are defined with the absolute value of the returns. Using the relation (7) we
obtain
$S^{n}(\tau;\pm\infty)=\lambda^{\zeta_{n}}S^{n}_{s}(\delta p_{s};\pm\infty),\
$ (9)
where the linear function $\zeta_{n}=\alpha n$ refer to the statistical self-
similarity, monoscaling case. On the contrary, in some cases, one may observe
multifractality scaling, in the sense that a nonlinear dependence is observed
on $n$ where $\zeta_{n}=n\alpha(n)$ is a convex function of $n$ and
$\zeta_{n+1}>\zeta_{n}\forall n$. This deviation from strict self-similarity
over all time scale $\tau$, also termed multifractal scaling, is caused by the
intermittent micro-scale structure of turbulence.
To test if the above-mentioned interesting observations in oil price are a
phenomenon related to inherent properties of stochastic processes, structure
functions $S^{n}(\tau)$ for different $\tau$, given by Eq. (8), are computed.
A difficulty that can arise in the experimental determination of the
$\zeta_{n}$ is that for a finite length times series, the integral Eq. (8) is
not sampled over the range $(-\infty;+\infty)$, rather the outlying measured
values of $y$ determine the limit, $[-y;+y]$. Fig. (6) shows some selected
$S^{n}(\tau)$, according to Eq. (8). The slope of the curve gives scaling
exponent $\zeta_{n}$ which can be obtained by fitting a straight line to a
log-log plot in the interval $l\in[1,100]$. Because of increasing of the
statistical errors at the higher orders, such a fitting becomes rather
arbitrary. In Fig. (7) we report the scaling exponents extracted from the
structure functions. The behavior of $\zeta_{n}$ against $n$ shows that
scaling exponents have nonlinear behavior at all scales , say, they are
different from the usual linear $\alpha n$ law.
Figure 6: The structure functions are depicted, as computed from Eq. (8). In
order to obtain the scaling exponents, we take the logarithmic slope of linear
least-square fits (solid line). Figure 7: The scaling exponents $\zeta_{n}$
of the structure functions are depicted versus the corresponding order $n$.
The nonlinear behavior is obvious in the figure.
To apply the rescaling procedure given by Eq. (7) the exponent $\alpha$ is
extracted from the underlying data by a independent technique, kian ; mom .
The standard deviation which is defined by the root of the second-order
structure function, $\sigma(\tau)=[S^{2}(\tau)]^{1/2}$ and has the minimum of
statistical error, exhibits power-law behavior with respect to the increment
distance, $\sigma(\tau)\sim\tau^{\alpha}$ as depicted in Fig. (8).
Figure 8: Standard deviation of oil price returns within the desired range.
A linear least-square fit is carried out to obtain $\alpha$. The
characteristic exponent deduced in this way is $\alpha=0.52\pm 0.031$ which is
in good agreement with the value of $H$ obtained in Section. (III). Fig. (9)
and Fig. (10) show the rescaled sets of PDFs on the micro- and macro-scales
respectively. Evidently the PDFs at micro-scales are self-similar and collapse
with weak scattering on the master PDF, $P_{s}$, when using the characteristic
exponents given above. The corresponding increment distances $\tau$ are all
shorter than 10 days. We may model this PDFs by a Lévy distribution, which
thus turns out to be a successful fit to the distribution of oil price
fluctuations. On the other hand, at macro-scale the PDFs do not show a self-
similar behavior and rather constitute a multicascade process. This occurs at
scales larger than the one month. Because of the resulting multifractal
scaling of the PDFs it is evident that they can not collapse onto a single
curve, see Fig. (10).
## V Lévy distribution model
Lévy-stable laws are a rich class of probability distributions that comprise
fat tails and have many intriguing mathematical properties. They have been
proposed as models for many types of physical and economic systems. There are
several reasons for using Lévy-stable laws to describe complex systems. First
of all, in some cases there are solid theoretical reasons for expecting a non-
Gaussian Lévy stable model, which can be a good fitting to experimental and
numerical results. The second reason is the Generalized Central Limit Theorem
which states that the only possible non-trivial limit of normalized sums of
independent identically distributed terms is Lévy-stable. (Recall that the
classical Central Limit Theorem states that the limit of normalized sums of
independent identically distributed terms with finite variance is Gaussian.)
The third argument for modeling with Lévy-stable distributions is empirical;
many large data sets exhibit fat tails (or heavy tails); for a review see jani
; silv . Such data sets were described by a Casting model based on the log
normal ansatz (in terms of the variance of the Gaussian distribution). To
confirm that model it is convenient to summarize the basic features of the
Lévy stable distribution.
Figure 9: Rescaled PDFs of oil price fluctuations in micro-scales . The Lévy
law is represented by the dashed line. Figure 10: Rescaled PDFs of oil price
fluctuations in a macro-scale. This shows that the collapse of the PDF’s to a
single curve is broken over the scales.
A Lévy process is a time-dependent or position-dependent process that at an
infinitesimal interval has the Lévy distribution of the process variable. The
characteristic function of the Lévy process is
$K_{\mu}(q,s)=\exp(-cs\mid q\mid^{\mu})\,,$ (10)
where $s$ can be a characteristic time or space scale. If $\mu=2$ the Lévy
collapses to the Gaussian distribution. If $\mu=1$ the Lévy becomes a Cauchy
distribution. The original Lévy process is given by its inverse Fourier
transform, i.e.
$P_{\mu}(x,s)=\int dqe^{iqx-cs\mid q\mid^{\mu}}\,,$ (11)
and the symmetric Lévy distribution becomes
$L_{\mu}(\delta x_{\Delta
s})\equiv\frac{1}{\pi}\int_{0}^{\infty}\exp(-\gamma\Delta
sq^{\mu})\cos(q\delta x_{\Delta s})dq\,,$ (12)
where the increment is $\delta x=x_{s}-x_{s-\Delta s}$; here, $0<\mu<2$, and
$\gamma>0$ is a scale factor. The maximum event probability leads to
$P({0})=L_{\mu}(0)=\frac{\Gamma(1/\mu)}{\pi\mu(\gamma\Delta s)^{1/\mu}}\,.$
(13)
The exponent $\mu$ of the best fits is constant at the micro-scales range and
amounts approximately to $\mu\sim 1.92$ which is $\mu=1/\alpha$. Natural
phenomena also investigated where similar findings have been reported include,
for example, financial systems e.g. the Tehran price stock market, where
$\mu\sim 1.36$ noroz , and also physical systems such as the solar wind, where
$\mu\sim 3.3$ hnat . From Fig. (9) we conclude that a central section of Lévy
distributions describe very well the dynamics of the PDFs of oil price
fluctuations at micro-scales. One can see that the rescaled PDFs are
definitely non-Gaussian.
## VI Summary
In this paper, we have presented a statistical analysis of oil price in the
United States for the period of 1986 to 2008. We have applied a generic MF-DFA
method to extract scaling exponent of the fluctuation functions, in
particular, relying on the second order $F_{2}(s)$ which was used in the
rescaling procedure. However, the simple scaling properties that we have found
via an analysis of the PDFs, allow us to detect mono(multi) fractality feature
over all time scales. The presence of intermittency in oil price fluctuations
is manifested by the leptokurtic nature of the PDFs which show increased
probability of large fluctuations compared to that of the Gaussian
distribution. Fluctuations on the macro temporal scales, $\tau>10$ day,
converge toward a Gaussian distribution and are an uncorrelated signal. The
reason may be due to unstable conditions in the Middle East or OPEC’s
decisions oil production. The critical macro scale which was obtained is
different for some financial and physical systems; see for example in Refs.
noroz ; ken . We have also obtained a good collapse onto a single curve for
$\tau<10$, according to the rescaling procedure (7). The proximity of the PDFs
to Lévy distributions is made plausible by a simple model mimicking nonlinear
spectral transfer.
## References
* (1) U. Frisch, Turbulence, (Cambridge University Press, Cambridge,1995).
* (2) D. Biskamp, Magnetohydrodynamic Turbulence, (Cambridge University Press, Cambridge, 2003).
* (3) A. N. Kolmogrov , J. Fluid. Mech . 13, 82 (1962).
* (4) P. Norouzzadeh and G.R. Jafari, Physica A 356, 609 (2005).
* (5) R.Mantegna and H.E. Stanley, An Introduction to Econophysics, (Cambridge University Press, Cambridge, 2000).
* (6) R.Mantegna and H.E. Stanley, Nuture, 376, 46 (1995).
* (7) J.-P. Bouchaud and M. Potters, Theory of Financial Risks, (Cambridge University Press, Cambridge, 2000).
* (8) Abraham C.-L. Chian, Erico L. Rempel, and Colin Rogers, Chaos, Solitons and Fractals 29, 1194 (2006).
* (9) F. Farahpour, Z. Eskandari, A. Bahraminasab, G.R. Jafari, F. Ghasemi, Muhammad Sahimi, and M. Reza Rahimi Tabar, Physica A 385, 601 (2007).
* (10) Ken Kiyono, Zbigniew R. Struzik, and Yoshiharu Yamamoto, Phys.Rev.Lett 96, 068701 (2006).
* (11) V.I. Keilis-Borok and A.A. Soloviev, Nonlinear Dynamics of the Lithosphere and Earthquake Prediction, (Springer, New York, 2003).
* (12) D. Vere-Jones, Math. Geol 9, 407 (1977).
* (13) B. Voight, Nature 332, 125 (1988); Science 243, 200 (1989)
* (14) C.G. Bufe and D.J. Varnes, J. Geophys. Res. 98, 9871 (1993).
* (15) A. Saichev and D. Sornette, Tectonophysics 431, 7 (2007).
* (16) E. Koscielny-Bunde, A. Bunde, S. Havlin, H.E. Roman, Y. Goldreich, and H.-J. Schellnhuber, Phys. Rev. Lett. 81 729 (1998).
* (17) S.M. Barbosa, M.J. Fernandes, and M.E. Silva, Physica A 371, 725 (2006)
* (18) K. Fraedrich, U. Luksch, and R. Blender, Phys. Rev. E 70,037301 (2004).
* (19) V. Plerou, P. Gopikrishnan, X. Gabaix, and H. E. Stanley, Phys. Rev. E 66, 027104 (2002).
* (20) S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, and Y. Dodge, Nature (London) 381, 767 (1996).
* (21) J. F. Muzy, J. Delour, and E. Bacry, Eur. Phys. J. B 17, 537 (2000).
* (22) M. Momeni and W.-C. Müller, Phys. Rev. E 77, 056401 (2008).
* (23) K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, and H.E. Stanley, Phys. Rev. E. 64, 011114 (2001).
* (24) C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger, Phys. Rev. E 49, 1685 (1994); S.M. Ossadnik, S.B. Buldyrev, A.L. Goldberger, S. Havlin, R.N. Mantegna, C.K. Peng, M. Simons, and H.E. Stanley, Biophys. J. 67, 64 (1994).
* (25) S.V. Buldyrev, A.L Goldberger, S. Havlin, R.N. Mantegna, M.E. Matsa, C.K. Peng, M. Simons, and H.E. Stanley, Phys. Rev. E 51,5084 (1995)
* (26) S.V. Buldyrev, N.V. Dokholyan, A.L. Goldberger, S. Havlin, C.K. Peng, H.E. Stanley, and G.M. Viswanathan, Physica A 249,430 (1998).
* (27) Ken Kiyono, Zbigniew R. Struzik, Naoko Aoyagi, Seiichiro Sakata, Junichiro Hayano, and Yoshiharu Yamamoto, Phys.Rev.Lett. 93,178103 (2004).
* (28) S. Blesic, S. Milosevic, D. Stratimirovic, and M. Ljubisavljevic, Physica A 268, 275 (1999).; S. Bahar, J.W. Kantelhardt, A. Neiman, H.A. Rego, D.F. Russell, L. Wilkens, A. Bunde,and F. Moss, Europhys. Lett. 56,454 (2001).
* (29) J.M. Hausdorff, S.L. Mitchell, R. Firtion, C.K. Peng, M.E. Cudkowicz, J.Y. Wei, and A.L. Goldberger, J. Appl. Physiology 82,262 (1997).
* (30) K. Ivanova and M. Ausloos, Physica A 274,349 (1999) ; P. Talkner and R.O. Weber, Phys. Rev. E 62,150 (2000).
* (31) K. Ivanova, M. Ausloos, E. Clothiaux, T.P. Ackerman, Europhys. Lett. 52, 40 (2000).
* (32) B.D Malamud, D.L. Turcotte, J. Stat. Plan. Infer. 80,173 (1999).
* (33) C.L. Alados, M.A. Huffman, Ethnology 106,105 (2000).
* (34) Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.K. Peng, and H.E. Stanley, Phys. Rev. E 60, 1390 (1999); N. Vandewalle, M. Ausloos, and P. Boveroux, Physica A 269,170 (1999).
* (35) J.W. Kantelhardt, R. Berkovits, S. Havlin, and A. Bunde, Physica A 266, 461 (1999); N. Vandewalle, M. Ausloos, M. Houssa, P.W. Mertens, and M. Heyns, Appl. Phys. Lett. 74,1579 (1999).
* (36) M. Sadegh Movahed, G.R. Jafari, F. Ghasemi, S. Rahvar, and M. Rahimi Tabar, J. Stat. Mech. P02003(2006).
* (37) K. Kiyani, S. C. Chapman and B. Hnat, Phys. Rev. E 74, 051122 (2006).
* (38) K. Kiyani, S. C. Chapman, B. Hnat and R. M. Nicol, Phys. Rev. Lett 98, 211101 (2007)
* (39) Nigel Goldenfeld, Phys. Rev. Lett. 96, 044503 (2006).
* (40) Data obtained from USA Energy Information Administration: http://tonto.eia.doe.gov/dnav/pet .
* (41) M. Sadegh Movahed, Evalds Hermanis, Physica A , doi:10.1016/j.physa.2007.10.007 (2007).
* (42) A. Janicki, and A. Weron , Simulation and Chaotic Behavior of a-Stable Stochastic Processes, (Marcel Dekker, New York, 1994).
* (43) S. D. Silva, R. Matsusita, I. Gleria, A. Figueiredo, and P. Rathie , Communications in Nonlinear Science and Numerical Simulation 10, 365 (2005).
* (44) B. Hnat, S. C. Chapman, G. Rowlands, N. W. Watkins, and W.M. Farrell, Geophys. Res. Lett. 29, 86 (2002).
|
arxiv-papers
| 2008-09-06T06:54:35
|
2024-09-04T02:48:57.674286
|
{
"license": "Public Domain",
"authors": "M. Momeni, I. Kourakis, K. Talebi",
"submitter": "Mehdi Momeni",
"url": "https://arxiv.org/abs/0809.1139"
}
|
0809.1144
|
# Bialgebra structures of 2-associative algebras
K. DEKKAR and A. MAKHLOUF Khadra DEKKAR, Université de Sétif, Faculté des
Sciences et Techniques, Algeria dekkar_na@yahoo.fr Abdenacer MAKHLOUF
(corresponding author), Université de Haute Alsace, Laboratoire de
Mathématiques, Informatique et Applications, Mulhouse, France
Abdenacer.Makhlouf@uha.fr
###### Abstract.
This work is devoted to study new bialgebra structures related to
2-associative algebras. A 2-associative algebra is a vector space equipped
with two associative multiplications. We discuss the notions of 2-associative
bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed
by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem,
the second was suggested by Loday and the third is a variation of the second
one. The main results of this paper are the construction of 2-associative
bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative
algebra and the classification of these structures in low dimensions.
## Introduction
The aim of this work is to study some new algebraic structures related to
bialgebras and 2-associative algebras. The motivation comes from an extension
to non-cocommutative situation of the Cartier–Milnor–Moore theorem. Namely, we
discuss 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. We provide,
using Kaplansky’s bialgebras (see [8]), a construction of $2$-associative
bialgebras and 2-bialgebras, starting from any two $n$-dimensional associative
algebras. Also, we establish some properties and classifications in low
dimensions.
A bialgebra is a $\mathbb{K}$-vector space $V$, where $\mathbb{K}$ is a field,
equipped with an algebra structure given by a multiplication $\mu$ and a unit
$\eta$ and a coalgebra structure given by a comultiplication $\Delta$ and a
counit $\varepsilon$, such that there is a compatibility condition between the
two structures expressed by the fact that $\Delta$ and $\varepsilon$ are
algebra morphisms, that is for $x,y\in V$
$\Delta(\mu(x\otimes
y))=\Delta(x)\bullet\Delta(y)\quad\text{and}\quad\varepsilon(\mu(x,y))=\varepsilon(x)\varepsilon(y).$
The multiplication $\bullet$ on $V\otimes V$ is the usual multiplication on
tensor product
$\left(x\otimes y\right)\bullet\left(x^{\prime}\otimes
y^{\prime}\right)=\mu\left(x\otimes x^{\prime}\right)\otimes\mu\left(y\otimes
y^{\prime}\right)$
We assume also that the unit is sent to the unit by the comultiplication. A
bialgebra is said to be a Hopf algebra if the identity on $V$ has an inverse
for the convolution product defined by
(0.1) $f\star g:=\mu\circ(f\otimes g)\circ\Delta\ $
Let $(C,\Delta,\varepsilon)$ be a graded coalgebra over a field $\mathbb{K}$,
that is $\mathcal{C}=\oplus_{k>0}\mathcal{C}^{k}$ such that
$\Delta(\mathcal{C}^{k})\subseteq\sum_{i+j=k}\mathcal{C}^{i}\otimes\mathcal{C}^{j}$
and $\varepsilon(C^{k})=0\ \forall k\neq 0.$ The coalgebra is said to be
graded connected if in addition $\mathcal{C}^{0}\cong\mathbb{K}.$ A graded
coalgebra $\mathcal{C}=\oplus_{k>0}\mathcal{C}^{k}$ is said to be cofree if it
satisfies the following universal property. Given a graded coalgebra
$\mathcal{Q}=\oplus_{k>0}\mathcal{Q}^{k}$ and a linear map
$\varphi:\mathcal{Q}\rightarrow\mathcal{C}^{1}$ with
$\varphi(\mathcal{Q}^{k})=0$ when $k\neq 1$, there is a unique morphism of
graded coalgebras $\widehat{\varphi}:\mathcal{Q}\rightarrow\mathcal{C}$ such
that $\pi\circ\widehat{\varphi}=\varphi$ where
$\pi:\mathcal{C}\rightarrow\mathcal{C}^{1}$ is the canonical projection. We
refer to (see [9], [14], [12]) for the basic knowledge about Hopf algebras.
The Cartier–Milnor–Moore theorem (see [13],[4]) states that, over a field of
characteristic zero, any connected cocommutative bialgebra $\mathcal{H}$ is of
the form $U(Prim(\mathcal{H}))$, where the primitive part $Prim(\mathcal{H})$
is viewed as a Lie algebra, and $U$ is the universal enveloping functor. Thus
there is an equivalence of categories between the cofree cocommutative
bialgebras and Lie algebras.
In (see [10]), J.-L. Loday and M. Ronco change the category of connected
cocommutative bialgebras for the category of 2-associative bialgebras, the Lie
algebras by a $B_{\infty}$-algebras and the universal enveloping functor by a
functor $U2$
$U2:B_{\infty}\text{-algebras}\longrightarrow\text{2-associative algebras.}$
A non-differential $B_{\infty}$-algebra is defined by $(p+q)$-ary operations
for any pair of positive integers $p,q$ satisfying some relations. It may be
viewed also as a deformation of the shuffle algebra $T^{sh}(V)$ where $V$ is
the underlying vector space. Loday and Ronco proved that if $\mathcal{H}$ is a
bialgebra over a field $\mathbb{K}$ the following assertions are equivalent
1. (1)
$\mathcal{H}$ is a connected $2$-associative bialgebra,
2. (2)
$\mathcal{H}$ is isomorphic to $U2(Prim\ \mathcal{H})$ as a $2$-associative
bialgebra,
3. (3)
$\mathcal{H}$ is cofree among the connected coalgebras.
Therefore, any cofree Hopf algebra is of the form $U2(R)$ where $R$ is
$B_{\infty}\text{-algebras}$. J.-L. Loday gives in (see [11]) a general
framework to work with triples of operads including the previous case.
The paper is organized as follows: Section 1 is dedicated to introduce the
definitions of 2-associative algebras, 2-associative bialgebras, infinitesimal
bialgebras, 2-bialgebras and 2-2-bialgebras, and to describe some basic
properties of these objects. In Section 2, we recall how I. Kaplansky (see
[8]) constructed from an associative algebra two bialgebras. We show that they
leads to a large class of 2-associative bialgebras, 2-bialgebras and
2-2-bialgebras. In Section 3, we establish classifications in dimensions 2 and
3 of bialgebra, bialgebras which carry also infinitesimal bialgebra structure.
Therefore, we enumerate and describe 2 and 3-dimensional 2-associative
bialgebras, 2-bialgebra and 2-2-bialgebras.
## 1\. Definitions and Properties
In this section, we introduce the definitions of bialgebras, 2-bialgebras and
2-2-bialgebras, which requires the notions of $2$-associative algebra,
bialgebra and infinitesimal bialgebra.
### 1.1. $2$-Associative algebras
We review first briefly the 2-associative algebra structure and some
properties, in particular concerning the free 2-associative algebra. A theory
of this structure was developed by J.-L. Loday and M. Ronco in (see [10]).
###### Definition 1.1.
A _2-associative algebra_ over $\mathbb{K}$ is a vector space equipped with
two associative operations. A 2-associative algebra is said to be unital if
there is a unit $1$ which is a unit for both operations.
One may define the notions of 2-associative monoid, 2-associative group,
2-associative monoidal category, 2-associative operad … The free 2-associative
algebra over the vector space $V$ is the 2-associative algebra $2as(V)$ such
that any linear map from $V$ to a 2-associative algebra $\mathcal{A}$ has a
unique extension to a homomorphism of 2-associative algebras from $2as(V)$ to
$\mathcal{A}$. An explicit description in terms of planar trees is given in
(see [10]). The free 2-associative algebra on one generator can be identified
with the non-commutative polynomials over the planar (rooted) trees. The
operad of 2-associative algebras is studied in (see [10]), and it turns out
that it is a Koszul operad.
### 1.2. Infinitesimal bialgebras
An infinitesimal bialgebra is a bialgebra where the compatibility condition is
modified. The comultiplication is no more an algebra morphism. The condition
is $\Delta\circ\mu=(\mu\otimes
id)\circ(id\otimes\Delta)+(id\otimes\mu)\circ(\Delta\otimes id).$ This
structure was introduced first by S. Joni and G.-C. Rota in (see [7]). The
basic theory was developed by M. Aguiar in (see [1] [2]). He also showed their
intimate link to Rota–Baxter algebras, Loday’s dendriform algebras, pre-Lie
structure and introduced the associative classical Yang–Baxter equation (see
[3]). The following definition was introduced by J.-L. Loday and M. Ronco.
###### Definition 1.2.
A _unital infinitesimal bialgebra_
$\left(V,\mu,\eta,\Delta,\varepsilon\right)$ is a vector space $V$ equipped
with a unital associative multiplication $\mu$ and a counital coassociative
comultiplication $\Delta$ which are related by the unital infinitesimal
relation
(1.1) $\Delta\left(\mu(x\otimes y)\right)=\left(x\otimes
1\right)\bullet\Delta\left(y\right)+\Delta\left(x\right)\bullet\left(1\otimes
y\right)-x\otimes y$
where $1=\eta(1)$ and $x,y\in V.$
###### Remark 1.3.
The unital infinitesimal relation 1.1 may be written as follows :
(1.2) $\Delta\circ\mu=(\mu\otimes
id_{V})\circ(id_{V}\otimes\Delta)+(id_{V}\otimes\mu)\circ(\Delta\otimes
id_{V})-id_{V}\otimes id_{V}$
or
$\Delta\circ\mu=\left(\mu\otimes\mu\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\mathit{\circ}\left(i_{1}\otimes\Delta+\Delta\otimes
i_{2}-i_{1}\otimes i_{2}\right)$
where $\tau$ is the usual flip and
$i_{1}:V\rightarrow V\otimes V\ \ \ x\rightarrow x\otimes 1$
$i_{2}:V\rightarrow V\otimes V\ \ y\rightarrow 1\otimes y$
The maps $i_{1}$ and $i_{2}$ are algebra morphisms.
###### Remark 1.4.
The condition 1.2 applied to $1\otimes 1$ implies that $\Delta(1)=1\otimes 1.$
The polynomial algebra and the tensor algebra may be endowed with
infinitesimal bialgebra structure, (see [10]).
###### Example 1.5.
The polynomial algebra $\mathbb{K}[x]$ is a unital infinitesimal bialgebra
with a comultiplication $\Delta$ defined by
$\Delta(x^{n})=\sum_{p=0}^{n}x^{p}\otimes x^{n-p}$.
###### Example 1.6.
The tensor algebra over a vector space $V$ which is the space
$T(V)=\mathbb{K}\oplus V\oplus V^{\otimes 2}\oplus\cdots\oplus V^{\otimes
n}\oplus\cdots$
equipped with the concatenation multiplication given by $v_{1}\cdots
v_{i}\otimes v_{i+1}\cdots v_{n}\mapsto v_{1}\cdots v_{i}v_{i+1}\cdots v_{n}\
,$ and the deconcatenation comultiplication given by $\Delta(v_{1}\cdots
v_{n})=\sum_{i=0}^{i=n}v_{1}\cdots v_{i}\otimes v_{i+1}\cdots v_{n}\ $ is a
unital infinitesimal bialgebra.
In (see [10]), the authors showed that any connected unital infinitesimal
bialgebra is isomorphic to $T(V)$ for some space $V$.
We may extend the unital infinitesimal bialgebra structure to a situation
where the multiplication $\mu$ is no more unital and the comultiplication
$\Delta$ is no more counital by considering a triple
$\left(V,\mu,\Delta\right)$ with the infinitesimal compatibility condition
(1.2).
The unital infinitesimal relation differs from the infinitesimal relation used
by S. Joni and G.-C. Rota by the presence of the term $-x\otimes y$. The
original definition assumes that the comultiplication is a derivation. One may
generalize the unital infinitesimal condition, with $\theta\in\mathbb{K}$ (see
[5]) to
(1.3) $\Delta\circ\mu=(\mu\otimes
id_{V})(id_{V}\otimes\Delta)+(id_{V}\otimes\mu)(\Delta\otimes id_{V})-\theta\
id_{V}\otimes id_{V}.$
The case $\theta=0$ is the relation defined by Joni and Rota, and $\theta=1$
recovers the notion of unital infinitesimal bialgebra.
The convolution product (0.1) defined by a unital infinitesimal bialgebra or a
generalized infinitesimal algebra (condition 1.3) still endows the set
$Hom(V,V)$ of $\mathbb{K}$-linear endomorphisms of $V$, with a structure of
associative algebra where $\varepsilon\circ\eta$ is the unit. Any
infinitesimal bialgebra induces over the algebra $(Hom(V,V),\circ)$, where
$\circ$ denotes the composition of maps, a Rota–Baxter structure (see [5]),
that is a $\mathbb{K}$-linear map $\phi:Hom(V,V)\rightarrow Hom(V,V)$
satisfying, for all $f,g\in Hom(V,V)$,
$\phi(f)\circ\phi(g)-\phi(f\circ g)=\phi(\phi(f)\circ g+f\circ\phi(g)).$
One may consider $\phi(f)=id_{V}\star f$ or $\phi(f)=f\star id_{V}$, where
$\star$ denotes the convolution product.
Aguiar showed also that to any infinitesimal bialgebra $(\theta=0)$,
corresponds a preLie algebra (see [1][theorem 3.2]). A preLie algebra is a
vector space $V$ together with a bilinear map $m$ satisfying, for all
$x,y,z\in V$, the following condition
$m(x,m(y,z))-m(m(x,y),z)=m(y,m(x,z))-m(m(y,x),z).$
The commutator $[x,y]=m(x,y)-m(y,x)$ defines a Lie algebra on $V$. The
infinitesimal algebra $(V,\mu,\Delta)$ induces the preLie multiplication
defined, using Sweedler notation, for $x,y\in V$ by
$m(x,y)=\sum_{(y)}{\mu(\mu(y_{(1)}\otimes x)\otimes y_{(2)})}.$
### 1.3. $2$-Associative Bialgebras
A 2-associative bialgebra is defined as follows.
###### Definition 1.7.
A _2-associative bialgebra_
$\mathcal{B}2as=\left(V,\mu_{1},\mu_{2},\eta,\Delta,\varepsilon\right)$ is a
vector space $V$ equipped with two multiplications $\mu_{1}$ and $\mu_{2}$, a
unit $\eta$, a comultiplication $\Delta$ and a counit $\varepsilon$ such that
1. (1)
$\left(V,\mu_{1},\eta,\Delta,\varepsilon\right)$ is a bialgebra,
2. (2)
$\left(V,\mu_{2},\eta,\Delta,\varepsilon\right)$ is a unital infinitesimal
bialgebra.
One may define in a similar way a 2-associative Hopf algebra and a unital
infinitesimal Hopf algebra.
Let $\left(V,\mu_{1},\mu_{2},\eta,\Delta,\varepsilon\right)$ and
$\left(V^{\prime},\mu_{1}^{\prime},\mu^{\prime},\eta_{2}^{\prime},\Delta^{\prime},\varepsilon^{\prime}\right)$
be two 2-associative bialgebras. A linear map $f:V\rightarrow V^{\prime}$ is a
morphism of 2-associative bialgebras if
$\mu_{1}^{\prime}\circ\left(f\otimes
f\right)=f\circ\mu_{1},\quad\quad\mu_{2}^{\prime}\circ\left(f\otimes
f\right)=f\circ\mu_{2},\quad\quad f\circ\eta=\eta^{\prime},\quad$ $(f\otimes
f)\circ\Delta=\Delta^{\prime}\circ
f\quad\text{and}\quad\varepsilon^{\prime}=\varepsilon\circ f$
### 1.4. $2$-Bialgebras and $2$-$2$-Bialgebras
In this section, we set up the definitions of $2$-bialgebra and
$2$-$2$-bialgebra, and we give some basic properties and examples.
###### Definition 1.8.
A _2-bialgebra_
$\mathcal{B}2=\left(V,\mu_{1},\mu_{2},\eta,\Delta_{1},\Delta_{2},\varepsilon_{1},\varepsilon_{2}\right)$
is a vector space V equipped with two multiplications $\mu_{1}$ , $\mu_{2}$,
one unit $\eta$, two comultiplications $\Delta_{1},$ $\Delta_{2},$ two counits
$\varepsilon_{1},\varepsilon_{2}$ such that
$\left(V,\mu_{1},\eta,\Delta_{1},\varepsilon_{1}\right)$,
$\left(V,\mu_{2},\eta,\Delta_{2},\varepsilon_{2}\right)$,
$\left(V,\mu_{1},\eta,\Delta_{2},\varepsilon_{2}\right)$ and
$\left(V,\mu_{2},\eta,\Delta_{1},\varepsilon_{1}\right)$ are bialgebras.
###### Remark 1.9.
The condition could be expressed by
1. (1)
$\mu_{1}$ is compatible with $\Delta_{1}$ and $\Delta_{2}.$
2. (2)
$\mu_{2}$ is compatible with $\Delta_{2}$ and $\Delta_{1}.$
3. (3)
$\mathit{\varepsilon_{1}\circ\mu_{1}=\varepsilon_{1}\otimes\varepsilon_{1}}$
and
$\mathit{\varepsilon_{1}\circ\mu_{2}=\varepsilon_{1}\otimes\varepsilon_{1}}$
4. (4)
$\mathit{\varepsilon}_{2}\mathit{\circ\mu_{1}=\varepsilon_{2}}\otimes\mathit{\varepsilon}_{2}$
and
$\mathit{\varepsilon}_{2}\mathit{\circ\mu_{2}=\varepsilon_{2}}\otimes\mathit{\varepsilon}_{2}$
Note that there is no relation assumed between $\Delta_{1}$ and
$\varepsilon_{2}$ (resp. $\Delta_{2}$ and $\varepsilon_{1}$).
A 2-bialgebra is called of _type_ (1-1) (resp. of type (2-2)) if the two
multiplications and the two comultiplications are identical (resp. distinct).
A 2-bialgebra is called of _type_ (1-2) (resp. of type (2-1)) if the two
multiplications are identical (resp. distinct) and the two comultiplications
are distinct (resp. identical).
Let
$\left(V,\mu_{1},\mu_{2},\eta,\Delta_{1},\Delta_{2},\varepsilon_{1},\varepsilon_{2}\right)$
and
$\left(V^{\prime},\mu_{1}^{\prime},\mu_{2}^{\prime},\eta^{\prime},\Delta_{1}^{\prime},\Delta_{2}^{\prime},\varepsilon_{1}^{\prime},\varepsilon_{2}^{\prime}\right)$
be two 2-bialgebras. A linear map $f:V\rightarrow V^{\prime}$ is a 2-bialgebra
morphism if
$\mu_{1}^{\prime}\circ\left(f\otimes
f\right)=f\circ\mu_{1},\quad\quad\mu_{2}^{\prime}\circ\left(f\otimes
f\right)=f\circ\mu_{2},\quad\quad f\circ\eta=\eta^{\prime},$ $(f\otimes
f)\circ\Delta_{1}=\Delta_{1}^{\prime}\circ
f,\quad\varepsilon_{1}=\varepsilon_{1}^{\prime}\circ f,\quad(f\otimes
f)\circ\Delta_{2}=\Delta_{2}^{\prime}\circ
f,\quad\varepsilon_{2}=\varepsilon_{2}^{\prime}\circ f.$
###### Example 1.10.
Let $\left(V,\mu_{1},\eta,\Delta,\varepsilon\right)$ and
$\left(V,\mu_{2},,\eta,\Delta,\varepsilon\right)$ be two bialgebras over the
vector space spanned by $\left\\{1,x,y\right\\}$, with $,\eta(1)=1$, and
defined by
$\mu_{1}\left(x\otimes x\right)=x,\quad\mu_{1}\left(y\otimes
y\right)=y,\quad\mu_{1}\left(x\otimes y\right)=\mu_{1}\left(y\otimes
x\right)=0,$ $\mu_{2}\left(x\otimes x\right)=x,\quad\mu_{2}\left(y\otimes
y\right)=\mu_{2}\left(x\otimes y\right)=\mu_{2}\left(y\otimes x\right)=0,$
$\Delta\left(1\right)=1\otimes 1,\quad\Delta\left(x\right)=x\otimes
x,\quad\Delta\left(y\right)=y\otimes 1+1\otimes y,$
$\varepsilon\left(1\right)=1,\quad\varepsilon\left(x\right)=1,\quad\varepsilon\left(y\right)=0.$
We assume that $\eta$, , $\eta(1)=1$, is a unit for both $\mu_{1}$ and
$\mu_{2}$.
Then $\left(V,\mu_{1},\mu_{2},\eta,\Delta,\Delta,\varepsilon\right)$ is a
2-bialgebra.
###### Remark 1.11.
Let $\left(V,\mu,\eta,\Delta,\varepsilon\right)$ be a bialgebra then
$\left(V,\mu,\mu,\eta,\Delta,\Delta,\varepsilon\right)\quad\text{ and
}\quad\left(V,\mu,\mu^{op},\eta,\Delta,\Delta^{cop},\varepsilon\right)$
are 2-bialgebras, where $\mu^{op}(x\otimes y)=\mu(y\otimes x)$ and
$\Delta^{cop}(x)=\tau\circ\Delta(x)$, with $\tau(x\otimes y)=y\otimes x.$ The
first 2-bialgebra is of type (1,1) and the second is of type (2,2) if the
bialgebra is neither commutative nor cocommutative.
We introduce here the notion of 2-2-bialgebra, which is a variation of the
previous structure.
###### Definition 1.12.
A _2-2-bialgebra_
$\mathcal{B}22=\left(V,\mu_{1},\mu_{2},\eta,\Delta_{1},\Delta_{2},\varepsilon_{1},\varepsilon_{2}\right)$
is a vector space V equipped with two multiplications $\mu_{1}$, $\mu_{2}$,
two comultiplications $\Delta_{1},$ $\Delta_{2}$ , two counits
$\varepsilon_{1},\varepsilon_{2}$ and one unit $\eta$, such that
1. (1)
$\left(V,\mu_{1},\eta,\Delta_{1},\varepsilon_{1}\right)$ and
$\left(V,\mu_{2},\eta,\Delta_{2},\varepsilon_{2}\right)$ are bialgebras,
2. (2)
$\left(V,\mu_{1},\eta,\Delta_{2},\varepsilon_{2}\right)$ and
$\left(V,\mu_{2},\eta,\Delta_{1},\varepsilon_{1}\right)$ are unital
infinitesimal bialgebras.
A 2-bialgebra is called of type (1-1) (resp. of type (2-2)) if the two
multiplications and the two comultiplications are identical (resp. distinct).
A 2-bialgebra is called of type (1-2) (resp. of type (2-1)) if the two
multiplications are identical (resp. distinct) and the two comultiplications
are distinct (resp. identical).
The definition of 2-2-bialgebra morphism is similar to 2-bialgebra morphism.
###### Example 1.13.
We provide the following 3-dimensional example of 2-2-bialgebra of type (2,2).
We consider a basis $\\{e_{1},e_{2},e_{3}\\}$ of $\mathbb{K}^{3}$. The 7-uple
$\left(\mathbb{K}^{3},\mu_{1},\mu_{2},\eta,\Delta_{1},\Delta_{2},\varepsilon_{1},\varepsilon_{2}\right),$
is defined as follows :
the unit for both multiplications $\eta(1)=e_{1}$,
the multiplications
$\displaystyle\mu_{1}\left(e_{1}\otimes e_{i}\right)$
$\displaystyle=\mu_{1}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3,$
$\displaystyle\mu_{1}\left(e_{j}\otimes e_{2}\right)$
$\displaystyle=\mu_{1}\left(e_{2}\otimes e_{j}\right)=e_{j}\ \ j=2,3,$
$\displaystyle\mu_{1}\left(e_{3}\otimes e_{3}\right)$ $\displaystyle=e_{3}.$
$\displaystyle\mu_{2}\left(e_{1}\otimes e_{i}\right)$
$\displaystyle=\mu_{2}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3,$
$\displaystyle\mu_{2}\left(e_{j}\otimes e_{2}\right)$
$\displaystyle=\mu_{2}\left(e_{2}\otimes e_{j}\right)=e_{j}\ \ j=2,3,~{}$
$\displaystyle\mu_{2}\left(e_{3}\otimes e_{3}\right)$ $\displaystyle=0,$
the comultiplications and counits
$\displaystyle\Delta_{1}\left(e_{1}\right)$ $\displaystyle=e_{1}\otimes
e_{1};$ $\displaystyle\Delta_{1}\left(e_{2}\right)$
$\displaystyle=e_{1}\otimes e_{2}+e_{2}\otimes e_{1}-e_{2}\otimes e_{2};$
$\displaystyle\Delta_{1}\left(e_{3}\right)$ $\displaystyle=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{2}\otimes e_{3};\ $
$\varepsilon_{1}\left(e_{1}\right)=1;\ \varepsilon_{1}\left(e_{2}\right)=0;\
\varepsilon_{1}\left(e_{3}\right)=0,$
$\displaystyle\Delta_{2}\left(e_{1}\right)$ $\displaystyle=e_{1}\otimes
e_{1};$ $\displaystyle\Delta_{2}\left(e_{2}\right)$
$\displaystyle=e_{1}\otimes e_{2}+e_{2}\otimes e_{1}-e_{2}\otimes e_{2};$
$\displaystyle\Delta_{2}\left(e_{3}\right)$ $\displaystyle=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2};\ $
$\varepsilon_{2}\left(e_{1}\right)=1;\ \varepsilon_{2}\left(e_{2}\right)=0;\
\varepsilon_{2}\left(e_{3}\right)=0.$
## 2\. Constructions
In this Section, we use Kaplansky’s constructions of bialgebras (see [8]) to
built 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. First we
recall briefly the definitions of these bialgebras.
###### Proposition 2.1.
Let $\mathcal{A}=(V,\mu,\eta)$ be a unital associative algebra (where
$e_{2}:=\eta(1)$ being the unit). Let $\widetilde{V}$ be the vector space
spanned by $V$ and $e_{1}$, $\widetilde{V}=span(V,e_{1})$.
We have the bialgebra
$\mathcal{K}_{1}(\mathcal{A}):=(\widetilde{V},\mu_{1},\eta_{1},\Delta_{1},\varepsilon_{1})$
where
the multiplication $\mu_{1}$ is defined by :
$\mu_{1}\left(e_{1}\otimes x\right)=\mu_{1}\left(x\otimes e_{1}\right)=x\ \ \
\ \ \forall x\in\widetilde{V}$ $\mu_{1}\left(x\otimes
y\right)=\mu\left(x\otimes y\right)\ \ \ \ \ \ \ \ \ \ \ \forall x,y\in V,$
the unit $\eta_{1}$ is given by $\eta_{1}(1)=e_{1}$,
the comultiplication $\Delta_{1}$ is defined by :
$\Delta_{1}\left(e_{1}\right)=e_{1}\otimes e_{1}$
$\Delta_{1}\left(x\right)=x\otimes e_{1}+e_{1}\otimes x-e_{2}\otimes x\ \ \ \
\ \ \ \forall x\in V$
and the counit $\varepsilon_{1}$ is defined by :
$\ \varepsilon_{1}\left(e_{1}\right)=1\textit{,
}\varepsilon_{1}\left(x\right)=0\ \ \ \ \ \forall x\in V.$
###### Proof.
Straightforward, (see [8]). ∎
###### Remark 2.2.
The previous construction may be done even with a nonunital algebra.
The second type bialgebra constructed by Kaplansky is given by the following
proposition.
###### Proposition 2.3.
Let $\mathcal{A}=(V,\mu,\eta)$ be a unital associative algebra (where
$e_{2}:=\eta(1)$ being the unit). Let $\widetilde{V}$ be the vector space
spanned by $V$ and $e_{1}$, $\widetilde{V}=span(V,e_{1})$.
We have the bialgebra
$\mathcal{K}_{2}(\mathcal{A}):=(\widetilde{V},\mu_{2},\eta_{2},\Delta_{2},\varepsilon_{2})$
where
the multiplication $\mu_{2}$ is defined by:
$\displaystyle\mu_{2}\left(e_{1}\otimes x\right)$ $\displaystyle=$
$\displaystyle\mu_{2}\left(x\otimes e_{1}\right)=x\quad\forall
x\in\widetilde{V}$ $\displaystyle\mu_{2}\left(x\otimes y\right)$
$\displaystyle=$ $\displaystyle\mu\left(x\otimes y\right)\quad\forall x,y\in
V$
the unit $\eta_{2}$ is given by $\eta_{2}(1)=e_{1}$,
the comultiplication $\Delta_{2}$ defined by :
$\displaystyle\Delta_{2}\left(e_{1}\right)$ $\displaystyle=$ $\displaystyle
e_{1}\otimes e_{1},$ $\displaystyle\Delta_{2}\left(e_{2}\right)$
$\displaystyle=$ $\displaystyle e_{2}\otimes e_{1}+e_{1}\otimes
e_{2}-e_{2}\otimes e_{2}$ $\displaystyle\Delta_{2}\left(x\right)$
$\displaystyle=$ $\displaystyle\left(e_{1}-e_{2}\right)\otimes
x+x\otimes\left(e_{1}-e_{2}\right)\quad\forall x\in V\setminus\\{e_{2}\\},$
and the counit defined by $\varepsilon$:
$\ \varepsilon_{2}\left(e_{1}\right)=1\textit{,
}\varepsilon_{2}\left(x\right)=0\ \ \ \ \ \forall x\in V$
###### Proof.
Straightforward, (see [8]). ∎
### 2.1. Construction of 2-Associative Bialgebras
We construct an $(n+1)$-dimensional 2-associative bialgebra structure from an
arbitrary $n$-dimensional associative algebras.
###### Lemma 2.4.
Let $\mathcal{A}=(V,\mu,\eta)$ be any unital associative algebra. The
bialgebra
$\mathcal{K}_{1}\left(\mathcal{A}\right)=(\widetilde{V},\mu_{1},\eta_{1},\Delta_{1},\varepsilon_{1})$
is a unital infinitesimal bialgebra.
###### Proof.
Since $\mathcal{K}_{1}\left(\mathcal{A}\right)$ is a bialgebra (prop. 2.1),
one has to check only the unital infinitesimal condition. Let $x,y\in V$ and
$\tau$ be the usual flip. We have
$\left(\mu_{1}\otimes\mu_{1}\right)\left(id_{V}\otimes\tau\otimes
id_{V}\right)\left(i_{1}\otimes\Delta_{1}+\Delta_{1}\otimes i_{2}-i_{1}\otimes
i_{2}\right)\left(x\otimes y\right)$
$=\left(\mu_{1}\otimes\mu_{1}\right)\left(id_{V}\otimes\tau\otimes
id_{V}\right)$
$\left(\left(x\otimes e_{1}\right)\otimes\left(y\otimes e_{1}+e_{1}\otimes
y-e_{2}\otimes y\right)+\left(x\otimes e_{1}+e_{1}\otimes x-e_{2}\otimes
x\right)\otimes\left(e_{1}\otimes y\right)-x\otimes y\right)$
$=\mu\left(x\otimes y\right)\otimes e_{1}+e_{1}\otimes\mu\left(x\otimes
y\right)-e_{2}\otimes\mu\left(x\otimes y\right)$
$=\Delta_{1}\left(\mu_{1}\left(x\otimes y\right)\right).$
With similar computation, one gets :
$\left(\mu_{1}\otimes\mu_{1}\right)\left(id_{V}\otimes\tau\otimes
id_{V}\right)\left(i_{1}\otimes\Delta_{1}+\Delta_{1}\otimes i_{2}-i_{1}\otimes
i_{2}\right)\left(e_{1}\otimes x\right)$
$=\Delta_{1}\left(\mu_{1}\left(e_{1}\otimes
x\right)\right)=\Delta_{1}\left(x\right),$
$\left(\mu_{1}\otimes\mu_{1}\right)\left(id_{V}\otimes\tau\otimes
id_{V}\right)\left(i_{1}\otimes\Delta_{1}+\Delta_{1}\otimes i_{2}-i_{1}\otimes
i_{2}\right)\left(e_{1}\otimes e_{1}\right)$
$=\Delta_{1}\left(\mu_{1}\left(e_{1}\otimes
e_{1}\right)\right)=\Delta_{1}\left(e_{1}\right).$
Then $\mathcal{K}_{1}\left(A\right)$ is a unital infinitesimal bialgebra. ∎
###### Remark 2.5.
Let $\mathcal{A}=(V,\mu,\eta)$ be any unital associative algebra. The
bialgebra $\mathcal{K}_{2}\left(\mathcal{A}\right)$ is not a unital
infinitesimal bialgebra, the unital infinitesimal condition is not fulfilled.
###### Remark 2.6.
Let $\mathcal{A}_{2}=(V,\mu_{1},\mu_{2},\eta)$ be a 2-associative algebra then
we have the same coalgebra structure in the associated bialgebra (resp. unital
infinitesimal bialgebra) attached to each unital associative algebra.
###### Proposition 2.7.
Let $\mathcal{A}=(V,\mu,\eta)$ and
$\mathcal{A}^{\prime}=(V,\mu^{\prime},\eta)$ be any two unital associative
algebras over an $n$-dimensional vector space $V$. Let
$\mathcal{K}_{1}\left(\mathcal{A}\right)=(\widetilde{V},\mu_{1},\eta_{1},\Delta_{1},\varepsilon_{1})$
and
$\mathcal{K}_{1}\left(\mathcal{A}^{\prime}\right)=(\widetilde{V},\mu_{1}^{\prime},\eta_{1},\Delta_{1},\varepsilon_{1})$
be the associated bialgebras defined above. Then
$\mathfrak{B}_{1}=(\widetilde{V},\mu_{1},\mu_{1}^{\prime},\eta_{1},\Delta_{1},\varepsilon_{1})$
is a $(n+1)$-dimensional 2-associative bialgebra over the vector space
$\widetilde{V}=span(V,e_{1})$ where $\eta_{1}(1)=e_{1}$.
###### Proof.
Since $\mathcal{K}_{1}\left(\mathcal{A}\right)$ is a bialgebra and by the
previous lemma $\mathcal{K}_{1}\left(\mathcal{A}^{\prime}\right)$ is an
infinitesimal bialgebra, then
$\mathfrak{B}_{1}=(\tilde{V},\mu_{1},\mu_{1}^{\prime},\eta_{1},\Delta_{1},\varepsilon_{1})$
is 2-associative bialgebra. ∎
###### Remark 2.8.
Let $B=\left(V,\mu,\eta,\Delta,\varepsilon\right)$ be a bialgebra, if the
comultiplication satisfies the unital infinitesimal condition 1.1 then
$\mathcal{B}2=\left(V,\mu,\mu,\eta,\Delta,\varepsilon\right)$ is a
2-associative bialgebra.
### 2.2. Construction of 2-Bialgebras
###### Proposition 2.9.
Let $V$ be an $n$-dimensional vector space over $\mathbb{K}$. Let
$\mathcal{A}_{1}=(V,\mu_{1},\eta_{1})$ and
$\mathcal{A}_{2}=(V,\mu_{2},\eta_{2})$ be two unital associative algebras.
Let
$\mathcal{K}_{i}\left(\mathcal{A}_{i}\right)=\left(\widetilde{V},\widetilde{\mu}_{i},\eta,\Delta_{i},\varepsilon\right)\
\ i=1,2$ be the associated bialgebras defined above.
Then
$\mathfrak{B}_{1}=\left(\widetilde{V},\widetilde{\mu}_{1},\widetilde{\mu}_{2},\eta,\Delta_{1},\Delta_{2},\varepsilon\right)\
\ \text{and}\ \
\mathfrak{B}_{2}=\left(\widetilde{V},\tilde{\mu}_{1},\widetilde{\mu}_{2},\eta,\Delta_{1}^{cop},\Delta_{2},\varepsilon\right)$
are two $(n+1)$-dimensional 2-bialgebras on $\widetilde{V}=span(V,e_{1})$,
where $e_{1}=\eta(1).$
###### Proof.
We show that $\mathfrak{B}_{1}$ is a 2-bialgebra. Since
$\mathfrak{B}_{i}\left(\mathcal{A}_{i}\right)$ $i=1,2$ are two bialgebras, one
has to prove the compatibility between $\widetilde{\mu}_{1}$ and $\Delta_{2}$,
then $\widetilde{\mu}_{2}$ and $\Delta_{1}.$
Let $x,y\in V$.
$\bullet\left(\widetilde{\mu}_{1}\otimes\widetilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{2}\otimes\Delta_{2}\right)\left(x\otimes
y\right)=\left(\widetilde{\mu}_{1}\otimes\tilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)$
$\left(\Delta_{2}\left(x\right)\otimes\Delta_{2}\left(y\right)\right)$
$=\left(\widetilde{\mu}_{1}\otimes\widetilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\left(\left(e_{1}-e_{2}\right)\otimes
x+x\otimes\left(e_{1}-e_{2}\right)\right)\otimes$
$\left(\left(e_{1}-e_{2}\right)\otimes
y+y\otimes\left(e_{1}-e_{2}\right)\right)\ $
$=\left(e_{1}-e_{2}\right)\otimes\widetilde{\mu}_{1}\left(x\otimes
y\right)+\widetilde{\mu}_{1}\left(x\otimes
y\right)\otimes\left(e_{1}-e_{2}\right)$
$=\Delta_{2}\circ\widetilde{\mu}_{1}\left(x\otimes y\right).$
Also
$\bullet\left(\widetilde{\mu}_{1}\otimes\widetilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{2}\otimes\Delta_{2}\right)\left(e_{2}\otimes
e_{2}\right)=\Delta_{2}\circ\widetilde{\mu}_{1}\left(e_{2}\otimes
e_{2}\right).$
$\bullet\left(\widetilde{\mu}_{1}\otimes\widetilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{2}\otimes\Delta_{2}\right)\left(e_{2}\otimes
x\right)=\Delta_{2}\circ\widetilde{\mu}_{1}\left(e_{2}\otimes x\right).$
$\bullet$
$\left(\widetilde{\mu}_{1}\otimes\widetilde{\mu}_{1}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{2}\otimes\Delta_{2}\right)\left(x\otimes
e_{2}\right)=\Delta_{2}\circ\widetilde{\mu}_{1}\left(x\otimes e_{2}\right).$
$\bullet\left(\widetilde{\mu}_{2}\otimes\widetilde{\mu}_{2}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{1}\otimes\Delta_{1}\right)\left(x\otimes
y\right)$
$=\left(\widetilde{\mu}_{2}\otimes\widetilde{\mu}_{2}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\left(\Delta_{1}\left(x\right)\otimes\Delta_{1}\left(y\right)\right)$
$=\left(\widetilde{\mu}_{2}\otimes\widetilde{\mu}_{2}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)(\left(e_{1}\otimes x+x\otimes e_{1}-e_{2}\otimes
x\right)\otimes$
$\left(e_{1}\otimes y+y\otimes e_{1}-e_{2}\otimes y\right))$
$=e_{1}\otimes\widetilde{\mu}_{2}\left(x\otimes
y\right)+\widetilde{\mu}_{2}\left(x\otimes y\right)\otimes
e_{1}-e_{2}\otimes\widetilde{\mu}_{2}\left(x\otimes y\right)$
$=\Delta_{1}\circ\widetilde{\mu}_{2}\left(x\otimes y\right)$.
One has also
$\bullet\left(\widetilde{\mu}_{2}\otimes\widetilde{\mu}_{2}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{1}\otimes\Delta_{1}\right)\left(x\otimes
e_{2}\right)=\Delta_{1}\circ\widetilde{\mu}_{2}\left(x\otimes e_{2}\right)$
$\bullet\left(\widetilde{\mu}_{2}\otimes\widetilde{\mu}_{2}\right)\circ\left(id_{V}\otimes\tau\otimes
id_{V}\right)\circ\left(\Delta_{1}\otimes\Delta_{1}\right)\left(e_{2}\otimes
x\right)=\Delta_{2}\circ\widetilde{\mu}_{1}\left(e_{2}\otimes x\right).$
Henceforth, the compatibility between $\widetilde{\mu}_{2}$ and $\Delta_{1}$.
Therefore $\mathfrak{B}_{1}$ is a 2-bialgebra.
Similar proof shows that $\mathfrak{B}_{2}$ is a 2-bialgebra. ∎
We have also the obvious following corollary which gives 2-2-bialgebra
starting with any two unital associative algebras.
###### Corollary 2.10.
Let $V$ be an $n$-dimensional vector space over $\mathbb{K}$. Let
$\mathcal{A}_{1}=(V,\mu_{1},\eta_{1})$ and
$\mathcal{A}_{2}=(V,\mu_{2},\eta_{2})$ be two unital associative algebras.
Let $\mathcal{K}_{1}\left(\mathcal{A}_{i}\right)\ \ i=1,2$ be the associated
bialgebras defined above.
Then
$\mathfrak{B}_{1}=\left(\widetilde{V},\widetilde{\mu}_{1},\widetilde{\mu}_{2},\eta,\Delta_{1},\Delta_{1},\varepsilon\right)$
is a two $(n+1)$-dimensional 2-2-bialgebras on $\widetilde{V}=span(V,e_{1})$,
where $e_{1}=\eta(1).$
###### Proof.
Straightforward. ∎
## 3\. Classification in low dimensions
In this section, we show that for fixed dimension $n$, the 2-associative
bialgebras, 2-bialgebras and 2-2-bialgebras are endowed with a structure of
algebraic variety and a natural structure transport action which describes the
set of isomorphic algebras. Solving such systems of polynomial equations leads
to classifications of such structures. We aim at classifying 2-associative
bialgebras and 2-bialgebras of dimension 2 and 3. First, we establish the
classification of bialgebras and infinitesimal bialgebras. Then, we describe
and enumerate 2 and 3-dimensional 2-associative bialgebras and 2-bialgebras.
Let $V$ be an $n$-dimensional vector space over $\mathbb{K}$. Setting a basis
$\left\\{e_{i}\right\\}_{i=\left\\{1,...,n\right\\}}$ of $V$, a multiplication
$\mu$ (resp. a comultiplication $\Delta$) is identified with its $n^{3}$
structure constants $C_{ij}^{k}\in\mathbb{K}$ (resp. $D_{i}^{jk}$), where
$\mu\left(e_{i}\otimes e_{j}\right)=\sum_{k=1}^{n}C_{ij}^{k}e_{k}$ and
$\Delta\left(e_{i}\right)=\sum_{j,k=1}^{n}D_{i}^{jk}e_{j}\otimes e_{k}$. The
counit $\varepsilon$ is identified to its $n$ structure constants $\xi_{i}$.
We assume that $e_{1}$ is the unit.
A collection
$\\{(C_{ij}^{k},\tilde{C}_{ij}^{k},D_{i}^{jk},\xi_{i}),...i,j,k\in\left\\{1,...,n\right\\}\\}$
represents a 2-associative bialgebra if the underlying multiplications,
comultiplication, and the counit satisfy the appropriate conditions which
translate to following polynomial equations.
$\left(A_{1}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(C_{ij}^{l}C_{lk}^{s}-C_{jk}^{l}C_{il}^{s}\right)=0\\\
C_{1i}^{j}=C_{i1}^{j}=\delta_{ij}\end{array}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(A_{2}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(\tilde{C}_{ij}^{l}\tilde{C}_{lk}^{s}-\tilde{C}_{jk}^{l}\tilde{C}_{il}^{s}\right)=0\\\
\tilde{C}_{1i}^{j}=\tilde{C}_{i1}^{j}=\delta_{ij}\end{array}\right.\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(A_{3}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(D_{s}^{lk}D_{l}^{ij}-D_{s}^{il}D_{l}^{jk}\right)=0\\\
\sum_{l=1}^{n}D_{i}^{jl}\zeta_{l}=\sum_{l=1}^{n}D_{i}^{lj}\zeta_{l}=\delta_{ij}\end{array}\right.\
\ \ \ \ \ \ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(A_{4}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\tilde{C}_{ij}^{l}D_{l}^{ks}-\sum_{r,t,p,q=1}^{n}D_{i}^{rt}D_{j}^{pq}\tilde{C}_{rp}^{k}\tilde{C}_{tq}^{s}=0\\\
D_{1}^{11}=1,D_{1}^{ij}=0\text{ \ \ }\left(i,j\right)\neq\left(1,1\right)\\\
\zeta_{1}=1,\sum_{l=1}^{n}\tilde{C}_{ij}^{l}\zeta_{l}=\zeta_{i}\zeta_{j}\end{array}\right.\
\forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(A_{5}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(D_{j}^{lj}C_{il}^{i}+D_{i}^{il}C_{lj}^{j}-D_{l}^{ij}C_{ij}^{l}\right)=1\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\\
\sum_{l=1}^{n}\left(D_{j}^{lk}C_{il}^{s}+D_{i}^{sl}C_{lj}^{k}-D_{l}^{sk}C_{ij}^{l}\right)=0\text{\
}\left(i,j\right)\neq\left(s,k\right)\end{array}\right.\forall
i,j,k,s\in\left\\{1,...,n\right\\}$
Then, the set of $n$-dimensional 2-associative bialgebras, which we denote by
$2Ass\mathcal{B}_{n}$, carries a structure of algebraic variety imbedded in
$\mathbb{K}^{3n^{3}+n}$ with its natural structure of algebraic variety.
Similarly, a collection
$\\{(C_{ij}^{k},\tilde{C}_{ij}^{k},D_{i}^{jk},\tilde{D}_{i}^{jk},\xi_{i},\tilde{\xi}_{i}),...i,j,k\in\left\\{1,...,n\right\\}\\}$
represents a 2-bialgebra if it satisfies the following system
$\left(B_{1}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(C_{ij}^{l}C_{lk}^{s}-C_{jk}^{l}C_{il}^{s}\right)=0\\\
C_{1i}^{j}=C_{i1}^{j}=\delta_{ij}\end{array}\right.$ $\ \ \forall
i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{2}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(\tilde{C}_{ij}^{l}\tilde{C}_{lk}^{s}-\tilde{C}_{jk}^{l}\tilde{C}_{il}^{s}\right)=0\\\
\tilde{C}_{1i}^{j}=\tilde{C}_{i1}^{j}=\delta_{ij}\end{array}\right.\ $ $\ \ \
\ \ \ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{3}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(D_{s}^{lk}D_{l}^{ij}-D_{s}^{il}D_{l}^{jk}\right)=0\\\
\sum_{l=1}^{n}D_{i}^{jl}\zeta_{l}=\stackrel{{\scriptstyle
n}}{{\sum{l=1}{\sum}}}D_{i}^{lj}\zeta_{l}=\delta_{ij}\end{array}\right.$ $\ \
\ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{4}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\left(\tilde{D}_{s}^{lk}\tilde{D}_{l}^{ij}-\tilde{D}_{s}^{il}\tilde{D}_{l}^{jk}\right)=0\\\
\sum_{l=1}^{n}\tilde{D}_{i}^{jl}\tilde{\zeta}_{l}=\sum_{l=1}^{n}\tilde{D}_{i}^{lj}\tilde{\zeta}_{l}=\delta_{ij}\end{array}\right.$
$\ \ \ \ \ \ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{5}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}C_{ij}^{l}D_{l}^{ks}-\sum_{r,t,p,q=1}^{n}D_{i}^{rt}D_{j}^{pq}C_{rp}^{k}C_{tq}^{s}=0\\\
D_{1}^{11}=1,D_{1}^{ij}=0\text{ \ \ \ \ \ \ \ \ \ \
}\left(i,j\right)\neq\left(1,1\right)\\\
\zeta_{1}=1,\sum_{l=1}^{n}C_{ij}^{l}\zeta_{l}=\zeta_{i}\zeta_{j}\text{ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\end{array}\right.\ \ \forall
i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{6}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\tilde{C}_{ij}^{l}\tilde{D}_{l}^{ks}-\sum_{r,t,p,q=1}^{n}\tilde{D}_{i}^{rt}\tilde{D}_{j}^{pq}\tilde{C}_{rp}^{k}\tilde{C}_{tq}^{s}=0\\\
\tilde{D}_{1}^{11}=1,\tilde{D}_{1}^{ij}=0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \
}\left(i,j\right)\neq\left(1,1\right)\\\
\tilde{\zeta}_{1}=1,\sum_{l=1}^{n}\tilde{C}_{ij}^{l}\tilde{\zeta}_{l}=\tilde{\zeta}_{i}\tilde{\zeta}_{j}\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\end{array}\right.\ \ \ \ \forall
i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{7}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\tilde{C}_{ij}^{l}D_{l}^{ks}-\sum_{r,t,p,q=1}^{n}D_{i}^{rt}D_{j}^{pq}\tilde{C}_{rp}^{k}\tilde{C}_{tq}^{s}=0\\\
\sum_{l=1}^{n}C_{ij}^{l}\tilde{D}_{l}^{ks}-\sum_{r,t,p,q=1}^{n}\tilde{D}_{i}^{rt}\tilde{D}_{j}^{pq}C_{rp}^{k}C_{tq}^{s}=0\end{array}\right.\
\ \ \ \ \ \ \forall i,j,k,s\in\left\\{1,...,n\right\\}$
$\left(B_{8}\right)\left\\{\begin{array}[]{c}\sum_{l=1}^{n}\tilde{C}_{ij}^{l}\zeta_{l}=\zeta_{i}\zeta_{j}\\\
\sum_{l=1}^{n}C_{ij}^{l}\tilde{\zeta}_{l}=\tilde{\zeta}_{i}\tilde{\zeta}_{j}\end{array}\right.\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall
i,j\in\left\\{1,...,n\right\\}$
Then, the set of $n$-dimensional 2-bialgebras, which we denote by
$2\mathcal{B}_{n}$, carries a structure of algebraic variety imbedded in
$\mathbb{K}^{4n^{3}+2n}$.
Similarly, we have an algebraic variety structure on the set of
$n$-dimensional 2-2-bialgebras, which we denote by $22\mathcal{B}_{n}.$
The ”structure transport” action is defined by the action of $GL_{n}(V)$ on
$2Ass\mathcal{B}_{n}$ (similarly on $2\mathcal{B}_{n}$ and
$22\mathcal{B}_{n}$). It corresponds to the change of basis.
Let
$\mathcal{B}=\left(V,\mu_{1},\eta_{1},\mu_{2},\eta_{2},\Delta,\varepsilon\right)$
be a 2-associative bialgebras and $f:V\rightarrow V$ be an invertible
endomorphism, then the action of $f$ on $\mathcal{B}$ transports the
2-associative bialgebra structure into a 2-associative bialgebra
$\mathcal{B}^{\prime}=\left(V,\mu_{1}^{\prime},\eta_{1}^{\prime},\mu_{2}^{\prime},\eta_{2}^{\prime},\Delta^{\prime},\varepsilon^{\prime}\right)$
defined by
$\displaystyle\mu_{1}^{\prime}=f\circ\mu_{1}\circ\left(f^{-1}\otimes
f^{-1}\right)\quad\text{and}\quad\eta_{1}^{\prime}=f\circ\eta_{1}$
$\displaystyle\mu_{2}^{\prime}=f\circ\mu_{2}\circ\left(f^{-1}\otimes
f^{-1}\right)\quad\text{and}\quad\eta_{2}^{\prime}=f\circ\eta_{2}$
$\displaystyle\Delta^{\prime}=(f\otimes f)\circ\Delta\circ
f^{-1}\quad\text{and}\quad\varepsilon^{\prime}=\varepsilon\circ
f^{-1}\mathit{\ }$
### 3.1. Classifications in dimension $2$
The set of 2-dimensional unital associative algebras yields two non-isomorphic
algebras (see [6]). Let $\\{e_{1},e_{2}\\}$ be a basis of $\mathbb{K}^{2}$,
then the algebras are given by the following non-trivial products.
$\bullet\mu_{1}^{2}\left(e_{1}\otimes
e_{i}\right)=\mu_{1}^{2}\left(e_{i}\otimes e_{1}\right)=e_{i},\ i=1,2,\
\mu_{1}^{2}\left(e_{2}\otimes e_{2}\right)=e_{2}\vskip 6.0pt plus 2.0pt minus
2.0pt$
$\bullet\mu_{2}^{2}\left(e_{1}\otimes
e_{i}\right)=\mu_{2}^{2}\left(e_{i}\otimes e_{1}\right)=e_{i},\ i=1,2,\
\mu_{1}^{2}\left(e_{2}\otimes e_{2}\right)=0\vskip 6.0pt plus 2.0pt minus
2.0pt$
In the sequel we consider that all the algebras are unital and the unit $\eta$
corresponds to $e_{1}$.
In the following, we list the coalgebras which, combined with $\mu_{1}$, give
bialgebra structures (up to isomorphism).
$\bullet\Delta_{1,1}^{2}\left(e_{1}\right)=e_{1}\otimes e_{1},\
\Delta_{1,1}^{2}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-2e_{2}\otimes e_{2},$
$\varepsilon_{1,1}^{2}\left(e_{1}\right)=1,\
\varepsilon_{1,1}^{2}\left(e_{2}\right)=0.$
$\bullet\Delta_{1,2}^{2}\left(e_{1}\right)=e_{1}\otimes e_{1},\
\Delta_{1,2}^{2}\left(e_{2}\right)=e_{2}\otimes e_{2},$
$\varepsilon_{1,2}^{2}\left(e_{1}\right)=1,\
\varepsilon_{1,2}^{2}\left(e_{2}\right)=1.$
$\bullet\Delta_{1,3}^{2}\left(e_{1}\right)=e_{1}\otimes
e_{1};\Delta_{1,3}^{2}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2},$
$\varepsilon_{1,3}^{2}\left(e_{1}\right)=1,\
\varepsilon_{1,3}^{2}\left(e_{2}\right)=0.$
We cannot associate to $\mu_{2}^{2}$ neither a bialgebra structure nor an
infinitesimal bialgebra structure.
Direct calculations show that there is only one bialgebra which carries also a
structure of infinitesimal bialgebra. This leads to the following
classification of 2-dimensional 2-associative bialgebras.
###### Proposition 3.1.
Every 2-dimensional 2-associative bialgebra is isomorphic to
$\left(\mathbb{K}^{2},\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,2}^{2},\varepsilon_{1,2}^{2}\right).$
The 2-dimensional 2-bialgebras are given by the following proposition.
###### Proposition 3.2.
Every 2-bialgebra of type (1,1) is isomorphic to one of the following
2-bialgebras
$\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,1}^{2},\Delta_{1,1}^{2},\varepsilon_{1,1}^{2}\right),\
\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,2}^{2},\Delta_{1,2}^{2},\varepsilon_{1,2}^{2}\right),\
\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,3}^{2},\Delta_{1,3}^{2},\varepsilon_{1,3}^{2}\right).$
Every 2-bialgebra of type (1,2) is isomorphic to one of the following
2-bialgebras
$\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,1}^{2},\Delta_{1,2}^{2},\varepsilon_{1,2}^{2}\right),\
\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,1}^{2},\Delta_{1,3}^{2},\varepsilon_{1,1}^{2}\right),\
\left(V,\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,2}^{2},\Delta_{1,3}^{2},\varepsilon_{1,3}^{2}\right).$
There is no 2-bialgebras of type (2,2) and (2,1) in dimension 2.
###### Remark 3.3.
There is only one 2-dimensional 2-2-bialgebra which is given by
$\left(\mathbb{K}^{2},\mu_{1}^{2},\mu_{1}^{2},\eta,\Delta_{1,2}^{2},\Delta_{1,2}^{2},\varepsilon_{1,2}^{2}\right).$
### 3.2. Classifications in dimension $3$
First, we recall the classification of 3-dimensional unital associative
algebras (see [6]). Let $\\{e_{1},e_{2},e_{3}\\}$ be a basis of
$\mathbb{K}^{3}$, then the algebras are given by the following non-trivial
products.
$\bullet\mu_{1}^{3}\left(e_{1}\otimes
e_{i}\right)=\mu_{1}^{3}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3,$
$\mu_{1}^{3}\left(e_{j}\otimes e_{2}\right)=\mu_{1}^{3}\left(e_{2}\otimes
e_{j}\right)=e_{j}\ \ j=2,3,\ \mu_{1}^{3}\left(e_{3}\otimes
e_{3}\right)=e_{3}.$
$\bullet\mu_{2}^{3}\left(e_{1}\otimes
e_{i}\right)=\mu_{2}^{3}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3,$
$\mu_{2}^{3}\left(e_{j}\otimes e_{2}\right)=\mu_{2}^{3}\left(e_{2}\otimes
e_{j}\right)=e_{j}\ \ j=2,3,~{}\mu_{2}^{3}\left(e_{3}\otimes e_{3}\right)=0.$
$\bullet\mu_{3}^{3}\left(e_{1}\otimes
e_{i}\right)=\mu_{3}^{3}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3,$
$\mu_{3}^{3}\left(e_{2}\otimes e_{2}\right)=e_{2}.$
$\bullet\mu_{4}^{3}\left(e_{1}\otimes
e_{i}\right)=\mu_{4}^{3}\left(e_{i}\otimes e_{1}\right)=e_{i},\quad i=1,2,3.$
$\bullet\mu_{5}^{3}\left(e_{1}\otimes
e_{i}\right)=\mu_{5}^{3}\left(e_{i}\otimes e_{1}\right)=e_{i}\ \ i=1,2,3;\ \
\mu_{5}^{3}\left(e_{2}\otimes e_{j}\right)=e_{j}\ \ j=2,3.$
Thanks to computer algebra, we obtain the following coalgebras associated to
the previous algebras in order to obtain a bialgebra structures. We denote the
comultiplications by $\Delta_{i,j}^{3}$ and the counits by
$\varepsilon_{i,j}^{3}$, where $i$ indicates the item of the multiplication
and $j$ the item of the comultiplication which combined with the
multiplication $i$ determine a bialgebra.
For the multiplication $\mu_{1}^{3}$, we have
1. (1)
$\Delta_{1,1}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,1}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,1}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-2e_{3}\otimes e_{3};\
\varepsilon_{1,1}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,1}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,1}^{3}\left(e_{3}\right)=0.$
2. (2)
$\Delta_{1,2}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,2}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,2}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{3};\
\varepsilon_{1,2}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,2}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,2}^{3}\left(e_{3}\right)=0.$
3. (3)
$\Delta_{1,3}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,3}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,3}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}-e_{2}\otimes e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2}-e_{3}\otimes
e_{3};\ \varepsilon_{1,3}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,3}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,3}^{3}\left(e_{3}\right)=0.$
4. (4)
$\Delta_{1,4}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,4}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,4}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}-e_{2}\otimes e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2};\
\varepsilon_{1,4}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,4}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,4}^{3}\left(e_{3}\right)=0$
5. (5)
$\Delta_{1,5}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,5}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,5}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{2}\otimes e_{3};\
\varepsilon_{1,5}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,5}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,5}^{3}\left(e_{3}\right)=0$
6. (6)
$\Delta_{1,6}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,6}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{1,6}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2};\
\varepsilon_{1,6}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,6}^{3}\left(e_{2}\right)=0;\
\varepsilon_{1,6}^{3}\left(e_{3}\right)=0.$
7. (7)
$\Delta_{1,7}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,7}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{1,7}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes
e_{2}-2e_{3}\otimes e_{3};\ \varepsilon_{1,7}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,7}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,7}^{3}\left(e_{3}\right)=0.$
8. (8)
$\Delta_{1,8}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,8}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{1,8}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes
e_{2}-e_{3}\otimes e_{3};\ \varepsilon_{1,8}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,8}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,8}^{3}\left(e_{3}\right)=0.$
9. (9)
$\Delta_{1,9}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,9}^{3}\left(e_{2}\right)=e_{1}\otimes e_{3}+e_{2}\otimes
e_{2}-e_{2}\otimes e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2};\
\Delta_{1,9}^{3}\left(e_{3}\right)=e_{1}\otimes e_{3}+e_{3}\otimes
e_{1}-e_{3}\otimes e_{3};\ \varepsilon_{1,9}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,9}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,9}^{3}\left(e_{3}\right)=0.$
10. (10)
$\Delta_{1,10}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,10}^{3}\left(e_{2}\right)=e_{1}\otimes e_{3}+e_{2}\otimes
e_{2}-e_{2}\otimes e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2}+e_{3}\otimes
e_{3};\ \Delta_{1,10}^{3}\left(e_{3}\right)=e_{1}\otimes e_{3}+e_{3}\otimes
e_{1}-2e_{3}\otimes e_{3};\ \varepsilon_{1,10}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,10}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,10}^{3}\left(e_{3}\right)=0.$
11. (11)
$\Delta_{1,11}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,11}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2}+e_{3}\otimes
e_{1}-e_{3}\otimes e_{2};\ \Delta_{1,11}^{3}\left(e_{3}\right)=e_{2}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{3};\
\varepsilon_{1,11}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,11}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,11}^{3}\left(e_{3}\right)=0.$
12. (12)
$\Delta_{1,12}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,12}^{3}\left(e_{2}\right)=e_{1}\otimes e_{3}+e_{2}\otimes
e_{2}-e_{2}\otimes e_{3};\ \Delta_{1,12}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{2}-e_{3}\otimes e_{3};\
\varepsilon_{1,12}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,12}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,12}^{3}\left(e_{3}\right)=0.$
13. (13)
$\Delta_{1,13}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,13}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}-e_{1}\otimes
e_{3}+e_{2}\otimes e_{1}-2e_{2}\otimes e_{2}+2e_{2}\otimes e_{3}-e_{3}\otimes
e_{1}+2e_{3}\otimes e_{2}-e_{3}\otimes e_{3};\
\Delta_{1,13}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes
e_{2}-2e_{3}\otimes e_{3};\ \varepsilon_{1,13}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,13}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,13}^{3}\left(e_{3}\right)=1.$
14. (14)
$\Delta_{1,14}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,14}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}-e_{1}\otimes
e_{3}+e_{2}\otimes e_{1}-e_{2}\otimes e_{2}+e_{2}\otimes e_{3}-e_{3}\otimes
e_{1}+e_{3}\otimes e_{2};\ \Delta_{1,14}^{3}\left(e_{3}\right)=e_{2}\otimes
e_{3}+e_{3}\otimes e_{2}-e_{3}\otimes e_{3};\
\varepsilon_{1,14}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,14}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,14}^{3}\left(e_{3}\right)=1.$
15. (15)
$\Delta_{1,15}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,15}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{1,15}^{3}\left(e_{3}\right)=e_{3}\otimes e_{3};\
\varepsilon_{1,15}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,15}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,15}^{3}\left(e_{3}\right)=1.$
16. (16)
$\Delta_{1,16}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,16}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{1,16}^{3}\left(e_{3}\right)=e_{2}\otimes e_{2}-e_{2}\otimes
e_{3}-e_{3}\otimes e_{2}+2e_{3}\otimes e_{3};\
\varepsilon_{1,16}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,16}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,16}^{3}\left(e_{3}\right)=1.$
17. (17)
$\Delta_{1,17}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,17}^{3}\left(e_{2}\right)=e_{2}\otimes e_{3}+e_{3}\otimes
e_{2}-e_{3}\otimes e_{3};\ \Delta_{1,17}^{3}\left(e_{3}\right)=e_{3}\otimes
e_{3};\ \varepsilon_{1,17}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,17}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,17}^{3}\left(e_{3}\right)=1.$
18. (18)
$\Delta_{1,18}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{1,18}^{3}\left(e_{2}\right)=e_{2}\otimes e_{1}-e_{3}\otimes
e_{1}+e_{3}\otimes e_{2};\ \Delta_{1,18}^{3}\left(e_{3}\right)=e_{3}\otimes
e_{3};\ \varepsilon_{1,18}^{3}\left(e_{1}\right)=1;\
\varepsilon_{1,18}^{3}\left(e_{2}\right)=1;\
\varepsilon_{1,18}^{3}\left(e_{3}\right)=1.$
For the multiplication $\mu_{2}^{3}$, we have
1. (1)
$\Delta_{2,1}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{2,1}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{2,1}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2};\
\varepsilon_{2,1}^{3}\left(e_{1}\right)=1;\
\varepsilon_{2,1}^{3}\left(e_{2}\right)=0;\
\varepsilon_{2,1}^{3}\left(e_{3}\right)=0.$
2. (2)
$\Delta_{2,2}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{2,2}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{2,2}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}+e_{2}\otimes e_{3}+e_{3}\otimes e_{1};\
\varepsilon_{2,2}^{3}\left(e_{1}\right)=1;\
\varepsilon_{2,2}^{3}\left(e_{2}\right)=0;\
\varepsilon_{2,2}^{3}\left(e_{3}\right)=0.$
3. (3)
$\Delta_{2,3}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{2,3}^{3}\left(e_{2}\right)=e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}-e_{2}\otimes e_{2};\ \Delta_{2,3}^{3}\left(e_{3}\right)=e_{1}\otimes
e_{3}-e_{2}\otimes e_{3}+e_{3}\otimes e_{1}-e_{3}\otimes e_{2}+\lambda
e_{3}\otimes e_{3};\ \varepsilon_{2,3}^{3}\left(e_{1}\right)=1;\
\varepsilon_{2,3}^{3}\left(e_{2}\right)=0;\
\varepsilon_{2,3}^{3}\left(e_{3}\right)=0.$
For the multiplication $\mu_{3}^{3}$, we have
1. (1)
$\Delta_{3,1}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{3,1}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{3,1}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes e_{2};\
\varepsilon_{3,1}^{3}\left(e_{1}\right)=1;\
\varepsilon_{3,1}^{3}\left(e_{2}\right)=1;\
\varepsilon_{3,1}^{3}\left(e_{3}\right)=0.$
2. (2)
$\Delta_{3,2}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{3,2}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{3,2}^{3}\left(e_{3}\right)=e_{1}\otimes e_{3}+e_{3}\otimes e_{2};\
\varepsilon_{3,2}^{3}\left(e_{1}\right)=1;\
\varepsilon_{3,2}^{3}\left(e_{2}\right)=1;\
\varepsilon_{3,2}^{3}\left(e_{3}\right)=0.$
3. (3)
$\Delta_{3,3}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{3,3}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{3,3}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes e_{1};\
\varepsilon_{3,3}^{3}\left(e_{1}\right)=1;\
\varepsilon_{3,3}^{3}\left(e_{2}\right)=1;\
\varepsilon_{3,3}^{3}\left(e_{3}\right)=0.$
For the multiplication $\mu_{4}^{3}$, there does not exist any bialgebras.
For the multiplication $\mu_{5}^{3}$, we have
1. (1)
$\Delta_{5,1}^{3}\left(e_{1}\right)=e_{1}\otimes e_{1};\
\Delta_{5,1}^{3}\left(e_{2}\right)=e_{2}\otimes e_{2};\
\Delta_{5,1}^{3}\left(e_{3}\right)=e_{2}\otimes e_{3}+e_{3}\otimes e_{2};\
\varepsilon_{5,1}^{3}\left(e_{1}\right)=1;\
\varepsilon_{5,1}^{3}\left(e_{2}\right)=1;\
\varepsilon_{5,1}^{3}\left(e_{3}\right)=0.$
In the sequel we consider that all the algebras are unital and the unit $\eta$
corresponds to $e_{1}$.
###### Proposition 3.4.
The bialgebras on $\mathbb{K}^{3}$ which are unital infinitesimal bialgebras
are given by the following pairs of multiplication and comultiplication with
the appropriate unit and counits.
$\left(\mu_{1}^{3},\Delta_{1,2}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,5}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,6}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,8}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,11}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,14}^{3}\right),\
\left(\mu_{1}^{3},\Delta_{1,15}^{3}\right),\ $
$\left(\mu_{1}^{3},\Delta_{1,18}^{3}\right),\
\left(\mu_{2}^{3},\Delta_{2,1}^{3}\right),\
\left(\mu_{2}^{3},\Delta_{2,2}^{3}\right),\
\left(\mu_{3}^{3},\Delta_{3,2}^{3}\right),\
\left(\mu_{3}^{3},\Delta_{3,3}^{3}\right),\
\left(\mu_{5}^{3},\Delta_{5,1}^{3}\right).$
###### Remark 3.5.
The unital infinitesimal bialgebras found here could be used to produce
examples of Rota-Baxter algebras.
We summarize, in the following table, the numbers of non-isomorphic bialgebras
and unital infinitesimal bialgebras associated to a given algebra.
algebra | bialgebras | infinitesimal bialgebras
---|---|---
$\mu_{1}^{3}$ | 18 | 8
$\mu_{2}^{3}$ | 3 | 2
$\mu_{3}^{3}$ | 3 | 2
$\mu_{4}^{3}$ | 0 | 0
$\mu_{5}^{3}$ | 1 | 1
###### Corollary 3.6.
* •
The number of 3-dimensional trivial 2-associative bialgebras (same
multiplication) is 13.
* •
We have the following 3-dimensional non-trivial 2-associative bialgebras
(different multiplication)
$\left(\mathbb{K}^{3},\mu_{3}^{3},\mu_{5}^{3},\eta,\Delta_{3,1}^{3},\varepsilon_{3,1}^{3}\right),\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{2,1}^{3},\varepsilon_{2,1}^{3}\right)$
and
$\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{2,2}^{3},\varepsilon_{2,2}^{3}\right).$
###### Proposition 3.7.
The number of 3-dimensional isomorphism classes of 2-bialgebras is
* •
type (1,1) : 25 .
* •
type (1,2) : 159.
* •
type (2,1) : 1. Namely
$\left(\mathbb{K}^{3},\mu_{3}^{3},\mu_{5}^{3},\eta,\Delta_{3,1}^{3},\Delta_{5,1}^{3},\varepsilon_{3,1}^{3},\varepsilon_{5,1}^{3}\right)$.
* •
type (2,2) : 3. Namely,
$\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{1,3}^{3},\Delta_{2,1}^{3},\varepsilon_{1,3}^{3},\varepsilon_{2,1}^{3}\right)$
,
$\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{1,4}^{3},\Delta_{2,1}^{3},\varepsilon_{1,4}^{3},\varepsilon_{2,1}^{3}\right)$
and
$\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{1,5}^{3},\Delta_{2,1}^{3},\varepsilon_{1,5}^{3},\varepsilon_{2,1}^{3}\right).$
###### Corollary 3.8.
There exists only one 2-2-bialgebra of dimension 3 which is given by
$\left(\mathbb{K}^{3},\mu_{1}^{3},\mu_{2}^{3},\eta,\Delta_{1,5}^{3},\Delta_{2,1}^{3},\varepsilon_{1,5}^{3},\varepsilon_{2,1}^{3}\right).$
## References
* [1] M. Aguiar, _Infinitesimal Hopf algebras_ , in New trends in Hopf algebra theory (La Falda 1999), Contemp. Math. 267, 1–29, (2000).
* [2] M. Aguiar, On the associative analog of Lie bialgebras, J. Algebra 244 (2), 492–532, (2001).
* [3] M. Aguiar, _Infinitesimal bialgebras, pre-Lie and dendriform algebras_ , Hopf algebras, Lecture Notes in Pure and Appl. Math., 237, Marcel Dekker, New York, 1–33 (2004).
* [4] P. Cartier, _Hyperalgèbres et groupes de Lie formels_ , Séminaire Sophus Lie, 2e année: 1955/56. Faculté des Sciences de Paris.
* [5] K. Ebrahimi-Fard, J.M. Gracia-Bondia and F. Patras, _Rota-Baxter algebras and new combinatorial identities_ , Letters in Mathematical Physics, 81, 61-75, (2007).
* [6] P. Gabriel, _Finite representation type is open_ in ”Proceedings of ICRAI, Ottawa 1974”, Lect. Notes in Math (1974).
* [7] S.A. Joni, and G.-C. Rota, _Coalgebras and bialgebras in combinatorics_ , Stud. Appl. Math. 61, no. 2, 93–139 (1979).
* [8] I. Kaplansky, _Bialgebras_ , University of Chicago, (1973).
* [9] C. Kassel, _Quantum groups_ , Graduate texts in Mathematics, vol 155, Springer-Verlager, New York, (1995).
* [10] J.-L. Loday and M. Ronco, _On the structure of the cofree Hopf algebras_ , J. Reine Angew. Math, 592, 123–155, (2006).
* [11] J.-L. Loday, _Generalized bialgebras and triples of operads_ , Preprint (2007)
* [12] A. Makhlouf, _Degeneration, rigidity and irreducible component of Hopf algebras_ , Algebra Colloquium 12(2), 241–254, (2005).
* [13] J.W. Milnor and J.C. Moore, _On the structure of Hopf algebras_ , Ann. of Math. 81 (2), 211–264 (1965).
* [14] S. Montgomery, _Hopf algebras and their actions on rings_ , American Mathematical Society(1992).
|
arxiv-papers
| 2008-09-06T07:52:01
|
2024-09-04T02:48:57.680357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Khadra Dekkar and Abdenacer Makhlouf",
"submitter": "Abdenacer Makhlouf",
"url": "https://arxiv.org/abs/0809.1144"
}
|
0809.1210
|
# Magnetic dilaton strings in anti-de Sitter spaces
Ahmad Sheykhi 111sheykhi@mail.uk.ac.ir Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran
###### Abstract
With an appropriate combination of three Liouville-type dilaton potentials, we
construct a new class of spinning magnetic dilaton string solutions which
produces a longitudinal magnetic field in the background of anti-de Sitter
spacetime. These solutions have no curvature singularity and no horizon, but
have a conic geometry. We find that the spinning string has a net electric
charge which is proportional to the rotation parameter. We present the
suitable counterterm which removes the divergences of the action in the
presence of dilaton potential. We also calculate the conserved quantities of
the solutions by using the counterterm method.
## I Introduction
The construction and analysis of black hole solutions in the background of
anti-de Sitter (AdS) spaces is a subject of much recent interest. This
interest is primarily motivated by the correspondence between the gravitating
fields in an AdS spacetime and conformal field theory living on the boundary
of the AdS spacetime Witt1 . This equivalence enables one to remove the
divergences of the action and conserved quantities of gravity in the same way
as one does in field theory. It was argued that the thermodynamics of black
holes in AdS spaces can be identified with that of a certain dual conformal
field theory (CFT) in the high temperature limit Witt2 . Having the AdS/CFT
correspondence idea at hand, one can gain some insights into thermodynamic
properties and phase structures of strong ’t Hooft coupling conformal field
theories by studying the thermodynamics of asymptotically AdS black holes.
On another front, scalar coupled black hole solutions with different
asymptotic spacetime structure is a subject of interest for a long time. There
has been a renewed interest in such studies ever since new black hole
solutions have been found in the context of string theory. The low energy
effective action of string theory contains two massless scalars namely dilaton
and axion. The dilaton field couples in a nontrivial way to other fields such
as gauge fields and results into interesting solutions for the background
spacetime. It was argued that with the exception of a pure cosmological
constant, no dilaton-de Sitter or anti-de Sitter black hole solution exists
with the presence of only one Liouville-type dilaton potential MW . Recently,
the dilaton potential leading to (anti)-de Sitter-like solutions of dilaton
gravity has been found Gao1 . It was shown that the cosmological constant is
coupled to the dilaton in a very nontrivial way. With the combination of three
Liouville-type dilaton potentials, a class of static dilaton black hole
solutions in (A)dS spaces has been obtained by using a coordinates
transformation which recast the solution in the schwarzschild coordinates
system Gao1 . More recently, a class of charged rotating dilaton black string
solutions in four-dimensional anti-de Sitter spacetime has been found in shey1
. Other studies on the dilaton black hole solutions in (A)dS spaces have been
carried out in Gao2 ; shey2 .
In this Letter, we turn to the investigation of asymptotically AdS spacetimes
generated by static and spinning string sources in four-dimensional Einstein-
Maxwell-dilaton theory which are horinzonless and have nontrivial external
solutions. The motivation for studying such kinds of solutions is that they
may be interpreted as cosmic strings. Cosmic strings are topological structure
that arise from the possible phase transitions to which the universe might
have been subjected to and may play an important role in the formation of
primordial structures. A short review of papers treating this subject follows.
The four-dimensional horizonless solutions of Einstein gravity have been
explored in Vil ; Ban . These horizonless solutions Vil ; Ban have a conical
geometry; they are everywhere flat except at the location of the line source.
The spacetime can be obtained from the flat spacetime by cutting out a wedge
and identifying its edges. The wedge has an opening angle which turns to be
proportional to the source mass. The extension to include the Maxwell field
has also been done Bon . Static and spinning magnetic sources in three and
four-dimensional Einstein-Maxwell gravity with negative cosmological constant
have been explored in Lem1 ; Lem2 . The generalization of these asymptotically
AdS magnetic rotating solutions to higher dimensions has also been done Deh2 .
In the context of electromagnetic cosmic string, it has been shown that there
are cosmic strings, known as superconducting cosmic strings, that behave as
superconductors and have interesting interactions with astrophysical magnetic
fields Wit2 . The properties of these superconducting cosmic strings have been
investigated in Moss . It is also of great interest to generalize the study to
the dilaton gravity theory Fer . While exact magnetic rotating dilaton
solution in three dimensions has been obtained in Dia , two classes of
magnetic rotating solutions in four Deh4 and higher dimensional dilaton
gravity in the presence of one Liouville-type potential have been constructed
SDR . Unfortunately, these solutions Deh4 ; SDR are neither asymptotically
flat nor (A)dS. The purpose of the present Letter is to construct a new class
of static and spinning magnetic dilaton string solutions which produces a
longitudinal magnetic field in the background of anti-de Sitter spacetime. We
will also present the suitable counterterm which removes the divergences of
the action, and calculate the conserved quantities by using the counterterm
method.
## II Basic Equations
Our starting point is the four-dimensional Einstein-Maxwell-dilaton action
$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(R\text{
}-2\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi)-e^{-2\alpha\Phi}F_{\mu\nu}F^{\mu\nu}\right)$
(1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\Theta(\gamma),$
where ${R}$ is the scalar curvature, $\Phi$ is the dilaton field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, and $A_{\mu}$ is the electromagnetic potential.
$\alpha$ is an arbitrary constant governing the strength of the coupling
between the dilaton and the Maxwell field. The last term in Eq. (1) is the
Gibbons-Hawking surface term. It is required for the variational principle to
be well-defined. The factor $\Theta$ represents the trace of the extrinsic
curvature for the boundary ${\partial\mathcal{M}}$ and $\gamma$ is the induced
metric on the boundary. While $\alpha=0$ corresponds to the usual Einstein-
Maxwell-scalar theory, $\alpha=1$ indicates the dilaton-electromagnetic
coupling that appears in the low energy string action in Einstein’s frame. For
arbitrary value of $\alpha$ in AdS space the form of the dilaton potential
$V(\Phi)$ is chosen as Gao1
$V(\Phi)=\frac{2\Lambda}{3(\alpha^{2}+1)^{2}}\left[{\alpha}^{2}\left(3\,{\alpha}^{2}-1\right){e^{-2\Phi/\alpha}}+\left(3-{\alpha}^{2}\right){e^{2\,\alpha\,\Phi}}+8\,{\alpha}^{2}{e^{\Phi(\alpha-1/\alpha)}}\right].$
(2)
Here $\Lambda$ is the cosmological constant. It is clear that the cosmological
constant is coupled to the dilaton field in a very nontrivial way. This type
of the dilaton potential was introduced for the first time by Gao and Zhang
Gao1 . They derived, by applying a coordinates transformation which recast the
solution in the Schwarzchild coordinates system, the static dilaton black hole
solutions in the background of (A)dS universe. For this purpose, they required
the existence of the (A)dS dilaton black hole solutions and extracted
successfully the form of the dilaton potential leading to (A)dS-like
solutions. They also argued that this type of derived potential can be
obtained when a higher dimensional theory is compactified to four dimensions,
including various supergravity models Gid . In the absence of the dilaton
field the action (1) reduces to the action of Einstein-Maxwell gravity with
cosmological constant. Varying the action (1) with respect to the
gravitational field $g_{\mu\nu}$, the dilaton field $\Phi$ and the gauge field
$A_{\mu}$, yields
${R}_{\mu\nu}=2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)+2e^{-2\alpha\Phi}\left(F_{\mu\eta}F_{\nu}^{\text{
}\eta}-\frac{1}{4}g_{\mu\nu}F_{\lambda\eta}F^{\lambda\eta}\right),$ (3)
$\nabla^{2}\Phi=\frac{1}{4}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{-2\alpha\Phi}F_{\lambda\eta}F^{\lambda\eta},$
(4) $\partial_{\mu}\left(\sqrt{-g}e^{-2\alpha\Phi}F^{\mu\nu}\right)=0.$ (5)
The conserved mass and angular momentum of the solutions of the above field
equations can be calculated through the use of the substraction method of
Brown and York BY . Such a procedure causes the resulting physical quantities
to depend on the choice of reference background. A well-known method of
dealing with this divergence for asymptotically AdS solutions of Einstein
gravity is through the use of counterterm method inspired by AdS/CFT
correspondence Mal . In this Letter, we deal with the spacetimes with zero
curvature boundary, $R_{abcd}(\gamma)=0$, and therefore the counterterm for
the stress energy tensor should be proportional to $\gamma^{ab}$. We find the
suitable counterterm which removes the divergences of the action in the form
(see also otha )
$I_{ct}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-\gamma}\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right).$
(6)
One may note that in the absence of a dilaton field where we have
$V(\Phi)=2\Lambda=-6/l^{2}$, the above counterterm has the same form as in the
case of asymptotically AdS solutions with zero-curvature boundary. Having the
total finite action $I=I_{G}+I_{\mathrm{ct}}$ at hand, one can use the
quasilocal definition to construct a divergence free stress-energy tensor BY .
Thus the finite stress-energy tensor in four-dimensional Einstein-dilaton
gravity with three Liouville-type dilaton potentials (2) can be written as
$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta\gamma^{ab}+\left(-\frac{1}{l}+\frac{\sqrt{-6V(\Phi)}}{2}\right)\gamma^{ab}\right].$
(7)
The first two terms in Eq. (7) are the variation of the action (1) with
respect to $\gamma_{ab}$, and the last two terms are the variation of the
boundary counterterm (6) with respect to $\gamma_{ab}$. To compute the
conserved charges of the spacetime, one should choose a spacelike surface
$\mathcal{B}$ in $\partial\mathcal{M}$ with metric $\sigma_{ij}$, and write
the boundary metric in ADM (Arnowitt-Deser-Misner) form:
$\gamma_{ab}dx^{a}dx^{b}=-N^{2}dt^{2}+\sigma_{ij}\left(d\varphi^{i}+V^{i}dt\right)\left(d\varphi^{j}+V^{j}dt\right),$
where the coordinates $\varphi^{i}$ are the angular variables parameterizing
the hypersurface of constant $r$ around the origin, and $N$ and $V^{i}$ are
the lapse and shift functions, respectively. When there is a Killing vector
field $\mathcal{\xi}$ on the boundary, then the quasilocal conserved
quantities associated with the stress tensors of Eq. (7) can be written as
$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\xi}^{b},$
(8)
where $\sigma$ is the determinant of the metric $\sigma_{ij}$, $\mathcal{\xi}$
and $n^{a}$ are, respectively, the Killing vector field and the unit normal
vector on the boundary $\mathcal{B}$. For boundaries with timelike
($\xi=\partial/\partial t$) and rotational ($\varsigma=\partial/\partial\phi$)
Killing vector fields, one obtains the quasilocal mass and angular momentum
$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b},$ (9)
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{2}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}.$
(10)
These quantities are, respectively, the conserved mass and angular momenta of
the system enclosed by the boundary $\mathcal{B}$. Note that they will both
depend on the location of the boundary $\mathcal{B}$ in the spacetime,
although each is independent of the particular choice of foliation
$\mathcal{B}$ within the surface $\partial\mathcal{M}$.
## III Static magnetic dilaton string
Here we want to obtain the four-dimensional solution of Eqs. (3)-(5) which
produces a longitudinal magnetic fields along the $z$ direction. We assume the
following form for the metric Lem1
$ds^{2}=-\frac{\rho^{2}}{l^{2}}R^{2}(\rho)dt^{2}+\frac{d\rho^{2}}{f(\rho)}+l^{2}f(\rho)d\phi^{2}+\frac{\rho^{2}}{l^{2}}R^{2}(\rho)dz^{2}.$
(11)
The functions $f(\rho)$ and $R(\rho)$ should be determined and $l$ has the
dimension of length which is related to the cosmological constant $\Lambda$ by
the relation $l^{2}=-3/\Lambda$. The coordinate $z$ has the dimension of
length and ranges $-\infty<z<\infty$, while the angular coordinate $\phi$ is
dimensionless as usual and ranges $0\leq\phi<2\pi$. The motivation for this
curious choice of the metric gauge $[g_{tt}\varpropto-\rho^{2}$ and
$(g_{\rho\rho})^{-1}\varpropto g_{\phi\phi}]$ instead of the usual
Schwarzschild gauge $[(g_{\rho\rho})^{-1}\varpropto g_{tt}$ and
$g_{\phi\phi}\varpropto\rho^{2}]$ comes from the fact that we are looking for
a magnetic solution instead of an electric one. It is well-known that the
electric field is associated with the time component, $A_{t}$, of the vector
potential while the magnetic field is associated with the angular component
$A_{\phi}$. From the above fact, one can expect that a magnetic solution can
be written in a metric gauge in which the components $g_{tt}$ and
$g_{\phi\phi}$ interchange their roles relatively to those present in the
Schwarzschild gauge used to describe electric solutions Lem1 . The Maxwell
equation (5) can be integrated immediately to give
$F_{\phi\rho}=\frac{qle^{2\alpha\Phi}}{\rho^{2}R^{2}},$ (12)
where $q$, an integration constant, is the charge parameter which is related
to the electric charge of the rotating string, as will be shown below.
Inserting the Maxwell fields (12) and the metric (11) in the field equations
(3) and (4), we can simplify these equations as
$\displaystyle
2\rho^{3}R^{4}f^{\prime}+2\rho^{4}R^{3}f^{\prime}R^{\prime}+2\rho^{2}R^{4}f+8\rho^{3}R^{3}fR^{\prime}+2\rho^{4}R^{2}fR^{\prime
2}$
$\displaystyle+2\rho^{4}R^{3}fR^{\prime\prime}+\rho^{4}R^{4}V\left(\Phi\right)-2q^{2}e^{2\alpha\Phi}=0,$
(13) $\displaystyle
2\rho^{3}R^{4}f^{\prime}+\rho^{4}R^{4}f^{\prime\prime}+8\rho^{3}R^{3}fR^{\prime}+4\rho^{4}R^{3}fR^{\prime\prime}+2\rho^{4}R^{3}R^{\prime}f^{\prime}$
$\displaystyle+4\rho^{4}R^{4}f\Phi^{\prime
2}+\rho^{4}R^{4}V\left(\Phi\right)+2q^{2}e^{2\alpha\Phi}=0,$ (14)
$\displaystyle
2\rho^{4}R^{3}R^{\prime}f^{\prime}+\rho^{4}R^{4}f^{\prime\prime}+2\rho^{3}R^{4}f^{\prime}+\rho^{4}R^{4}V(\Phi)+2q^{2}e^{2\alpha\Phi}=0,$
(15)
$\displaystyle\rho^{4}R^{4}{\Phi^{\prime}}{f^{\prime}}+\rho^{4}R^{4}{\Phi^{\prime\prime}}{f}+2\rho^{3}R^{4}\Phi^{\prime}f+2\rho^{4}R^{3}R^{\prime}{\Phi^{\prime}}{f}-\rho^{4}R^{4}\frac{\partial{V}}{4\partial{\Phi}}+\alpha
q^{2}e^{2\alpha\Phi}=0,$ (16)
where the “prime” denotes differentiation with respect to $\rho$. Subtracting
Eq. (15) from Eq. (14) we get
$\displaystyle 2R^{\prime}+\rho R^{\prime\prime}+\rho R\Phi^{\prime 2}=0.$
(17)
Then we make the ansatz shey1
$R(\rho)=e^{\alpha\Phi}.$ (18)
Substituting this ansatz in Eq. (17), it reduces to
$\displaystyle\rho\alpha\Phi^{\prime\prime}+2\alpha\Phi^{\prime}+\rho(1+{\alpha}^{2})\Phi^{\prime
2}=0,$ (19)
which has a solution of the form
$\Phi(\rho)=\frac{\alpha}{\alpha^{2}+1}\ln(1-\frac{b}{\rho}),$ (20)
where $b$ is a constant of integration related to the mass of the string, as
will be shown. Inserting (20), the ansatz (18), and the dilaton potential (2)
into the field equations (III)-(16), one can show that these equations have
the following solution
$f(\rho)=\frac{c}{\rho}\left(1-{\frac{b}{\rho}}\right)^{{\frac{1-{\alpha}^{2}}{1+{\alpha}^{2}}}}-\frac{\Lambda}{3}\,{\rho}^{2}\left(1-{\frac{b}{\rho}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}},$
(21)
where $c$ is an integration constant. The two constants $c$ and $b$ are
related to the charge parameter via $q^{2}(1+\alpha^{2})=bc$. It is apparent
that this spacetime is asymptotically AdS. In the absence of a nontrivial
dilaton ($\alpha=0$), the solution reduces to the asymtotically AdS
horizonless magnetic string for $\Lambda=-3/l^{2}$ Lem2 .
Then we study the general structure of the solution. It is easy to show that
the Kretschmann scalar $R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$
diverges at $\rho=0$ and therefore one might think that there is a curvature
singularity located at $\rho=0$. However, as we will see below, the spacetime
will never achieve $\rho=0$. Second, we look for the existence of horizons
and, in particular, we look for the possible presence of magnetically charged
black hole solutions. The surface $r=b$ is a curvature singularity for
$\alpha\neq 0$. The horizons, if any exist, are given by the zeros of the
function $f(\rho)=(g_{\rho\rho})^{-1}$. Let us denote the largest positive
root of $f(\rho)=0$ by $r_{+}$. The function $f(\rho)$ is negative for
$\rho<r_{+}$, and therefore one may think that the hypersurface of constant
time and $\rho=r_{+}$ is the horizon. However, the above analysis is wrong.
Indeed, we first notice that $g_{\rho\rho}$ and $g_{\phi\phi}$ are related by
$f(\rho)=g_{\rho\rho}^{-1}=l^{-2}g_{\phi\phi}$, and therefore when
$g_{\rho\rho}$ becomes negative (which occurs for $\rho<r_{+}$) so does
$g_{\phi\phi}$. This leads to an apparent change of signature of the metric
from $+2$ to $-2$. This indicates that we are using an incorrect extension. To
get rid of this incorrect extension, we introduce the new radial coordinate
$r$ as
$r^{2}=\rho^{2}-r_{+}^{2}\Rightarrow
d\rho^{2}=\frac{r^{2}}{r^{2}+r_{+}^{2}}dr^{2}.$ (22)
With this coordinate change, the metric (11) is
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{r^{2}+r_{+}^{2}}{l^{2}}R^{2}(r)dt^{2}+l^{2}f(r)d\phi^{2}$
(23)
$\displaystyle+\frac{r^{2}}{(r^{2}+r_{+}^{2})f(r)}dr^{2}+\frac{r^{2}+r_{+}^{2}}{l^{2}}R^{2}(r)dz^{2},$
where the coordinates $r$ assumes the values $0\leq r<\infty$, and $f(r)$,
$R(r)$, and $\Phi(r)$ are now given as
$f(r)=\frac{c}{\sqrt{r^{2}+r_{+}^{2}}}\left(1-{\frac{b}{\sqrt{r^{2}+r_{+}^{2}}}}\right)^{{\frac{1-{\alpha}^{2}}{1+{\alpha}^{2}}}}-\frac{\Lambda}{3}\,({r^{2}+r_{+}^{2}})\left(1-{\frac{b}{\sqrt{r^{2}+r_{+}^{2}}}}\right)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}},$
(24)
$R(r)=\left(1-\frac{b}{\sqrt{r^{2}+r_{+}^{2}}}\right)^{\frac{\alpha^{2}}{1+\alpha^{2}}},\hskip
19.91684pt\Phi(r)=\frac{\alpha}{1+\alpha^{2}}\ln\left(1-\frac{b}{\sqrt{r^{2}+r_{+}^{2}}}\right).$
(25)
One can easily show that the Kretschmann scalar does not diverge in the range
$0\leq r<\infty$. However, the spacetime has a conic geometry and has a
conical singularity at $r=0$, since:
$\lim_{r\rightarrow 0}\frac{1}{r}\sqrt{\frac{g_{\phi\phi}}{g_{rr}}}\neq 1.$
(26)
That is, as the radius $r$ tends to zero, the limit of the ratio
“circumference/radius” is not $2\pi$ and therefore the spacetime has a conical
singularity at $r=0$. The canonical singularity can be removed if one
identifies the coordinate $\phi$ with the period
$\textrm{Period}_{\phi}=2\pi\left(\lim_{r\rightarrow
0}\frac{1}{r}\sqrt{\frac{g_{\phi\phi}}{g_{rr}}}\right)^{-1}=2\pi(1-4\mu),$
(27)
where
$\displaystyle 1-4\mu$ $\displaystyle=$ $\displaystyle\Bigg{\\{}\frac{\Lambda
l(r_{+}-b)^{{\frac{2{\alpha}^{2}}{{\alpha}^{2}+1}}}\left[r_{+}(\alpha^{2}+1)-b\right]}{3(\alpha^{2}+1)(b-r_{+})r_{+}^{\frac{\alpha^{2}-1}{1+\alpha^{2}}}}+\frac{lc(r_{+}-b)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}\left[r_{+}(\alpha^{2}+1)-2b\right]}{2(\alpha^{2}+1)(b-r_{+})r_{+}^{\alpha^{2}+3}}\Bigg{\\}}^{-1}.$
(28)
The above analysis shows that near the origin $r=0$, the metric (23) describes
a spacetime which is locally flat but has a conical singularity at $r=0$ with
a deficit angle $\delta\phi=8\pi\mu$. Since near the origin the metric (23) is
identical to the spacetime generated by a cosmic string, by using the Vilenkin
procedure, one can show that $\mu$ in Eq. (28) can be interpreted as the mass
per unit length of the string Vil2 .
## IV Spinning magnetic dilaton string
Now, we would like to endow the spacetime solution (11) with a rotation. In
order to add an angular momentum to the spacetime, we perform the following
rotation boost in the $t-\phi$ plane
$t\mapsto\Xi t-a\phi,\hskip 14.22636pt\phi\mapsto\Xi\phi-\frac{a}{l^{2}}t,$
(29)
where $a$ is a rotation parameter and $\Xi=\sqrt{1+a^{2}/l^{2}}$. Substituting
Eq. (29) into Eq. (23) we obtain
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{r^{2}+r_{+}^{2}}{l^{2}}R^{2}(r)\left(\Xi dt-
ad\phi\right)^{2}+\frac{r^{2}dr^{2}}{(r^{2}+r_{+}^{2})f(r)}$ (30)
$\displaystyle+l^{2}f(r)\left(\frac{a}{l^{2}}dt-\Xi
d\phi\right)^{2}+\frac{r^{2}+r_{+}^{2}}{l^{2}}R^{2}(r)dz^{2},$
where $f(r)$ and $R(r)$ are given in Eqs. (24) and (25). The non-vanishing
electromagnetic field components become
$\displaystyle F_{\phi r}=\frac{q\Xi l}{r^{2}+r_{+}^{2}},\hskip
19.91684ptF_{tr}=-\frac{a}{\Xi l^{2}}F_{\phi r}.$ (31)
The transformation (29) generates a new metric, because it is not a permitted
global coordinate transformation. This transformation can be done locally but
not globally. Therefore, the metrics (23) and (30) can be locally mapped into
each other but not globally, and so they are distinct. Note that this
spacetime has no horizon and curvature singularity. However, it has a conical
singularity at $r=0$. It is notable to mention that for $\alpha=0$, this
solution reduces to the asymtotically AdS magnetic rotating string solution
presented in Lem2 .
The mass and angular momentum per unit length of the string when the boundary
$\mathcal{B}$ goes to infinity can be calculated through the use of Eqs. (9)
and (10). We obtain
$\displaystyle{M}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}(\alpha^{2}-1)b^{3}}{24\pi
l^{3}(\alpha^{2}+1)^{3}}+\frac{(3\Xi^{2}-2)c}{16\pi l},$ (32)
$\displaystyle{J}$ $\displaystyle=$ $\displaystyle\frac{3\Xi
c\sqrt{\Xi^{2}-1}}{16\pi}.$ (33)
For $a=0$ ($\Xi=1$), the angular momentum per unit length vanishes, and
therefore $a$ is the rotational parameter of the spacetime.
Finally, we compute the electric charge of the solutions. To determine the
electric field one should consider the projections of the electromagnetic
field tensor on special hypersurface. The normal vectors to such hypersurface
for the spacetime with a longitudinal magnetic field are
$u^{0}=\frac{1}{N},\text{ \ }u^{r}=0,\text{ \ }u^{i}=-\frac{V^{i}}{N},$
and the electric field is
$E^{\mu}=g^{\mu\rho}e^{-2\alpha\Phi}F_{\rho\nu}u^{\nu}$. Then the electric
charge per unit length ${Q}$ can be found by calculating the flux of the
electric field at infinity, yielding
${Q}=\frac{q\sqrt{\Xi^{2}-1}}{4\pi l}.$ (34)
It is worth noting that the electric charge is proportional to the rotation
parameter, and is zero for the case of static solution. This result is
expected since now, besides the magnetic field along the $\phi$ coordinate,
there is also a radial electric field ($F_{tr}\neq 0$). To give a physical
interpretation for the appearance of the net electric charge, we first
consider the static spacetime. The magnetic field source can be interpreted as
composed of equal positive and negative charge densities, where one of the
charge density is at rest and the other one is spinning. Clearly, this system
produce no electric field since the net electric charge density is zero, and
the magnetic field is produced by the rotating electric charge density. Now,
we consider the rotating solution. From the point of view of an observer at
rest relative to the source ($S$), the two charge densities are equal, while
from the point of view of an observe $S^{\prime}$ that follows the intrinsic
rotation of the spacetime, the positive and negative charge densities are not
equal, and therefore the net electric charge of the spacetime is not zero.
## V Conclusion and discussion
In conclusion, with an appropriate combination of three Liouville-type dilaton
potentials, we constructed a class of four-dimensionl magnetic dilaton string
solutions which produces a longitudinal magnetic field in the background of
anti-de Sitter universe. These solutions have no curvature singularity and no
horizon, but have conic singularity at $r=0$. In fact, we showed that near the
origin $r=0$, the metric (23) describes a spacetime which is locally flat but
has a conical singularity at $r=0$ with a deficit angle $\delta\phi=8\pi\mu$,
where $\mu$ can be interpreted as the mass per unit length of the string. In
these static spacetimes, the electric field vanishes and therefore the string
has no net electric charge. Then we added an angular momentum to the spacetime
by performing a rotation boost in the $t-\phi$ plane. For the spinning string,
when the rotation parameter is nonzero, the string has a net electric charge
which is proportional to the magnitude of the rotation parameter. We found the
suitable counterterm which removes the divergences of the action in the
presence of three Liouville-type dilaton potentials. We also computed the
conserved quantities of the solutions through the use of the conterterm method
inspired by the AdS/CFT correspondence.
It is worth comparing the solutions obtained here to the electrically charged
rotating dilaton black string solutions presented in shey1 . In the present
work I have studied the magnetic spinning dilaton string which produces a
longitudinal magnetic field in AdS spaces which is the correct one
generalizing of the magnetic string solution of Dias and Lemos in dilaton
theory Lem2 , while in shey1 I constructed charged rotating dilaton black
string in AdS spaces which is the generalization of the charged rotating
string solutions of Lem3 in dilaton gravity. Although solution (21) of the
present paper is similar to Eq. (16) of Ref. shey1 (except the sign of $c$)
and both solutions represent dilaton string, however, there are some different
between the magnetic string and the electrically charged dilaton black string
solutions. First, the choice of the metric gauge $[g_{tt}\varpropto-\rho^{2}$
and $(g_{\rho\rho})^{-1}\varpropto g_{\phi\phi}]$ in the magnetic case which
is quite different from the Schwarzschild gauge
$[(g_{\rho\rho})^{-1}\varpropto g_{tt}$ and $g_{\phi\phi}\varpropto\rho^{2}]$
proposed in shey1 . Second, the electrically charged dilaton black strings
have an essential singularity located at $r=0$ and also have horizons, while
the magnetic strings version presented here have no curvature singularity and
no horizon, but have a conic geometry. Third, when the rotation parameter is
nonzero, the magnetic string has a net electric charge which is proportional
to the rotation parameter, while charged dilaton black string has always an
electric charge regardless of the rotation parameter.
The generalization of the present work to higher dimensions, that is the
magnetic rotating dilaton branes in AdS spaces with complete set of rotation
parameters and arbitrary dilaton coupling constant is now under investigation
and will be addressed elsewhere.
###### Acknowledgements.
This work has been supported financially by the Research Institute for
Astronomy and Astrophysics of Maragha, Iran. The author would like to thank
the anonymous referee for his/her useful comments.
## References
* (1) E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253.
* (2) E. Witten, Adv. Theor. Math. Phys. 2 (1998) 505.
* (3) S. J. Poletti and D. L. Wiltshire, Phys. Rev. D 50 (1994) 7260 ;
S. J. Poletti, J. Twamley and D. L. Wiltshire, Phys. Rev. D 51 (1995) 5720.
* (4) C. J. Gao and S. N. Zhang, Phys. Rev. D 70 (2004) 124019.
* (5) A. Sheykhi, Phys. Rev. D 78 (2008) 064055\.
* (6) C. J. Gao and S. N. Zhang, Phys. Lett. B 605 (2005) 185;
C. J. Gao and S. N. Zhang, Phys. Lett. B 612 (2005) 127.
* (7) A. Sheykhi and M. Allahverdizadeh, Phys. Rev. D 78 (2008) 064073.
* (8) A. Vilenkin, Phys. Rev. D 23 (1981) 852;
W. A. Hiscock, Phys. Rev. D. 31 (1985) 3288;
D. Harari and P. Sikivie, Phys. Rev. D 37 (1988) 3438;
A. D. Cohen and D. B. Kaplan, Phys. Lett. B 215 (1988) 65;
R. Gregory, Phys. Rev. D. 215 (1988) 663.
* (9) A. Banerjee, N. Banerjee, and A. A. Sen, Phys. Rev. D 53 (1996) 5508;
M. H. Dehghani and T. Jalali, Phys. Rev. D 66 (2002) 124014 ;
M. H. Dehghani and A. Khodam-Mohammadi, Can. J. Phys. 83 (2005) 229.
* (10) W. B. Bonnor, Proc. Roy. S. London A 67 (1954) 225;
A. Melvin, Phys. Lett. 8 (1964) 65.
* (11) O. J. C. Dias and J. P. S. Lemos, J. High Energy Phys. 01 (2002) 006.
* (12) O. J. C. Dias and J. P. S. Lemos, Class. Quantum Grav. 19 (2002) 2265.
* (13) M. H. Dehghani, Phys. Rev. D 69 (2004) 044024 .
* (14) E. Witten, Nucl. Phys. B 249 (1985) 557;
P. Peter, Phys. Rev. D 49 (1994) 5052.
* (15) I. Moss and S. Poletti, Phys. Lett. B 199 (1987) 34.
* (16) C. N. Ferreira, M. E. X. Guimaraes and J. A. Helayel-Neto, Nucl. Phys. B 581 (2000) 165.
* (17) O. J. C. Dias and J. P. S. Lemos, Phys. Rev. D 66 (2002) 024034.
* (18) M. H. Dehghani, Phys. Rev. D 71 (2005) 064010.
* (19) A. Sheykhi, M. H. Dehghani, N. Riazi, Phys. Rev. D 75 (2007) 044020;
M. H. Dehghani, A. Sheykhi and S. H. Hendi, Phys. Lett. B 659 (2008) 476.
* (20) S. B. Giddings, Phys. Rev. D 68 026006 (2003);
E. Radu, D. H. Tchrakian, Class. Quantum Grav. 22 (2005) 879.
* (21) J. Brown and J. York, Phys. Rev. D 47 (1993) 1407.
* (22) J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231.
* (23) R. G. Cai and N. Ohta, Phys. Rev. D 62 (2000) 024006 .
* (24) A. Vilenkin, Phys. Rep. 121 (1985) 263.
* (25) J. P. S. Lemos, Class. Quantum Grav. 12, 1081 (1995);
J. P. S. Lemos, Phys. Lett. B 353, 46 (1995).
|
arxiv-papers
| 2008-09-07T06:13:46
|
2024-09-04T02:48:57.686741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0809.1210"
}
|
0809.1222
|
# Quark deconfinement phase transition for improved quark mass density-
dependent model
Chen Wu1 and Ru-Keng Su1,2,3111rksu@fudan.ac.cn 1\. Department of Physics,
Fudan University,Shanghai 200433, P.R. China
2\. CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China
3\. Center of Theoretical Nuclear Physics,
National Laboratory of Heavy Ion Collisions, Lanzhou 730000, P.R.China
###### Abstract
By using the finite temperature quantum field theory, we calculate the finite
temperature effective potential and extend the improved quark mass density-
dependent model to finite temperature. It is shown that this model can not
only describe the saturation properties of nuclear matter, but also explain
the quark deconfinement phase transition successfully. The critical
temperature is given and the effect of $\omega$\- meson is addressed.
## I Introduction
Quark- meson coupling(QMC) model suggested by Guichon [1] is a famous hybrid
quark meson model, which can describe the saturation properties of nuclear
matter and many other properties of nuclei successfully. In this model, the
nuclear system was suggested as a collection of MIT bag, $\omega$\- meson and
$\sigma$\- meson and the interactions between quarks and mesons are limited
within the MIT bag regions because the quark cannot escape from the MIT bag.
This model had been extended by many authors, for example, including $\rho$
meson to discuss the neutron matter [2], adding hyperons to study the strange
hadronic matter [3], suggesting the density- dependent vacuum energy to
investigate the effect of environment [4] and etc. [5].
Although the QMC model is successful in describing many physical properties of
nuclear systems, but as was pointed in our previous paper [6], it has two
major shortcomings: (1) It is a permanent quark confinement model because the
MIT bag boundary condition cannot be destroyed by temperature and density. It
cannot describe the quark deconfinement phase transition. (2) It is difficult
to do nuclear many-body calculation beyond mean field approximation(MFA) by
means of QMC model, because we cannot find the free propagators of quarks and
mesons easily. The reason is that the interactions between quarks and mesons
are limited within the bag regions, the multireflection of quarks and mesons
by MIT bag boundary must be taken into account for getting the free
propagators. These two shortcomings actually inherited from the MIT bag.
To overcome these two shortcomings, in our previous paper [6], we suggested an
improved quark mass density- dependent(QMDD) model. The QMDD model was
suggested by Fowler, Raha and Weiner [7] many years ago. According to the QMDD
model, the masses of u, d, s quarks and corresponding antiquarks are given by
$\displaystyle m_{q}=\frac{B}{3n_{B}}(i=u,d,\bar{u},\bar{d})$ (1)
$\displaystyle m_{s,\bar{s}}=m_{s0}+\frac{B}{3n_{B}}$ (2)
where $m_{s0}$ is the current mass of the strange quark, $B$ is the bag
constant, $n_{B}$ is the baryon number density
$\displaystyle n_{B}=\frac{1}{3}(n_{u}+n_{d}+n_{s}),$ (3)
$n_{u},n_{d},n_{s}$ represent the density of u quark, d quark, and s quark,
respectively. As was explained and proved in Ref. [8, 9], the basic hypothesis
(1), (2) correspond to a confinement mechanism of quarks because when volume
$V\rightarrow\infty$, $n_{B}\rightarrow 0$, and then $m_{q}\rightarrow\infty$,
quark will be confined. Using QMDD model, many authors investigated the
dynamical and thermodynamical properties of strange quark matter and strange
star and found that the results given by QMDD model are nearly the same as
those obtained in MIT bag model [8-16]. But a nice advantage for QMDD model is
that the MIT bag boundary condition has been given up. The ansatz Eq. (1) is
employed to replace the MIT bag boundary.
But QMDD model still has two shortcomings: (1) It is still an ideal quark gas
model. No interactions between quarks exist except a confinement ansatz Eqs.
(1-2). (2) It still cannot explain the quark deconfinement phase transition
and give us a correct phase diagram as that given by lattice QCD because when
$n_{B}\rightarrow 0$,$m_{q}\rightarrow\infty$, $T\rightarrow\infty$ [8]. To
overcome the difficulty(2), we introduced a new ansatz that $m_{q}$ is not
only a function of density, but also depends on temperature T [8,12,15], we
suggested
$\displaystyle B=B_{0}[1-(T/T_{C})^{2}],0\leq T\leq T_{C},$ (4) $\displaystyle
B=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ ,T>T_{C}$ (5)
and extended the QMDD model to a quark mass density- and temperature-
dependent model(QMDTD) model. The ansatz(4) guarantees that $m_{q}\rightarrow
0$ when $T\rightarrow T_{C}$. It changes the permanent confinement
mechanism(MIT bag) to a nonpermanent confinement mechanism (Friedberg-Lee(FL)
soliton bag) in the QMDTD model. Since the vacuum density B equals to the
different value between the local false vacuum minimum and the absolute real
vacuum minimum of nonlinear scalar field in FL model, we introduced a
nonlinear scalar field to improve the QMDD model in Refs. [17, 22] and changed
the ad-hoc ansatz(4) to an output B(T) curve which can be calculated from the
improved QMDD model.
To improve the shortcoming(1) of QMDD model, we introduced the $\omega-$ meson
and $\sigma-$ meson in QMDD model to mimic the repulsive and attractive
interactions between quarks in Refs. [6, 23]. The interaction between quarks
and nonlinear $\sigma$ field forms a FL soliton bag [17, 22]. The $qq\omega$
and $qq\sigma$ interaction guarantees that we can get valid saturation
properties, the equation of state and the compressibility of nuclear matter
[6]. But the deconfinement phase transition has not yet been studied in Ref.
[6]. This motivates us to study the deconfinement properties of the improved
QMDD(IQMDD) model in this paper. We would like to emphasize that this is the
basic important advantage for IQMDD model, because the saturation properties
can be explained not only by IQMDD model but also by QMC model. The reason for
the explaination of quark deconfinement by IQMDD model is that MIT boundary
constraint has been dropped and interactions between quark and mesons have
been extended to the whole space. Since instead of the MIT bag in QMC model, a
FL soliton bag is introduced in IQMDD model, we can use our model to discuss
the quark deconfinement phase transition. The spontaneous breaking symmetry of
nonlinear $\sigma$ field will be restored and the soliton bag will disappear
at critical temperature [24-27].
The organization of this paper is as follows. In the next section, we give the
main formulae of the IQMDD model and effective potential at finite
temperature. The soliton solutions of IQMDD model at different temperature and
other numerical results are presented in the third section. The last section
contains a summary and discussions.
## II The IQMDD model at zero and finite temperature
The effective Lagrangian density of the IQMDD model is given by
$\displaystyle\mathcal{L}=\overline{\psi}[i\gamma^{\mu}\partial_{\mu}-m_{q}-f\sigma-g\gamma^{\mu}\omega_{\mu}]\psi\hskip
57.81621pt$
$\displaystyle+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-U(\sigma)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu},$
(6)
where $F_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$,
$\psi$ represents the quark field, $m_{q}=\frac{B}{3n_{B}}$ is the mass of
$u(d)$ quark, the $\sigma$ and $\omega$ field are not dependent on time, $f$
is the coupling constant between the quark field $\psi$ and the scalar meson
field $\sigma$, $g$ is the coupling constant between the quark field $\psi$
and the vector meson field $\omega_{\mu}$, $U(\sigma)$ is the self interaction
potential for $\sigma$ field. We omit the contribution of the s quark and
consider the nuclear system only in this paper. The potential field
$U(\sigma)$ is chosen as [18]
$\displaystyle
U(\sigma)=\frac{a}{2!}\sigma^{2}+\frac{b}{3!}\sigma^{3}+\frac{c}{4!}\sigma^{4}+B,$
(7) $b^{2}>3ac$ (8)
The condition (8) ensures that the absolute minimum of $U(\sigma)$ is at
$\sigma=\sigma_{v}\neq 0$. The potential $U(\sigma)$ has two minima: one is
the absolute minimum $\sigma_{v}$
$\displaystyle\sigma_{v}=\frac{3|b|}{2c}\left[1+\left[1-\frac{8ac}{3b^{2}}\right]^{\frac{1}{2}}\right],$
(9)
it corresponds to the physical vacuum, and the other is at $\sigma_{0}=0$, it
represents a metastable local false vacuum. We take $U(\sigma_{v})=0$ and the
bag constant $B$ can be expressed as
$\displaystyle-B=\frac{a}{2!}\sigma^{2}_{v}+\frac{b}{3!}\sigma^{3}_{v}+\frac{c}{4!}\sigma^{4}_{v}.$
(10)
From Eq. (6), we obtain the equation of motion for quark as
$(i\gamma^{\mu}\partial_{\mu}-m_{q}-f\sigma-g\gamma^{\mu}\emph{}\omega_{\mu})\psi=0,$
(11)
and the equations for the scalar meson field and vector meson field as
$\displaystyle\partial_{\mu}\partial^{\mu}\sigma+\frac{dU(\sigma)}{d\sigma}=-f\bar{\psi}\psi,$
(12)
$\partial_{\nu}F^{\nu\mu}+m_{\omega}\omega^{2}=g\bar{\psi}\gamma^{\mu}\psi.$
(13)
respectively. Using an approximation as that of the QMC model, we replace
$\sigma(\mathbf{r},t)\rightarrow\sigma(r),$
$\omega_{\mu}(\mathbf{r},t)\rightarrow\delta_{\mu 0}\omega(r)$ and consider a
fixed occupation number of valence quarks (3 quarks for nucleons, and quark-
antiquark pair for mesons)only. In the following, we will discuss the ground
state solution of the system. The Hamiltonian density is
$\displaystyle\mathcal{H}=\psi^{+}[\frac{1}{i}\vec{\alpha}\cdot\vec{\nabla}+\beta(m_{q}+f\sigma)+g\omega]\psi+\frac{1}{2}\Pi_{\sigma}^{2}$
$\displaystyle+\frac{1}{2}(\nabla\sigma)^{2}+U(\sigma)-\frac{1}{2}(\nabla\omega)^{2}-\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}.$
(14)
where $\vec{\alpha}$ and $\beta$ are the Dirac matrices $\Pi_{\sigma}$ is
conjugate field of the scalar meson field. One can construct a Fock space of
quark states and expand the operator $\psi$ in terms of annihilation and
creation operators on the space with spinor function $\varphi^{\pm}_{n}$,
which satisfies the Dirac equation [17]:
$[\vec{\alpha}\cdot\vec{p}+\beta(m_{q}+f\sigma)+g\omega]\varphi^{\pm}_{n}=\pm\epsilon_{n}\varphi^{\pm}_{n}.$
(15)
The functions $\varphi_{n}$ satisfies the normalized condition
$\int\varphi_{n}^{+}\varphi_{n}d^{3}r=1$. From Eq. (14), the total energy of
the system is given by
$E(\sigma,\omega)=\sum_{n}\epsilon_{n}+\int[\frac{1}{2}(\nabla\sigma)^{2}+U(\sigma)-\frac{1}{2}(\nabla\omega)^{2}-\frac{1}{2}m_{\omega}^{2}\omega^{2}]d^{3}r.$
(16)
Substituting Eq. (15) into Eqs. (12, 13), and using the variational principle
under the spherical symmetric condition, we find when $\sigma$ and $\omega$
satisfy the follow equations
$-\nabla_{r}^{2}\sigma+\frac{dU(\sigma)}{d\sigma}=-f\sum_{n}\bar{\varphi_{n}}\varphi_{n},$
$-\nabla_{r}^{2}\omega+m_{\omega}^{2}\omega=g\sum_{n}\varphi_{n}^{+}\varphi_{n}.$
(17)
respectively where $\overline{\varphi}_{n}=\varphi_{n}^{+}\gamma_{0}$, we have
the minimum of $E(\sigma,\omega)$.
We discuss the ground state solution of the system now. The quark spinor in
the lowest state is assumed [17-19]:
$\varphi=\left(\begin{array}[]{c}u(r)\\\
i(\frac{\vec{\sigma}\cdot\vec{r}}{r})v(r)\\\ \end{array}\right)\chi_{m},\ \
\chi_{m}=\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)or\left(\begin{array}[]{c}0\\\ 1\\\ \end{array}\right),$
(18)
where $\vec{\sigma}$ are the Pauli matrices. Substituting Eq. (18) into Eq.
(15), we get the equations of spinor components $u$ and $v$ as
$\frac{du(r)}{dr}=-[\epsilon+m_{q}+f\sigma(r)-g\omega]v(r),$
$\frac{dv(r)}{dr}=-\frac{2}{r}v(r)+[\epsilon-m_{q}-f\sigma(r)-g\omega]u(r).$
(19)
respectively. The normalized condition then reads as
$4\pi\int^{\infty}_{0}[u^{2}(r)+v^{2}(r)]r^{2}dr=1$.
From Eq. (17) and Eq. (18), after summation of the quark states, we obtain the
equation of motion of the $\sigma,\omega$ field
$\frac{d^{2}\sigma}{dr^{2}}+\frac{2}{r}\frac{d\sigma}{dr}-\frac{dU(\sigma)}{d\sigma}=Nf(u^{2}-v^{2}),$
(20)
$\frac{d^{2}\omega}{dr^{2}}+\frac{2}{r}\frac{d\omega}{dr}-m_{\omega}^{2}\omega=-Ng(u^{2}+v^{2})\equiv
F(r).$ (21)
where, the number of quarks is $N=3$ for baryons and $N=2$ for mesons. In the
following discussions, we only consider the case $N=3$. To get a self-
consistent solution of Eqs. (19, 20, 21), we select the boundary conditions
for quark field and $\sigma$ field as [17, 22]
$\displaystyle v(r=0)=0,u(r=\infty)=0,$
$\displaystyle\sigma^{\prime}(r=0)=0,\sigma(r=\infty)=\sigma_{v}.$ (22)
respectively. Noting that the $r\rightarrow\infty$ asymptotic behavior of the
$\omega$ field given by Eq. (21) tends to an exponent decay wave because
$F(r\rightarrow\infty)\rightarrow 0$ for a soliton bag, we can find the
corresponding Green function $G(r,r^{\prime})$ easily and obtain the $\omega$
field by integral [20]:
$\displaystyle\omega(r)=\int_{0}^{\infty}r^{\prime
2}dr^{\prime}F(r^{\prime})G(r,r^{\prime})$ (23)
The numerical results of $u(r),v(r),\sigma(r)$ and $\omega(r)$ will be shown
in next section.
In order to study the deconfinement phase transition, we turn to extend IQMDD
model to finite temperature. The appropriate framework is the finite
temperature quantum field theory. The finite temperature effective potential
plays a central role within this framework. Under mean field approximation,
the meson field operators can be replaced by their expectation values,
$\omega_{\mu}\rightarrow\bar{\omega}_{\mu}=\delta_{\mu
0}\bar{\omega}=\delta_{\mu 0}\frac{g}{m_{\omega}^{2}}\rho_{B0}$[20, 21]. Using
the method of Dolan and Jackiw [24], up to one-loop approximation, the
effective potential reads:
$\displaystyle
V(\sigma;T;\mu;V_{\omega})=U(\sigma)+V_{B}(\sigma;T)+V_{F}(\sigma;T;\mu;V_{\omega}),$
(24)
where
$\displaystyle V_{\omega}=g\bar{\omega}=\frac{g^{2}}{m_{\omega}^{2}}\rho_{B0}$
(25)
is the contribution of $\omega$-field, $\rho_{B0}$ is saturation density of
nuclear matter, T is temperature and
$\displaystyle
V_{B}(\sigma;T)=\frac{T^{4}}{2\pi^{2}}\int^{\infty}_{0}dxx^{2}\mathrm{ln}\left(1-e^{-\sqrt{(x^{2}+m_{\sigma}^{2}/T^{2})}}\right),$
(26) $\displaystyle
V_{F}(\sigma;T;\mu;V_{\omega})=-12\sum_{n}\frac{T^{4}}{2\pi^{2}}\int^{\infty}_{0}dxx^{2}\mathrm{ln}\left(1+e^{-(\sqrt{(x^{2}+m_{qn}^{2}/T^{2})}-\mu_{n}/T+V_{\omega}/T)}\right),$
(27)
where the minus sign of Eq. (27) is the consequence of Fermi-Dirac statistics.
The degenerate factor 12 comes from: 2(particle and antiparticle), 2(spin),
3(color). $m_{\sigma}$ and $m_{q}$ are the effective masses of the scalar
field $\sigma$ and the quark field respectively:
$\displaystyle m^{2}_{\sigma}$ $\displaystyle=$ $\displaystyle
a+b\sigma(T)+\frac{c}{2}\sigma^{2}(T)$ (28) $\displaystyle m_{q}$
$\displaystyle=$ $\displaystyle\frac{B(T)}{n_{q}}+f\sigma(T).$ (29)
We see from Eqs. (24-27) that the scalarlike interaction $f\psi^{+}\sigma\psi$
gives contribution to effective masses of quark and $\sigma$ meson and then
forms a confined soliton bag, the vectorlike interaction
$g\psi^{+}\gamma^{\mu}\omega_{\mu}\psi$ gives contribution to an effective
chemical potential of quarks. In fact, this finite temperature effective
potential for FL model had been calculated by many others authors including us
[22, 25-27] , but without $\omega$\- field.
In the soliton bag model, the finite temperature vacuum energy density $B(T)$
is defined as
$\displaystyle
B(T;\mu;V_{\omega})=V(\sigma_{0};T;\mu;V_{\omega})-V(\sigma_{v};T;\mu;V_{\omega}).$
(30)
It is the different from the values at the perturbative false vacuum state and
the values at the physical real vacuum state of the finite temperature
effective potential. At critical temperature $T_{C}$ of quark deconfinement
phase transition, $B$ equal to zero: $B(T_{C})=0$.
## III results
Before numerical calculations, we consider the parameters of IQMDD model at
first. To guarantee our model can be used not only to explain the saturation
properties of nuclear matter, but also to describe the quark deconfinement
phase transition, we fix our parameters as those in Ref. [6]. Hereafter we fix
the parameters as: at zero temperature, the bag constant $B$=174 MeV$fm^{-3}$,
the masses of $\omega$\- meson and $\sigma$\- meson $m_{\omega}=783$ MeV and
$m_{\sigma}=509$ MeV, the coupling constant $f=5.45$ and $g=3.37$ respectively
and the mass of nucleon $M_{N}=939$ MeV, $b=-8400$ MeV. As was shown in Ref.
[6], this set of parameters can give us reasonable properties of nuclear
matter, including the value of saturation point, the equation of state and the
compressibility. we will prove in this section that we can also obtain a
reasonable quark deconfinement critical temperature $T_{C}$ by means of this
set of parameters.
The set of coupled differential equations (19), (20) with boundary conditions
Eq. (22) can be solved numerically at zero temperature. Our results at zero
temperature are shown in Figs. (1-4). The variation of the scalar field
$\sigma$ as a function of the radius r is presented in Fig. 1. We see that the
value of $\sigma$ inside the hadron is very different from that of outside:
inside $\sigma$ is less than zero, and outside $\sigma\rightarrow\sigma_{v}$.
The transitional values of $\sigma$ field through the surface from inside to
outside is abrupt. The curve in Fig. 1 is very similar as that given by Refs.
[17, 22] where the $\omega$\- meson is omitted. This is reasonable because the
main contribution to form a soliton bag comes from the nonlinear $\sigma$
field. The variation of wave function of quark field is shown in Fig. 2 and
Fig. 3. In Fig. 2, we plot the curves of quark wave function $u,v$ vs. $r$
respectively. In Fig. 3, we plot the curve of quark density $u^{2}-v^{2}$ vs.
$r$. The soliton bag is exhibited in this figure transparently. The curve of
$\omega$\- field vs. $r$ given by Eq. (21) is shown in Fig. 4. This solution
is obtained from Green function method and Eq. (23). We see from this Figure
that the curve of $\omega$\- field decays considerably when $r$ is large than
the soliton bag radius.
Now we turn to investigate the case of finite temperature. The effective
potential at finite temperature can be obtained by numerical calculations from
the set of Eqs. (24-27). The curves of effective potential
$U(\sigma;T;\mu;V_{\omega})$ at different temperatures $T=0$ MeV, $T=80$ MeV,
$T=127$ MeV and $T=204$ MeV are shown in Fig. 5 respectively. The shape of the
effective potential confirms that a first order phase transition will take
place [22, 25-27]. We see from Fig. 5 there are two vacua where one
corresponds to the physical vacuum and the other corresponds to the false
vacuum when $T<127$ MeV. But when $T$ increases to a critical temperature
$T_{C}=127$ MeV, these two vacua degenerate and the values of bag constant
tends to zero.
To show this behaviour more clearly, we plot the bag constant $B$ vs. $T$
curve by solid curve in Fig. 6, we see that the bag constant $B$ decreases as
temperature $T$ increases. When $T$ approaches to $T_{C}$, $B$ approaches to
zero and quark deconfined phase transition happens.
When temperature increases to the regions 127 MeV $<T<204$ MeV, as shown in
Fig. 5 the physical vacuum becomes instable and the purturbative vacuum
becomes stable. The soliton solution tends to disappear [22]. When $T$
approached to 204 MeV, the effective potential becomes to have an unique
minimum only. Such potential no longer ensures the existence of the soliton
bag anymore because the spontaneous breaking symmetry has been restored. We
have zero solution $\sigma=0$ in the regions $T\geq 204$ MeV only.
To illustrate our soliton solutions more transparently, we plot the soliton
solutions by fixed the temperature $T=0$ MeV, $T=80$ MeV and $T=125$ MeV in
Fig. 7. It is shown in Fig. 7 that the curves stretch slowly to infinite with
increasing temperature. The radius of soliton increases and the skin of the
soliton becomes thicker when the temperature increases. It is of course very
reasonable.
All characters of the soliton solutions found by IQMDD model are almost the
same as those given by Ref. [22] where the $\omega$ field has not been take
into account. But we hope to emphasize although the $\omega$ field will not
affect the main characters of the soliton solutions, but it will change their
detailed behavior. To address the effect of $\omega$ field on the
deconfinement transition, as an example, we plot the $B(T)$ curve without
$\omega$ field by dotted line in Fig. 6. We find the critical temperature from
the condition $B(T_{C})=0$ changes to $T_{C}=140$ MeV which is higher than 127
MeV where the $\omega$ field existed. If we notice that the interaction of
$qq\omega$ is repulsive, and then it can help to deconfine quarks, the
decrease of $T_{C}$ due to $\omega$\- field is natural.
## IV Summary and discussion
By using the finite temperature quantum field theory, we calculate the finite
temperature effective potential and extend our previous discussions to finite
temperature. It is shown that the IQMDD model can not only describe the
saturation properties of nuclear matter, but also explain the quark
deconfinement phase transition successfully. We have shown the soliton
solution curves for different temperatures and found the critical temperature
$T_{C}=127$ MeV. The $\omega$\- field in IQMDD model is important. The
repulsive $qq\omega$ interaction plays the central role to describe the
saturation properties, and affect the critical temperature remarkably.
Comparing to naive QMC model, the advantage of IQMDD model is obvious because
instead of MIT bag, a Friedberg-Lee soliton bag exists in this model.
Of course, there are still shortcomings in the IQMDD model. For example, the
interactions between quarks and mesons are still isospin independent. The
chiral phase transition cannot be discussed by this model because it is lack
of chiral symmetry. To Overcome these shortcomings, we hope to add the
$\rho$\- meson and $\pi$\- meson in IQMDD model in the near future.
###### Acknowledgements.
This work is supported in part by the National Natural Science Foundation of
People’s Republic of China. C Wu is extremely grateful to Dr. H Mao for useful
discussions and correspondence.
## References
* (1) Guichon P A M 1988 Phys. Lett. B 200 235
* (2) Saito K and Thomas A W 1994 Phys. Lett. B 327 9; Saito K and Thomas A W 1995 Phys. Rev. C 52 2789
* (3) Wang P, Su R K, Song H Q and Zhang L L 1999 Nucl. Phys. A 653 166
* (4) Jin X and Jennings B K 1996 Phys. Lett. B 374 13; Jin X and Jennings B K 1996 Phys. Rev. C 54 1427
* (5) Song H Q and Su R K 1994 Phys. Lett. B 328 179; Song H Q and Su R K 1996 J. Phys. G: Nucl. Part. Phys. 22 1025
* (6) Wu C, Qian W L and Su R K 2008 Phys. Rev. C 77 015203
* (7) Fowler G N, Raha S and Weiner R M 1981 Z. Phys. C 9 271
* (8) Zhang Y and Su R K 2002 Phys. Rev. C 65 035202 ; Zhang Y and Su R K 2003 Phys. Rev. C 67 015202
* (9) Benrenuto O G and Lugones G 1995 Phys. Rev. D 51 1989; Lugones G and Benrenuto O G 1995 Phys. Rev. D 52 1276
* (10) Wen X J, Zhong X H, Peng G X, Shen P N and Ning P Z 2005 Phys. Rev. C 72 015204
* (11) Peng G X, Chiang H C and Ning P Z 2000 Phys. Rev. C 62 025801
* (12) Zhang Y and Su R K 2004 J. Phys. G: Nucl. Part. Phys. 30 811
* (13) Gupta V K, Gupta A, Singh S and Anand J D 2003 Int. J. Mod. Phys. D 12 583
* (14) Shen J Y, Zhang Y, Wang B and Su R K 2005 Int. J. Mod. Phys. A 20 7547
* (15) Zhang Y, Su R K, Ying S Q and Wang P 2001 Europhys. Lett. 53 361
* (16) Yin S and Su R K 2008 Phys. Rev. C 77 055204
* (17) Wu C, Qian W L and Su R K 2005 Phys. Rev. C 72 035205; Wu C, Qian W L and Su R K 2005 Chin. Phys. Lett. 22 1866
* (18) Saly R and Sundaresan M K 1984 Phys. Rev. D 29 525
* (19) Walecka J D 1995 Theoretical Nuclear and Subnuclear Physics (New York: Oxford university press)
* (20) Serot B D and Walecka J D 1986 Advances in Nuclear Physics 16 ed J W Negele and E Vagt (New York: Plenum)
* (21) Saito K and Thomas A W 1994 Phys. Lett. B 327 9
* (22) Mao H, Su R K and Zhao W Q 2006 Phys. Rev. C 74 055204
* (23) Qian W L and Su R K 2005 Int. J. Mod. Phy. A 20 1931
* (24) Dolan L and Jackiw R 1974 Phys. Rev. D 9 3320
* (25) Gao S, Wang E K and Li J R 1992 Phys. Rev. D 46 3211
* (26) Li M, Birse M C and Wilets L 1987 J. Phys. G: Nucl. Part. Phys. 13 1
* (27) Holman R, Hsu S, Vachaspati T and Watkins R 1992 Phys. Rev. D 46 5352
Figure 1: The $\sigma$ field as a functions of r at zero temperature. Figure
2: Quark wave functions in arbitrary unit as a functions of r. Figure 3: The
quark density $u^{2}(r)-v^{2}(r)$ in arbitrary unit as a functions of r.
Figure 4: The $\omega$ field as a functions of r. Figure 5: The temperature-
dependent effective potential. Figure 6: The bag constant B(T) as functions of
T, the solid line corresponds to IQMDD model with $\omega$-meson and the
dotted line corresponds to the same model without $\omega$-meson. Figure 7:
The soliton solutions for different temperature T=0 MeV, 80 MeV, 125 MeV
|
arxiv-papers
| 2008-09-07T13:04:59
|
2024-09-04T02:48:57.691949
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Wu, Ru-Keng Su",
"submitter": "Chen Wu",
"url": "https://arxiv.org/abs/0809.1222"
}
|
0809.1250
|
# Relativistic gravitational deflection of light and its impact on the
modeling accuracy for the Space Interferometry Mission
Slava G. Turyshev Jet Propulsion Laboratory, California Institute of
Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA Sternberg Astronomical
Institute, 13 Universitetskij Prospect, 119992 Moscow, Russia111Email:
turyshev@jpl.nasa.gov, turyshev@sai.msu.ru
###### Abstract
We study the impact of relativistic gravitational deflection of light on the
accuracy of future Space Interferometry Mission (SIM). We estimate the
deflection angles caused by the monopole, quadrupole and octupole components
of gravitational fields for a number of celestial bodies in the solar system.
We observe that, in many cases, the magnitude of the corresponding effects is
significantly larger than the $1~{}\mu$as accuracy expected from SIM. This
fact argues for the development of a relativistic observational model for the
mission that would account for the influence of both static and time-varying
effects of gravity on light propagation. Results presented here are different
from the ones obtained elsewhere by the fact that we specifically account for
the differential nature of the future SIM astrometric measurements. We also
obtain an estimate for the accuracy of possible determination of the
Eddington’s parameter $\gamma$ via SIM global astrometric campaign; we
conclude that accuracy of $\sim 7\times 10^{-6}$ is achievable via
measurements of deflection of light by solar gravity.
Interferometric astrometry; SIM; tests of general relativity; solar system.
## I Introduction
The last quarter of the 20th century has changed the status of Einstein’s
general theory of relativity from a purely theoretical discipline to a
practically important science. Today general relativity is the standard theory
of gravity, especially where the needs of astronomy, astrophysics, cosmology
and fundamental physics are concerned Turyshev-etal-2007 ; Turyshev-2008 ;
Soffel-etal-2003 . As such, this theory is used for many practical purposes
involving spacecraft navigation, geodesy and time transfer. Present accuracy
of astronomical observations already requires relativistic description of
light propagation as well as the relativistically correct treatment of the
dynamics of the extended celestial bodies Kopeikin-Makarov-2007 . As a result,
some of the leading static-field post-Newtonian perturbations in the dynamics
of the planets, the Moon and artificial satellites have been included in the
equations of motion, and in time and position transformation. It is also well
understood that effects due to non-stationary behavior of the solar system
gravitational field as well as its deviation from spherical symmetry should be
also considered Kopeikin-1997 and implemented in the appropriate models.
Space-based astrometry has brought about a renaissance in the entire field of
astrometry that is perhaps the most fundamental, and oldest of all areas in
astronomy Unwin-etal-2008 . The ESA Hipparcos mission, which operated from
1989-1993, yielded an astrometric catalog of 118,000 stars down to 12.5
magnitude, with positional accuracy of 1 mas for stars brighter than V = 11.
The European Space Agency (ESA) is now developing the Gaia mission as a next
generation astrometric survey mission Perryman-2001 ; Perryman-2002 , which is
expected to produce a catalog of $\sim 10^{9}$ stars, with accuracy $\simeq$
20–25 microarcsec ($\mu$as) for stars brighter than V = 15. Precision
astrometry remains a cornerstone of the field and is poised to make a major
impact on many fields of modern astronomy, astrophysics, and cosmology Unwin-
etal-2008 .
NASA’s SIM PlanetQuest mission, hereinafter SIM, as another example of a
space-based facility instrument for astrometry. The acronym SIM stands for
Space Interferometry Mission. SIM will be the first space-based Michelson
interferometer for astrometry. The instrument will operate in the optical
waveband using a 9-m baseline between the apertures. With a global astrometry
accuracy of $3~{}\mu$as for stars brighter than V = 20, it will measure
parallaxes and proper motions of stars throughout the Galaxy with
unprecedented accuracy. Operating in a narrow-angle mode, it will achieve a
positional accuracy of $0.6~{}\mu$as for a single measurement, equivalent to a
differential positional accuracy at the end of the nominal 5-year mission of =
$0.1~{}\mu$as. This performance is about 1000 times better than existing
capabilities on the ground or in space, and about 100 times better than the
upcoming Gaia mission, for differential measurements. Such high accuracy will
allow SIM to detect and measure masses of terrestrial planets around stars in
our Galactic neighborhood (see Unwin-etal-2008 for review).
SIM is a targeted mission which measures the astrometric positions of stars,
referencing the measurements to a grid of 1302 stars covering the entire sky.
Its scheduling is highly flexible, in both the order of observations, their
cadence, and the accuracy of each individual measurement. This contrasts with
the Hipparcos and Gaia missions, which scan the entire sky according to a pre-
determined scanning pattern. Many astrometry experiments can make effective
use, or in some cases require, this pointing capability for instance, searches
for terrestrial planets (especially in multiple planet systems), stellar
microlensing events, orbits of eccentric binary systems, and variable targets
such as X-ray binaries and active galactic nuclei. Currently, the ICRF,
defined by the locations of 212 extragalactic radio sources Johnston-etal-1995
; Ma-1998 with most having errors less than 1 mas, is the standard frame for
astrometry. SIM is expected to yield an optical reference frame at a level of
about $3~{}\mu$as; it will be tied’ to the ICRF by observing a number of
radio-loud quasars in common.
Figure 1: Geometry of gravitational deflection of starlight by the Sun.
In this paper, we discuss the rapidly forming field of applied general
relativity to demonstrate its influence on the high-precision astrometry.
Recent advances in the accuracy of astrometric observations have demonstrated
importance of taking into account the relativistic effects introduced by the
solar system’s gravitational environment. It is known that the reduction of
the Hipparcos data has necessitated the inclusion of stellar aberration up to
the terms of the second order in $v/c$, and the general relativistic treatment
of light bending due to the gravitational field of the Sun Perryman92 and
Earth Gould93 . Even higher modeling accuracy is anticipated for Gaia
Klioner-2004 .
Prediction of the gravitational deflection of light was one of the first
successes of general relativity. Since the first confirmation by the
Eddington’s expedition in 1919, the effect of gravitational deflection has
been studied quite extensively and currently analysis of almost every precise
astronomical measurement must take this effect into account Modest96 ;
Brumberg-1972 ; Brumberg-1991 ; Turyshev-etal-2007 ; Turyshev-2008 . According
to general relativity, the light rays propagating near a gravitating body are
achromatically deflected by the body’s relativistic gravity field. The whole
trajectory of the light ray is bent towards the body by an angle depending on
the strength of the body’s gravity. The solar gravity field produces the
largest effect on the light traversing the solar system. To first order in the
gravitational constant, $G$, the solar deflection angle $\theta^{\odot}_{\tt
gr}$ depends only on the solar mass $M_{\odot}$ and the impact parameter $d$
relative to the Sun Will_book93 :
$\theta^{\odot}_{\tt gr}={4GM_{\odot}\over
c^{2}d}\cdot\frac{1+\cos\chi}{2}=1^{\prime\prime}.751\Big{[}\frac{R_{\odot}}{d}\Big{]}\frac{1+\cos\chi}{2},$
(1)
where $R_{\odot}$ solar radus. The absolute magnitude for the light deflection
angle is maximal for the rays grazing the sun, e.g. $\theta^{\odot}_{\tt
gr}=1.751$ seconds of arc. Most of the measurements of the gravitational
deflection to date involved the solar gravity field, planets in the solar
system or gravitational lenses. For the future astrometric observations with
SIM, in addition to the Sun, effect of planetary gravitatiobal deflections of
light must also be considered. The most precise measurement of the light
deflection with the planet Jupiter was done in Fomalont-Kopeikin03 .
Relativistic deflection of light has been observed, with various degrees of
precision, on distance scales of $10^{9}$ to $10^{21}$ m, and on mass scales
from $10^{-3}$ to $10^{13}$ solar masses, the upper ranges determined from the
gravitational lensing of quasars Dar92 ; TreuhaftLowe91 .
In the case of SIM, the star is assumed to be at a very large distance
compared to the Sun, and $\chi$ is the angular separation between the
deflector and the star. With the space observations carried out by SIM, $\chi$
is not necessarily a small angle. The relevant geometry and notations are
shown in Fig. 1. In this figure we emphasized the fact that the difference of
the apparent position of the source from it’s true position depends on the
impact parameter of the incoming light with respect to the deflector. For the
astrometric accuracy of a few $\mu$as and, in the case when the Sun is the
deflector, positions of all observed sources experiencing such a displacement.
This is why, in order to correctly account for the effect of gravitational
deflection, it is important to process together the data taken with the
different separation angles from the deflector. In the wide-angle astrometry
mode SIM, will be observing the sky in a 15∘ patches of sky (called field of
regard or FoR) making a set of differential observations within the FoR.
Therefore, this differential nature of the measurements would result in
minimizing the contribution of the gravitational deflection on the single
measurement. To reflect this fact, we will present results for two types of
astrometric measurements, namely for the absolute (single ray deflection) and
differential (two sources separated by the 15∘ field of regard) observations.
A major objective of this paper is to show that, before microarcsecond-level
astrometry will become a powerful tool for 21-st century astronomy, there is a
need for an adequate modeling necessary to match this new frontier of
astrometric accuracy. The prospect of new high precision astrometric
measurements from space with SIM requires inclusion of relativistic effects at
the $(v/c)^{3}$ level Turyshev98 ; Unwin-etal-2008 . At the level of accuracy
expected from SIM, even more subtle gravitational effects on astrometry from
within the solar system will start to become apparent, such as the monopole
and the quadrupole components of the gravitational fields of the planets
Sovers98 and the gravito-magnetic effects caused by their motions and
rotations. Thus, the identification of all possible sources of “astrophysical”
noise that may contribute to the future SIM astrometric campaign, is well
justified.
This work is organized as follows: Section II discusses the influence of the
relativistic deflection of light by the monopole components of the
gravitational fields of the solar system’s bodies. We present the model and
our estimates for the most important effects that will be influencing
astrometric observations of a few $\mu$as accuracy, that will be made from
within the solar system. Section III will specifically address three most
intense gravitational environments in the solar system, namely the vicinities
of the Sun, Jupiter and Earth. In Section IV we will discuss the effects of
the gravitational deflection of light by the higher gravitational multipoles
(both mass and current ones) of some of the bodies in the solar system. We
derive constraints on the navigation of the spacecraft and the accuracy of the
solar system ephemerides. In Section V we investigate the possibility of
improving the accuracy of the Eddington’s parameter $\gamma$ via astrometric
tests of general relativity in the solar system. We also discuss the
opportunity to measure the solar acceleration towards the Galactic Center with
SIM. We will conclude the paper with the discussion of the results obtained
and our recommendations for future studies.
## II Gravity Contributions to the Local Astrometric Environment
In this Section we develop a model for light propagation that will be used to
estimate various relativistic effects due to gravitational deflection of light
by the solar system’s bodies.
### II.1 Relativistic deflection of light by the gravity monopole
The first step into a relativistic modeling of a light path consists of
determining the direction of the incoming photon as measured by an observer
located in the solar system as a function of the barycentric coordinate
position of the light source. Apart from second and third orders of velocity
aberration the only other sizable effect is due to the bending of light rays
in the gravitational field of solar system bodies Brumberg-Klioner-
Kopeikin-1990 ; Turyshev98 . Effects of the gravitational monopole deflection
of light are the largest among those in the solar system.
Generalizing on a phenomenological parameterization of the gravitational
metric tensor field, which Eddington originally developed for a special case,
a method called the parameterized post-Newtonian (PPN) formalism has been
developed (see Turyshev-2008 for discussion). This method represents the
gravity tensor’s potentials for slowly moving bodies and weak inter-body
gravity, and is valid for a broad class of metric theories, including general
relativity as a unique case. The several parameters in the PPN metric
expansion vary from theory to theory, and they are individually associated
with various symmetries and invariance properties of the underlying theory
(see Will_book93 for details).
If (for the sake of simplicity) one assumes that Lorentz invariance, local
position invariance and total momentum conservation hold, the metric tensor
for a single, slowly-rotating gravitational source is given by:
$\displaystyle g_{00}$ $\displaystyle=$ $\displaystyle
1-2\frac{GM}{c^{2}r}\Big{(}1-J_{2}\frac{R^{2}}{r^{2}}\frac{3\cos^{2}\theta-1}{2}\Big{)}+{\cal
O}(c^{-4}),$ $\displaystyle g_{0i}$ $\displaystyle=$ $\displaystyle
2(\gamma+1)\frac{G[\vec{\cal S}\times\vec{r}]_{i}}{c^{3}r^{3}}+{\cal
O}(c^{-5}),$ (2) $\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle-\delta_{ij}\Big{[}1+2\gamma\frac{GM}{c^{2}r}\Big{(}1-J_{2}\frac{R^{2}}{r^{2}}\frac{3\cos^{2}\theta-1}{2}\Big{)}+{\cal
O}(c^{-5}),$
where $M$ and $\vec{\cal S}$ being the mass and angular momentum of the body,
$J_{2}$ and $R$ are the body’s quadrupole moment and its radius, and $G$ is
the universal gravitational constant, $r$ is the distance from the center of
the body to a particular point. The $1/c^{2}$ term in $g_{00}$ is the
Newtonian limit; the $1/c^{3}$ term in $g_{0i}$ and the $1/c^{2}$ term in
$g_{ij}$, are post-Newtonian corrections. All of these terms are required to
describe light propagation phenomena to the first post-Newtonian order.
The Eddington parameter $\gamma$ in the Eqs. (II.1) represents the measure of
the curvature of the space created by a unit rest mass Will_book93 . Note that
general relativity, when analyzed in standard gauge of the PPN formalism (see
Will_book93 ; Will-lrr-2006-3 for details), gives: $\gamma=1$. The Brans-
Dicke theory is the most famous among the alternative theories of gravity. It
contains, besides the metric tensor, a scalar field $\phi$ and an arbitrary
coupling constant $\omega$, related to this PPN parameter as
$\gamma={1+\omega\over 2+\omega}$. The stringent observational bound resulting
from the 2003 experiment with the Cassini spacecraft require that
$|\omega|\gtrsim 40000$ Bertotti-Iess-Tortora-2003 ; Will-lrr-2006-3 . There
exist additional alternative theories that provide guidance for gravitational
experiments Will-lrr-2006-3 .
Metric tensor Eq. (II.1) can be used to derive expressions need to describe
propagation of electro-magnetic signals between any of the two points in
space. Following the standard procedure of integrating light geodesics (see
Brumberg-1972 ; Brumberg-1991 for details), the corresponding light-time
equation for a single deflecting body can be derived in the following form
$t_{2}-t_{1}=\frac{r_{12}}{c}+(1+\gamma)\frac{GM}{c^{3}}\ln\left[\frac{r_{1}+r_{2}+r_{12}}{r_{1}+r_{2}-r_{12}}\right]+{\cal
O}(c^{-5}),$ (3)
where $t_{1}$ refers to the signal transmission time, and $t_{2}$ refers to
the reception time. $r_{1,2}$ are the barycentric positions of the transmitter
and receiver, and $r_{12}$ is their spatial separation (see Moyer-2003 for
details). Also, for the moment, we neglected the presence of $J_{2}$ and
$\vec{\cal S}$ in the Eqs. (II.1), but will investigate contributions of these
and other multipoles in Sec. IV.
For a realistic observing scenario with the SIM, the sources of light are
located far out side the solar system, $r_{2}\ll r_{1}\equiv r_{S}$ and
$r_{12}$ can be approximated as $r_{12}=|\vec{r}_{1}-\vec{r}_{2}|\simeq
r_{1}-(\vec{n}_{S}\cdot\vec{r}_{2})$, where we introduced a notation
$\vec{n}_{S}=\vec{r}_{1}/r_{1}$. This approximation allows one to represent
the expression in the square brackets in Eq. (3) as follows:
$\frac{r_{1}+r_{2}+r_{12}}{r_{1}+r_{2}-r_{12}}\simeq\frac{2r_{1}+r_{2}-(\vec{n}_{S}\cdot\vec{r}_{2})}{r_{2}+(\vec{n}_{S}\cdot\vec{r}_{2})}.$
(4)
Figure 2: Basic geometry of light propagation in the stellar interferometry.
The SIM instrument does not directly measure the angular separation between
stars, but the projection of each star direction vector onto the
interferometer baseline by measuring the pathlength delay of starlight as it
passes through the two arms of the interferometer. The SIM instrument will
precisely measure optical path difference (OPD) between the wavefronts of
light received by the two telescopes forming the interferometric baseline (see
Fig. 2). (In fact, for SIM, with its 10 m baseline, such a measurement will be
done with a precision at the level of 1 picometer.) This difference will
result in the different internal OPDs needed to apply in order to coherently
add the signals. The delay measurement is made by a combination of internal
metrology measurements to determine the distance the starlight travels through
each arm, external metrology measurements that determine the length and local
orientation of the baseline, and a measurement of the central white light
fringe to determine the point of equal optical pathlength Unwin-etal-2008 .
Therefore, the OPD is the main observable that the interferometric instrument
will measure; relativistic modeling this delay will be among the main
objectives of the upcoming SIM modeling effort.
An interferometer measures optical path difference (OPD) between the
wavefronts of light received by the two telescopes forming the interferometric
baseline $\vec{b}$ at points $\vec{r}_{2}$ and
$\vec{r}^{\prime}_{2}=\vec{r}_{2}+\vec{b}$. To account for that fact we need
to determine the temporal difference between the signals received at these
telescopes which is
$\ell=c(t^{\prime}_{2}-t_{1})-c(t_{2}-t_{1})=c(t^{\prime}_{2}-t_{2})$. The
first term in Eq. (3) is the geometric delay. Using approximations $(b\ll
r_{2}\ll r_{1})$, it is easy to see that this term leads to the approximate
expression for geometric delay, given as $\ell_{\rm
geom}=r_{12}^{\prime}-r_{12}=-(\vec{b}\cdot\vec{n}_{S})/(1-\vec{n}_{S}\cdot\dot{\vec{r}}_{2}/c)$.
In this paper, we concern with only the largest contributions from the
gravitational defection of light, thus most of the velocity-dependent terms
will discarded (see Sovers98 ; Kopeikin-Schafer-1999 for details).
The second term Eq. (3) is the relativistic delay $\ell_{\rm gr}$, the focus
of this work. Using Eq. (4) one can present expression in the square brackets
of Eq. (3) as below
$\frac{2r_{1}+{r_{2}}^{\prime}-(\vec{n}_{S}\cdot\vec{r}_{2}{}^{\prime})}{2r_{1}+r_{2}-(\vec{n}_{S}\cdot\vec{r}_{2})}~{}\frac{r_{2}+(\vec{n}_{S}\cdot\vec{r}_{2})}{{r_{2}}{}^{\prime}+(\vec{n}_{S}\cdot\vec{r}_{2}{}^{\prime})}\simeq
1-\frac{1}{r_{2}}\,\frac{\vec{b}(\vec{n}_{S}+\vec{n}_{2})}{1+(\vec{n}_{S}\cdot\vec{n}_{2})}.$
(5)
In the first order in gravitational constant, one can add individual
interferometric delays due to the gravity of the bodies along the light path.
As a result, the general relativistic contribution to the OPD $\ell_{\tt
gr}=c\tau_{\tt gr}$ takes the following approximate form:
$\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\Big{[}{\vec{b}(\vec{n}_{S}+\vec{n}_{B})\over
1+(\vec{n}_{S}\cdot\vec{n}_{B})}\Big{]},$ (6)
where $r_{B}$ is the distance from SIM to a deflecting body $B$,
$\vec{n}_{B}={\vec{r}}_{B}/r_{B}$ is the unit vector in this direction. This
OPD is the leading general relativistic observable that the interferometric
instrument will measure; a complete relativistic modeling this delay should be
the main objective of the upcoming SIM modeling effort.
In general, a three-dimensional approach must be used in order to work out a
practical model of the interferometric time delay. Howevere, for the purpose
of this paper, it is sufficient to confine our analysis to a plane and
parameterize the quantities involved as follows (see Fig. 3):
$\vec{b}=b\,(\cos\epsilon,\,\sin\epsilon),~{}~{}~{}~{}~{}\vec{r}_{B}=r_{B}\,(\cos\alpha_{B},\,\sin\alpha_{B}),~{}~{}~{}~{}~{}\vec{n}_{S}=(\cos\theta,\,\sin\theta),$
(7)
where $\epsilon$ is the angle of the baseline’s orientation with respect to
the instantaneous body-centric coordinate frame, $\alpha_{B}$ is the right
assention angle of the interferometer as seen from the this frame and $\theta$
is the direction to the observed source correspondingly. The geometry of the
problem and notations are presented in the Figure 3.
Figure 3: Geometry and notations for the gravitational deflection of light.
It is convenient to express the gravitational contribution to the total OPD
Eq.(6) in terms of the deflector and the source separation angle $\chi_{B}$ as
observed by the interferometer. In our approximation, the following relations
$d=r_{S}\sin\chi_{S}=r_{B}\sin\chi_{B}$ and
$\chi_{S}+\chi_{B}+\theta-\alpha=\pi$ are valid; this allows one to eliminate
angle $\chi_{S}$ by expressing the source’s position angle $\theta$ via the
separation angle $\chi_{B}$ as below:
$\theta=\pi+\alpha_{B}-\chi_{B}-\arcsin\Big{[}\,\frac{r_{B}}{r_{S}}\,\sin\chi_{B}\,\Big{]}.$
(8)
As the sources will be located at a very large distance, $r_{S}$, compares to
the distance between the interferometer and the deflector ($r_{B}\ll r_{S}$),
we can neglect the presence of the last term in the equation Eq.(8), so that
$\theta\simeq\pi+\alpha_{B}-\chi_{B}$. After substituting expressions (8) and
(7) into Eq. (6), we rewrite the contribution of the gravitational deflection
to the total OPD, Eq.(6), in the following form:
$\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\Big{[}\cos(\epsilon-\alpha_{B})+\sin(\epsilon-\alpha_{B})\frac{1+\cos\chi_{B}}{\sin\chi_{B}}\Big{]}.$
(9)
This expression describes gravitational delay as measured by an
interferometer; we will use it for estimation purposes.
#### II.1.1 Absolute Astrometric Measurements
Eq. (9) is appropriate for estimation the magnitudes of the gravitational
bending effects measured by an interferometer. It depends on the angle between
the baseline and deflector-instrument vectors, $\epsilon-\alpha_{B}$. As the
main objective of this paper is to estimate the magnitudes of the effects
involved, we choose $\epsilon-\alpha_{B}=\frac{\pi}{2}$ that maximizes
contribution of each individual deflector for a particular orbital position of
the spacecraft and the baseline orientation. As a result, in the SIM proper
reference frame Eq. (9) may be re-written as $\ell_{\tt
gr}=-\sum_{B}\,\ell_{\tt gr}^{B}$, with the individual contributions of the
deflecting bodies to gravitational delay $\ell_{\tt gr}^{B}$ and deflection
angle $\theta^{B}_{\tt gr}\simeq{\ell^{B}_{\tt gr}}/{b}$ in the following form
$\ell_{\tt
gr}^{B}=-(\gamma+1)\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\frac{1+\cos\chi_{B}}{\sin\chi_{B}}\qquad{\rm
and}\qquad\theta^{B}_{\tt
gr}=-(\gamma+1)\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\frac{1+\cos\chi_{B}}{\sin\chi_{B}}.$
(10)
The two expressions $\ell_{\tt gr}^{B}$ and $\theta^{B}_{\tt gr}$ will be used
interchangeably throughout the paper.
For complete analysis of the gravitational deflection of light we will have to
account for the time dependency in all the quantities involved. Thus, one will
have to use the knowledge of the position of the spacecraft in the solar
system’s barycentric reference frame, the instrument’s orientation in the
proper coordinate frame Klioner03 , the time that was spent in a particular
orientation, the history of all the maneuvers and re-pointings of the
instrument, etc. These issues are closely related to the principles of the
operational mode of the instrument that is currently still being developed.
#### II.1.2 Differential Astrometric Measurements
SIM will perform its astrometric campaign working in differential mode either
within FoR=15∘ for the wide angle astrometry or within FoR=1∘ for the narrow
angle astrometry. To evaluate the impact of gravitational delay of light on
these measurements, we need to derive the appropriate expressions reflecting
the differential nature of astrometric measurements with SIM.
Within the accepted approximation, the necessary expression for the
differential OPD may be obtained by subtracting OPDs for the different sources
one from one another. Using Eq. (9), this results in the following expression:
$\delta\ell^{B}_{\tt gr}=\ell^{B}_{1\tt gr}-\ell^{B}_{2\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}\Big{[}\frac{\vec{b}(\vec{n}_{S1}+\vec{n}_{B})}{1+(\vec{n}_{S1}\vec{n}_{B})}-\frac{\vec{b}(\vec{n}_{S2}+\vec{n}_{B})}{1+(\vec{n}_{S2}\vec{n}_{B})}\Big{]},$
(11)
where ${\vec{n}}_{S1}$ and $\vec{n}_{S2}$ are the barycentric positions of the
primary and the secondary objects. By using parameterization for the
quantities involved similar to that above ($b\ll r_{B}\ll r_{S1,S2}$), this
expression may be presented in terms of the deflector-source separation
angles, $\chi_{1B},~{}\chi_{2B}$, as follows:
$\delta\ell_{\tt
gr}=-(\gamma+1)\sum_{B}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}\sin(\epsilon-\alpha_{B})\,\,\frac{\sin\frac{1}{2}(\chi_{B2}-\chi_{B1})}{\sin\frac{1}{2}\chi_{B1}\sin\frac{1}{2}\chi_{B2}}.$
(12)
Similar to the discussion of absolute defection angles, we choose
$\epsilon-\alpha_{B}=\frac{\pi}{2}$ that maximizes contribution of each
individual deflector for a particular orbital position of the spacecraft and
the baseline orientation. Therefore, in the SIM proper reference frame Eq.
(12) may be re-written as $\delta\ell_{\tt gr}=-\sum_{B}\,\delta\ell_{\tt
gr}^{B}$, with the individual contributions for gravitational delay
$\delta\ell_{\tt gr}^{B}$ and corresponding deflection angle
$\delta\theta^{B}_{\tt gr}\simeq\delta{\ell^{B}_{\tt gr}}/{b}$ in the
following form
$\delta\ell^{B}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{B}b}{r_{B}}~{}\frac{\sin\frac{1}{2}(\chi_{2B}-\chi_{1B})}{\sin\frac{1}{2}\chi_{1B}\cdot\sin\frac{1}{2}\chi_{2B}}\qquad{\rm
and}\qquad\delta\theta^{B}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{B}}{r_{B}}~{}\frac{\sin\frac{1}{2}(\chi_{2B}-\chi_{1B})}{\sin\frac{1}{2}\chi_{1B}\cdot\sin\frac{1}{2}\chi_{2B}}.$
(13)
The two expressions $\delta\ell^{B}_{\tt gr}$ and $\delta\theta^{B}_{\tt gr}$
will be used interchangeably throughout the paper.
### II.2 Deflection of Grazing Rays by the Bodies of the Solar System
Table 1: Relativistic monopole deflection of by the solar system bodies at the SIM’s location. Solar | Angular size | Deflection of grazing rays
---|---|---
system’s | at SIM pos., | absolute | diff. $[15^{\circ}]$ | diff. $[1^{\circ}]$
object | ${\cal R}_{B}$, arcsec | $\theta^{B}_{\tt gr},\mu$as | $\delta\theta^{B}_{\tt gr},~{}\mu$as | $\delta\theta^{B}_{\tt gr},~{}\mu$as
Sun | 0∘.26656 | 1′′.75064 | 1′′.72025 | 1′′.38221
Sun at 45∘ | 45∘ | 9 831.39 | 2 777.97 | 237.66
Moon | 47.92690 | 25.91 | 25.87 | 25.56
Mercury | 5.48682 | 82.93 | 82.92 | 82.81
Venus | 30.15040 | 492.97 | 492.69 | 488.88
Earth | 175.88401 | 573.75 | 571.90 | 547.03
Mars | 8.93571 | 115.85 | 115.83 | 115.57
Jupiter | 23.23850 | 16 419.61 | 16 412.60 | 16 314.30
Jupiter at 30′′ | 30.0 | 12 719.12 | 12 712.03 | 12 614.21
Saturn | 9.64159 | 5 805.31 | 5 804.27 | 5 789.79
Uranus | 1.86211 | 2 171.38 | 2 171.30 | 2 170.26
Neptune | 1.18527 | 2 500.35 | 2 500.29 | 2 499.52
Pluto | 0.11478 | 2.82 | 2.82 | 2.82
We are now ready to evaluate the influence of the solar system’s gravity field
on the future high-accuracy astrometric observations. In particular, we
estimate the magnitudes of the angles of gravitational deflection for those
light rays that are grazing the surfaces of celestial bodies.
Table 1 shows magnitude of the angles characterizing relativistic monopole
deflection of grazing (e.g. $\chi_{1B}={\cal R}_{B}$) light rays by the bodies
of the solar system at the SIM’s location (i.e., the solar Earth-trailing
orbit Unwin-etal-2008 ). Results for absolute deflection angles agree with
values obtained by other authors (for instance, Brumberg-Klioner-Kopeikin-1990
). The results presented in the terms of the following quantities:
* i).
for absolute astrometry results are given in terms of the absolute
measurements $\ell^{B}_{\tt gr}$ and $\theta^{B}_{\tt gr}$ from Eq. (10);
* ii).
for differential astrometry results are given in terms of the absolute
measurements $~{}\delta\ell^{B}_{\tt gr}$ and $\delta\theta^{B}_{\tt gr}$ from
Eq. (13).
For the differential observations the two stars are assumed to be separated by
the size of the instrument’s field of regard. For the grazing rays, position
of the primary star is assumed to be on the limb of the deflector. Moreover,
results are given for the smallest distances from SIM to the bodies (e.g.when
the gravitational deflection effect is largest). For the Earth-Moon system we
took the SIM’s position at the end of the first half of the first year mission
at the distance of 0.05 AU from the Earth. Presented in the right column of
Table 1 are the magnitudes of the body’s individual contributions to the
gravitational delay of light at the SIM’s location.
Note, that the angular separation of the secondary star will always be taken
larger than that for the primary. It is convenient to study the case of the
most distant available separations of the sources. In the case of SIM, this is
the size of the field of regard (FoR). Thus for the wide-angle astrometry the
size of FoR will be $15^{\circ}\equiv\frac{\pi}{12}$ rad, thus
$\chi_{2B}=\chi_{1B}+\frac{\pi}{12}$. For the narrow-angle observations this
size is FoR $=1^{\circ}\equiv\frac{\pi}{180}$ rad, thus for this type of
astrometric observations we will use $\chi_{2B}=\chi_{1B}+\frac{\pi}{180}$;
finally, $b=10$ m is the baseline length used in the estimates.
#### II.2.1 Critical Impact Parameter for High Accuracy Astrometry
The estimates, presented in the Table 1 have demonstrated that it is very
important to correctly model and account for gravitational influence of the
bodies of the solar system. Depending on the impact parameter $d_{B}$ (or
planet-source separation angle, $\chi_{B}$), one will have to account for the
post-Newtonian deflection of light by a particular planet. Most important is
that one will have to permanently monitor the presence of some of the bodies
of the solar system during all astrometric observations, independently on the
position of the spacecraft in it’s solar orbit and the observing direction.
The bodies that introduce the biggest astrometric inhomogeneity are the Sun,
Jupiter and the Earth (especially at the beginning of the mission, when the
spacecraft is in the Earth’ immediate proximity).
Table 2: Relativistic monopole deflection of light: the angles and the critical distances for $\Delta\theta_{0}=1~{}\mu$as astrometric accuracy. Solar defection of light must be always taken into account, as at the SIM’s position at 1 AU from the Sun, the solar gravitational deflection effects are always larger than 1 $\mu$as. The critical distances for the Earth are given for two distances, namely for 0.05 AU (27∘.49) and 0.01 AU (78∘.54). Object | $\theta^{B}_{\tt gr},~{}\mu$as | Critical distances for accuracy of $1~{}\mu$as
---|---|---
| | $d^{B}_{\tt crit}$, km | $d^{B}_{\tt crit}$, deg | $d^{B}_{\tt crit}$, ${\cal R}_{B}$
Sun | 1′′.75064 | always | always | always
Moon | 25.91 | $4.501\times 10^{4}$ | $0^{\circ}.34-1^{\circ}.72$ | $25.9\cdot{\cal R}_{m}$
Mercury | 82.93 | $2.023\times 10^{4}$ | $0^{\circ}.06-0^{\circ}.13$ | $82.9\cdot{\cal R}_{Me}$
Venus | 492.97 | $2.982\times 10^{6}$ | $0^{\circ}.66-4^{\circ}.13$ | $492.9\cdot{\cal R}_{V}$
Earth | 573.75 | $3.453\times 10^{6}$ | 27${}^{\circ}.49-78^{\circ}$.54 | $541.4\cdot{\cal R}_{\oplus}$
Mars | 115.85 | $3.931\times 10^{5}$ | 0${}^{\circ}.06-0^{\circ}$.29 | $115.9\cdot{\cal R}_{Ms}$
Jupiter | 16 419.61 | $6.270\times 10^{8}$ | $64^{\circ}.06-88^{\circ}.51$ | $8\,849\cdot{\cal R}_{J}$
Saturn | 5 805.31 | $3.420\times 10^{8}$ | $12^{\circ}.56-15^{\circ}.45$ | $5\,700\cdot{\cal R}_{S}$
Uranus | 2 171.38 | $5.319\times 10^{7}$ | $1^{\circ}.01-1^{\circ}.12$ | $2\,171\cdot{\cal R}_{U}$
Neptune | 2 500.35 | $6.276\times 10^{7}$ | $0^{\circ}.77-0^{\circ}.82$ | $2\,500\cdot{\cal R}_{N}$
Pluto | 2.82 | $9.025\times 10^{3}$ | $0^{\prime\prime}.31-0^{\prime\prime}.32$ | $2.8\cdot{\cal R}_{P}$
Let us introduce a measure of such a gravitational inhomogeneity due to a
particular body in the solar system. To do this, suppose that future
astrometric experiments with SIM will be capable to measure astrometric
parameters with accuracy of $\Delta\theta_{0}=\Delta k~{}\mu$as, where $\Delta
k$ is some number characterizing the accuracy of the instrument (e.g. for a
single measurement accuracy $\Delta k=3$ for stars brighter than $V=20$ and
for the mission accuracy $\Delta k=0.1$, see Unwin-etal-2008 ). Then, there
will be a critical distance from the body, beginning from which, it is
important to account for the presence of the body’s gravity in the vicinity of
the observed part of the sky. We call this distance – critical impact
parameter, $d^{B}_{\tt crit}$, the closest distance between the body and the
light ray that is gravitationally deflected to the angle
$\theta^{\tt c}_{\tt gr}(d^{B}_{\tt crit})=\Delta\theta_{0}=\Delta k~{}\mu{\sf
as}.$ (14)
The necessary expression for $d^{B}_{\tt crit}$ is obtained from Eq. (10).
Assuming that the angle $\chi_{B}$ is small and noting that
$r_{B}\sin\chi_{B}=d$, we can write this equation as follows $\theta^{B}_{\tt
gr}\simeq{2\mu_{B}}/{d}$, where $\mu_{B}=2GM_{B}/c^{2}$ being relativistic
gravitational radius of the body. As the effect of gravitational deflection
light is inversely proportional to the impact parameter, then beginning from a
certain value of the parameter, $d^{B}_{\tt crit}$, the deflection angle will
be larger $\Delta\theta_{0}$; this value is given by the following expression:
$d^{B}_{\tt crit}=\frac{2\,\mu_{B}}{\Delta\theta_{0}},$ (15)
Different forms of the critical impact parameters $d^{B}_{\tt crit}$ for
$\Delta\theta_{0}=1~{}\mu$as are given in the Table 2. With the help of Eq.
(15), the results given in this table are easily scaled for any astrometric
accuracy $\Delta\theta_{0}$.
#### II.2.2 Deflection of Light by Planetary Satellites
One may expect that the planetary satellites will affect the astrometric
studies if a light ray would pass in their vicinities. Just for completeness
of our study we would like to present the estimates for the gravitational
deflection of light by the planetary satellites and the small bodies in the
solar system. The corresponding estimates for deflection angles,
$\theta^{B}_{\tt gr}$, and critical distances, $d_{\tt crit}$ are presented in
the Table 3. Due, to the fact that the angular sizes for those bodies are much
less than the smallest field of regard of the SIM instrument (e.g. FoR=1∘),
the results for the differential observations will be effectively insensitive
to the size of the the two available FoRs. The obtained results demonstrate
the fact that observations of these objects with that size of FoR will
evidently have the effect from the relativistic bending of light. Thus, in
Table 3 we have presented there only the angle for the absolute gravitational
deflection in terms of quantities $\theta^{B}_{\tt gr}$. As a result, the
major satellites of Jupiter, Saturn and Neptune should also be included in the
model if the light ray passes close to these bodies.
Table 3: Relativistic deflection of light by some planetary satellites. Object | Mass, | Radius, | Angular size, | Grazing | 1 $\mu$as critical radius
---|---|---|---|---|---
| $10^{25}$ g | ${\cal R}_{B}$, km | ${\cal R}_{B}$, arcsec | $\theta^{B}_{\tt gr},~{}\mu$as | $d_{\tt crit},$ km | $d_{\tt crit},$ ${\cal R}_{planet}$
Io | 7.23 | 1 738 | 0.570056 | 25.48 | 44 291 | $0.63\cdot{\cal R}_{J}$
Europa | 4.7 | 1 620 | 0.531353 | 17.77 | 28 793 | $0.41\cdot{\cal R}_{J}$
Ganymede | 15.5 | 1 415 | 0.464114 | 67.11 | 94 954 | $1.34\cdot{\cal R}_{J}$
Callisto | 9.66 | 2,450 | 0.803589 | 24.15 | 59 178 | $0.84\cdot{\cal R}_{J}$
Rhea | 0.227 | 675 | 0.108468 | 2.06 | 1 391 | $0.02\cdot{\cal R}_{S}$
Titan | 14.1 | 2 475 | 0.397715 | 34.90 | 86 378 | $1.44\cdot{\cal R}_{S}$
Triton | 13 | 1 750 | 0.082638 | 45.51 | 79 639 | $3.17\cdot{\cal R}_{N}$
#### II.2.3 Gravitational Influence of Small Bodies
Additionally, for the astrometric accuracy at the level of few $\mu$as (i.g.,
$\Delta\theta_{0}=\Delta k~{}\mu{\rm as}$), one needs to account for the post-
Newtonian deflection of light due to rather a large number of small bodies in
the solar system having a mean radius
${\cal R}_{B}\geq 624~{}\sqrt{\Delta k\over\rho_{B}}\hskip 8.0pt~{}{\rm km}.$
(16)
The deflection angle for the largest asteroids Ceres, Pallas and Vesta for
$\Delta k=1$ are given in the Table 4. The quoted properties of the asteroids
were taken from StandishHellings89 ; Mouret-Hestroffer-Mignard-2007 .
Positions of these asteroids are known and they are incorporated in the JPL
ephemerides. Other small bodies (e.g. asteroids, Kuiper belt objects, etc.)
may produce a stochastic noise in the future astrometric observations with
SIM; therefore, they should also be properly modeled.
Table 4: Relativistic deflection of light by the asteroids. Object | $\rho_{B},$ g/cm3 | Radius, km | $\theta^{B}_{\tt gr},~{}\mu$as
---|---|---|---
Ceres | 2.3 | 470 | 1.3
Pallas | 3.4 | 269 | 0.6
Vesta | 3.6 | 263 | 0.6
Class S | 2.1 $\pm$ 0.2 | TBD | $\leq 0.3$
Class C | 1.7 $\pm$ 0.5 | TBD | $\leq 0.3$
## III Regions with the most gravitationally intense environments for SIM
The properties of the solar system’s gravity field presented in the Tables 1
and 2 suggest that the most intense gravitational environments in the solar
system are those offered by the Sun and two planets, namely the Earth and
Jupiter. In this Section we will analyze these regions in more details.
### III.1 Gravitational Deflection of Light by the Sun
From the expressions Eq. (9) and Eq. (13) we obtain the relations for
relativistic deflection of light by the solar gravitational monopole. The
expression for the absolute astrometry takes the form:
$\theta^{\odot}_{\tt gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{\odot}}{r_{\rm
AU}}~{}\frac{1+\cos\chi_{1\odot}}{\sin\chi_{1\odot}}=4.072\cdot\frac{1+\cos\chi_{1\odot}}{\sin\chi_{1\odot}}~{}~{}~{}~{}{\rm
mas},$ (17)
where $\chi_{1\odot}$ is the Sun-source separation angle, $r_{\rm AU}=1$ AU,
and $\gamma=1$. Similarly, for differential astrometric observations one
obtains:
$\delta\theta^{\odot}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{\odot}}{r_{\rm
AU}}\frac{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})}{\sin\frac{1}{2}\chi_{1\odot}\cdot\sin\frac{1}{2}\chi_{2\odot}}=4.072\cdot\frac{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})}{\sin\frac{1}{2}\chi_{1\odot}\cdot\sin\frac{1}{2}\chi_{2\odot}}~{}~{}~{}~{}{\rm
mas},$ (18)
with $\chi_{1\odot},\chi_{2\odot}$ being the Sun-source separation angles for
the primary and the secondary stars correspondingly. We use two stars
separated by the SIM’s field of regard, namely
$\chi_{2\odot}=\chi_{1\odot}+\frac{\pi}{12}$. The solar angular dimensions
from the Earth’ orbit are calculated to be ${\cal R}_{\odot}=0^{\circ}.26656$.
This angle corresponds to a deflection of light to $1.75065$ arcsec on the
limb of the Sun. Results for the most interesting range of $\chi_{1\odot}$ are
given in the Table 5.
Table 5: Magnitudes of the gravitational deflection angle vs. the Sun-source
separation angle $\chi_{1\odot}$.
Solar | small $\chi_{1\odot},$ deg
---|---
deflection | $0^{\circ}.27$ | $0^{\circ}.5$ | $1^{\circ}$ | $2^{\circ}$ | $5^{\circ}$ | $10^{\circ}$ | $15^{\circ}$
$~{}\theta^{\odot}_{\tt gr}$, mas | 1 728 | 933.295 | 466.639 | 233.302 | 93.271 | 46.547 | 30.932
$\delta\theta^{\odot}_{\tt gr}~{}[15^{\circ}]$, mas | 1 698 | 903.372 | 437.663 | 206.053 | 70.176 | 28.178 | 15.734
$\delta\theta^{\odot}_{\tt gr}~{}[1^{\circ}]$, mas | 1 361 | 622.212 | 233.337 | 77.787 | 15.567 | 4.254 | 1.956
Solar | large $\chi_{1\odot},$ deg
---|---
deflection | $20^{\circ}$ | $40^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $60^{\circ}$ | $70^{\circ}$ | $80^{\circ}$ | $90^{\circ}$
$~{}\theta^{\odot}_{\tt gr}$, mas | 23.095 | 11.189 | 9.832 | 8.733 | 7.053 | 5.816 | 4.853 | 4.072
$\delta\theta^{\odot}_{\tt gr}~{}[15^{\circ}]$, mas | 10.180 | 3.366 | 2.778 | 2.341 | 1.746 | 1.372 | 1.122 | 0.948
$\delta\theta^{\odot}_{\tt gr}~{}[1^{\circ}]$, mas | 1.123 | 0.297 | 0.238 | 0.195 | 0.140 | 0.107 | 0.085 | 0.071
Sun-source separation angle, $\chi_{1\odot}$ [deg]
Deflection angle $\log_{10}[\theta^{\odot}_{\tt gr}],~{}[\mu{\rm as}]$
Deflection angle $\log_{10}[\theta^{\odot}_{\tt gr}],~{}[\mu{\rm as}]$
Sun-source separation angle, $\chi_{1\odot}$ [deg]
Figure 4: Solar gravitational deflection of light. On all plots: the upper
thick line is for the absolute astrometric measurements, while the other two
are for the differential astrometry. Thus, the dashed line is for the
observations over field of regard of FoR$\,\,=15^{\circ}$, the lower thick
line is for FoR $=1^{\circ}$.
Figure 4 shows a qualitative presentation of the solar gravitational
deflection. The upper thick line on both plots represents the absolute
astrometric measurements, while the other two are for the differential
astrometry. Thus, the middle dashed line is for the observations over the
maximal field of regard of the instrument FoR$\,\,=15^{\circ}$, the lower
thick line is for FoR $=1^{\circ}$.
One can also account for the post-post-Newtonian (post-PN) terms (e.g.
$\propto G^{2}$) as well as the contributions due to other PPN parameters
Will_book93 . Thus, in the weak gravity field approximation the total
deflection angle $\theta_{\tt gr}$ has an additional contribution due the
post-post-Newtonian terms in the metric tensor. For the crude estimation
purposes this effect could be given by the following expression post-PPN :
$\delta\theta_{\tt post-
PN}=\frac{1}{4}(\gamma+1)^{2}\left(\frac{2GM}{c^{2}d}\right)^{2}\left(\frac{15\pi}{16}-1\right)\left(\frac{1+\cos\chi}{2}\right)^{2}.$
(19)
However, a quick look on the magnitudes of these terms for the solar system’s
bodies suggested that SIM astrometric data will be insensitive to the post-PN
effects. The post-PN effects due to the Sun are the largest among those in the
solar system. However, even for the absolute astrometry with the Sun-grazing
rays the post-PN terms were estimated to be of order $\delta\theta_{\tt post-
PN}^{\odot}=11~{}\mu$as. Note that the SIM solar avoidance angle is
constraining the Sun-source separation angle as $\chi_{1\odot}\geq$ 45∘. The
post-PN effect is inversely proportional to the square of the impact
parameter, thus reducing the effect to $\delta\theta^{\odot}_{\tt post-PN}\leq
4.9$ nanoarcseconds at the edge of the solar avoidance angle. Therefore, the
post-PN effects will not be accessible with SIM.
### III.2 Gravitational Deflection of Light by Jupiter
Astrometric measurement with SIM would have to account for the light bending
by Jupiter Crosta-Mignard-2006 ; Fomalont-Kopeikin07 . One may obtain the
expression, similar to Eq. (17) for the relativistic deflection of light by
the Jovian gravitational monopole in the following form:
$\theta^{J}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{J}}{r_{J}}~{}\frac{1+\cos\chi_{1J}}{\sin\chi_{1J}}=0.924944\cdot\frac{1+\cos\chi_{1J}}{\sin\chi_{1J}}~{}~{}~{}~{}\mu{\sf
as},$ (20)
with $\chi_{1J}$ being Jupiter-source separation angle as seen by the
interferometer at the distance $r_{J}$ from Jupiter. For the differential
observations one will have expression, similar to that Eq. (18) for the Sun:
$\delta\theta^{J}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{J}}{r_{J}}\frac{\sin\frac{1}{2}(\chi_{2J}-\chi_{1J})}{\sin\frac{1}{2}\chi_{1J}\cdot\sin\frac{1}{2}\chi_{2J}}=0.924944\cdot\frac{\sin\frac{1}{2}(\chi_{2J}-\chi_{1J})}{\sin\frac{1}{2}\chi_{1J}\cdot\sin\frac{1}{2}\chi_{2J}}~{}~{}~{}~{}\mu{\sf
as},$ (21)
where again $\chi_{1J},\chi_{2J}$ are Jupiter-source separation angles for the
primary and secondary stars correspondingly,
$\chi_{2J}=\chi_{1J}+\frac{\pi}{12}$ (and
$\chi_{2J}=\chi_{1J}+\frac{\pi}{180}$ for the narrow angle astrometry). The
largest effect will come when SIM and Jupiter are at the closest distance from
each other $\sim 4.2~{}$AU. Jupiter’s angular dimensions from the Earth’ orbit
for this situation are calculated to be ${\cal R}_{J}=23.24$ arcsec, which
correspond to a deflection angle of 16.419 mas. Results for some $\chi_{1J}$
are given in the Table 6. Note that for the light rays coming perpendicular to
the ecliptic plane the Jovian deflection will be in the range:
$\delta\alpha_{1J}\sim(0.7\\--1.0)~{}\mu$as!
A qualitative behavior of the effect of the gravitational deflection of light
by the Jovian gravity field is plotted in the Figure 5. As in the case of the
solar deflection, the upper thick line on both plots represents the absolute
astrometric measurements, while the other two are for the differential
astrometry (the dashed line is for the observations over FoR$\,\,=15^{\circ}$
and the lower thick line is for FoR $=1^{\circ}$).
Table 6: Jovian gravitational monopole deflection vs. the Jupiter-source sky separation angle $\chi_{1J}$. Jovian | Jupiter-source separation angles $\chi_{1J},~{}~{}$arcsec
---|---
deflection | $23.24^{\prime\prime}$ | $26^{\prime\prime}$ | $30^{\prime\prime}$ | $60^{\prime\prime}$ | $120^{\prime\prime}$ | $180^{\prime\prime}$ | $360^{\prime\prime}$ | $90^{\circ}$
$~{}\theta^{J}_{\tt gr},~{}$mas | 16.419 | 14.676 | 12.719 | 6.360 | 3.180 | 2.120 | 1.060 | $0.9~{}\mu$as
$\delta\theta^{J}_{\tt gr}[15^{\circ}],~{}$mas | 16.412 | 14.669 | 12.712 | 6.352 | 3.173 | 2.113 | 1.053 | $0.2~{}\mu$as
$\delta\theta^{J}_{\tt gr}[1^{\circ}],~{}$mas | 16.313 | 14.570 | 12.614 | 6.255 | 3.077 | 2.019 | 0.964 | $0.0~{}\mu$as
Jupiter-source separation angle, $\chi_{1J}$ [arcsec]
Deflection angle $\log_{10}[\theta^{J}_{\tt gr}],~{}[\mu{\rm as}]$
Deflection angle $\log_{10}[\theta^{J}_{\tt gr}],~{}[\mu{\rm as}]$
Jupiter-source separation angle, $\chi_{1J}$ [deg]
Figure 5: Jovian gravitational deflection of light.
### III.3 Gravitational Deflection of Light by the Earth
The deflection of light rays by the Earth’s gravity field may also be of
interest. The expressions, describing the relativistic deflection of light by
the Earth’ gravitational monopole are given below:
$\theta^{\oplus}_{\tt
gr}=(\gamma+1)~{}\frac{G}{c^{2}}\frac{M_{\oplus}}{r_{\oplus}}~{}\frac{1+\cos\chi_{1\oplus}}{\sin\chi_{1\oplus}}=0.2446\cdot\frac{1+\cos\chi_{1\oplus}}{\sin\chi_{1\oplus}}~{}~{}~{}~{}\mu{\sf
as},$ (22)
with $\chi_{1\oplus}$ being the Earth-source separation angle as seen by the
interferometer at the distance $r_{\odot}$ from the Earth. Relation for the
differential astrometric measurements was obtained in the form:
$\delta\theta^{\oplus}_{\tt
gr}=(\gamma+1)\frac{G}{c^{2}}~{}\frac{M_{\oplus}}{r_{\oplus}}\frac{\sin\frac{1}{2}(\chi_{2\oplus}-\chi_{1\oplus})}{\sin\frac{1}{2}\chi_{1\oplus}\cdot\sin\frac{1}{2}\chi_{2\oplus}}=0.2446\cdot\frac{\sin\frac{1}{2}(\chi_{2\oplus}-\chi_{1\oplus})}{\sin\frac{1}{2}\chi_{1\oplus}\cdot\sin\frac{1}{2}\chi_{2\oplus}}~{}~{}~{}~{}\mu{\sf
as},$ (23)
where, as before, $\chi_{1\oplus},\chi_{2\oplus}$ are the Earth-source
separation angles for the primary and secondary stars correspondingly,
$\chi_{2\oplus}=\chi_{1\oplus}+\frac{\pi}{12}$ (and
$\chi_{2\oplus}=\chi_{1\oplus}+\frac{\pi}{180}$ for the narrow angle
astrometry). The largest effect will come when SIM and the Earth are at the
closest distance, say at the end of the first half of the first year of the
mission, $r_{\oplus}=0.05$ AU. The Earth’s angular dimensions being measured
from the spacecraft from that distance are calculated to be ${\cal R}^{\tt
SIM}_{\oplus}=175.88401$ arcsec, which correspond to a deflection angle of
573.75 $\mu$as. The summary of the deflection angles for sevral
$\chi_{1\oplus}$ are given in the Table 7.
Table 7: Solar relativistic deflection angle as a function of the Earth-source separation angle. SIM | $\chi^{\tt SIM}_{1\oplus},$ arcsec
---|---
mission | 175.88 | 200 | 360 | $1^{\circ}$ | $5^{\circ}$ | $10^{\circ}$ | $15^{\circ}$
$~{}~{}\theta_{1\oplus},~{}\mu$as | 573.8 | 504.7 | 280.3 | 28.0 | 5.6 | 2.8 | 1.9
$\delta\theta_{1\oplus}[15^{\circ}],~{}\mu$as | 571.9 | 502.7 | 278.5 | 26.3 | 4.2 | 1.7 | 1.0
$\delta\theta_{1\oplus}[1^{\circ}],~{}\mu$as | 547.0 | 478.0 | 254.8 | 14.0 | 0.9 | 0.3 | 0.1
At the distance of 0.05 AU from the Earth
Earth-source separation angle, $\chi_{1\oplus}$ [deg]
Deflection angle $\log_{10}[\theta^{\oplus}_{\tt gr}],~{}[\mu{\rm as}]$
Deflection angle $\log_{10}[\theta^{\oplus}_{\tt gr}],~{}[\mu{\rm as}]$
At the distance of 0.5 AU from the Earth
Earth-source separation angle, $\chi_{1\oplus}$ [deg]
Figure 6: Gravitational deflection of light in the proximity of the Earth.
Figure 6 shows the expected variation in the magnitude of the Earth’ gravity
influence as mission progresses. The left plot presents results for the end of
the first half of the year of the mission, when the spacecraft is at the
distance of 0.05 AU from the Earth (the drift rate is 0.1 AU per year). The
plot on the right side is for the end of the 5-th year of the mission, when
SIM is at 0.5 AU from Earth.
### III.4 Constraints Derived From the Monopole Deflection of Light
While analyzing the solar gravity field’s influence on the future astrometric
observations with SIM, we found several interesting situations, that may
potentially put an additional navigational requirements. In this section we
will consider these situations in a more detailed way.
To measure gravitational deflection of light with an accuracy of
$\Delta\theta_{0}=\Delta k~{}\mu{\rm as}$, one needs to precisely determine
the value of impact parameter of photon’s trajectory with respect to the
deflector. As before, we will present two types of necessary expressions,
namely for absolute and differential observations.
In the case of absolute astrometry we use expression for the deflection angle
$\theta^{B}_{\tt gr}$ from Eq. (10) and present it as $\theta^{B}_{\tt
gr}=({\mu_{B}}/{d})(1+\cos\chi_{B})$, where again $d=r_{B}\sin\chi_{B}$ and
quantity $\mu_{B}=2GM_{B}/c^{2}$ being the relativistic gravitational radius
of the body at question. One may ask a question – what uncertainty in the
knowledge of the impact parameter, $\Delta d$, will result in the astrometric
error of $\Delta\theta_{0}$? The answer is given the following expression
$\Delta
d_{B}=\Delta\theta_{0}\,\,\frac{r^{2}_{B}}{\mu_{B}}\cdot\frac{\sin^{2}\chi_{1B}}{1+\cos\chi_{1B}}.$
(24)
The corresponding result for differential observations may be obtained with
the help of Eq. (13) as:
$\Delta d^{\tt
diff}_{B}=\Delta\theta_{0}\,\,\frac{r^{2}_{B}\sin^{2}\chi_{1B}}{2\mu_{B}}\cdot\Big{[}1+\tan\frac{\chi_{1B}}{2}\cdot\cot\frac{1}{2}(\chi_{2B}-\chi_{1B})\Big{]}.$
(25)
Similarly, the uncertainty in determining the barycentric distance $r_{B}$ is
obtained from Eq. (10) leading to expression:
$\Delta
r_{B}=\Delta\theta_{0}\,\frac{r^{2}_{B}}{\mu_{B}}\cdot\frac{\sin\chi_{1B}}{1+\cos\chi_{1B}}.$
(26)
Note that, when differential observations are concerned, uncertainty in
barycentric position $\Delta r^{\tt diff}_{B}$ does not produce new
constraints significantly different from those derived from Eq. (26). Looking
at the results presented in the Table 8, one may see that for an accuracy of
$\Delta\theta_{0}=1~{}\mu$as our estimates require the knowledge of the solar
impact parameter with the accuracy of $\sim 0.4$ km (grazing rays), that for
Jupiter with the accuracy of $\sim 4~{}$km and other big planets with the
accuracy of about 10 km. Table 9 shows a comparison of these derived
requirements on the barycentric positions of the solar system’s bodies with
the accuracy of their current determination.
Table 8: Required accuracy of barycentric positions and impact parameters for astrometric observations with accuracy of 1 $\mu$as. The Earth is taken at the distance of 0.05 AU from the spacecraft; moon’s position accuracy is for the geocentric frame. Solar | Required knowledge: grazing rays | Required knowledge: differential astrometry
---|---|---
system’s | Distance, | Impact parameter | Impact param. [15∘] | Impact param. [1∘]
object | $\sigma_{r_{B}},$ km | $\sigma_{d_{B}},$ km | $\sigma_{d_{B}},$ mas | $\sigma_{r_{B}}$, km | $\sigma_{r_{B}}$, mas | $\sigma_{r_{B}}$, km | $\sigma_{r_{B}}$, mas
Sun | 85.45 | 0.39 | 0.55 | 0.40 | 0.55 | 0.50 | 0.69
Sun at 45∘ | 1.5$\times 10^{4}$ | 7.6$\times 10^{3}$ | 10′′.49 | 3.81$\times 10^{4}$ | 52′′.53 | 4.45$\times 10^{5}$ | 613′′.66
Moon | 2.8$\times 10^{5}$ | 67.14 | 1′′.85 | 67.17 | 1′′.85 | 68.00 | 1′′.88
Mercury | 1.1$\times 10^{6}$ | 29.39 | 66.16 | 29.41 | 66.16 | 29.45 | 66.25
Venus | 8.4$\times 10^{4}$ | 12.18 | 61.00 | 12.28 | 61.20 | 12.38 | 62.70
Earth-Moon | 1.3$\times 10^{4}$ | 11.14 | 306.55 | 11.15 | 307.47 | 11.66 | 321.54
Mars | 6.8$\times 10^{5}$ | 29.29 | 77.11 | 29.30 | 77.14 | 29.37 | 77.33
Jupiter | 3.8$\times 10^{4}$ | 4.31 | 1.42 | 4.32 | 1.42 | 4.34 | 1.42
Jupiter at 30′′ | 4.9$\times 10^{4}$ | 7.14 | 2.34 | 7.20 | 2.36 | 7.25 | 2.38
Saturn | 2.2$\times 10^{5}$ | 10.32 | 1.66 | 10.34 | 1.66 | 10.36 | 1.66
Uranus | 1.2$\times 10^{6}$ | 11.27 | 0.86 | 11.28 | 0.86 | 11.29 | 0.86
Neptune | 1.7$\times 10^{6}$ | 10.04 | 0.47 | 10.04 | 0.47 | 10.04 | 0.47
Pluto | 2.0$\times 10^{9}$ | 1 133.92 | 40.7 | 1 133.93 | 40.7 | 1133.96 | 40.7
One may see that the present accuracy of knowledge of the inner planets’
positions from the Table 9 is given by the radio observations and it is even
better than the level of relativity requirements given in the Table 8.
However, the positional accuracy for the outer planets is below the required
level. The SIM observation program should include the astrometric studies of
the outer planets in order to minimize the errors in their positional accuracy
determination. Thus, in order to get the radial uncertainty in Pluto’s
ephemeris with accuracy below 1000 km, it is necessary only 4 measurements of
Pluto’s position, taken sometime within a week of the stationary points,
spread over 3 years. Each measurement could be taken with an accuracy of about
$200~{}\mu$as, as suggested by Standish95 . Additionally, one will have to
significantly lean on the radio observations in order to conduct the reduction
of the optical data with an accuracy of a few $\mu$as. For this reason one
will have to use the precise catalog of the radio-sources and to study the
problem of the radio and optical reference frame ties Standish95 ;
Standish_etal95 ; Folkner94 ; Ma-1998 .
The estimates, presented here were given for static gravitational field.
Analysis of a real experimental situation should consider a non-static
gravitational environment of the solar system and should include the
description of light propagation in a different reference frames involved in
the experiment Kopeikin-Schafer-1999 . Additionally, the observations will be
affected by the relativistic orbital dynamics of the spacecraft Turyshev98 .
Table 9: The best known accuracies of barycentric positions and masses for the solar system’s objects Yoder95 ; Pitjeva-2005 ; Folkner-Williams-Boggs-2008 . Solar | Knowledge of barycentric position | Knowledge of
---|---|---
system’s | Best known, | Method used | planetary masses,
object | $\sigma_{r_{B}},$ km | $\sigma_{r_{B}},$ mas | for determination | $\Delta M_{B}/M_{B}$
Sun | 362/725 | $0^{\prime\prime}.5/1^{\prime\prime}.0$ | Optical meridian transits | 3.77$\times 10^{-10}$
Moon | 27 cm | 7.4 $\mu$as | LLR, 1995 | 1.02$\times 10^{-6}$
Mercury | 1 | 2.25 | Radar ranging | 4.13$\times 10^{-5}$
Venus | 1 | 4.98 | Radar ranging | 1.23$\times 10^{-7}$
Earth | 1 | 27.58 | Radar ranging | TBD$\times 10^{-6}$
Mars | 1 | 2.63 | Radar ranging | 2.33$\times 10^{-6}$
Jupiter | 30 | 9.84 | Radar ranging | 7.89$\times 10^{-7}$
Saturn | 350 | 56.24 | Optical astrometry | 2.64$\times 10^{-6}$
Uranus | 750 | 57.00 | Optical astrometry | 3.97$\times 10^{-6}$
Neptune | 3 000 | 141.67 | Optical astrometry | 2.19$\times 10^{-6}$
Pluto/Charon | 20 000 | 717.40 | Photographic astrometry | 0.014
## IV Deflection of light by higher gravity multipoles
In order to carry out a complete analysis of the relativistic light deflection
one should account for other possible terms in the expansion (1) that may
potentially contribute to this effect. These terms are due to non-sphericity
and non-staticity of the body’s gravity field Kopeikin-1997 . Here we will
consider several of them, namely those due to higher gravitational multipoles
of the celestial bodies.
### IV.1 Gravitational quadrupole deflection of light
Although a complete three-dimensional deflection of light must be considered
for a real experiment, for the purposes of this paper, we consider only two
dimensional case. In this case, quadrupole term may be given as Turyshev98 ;
Klioner-1991 ; Kopeikin-Makarov-2007 :
$\theta_{J_{2}}=\frac{1}{2}(\gamma+1)\frac{4GM_{B}}{c^{2}{\cal
R}_{B}}J_{2B}\Big{(}1-s_{z}^{2}-2u_{z}^{\hskip 2.0pt2}\Big{)}\left({{\cal
R}_{B}\over d}\right)^{3},$ (27)
where $J_{2}$ is the second zonal harmonic of the body under question,
$\vec{s}=(s_{x},s_{y},s_{z})$ is the unit vector in the direction of the light
ray propagation and vector $\vec{d}=d(u_{x},u_{y},u_{z})$ is the impact
parameter. A similar expression may also be obtained for differential
observations. For estimation purposes, this formula may be given as follows
($d=r_{B}\sin\chi_{B}$):
$\delta\theta_{J_{2}}\approx\frac{4\mu_{B}J_{2B}{\cal
R}^{2}_{B}}{r^{3}_{B}}~{}\Big{[}\,\frac{1}{\sin^{3}\chi_{1B}}-\,\frac{1}{\sin^{3}\chi_{2B}}\Big{]}.$
(28)
The corresponding effects for the deflection of light by the quadrupole mass
moments within the planets of the solar system are given in the Table 11. The
effect depends on a number of different instantaneous geometric parameters
defining the mutual orientation of the vector of the light propagation,
position of the planet in orbit, the orientation of the axes defining it’s
figure, etc. A mission-independent modeling in the static gravitational regime
has been done Crosta-Mignard-2006 ; the modeling of the dynamic regime was
presented in Kopeikin-Makarov-2007 . An effort to develop a SIM-specfic model
is well justified.
The quadruple deflection of light depends on the third power of the inverse
impact parameter with respect to the deflecting body, Eq. (27). SIM will
measure this effect directly for many celestial bodies. At the expected level
of accuracy the knowledge of jovian atmosphere, the magnetic field
fluctuations, etc., may contribute to the errors in the experiment
TreuhaftLowe91 . A detailed study of these effects is given in Kopeikin-
Fomalont-02 .
Table 10: Higher gravitational coefficients for solar system bodies (http://nssdc.gsfc.nasa.gov/planetary/factsheet/). Solar system’s | $J^{B}_{2},$ | $J^{B}_{4},$ | $J^{B}_{6},$
---|---|---|---
object | $\times~{}10^{-6}$ | $\times~{}10^{-6}$ | $\times~{}10^{-6}$
Sun | $0.17\pm 0.017$ | — | —
Sun at 45∘ | — | — | —
Moon | 202.2 | $-0.1$ | —
Mercury | 60. | — | —
Venus | 4.5 | $-2.1$ | —
Earth | 1 082.6 | $-1.6$ | 0.5
Mars | 1 960.45 | — | —
Jupiter | 14 738$\pm$1 | $-$587$\pm$5 | 34$\pm$50
Saturn | 16 298$\pm 50$ | $-$915$\pm$80 | $103.0$
Uranus | 3 343.43 | — | —
Neptune | 3 411. | — | —
Pluto | — | — | —
Table 11: Relativistic deflection of light by the planetary quadrupole mass moments for solar system bodies. Solar system’ | $J^{B}_{2},$ | $\theta^{B}_{J_{2}},$ | $d^{\tt crit}_{{J_{2}}}$ | $\delta\theta^{B}_{J_{2}}[15^{\circ}],$ | $\delta\theta^{B}_{J_{2}}[1^{\circ}],$
---|---|---|---|---|---
object | $\times~{}10^{-6}$ | $\mu$as | | $\mu$as | $\mu$as
Sun | $0.17\pm 0.017$ | 0.3 | — | 0.3 | 0.3
Moon | 202.2 | 2$\times 10^{-2}$ | — | 2$\times 10^{-2}$ | $2\times 10^{-2}$
Mercury | 60. | 5$\times 10^{-3}$ | — | — | —
Venus | 4.5 | 2$\times 10^{-3}$ | — | — | —
Earth | 1 082.6 | 0.6 | — | 0.6 | 0.6
Mars | 1 960.45 | 0.2 | — | 0.2 | 0.2
Jupiter | 14 738$\pm$1 | 242.0 | 98′′.12 – 144′′.81 | 242.0 | 242.0
| | | 6.23 ${\cal R}_{J}$ | |
Saturn | 16 298$\pm$ 50 | 94.6 | 35′′.62 – 43′′.93 | 94.6 | 94.6
| | | 4.56 ${\cal R}_{S}$ | |
Uranus | 3 343.43 | 7.3 | 3′′.25 – 3′′.61 | 7.3 | 7.3
| | | 1.94 ${\cal R}_{U}$ | |
Neptune | 3 411. | 8.5 | 2′′.23 – 2′′.42 | 8.5 | 8.5
| | | 2.04 ${\cal R}_{N}$ | |
Pluto | — | — | — | — | —
Table 11 presents estimates of the magnitudes of the relativistic deflection
of light by the planetary quadrupole mass moments for solar system bodies.
Based on the sizes of these effects, one would have to account for the
quadrupole component of the gravity fields when observations will be conducted
in the vicinity of the outer planets. In addition, the influence of the higher
harmonic may be also of interest. Thus, Table 11 shows the estimates of some
higher gravitational multipole moments of Jupiter and Saturn. We will discuss
the deflection by the $J_{2}$ and $J_{4}$ coefficients of the jovian gravity
field in terms of Jupiter-source separation angle $\chi_{1J}$. An expression,
similar to that of Eq. (20) for the monopole deflection, may be given as:
$\theta^{\tt max}_{J_{2}}={3.46058\times
10^{-10}}~{}\frac{1}{\sin^{3}\chi_{1J}}~{}~{}\mu{\sf as}.$ (29)
Jupiter’s angular dimensions from the Earth are calculated to be ${\cal
R}_{J}=23.24~{}$arcsec, which correspond to a deflection angle of
$242~{}\mu$as. The deflection on the multipoles for some $\chi_{1J}$ is given
in the Table 13.
Table 12: Deflection of light by the Jovian higher gravitational coefficients.
Jovian | $\chi_{1J}$, arcsec
---|---
deflection | $23^{\prime\prime}.24$ | 26′′ | 30′′ | 35′′ | 40′′ | 50′′ | 120′′
$\theta^{J}_{J_{2}},~{}\mu$as | 242 | 173 | 112 | 71 | 47 | 24 | 1.8
$\delta\theta^{J}_{J_{2}}[15^{\circ}],~{}\mu$as | 242 | 173 | 112 | 71 | 47 | 24 | 1.8
$\theta^{J}_{J_{4}},~{}\mu$as | 9.6 | 5.5 | 2.7 | 1.3 | 0.6 | 0.2 | 0.0
Table 13: Deflection of light by the Saturnian higher gravitational
coefficients.
Saturnian | $\chi_{1S}$, arcsec
---|---
deflection | $9^{\prime\prime}.64$ | 12′′ | 15′′ | 20′′ | 25′′ | 30′′ | 35′′
$\theta^{S}_{J_{2}},~{}\mu$as | 94.7 | 49.1 | 25.1 | 10.6 | 5.4 | 3.1 | 2
$\delta\theta^{S}_{J_{2}}[15^{\circ}],~{}\mu$as | 94.7 | 49.1 | 25.1 | 10.6 | 5.4 | 3.1 | 2
$\theta^{S}_{J_{4}},~{}\mu$as | 5.3 | 1.8 | 0.6 | 0.1 | — | — | —
Similar studies are important for Saturn (see Kopeikin-Makarov-2007 for
details). In terms of the Saturn-source separation angle $\chi_{1S}$ the
saturnian quadrupole deflection mat be estimated with the help of the
following expression:
$\theta^{\tt max}_{J_{2}}=9.66338\times
10^{-12}~{}\frac{1}{\sin^{3}\chi_{1S}}~{}~{}\mu{\sf as}.$ (30)
The Saturn’s angular dimensions from the Earth’ orbit are calculated to be
${\cal R}_{S}=9.64~{}$arcsec, which correspond to a deflection angle of
$94.7~{}\mu$as. The corresponding estimates for the deflection angles are
presented in the Table 13.
As a result, for astronomical observations with accuracy of about $1~{}\mu{\rm
as}$, one will have to account for the quadrupole gravitational fields of the
Sun, Jupiter, Saturn, Neptune, and Uranus. In addition, the influence of the
higher harmonics may be of interest. For example some of the moments for
Jupiter and Saturn are given in the Table 10.
Higher multipoles may also influence the astrometric observations taken close
to these planets. Thus, for both Jupiter and Saturn the rays, grazing their
surface, will be deflected by the fourth zonal harmonic $J_{4}$ as follows:
$\delta\theta^{J}_{J_{4}}\approx 9.6~{}\mu$as,
$\delta\theta^{S}_{J_{4}}\approx 5.3~{}\mu$as. In addition, the contribution
of the $J_{6}$ for Jupiter and Saturn will deflect the grazing rays on the
angles $\delta\theta^{J}_{J_{6}}\approx 0.8~{}\mu$as,
$\delta\theta^{S}_{J_{6}}\approx 0.6~{}\mu$as. The contribution of $J_{4}$ is
decreasing with the distance from the body as $d^{-5}$ and contribution of
$J_{6}$ as $d^{-7}$. As a result the deflection angle will be less then
$1~{}\mu$as when $\hskip 2.0ptd>1.6~{}{\cal R}$, where ${\cal R}$ is the
radius of the planet.
Using Eq. (27), one can derive the expression for the critical distance
$d^{\tt crit}_{J_{2}}$ for the astrometric observations in the regime of
quadrupole deflection of light with accuracy of $\Delta\theta_{0}=\Delta
k~{}\mu$as. Indeed, approximating this equation as
$\theta_{J_{2}}\simeq({4\mu_{B}}/{d^{3}})J_{2}{\cal R}^{2}$ and solving it for
$d$ one obtaines the following result:
$d^{\tt crit}_{J_{2}}={\cal R}_{B}\Big{[}\frac{4\mu_{B}}{{\cal
R}_{B}}\frac{J^{B}_{2}}{\Delta\theta_{0}}\Big{]}^{\frac{1}{3}}.$ (31)
The critical distances for the relativistic quadrupole deflection of light by
the solar system’s bodies for the case of $\Delta k=1$ presented in the Table
11.
### IV.2 Gravito-Magnetic Deflection of Light
Besides the gravitational deflection of light by the monopole and the
quadrupole components of the static gravity field of the bodies, the light ray
trajectories will also be affected by the non-static contributions from this
field. It is easy to demonstrate that a rotational motion of a gravitating
body contributes to the total curvature of the space-time generated by this
same body. This contribution produces an additional deflection of light rays
Klioner-1991 ; Ciufolini-etal-2003 ; Kopeikin-Mashhoon-2003 on the angle
$\delta\theta_{\vec{\cal
S}}=\frac{1}{2}(\gamma+1)\frac{4G}{c^{3}d^{3}}\vec{\cal
S}(\vec{s}\cdot\vec{d}),$ (32)
where ${\vec{\cal S}}$ is the body’s angular momentum.
The most significant contributions of gravito-magnetic deflection of light by
the bodies of the solar system are the following ones: i) the solar deflection
amounts to $\delta\theta^{\odot}_{\vec{\cal S}}=\pm(0.7~{}-1.3)\mu$as (the
first term listed is for a uniformly rotating Sun; the second is for the
Dicke’s model Dicke-1974 ); ii) jovian rotation contributes
$\delta\theta^{J}_{\vec{\cal S}}=\pm 0.2~{}\mu$as; and iii) Saturn’s rotation
$\delta\theta^{Sa}_{\vec{\cal S}}=\pm 0.04~{}\mu$as. Thus, depending on the
model for the solar interior, solar rotation may produce a noticeable
contribution for the grazing rays. The estimates of magnitude of deflection of
light ray’s trajectory, caused by the rotation of gravitating bodies
demonstrate that for precision of observations of $1~{}\mu$as it is sufficient
to account for influence of the Sun and Jupiter only.
The relativistic gravito-magnetic deflection of light has never been directly
tested before. However, because of the smallness of the magnitudes of
corresponding effects in the solar system and SIM’s operational mode that
limits the viewing angle for a sources as $\chi_{1\odot}\geq 45^{\circ}$, SIM
will not be sensitive to this effect.
## V Astrophysics Investigations with SIM
### V.1 Astrometric Test of General Relativity
The Eddington parameter $\gamma$ in Eq. (II.1), whose value in general
relativity is unity, is perhaps the most fundamental PPN parameter Will-
lrr-2006-3 , in that $\frac{1}{2}(1-\gamma)$ is a measure, for example, of the
fractional strength of the scalar gravity interaction in scalar-tensor
theories of gravity Damour-Nordtvedt-1993 . Currently, the most precise value
for this parameter, $\gamma-1=(2.1\pm 2.3)\times 10^{-5}$, was obtained using
radio-metric tracking data received from the Cassini spacecraft Bertotti-Iess-
Tortora-2003 during a solar conjunction experiment. This accuracy approaches
the region where multiple tensor-scalar gravity models, consistent with the
recent cosmological observations Spergel-etal-2007 , predict a lower bound for
the present value of this parameter at the level of $(1-\gamma)\sim
10^{-6}-10^{-7}$ Damour-Nordtvedt-1993 . Therefore, improving the measurement
of this parameter would provide the crucial information separating modern
scalar-tensor theories of gravity from general relativity, probe possible ways
for gravity quantization, and test modern theories of cosmological evolution
Turyshev-etal-2007 ; Turyshev-2008 .
The reasons above led to a number of specific space experiments dedicated to
measurement of the parameter $\gamma$ with a precision better than $10^{-5}$
to $10^{-6}$ Turyshev-etal-2007 ; Turyshev-2008 . Note that SIM will operate
at this level of accuracy and, therefore, the Eddington’s parameter $\gamma$
will have to be included into the future SIM’s astrometric model and the
corresponding data analysis.
#### V.1.1 Solar Gravity Field as a Deflector
To model the astrometric data to the nominal measurement accuracy will require
including the effect of general relativity on the propagation of light. In the
PPN framework, the parameter $\gamma$ would be part of this model and could be
estimated in global solutions. The astrometric residuals may be tested for any
discrepancies with the prescriptions of general relativity. To address this
problem in a more detailed way, one will have to use the astrometric model for
the instrument including the information about it’s position in the solar
system, it’s attitude orientation in the proper reference frame, the time
history of different pointings and their durations, etc. This information then
should be folded into the parameter estimation program that will use a model
based on the expression, similar to that given by Eq. (12).
To estimate the expected accuracy of the parameter $\gamma$, we use Eq. (13)
to assume that the single astrometric measurement may be able to determine
this parameter with accuracy:
$\Delta\gamma=\Delta\theta_{0}~{}\frac{r^{\tt
SIM}_{\odot}}{\mu_{\odot}}~{}\frac{\sin\frac{1}{2}\chi_{1\odot}\,\sin\frac{1}{2}\chi_{2\odot}}{\sin\frac{1}{2}(\chi_{2\odot}-\chi_{1\odot})},$
(33)
where $\Delta\theta_{0}$ is the astrometric error of the measurement.
The relativity test will be enhanced by scheduling measurements of stars as
close to the Sun as possible. Although SIM will never be able observe closer
to the Sun than $45^{\circ}$, it will allow for an accurate determination of
this PPN parameter. For the accuracy of $\Delta\theta_{0}=1~{}\mu$as at the
rim of the solar avoidance angle of $\chi_{1\odot}=45^{\circ}$, one could
determine this parameter with an accuracy $\sigma_{\gamma}\sim 7.19\times
10^{-4}$ in a single measurement. Assuming Gaussian error distribution, the
accuracy of this experiment will improve as $1/\sqrt{N}$, where $N$ is the
number of independent observations. Therefore, by performing differential
astrometric measurements with an accuracy of $\Delta\theta_{0}=1~{}\mu$as over
the instrument’s FoR=15∘, at the end of the mission (with $N\sim 10000$
observations) SIM may reach the accuracy of $\sigma_{\gamma}\sim 7.2\times
10^{-6}$ in astrometric test of general relativity in the solar gravity field.
SIM will provide this precision as a by-product of its astrometric program,
thus allowing for a factor of 3 improvement of the currently best Cassini’s
2003 result Bertotti-Iess-Tortora-2003 . Such a measurement improves the
accuracy of the search for cosmologically relevant scalar-tensor theories of
gravity by looking for a remnant scalar field in today’s solar system.
#### V.1.2 GR Test in the Jovian and Earth’ Gravity Fields
One could also perform a relativity experiment with Jupiter and the Earth. In
fact, for the proposed SIM’s observing mode, the accuracy of determining of
the parameter $\gamma$ may be even better than that achievable with the Sun.
Indeed, with the same assumptions as above, one may achieve a single
measurement the accuracy of $|\gamma-1|\sim 4.0\times 10^{-4}$ determined via
deflection of light by Jupiter. As the astrometric observations in the
Jupiter’s vicinity will require careful planning thereby minimizing the number
of possible independent observations. As a result, the PPN parameter $\gamma$
may be obtained with accuracy of about $\sigma_{\gamma}\sim 1.3\times 10^{-5}$
with astrometric experiments in Jupiter’s gravity field (note that only
$N\sim$ 1000 needed).
Lastly, let us mention that the experiments conducted in the Earth’s gravity
field, could also determine this parameter to an accuracy $|\gamma-1|\sim
8.9\times 10^{-3}$ in a single measurement (which in return extends the
measurement of the gravitational bending of light to a different mass and
distances scale, as shown by Gould93 ). One may expect a large statistics
gained from both the astrometric observations and the telecommunications with
the spacecraft. This, in return, will significantly enhance the overall
solution for $\gamma$ obtained in the Earth’ gravitational environment.
### V.2 Solar Acceleration Towards the Galactic Center
The Sun’s absolute velocity with respect to a cosmological reference frame was
measured photometrically: it shown up as the dipole anisotropy of the cosmic
microwave background Spergel-etal-2007 . The Sun’s absolute acceleration with
respect to galactic frame can be measured astrometrically: it will show up as
a dipole vector harmonic in the global pattern of proper motion of quasars.
The aberration due to the solar system’s galactocentric motion will not be
observable because its main contribution is static. However, the rate of this
aberration will produce an apparent proper motion for the observed sources
Kopeikin-Makarov-2006 . Indeed, the solar system’s orbital velocity around the
galactic center causes an aberrational affect of the order of 2.5 arcmin. All
measured star and quasar positions are shifted towards the point on the sky
having galactic coordinates $l=90^{\circ},~{}b=0^{\circ}$. For an arbitrary
point on the sky the size of the effect is $2.5~{}\sin\eta$ arcmin, where
$\eta$ is the angular distance to the point $l=90^{\circ},~{}b=0^{\circ}$. The
acceleration of the solar system towards the galactic center causes this
aberrational effect to change slowly. This leads to a slow change of the
apparent position of distant celestial objects, i.e. to an apparent proper
motion.
Let us assume a solar velocity of $220$ km/sec and a distance of 8.5 kpc to
the galactic center. The orbital period of the Sun is then 250 million years,
and the galactocentric acceleration takes a value of about $1.75\times
10^{-13}$ km/sec2. Expressed in a more useful units it is 5.5 mm/s/yr. A
change in velocity by 5.5 mm/sec causes a change in aberration of the order of
4 $\mu$as. The apparent proper motion of a celestial object caused by this
effect always points towards the direction of the galactic center. Its size is
$~{}4\sin\eta~{}\mu$as/yr, where $\eta$ is now the angular distance between
the object and the galactic center.
The above holds in principle for quasars, for which it can be assumed that the
intrinsic proper motions (i.e. those caused by real transverse motions) are
negligible. A proper motion of $4~{}\mu$as/yr corresponds to a transverse
velocity of $2\times 10^{4}$ km/sec at $z=0.3$ for $H_{0}$=100 km/sec/Mpc, and
to $4\times 10^{4}$ km/sec for $H_{0}=50$ km/sec/Mpc. Thus, all quasars will
exhibit a distance-independent steering motion towards the galactic center.
Within the Galaxy, on the other hand, the effect is drowned in the local
kinematics: at 10 pc it corresponds to only 200 m/sec.
However, for a differential astrometry with SIM this effect will have to be
scaled down to account for the size of the field of regard Turyshev98 , namely
$2\sin\frac{\tt FoR}{2}=2\sin\frac{\pi}{24}=0.261$. This fact is reducing the
total effect of the galactocentric acceleration to only $~{}\sim
1\sin\eta~{}\mu$as/yr and, thus, it makes the detection of the solar system’s
galactocentric acceleration with SIM to be a quite problematic issue.
## Discussion
General relativistic deflection of light produces a significant contribution
to the future astrometric observations with accuracy of about a few $\mu$as.
In this paper we addressed the problem of light propagation on the
gravitational field of the solar system. It was shown that for high accuracy
observations it is necessary to correct for the post-Newtonian deflection of
light by the monopole components of gravitational fields of a large number of
celestial bodies in the solar system, namely the Sun and the nine planets,
together with the planetary satellites and the largest asteroids (important
only if observations are conducted in their close proximity). The most
important fact is that the gravitational presence of the Sun, Jupiter and the
Earth should be always taken into account, independently on the positions of
these bodies relative to the interferometer. It is worth noting that the post-
post Newtonian effects due to the solar gravity are unlikely to be accessible
with SIM. This effect as well as the effect of gravitational deflection of
light caused by the mass quadrupole term of the Sun are negligible at the
level of expected accuracy. However, deflection of light by some planetary
quadrupoles may have a big impact on the astrometric accuracy. Thus, the
higher gravitational multipoles should be taken into account when observations
are conducted in the close proximity of two bodies of the solar system,
notably Jupiter and Saturn.
We emphasized the need of development of a general relativistic model for SIM
observables to enable the mission to improve the current astrometric accuracy
by a factor of over 1000. This model would have to account for a number of
dynamical effects both external to the spacecraft (e.g., motion with respect
to the solar system barycentric reference frame, effects of time-varying
gravitational field in the solar system (due to planetary motion nd rotation)
on light propagation, various interplanetary media effects, etc.) and internal
to the spacecraft (e.g., systematic effects introduced by the spacecraft
itself). Some of this work has already begun in the context of the development
relativistic reference frames for the need of future high-precision
observations Kopeikin-Makarov-2007 . However, a lot more efforts is needed;
this paper intends to motivate initiation of such a work in the near future.
## Acknowledgments
The reported research has been done at the Jet Propulsion Laboratory,
California Institute of Technology, which is under contract to the National
Aeronautic and Space Administration.
## References
* (1) Turyshev S. G., Israelsson U. E., Shao M., Yu N., Kusenko A., Wright E. L., Everitt C. W. F., Kasevich M., Lipa J. A., Mester J. C., Reasenberg R. D., Walsworth R. L., Ashby N., Gould H., Paik H. J., “Space-based research in fundamental physics and quantum technologies,” Inter. J. Modern Phys. D 16(12a), 1879-1925 (2007), arXiv:0711.0150 [gr-qc].
* (2) Turyshev S. G., “Experimental Tests of General Relativity,” in print, Ann. Rev. Nucl. Part. Sci. 58 (2008), arXiv:0806.1731 [gr-qc].
* (3) Soffel M., Klioner S. A., Petit G., Wolf P., Kopeikin S. M., Bretagnon P., Brumberg V. A., Capitaine N., Damour T., Fukushima T., Guinot B., Huang T.-Y., Lindegren L., Ma C., Nordtvedt K., Ries J. C., Seidelmann P. K., Vokrouhlický D., Will C. M., Xu C., “The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement,” Astron. J. 126(6), 2687-2706 (2003).
* (4) Kopeikin S. M., Makarov V. V., “Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers,” Phys. Rev D. 75(6), 062002 (2007).
* (5) Kopeikin S. M., “Propagation of light in the stationary field of multipole gravitational lens,” Journal of Mathematical Physics 38, 2587-2601 (1997).
* (6) Unwin S. C., Shao M., Tanner A. M., Allen R. J., Beichman C. A., Boboltz D., Catanzarite J. H., Chaboyer B. C., Ciardi D. R., Edberg S. J., Fey A. L., Fischer D. A., Gelino C. R., Gould A. P., Grillmair C., Henry T. J., Johnston K. V., Johnston K. J., Jones D. L., Kulkarni S. R., Law N. M., Majewski S. R., Makarov V. V., Marcy G. W., Meier D. L., Olling R. P., Pan X., Patterson R. J., Pitesky J. E., Quirrenbach A., Shaklan S. B., Shaya E. J., Strigari L. E., Tomsick J. A., Wehrle A. E., and Worthey G., “Taking the Measure of the Universe: Precision Astrometry with SIM PlanetQuest,” Publ.Astron. Soc. Pacific 120, 38-88 (2008), arXiv:0708.3953 [astro-ph].
* (7) Perryman M.A.C., de Boer K.S., Gilmore G., Høg E., Lattanzi M.G. , Lindegren L., Luri X., Mignard F., Pace O., and de Zeeuw, P.T., “Gaia: Composition, Formation and Evolution of the Galaxy,” A&A 369, 339-363 (2001).
* (8) Perryman M.A.C., “GAIA: An Astrometric and Photometric Survey of our Galaxy,” Ap&SS 280, 1 (2002).
* (9) Johnston K. J., Fey A. L., Zacharias N., Russell J. L., Ma C., de Vegt C., Reynolds J. E., Jauncey D. L., Archinal B. A., Carter M. S., Corbin T. E., Eubanks T. M., Florkowski D. R., Hall D. M., McCarthy D. D., McCulloch P. M., King E. A., Nicolson G., Shaffer D. B., “A Radio Reference Frame,” Astron. J 110, 880 (1995).
* (10) Ma C., Arias E. F., Eubanks T. M., Fey A. L., Gontier A., Jacobs C. S., Sovers O. J., Archinal B. A., Charlot P., “The International Celestial Reference Frame As Realized by Very Long Baseline Interferometry,” AJ 116, 516-546 (1998).
* (11) Perryman M. A. C., Hög E., Kovalevsky J., Lindegren L., Turon C., Bernacca P. L., Creze M., Donati F., Grenon M., Grewing M., “In-orbit performance of the HIPPARCOS astrometry satellite,” Astron. & Astrophys. 258, 1 (1992).
* (12) Gould A., “Deflection of light by the earth,” ApJ 414, L37 (1993).
* (13) Klioner S. A., “Physically adequate reference system of a massless observer and relativistic description of the GAIA attitude,” Phys. Rev. D 69, 124001 (2004), astro-ph/0311540.
* (14) Brumberg V. A., Relativistic Celestial Mechanics, Nauka, Moscow (1972).
* (15) Brumberg V. A., Essential Relativistic Celestial Mechanics. Adam Hilger, London (1991).
* (16) Sovers O. J., Jacobs C. S., in Observation Model and Parameter Partials for the JPL VLBI Parameter Estimation Software ”MODEST” - 1996, JPL Technical Report 83-39, Rev. 6, Pasadena, CA (1996).
* (17) Will C. M., Theory and Experiment in Gravitational Physics, (Rev. Ed.), Cambridge Univ. Press, England (1993).
* (18) Fomalont E. B., Kopeikin S. M., “The Measurement of the Light Deflection from Jupiter: Experimental Results,” ApJ 598, 704-711 (2003).
* (19) Dar A., “Tests of general relativity and Newtonian gravity at large distances and the dark matter problem,” Nucl. Phys. B (Suppl.) 28A, 321 (1992)
* (20) Treuhaft R. N., Lowe S. T., “A Measurement of Planetary Relativistic Deflection,” Astron. J. 102, 1879 (1991).
* (21) Turyshev S. G., “Relativistic Effects in the SIM Astrometric Campaign.” BAAS 29(5), 1223 (1998); Turyshev S. G., gr-qc/0205061; gr-qc/0205062; gr-qc/0205063.
* (22) Sovers O. J., Fanselow, J. L., and Jacobs, C. S., “Astrometry and geodesy with radio interferometry: experiments, models, results”, Reviews of Mod. Phys.70(4), 1393-1454 (1998).
* (23) Brumberg V. A., Klioner S. A., Kopeikin S. M., “Relativistic reduction of astrometric observations at POINTS level of accuracy”, IAU Symposium 141, eds. Lieske, J. H. and Abalakin, V. K., 229-239 (1990).
* (24) Will C. M., “The Confrontation between General Relativity and Experiment,” Liv. Rev. Relativity 9 (2006), gr-qc/0510072.
* (25) Bertotti B., Iess L., Tortora P., “A test of general relativity using radio links with the Cassini spacecraft,” Nature 425, 374 (2003).
* (26) Moyer T. D., Formulation for Observed and Computed Values of Deep Space Network Data Types for NavigationJPL Deep-Space Communications and Navigation Series (John Wiley & Sons, Inc., Hoboken, New Jersey, 2003).
* (27) Kopeikin, S M., Schäfer, G., “Lorentz covariant theory of light propagation in gravitational fields of arbitrary-moving bodies,” Phys. Rev. D 60(12), 124002 (1999).
* (28) Klioner S. A., “A Practical Relativistic Model for Microarcsecond Astrometry in Space,” ApJ 125, 1580-1597 (2003).
* (29) Standish E. M. Jr., Hellings, R. W., “A determination of the masses of Ceres, Pallas, and Vesta from their perturbations upon the orbit of Mars,” Icarus 80, 326-333 (1989).
* (30) Mouret S., Hestroffer D., and Mignard F., “Asteroid masses and improvement with Gaia,” Astron. & Astrophys. 472, 1017-1027 (2007).
* (31) Epstein R., Shapiro I. I., “Post-post-Newtonian deflection of light by the Sun,” Phys. Rev. D 22, 2947 (1980); Fischbach E., Freeman B. S., “Second-order contribution to the gravitational deflection of light,” Phys. Rev. D 22, 2950 (1980); Richter G. W., Matzner R. A., “2nd-order contributions to relativistic time delay in the parametrized post-Newtonian formalism,” Phys. Rev. D 26, 1219 (1982); “Second-order contributions to gravitational deflection of light in the parametrized post-Newtonian formalism. II. Photon orbits and deflections in three dimensions,” ibid, Phys. Rev. D 26, 2549 (1982); Richter, G. W., Matzner, R. A., ibid, Phys. Rev. D 28, 3007 (1983).
* (32) Crosta M. T., and Mignard F., “Microarcsecond light bending by Jupiter,” Class. Quant. Grav. 23(15) 4853 (2006).
* (33) Fomalont E. B., Kopeikin S. M., “Radio interferometric tests of general relativity,” Proc. IAU Symposium No. 248, eds. Jin et al., 383 (2007).
* (34) Yoder C. F., Astrometric and Geodetic Properties of Earth and the Solar System. Global Earth Physics. A Handbook of Physical Constants, AGU Reference Shelf 1. (1995).
* (35) Pitjeva E. V., “High-Precision Ephemerides of Planets—EPM and Determination of Some Astronomical Constants,” Solar System Res. 39, 176-186 (2005).
* (36) Folkner W. M., Williams J. G., Boggs D. H., “The Planetary and Lunar Ephemeris DE 421,” JPL Memorandum IOM 343R-08-003, 31 March 2008, ftp://ssd.jpl.nasa.gov/pub/eph/planets/ascii/de421
* (37) Standish E. M. Jr., Astronomical and Astrophysical Objectives of Sub-Milliarcsecond Optical Astrometry. IAU-SYMP, 166, eds. E. Hög and P. K. Seidelmann., 109 (1995).
* (38) Standish E. M. Jr., Newhall X X, Williams J. G., and Folkner W. M., JPL Planetary and Lunar Ephemeris, DE403/LE403, Jet Propulsion Laboratory IOM # 314.10-127 (1995).
* (39) Folkner W. M., Charlot P., Finger M. H., Williams J. G., Sovers O. J., Newhall X X, Standish E. M. Jr., “Determination of the extragalactic-planetary frame tie from joint analysis of radio interferometric and lunar laser ranging measurements,” Astron. & Astrophys. 287, 279 (1994).
* (40) Klioner S. A., “Influence of the Quadrupole Field and Rotation of Objects on Light Propagation,” Soviet Astron. 35, 523 (1991).
* (41) Kopeikin, S. M., Fomalont, E. B., “General relativistic model for experimental measurement of the speed of propagation of gravity by VLBI,” in Proc. of the 6th EVN Symposium, 2002, eds. Ros E. et al., 49 (2002).
* (42) Ciufolini I., Kopeikin S., Mashhoon B., Ricci F., “On the gravitomagnetic time delay,” Phys. Lett. A 308, 101-109 (2003).
* (43) Kopeikin S., Mashhoon B., “Gravitomagnetic effects in the propagation of electromagnetic waves in variable gravitational fields of arbitrary-moving and spinning bodies,” Phys. Rev. D 65, 064025 (2002).
* (44) Dicke R. H., “The Oblateness of the Sun and Relativity,” Science 184, 419-429 (1974).
* (45) Damour T., Nordtvedt K., Phys. Rev. D 48, 3436 (1993); Damour T., Polyakov A. M., Nucl. Phys. B 423, 532 (1994); Damour T., Piazza F., and Veneziano G., Phys. Rev. D 66, 046007 (2002) [arXiv:hep-th/0205111].
* (46) Spergel D. N., Bean R., Doré O., Nolta M. R., Bennett C. L., Hinshaw G., Jarosik N., Komatsu E., Page L., Peiris H. V., Verde L., Barnes C., Halpern M., Hill R. S., Kogut A., Limon M., Meyer S. S., Odegard N., Tucker G. S., Weiland J. L., Wollack E., Wright E. L., “Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology,” Astrophys. J. Suppl. 170, 377 (2007), [arXiv:astro-ph/0603449].
* (47) Kopeikin S. M., Makarov V. V., “Astrometric Effects of secular aberration”, Astron. J. 131(3), 1471-1478 (2006).
|
arxiv-papers
| 2008-09-07T20:50:51
|
2024-09-04T02:48:57.698454
|
{
"license": "Public Domain",
"authors": "Slava G. Turyshev",
"submitter": "Slava G. Turyshev",
"url": "https://arxiv.org/abs/0809.1250"
}
|
0809.1303
|
Current conservation, screening and the magnetic moment of the $\Delta$
resonance. 111 Supported by the ”Deutsche Forschungsgemeinschaft” under
contract GRK683
2\. Formulation with quark degrees of freedom
3\. Magnetic moment of the $\Delta^{o}$ and $\Delta^{-}$ resonances.
A. I. Machavariania b c and Amand Faessler a
a Institute für Theoretische Physik der Univesität Tübingen,
Tübingen D-72076, Germany
b Joint Institute for Nuclear Research, Dubna, Moscow region 141980, Russia
c Tbilisi State University, University str. 9, Tbilisi, Georgia
###### Abstract
Our previous paper [1] is generalized within the field theoretical formulation
with the quark degrees of freedom [2, 3, 4, 5], where pions and nucleons are
treated as the bound systems of quarks. It is shown that relations generated
by current conservation for the on shell $\pi N$ bremsstrahlung amplitude with
composite nucleons and pions have the same form as in the usual quantum field
theory [6, 7] without quark degrees of freedom [1]. Consequently, the model
independent relations for the magnetic dipole moments of the $\Delta^{+}$ and
$\Delta^{++}$ resonances in [1] remain be the same in the quantum field theory
with the quark degrees of freedom. These relations are extended for the
magnetic dipole moments of the $\Delta^{o}$ and $\Delta^{-}$ resonances which
are determined via the anomalous magnetic moment of the neutron $\mu_{n}$ as
$\mu_{\Delta^{o}}={{M_{\Delta}}\over{m_{p}}}\mu_{n}$ and
$\mu_{\Delta^{-}}={3\over 2}\mu_{\Delta^{o}}$.
1\. Introduction
The self-consistent generalization of the conventional quantum field theory
for composite particles was developed in refs. [2, 3, 4, 5]. In this approach
the quark-hadron bound state functions satisfy the appropriate Bethe-Salpeter
equations and determine the transitions from the quark-gluon degrees of
freedom into hadron degrees of freedom. The general Bethe-Salpeter equations
for the quark-hadron wave functions were derived by Huang and Weldon [2]. The
basic objects in this approach as well as in the Haag-Nishijima-Zimmermann
approaches [3, 4, 5] are the creation and annihilation operators of composite
particles which allows to obtain the $S$-matrix reduction formula for
composite particles. The equivalent three dimensional field-theoretical
equations for composite hadron interaction amplitudes are given in ref. [8, 9,
10]. The advantages of this three-dimensional formulation may be summarized as
follows:
* •
The poles of the intermediate quark and gluon propagators do not contribute to
the unitarity condition of the hadron-hadron scattering amplitudes. Therefore,
the quark-gluon and hadron degrees of freedom are unambiguously separated and
the problems with the double counting do not appear.
* •
The resulting equations for the hadron-hadron scattering amplitudes with and
without quark degrees of freedom have the same form. Therefore, one can easily
extend the field-theoretical relations without quarks to the formulation with
the quark-gluon degrees of freedom.
The usual $\pi N$ bremstrahlung amplitude
$<out;{N^{\prime}},{\pi^{\prime}}|{\cal J}^{\mu}(0)|{\pi},N;in>$ with on mass
shell pions and nucleons in the ${}^{\prime\prime}in^{\prime\prime}$ and
${}^{\prime\prime}out^{\prime\prime}$ asymptotic states and with the photon
current operator ${\cal J}^{\mu}(x)$ is depicted by the left diagram in Fig.1.
In the generalized quark-gluon approach [2, 3, 4, 5] this amplitude is
determined through the Green function of the transition between the
$4quark+antiquark$ systems and the quark-hadron bound-state wave functions.
The graphical representation of the $\pi N$ radiation amplitude in the quark-
gluon approach [2, 3, 4, 5] is given by the diagram on the right-side of Fig.
1. The construction of the creation and annihilation operators of the
asymptotic nucleons and pions via quarks allows to operate with the usual
expression of the $\pi N$ radiation amplitude
$<out;{N^{\prime}},{\pi^{\prime}}|{\cal J}^{\mu}(0)|{\pi},N;in>$ because
$<out;{N^{\prime}},{\pi^{\prime}}|{\cal J}^{\mu}(0)|{\pi},N;in>\equiv<0|{\cal
B}_{out}(N^{\prime}){\sf a}_{out}(\pi^{\prime}){\cal J}^{\mu}(0){\sf
a^{+}}_{in}(\pi){\cal B^{+}}_{in}(N)|0>,$ $None$
where ${\cal B}_{out(in)}(N)$ and ${\sf a}_{out(in)}(\pi)$ denote the creation
or annihilation operators of the composed nucleon and pion in the asymptotic
$"out"$ or $"in"$ states.
Figure 1: The $\pi N$ bremsstrahlung amplitude for composite nucleons and
pions.
In this paper we shall show that current conservation on the quark level for
the on shell $\pi N$ bremsstrahlung amplitude (1.1) takes again the form of
the modified Ward-Takahashi identity for the on shell external particle
radiation amplitude ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}$
$k^{\prime}_{\mu}<out;{N^{\prime}},{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\pi},N;in>=k^{\prime}_{\mu}{\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}+{\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}=0,$ $None$
where $k^{\prime}_{\mu}$ is the four momentum of the final photon, ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$ stands for a sum of the off shell elastic
$\pi N$ scattering amplitudes.
The external particle radiation diagrams are depicted in Fig. 2. The only
difference between the external particle radiation amplitude in Fig. 2 and in
the formulation without quark degrees of freedom (see Fig. 1 in [1]) is the
off shell $\pi N$ amplitudes, which contains the nonlocal momentum-depending
source operators of the pion or nucleon.
Figure 2: The external particle radiation diagrams of the $\pi N$
bremsstrahlung amplitude.
Current conservation (1.2a) indicates a connection between the four-divergence
of the external ${\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$
and internal ${\cal I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$
particle radiation amplitudes ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$
$k^{\prime}_{\mu}{\cal E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=-k^{\prime}_{\mu}{\cal I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}=-{\cal B}_{\pi^{\prime}N^{\prime}-\pi N},$ $None$
where $<out;{N^{\prime}},{\pi^{\prime}}|{\cal J}^{\mu}(0)|{\pi},N;in>={\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}+{\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$.
Figure 3: The double on mass shell $\Delta$ exchange diagram with the
intermediate $\Delta$ radiation vertex. The $\Delta-\gamma\Delta$ vertex
contains the dipole magnetic moment of the $\Delta$.
One can decompose (1.2a,b) into a set of independent current conservation for
the longitudinal part of the on shell $\pi N$ radiation amplitude [1]. In
particular, one can unambiguously separate the $\Delta$-pole parts of the $\pi
N$ amplitudes which are contained in ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ and ${\cal
B}_{\pi^{\prime}N^{\prime}-\pi N}$. The $\Delta$-pole part of the full $\pi N$
Green function determines the $\pi N-\Delta$ wave function with the on mass
shell $\Delta$ and the effective mass of the $\Delta$ [1, 10, 11, 12] ${\sf
m}_{\Delta}(s)=M_{\Delta}(s)-i/2\Gamma_{\Delta}(s)$ which generally depends on
the Mandelstam variable $s$. This $\pi N-\Delta$ wave function and ${\sf
m}_{\Delta}(s)$ define the intermediate on mass shell $\Delta$ state with the
four momentum $P_{\Delta}=(\sqrt{{\sf m}^{2}_{\Delta}(s)+{\bf
P}^{2}_{\Delta}},{\bf P}_{\Delta})$ also in the present formulation with quark
degrees of freedom. The considered field theoretical $\pi N$ bremsstrahlung
amplitudes are not depending on the model of ${\sf m}_{\Delta}(s)$ which must
be determined separately. The sum of the $\Delta$-pole parts of the off shell
$\pi N$ amplitudes in ${\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ (Fig. 2) reproduces
the double on mass shell $\Delta$ exchange amplitude $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ which contains the $\Delta-\gamma\Delta$ vertex with
the anomalous magnetic moment of the proton instead of the magnetic dipole
moment of the $\Delta$. $({\cal
E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$ has the same analytical structure as the intermediate
$\Delta$ radiation diagram ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
in Fig. 3. This amplitude is unambiguously separated from the internal
particle radiation diagrams based on the $\Delta$-pole terms of the
intermediate $\pi N$ Green function [1]. Thus
$k^{\prime}_{\mu}({{\cal
E_{L}}^{3/2}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=-{k^{\prime}}_{\mu}{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta),$ $None$
where the lower index L and the upper index 3/2 denotes the longitudinal and
the spin-isospin $(3/2,3/2)$ part of the corresponding expressions. The
identical structure of
$({\cal E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ and ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}({\Delta}-\gamma\Delta)$
in Fig. 3 allows to obtain an analytical and model independent relations
between the magnetic dipole moments of the $\Delta^{+}$ and $\Delta^{++}$
resonances and the anomalous magnetic moment of the proton. In this paper we
generalize this relation for the magnetic dipole moments of the $\Delta^{o}$
and $\Delta^{-}$ resonances.
This paper consists of four Sections. The creation and annihilation operators
of composite particle and corresponding Ward-Takahashi identity are
constructed in the next Section. The model-independent relation between the
magnetic dipole moments of the $\Delta^{o}$ and $\Delta^{-}$ resonances and
the anomalous magnetic moment of the neutron is given in Section 3. The
conclusions and comparison of the suggested relations for the magnetic dipole
moments of ${\Delta^{o}}$ and ${\Delta^{-}}$ with the numerical values of
other authors are presented in Sect. 4.
2\. The Ward-Takahashi identity for the on shell $\pi N$ bremstrahlung
amplitude in the field theoretical approach with the quark degrees of freedom
The creation and annihilation operators of the hadrons as the quark cluster
operators were constructed in ref. [2] in the framework of the usual quantum
field theory. The corresponding nucleon and pion field operators
$\Psi_{p_{N}}(Y)$ and $\Phi_{p_{\pi}}(X)$ are composed through the local quark
field operators $q(x)$. $\Psi_{p_{N}}(Y)$ and $\Phi_{p_{\pi}}(X)$ are nonlocal
because they depend on the nucleon and pion four moments $p_{N}$ and $p_{\pi}$
correspondingly
$\Psi_{\bf p_{N}}(Y)=\int
d^{4}r_{3}d^{4}r_{1,2}{\widetilde{\chi}}^{{\dagger}}_{\bf
p_{N}}(Y=0,r_{1,2}.r_{3}){\sf
T}\biggl{(}q_{1}(y_{1})q_{2}(y_{2})q_{3}(y_{3})\biggr{)},$ $None$
where $Y$, $r_{1,2}$ and $r_{3}$ are the Jacobi coordinates
$y_{1}=Y-\eta_{3}r_{3}+\eta_{2}r_{1,2}$,
$y_{2}=Y-\eta_{3}r_{3}-\eta_{1}r_{1,2}$, $y_{3}=Y+\eta_{1,2}r_{3}$ with
$\eta_{3}+\eta_{1,2}=1$ and $\eta_{1}+\eta_{2}=1$,
$\chi_{\bf p_{N}}(y_{1},y_{2},y_{3})=<0|{\sf
T}\biggl{(}q_{i}(y_{1})q_{j}(y_{2})q_{k}(y_{3})|{\bf
p}_{N},s_{N},i_{N};in>=e^{-ip_{N}Y}\chi_{\bf p_{N}}(Y=0,r_{1,2},r_{3})$ $None$
is a solution of the Bethe-Salpeter equation for the three quark bound state
with the nucleon mass $m_{N}$ and four momentum $p_{N}=(\sqrt{{\bf
p_{N}}^{2}+m_{N}^{2}},{\bf p_{N}})$; $s_{N}$ and $i_{N}$ denotes the spin-
isospin projections of the nucleon. ${\widetilde{\chi}}^{{\dagger}}_{p}$
satisfies the normalization condition [6]
${1\over{2im_{N}^{2}}}<\chi_{\bf
p^{\prime}_{N}}|p_{N}^{\mu}{{\partial}\over{\partial
p_{N}^{\mu}}}\Bigl{[}G^{-1}(3q)\Bigr{]}|\chi_{\bf
p_{N}}>\equiv<{\widetilde{\chi}}_{\bf p^{\prime}_{N}}|\chi_{\bf p_{N}}>=<{\bf
p^{\prime}_{N}},s^{\prime}_{N},i^{\prime}_{N};m_{N}|{\bf
p_{N}},s_{N},i_{N};m_{N}>,$ $None$
where $G(3q)$ is the full Green function of three interacting quarks222 The on
mass shell $\Delta$ state with the complex mass
$m_{\Delta}=M_{\Delta}+i\Gamma_{\Delta}/2$ can be constructed through the
intermediate three quark state in the same manner as the one nucleon state. In
particular, it is necessary to find the solution of the Bethe-Salpeter
equation for the $\Delta$-pole state of the $3quark-3quark$ Green function:
$\Psi_{{\bf
p}_{\Delta}}(x_{1},x_{2},x_{3})=<0|T\bigl{(}q_{1}(x_{1})q_{2}(x_{2})q_{3}(x_{3})\bigr{)}|\Psi_{{\bf
p}_{\Delta}}>$..
The asymptotic nucleon annihilation (creation) operator ${\cal
B}^{in(out)}({\bf p}_{N})$ for an on mass-shell nucleon is determined as
${\cal B}_{in(out)}({\bf p}_{N})=\lim_{x^{o}\to-\infty(+\infty)}{\cal B}_{\bf
p_{N}}(x^{o}),$ $None$
where the weak limit $\lim_{x^{o}\to-\infty(+\infty)}$ is assumed. The
Heisenberg operator ${\cal B}_{\bf p_{N}}(X^{o})$ is given in the same form as
in conventional quantum field theory
${\cal B}_{\bf p_{N}}(x^{o})=\int d^{3}{\bf
x}\exp{(ip_{N}x)}{\overline{u}}({\bf p}_{N})\gamma_{o}\Psi_{\bf p_{N}}(x).$
$None$
The composite meson fields are constructed using the quark-antiquark operator
$\Phi_{{\bf p}_{\pi}}(X)=\int d^{4}\rho_{1,2}{\widetilde{\phi}}^{+}_{\bf
p_{\pi}}(X=0,\rho_{1,2}){\sf
T}\biggl{(}q_{i}(x_{1}){\overline{q}}_{i}(x_{2})\biggr{)},$ $None$
where $p_{\pi}=(\sqrt{{\bf p}_{\pi}^{2}+m_{\pi}^{2}},{\bf p}_{\pi})$,
$x_{1}=X+\mu_{2}\rho_{1,2}$, $x_{2}=X-\mu_{1}\rho_{1,2}$ with
$\mu_{1}+\mu_{2}=1$ and
$\phi_{\bf p_{\pi}}(x,y)=<0|{\sf
T}\biggl{(}q_{i}(x){\overline{q}}_{i}(y)\biggr{)}|{\bf
p_{\pi}},i_{\pi};m_{\pi}>$ $None$
is the solution of the Bethe-Salpeter equation of the quark-antiquark bound
state. This function satisfies the normalization condition
${!\over{2im_{\pi}^{2}}}<\phi_{{\bf
p^{\prime}}_{\pi}}|p^{\mu}_{\pi}{{\partial}\over{\partial
p^{\mu}_{\pi}}}g^{-1}(q{\overline{q}})|\phi_{{\bf
p}_{\pi}}>\equiv<{\widetilde{\phi}}_{{\bf p^{\prime}}_{\pi}}|\phi_{{\bf
p}_{\pi}}>=<{\bf p^{\prime}_{\pi}},i^{\prime}_{\pi};m_{\pi}|{\bf
p_{\pi}},i_{\pi};m_{\pi}>,$ $None$
where $g(q{\overline{q}})$ is the full quark-antiquark Green function.
The asymptotic meson creation or annihilation operator is
${\sf a}_{in(out)}({\bf p}_{\pi})=\lim_{x^{o}\to-\infty(+\infty)}{\sf a}_{{\bf
p}_{\pi}}(x^{o}),$ $None$
where
${\sf a}_{{\bf p}_{\pi}}(x^{o})=\int d^{3}{\bf
x}\exp{(ip_{\pi}x)}\Bigl{[}{{\partial}\over{\partial
x^{o}}}-ip_{\pi}^{o}\Bigr{]}\Phi_{p_{\pi}}(x).$ $None$
The composite operators (2.3a) and (2.5a) satisfy the same commutation
relations as the ordinary local field operators of the asymptotic nucleons and
pions in the usual quantum field theory [6, 7]
$\biggl{\\{}{\cal B}_{in(out)}^{+}({\bf p^{\prime}}),{\cal B}_{in(out)}({\bf
p})\biggl{\\}}=(2\pi)^{3}{{p_{N}^{o}}\over{m_{N}}}\delta({\bf p^{\prime}-p});$
$\biggl{\\{}{\cal B}_{in(out)}({\bf p^{\prime}}),{\cal B}_{in(out)}({\bf
p})\biggr{\\}}=\biggl{\\{}{\cal B}_{in(out)}^{+}({\bf p^{\prime}}),{\cal
B}^{+}_{in(out)}({\bf p})\biggr{\\}}=0,$ $None$
$\biggl{[}{\sf a}_{in(out)}^{+}({\bf p^{\prime}}_{\pi}),{\sf a}_{in(out)}({\bf
p}_{\pi})\biggr{]}=(2\pi)^{3}2p^{o}_{\pi}\delta({\bf
p^{\prime}_{\pi}-p_{\pi}});$ $\biggl{[}{\sf a}_{in(out)}({\bf
p^{\prime}}_{\pi}),{\sf a}_{in(out)}({\bf p}_{\pi})\biggr{]}=\biggl{[}{\sf
a}_{in(out)}^{+}({\bf p^{\prime}}_{\pi}),{\sf a}_{in(out)}^{+}({\bf
p}_{\pi})\biggr{]}=0.$ $None$
The relations (2.6a,b) allow to build any $"in"$ or $"out"$ hadron states
through the intermediate quark-cluster states. These operators form the usual
completeness condition for the asymptotic $"in"$ or $"out"$ hadron fields
$\sum_{n}|n;in(out)>$ $<(out)in;n|={\widehat{\bf 1}}$ and the well-known
${\cal S}$-matrix element as ${\cal S}_{nm}=<out;n|m;in>$. In contrast to
local quantum field theory, the Heisenberg fields ${\cal B}_{\bf p}(x^{o})$
(2.3b) and ${\sf a}_{\bf p}(x^{o})$ (2.5b) do not satisfy the equal-time
commutation relations (2.6a,b) $\biggl{\\{}{\cal B}_{\bf p^{\prime}}(x_{o})$,
${\cal B}_{\bf p}(x_{o})\biggr{\\}}\neq 0$, $\biggl{[}{\cal B}_{\bf
p^{\prime}}(x_{o}),{\sf a}_{{\bf p}_{\pi}}(x_{o})\biggr{]}\neq 0$, etc.
Nevertheless, the basic relations of the usual quantum field theory remain the
same in the field theoretical approach with quarks. In particular, for the
$\pi N$ radiation amplitude $A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}$ one has
$A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=i\int
d^{4}ze^{ik^{\prime}z}<0|{\cal B}_{out}({\bf p^{\prime}}_{N}){\sf
a}_{out}({\bf p^{\prime}}_{\pi}){\cal J}^{\mu}(z){\sf a^{+}}_{in}({\bf
p}_{\pi}){\cal B^{+}}_{in}({\bf p}_{N})|0>,$ $None$
where $k^{\prime}_{\mu}$ denotes the four momentum of the on shell final
photon $k^{\prime}_{\mu}{k^{\prime}}^{\mu}=0$ and
$k^{\prime}_{\mu}=(p_{N}+p_{\pi}-p^{\prime}_{N}-p^{\prime}_{\pi})_{\mu}\equiv(P-P^{\prime})_{\mu}$;
${\cal
J}^{\mu}(z)={\overline{q}}(z)\Bigl{(}{{\lambda^{3}}\over{2}}+{{\lambda^{8}}\over{2\sqrt{3}}}\Bigr{)}\gamma^{\mu}q(z)$
$None$
denotes the photon source operator with the Gell-Mann flavor matrices
$\lambda$ [6].
A symbolical picture of the $\pi N$ bremsstrahlung amplitude (2.7a) with the
intermediate quark-clusters states is given in Fig. 1. The triangles in Fig. 1
describe the quark-hadron bound state wave functions (2.2a), (2.4a) and their
orthogonal expressions. These vertices play the role of the hadronization
functions. Consequently, the $\pi
N\Longrightarrow\gamma^{\prime}\pi^{\prime}N^{\prime}$ amplitude (2.7a) is
replaced by the
$4q{\overline{q}}-\gamma^{\prime}4q^{\prime}{\overline{q}}^{\prime}$
transition amplitude. Using the generalized ${\cal S}$-matrix reduction
formula [2] we obtain
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}={\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\
{k^{\prime}}_{\mu}{\sc G}^{\mu}\
(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})({p_{\pi}}^{2}-m_{\pi}^{2})u({\bf p}_{N}),$
$None$
where
${k^{\prime}}_{\mu}{\sc G}^{\mu}=i\int
d^{4}y^{\prime}_{1}d^{4}y^{\prime}_{2}d^{4}y^{\prime}_{3}d^{4}x^{\prime}_{1}d^{4}x^{\prime}_{2}d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}d^{4}x_{1}d^{4}x_{2}e^{ik^{\prime}z}d^{4}z{\widetilde{\chi}}^{+}_{\bf
p^{\prime}_{N}}(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3}){\widetilde{\phi}}^{+}_{\bf
p^{\prime}_{\pi}}(x^{\prime}_{1},x^{\prime}_{2}){{\partial}\over{\partial
z^{\mu}}}$ $<0|{\sf T}\biggl{(}{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}{\cal
J}^{\mu}(z){\sf
T}\Bigl{(}{\overline{q}}_{1}(y_{1}){\overline{q}}_{2}(y_{2}){\overline{q}}_{3}(y_{3})\Bigr{)}{\sf
T}\Bigl{(}{\overline{q}}_{1}(x_{1})q_{2}(x_{2})\Bigr{)}\biggr{)}|0>$
$\chi_{\bf p_{N}}(y_{1},y_{2},y_{3})\phi_{\bf p_{\pi}}(x_{1},x_{2}),$ $None$
where the double time-ordered product is defined as ${\sf T}\biggl{(}{\sf
T}\Bigl{(}{\overline{q}}_{1}(x_{1})q_{2}(x_{2})\Bigr{)}{\cal
J}^{\mu}(z)\biggr{)}=$
$\Bigl{(}{\overline{q}}_{1}(x_{1})\theta(x_{1}^{o}-x_{2}^{o})q_{2}(x_{2})\theta(x_{2}^{o}-z^{o}){\cal
J}^{\mu}(z)-{\overline{q}}_{2}(x_{2})\theta(x_{2}^{o}-x_{1}^{o})q_{1}(x_{1})\theta(x_{1}^{o}-z^{o}){\cal
J}^{\mu}(z)\Bigr{)}$
$+\Bigl{(}{\cal
J}^{\mu}(z)\theta(z^{o}-x_{1}^{o}){\overline{q}}_{1}(x_{1})\theta(x_{1}^{o}-x_{2}^{o})q_{2}(x_{2})-{\cal
J}^{\mu}(z)\theta(z^{o}-x_{2}^{o}){\overline{q}}_{2}(x_{2})\theta(x_{2}^{o}-x_{1}^{o})q_{1}(x_{1})\Bigr{)}$.
The choice of the average $X=x_{1}+x_{2}$, $Y=y_{1}+y_{2}+y_{3}$ or the c.m.
coordinates $X=\mu_{1}x_{1}+\mu_{2}x_{2}$, $\rho_{1,2}=x_{1}-x_{2}$ and
$Y=\eta_{3}y_{3}+\eta_{1,2}Y_{1,2}$, $\rho_{3}=y_{3}-Y_{1,2}$,
$Y_{1,2}=\eta_{1}y_{1}+\eta_{2}y_{2}$ in (2.1) and in (2.4a) is not unique.
But the $S$-matrix reduction formula and (2.8a,b) are not depend on the choice
of $X$ and $Y$.
The important property of the on shell amplitude (2.8a) is that only the
operators ${\overline{u}}({\bf p}_{N})(\gamma_{\nu}{\bf
p_{N}}^{\nu}-m_{N}){\widehat{\sc P}}_{N^{\prime}}\equiv{\overline{u}}({\bf
p}_{N})(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})\int
d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}{\widetilde{\chi}}^{+}_{\bf
p_{N}}(y_{1},y_{2},y_{3}){\sf
T}\Bigl{(}q_{1}(y_{1})q_{2}(y_{2})q_{3}(y_{3})\Bigr{)}$,
$({p_{\pi}}^{2}-m_{\pi}^{2}){\widehat{\sc
P}}_{\pi}\equiv({p_{\pi}}^{2}-m_{\pi}^{2})\int
d^{4}x_{1}d^{4}x_{2}{\widetilde{\phi}}^{+}_{\bf p_{\pi}}(x_{1},x_{2}){\sf
T}\Bigl{(}q_{1}(x_{1}){\overline{q}}_{2}(x_{2})\Bigr{)}$ and their Hermitian
conjugate produce the asymptotic one-nucleon and one-pion states. Therefore,
the zeros of the Dirac $(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})$,
$(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})$ and the Klein-Gordon operators
$({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})$, $({p_{\pi}}^{2}-m_{\pi}^{2})$ in the
on shell amplitude (2.8a) can be compensated only by ${\widehat{\sc
P}}_{N^{\prime}}$, ${\widehat{\sc P}}_{\pi^{\prime}}$ and their Hermitian
conjugate operators. But ${\widehat{\sc P}}_{N^{\prime}}$, ${\widehat{\sc
P}}_{\pi^{\prime}}$ and their conjugate are included in ${\sc G}^{\mu}$
(2.8b). The remaining part of the full Green function
${\tau}^{\mu}={\sc G}^{\mu}+{\em g}^{\mu}=$ $\int
d^{4}y^{\prime}_{1}d^{4}y^{\prime}_{2}d^{4}y^{\prime}_{3}d^{4}x^{\prime}_{1}d^{4}x^{\prime}_{2}d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}d^{4}x_{1}d^{4}x_{2}e^{ik^{\prime}z}d^{4}z{\widetilde{\chi}}^{+}_{\bf
p^{\prime}_{N}}(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3}){\widetilde{\phi}}^{+}_{\bf
p^{\prime}_{\pi}}(x^{\prime}_{1},x^{\prime}_{2})$ $<0|{\sf
T}\biggl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2}){\cal
J}^{\mu}(z){\overline{q}}_{1}(y_{1}){\overline{q}}_{2}(y_{2}){\overline{q}}_{3}(y_{3}){\overline{q}}_{1}(x_{1})q_{2}(x_{2})\biggr{)}|0>$
$\chi_{\bf p_{N}}(y_{1},y_{2},y_{3})\phi_{\bf p_{\pi}}(x_{1},x_{2})$ $None$
involves all possible exchanges of the quark operators between ${\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}$,
${\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}$,
${\sf T}\Bigl{(}q_{1}(y_{1})q_{2}(y_{2})q_{3}(y_{3})\Bigr{)}$ and ${\sf
T}\Bigl{(}q_{1}(x_{1}){\overline{q}}_{2}(x_{2})\Bigr{)}$. Therefore, ${\em
g}^{\mu}$ does not contribute into the on shell amplitude (2.7a), because
${\em g}^{\mu}$ does not contain ${\widehat{\sc P}}_{N^{\prime}}$,
${\widehat{\sc P}}_{N}$, ${\widehat{\sc P}}_{\pi^{\prime}}$, ${\widehat{\sc
P}}_{\pi^{\prime}}$.333 This can be also verified using the limits over the
coordinates ${{X^{\prime}}^{o}}\Longrightarrow\infty$,
${{X}^{o}}\Longrightarrow-\infty$ ${{Y^{\prime}}^{o}}\Longrightarrow\infty$,
${{Y}^{o}}\Longrightarrow-\infty$ in the quark and composite particle
operators (2.3a,b), (2.5a,b). Correspondingly, one can replace ${\sc G}^{\mu}$
(2.8b) with ${\tau}^{\mu}$ (2.8c) in (2.8a)
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}={\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\
{k^{\prime}}_{\mu}{\tau}^{\mu}\
(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})({p_{\pi}}^{2}-m_{\pi}^{2})u({\bf p}_{N}),$
$None$
Based on the equal-time commutation relations between the quark operators it
is easy to get the equal-time commutation rules for the photon source operator
(2.7b) and the quark operators
$\biggl{[}{\cal
J}^{o}(z),q_{j}(y^{\prime})\biggr{]}\delta(z_{o}-y^{\prime}_{o})=-e_{j}\delta^{(4)}(z-y^{\prime})q_{j}(y^{\prime});\
\ \ \biggl{[}{\cal
J}^{o}(z),{\overline{q}}_{J}(y)\biggr{]}\delta(z_{o}-y_{o})=e_{J}\delta^{(4)}(z-y){\overline{q}}_{J}(y),$
$None$
where $e_{j}$ denotes the charge of the quark $j$. In particular, $e_{j}=2/3\
e$ for the $u$-quark and $e_{j}=-1/3\ e$ for the $d$ quarks. The equal-time
commutators (2.9) and integration over $z$ in (2.8c) allows to rewrite (2.8d)
as
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-i{\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})\int
d^{4}y^{\prime}_{1}d^{4}y^{\prime}_{2}d^{4}y^{\prime}_{3}{\widetilde{\chi}}^{+}_{\bf
p^{\prime}_{N}}(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3})$
$\Bigl{[}e_{1}^{N^{\prime}}e^{ik^{\prime}y^{\prime}_{1}}+e_{2}^{N^{\prime}}e^{ik^{\prime}y^{\prime}_{2}}+e_{3}^{N^{\prime}}e^{ik^{\prime}y^{\prime}_{3}}\Bigr{]}<out;{\bf
p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}|{\bf
p}_{\pi}{\bf p}_{N};in>$ $-i({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\int
d^{4}x^{\prime}_{1}d^{4}x^{\prime}_{2}{\widetilde{\phi}}^{+}_{\bf
p^{\prime}_{\pi}}(x^{\prime}_{1},x^{\prime}_{2})\Bigl{[}e_{1}^{\pi^{\prime}}e^{ik^{\prime}x^{\prime}_{1}}+e_{2}^{\pi^{\prime}}e^{ik^{\prime}x^{\prime}_{2}}\Bigr{]}<out;{\bf
p^{\prime}}_{N}|{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}|{\bf
p}_{\pi}{\bf p}_{N};in>$ $+i\int
d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}\Bigl{[}e_{1}^{N}e^{ik^{\prime}y_{1}}+e_{2}^{N}e^{ik^{\prime}y_{2}}+e_{3}^{N}e^{ik^{\prime}y_{3}}\Bigr{]}<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}{\overline{q}}_{1}(y_{1}){\overline{q}}_{2}(y_{2}){\overline{q}}_{3}(y_{3})\Bigr{)}|{\bf
p}_{\pi};in>$ $\chi_{\bf
p_{N}}(y_{1},y_{2},y_{3})(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})({p_{\pi}}^{2}-m_{\pi}^{2})u({\bf
p}_{N})$ $+i({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\int
d^{4}x_{1}d^{4}x_{2}\Bigl{[}e_{1}^{\pi}e^{ik^{\prime}x_{1}}+e_{2}^{\pi}e^{ik^{\prime}x_{2}}\Bigr{]}<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}{\overline{q}}_{1}(x_{1})q_{2}(x_{2})\Bigl{)}|{\bf
p}_{N};in>\phi_{\bf p_{\pi}}(x_{1},x_{2}).$ $None$
After integration over the coordinates $X^{\prime}$, $X$, $Y^{\prime}$ and $Y$
we obtain
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}({\bf
p^{\prime}_{\pi},p^{\prime}_{N},k^{\prime};p_{\pi},p_{N}})=-i(2\pi)^{4}\
\delta^{(4)}(p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime}-p_{\pi}-p_{N})$
$\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N}){{e_{N^{\prime}}}\over{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}-m_{N}}}<out;{\bf
p^{\prime}}_{\pi}|J_{\bf p^{\prime}_{N},k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$
$+({p^{\prime}}_{\pi}^{2}-m_{\pi}^{2}){{e_{\pi^{\prime}}}\over{(p^{\prime}_{\pi}+k^{\prime})^{2}-m_{\pi}^{2}}}<out;{\bf
p^{\prime}}_{N}|j_{\bf p^{\prime}_{\pi},k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}_{\bf p_{N},k^{\prime}}(0)|{\bf
p}_{\pi};in>{{e_{N}}\over{\gamma_{\nu}(p_{N}-k^{\prime})^{\nu}-m_{N}}}(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})u({\bf
p}_{N})$ $-<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\bf
p_{\pi},k^{\prime}}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{(p_{\pi}-k^{\prime})^{2}-m_{\pi}^{2}}}(p_{\pi}^{2}-m_{\pi}^{2})\Biggr{]},$
$None$
where the source operators of the nucleon and pion are constructed through the
nonlocal nucleon and pion fields (2.1) and (2.4a) as
$J_{\bf
p^{\prime}_{N},k^{\prime}}(Y)=\bigl{(}i\gamma_{\sigma}\nabla_{Y}^{\sigma}-m_{N}\bigr{)}\Psi_{{\bf
p}_{N},k^{\prime}}(Y);\ \ \Psi_{{\bf
p^{\prime}}_{N},k^{\prime}}(Y^{\prime})=\int
d^{4}r^{\prime}_{3}d^{4}r^{\prime}_{1,2}{\widetilde{\chi}}^{+}_{\bf
p^{\prime}_{N}}(Y^{\prime}=0,r^{\prime}_{1.2},r^{\prime}_{3})$
$\Bigl{[}{{e_{1}^{N^{\prime}}}\over{e_{N}^{\prime}}}e^{ik^{\prime}(-\eta_{3}r^{\prime}_{3}+\eta_{2}r^{\prime}_{1,2})}+{{e_{2}^{N^{\prime}}}\over{e_{N}^{\prime}}}e^{-ik^{\prime}(\eta_{3}r^{\prime}_{3}+\eta_{1}r^{\prime}_{1,2})}+{{e_{3}^{N^{\prime}}}\over{e_{N}^{\prime}}}e^{ik^{\prime}\eta_{1,2}r^{\prime}_{3}}\Bigr{]}{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}$
$None$
$j_{\bf{p^{\prime}}_{\pi},k^{\prime}}(X)=\bigl{(}\Box_{X}+m_{\pi}^{2}\bigr{)}\phi_{p_{\pi}}(X);\
\ \phi_{\bf p_{\pi},k^{\prime}}(X^{\prime})=\int
d^{4}\rho^{\prime}_{1,2}{\widetilde{\phi}}^{+}_{\bf
p^{\prime}_{\pi}}(X^{\prime}=0,\rho^{\prime}_{1,2})$
$\Bigl{[}{{e_{1}^{\pi^{\prime}}}\over{e_{\pi^{\prime}}}}e^{ik^{\prime}\mu_{2}\rho^{\prime}_{1,2}}+{{e_{2}^{\pi^{\prime}}}\over{e_{\pi^{\prime}}}}e^{-ik^{\prime}\mu_{1}\rho^{\prime}_{1,2}}\Bigr{]}{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}.$
$None$
The charge factors in front of the off shell $\pi N$ amplitudes in (2.12) are
extracted in analogue to the formulation without quark degrees of freedom [1],
where the charge of the nucleon and pion arise in the equal time commutators
due to charge conservation. In the source operators (2.13a,b) the quark
charges are distributed according to the commutators (2.9). This distribution
is a result of the choice of the field operators (2.1) and (2.4a). Other
choices of the source operators of the composite nucleons and pions are
considered in Appendix A. Unlike (2.13a,b) other source operators do not
contain the quark charge distributions.
Because of the zeros of the free Dirac and the Klein-Gordon operators
$(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})$,
$(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})$, $({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})$,
$({p_{\pi}}^{2}-m_{\pi}^{2})$ equation (2.12) corresponds to current
conservation ${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=0$ for any $k^{\prime}$. For $k^{\prime}=0$
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}=0$
according to cancellations of the on shell $\pi N$ amplitudes. Therefore,
(2.12) represents current conservation for the on-mass shell $\pi N$
bremsstrahlung amplitude
${k^{\prime}}_{\mu}\Biggl{[}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\bf
p^{\prime}_{\pi},p^{\prime}_{N},k^{\prime};p_{\pi},p_{N}})\Biggr{]}_{on\ mass\
shell\ \pi^{\prime},\ N^{\prime},\ \pi,\ N}=0.$ $None$
Following [1] we extract the full energy-momentum conservation $\delta$
function from the radiative $\pi N$ scattering amplitude
$A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ and introduce the
corresponding non-singular amplitude $<out;{\bf p^{\prime}}_{N}{\bf
p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-i(2\pi)^{4}\
\delta^{(4)}(p^{\prime}_{N}+p^{\prime}_{\pi}+k^{\prime}-p_{\pi}-p_{N}){k^{\prime}}_{\mu}<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal J}^{\mu}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>.$ $None$
The identity $a/(a+b)\equiv 1-b/(a+b)$ and (2.15) allows to rewrite (2.12) as
${k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf p}_{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>={\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}+{k^{\prime}}_{\mu}{\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=0,$ $None$
where
${\cal B}_{\pi^{\prime}N^{\prime}-\pi N}=e_{N^{\prime}}{\overline{u}}({\bf
p^{\prime}}_{N})<out;{\bf p^{\prime}}_{\pi}|J_{\bf
p^{\prime}_{N}k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>+e_{\pi^{\prime}}<out;{\bf
p^{\prime}}_{N}|j_{\bf{p^{\prime}}_{\pi}k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}_{\bf p_{N}k^{\prime}}(0)|{\bf
p}_{\pi};in>u({\bf p}_{N})-e_{\pi}<out;{\bf p^{\prime}}_{{p}_{\pi}}{\bf
p^{\prime}}_{N}|j_{\bf p_{\pi}k^{\prime}}(0)|{\bf p}_{N};in>,$ $None$
${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-\Biggl{[}{\overline{u}}({\bf
p^{\prime}}_{N})\gamma^{\mu}{{\gamma_{\nu}(p^{\prime}_{N}+k^{\prime})^{\nu}+m_{N}}\over{2p^{\prime}_{N}k^{\prime}}}e_{N^{\prime}}<out;{\bf
p^{\prime}}_{\pi}|J_{\bf p^{\prime}_{N}k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$
$+(2{p^{\prime}}_{\pi}+k^{\prime})^{\mu}{{e_{\pi^{\prime}}}\over{{2p^{\prime}_{\pi}k^{\prime}}}}<out;{\bf
p^{\prime}}_{N}|j_{\bf p^{\prime}_{\pi}k^{\prime}}(0)|{\bf p}_{\pi}{\bf
p}_{N};in>$ $-e_{N}<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|{\overline{J}}_{\bf p^{\prime}_{N}k^{\prime}}(0)|{\bf
p}_{\pi};in>{{\gamma_{\nu}(p_{N}-k^{\prime})^{\nu}+m_{N}}\over{2p_{N}k^{\prime}}}\gamma^{\mu}u({\bf
p}_{N})$ $-<out;{\bf p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|j_{\bf
p_{\pi}k^{\prime}}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{2p_{\pi}k^{\prime}}}(2p_{\pi}-k^{\prime})^{\mu}\Biggr{]}$
$None$
The relations (2.16) and (2.17a,b) have the same form as (2.6) and (2.8a,b) in
the formulation without quarks [1]. The only differences are in the source
operators of the nucleons and pions. The off shell $\pi N$ amplitudes in
(2.17a,b) contains the nonlocal source operators (2.13a,b) of composite
particles which in contrast to the local sources
$J(x)=(i\gamma_{\nu}\partial/\partial x_{\nu}-m_{N})\Psi(x)$ and
$j_{\pi}(x)=(\Box_{x}+m_{\pi}^{2})\Phi(x)$ depends on the four moments of the
composed particle and on ${\bf k^{\prime}}$. Consequently, the off mass shell
$\pi N$ amplitudes in (2.17a,b) have an additional dependence on the
Mandelstam variables.
The Ward-Takahashi identity (2.16) presents the general scheme of current
conservation for the $\pi N$ bremsstrahlung reaction with the composed on mass
shell pions and nucleons. According to this scheme it is necessary to find a
special part of the internal particle radiation amplitude ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ which insures current
conservation because
$k^{\prime}_{\mu}{\cal I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}={\cal B}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}$ $None$
and consequently
$k^{\prime}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>=k^{\prime}_{\mu}\biggl{(}{\cal
E}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}+{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}\biggr{)}=0.$ $None$
An example of such an internal particle radiation amplitude is the
intermediate on mass shell $\Delta$ radiation amplitude depicted in Fig. 3
[1]. The $\Delta$ radiation amplitude in Fig. 3 ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
does not satisfy current conservation separately. ${\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(\Delta-\gamma\Delta)$
satisfy current conservation together with $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ which is extracted from ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (2.17b) in Fig.2 after
the set of the decompositions.
$k^{\prime}_{\mu}({{\cal
E_{L}}^{3/2}})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=-{k^{\prime}}_{\mu}{\cal
I}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(\Delta-\gamma\Delta)=-{\cal B}^{3/2}_{\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta),$ $None$
where the lower index L and the upper index 3/2 denotes the longitudinal and
the spin-isospin $(3/2,3/2)$ part of the corresponding expressions. From the
same structure of $({\cal
E_{L}}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ and $({\cal
I}^{3/2})_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}({\Delta}-\gamma\Delta)$ follows
$({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)=-{\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
$None$
which allows to equate the $\Delta-\gamma\Delta$ vertex functions in ${\cal
I}^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}(\Delta-\gamma\Delta)$
and in $({\cal E_{L}}^{3/2})^{\mu}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}(\Delta-\gamma\Delta)$
$G_{C0}({k^{\prime}},s,s^{\prime})=-2M_{\Delta}\biggl{[}{{|{\bf
k^{\prime}}|}\over{s-s^{\prime}}}-{{{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}\over{s-s^{\prime}}}\biggr{]}$
$\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}e_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}+e_{\pi}{{{\cal R}_{\pi^{\prime}}+{\cal
R}_{\pi}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi N}(s)\Bigr{]}^{-1},$
$None$
$G_{M1}({k^{\prime}}_{\Delta},s,s^{\prime})=-2M_{\Delta}\biggl{[}{{|{\bf
k^{\prime}}|}\over{s-s^{\prime}}}-{{{P}^{o}_{\Delta}(s)-{P^{\prime}}^{o}_{\Delta}(s^{\prime})}\over{s-s^{\prime}}}\biggr{]}\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}\mu_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s)\Bigr{]}^{-1},$ $None$
where $G_{C}$ and $G_{M1}$ denote the electric and magnetic dipole form
factors of the $\Delta$’s,
$k^{\prime}_{\Delta}=P_{\Delta}-P^{\prime}_{\Delta}$ and we have considered
the $\pi N$ bremsstrahlung reactions with only $e_{N}=e_{N^{\prime}}$ and
$\mu_{N}=\mu_{N^{\prime}}$. Equations (2.21a,b) present a relationship between
$G_{C0}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$,
$G_{M1}({k^{\prime}}_{\Delta}^{2},s,s^{\prime})$ and the residues of the $\pi
N$ amplitudes ${\cal R}$ (see (A.9a,b,c,d) in [1]).
At the threshold $k^{\prime}=0$ one obtains the same model-independent
relation of the magnetic dipole moments of the $\Delta^{+}$ and $\Delta^{++}$
resonances as in [1] $\mu_{\Delta^{+}}={{M_{\Delta}}\over{m_{p}}}\mu_{p}$ and
$\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^{+}}$.
3\. Magnetic dipole moments of the $\Delta^{o}$ and $\Delta^{-}$ resonances.
The external particle radiation amplitude ${\cal
E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}$ (2.17b) can be
replaced by the one on mass shell particle exchange amplitude
${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)=-\Biggl{[}(2{p^{\prime}}_{\pi}+k^{\prime})^{\mu}{{e_{\pi^{\prime}}}\over{{2p^{\prime}_{\pi}k^{\prime}}}}<out;{\bf
p^{\prime}}_{N}|j_{\bf p^{\prime}_{\pi^{\prime}}k^{\prime}}(0)|{\bf
p}_{\pi}{\bf p}_{N};in>$ $+{{{\overline{u}}({\bf
p^{\prime}}_{N})\Bigl{[}(2p^{\prime}_{N}+k^{\prime})^{\mu}\tau_{+}-i\mu_{N^{\prime}}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}}\over{2p^{\prime}_{N}k^{\prime}}}u({\bf
p_{N}^{\prime}+k^{\prime}}){\overline{u}}({\bf
p_{N}^{\prime}+k^{\prime}})e_{N^{\prime}}<out;{\bf p^{\prime}}_{\pi}|J_{\bf
p^{\prime}_{N}k^{\prime}}(0)|{\bf p}_{\pi}{\bf p}_{N};in>$ $-e_{N}<out;{\bf
p^{\prime}}_{\pi}{\bf p^{\prime}}_{N}|{\overline{J}}_{\bf
p_{N}k^{\prime}}(0)|{\bf p}_{\pi};in>u({\bf
p_{N}-k^{\prime}}){\overline{u}}({\bf
p_{N}-k^{\prime}}){{\Bigl{[}(2p_{N}-k^{\prime})^{\mu}-i\mu_{N}\sigma^{\mu\nu}k^{\prime}_{\nu}\Bigr{]}u({\bf
p}_{N})}\over{2p_{N}k^{\prime}}}$ $-<out;{\bf p^{\prime}}_{\pi}{\bf
p^{\prime}}_{N}|j_{\bf p_{\pi^{\prime}}k^{\prime}}(0)|{\bf
p}_{N};in>{{e_{\pi}}\over{2p_{\pi}k^{\prime}}}(2p_{\pi}-k^{\prime})^{\mu}\Biggr{]},$
$None$
where the antiparticle contributions are separated as it was done in [1]. Thus
starting from the generalized $S$-matrix reduction formulas (2.8a,b) for the
$\pi N$ radiation amplitude with composite pions and nucleons one obtains
again the modified Ward-Takahashi identity
${k^{\prime}}_{\mu}<out;{\bf p^{\prime}}_{N}{\bf p}_{\pi^{\prime}}|{\cal
J}^{\mu}(0)|{\bf p}_{\pi}{\bf p}_{N};in>={\cal B}_{\pi^{\prime}N^{\prime}-\pi
N}+{k^{\prime}}_{\mu}{\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}(N)=0,$ $None$
with the external particle radiation amplitude (3.1a), where $e_{p}=e$,
$\mu_{p}=1$ for the protons and $e_{n}=0$, $\mu_{n}=0$ for the neutrons.
In order to take into account the anomalous magnetic moments of nucleons it is
necessary to consider the loop corrections of the $\gamma NN$ vertex. The
corresponding contributions can be extracted from the transverse part of
${\cal E}_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi N}^{\mu}(N)$. Then one
obtains again the expression (3.1a) with $\mu_{p}=2.79\mu_{B}$ for protons and
$\mu_{n}=-1.91\mu_{B}$ for neutrons in the units of the nuclear magneton
$\mu_{B}=e/2m_{p}$.
The external particle radiation part of the $\pi N$ bremstrahlung amplitude
with the complete $\gamma^{\prime}NN$ and $\gamma^{\prime}\pi\pi$ vertices is
depicted in Fig.2. At the threshold $k^{\prime}=0$ (2.21a,b) presents the
exact relationship between $e_{\Delta}$, $\mu_{\Delta}$ and the $\Delta$ pole
residues ${\cal R}$ of the off shell $\pi N$ amplitudes (see (A.9a,b,c,d) in
[1])
$e_{\Delta}=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}e_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}+e_{\pi}{{{\cal R}_{\pi^{\prime}}+{\cal
R}_{\pi}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s,k^{\prime})\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}},$
$None$
$\mu_{\Delta}=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}\biggl{(}\mu_{N}{{{\cal
R}_{N^{\prime}}+{\cal R}_{N}}\over 2}\biggr{)}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s,k^{\prime})\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}},$
$None$
where ${\cal
N}(s)=1/(d{\sqrt{s}}/dk^{\prime})-d{P}^{o}_{\Delta}(s)/d{\sqrt{s}}$ and ${\rm
g}_{\Delta-\pi N}$ and ${\rm g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}$
denotes the $\Delta-\pi N$ form factors.
Now we assume that the charge of the neutron is an auxiliary parameter $e_{n}$
which will be fixed in the finally relations as $e_{n}=0$. Then for the $\pi
N$ bremsstrahlung with the intermediate $\Delta^{o}$ we have
$e_{\Delta^{o}}=-e_{n}\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}{{{\cal
R}_{n^{\prime}}+{\cal R}_{n}}\over 2}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s,k^{\prime})\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s,k^{\prime}}=M_{\Delta}}.\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}$
$None$
But $e_{\Delta^{o}}=e_{n}$ for the reaction
$\pi^{o}n\to\gamma^{\prime}{\pi^{o}}^{\prime}n^{\prime}$. The cancellation of
$e_{n}$ from both sides of (3.3a) gives the normalization condition for ${\cal
R}_{n}$
$1=-\Biggl{[}{\cal N}(s)\Bigl{[}{\rm
g}_{\pi^{\prime}N^{\prime}-\Delta^{\prime}}(s^{\prime},k^{\prime})\Bigr{]}^{-1}{{{\cal
R}_{n^{\prime}}+{\cal R}_{n}}\over 2}\Bigl{[}{\rm g}_{\Delta-\pi
N}(s,k^{\prime})\Bigr{]}^{-1}\Biggr{]}^{k^{\prime}=0}_{\sqrt{s^{\prime}}=\sqrt{s}=M_{\Delta}}.$
$None$
Substituting (3.3b) into (3.2b) we obtain
$\mu_{\Delta^{o}}=\mu_{n}{{M_{\Delta}}\over{m_{p}}},$ $None$
where the different units of $\mu_{\Delta}$ and $\mu_{N}$ generates the factor
${{M_{\Delta}}/{m_{p}}}$. The isospin symmetry between the $\pi N$ amplitudes
of the reactions $\pi^{o}n\to\pi^{o}n$ and $\pi^{-}n\to\pi^{-}n$ in (3.2b)
allows to estimate $\mu_{\Delta^{-}}$
$\mu_{\Delta^{-}}={3\over 2}\mu_{\Delta^{o}}={3\over
2}\mu_{n}{{M_{\Delta}}\over{m_{p}}}$ $None$
4\. Conclusion
The main result of this paper is that the magnetic dipole moments of the
$\Delta$ resonances are the same in the quantum field theories with and
without quark degrees of freedom. This follows from the same structure of
current conservation for the on shell $\pi N$ radiation amplitudes in the
formulations with and without quark degrees of freedom, and the simple
relationship between the magnetic dipole moments of the $\Delta$’s and the
anomalous magnetic moment of the nucleons $\mu_{\Delta}=M_{\Delta}/m_{p}\
\mu_{N}$. According to this formula the $\mu_{\Delta}$ are dependent only on
the anomalous magnetic moment of the nucleons $\mu_{N}$ and the $\Delta$
resonance pole position $M_{\Delta}=1232\ MeV$. The present approach allows to
connect analytically the electric and the magnetic dipole form factors $G_{C}$
and $G_{M1}$ with the $\Delta$ pole residues ${\cal R}$ of the off shell $\pi
N$ amplitudes according to (2.21a,b). The $\Delta$ pole residues ${\cal R}$ as
well as the source operators (2.13a,b) and (A.6a,b) are different in the
different models. Correspondingly, the dependence on $k^{\prime}$ of the off
shell $\pi N$ amplitudes is different and model-dependent. But at threshold
$k^{\prime}=0$ the expressions for $e_{\Delta}$, $\mu_{\Delta}$ (3.2a,b) as
well as the normalization condition (3.3b) are the same for any model with the
fixed charge of the particles. Therefore, the formulas (3.4) and (3.5) for the
magnetic dipole moment of the $\Delta$’s are unique and model-independent.
Table 1
Magnetic moments of $\Delta^{o}$ and $\Delta^{-}$ in nuclear magneton
$\mu_{B}={e/{2m_{p}}}$.
models | This | $SU(6)$ | Skyrme |
---|---|---|---|---
| work | and Bag | | quark
| | 0\. [13, 14] | | 0.[18]
${\mu_{\Delta^{o}}}$ | -2.504 | 0.[15] | -1.33$\sim$-0.19[21] | 0.375[19]
| | 0.[16] | | -0.3$\sim$0.[20]
| | 0.[17] | |
| | -2.79 [13, 14] | | -3.49[18]
${\mu_{\Delta^{-}}}$ | -3.759 | -2.13[15] | -5.62$\sim$-2.38[21] | -2.1[19]
| | -2.20-2.45[16] | | -2.72-3.06[20]
| | -3.27[17] | |
The present relations for $\mu_{\Delta}$ requires proportionality of
$\mu_{\Delta}$ and the anomalous magnetic moment of the nucleon $\mu_{N}$. In
particular, $\mu_{\Delta^{o}}$ and $\mu_{\Delta^{-}}$ are determined via the
anomalous magnetic moment of the neutron. A comparison our numerical values
for $\mu_{\Delta^{o}}$ and $\mu_{\Delta^{-}}$ with the calculations of other
authors is given in Table 1. In the $SU(6)$ symmetry quark models and their
modifications [13]-[17] $\mu_{\Delta}$ is proportional to the charge of the
$\Delta$. Therefore in these models[13, 14, 15, 16, 17] $\mu_{\Delta^{o}}=0$
and $\mu_{\Delta^{-}}=-\mu_{\Delta^{+}}$ and $\mu_{\Delta^{+}}=1/2\
\mu_{\Delta^{++}}$. This property is preserved in the constituent quark model
[18]. But it is slightly broken in the Skyrme model [21], chiral quark model
[20], chiral quark-soliton model [19] and effective quark model. The crucial
difference between our result and the other estimations is in
$\mu_{\Delta^{o}}$ which is larger than the predictions of other authors.
Appendix A: Alternative field operators of composite particles
The source operators $J_{p^{\prime}_{N},k^{\prime}}(Y)$ (2.13a) and
$j_{{p^{\prime}}_{\pi},k^{\prime}}(X)$ (2.13b) can be constructed in the
independent over the quark charges $e_{j}$ form. For this aim one can
introduce other quark cluster operators
$\Psi_{p_{N}}(Y)={1\over 6}\int d^{4}r_{3}d^{4}r_{1,2}\Biggl{\\{}$
$\biggl{[}{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{1},y_{2}.y_{3})|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{1})q_{2}(y_{2})q_{3}(y_{3})\biggr{)}+{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{2},y_{1}.y_{3})|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{2})q_{2}(y_{1})q_{3}(y_{3})\biggr{)}\biggr{]}$
$+\biggl{[}{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{3},y_{2}.y_{1}))|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{3})q_{2}(y_{2})q_{3}(y_{1})\biggr{)}+{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{2},y_{3}.y_{1}))|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{2})q_{2}(y_{3})q_{3}(y_{1})\biggr{)}\biggr{]}$
$+\biggl{[}{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{1},y_{3}.y_{2}))|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{1})q_{2}(y_{3})q_{3}(y_{2})\biggr{)}+{\widetilde{\chi}}^{{\dagger}}_{p_{N}}(y_{1},y_{1}.y_{2})|_{Y=0}{\sf
T}\biggl{(}q_{1}(y_{3})q_{2}(y_{1})q_{3}(y_{2})\biggr{)}\biggr{]}\Biggr{\\}}$
$None$
and
$\Phi_{{p}_{\pi}}(X)={1\over 2}\Biggl{\\{}\int
d^{4}\rho_{1,2}{\widetilde{\phi}}^{+}_{p_{\pi}}(X=0,\rho_{1,2}){\sf
T}\biggl{(}q_{i}(x_{1}){\overline{q}}_{i}(x_{2})\biggr{)}+\int
d^{4}\rho_{1,2}{\widetilde{\phi}}^{+}_{p_{\pi}}(X=0,-\rho_{1,2}){\sf
T}\biggl{(}q_{i}(x_{2}){\overline{q}}_{i}(x_{1})\biggr{)}\Biggr{\\}},$ $None$
Unlike to (2.1) and (2.4a), the field operators (A.1) and (A.2) contains all
transpositions of the integration variables $y_{1}$, $y_{2}$, $y_{3}$ and
$x_{1}$, $x_{2}$. Therefore, instead of (2.8b) we obtain
${k^{\prime}}_{\mu}{\sc G}^{\mu}=i\int
d^{4}[y^{\prime}]d^{4}[x^{\prime}]d^{4}[y]d^{4}[x]{\cal
P}_{x^{\prime}_{1}x^{\prime}_{2}}{\cal P}_{x_{1}x_{2}}{\cal
P}_{y^{\prime}_{1}y^{\prime}_{2}y^{\prime}_{3}}{\cal
P}_{y_{1}y_{2}y_{3}}e^{ik^{\prime}z}d^{4}z{\widetilde{\chi}}^{+}_{p^{\prime}_{N}}(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3}){\widetilde{\phi}}^{+}_{p^{\prime}_{\pi}}(x^{\prime}_{1},x^{\prime}_{2}){{\partial}\over{\partial
z^{\mu}}}$ $<0|{\sf T}\biggl{(}{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}{\cal
J}^{\mu}(z){\sf
T}\Bigl{(}{\overline{q}}_{1}(y_{1}){\overline{q}}_{2}(y_{2}){\overline{q}}_{3}(y_{3})\Bigr{)}{\sf
T}\Bigl{(}{\overline{q}}_{1}(x_{1})q_{2}(x_{2})\Bigr{)}\biggr{)}|0>$
$\chi_{p_{N}}(y_{1},y_{2},y_{3})\phi_{p_{\pi}}(x_{1},x_{2})$ $None$
where $d^{4}[y]\equiv
1/6\Bigl{\\{}[d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}+d^{4}y_{2}d^{4}y_{1}d^{4}y_{3}]+[d^{4}y_{3}d^{4}y_{1}d^{4}y_{2}+d^{4}y_{1}d^{4}y_{3}d^{4}y_{2}]+[d^{4}y_{1}d^{4}y_{2}d^{4}y_{3}+d^{4}y_{2}d^{4}y_{3}d^{4}y_{1}]\Bigr{\\}}$,
$d^{4}[x]\equiv
1/2\Bigl{\\{}d^{4}x_{1}d^{4}x_{2}+d^{4}x_{2}d^{4}x_{1}\Bigr{\\}}$ and ${\cal
P}_{x_{1}x_{2}}$ and ${\cal P}_{y_{1}y_{2}y_{3}}$ are defined via
transposition operator ${\sf P}_{x_{1}x_{2}}$ of the variables $x_{1}$ and
$x_{2}$ as
${\cal P}_{x_{1}x_{2}}={{1+{\sf P}_{x_{1}x_{2}}}\over 2};\ \ \ {\cal
P}_{y_{1}y_{2}y_{3}}={{{\sf P}_{y_{1}y_{2}}+{\sf P}_{y_{2}y_{3}}+{\sf
P}_{y_{3}y_{1}}}\over 3}$ $None$
The symmetry over the rearrangement of the integration variables in (A.3) and
$\int dx_{1}dx_{2}e^{ik^{\prime}x_{1}}[f(x_{1},x_{2})+f(x_{2},x_{1})]=\int
dx_{2}dx_{1}e^{ik^{\prime}x_{2}}[f(x_{2},x_{1})+f(x_{1},x_{2})]$ allows to
modify (2.11a) as
${k^{\prime}}_{\mu}A_{\gamma^{\prime}\pi^{\prime}N^{\prime}-\pi
N}^{\mu}=-ie_{N^{\prime}}{\overline{u}}({\bf
p^{\prime}}_{N})(\gamma_{\nu}{p^{\prime}_{N}}^{\nu}-m_{N})\int
d^{4}[y^{\prime}]{\cal
P}_{y^{\prime}_{1}y^{\prime}_{2}y^{\prime}_{3}}{\widetilde{\chi}}^{+}_{p^{\prime}_{N}}(y^{\prime}_{1},y^{\prime}_{2},y^{\prime}_{3})$
${{e^{ik^{\prime}y^{\prime}_{1}}+e^{ik^{\prime}y^{\prime}_{2}}+e^{ik^{\prime}y^{\prime}_{3}}}\over
3}<out;{\bf p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}|{\bf
p}_{\pi}{\bf p}_{N};in>$
$-ie_{\pi^{\prime}}({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\int
d^{4}[x^{\prime}]{\cal
P}_{x^{\prime}_{1}x^{\prime}_{2}}{\widetilde{\phi}}^{+}_{p^{\prime}_{\pi}}(x^{\prime}_{1},x^{\prime}_{2}){{e^{ik^{\prime}x^{\prime}_{1}}+e^{ik^{\prime}x^{\prime}_{2}}}\over
2}<out;{\bf p^{\prime}}_{N}|{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}|{\bf
p}_{\pi}{\bf p}_{N};in>$ $+ie_{N}\int d^{4}[y]{\cal
P}_{y_{1}y_{2}y_{3}}{{e^{ik^{\prime}y_{1}}+e^{ik^{\prime}y_{2}}+e^{ik^{\prime}y_{3}}}\over
3}<out;{\bf p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}{\overline{q}}_{1}(y_{1}){\overline{q}}_{2}(y_{2}){\overline{q}}_{3}(y_{3})\Bigr{)}|{\bf
p}_{\pi};in>$
$\chi_{p_{N}}(y_{1},y_{2},y_{3})(\gamma_{\nu}{p_{N}}^{\nu}-m_{N})({p_{\pi}}^{2}-m_{\pi}^{2})u({\bf
p}_{N})$ $+ie_{\pi}({p^{\prime}_{\pi}}^{2}-m_{\pi}^{2})\int d^{4}[x]{\cal
P}_{x_{1}x_{2}}{{e^{ik^{\prime}x_{1}}+e^{ik^{\prime}x_{2}}}\over 2}<out;{\bf
p^{\prime}}_{N}{\bf p^{\prime}}_{\pi}|{\sf
T}\Bigl{(}{\overline{q}}_{1}(x_{1})q_{2}(x_{2})\Bigl{)}|{\bf
p}_{N};in>\phi_{p_{\pi}}(x_{1},x_{2}),$ $None$
Consequently, after integration over $X^{\prime}$, $X$, $Y^{\prime}$ and $Y$
we obtain again (2.12) with independent on the quark charges $e_{j}$ source
operators
$J_{p^{\prime}_{N},k^{\prime}}(Y)=\bigl{(}i\gamma_{\sigma}\nabla_{Y}^{\sigma}-m_{N}\bigr{)}\Psi_{{\bf
p}_{N},k^{\prime}}(Y);\ \ \Psi_{{\bf
p^{\prime}}_{N},k^{\prime}}(Y^{\prime})=\int
d^{4}r^{\prime}_{3}d^{4}r^{\prime}_{1,2}{\widetilde{\chi}}^{+}_{p^{\prime}_{N}}(Y^{\prime}=0,r^{\prime}_{1.2},r^{\prime}_{3})$
${{e^{ik^{\prime}y^{\prime}_{1}}+e^{ik^{\prime}y^{\prime}_{2}}+e^{ik^{\prime}y^{\prime}_{3}}}\over
3}|_{Y^{\prime}=0}{\sf
T}\Bigl{(}q_{1}(y^{\prime}_{1})q_{2}(y^{\prime}_{2})q_{3}(y^{\prime}_{3})\Bigr{)}$
$None$
$j_{{p^{\prime}}_{\pi},k^{\prime}}(X)=\bigl{(}\Box_{X}+m_{\pi}^{2}\bigr{)}\phi_{p_{\pi}}(X);\
\ \phi_{p_{\pi},k^{\prime}}(X^{\prime})=\int
d^{4}\rho^{\prime}_{1,2}{\widetilde{\phi}}^{+}_{p^{\prime}_{\pi}}(X^{\prime}=0,\rho^{\prime}_{1,2})$
${{e^{ik^{\prime}x^{\prime}_{1}}+e^{ik^{\prime}x^{\prime}_{2}}}\over
2}|_{X^{\prime}=0}{\sf
T}\Bigl{(}q_{1}(x^{\prime}_{1}){\overline{q}}_{2}(x^{\prime}_{2})\Bigr{)}.$
$None$
## References
* [1] A. I. Machavariani and Amand Faessler. Preprint arXiv:nucl-th/0804.1322 2008(to be publ.)
* [2] K. Huang and H. A. Weldon, Phys. Rev. D11 (1975) 257.
* [3] R. Haag, Phys. Rev. 112 (1958) 669.
* [4] K. Nishijima, Phys. Rev. 111 (1958) 995.
* [5] W. Zimmermann, Nuovo Cim. 10 (1958) 598.
* [6] C. Itzykson and C. Zuber. Quantum Field Theory. (New York, McGraw-Hill) 1980.
* [7] J. D. Bjorken and S.D.Drell, Relativistic Quantum Fields. (New York, Mc Graw-Hill) 1965.
* [8] A. I. Machavariani, Fiz. Elem. Chastits At Yadra 24 (1993) 731; A. I. Machavariani, Few-Body Phys. 14 (1993) 59.
* [9] A. I. Machavariani, A. J. Buchmann, Amand Faessler, and G. A. Emelyanenko, Ann. of. Phys. 253 (1997) 149.
* [10] A. I. Machavariani, Amand Faessler and A. J. Buchmann. Nucl. Phys. A646 (1999) 231; (Erratum A686 (2001) 601).
* [11] A. I. Machavariani and Amand Faessler. Ann. Phys. 309 (2004) 49.
* [12] A. I. Machavariani, and Amand Faessler. Phys. Rev.C72 (2005) 024002.
* [13] M. A.B. Beg, B.W. Lee, and A. Pais, Phys. Rev. Lett. 13 (1964) 514,
* [14] H. Georgi. Lie Algebras in Particle Physics (Reading) 1982.
* [15] M. A.B. Beg, and A. Pais, Phys. Rev. 137 (1965) B1514,
* [16] G. E. Brown, M. Rho, and V. Vento, Phys. Lett. B97 (1980) 423.
* [17] M. I. Krivoruchenko, Sov. J. Nucl. Phys. 45 (1987) 109.
* [18] A. J. Buchmann, E. Hernández and Amand Faessler, Phys. Rev. C55 (1997) 448.
* [19] H.-C. Kim, M. Praszalowicz, and K. Goeke, Phys. Rev. D57 (1998) 2859.
* [20] J. Linde, T. Ohlsson, and H. Snellman, Phys. Rev. D57 (1998) 5916.
* [21] A. Acus, E. Norvai${\check{\rm s}}$as, and D. O. Riska, Phys. Rev. C57 (1998) 2597.
|
arxiv-papers
| 2008-09-08T13:59:40
|
2024-09-04T02:48:57.707056
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.I. Machavariani and Amand Faessler",
"submitter": "Alexander Machavariani",
"url": "https://arxiv.org/abs/0809.1303"
}
|
0809.1305
|
# Semistar-Krull and valuative dimension of integral domains
Parviz Sahandi Department of Mathematics, University of Tabriz, Tabriz, Iran
and School of Mathematics, Institute for Research in Fundamental Sciences
(IPM), Tehran Iran. sahandi@ipm.ir, sahandi@tabrizu.ac.ir
###### Abstract.
Given a stable semistar operation of finite type $\star$ on an integral domain
$D$, we show that it is possible to define in a canonical way a stable
semistar operation of finite type $\star[X]$ on the polynomial ring $D[X]$,
such that, if $n:=\star$-$\operatorname{dim}(D)$, then
$n+1\leq\star[X]\text{-}\operatorname{dim}(D[X])\leq 2n+1$. We also establish
that if $D$ is a $\star$-Noetherian domain or is a Prüfer
$\star$-multiplication domain, then
$\star[X]\text{-}\operatorname{dim}(D[X])=\star\text{-}\operatorname{dim}(D)+1$.
Moreover we define the semistar valuative dimension of the domain $D$, denoted
by $\star$-$\operatorname{dim}_{v}(D)$, to be the maximal rank of the
$\star$-valuation overrings of $D$. We show that
$\star$-$\operatorname{dim}_{v}(D)=n$ if and only if
$\star[X_{1},\cdots,X_{n}]$-$\operatorname{dim}_{v}(D[X_{1},\cdots,X_{n}])=2n$,
and that if $\star$-$\operatorname{dim}_{v}(D)<\infty$ then
$\star[X]$-$\operatorname{dim}_{v}(D[X])=\star$-$\operatorname{dim}_{v}(D)+1$.
In general
$\star$-$\operatorname{dim}(D)\leq\star$-$\operatorname{dim}_{v}(D)$ and
equality holds if $D$ is a $\star$-Noetherian domain or is a Prüfer
$\star$-multiplication domain. We define the $\star$-Jaffard domains as
domains $D$ such that $\star$-$\operatorname{dim}(D)<\infty$ and
$\star$-$\operatorname{dim}(D)=\star$-$\operatorname{dim}_{v}(D)$. As an
application, $\star$-quasi-Prüfer domains are characterized as domains $D$
such that each $(\star,\star^{\prime})$-linked overring $T$ of $D$, is a
$\star^{\prime}$-Jaffard domain, where $\star^{\prime}$ is a stable semistar
operation of finite type on $T$. As a consequence of this result we obtain
that a Krull domain $D$, must be a $w_{D}$-Jaffard domain.
###### Key words and phrases:
Semistar operation, star operation, Krull dimension, valuative dimension,
Prüfer domain, quasi-Prüfer domain, Prüfer $\star$-multiplication domain,
Noetherian domain, Jaffard domain
###### 2000 Mathematics Subject Classification:
Primary 13G05, 13A15, 13C15, 13M10.
## 1\. Introduction
Throughout this paper, $D$ denotes a (commutative integral) domain with
identity and $K$ denotes the quotient field of $D$. Let $X$ be an
algebraically independent indeterminate over $D$. Seidenberg proved in [35,
Theorem 2], that if $D$ has finite Krull dimension, then
$\operatorname{dim}(D)+1\leq\operatorname{dim}(D[X])\leq
2(\operatorname{dim}(D))+1.$
Moreover, Krull [27] has shown that if $D$ is any finite-dimensional
Noetherian ring, then $\operatorname{dim}(D[X])=1+\operatorname{dim}(D)$ (cf.
also [35, Theorem 9]). Seidenberg subsequently proved the same equality in
case $D$ is any finite-dimensional Prüfer domain. To unify and extend such
results on Krull-dimension, Jaffard [23] introduced and studied the _valuative
dimension_ denoted by $\operatorname{dim}_{v}(D)$, for a domain $D$. This is
the maximum of the ranks of the valuation overrings of $D$. Jaffard proved in
[23, Chapitre IV] (see also Arnold [2]), that if $D$ has finite valuative
dimension, then $\operatorname{dim}_{v}(D[X])=1+\operatorname{dim}_{v}(D)$ and
that if $D$ is a Noetherian or a Prüfer domain, then
$\operatorname{dim}(D)=\operatorname{dim}_{v}(D)$. Also he showed that
$\operatorname{dim}_{v}(D)=n$ if and only if
$\operatorname{dim}(D[X_{1},\cdots,X_{n}])=2n$, where $X_{1},\cdots,X_{n}$ are
indeterminates over $D$. In [1] the authors introduced the notion of Jaffard
domains, as integral domains $D$ such that
$\operatorname{dim}(D)=\operatorname{dim}_{v}(D)$. The class of Jaffard
domains contains most of the well-known classes of finite dimensional rings
involved in dimension theory of commutative rings, such as Noetherian domains,
Prüfer domains, universally catenarian domains [4], and stably strong
S-domains [28, 24]. A good and available reference for the dimension theory of
commutative rings is Gilmer [17, Section 30].
For several decades, star operations, as described in [17, Section 32], have
proven to be an essential tool in _multiplicative ideal theory_ , for studying
various classes of domains. In [30], Okabe and Matsuda introduced the concept
of a semistar operation to extend the notion of a star operation. Since then,
semistar operations have been extensively studied and, because of a greater
flexibility than star operations, have permitted a finer study and new
classifications of special classes of integral domains. For instance,
semistar-theoretic analogues of the classical notions of Krull dimension,
Noetherian and Prüfer domains have been introduced: see [10] and [13] for the
basics on $\star$-Krull dimension, $\star$-Noetherian domains and Prüfer
$\star$-multiplication domains (for short P$\star$MD), respectively.
Now it is natural to ask:
###### Question 1.1.
Given a semistar operation of finite type $\star$ on $D$, is it possible to
define in a canonical way a semistar operation of finite type $\star[X]$ on
$D[X]$, such that
$\star\text{-}\operatorname{dim}(D)+1\leq\star[X]\text{-}\operatorname{dim}(D[X])\leq
2(\star\text{-}\operatorname{dim}(D))+1$, and that if $D$ is a
$\star$-Noetherian domain or a P$\star$MD, then
$\star[X]\text{-}\operatorname{dim}(D[X])=\star\text{-}\operatorname{dim}(D)+1$?
In this paper, we answer this question, in case that $\star$ is a stable
semistar operation of finite type on $D$. More precisely, in Section 2, using
the technique introduced by Chang and Fontana in [6], we define in a canonical
way a semistar operation stable and of finite type $\star[X]$ on $D[X]$: see
Theorem 2.1. In Section 3 we show among other things that this question has an
affirmative answer: see Theorems 3.1, 3.2, and 3.3.
Let $\star$ be a semistar operation on the integral domain $D$ and let
$\widetilde{\star}$ be the stable semistar operation of finite type
canonically associated to $\star$ (the definitions are recalled later in this
section). We define in Section 4, what it means the semistar valuative
dimension of $D$, denoted by $\star$-$\operatorname{dim}_{v}(D)$. It extends
the “classical” valuative dimension of P. Jaffard [23], denoted by
$\operatorname{dim}_{v}(D)$ to the setting of semistar operations. We show
that the semistar valuative dimension of $D$ has various nice properties, like
the classical valuative dimension. For example we show that if
$\widetilde{\star}$-$\operatorname{dim}_{v}(D)<\infty$ then
$\star[X]\text{-}\operatorname{dim}_{v}(D[X])=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)+1$:
see Theorem 4.8. Also we established that
$\widetilde{\star}\text{-}\operatorname{dim}(D)\leq\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)$,
and equality holds if $D$ is a $\widetilde{\star}$-Noetherian domain or a
P$\star$MD: see Corollaries 4.6 and 4.11. In relation with the $\star$-Nagata
ring $\operatorname{Na}(D,\star)$, it is shown that
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(\operatorname{Na}(D,\star))$:
see Theorem 4.17. If $\widetilde{\star}\text{-}\operatorname{dim}(D)<\infty$
and
$\widetilde{\star}\text{-}\operatorname{dim}(D)=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)$,
we say that, $D$ is a $\widetilde{\star}$-Jaffard domain. We establish that
$D$ is a $\widetilde{\star}$-quasi-Prüfer domain if and only if each
$(\star,\star^{\prime})$-linked overring $T$ of $D$ is a
$\widetilde{\star^{\prime}}$-Jaffard domain, where $\star^{\prime}$ is a
semistar operation on $T$: see Theorem 4.14. As a consequence of this result
we obtain that a Krull domain $D$, must be a $w_{D}$-Jaffard domain.
To facilitate the reading of the introduction and of the paper, we first
review some basic facts on semistar operations. Let
$\overline{\mathcal{F}}(D)$ denote the set of all nonzero $D$-submodules of
$K$. Let $\mathcal{F}(D)$ be the set of all nonzero _fractional_ ideals of
$D$; i.e., $E\in\mathcal{F}(D)$ if $E\in\overline{\mathcal{F}}(D)$ and there
exists a nonzero element $r\in D$ with $rE\subseteq D$. Let $f(D)$ be the set
of all nonzero finitely generated fractional ideals of $D$. Obviously,
$f(D)\subseteq\mathcal{F}(D)\subseteq\overline{\mathcal{F}}(D)$. As in [30], a
semistar operation on $D$ is a map
$\star:\overline{\mathcal{F}}(D)\rightarrow\overline{\mathcal{F}}(D)$,
$E\mapsto E^{\star}$, such that, for all $x\in K$, $x\neq 0$, and for all
$E,F\in\overline{\mathcal{F}}(D)$, the following three properties hold:
* $\star_{1}$
: $(xE)^{\star}=xE^{\star}$;
* $\star_{2}$
: $E\subseteq F$ implies that $E^{\star}\subseteq F^{\star}$;
* $\star_{3}$
: $E\subseteq E^{\star}$ and $E^{\star\star}:=(E^{\star})^{\star}=E^{\star}$.
Recall from [30, Proposition 5] that if $\star$ is a semistar operation on
$D$, then, for all $E,F\in\overline{\mathcal{F}}(D)$, the following basic
formulas follow easily from the above axioms:
* (1)
$(EF)^{\star}=(E^{\star}F)^{\star}=(EF^{\star})^{\star}=(E^{\star}F^{\star})^{\star}$;
* (2)
$(E+F)^{\star}=(E^{\star}+F)^{\star}=(E+F^{\star})^{\star}=(E^{\star}+F^{\star})^{\star}$;
* (3)
$(E:F)^{\star}\subseteq(E^{\star}:F^{\star})=(E^{\star}:F)=(E^{\star}:F)^{\star}$,
if $(E:F)\neq(0)$;
* (4)
$(E\cap F)^{\star}\subseteq E^{\star}\cap F^{\star}=(E^{\star}\cap
F^{\star})^{\star}$ if $(E\cap F)\neq(0)$.
It is convenient to say that a _(semi)star operation on_ $D$ is a semistar
operation which, when restricted to $\mathcal{F}(D)$, is a star operation (in
the sense of [17, Section 32]). It is easy to see that a semistar operation
$\star$ on $D$ is a (semi)star operation on $D$ if and only if $D^{\star}=D$.
Let $\star$ be a semistar operation on the domain $D$. For every
$E\in\overline{\mathcal{F}}(D)$, put $E^{\star_{f}}:=\cup F^{\star}$, where
the union is taken over all finitely generated $F\in f(D)$ with $F\subseteq
E$. It is easy to see that $\star_{f}$ is a semistar operation on $D$, and
${\star_{f}}$ is called _the semistar operation of finite type associated to_
$\star$. Note that $(\star_{f})_{f}=\star_{f}$. A semistar operation $\star$
is said to be of _finite type_ if $\star=\star_{f}$; in particular
${\star_{f}}$ is of finite type. We say that a nonzero ideal $I$ of $D$ is a
_quasi- $\star$-ideal_ of $D$, if $I^{\star}\cap D=I$; a _quasi-
$\star$-prime_ (ideal of $D$), if $I$ is a prime quasi-$\star$-ideal of $D$;
and a _quasi- $\star$-maximal_ (ideal of $D$), if $I$ is maximal in the set of
all proper quasi-$\star$-ideals of $D$. Each quasi-$\star$-maximal ideal is a
prime ideal. It was shown in [11, Lemma 4.20] that if $D^{\star}\neq K$, then
each proper quasi-$\star_{f}$-ideal of $D$ is contained in a
quasi-$\star_{f}$-maximal ideal of $D$. We denote by
$\operatorname{QMax}^{\star}(D)$ (resp., $\operatorname{QSpec}^{\star}(D)$)
the set of all quasi-$\star$-maximal ideals (resp., quasi-$\star$-prime
ideals) of $D$. When $\star$ is a (semi)star operation, it is easy to see that
the notion of quasi-$\star$-ideal is equivalent to the classical notion of
$\star$-ideal (i.e., a nonzero ideal $I$ of $D$ such that $I^{\star}=I$).
If $\star_{1}$ and $\star_{2}$ are semistar operations on $D$, one says that
$\star_{1}\leq\star_{2}$ if $E^{\star_{1}}\subseteq E^{\star_{2}}$ for each
$E\in\overline{\mathcal{F}}(D)$ (cf. [30, page 6]). This is equivalent to
saying that
$(E^{\star_{1}})^{\star_{2}}=E^{\star_{2}}=(E^{\star_{2}})^{\star_{1}}$ for
each $E\in\overline{\mathcal{F}}(D)$ (cf. [30, Lemma 16]). Obviously, for each
semistar operation $\star$ defined on $D$, we have $\star_{f}\leq\star$. Let
$d_{D}$ (or, simply, $d$) denote the identity (semi)star operation on $D$.
Clearly, $d_{D}\leq\star$ for all semistar operations $\star$ on $D$.
If $\Delta$ is a set of prime ideals of a domain $D$, then there is an
associated semistar operation on $D$, denoted by $\star_{\Delta}$, defined as
follows:
$E^{\star_{\Delta}}:=\cap\\{ED_{P}|P\in\Delta\\}\text{, for each
}E\in\overline{\mathcal{F}}(D).$
If $\Delta=\emptyset$, let $E^{\star_{\Delta}}:=K$ for each
$E\in\overline{F}(D)$. Note that $E^{\star_{\Delta}}D_{P}=ED_{P}$ for each
$E\in\overline{\mathcal{F}}(D)$ and $P\in\Delta$ by [11, Lemma 4.1 (2)]. One
calls $\star_{\Delta}$ the _spectral semistar operation associated to_
$\Delta$. A semistar operation $\star$ on a domain $D$ is called a _spectral
semistar operation_ if there exists a subset $\Delta$ of the prime spectrum of
$D$, Spec$(D)$, such that $\star=\star_{\Delta}$. When
$\Delta:=\operatorname{QMax}^{\star_{f}}(D)$, we set
$\widetilde{\star}:=\star_{\Delta}$; i.e.,
$E^{\widetilde{\star}}:=\cap\\{ED_{P}|P\in\operatorname{QMax}^{\star_{f}}(D)\\}\text{,
for each }E\in\overline{\mathcal{F}}(D).$
It has become standard to say that a semistar operation $\star$ is stable if
$(E\cap F)^{\star}=E^{\star}\cap F^{\star}$ for all $E$,
$F\in\overline{\mathcal{F}}(D)$. (“Stable” has replaced the earlier usage,
“quotient”, in [30, Definition 21].) All spectral semistar operations are
stable [11, Lemma 4.1(3)]. In particular, for any semistar operation $\star$,
we have that $\widetilde{\star}$ is a stable semistar operation of finite type
[11, Corollary 3.9].
Let $D$ be a domain, $\star$ a semistar operation on $D$, $T$ an overring of
$D$, and $\iota:D\hookrightarrow T$ the corresponding inclusion map. In a
canonical way, one can define an associated semistar operation $\star_{\iota}$
on $T$, by $E\mapsto E^{\star_{\iota}}:=E^{\star}$, for each
$E\in\overline{\mathcal{F}}(T)(\subseteq\overline{\mathcal{F}}(D))$.
The most widely studied (semi)star operations on $D$ have been the identity
$d_{D}$ and $v_{D}$, $t_{D}:=(v_{D})_{f}$, and $w_{D}:=\widetilde{v_{D}}$
operations, where $E^{v_{D}}:=(E^{-1})^{-1}$, with $E^{-1}:=(D:E):=\\{x\in
K|xE\subseteq D\\}$.
Let $D$ be a domain with quotient field $K$, and let $X$ be an indeterminate
over $K$. For each $f\in K[X]$, we let $c_{D}(f)$ denote the content of the
polynomial $f$, i.e., the (fractional) ideal of $D$ generated by the
coefficients of $f$. Let $\star$ be a semistar operation on $D$. If
$N_{\star}:=\\{g\in D[X]|g\neq 0\text{ and }c_{D}(g)^{\star}=D^{\star}\\}$,
then
$N_{\star}=D[X]\backslash\bigcup\\{P[X]|P\in\operatorname{QMax}^{\star_{f}}(D)\\}$
is a saturated multiplicative subset of $D[X]$. The ring of fractions
$\operatorname{Na}(D,\star):=D[X]_{N_{\star}}$
is called the $\star$-Nagata domain (of $D$ with respect to the semistar
operation $\star$). When $\star=d$, the identity (semi)star operation on $D$,
then $\operatorname{Na}(D,d)$ coincides with the classical Nagata domain
$D(X)$ (as in, for instance [29, page 18], [17, Section 33] and [14]).
## 2\. Semistar operations on polynomial rings
In [6], Chang and Fontana introduced a new technique for defining new semistar
operations on integral domains. Let $D$ be an integral domain with quotient
field $K$, and let $X$ be an indeterminate over $K$. For a given
multiplicative subset $\mathcal{S}$ of $D[X]$, set
$E^{\circlearrowleft_{\mathcal{S}}}:=E[X]_{\mathcal{S}}\cap K,\text{ for all
}E\in\overline{\mathcal{F}}(D).$
Then it is proved in [6, Theorem 2.1] among other things that, the mapping
$\circlearrowleft_{\mathcal{S}}:\overline{\mathcal{F}}(D)\to\overline{\mathcal{F}}(D)$,
$E\mapsto E^{\circlearrowleft_{\mathcal{S}}}$ is a stable semistar operation
of finite type on $D[X]$, i.e.,
$\widetilde{\circlearrowleft_{\mathcal{S}}}=\circlearrowleft_{\mathcal{S}}$,
and $\operatorname{QMax}^{\circlearrowleft_{\mathcal{S}}}(D)=$ the set of
maximal elements of
$\Delta(\mathcal{S}):=\\{P\in\operatorname{Spec}(D)|P[X]\cap\mathcal{S}=\emptyset\\}$.
Let $D$ be an integral domain, and $\star$ a semistar operation on $D$. Using
the technique discussed in the first paragraph, Chang and Fontana defined
canonically a semistar operation denoted by $[\star]$ on the polynomial ring
$D[X]$. More precisely suppose that $X$, $Y$ are two indeterminates over $D$,
and set $D_{1}:=D[X]$, $K_{1}:=K(X)$. Take the following subset of
$\operatorname{Spec}(D_{1})$:
$\Delta_{1}^{\star}:=\\{Q_{1}\in\operatorname{Spec}(D_{1})|\text{ }Q_{1}\cap
D=(0)\text{ or }Q_{1}=(Q_{1}\cap D)[X]\text{ and }(Q_{1}\cap
D)^{\star_{f}}\subsetneq D^{\star}\\}.$
Set
$\mathcal{S}_{1}^{\star}:=\mathcal{S}(\Delta_{1}^{\star}):=D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\Delta_{1}^{\star}\\})$
and $[\star]:=\circlearrowleft_{\mathcal{S}_{1}^{\star}}$, that is:
$E^{[\star]}:=E[Y]_{\mathcal{S}_{1}^{\star}}\cap K_{1},\text{ for all
}E\in\overline{\mathcal{F}}(D_{1}).$
They proved answering their question [7, Question], that $D$ is a
$\widetilde{\star}$-quasi-Prüfer domain if and only if each upper to zero, is
a quasi-$[\star]$-maximal ideal of $D[X]$. Recall that $D$ is said to be a
$\star$-quasi-Prüfer domain, in case, if $Q$ is a prime ideal in $D[X]$, and
$Q\subseteq P[X]$, for some $P\in\operatorname{QSpec}^{\star}(D)$, then
$Q=(Q\cap D)[X]$. This notion is the semistar analogue of the classical notion
of the quasi-Prüfer domains [12, Section 6.5] (that is among other equivalent
conditions, the domain $D$ is said to be a _quasi-Prüfer domain_ if it has
Prüferian integral closure).
Now by the same technique, we define canonically a semistar operation denoted
by $\star[X]$ on the polynomial ring $D[X]$, which has desired semistar
(Krull) dimension theoretic properties.
###### Theorem 2.1.
Let $D$ be an integral domain with quotient field $K$, let $X$, $Y$ be two
indeterminates over $D$ and let $\star$ be a semistar operation on D. Set
$D_{1}:=D[X]$, $K_{1}:=K(X)$ and take the following subset of
$\operatorname{Spec}(D_{1})$:
$\Theta_{1}^{\star}:=\\{Q_{1}\in\operatorname{Spec}(D_{1})|\text{ }Q_{1}\cap
D=(0)\text{ or }(Q_{1}\cap D)^{\star_{f}}\subsetneq D^{\star}\\}.$
Set
$\mathfrak{S}_{1}^{\star}:=\mathcal{S}(\Theta_{1}^{\star}):=D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\Theta_{1}^{\star}\\})$
and:
$E^{\circlearrowleft_{\mathfrak{S}_{1}^{\star}}}:=E[Y]_{\mathfrak{S}_{1}^{\star}}\cap
K_{1},\text{ for all }E\in\overline{\mathcal{F}}(D_{1}).$
* (a)
The mapping
$\star[X]:=\circlearrowleft_{\mathfrak{S}_{1}^{\star}}:\overline{\mathcal{F}}(D_{1})\to\overline{\mathcal{F}}(D_{1})$,
$E\mapsto E^{\circlearrowleft_{\mathfrak{S}_{1}^{\star}}}$ is a stable
semistar operation of finite type on $D[X]$, i.e.,
$\widetilde{\star[X]}=\star[X]$.
* (b)
$\widetilde{\star}[X]=\star_{f}[X]=\star[X]$.
* (c)
$\star[X]\leq[\star]$. In particular, if $\star$ is a (semi)star operation on
$D$, then $\star[X]$ is a (semi)star operation on $D[X]$.
* (d)
$d_{D}[X]=d_{D[X]}$.
###### Proof.
Note that, if $Q_{1}\in\operatorname{Spec}(D[X])$ is not an upper to zero and
$(Q_{1}\cap D)^{\star_{f}}\subsetneq D^{\star}$, then the prime ideal
$Q_{1}\cap D$ is contained in a quasi-$\star_{f}$-maximal ideal of $D$.
Moreover if $Q_{1}\cap D=(0)$ and $c_{D}(Q_{1})^{\star_{f}}\subsetneq
D^{\star}$ then $c_{D}(Q_{1})^{\star_{f}}$ is contained in a
quasi-$\star_{f}$-prime ideal $P$ of $D$ and hence $Q_{1}\subseteq P[X]$ with
$P^{\star_{f}}\subsetneq D^{\star}$. Set
$\circleddash_{1}^{\star}:=\\{Q_{1}\in\operatorname{Spec}(D_{1})|\text{
}Q_{1}\cap D=(0)\text{ and }c_{D}(Q_{1})^{\star_{f}}=D^{\star}\text{ or
}Q_{1}\cap D\in\operatorname{QMax}^{\star_{f}}(D)\\}.$ Now we show that:
$\mathfrak{S}_{1}^{\star}:=D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\Theta_{1}^{\star}\\})=D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\circleddash_{1}^{\star}\\})=\mathcal{S}(\circleddash_{1}^{\star}).$
Since $\circleddash_{1}^{\star}\subseteq\Theta_{1}^{\star}$ one has
$\mathfrak{S}_{1}^{\star}\subseteq\mathcal{S}(\circleddash_{1}^{\star})$. For
the other inclusion suppose that $f\in\mathcal{S}(\circleddash_{1}^{\star})$.
So that $f\notin Q_{1}[Y]$ for each $Q_{1}\in\circleddash_{1}^{\star}$. We
want to show that $f\in\mathfrak{S}_{1}^{\star}$. Suppose the contrary, hence
$f\in Q_{1}[Y]$ for some $Q_{1}\in\Theta_{1}^{\star}$. Therefore there are two
cases to consider:
1) If $Q_{1}\cap D=(0)$, we have $c_{D}(Q_{1})^{\star_{f}}\neq D^{\star}$ as
$Q_{1}\notin\circleddash_{1}^{\star}$. Thus $Q_{1}\subseteq P[X]$ for some
quasi-$\star_{f}$-prime ideal $P$ of $D$. Choose
$M\in\operatorname{QMax}^{\star_{f}}(D)$ such that $P\subseteq M$. So that
$Q_{1}\subseteq M[X]$ and hence $f\in M[X][Y]$ while
$M[X]\in\circleddash_{1}^{\star}$, which is a contradiction.
2) If $(Q_{1}\cap D)^{\star_{f}}\subsetneq D^{\star}$, then $Q_{1}\cap
D\subseteq M$ for some quasi-$\star_{f}$-maximal ideal of $D$. We have
$Q_{1}\cap D\neq M$, since otherwise $Q_{1}\in\circleddash_{1}^{\star}$ and
$f\in Q_{1}[Y]$ which is a contradiction. Note that $(Q_{1}+M[X])\cap
D=(M[X]+(X))\cap D=M$ and $Q_{1}\subseteq Q_{1}+M[X]\subseteq M[X]+(X)$.
Therefore $f\in(M[X]+(X))[Y]$ while $M[X]+(X)\in\circleddash_{1}^{\star}$
which is again a contradiction.
So that we have $f\notin Q_{1}[Y]$ for each $Q_{1}\in\Theta_{1}^{\star}$. Thus
$f\in\mathfrak{S}_{1}^{\star}$, that is
$\mathfrak{S}_{1}^{\star}=\mathcal{S}(\circleddash_{1}^{\star})$.
$(a)$ It follows from [6, Theorem 2.1 (a) and (b)], that $\star[X]$ is a
stable semistar operation of finite type on $D[X]$.
$(b)$ Since
$\operatorname{QMax}^{\star_{f}}(D)=\operatorname{QMax}^{\widetilde{\star}}(D)$,
the conclusion follows easily from the fact that
$\mathfrak{S}_{1}^{\widetilde{\star}}=\mathfrak{S}_{1}^{\star_{f}}=\mathfrak{S}_{1}^{\star}$.
$(c)$ It is easily seen that
$\mathfrak{S}_{1}^{\star}\subseteq\mathcal{S}_{1}^{\star}$. Then
$E^{\star[X]}=E[Y]_{\mathfrak{S}_{1}^{\star}}\cap
K_{1}\subseteq(E[Y]_{\mathfrak{S}_{1}^{\star}})_{\mathcal{S}_{1}^{\star}}\cap
K_{1}=E[Y]_{\mathcal{S}_{1}^{\star}}\cap K_{1}=E^{[\star]}.$
This means that $\star[X]\leq[\star]$ by definition. Now if $\star$ is a
(semi)star operation on $D$, then $[\star]$ is a (semi)star operation on
$D[X]$ by [6, Theorem 2.3 (a)]. So that $D_{1}\subseteq
D_{1}^{\star[X]}\subseteq D_{1}^{[\star]}=D_{1}$, that is
$D_{1}^{\star[X]}=D_{1}$. Hence $\star[X]$ is a (semi)star operation on
$D[X]$.
$(d)$ Note that we have:
$\displaystyle\mathfrak{S}_{1}^{d_{D}}=$ $\displaystyle
D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\Theta_{1}^{d_{D}}\\})$
$\displaystyle=$ $\displaystyle
D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\operatorname{Spec}(D_{1})\text{
and }Q_{1}\cap D\neq D\\})$ $\displaystyle=$ $\displaystyle
D_{1}[Y]\backslash(\bigcup\\{Q_{1}[Y]|Q_{1}\in\operatorname{Max}(D_{1})\\}).$
So for an element $E\in\overline{\mathcal{F}}(D_{1})$ we have:
$E=E^{d_{D[X]}}\subseteq E^{d_{D}[X]}=E[Y]_{\mathfrak{S}_{1}^{d_{D}}}\cap
K_{1}=ED_{1}(Y)\cap K_{1}=E.$
The last equality follows from [14, Proposition 3.4 (3)]. Thus
$E^{d_{D[X]}}=E^{d_{D}[X]}$, that is $d_{D}[X]=d_{D[X]}$. ∎
A different approach to the semistar operations on polynomial rings is
possible by using the notion of localizing system. Recall that a _localizing
system of ideals_ $\mathcal{F}$ of $D$ is a set of (integral) ideals of $D$
verifying the following conditions $(a)$ if $I\in\mathcal{F}$ and if
$I\subseteq J$ , then $J\in\mathcal{F}$; $(b)$ if $I\in\mathcal{F}$ and if $J$
is an ideal of $D$ such that $(J:_{D}iD)\in\mathcal{F}$, for each $i\in I$,
then $J\in\mathcal{F}$. The relation between stable semistar operations and
localizing systems has been deeply investigated by M. Fontana and J. Huckaba
in [11] and by F. Halter-Koch in the context of module systems [25]. If
$\star$ is a semistar operation on $D$, then $\mathcal{F}^{\star}:=\\{I$ ideal
of $D|I^{\star}=D^{\star}\\}$ is a localizing system (called the _localizing
system associated to $\star$_) of $D$. And if $\mathcal{F}$ is a localizing
system of $D$, then the map $E\mapsto
E^{\star_{\mathcal{F}}}:=\bigcup\\{(E:J)|J\in\mathcal{F}\\}$, for each
$E\in\overline{\mathcal{F}}(D)$, is a stable semistar operation on $D$. It is
proved in [32, Proposition 3.1] that if $\mathcal{F}$ is a localizing system
of $D$, then $\mathcal{F}[X]:=\\{A$ ideal of $D[X]|A\cap D\in\mathcal{F}\\}$
is a localizing system of the polynomial ring $D[X]$. Now let $\star$ be a
stable semistar operation on $D$ and let $\mathcal{F}^{\star}$ be the
localizing system of $D$ associated to $\star$. Consider the localizing system
$\mathcal{F}^{\star}[X]$ of $D[X]$. Then G. Picozza [32, Page 426] introduced
a semistar operation denoted by $\star^{\prime}$ on the polynomial ring $D[X]$
as $\star_{\mathcal{F}^{\star}[X]}$. He used the semistar operation
$\star^{\prime}$ to provide the semistar version of the Hilbert basis Theorem
[32, Theorem 3.3].
###### Proposition 2.2.
If $\star$ is a stable semistar operation of finite type on $D$, that is if,
$\star=\widetilde{\star}$ then $\star^{\prime}=\star[X]$.
###### Proof.
Adapt the notation in the paragraph before the proposition. Recall from [6,
Corollary 2.2] that if $\mathcal{F}$ is a localizing system of $D$, $Y$ is an
indeterminate over $D$, and
$\mathcal{S}(\mathcal{F}):=D[Y]\backslash\bigcup\\{Q[Y]|Q\in\operatorname{Spec}(D)$
and $Q\notin\mathcal{F}\\}$ which is a saturated multiplicatively closed
subset of $D[Y]$, then
$\star_{\mathcal{F}}=\circlearrowleft_{\mathcal{S}(\mathcal{F})}$. Now let
$\mathcal{F}:=\mathcal{F}^{\star}[X]=\\{A\text{ ideal of }D_{1}|(A\cap
D)^{\star}=D^{\star}\\}$. Then
$\displaystyle\mathcal{S}(\mathcal{F})=$ $\displaystyle
D_{1}[Y]\backslash\bigcup\\{Q_{1}[Y]|Q_{1}\in\operatorname{Spec}(D_{1})\text{
and }Q_{1}\notin\mathcal{F}\\}$ $\displaystyle=$ $\displaystyle
D_{1}[Y]\backslash\bigcup\\{Q_{1}[Y]|Q_{1}\in\operatorname{Spec}(D_{1})\text{
s.t. }Q_{1}\cap D=(0)\text{ or }(Q_{1}\cap D)^{\star}\subsetneq D^{\star}\\}$
$\displaystyle=$ $\displaystyle
D_{1}[Y]\backslash\bigcup\\{Q_{1}[Y]|Q_{1}\in\Theta^{\star}_{1}\\}$
$\displaystyle=$ $\displaystyle\mathfrak{S}^{\star}_{1}.$
Consequently
$\star^{\prime}=\star_{\mathcal{F}}=\circlearrowleft_{\mathcal{S}(\mathcal{F})}=\circlearrowleft_{\mathfrak{S}^{\star}_{1}}=\star[X]$
which ends the proof. ∎
Note that the semistar operation $[\star]$ has a main difference with
$\star[X]$ and $\star^{\prime}$. Indeed let $\star=d_{D}$. Then one has
$d_{D}^{\prime}=d_{D}[X]=d_{D[X]}$ by Theorem 2.1 $(d)$ and Proposition 2.2.
But $[d_{D}]\neq d_{D[X]}$. Since if $[d_{D}]=d_{D[X]}$, then [7, Corollary
2.5 (1)] implies that if $D$ is a Prüfer domain then $D[X]$ should be a Prüfer
domain which is absurd.
###### Remark 2.3.
Note that the set of quasi-$\star[X]$-prime ideals of $D[X]$, coincides with
the set $\Theta_{1}^{\star}\backslash\\{0\\}$. Indeed let $Q$ be an element of
$\Theta_{1}^{\star}\backslash\\{0\\}$. Then we have
$Q[Y]\cap\mathfrak{S}_{1}^{\star}=\emptyset$. Hence
$\displaystyle Q^{\star[X]}\cap D[X]=$
$\displaystyle(Q[Y]_{\mathfrak{S}_{1}^{\star}}\cap K(X))\cap D[X]$
$\displaystyle=$ $\displaystyle(Q[Y]_{\mathfrak{S}_{1}^{\star}}\cap
D[X,Y])\cap D[X]$ $\displaystyle=$ $\displaystyle Q[Y]\cap D[X]=Q.$
Therefore $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$; i.e.,
$\Theta_{1}^{\star}\backslash\\{0\\}\subseteq\operatorname{QSpec}^{\star[X]}(D[X])$.
Since the other inclusion is trivial, we obtain that
$\operatorname{QSpec}^{\star[X]}(D[X])=\Theta_{1}^{\star}\backslash\\{0\\}$.
In the rest of the paper for every semistar operation $\star$ on an integral
domain $D$, we let $\star[X]$, to be the stable semistar operation of finite
type on $D[X]$ canonically associated to $\star$ as in Theorem 2.1(a).
Let $\star$ be a semistar operation on a domain $D$. As in [13] and [9] (cf.
also [20] for the case of a star operation), $D$ is called a _Prüfer
$\star$-multiplication domain_ (for short, a P$\star$MD) if each finitely
generated ideal of $D$ is $\star_{f}$-invertible; i.e., if
$(II^{-1})^{\star_{f}}=D^{\star}$ for all $I\in f(D)$. When $\star=v$, we
recover the classical notion of P$v$MD; when $\star=d_{D}$, the identity
(semi)star operation, we recover the notion of Prüfer domain.
###### Remark 2.4.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$D[X]$ is a P$\star[X]$MD (resp. a $\star[X]$-quasi-Prüfer domain). Since
$\star[X]\leq[\star]$ by Theorem 2.1(c), we obtain that $D[X]$ is a
P$[\star]$MD by [13](resp. a $[\star]$-quasi-Prüfer domain by [7, Corollary
2.4]). So that $D$ is a P$\star$MD by [6, Corollary 2.5 (1)] (resp. a
$\widetilde{\star}$-quasi-Prüfer domain by [6, Corollary 2.4]).
In [10, Section 3], El Baghdadi, Fontana and Picozza defined and studied the
semistar Noetherian domains, i.e., domains having the ascending chain
condition on quasi-semistar-ideals.
###### Remark 2.5.
(Cf. [32, Theorem 3.6]) Let $\star$ be a semistar operation on an integral
domain $D$. Then $D$ is a $\widetilde{\star}$-Noetherian domain if and only if
$D[X]$ is a $\star[X]$-Noetherian domain. In fact if $D[X]$ is a
$\star[X]$-Noetherian domain, since $\star[X]\leq[\star]$ by Theorem 2.1(c),
we obtain that $D[X]$ is a $[\star]$-Noetherian domain. So that $D$ is
$\widetilde{\star}$-Noetherian by [6, Corollary 2.5 (2)]. For the other
implication use Remark 2.2 together with [32, Theorem 3.3].
## 3\. Semistar-Krull dimension
Let $\star$ be a semistar operation on an integral domain $D$. In this section
we make use of the semistar operation $\star[X]$ on $D[X]$, canonically
associated to the given semistar operation $\star$ on $D$, to provide an
answer to the question raised in the introduction. First we recall some
definitions and properties of $\star$-dimension. For each quasi-$\star$-prime
$P$ of $D$, the $\star$-height of $P$ (for short,
$\star$-$\operatorname{ht}(P)$) is defined to be the supremum of the lengths
of the chains of quasi-$\star$-prime ideals of $D$, between prime ideal $(0)$
(included) and $P$. Obviously, if $\star=d_{D}$ is the identity (semi)star
operation on $D$, then $\star$-$\operatorname{ht}(P)=\operatorname{ht}(P)$,
for each prime ideal $P$ of $D$. If the set of quasi-$\star$-prime of $D$ is
not empty, the $\star$-dimension of $D$ is defined as follows:
$\star\text{-}\operatorname{dim}(D):=\sup\\{\star\text{-}\operatorname{ht}(P)|P\text{
is a quasi-}\star\text{-prime of }D\\}.$
If the set of quasi-$\star$-primes of $D$ is empty, then pose
$\star\text{-}\operatorname{dim}(D):=0$. Thus, if $\star=d_{D}$, then
$\star\text{-}\operatorname{dim}(D)=\operatorname{dim}(D)$, the usual (Krull)
dimension of $D$.
Note that, the notions of $t$-dimension and of $w$-dimension have received a
considerable interest by several authors (cf. for instance, [37, 38, 19]).
It is known (see [10, Lemma 2.11]) that
$\displaystyle\widetilde{\star}\text{-}\operatorname{dim}(D)=$
$\displaystyle\sup\\{\operatorname{ht}(P)\mid P\text{ is a
quasi-}\widetilde{\star}\text{-prime ideal of }D\\}$ $\displaystyle=$
$\displaystyle\sup\\{\operatorname{ht}(P)\mid P\text{ is a
quasi-}\widetilde{\star}\text{-maximal ideal of }D\\}.$
We answer to the Question 1.1, in the results 3.1, 3.2 and 4.11. The following
result is the semistar version of the classical theorem of Seidenberg [35,
Theorem 2].
###### Theorem 3.1.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$n:=\widetilde{\star}$-$\operatorname{dim}(D)$. Then
$n+1\leq\star[X]\text{-}\operatorname{dim}(D[X])\leq 2n+1.$
###### Proof.
Consider a chain $P_{1}\subseteq\cdots\subseteq P_{n}$ of
quasi-$\widetilde{\star}$-prime ideals of $D$. Let $Q:=P_{n}[X]+(X)$. Since
$Q\cap D=(P_{n}[X]+(X))\cap
D=P_{n}\in\operatorname{QSpec}^{\widetilde{\star}}(D)$, we have, using Remark
2.3 that, $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$. Then
$P_{1}[X]\subseteq\cdots\subseteq P_{n}[X]\subseteq P_{n}[X]+(X),$
is a chain of $n+1$ quasi-$\star[X]$-prime ideals of $D[X]$. Hence
$n+1\leq\star[X]\text{-}\operatorname{dim}(D[X])$.
For the second inequality suppose that
$Q\in\operatorname{QMax}^{\star[X]}(D[X])$ is such that
$\operatorname{ht}_{D[X]}Q=\star[X]\text{-}\operatorname{dim}(D[X]).$
Hence by [26, Theorem 38] we obtain that $\operatorname{ht}_{D[X]}Q\leq
2(\operatorname{ht}_{D}(Q\cap D))+1\leq 2n+1$. Consequently we have
$\star[X]\text{-}\operatorname{dim}(D[X])\leq 2n+1$. ∎
In [36, Theorem 3], Seidenberg showed that for any pair of positive integers
$(n,m)$ with $n+1\leq m\leq 2n+1$, there exists a domain $D$ such that
$\operatorname{dim}(D)=d_{D}$-$\operatorname{dim}(D)=n$ and
$\operatorname{dim}(D[X])=d_{D[X]}$-$\operatorname{dim}(D[X])=d_{D}[X]$-$\operatorname{dim}(D[X])=m$.
If $X_{1},\cdots,X_{r}$ are indeterminates over $D$, for $r\geq 2$, we let
$\star[X_{1},\cdots,X_{r}]:=(\star[X_{1},\cdots,X_{r-1}])[X_{r}],$
where $\star[X_{1},\cdots,X_{r-1}]$ is a stable semistar operation of finite
type on $D[X_{1},\cdots,X_{r-1}]$.
###### Theorem 3.2.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$D$ is a $\widetilde{\star}$-Noetherian domain of $\widetilde{\star}$-Krull
dimension $n$. Then
$\star[X_{1},\cdots,X_{m}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{m}])=n+m.$
###### Proof.
Since $D[X_{1},\cdots,X_{m-1}]$ is $\star[X_{1},\cdots,X_{m-1}]$-Noetherian
domain, it suffices to prove the theorem for the case $m=1$. By Theorem 3.1,
we have $n+1\leq\star[X]$-$\operatorname{dim}(D[X])$. Now let $M$ be an
arbitrary quasi-$\star[X]$-maximal ideal of $D[X]$. Then $M$ is either an
upper to zero, or $P:=M\cap D\in\operatorname{QSpec}^{\widetilde{\star}}(D)$.
Note that in either case $D_{P}$ is a Noetherian domain ([10, Proposition
3.8]). Hence:
$\displaystyle\operatorname{ht}_{D[X]}M=$
$\displaystyle\operatorname{dim}(D[X]_{M})=\operatorname{dim}(D_{P}[X]_{MD_{P}[X]})$
$\displaystyle\leq$
$\displaystyle\operatorname{dim}(D_{P}[X])=\operatorname{dim}(D_{P})+1$
$\displaystyle\leq$ $\displaystyle n+1.$
The third equality holds since $D_{P}$ is a Noetherian domain and [17, Theorem
30.5], and the second inequality holds by [10, Lemma 2.11]. So that by [10,
Lemma 2.11] we obtain that
$\star[X]\text{-}\operatorname{dim}(D[X])=\sup\\{\operatorname{ht}_{D[X]}M|M\in\operatorname{QMax}^{\star[X]}(D[X])\\}\leq
n+1,$
which ends the proof. ∎
###### Theorem 3.3.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$D$ is a P$\star$MD of $\widetilde{\star}$-Krull dimension $n$. Then
$\star[X]\text{-}\operatorname{dim}(D[X])=n+1$.
###### Proof.
Use the fact that if $D$ is a Prüfer domain, then
$\operatorname{dim}(D[X])=\operatorname{dim}(D)+1$ [36, Corollary] and by the
same argument as Theorem 3.2 the proof is complete. ∎
In Corollary 4.11, we show that if $D$ is a P$\star$MD then
$\star[X_{1},\cdots,X_{m}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{m}])=\widetilde{\star}\text{-}\operatorname{dim}(D)+m.$
One of the key concepts of Jaffard in [23], is that of a _special chain_ ,
defined as follows. A chain $\mathcal{C}=\\{P_{i}\\}_{i=0}^{m}$ of primes in a
polynomial ring $D[X_{1},\cdots,X_{m}]$ is called a _special chain_ if, for
each $P_{i}\in\mathcal{C}$, the ideal $(P_{i}\cap D)[X_{1},\cdots,X_{m}]$ is a
member of $\mathcal{C}$. Jaffard’s _special chain theorem_ asserts that, if
$Q$ is a prime ideal of $D[X_{1},\cdots,X_{m}]$ of finite height, then
$\operatorname{ht}(Q)$ can be realized as the length of a special chain of
primes in $D[X_{1},\cdots,X_{m}]$ with terminal element $Q$. In particular, if
$D$ is a finite dimensional domain, then
$\operatorname{dim}(D[X_{1},\cdots,X_{m}])$ can be realized as the length of a
special chain of prime ideals of $D[X_{1},\cdots,X_{m}]$ (see [17, Corollary
30.19] for a simple proof). So we make the following remark.
###### Remark 3.4.
Let $\star$ be a semistar operation on an integral domain $D$. If
$\widetilde{\star}$-$\operatorname{dim}(D)$ is finite, then
$\star[X_{1},\cdots,X_{m}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{m}])$
can be realized as the length of a special chain of
quasi-$\star[X_{1},\cdots,X_{m}]$-prime ideals of $D[X_{1},\cdots,X_{m}]$. In
fact there exists a quasi-$\star[X_{1},\cdots,X_{m}]$-maximal ideal $Q$ of
$D[X_{1},\cdots,X_{m}]$ such that
$\star[X_{1},\cdots,X_{m}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{m}])=\operatorname{ht}Q.$
Now by Jaffard’s special chain theorem [17, Corollary 30.19],
$\operatorname{ht}(Q)$ can be realized as the length of a special chain
$(0)=Q_{0}\subseteq Q_{1}\subseteq\cdots\subseteq Q_{n}$ of prime ideals in
$D[X_{1},\cdots,X_{m}]$ with $Q_{n}=Q$. Since $Q_{n}$ is a
quasi-$\star[X_{1},\cdots,X_{m}]$-prime ideal of $D[X_{1},\cdots,X_{m}]$, then
each of $Q_{1},\cdots,Q_{n-1}$ is a quasi-$\star[X_{1},\cdots,X_{m}]$-prime
ideal of $D[X_{1},\cdots,X_{m}]$ by Theorem 2.1(a) and [11, Lemma 4.1, and
Remark 4.5].
As an application of Theorem 3.1 is the following result, which is the
semistar version of [35, Theorem 8].
###### Theorem 3.5.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}(D)=1$. Then
$\star[X]$-$\operatorname{dim}(D[X])=2$ if and only if $D$ is a
$\widetilde{\star}$-quasi-Prüfer domain.
###### Proof.
$(\Rightarrow)$. Suppose the contrary. Hence by [7, Lemma 2.3], there exists
an upper to zero $Q$ of $D[X]$ such that
$c_{D}(Q)^{\widetilde{\star}}\varsubsetneq D^{\widetilde{\star}}$. Then
$c_{D}(Q)^{\widetilde{\star}}$ is contained in a
quasi-$\widetilde{\star}$-prime ideal $P$ of $D$ and hence $Q\varsubsetneq
P[X]$. So that
$2\leq\operatorname{ht}_{D[X]}(P[X])\leq\star[X]\text{-}\operatorname{dim}(D[X])=2$,
that is $\operatorname{ht}_{D[X]}(P[X])=2$. This means that $P[X]$ is a
quasi-$\star[X]$-maximal ideal of $D[X]$. Therefore since $(P[X]+(X))\cap
D=P\in\operatorname{QSpec}^{\widetilde{\star}}(D)$, we obtain that
$P[X]+(X)\in\operatorname{QSpec}^{\star[X]}(D[X])$. Hence $P[X]=P[X]+(X)$
since $P[X]$ is a quasi-$\star[X]$-maximal ideal of $D[X]$. So that
$(X)\subseteq P[X]$. Consequently $D=c_{D}((X))\subseteq c_{D}(P[X])\subseteq
P$, which is a contradiction.
$(\Leftarrow)$. By Theorem 3.1 we have
$2\leq\star[X]\text{-}\operatorname{dim}(D[X])\leq 3$. If
$\star[X]\text{-}\operatorname{dim}(D[X])=3$, then
$\operatorname{ht}_{D[X]}(M)=3$ for some
$M\in\operatorname{QMax}^{\star[X]}(D[X])$. By [17, Corollary 30.2], $M$ can
not be an upper to zero. So that $P:=M\cap
D\in\operatorname{QMax}^{\widetilde{\star}}(D)$. From [7, Lemma 2.1] and the
hypothesis, we obtain that $D_{P}$ is a quasi-Prüfer domain of dimension $1$.
Hence $\operatorname{dim}(D_{P}[X])=2$ by [17, Proposition 30.14]. So we have:
$3=\operatorname{ht}_{D[X]}(M)=\operatorname{dim}(D[X]_{M})=\operatorname{dim}(D_{P}[X]_{MD_{P}[X]})\leq\operatorname{dim}(D_{P}[X])=2,$
which is a contradiction. Hence $\star[X]\text{-}\operatorname{dim}(D[X])=2$.
∎
Recall that an integral domain $D$ is called a UM$t$-domain (UM$t$ means
“uppers to zero are maximal $t$-ideals”) if every upper to zero in $D[X]$ is a
maximal $t$-ideal [21, Section 3]. It is observed in [7, Corollary 2.4 (b)]
that $D$ is a $w$-quasi-Prüfer domain if and only if $D$ is a UM$t$-domain.
###### Corollary 3.6.
Let $D$ be an integral domain. Suppose that $w$-$\operatorname{dim}(D)=1$.
Then $w[X]$-$\operatorname{dim}(D[X])=2$ if and only if $D$ is a UM$t$ domain.
###### Corollary 3.7.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}(D)=1$. The following statements are
equivalent:
* (1)
$D$ is a P$\star$MD.
* (2)
$D^{\widetilde{\star}}$ is integrally closed and
$\star[X]$-$\operatorname{dim}(D[X])=2$.
###### Proof.
The equivalence follows easily from Theorem 3.5 and from the fact that $D$ is
a P$\star$MD if and only if, $D$ is a $\widetilde{\star}$-quasi-Prüfer domain
and $D^{\widetilde{\star}}$ is integrally closed, [7, Lemma 2.17]. ∎
In the following result we collect the semistar (Krull) dimension properties
of $[\star]$.
###### Proposition 3.8.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$n:=\widetilde{\star}\text{-}\operatorname{dim}(D)$. Then
$n\leq[\star]\text{-}\operatorname{dim}(D[X])\leq 2n$. Moreover if $D$ is a
$\widetilde{\star}$-Noetherian domain or a P$\star$MD, then
$[\star]\text{-}\operatorname{dim}(D[X])=\widetilde{\star}\text{-}\operatorname{dim}(D)$.
###### Proof.
Consider a chain $P_{1}\subseteq\cdots\subseteq P_{n}$ of
quasi-$\widetilde{\star}$-prime ideals of $D$. Since
$P_{1}[X]\subseteq\cdots\subseteq P_{n}[X]$ is a chain of $n$
quasi-$[\star]$-prime ideals of $D[X]$, we have
$n\leq[\star]\text{-}\operatorname{dim}(D[X])$. For the second inequality
suppose that $Q\in\operatorname{QMax}^{[\star]}(D[X])$ is such that
$\operatorname{ht}_{D[X]}Q=[\star]\text{-}\operatorname{dim}(D[X]).$
If $Q$ is an upper to zero, then $\operatorname{ht}_{D[X]}Q\leq 1\leq 2n$.
Otherwise by [6, Theorem 2.3 (e)], there exists a
quasi-$\widetilde{\star}$-maximal ideal $P$ of $D$ such that $Q=P[X]$. Hence
by [26, Theorem 38] we obtain that $\operatorname{ht}_{D[X]}Q\leq
2(\operatorname{ht}_{D}(P))\leq 2n$. Consequently we have
$[\star]\text{-}\operatorname{dim}(D[X])\leq 2n$.
Now suppose that $D$ is a $\widetilde{\star}$-Noetherian domain or a
P$\star$MD. We know that
$\widetilde{\star}\text{-}\operatorname{dim}(D)\leq[\star]$-$\operatorname{dim}(D[X])$.
Let $M$ be an arbitrary quasi-$[\star]$-maximal ideal of $D[X]$. Then $M$ is
either an upper to zero, or $M=P[X]$ for some
$P\in\operatorname{QMax}^{\widetilde{\star}}(D)$ by [6, Theorem 2.3 (e)]. Note
that in either case $D_{P}$ is a Noetherian domain by [10, Proposition 3.8]
(resp. a valuation domain by [13, Theorem 3.1]). Hence:
$\displaystyle\operatorname{ht}_{D[X]}P[X]=$
$\displaystyle\operatorname{dim}(D[X]_{P[X]})=\operatorname{dim}(D_{P}[X]_{PD_{P}[X]})$
$\displaystyle\leq$
$\displaystyle\operatorname{dim}(D_{P}[X])-\operatorname{dim}(D_{P}[X]/PD_{P}[X])$
$\displaystyle=$
$\displaystyle\operatorname{dim}(D_{P}[X])-\operatorname{dim}((D_{P}/PD_{P})[X])$
$\displaystyle=$
$\displaystyle\operatorname{dim}(D_{P})\leq\widetilde{\star}\text{-}\operatorname{dim}(D).$
The fourth equality holds since $D_{P}$ is a Noetherian domain and [17,
Theorem 30.5] (resp. a valuation domain and [36, Theorem 4]) and the second
inequality holds by [10, Lemma 2.11]. So that by [10, Lemma 2.11] we obtain
that
$[\star]\text{-}\operatorname{dim}(D[X])\leq\widetilde{\star}\text{-}\operatorname{dim}(D)$,
which ends the proof. ∎
Analogous to Seidenberg, in [38, Theorem 2.10], Wang, showed that for any pair
of positive integers $(n,m)$ with $1\leq n\leq m\leq 2n$, there exists a
domain $D$ such that $w_{D}$-$\operatorname{dim}(D)=n$ and
$w_{D[X]}$-$\operatorname{dim}(D[X])=m$. Note that $[w_{D}]=w_{D[X]}$ by [6,
Theorem 2.3].
###### Remark 3.9.
Let $D$ be an integral domain which is $w_{D}$-Noetherian and of
$w_{D}$-dimension $n$. Then
$[w_{D}]\text{-}\operatorname{dim}(D[X])=w_{D[X]}\text{-}\operatorname{dim}(D[X])=n$
by Proposition 3.8, while $w_{D}[X]\text{-}\operatorname{dim}(D[X])=n+1$ by
Theorem 3.2. This means that $w_{D}[X]\neq w_{D[X]}(=[w_{D}])$. Actually
noting Part $(c)$ of Theorem 2.1, we have $w_{D}[X]\lneq[w_{D}]$.
## 4\. Semistar-valuative dimension
It is worth reminding the reader of the nice behavior of the valuative
dimension with respect to polynomial rings, in the sense that
$\operatorname{dim}_{v}(D[X_{1},\cdots,X_{n}])=\operatorname{dim}_{v}(D)+n$
for each positive integer $n$ and each ring $D$ ([23, Theorem 2]). In this
section we define the _semistar-valuative dimension_ of integral domains and
derive its properties.
For this section we need to recall the notion of $\star$-valuation overring (a
notion due essentially to P. Jaffard [22, page 46]). For a domain $D$ and a
semistar operation $\star$ on $D$, we say that a valuation overring $V$ of $D$
is a _$\star$ -valuation overring of $D$_ provided $F^{\star}\subseteq FV$,
for each $F\in f(D)$. Note that, by definition, the $\star$-valuation
overrings coincide with the $\star_{f}$-valuation overrings. By [14, Theorem
3.9], $V$ is a $\widetilde{\star}$-valuation overring of $D$ if and only if
$V$ is a valuation overring of $D_{P}$ for some quasi-$\star_{f}$-maximal
ideal $P$ of $D$. Also $V$ is a $\star$-valuation overring of $D$ if and only
if $V^{\star_{f}}=V$, (cf. [10, Page 34]).
Let $R$ be a Bézout domain. Then each (nonzero) finitely generated ideal of
$R$ is principal. So that if $J$ is a nonzero finitely generated ideal of $R$,
then $J=J^{t}$ , and hence each nonzero ideal of $R$ is a $t$-ideal. This
implies that the $d_{R}$-operation on $R$ is a unique (semi)star operation of
finite type on $R$. Therefore every (semi)star operation of finite type on a
valuation domain, is the trivial identity operation. The following result is
the key lemma in this section.
###### Lemma 4.1.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$W$ is a valuation overring of $D[X]$. Then $W$ is a $\star[X]$-valuation
overring of $D[X]$, if and only if $W\cap K$ is a
$\widetilde{\star}$-valuation overring of $D$.
###### Proof.
$(\Rightarrow)$. Suppose that $W$ is a $\star[X]$-valuation overring of
$D[X]$. Then by [14, Theorem 3.9], there exists a
$Q\in\operatorname{QMax}^{\star[X]}(D[X])$, such that $D[X]_{Q}\subseteq W$.
Put $P:=Q\cap D$. Note that $D[X]_{Q}=D_{P}[X]_{QD_{P}[X]}$. Therefore
$D_{P}[X]\subseteq W$, and hence $D_{P}\subseteq W\cap K$. If $P=0$, then
$K=W\cap K$, and hence clearly $W\cap K$ is a $\widetilde{\star}$-valuation
overring of $D$. If $P\neq 0$, then $P^{\widetilde{\star}}=(Q\cap
D)^{\widetilde{\star}}\varsubsetneq D^{\widetilde{\star}}$ by Remark 2.3.
Hence $P\in\operatorname{QSpec}^{\widetilde{\star}}(D)$. Choose a
quasi-$\widetilde{\star}$-maximal ideal $M$ of $D$ containing $P$ by [14,
Lemma 2.3 (1)]. So that $D_{M}\subseteq D_{P}\subseteq W\cap K$. Therefore
$W\cap K$ is a $\widetilde{\star}$-valuation overring of $D$ by [14, Theorem
3.9].
$(\Leftarrow)$. Let $M$ be the maximal ideal of $W$, and set $Q:=M\cap D[X]$.
We need to show that $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$. Note
that $M\cap K$ is the maximal ideal of $W\cap K$ by [17, Theorem 19.16]. Since
$W\cap K$ is a $\widetilde{\star}$-valuation overring of $D$, we have $(W\cap
K)^{\widetilde{\star}}=W\cap K$ by [10, Page 34]. Thus
$\widetilde{\star}_{\iota}$ is a (semi)star operation of finite type by [33,
Proposition 3.4], on $W\cap K$, where $\iota$ is the canonical inclusion of
$D$ to $W\cap K$. So that since $W\cap K$ is a valuation domain it is the
identity operation. Put $P:=Q\cap D=(M\cap K)\cap D$. If $P=0$ then by
construction of $\star[X]$, $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$.
So assume that $P\neq 0$. Now we show that $P^{\widetilde{\star}}\neq
D^{\widetilde{\star}}$. If not
$D^{\widetilde{\star}}=P^{\widetilde{\star}}=((M\cap K)\cap
D)^{\widetilde{\star}}=(M\cap K)^{\widetilde{\star}}\cap
D^{\widetilde{\star}}=(M\cap K)\cap D^{\widetilde{\star}}.$
Hence $D^{\widetilde{\star}}\subseteq M\cap K$ and therefore, intersecting
with $D$ we find that $D=M\cap D$, which is a contradiction. Now using Remark
2.3, we see that $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$. Now choose a
quasi-$\star[X]$-maximal ideal $\mathcal{M}$ of $D[X]$ containing $Q$. Thus we
have $D[X]_{\mathcal{M}}\subseteq D[X]_{Q}\subseteq W$. Consequently by [14,
Theorem 3.9], we obtain that $W$ is a
$(\widetilde{\star[X]}=)\star[X]$-valuation overring of $D[X]$. ∎
The following theorem is one of the main results of this section, whose proof
is based on that of [17, Theorem 30.8]. First, we need the following
definition. Let $D$ be a domain and $T$ an overring of $D$. Let $\star$ and
$\star^{\prime}$ be semistar operations on $D$ and $T$, respectively. One says
that $T$ is _$(\star,\star^{\prime})$ -linked to_ $D$ (or that $T$ is a
$(\star,\star^{\prime})$-linked overring of $D$) if
$F^{\star}=D^{\star}\Rightarrow(FT)^{\star^{\prime}}=T^{\star^{\prime}}$
for each nonzero finitely generated ideal $F$ of $D$. It was proved in [9,
Theorem 3.8] that $T$ is $(\star,\star^{\prime})$-linked to $D$ if and only if
$\operatorname{Na}(D,\star)\subseteq\operatorname{Na}(T,\star^{\prime})$.
###### Theorem 4.2.
Let $\star$ be a semistar operation on an integral domain $D$, and let $n$ be
an integer. Then the following statements are equivalent:
* (1)
Each $(\star,\star^{\prime})$-linked overring $T$ of $D$ has
$\widetilde{\star^{\prime}}$-dimension at most $n$, whenever $\star^{\prime}$
is a semistar operation on $T$.
* (2)
Each $(\star,w_{T})$-linked overring $T$ of $D$ has $w_{T}$-dimension at most
$n$.
* (3)
Each overring $T$ of $D$ has $\widetilde{\star}_{\iota}$-dimension at most
$n$, where $\iota:D\to T$ is the canonical inclusion.
* (4)
Each $\widetilde{\star}$-valuation overring of $D$ has dimension at most $n$.
* (5)
For each finite subset $\\{t_{i}\\}_{i=1}^{n}$ of $K$,
$\widetilde{\star}_{\iota}$-$\operatorname{dim}(D[t_{1},\cdots,t_{n}])\leq n$,
where $\iota:D\to D[t_{1},\cdots,t_{n}]$ is the canonical inclusion.
* (6)
For each finite subset $\\{t_{i}\\}_{i=1}^{n}$ of $K$, such that
$D[t_{1},\cdots,t_{n}]$ is a $(\star,\star^{\prime})$-linked overring of $D$,
$\widetilde{\star^{\prime}}$-$\operatorname{dim}(D[t_{1},\cdots,t_{n}])\leq
n$, whenever $\star^{\prime}$ is a semistar operation on
$D[t_{1},\cdots,t_{n}]$.
* (7)
$\star[X_{1},\cdots,X_{n}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{n}])\leq
2n$.
###### Proof.
$(1)\Rightarrow(2)$, $(1)\Rightarrow(3)$, $(1)\Rightarrow(6)$,
$(3)\Rightarrow(5)$ and $(6)\Rightarrow(5)$ are trivial.
$(2)\Rightarrow(4)$. By [10, Lemma 2.7], $V$ is a
${\widetilde{\star}}$-valuation overring of $D$ if and only if $V$ is a
$({\widetilde{\star}},d_{V})$-linked valuation overring of $D$. The assertion
therefore follows since $w_{V}=d_{V}$ for a valuation domain.
$(4)\Rightarrow(3)$. Suppose the contrary. So there exists an overring $T$ of
$D$ containing $P_{0}\subset P_{1}\subset\cdots\subset P_{n}$, of
quasi-$\widetilde{\star}_{\iota}$-prime ideals of $T$, where $\iota:D\to T$ is
the canonical inclusion. Actually one can choose $P_{n}$ so that
$P_{n}\in\operatorname{QMax}^{\widetilde{\star}_{\iota}}(T)$. Consider the
chain $P_{0}T_{P_{n}}\subset P_{1}T_{P_{n}}\subset\cdots\subset
P_{n}T_{P_{n}}$ of distinct prime ideals of $T_{P_{n}}$. Using [17, Corollary
19.7], there exists a valuation overring $V$ of $T_{P_{n}}$, such that $V$
contains a chain $M_{0}\subset M_{1}\subset\cdots\subset M_{n}$ of prime
ideals of $V$ and $M_{i}\cap T_{P_{n}}=P_{i}T_{P_{n}}$. Since
$P_{n}\in\operatorname{QMax}^{\widetilde{\star}_{\iota}}(T)$ and $V$ is an
overring of $T_{P_{n}}$, we obtain that $V$ is a
$\widetilde{\star}_{\iota}$-valuation overring of $T$, by [14, Theorem 3.9].
So that $V^{\widetilde{\star}_{\iota}}=V$, (see [10, Page 34]). Hence
$V^{\widetilde{\star}}=V$. Therefore $V$ is a $\widetilde{\star}$-valuation
overring of $D$ (see [10, Page 34]) and $\operatorname{dim}(V)>n$, which is
impossible.
$(5)\Rightarrow(3)$. Suppose there exists an overring $T$ of $D$ containing a
chain $P_{0}\subset P_{1}\subset\cdots\subset P_{n}$ of
quasi-$\widetilde{\star}_{\iota}$-prime ideals of $T$, where $\iota:D\to T$ is
the canonical inclusion. Choose $t_{i}\in P_{i}\backslash P_{i-1}$, for each
$i=1,\cdots,n$. If $D^{\prime}=D[t_{1},\cdots,t_{n}]$, then
$(0)\subseteq P_{0}\cap D^{\prime}\subset P_{1}\cap
D^{\prime}\subset\cdots\subset P_{n}\cap D^{\prime}\subset D^{\prime}.$
And since $T$ is an overring of $D^{\prime}$, $P_{0}\cap D^{\prime}\neq 0$.
Indeed let $r/s\in P_{0}$, where $r,s\in D\backslash\\{0\\}$. Then $r=s(r/s)$
is an element of $P_{0}\cap D^{\prime}$. On the other hand each $P_{i}\cap
D^{\prime}$ is a quasi-$\widetilde{\star}_{\iota}$-prime ideals of
$D^{\prime}$, where $\iota:D\to D^{\prime}$ is the canonical inclusion. More
precisely
$(P_{i}\cap D^{\prime})^{\widetilde{\star}_{\iota}}\cap D^{\prime}=(P_{i}\cap
D^{\prime})^{\widetilde{\star}}\cap D^{\prime}=P_{i}^{\widetilde{\star}}\cap
D^{\prime\widetilde{\star}}\cap D^{\prime}=P_{i}^{\widetilde{\star}}\cap
D^{\prime}=$ $P_{i}^{\widetilde{\star}}\cap(T\cap
D^{\prime})=(P_{i}^{\widetilde{\star}}\cap T)\cap
D^{\prime}=(P_{i}^{\widetilde{\star}_{\iota}}\cap T)\cap D^{\prime}=P_{i}\cap
D^{\prime}.$
Therefore
$\widetilde{\star}_{\iota}$-$\operatorname{dim}(D[t_{1},\cdots,t_{n}])>n$,
which is a contradiction.
$(3)\Rightarrow(4)$. Let $V$ be a $\widetilde{\star}$-valuation overring of
$D$. Hence we have $V^{\widetilde{\star}}=V$ by [10, Page 34]. This means that
$\widetilde{\star}_{\iota}$ is a (semi)star operation on $V$, where
$\iota:D\to V$ is the canonical inclusion. Note that since
$\widetilde{\star}_{\iota}$ is of finite type, then it is the identity
operation on the valuation domain $V$. Thus
$\operatorname{dim}(V)=\widetilde{\star}_{\iota}$-$\operatorname{dim}(V)\leq
n$.
$(4)\Rightarrow(1)$. Suppose the contrary. So there exists a
$(\star,\star^{\prime})$-linked overring $T$ of $D$ containing a chain
$P_{0}\subset P_{1}\subset\cdots\subset P_{n}$ of
quasi-$\widetilde{\star^{\prime}}$-prime ideals of $T$. By the same reasoning
as in the proof of $(4)\Rightarrow(3)$, there exists a
$\widetilde{\star^{\prime}}$-valuation overring $V$ of $T$ with
$\operatorname{dim}(V)>n$. Thus, by [10, Lemma 2.7], $V$ is a
$(\widetilde{\star^{\prime}},d_{V})$-linked overring of $T$. Since linked-ness
is a transitive relation ([9, Theorem 3.8]), $V$ is a
$({\widetilde{\star}},d_{V})$-linked overring of $D$. Consequently $V$ is a
$\widetilde{\star}$-valuation overring of $D$, which is impossible.
So we showed that $(1)-(6)$ are equivalent.
$(4)\Rightarrow(7)$. To prove
$\star[X_{1},\cdots,X_{n}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{n}])\leq
2n$, it suffices in view of what we have just shown, to prove that each
$\star[X_{1},\cdots,X_{n}]$-valuation overring $W$ of $D[X_{1},\cdots,X_{n}]$
has dimension at most $2n$. Thus by Lemma 4.1, $W\cap K$ is a
$\widetilde{\star}$-valuation overring of $D$. So that
$\operatorname{dim}(W\cap K)\leq n$. Then by [17, Theorem 20.7], we have
$\operatorname{dim}(W)\leq 2n$.
$(7)\Rightarrow(5)$. We consider a subset $\\{t_{i}\\}_{i=1}^{n}$ of $K$. If
$Q_{0}$ is the kernel of the $D$-homomorphism
$\varphi:D[X_{1},\cdots,X_{n}]\to D[t_{1},\cdots,t_{n}]$, sending $X_{i}$ to
$t_{i}$, then [17, Lemma 30.7], shows that $\operatorname{ht}(Q_{0})=n$. Note
that $D[t_{1},\cdots,t_{n}]\cong D[X_{1},\cdots,X_{n}]/Q_{0}$. Suppose that
$\beta\in\operatorname{QSpec}^{\widetilde{\star}_{\iota}}(D[t_{1},\cdots,t_{n}])$
is such that
$\operatorname{ht}(\beta)=\widetilde{\star}_{\iota}\text{-}\operatorname{dim}(D[t_{1},\cdots,t_{n}]),$
where $\iota:D\to D[t_{1},\cdots,t_{n}]$ is the canonical inclusion. There
exists a prime ideal $Q$ of $D[X_{1},\cdots,X_{n}]$, such that
$\beta=\varphi(Q)\cong Q/Q_{0}$. We claim that $Q$ is a
quasi-$\star[X_{1},\cdots,X_{n}]$-prime ideal of $D[X_{1},\cdots,X_{n}]$. To
this end set $P:=\beta\cap D$, which, by the same argument as in the proof of
part $(5)\Rightarrow(3)$, is a quasi-$\widetilde{\star}$-prime ideal of $D$.
Note that $Q\cap D=\beta\cap D=P$. Therefore $(Q\cap
D)^{\widetilde{\star}}=P^{\widetilde{\star}}\subsetneq D^{\widetilde{\star}}$.
Then by repeated applications of Remark 2.3, we claim that $Q$ is a
quasi-$\star[X_{1},\cdots,X_{n}]$-prime ideal of $D[X_{1},\cdots,X_{n}]$. This
means that $\operatorname{ht}(Q)\leq 2n$ by the hypothesis. Thus we have
$\widetilde{\star}_{\iota}\text{-}\operatorname{dim}(D[t_{1},\cdots,t_{n}])=\operatorname{ht}(\beta)=\operatorname{ht}(Q/Q_{0})\leq\operatorname{ht}(Q)-\operatorname{ht}(Q_{0})\leq
2n-n=n,$
which ends the proof. ∎
In [23] Jaffard defines the _valuative dimension_ , denoted
$\operatorname{dim}_{v}(D)$, of the domain $D$ to be the maximal rank of the
valuation overrings of $D$. Now we make the following definition.
###### Definition 4.3.
Let $\star$ be a semistar operation on an integral domain $D$. We say that $D$
has _$\star$ -valuative dimension_ $n$, and we write
$\star$-$\operatorname{dim}_{v}(D)=n$, if each $\star$-valuation overring of
$D$ has dimension at most $n$ and if there exists a $\star$-valuation overring
of $D$ of dimension $n$. If no such integer exists, we say that the
$\star$-valuative dimension of $D$ is infinite.
Note that $d_{D}$-$\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(D)$. Since
by definition, the $\star$-valuation overrings coincide with the
$\star_{f}$-valuation overrings we have
$\star_{f}$-$\operatorname{dim}_{v}(D)=\star$-$\operatorname{dim}_{v}(D)$. In
particular
$t_{D}$-$\operatorname{dim}_{v}(D)=v_{D}$-$\operatorname{dim}_{v}(D)$. Suppose
that $\star_{1}$ and $\star_{2}$ are two semistar operations on an integral
domain $D$, such that $\star_{1}\leq\star_{2}$. If $V$ is a
$\star_{2}$-valuation overring of $D$, then for each $F\in f(D)$ we have
$F^{\star_{1}}\subseteq F^{\star_{2}}\subseteq FV$. Hence $V$ is a
$\star_{1}$-valuation overring of $D$ by definition. So we have:
$\star_{2}\text{-}\operatorname{dim}_{v}(D)\leq\star_{1}\text{-}\operatorname{dim}_{v}(D).$
Using [17, Corollary 19.7] together with [14, Theorem 3.9], one can easily see
that
$\widetilde{\star}$-$\operatorname{dim}(D)\leq\widetilde{\star}$-$\operatorname{dim}_{v}(D)$.
The following example shows that this inequality is not true in general.
###### Example 4.4.
Let $(D,M)$ be a two dimensional local Noetherian domain and suppose that
$0\subsetneq P\subsetneq M$ be the corresponding chain of prime ideals. Let
$(T_{1},N_{1})$ and $(T_{2},N_{2})$ be two rank one discrete valuation rings
[8] dominating the local rings $D_{P}$ and $D$ respectively. Let $\star$ be a
semistar operation on $D$ defined by $E^{\star}=ET_{1}\cap ET_{2}$ for each
$E\in\overline{\mathcal{F}}(D)$. Then clearly $\star=\star_{f}$. We show that
$P,M\in\operatorname{QSpec}^{\star}(D)$. Indeed there exists a positive
integer $k$ such that $PT_{1}=N_{1}^{k}$. Hence $P\subseteq P^{\star}\cap
D=PT_{1}\cap PT_{2}\cap D\subseteq PT_{1}\cap D=N_{1}^{k}\cap D\subseteq
N_{1}\cap D=P$. Therefore $P^{\star}\cap D=P$. By the same way $M^{\star}\cap
D=M$. Therefore we have $\star$-$\operatorname{dim}(D)=2$. Now we compute
$\star$-$\operatorname{dim}_{v}(D)$. Suppose that $V$ is a $\star$-valuation
overring of $D$. Thus in particular we have $D^{\star}\subseteq DV$ that is
$T_{1}\cap T_{2}\subseteq V$. Using [17, Theorem 26.1] we obtain that
$T_{1}\subseteq V$ or $T_{2}\subseteq V$. Consequently
$\operatorname{dim}V\leq 1$. This means that
$\star$-$\operatorname{dim}_{v}(D)=1$. Thus we have
$2=\star\text{-}\operatorname{dim}(D)>\star\text{-}\operatorname{dim}_{v}(D)=1.$
Note that $\widetilde{\star}=d_{D}$. So that we have
$\widetilde{\star}\lneq\star$ and
$1=\star\text{-}\operatorname{dim}_{v}(D)<\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=2$.
By a slight modification of Theorem 4.2, we have:
###### Theorem 4.5.
Let $\star$ be a semistar operation on an integral domain $D$, and let $n$ be
an integer. Then the following statements are equivalent:
* (1)
Each $(\star,\star^{\prime})$-linked overring $T$ of $D$ has
$\widetilde{\star^{\prime}}$-dimension at most $n$, and $n$ is minimal,
whenever $\star^{\prime}$ is a semistar operation on $T$.
* (2)
Each $(\star,w_{T})$-linked overring $T$ of $D$ has $w_{T}$-dimension at most
$n$, and $n$ is minimal.
* (3)
Each overring $T$ of $D$ has $\widetilde{\star}_{\iota}$-dimension at most
$n$, and $n$ is minimal, where $\iota:D\to T$ is the canonical inclusion.
* (4)
$\widetilde{\star}$-$\operatorname{dim}_{v}(D)=n$.
* (5)
For each finite subset $\\{t_{i}\\}_{i=1}^{n}$ of $K$,
$\widetilde{\star}_{\iota}$-$\operatorname{dim}(D[t_{1},\cdots,t_{n}])\leq n$,
and $n$ is minimal, where $\iota:D\to D[t_{1},\cdots,t_{n}]$ is the canonical
inclusion.
* (6)
For each finite subset $\\{t_{i}\\}_{i=1}^{n}$ of $K$, such that
$D[t_{1},\cdots,t_{n}]$ is a $(\star,\star^{\prime})$-linked overring of $D$,
$\widetilde{\star^{\prime}}$-$\operatorname{dim}(D[t_{1},\cdots,t_{n}])\leq
n$, and $n$ is minimal, whenever $\star^{\prime}$ is a semistar operation on
$D[t_{1},\cdots,t_{n}]$.
* (7)
$\star[X_{1},\cdots,X_{n}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{n}])=2n$.
###### Corollary 4.6.
Let $\star$ be a semistar operation on an integral domain $D$. If $D$ is a
$\widetilde{\star}$-Notherian domain of $\widetilde{\star}$-dimension $n$,
then $\widetilde{\star}$-$\operatorname{dim}_{v}(D)=n$.
###### Proof.
By Theorem 3.2, we know
$\star[X_{1},\cdots,X_{n}]\text{-}\operatorname{dim}(D[X_{1},\cdots,X_{m}])=2n$.
Hence $\widetilde{\star}$-$\operatorname{dim}_{v}(D)=n$. ∎
Let $D$ be a P$\star$MD. Since for each
$M\in\operatorname{QMax}^{\star_{f}}(D)$, $D_{M}$ is a valuation domain by
[13, Theorem 3.1], we have
$\widetilde{\star}$-$\operatorname{dim}(D)=\widetilde{\star}$-$\operatorname{dim}_{v}(D)$.
For an integer $r$, it is convenient to put $\star[r]$ to denote
$\star[X_{1},\cdots,X_{r}]$ and $D[r]$ to denote $D[X_{1},\cdots,X_{r}]$,
where $X_{1},\cdots,X_{r}$ are indeterminates over $D$.
###### Corollary 4.7.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}_{v}(D)=k$. Then
$\star[r]$-$\operatorname{dim}(D[r])=\star[r]$-$\operatorname{dim}_{v}(D[r])$,
for each $r\geq k$.
###### Proof.
Theorem 4.5 shows that $\star[k]$-$\operatorname{dim}(D[k])=2k$. Since
$D[r]=D[k][X_{k+1},\cdots,X_{r}]$, it follows that:
$\star[r]\text{-}\operatorname{dim}(D[r])\geq\operatorname{dim}(D[k])+r-k=2k+r-k=r+k.$
If $V$ is a $\star[r]$-valuation overring of $D[r]$, then $V\cap K$ is a
$\widetilde{\star}$-valuation overring of $D$ by Lemma 4.1. So that by [17,
Theorem 20.7], we have $\operatorname{dim}(V)\leq\operatorname{dim}(V\cap
K)+r\leq k+r$. Consequently $\star[r]$-$\operatorname{dim}_{v}(D[r])\leq k+r$.
Since
$\star[r]$-$\operatorname{dim}(D[r])\leq\star[r]$-$\operatorname{dim}_{v}(D[r])$
is always valid, we obtain that
$\star[r]$-$\operatorname{dim}(D[r])=\star[r]$-$\operatorname{dim}_{v}(D[r])=k+r=\widetilde{\star}$-$\operatorname{dim}_{v}(D)+r$.
∎
###### Theorem 4.8.
Let $\star$ be a semistar operation on an integral domain $D$. Then:
$\star[m]\text{-}\operatorname{dim}_{v}(D[m])=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)+m.$
###### Proof.
Put $n:=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)$. If $W$ is a
$\star[m]$-valuation overring of $D[m]$, then by Lemma 4.1, $W\cap K$ is a
$\widetilde{\star}$-valuation overring of $D$. So that
$\operatorname{dim}(W\cap K)\leq n$. Therefore [17, Theorem 20.7], shows that
$\operatorname{dim}(W)\leq n+m$. Consequently
$\star[m]\text{-}\operatorname{dim}_{v}(D[m])\leq n+m$.
But by assumption, there exists a $\widetilde{\star}$-valuation overring $V$
of $D$ of rank $n$. So that by [17, Remark 20.4], $V$ has an extension to a
valuation domain $W$ on $K(X_{1},\cdots,X_{m})$, with
$\operatorname{dim}(W)=n+m$ and such that $\\{X_{1},\cdots,X_{m}\\}$ is
contained in the maximal ideal of $W$. Therefore $W$ is a valuation overring
of $D[m]$ of dimension $n+m$. Since $V=W\cap K$ is a
$\widetilde{\star}$-valuation overring of $D$, Lemma 4.1 shows that $W$ is a
$\star[m]$-valuation overring of $D[m]$. So that
$\star[m]\text{-}\operatorname{dim}_{v}(D[m])\geq n+m$. Thus we have
$\star[m]\text{-}\operatorname{dim}_{v}(D[m])=n+m,$
which is the desired equality. ∎
###### Corollary 4.9.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}(D)=\widetilde{\star}$-$\operatorname{dim}_{v}(D)<\infty$.
If $n$ is a positive integer, then
$\star[n]\text{-}\operatorname{dim}(D[n])=\star[n]\text{-}\operatorname{dim}_{v}(D[n])=n+\widetilde{\star}\text{-}\operatorname{dim}(D).$
Let $\star$ be a semistar operation on an integral domain $D$. Recall that the
$\star$-closure of $D$, defined by:
$D^{cl^{\star}}:=\bigcup\\{(F^{\star}:F^{\star})|F\in f(D)\\}$
is an integrally closed overring of $D$ and, more precisely,
$D^{cl^{\star}}=\bigcap\\{V|V\text{ is a }\star\text{-valuation overring of
}D\\}$. For more details on this subject and for the proof of the result
recalled above, see [31], [18], [15, Proposition 3.2 and Corollary 3.6]. Set
$\widetilde{D}:=D^{cl^{\widetilde{\star}}}$ and
$*:=\widetilde{\star}_{\iota}$, where $\iota:D\to\widetilde{D}$ is the
canonical embedding. Note that $\widetilde{*}=*$ by [33, Proposition 3.1].
###### Proposition 4.10.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$D$ is a $\widetilde{\star}$-quasi-Prüfer domain. Then
$\widetilde{\star}\text{-}\operatorname{dim}(D)=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D).$
###### Proof.
Recall from [7, Theorem 2.16] that, $D$ is a $\widetilde{\star}$-quasi-Prüfer
domain if and only if $\operatorname{Na}(\widetilde{D},*)$ is a Prüfer domain,
that is $\widetilde{D}$ is a P$*$MD by [13, Theorem 3.1]. Since
$\widetilde{D}$ is a P$*$MD, we have
$*$-$\operatorname{dim}(\widetilde{D})=*$-$\operatorname{dim}_{v}(\widetilde{D})$.
Also an easy application of [7, Lemma 2.15], yields us that
$\widetilde{\star}$-$\operatorname{dim}(D)=*$-$\operatorname{dim}(\widetilde{D})$.
So
$\widetilde{\star}\text{-}\operatorname{dim}(D)=*\text{-}\operatorname{dim}(\widetilde{D})=*\text{-}\operatorname{dim}_{v}(\widetilde{D})=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D).$
The last equality holds true since by [15, Corollary 3.6] a valuation domain
is a $\widetilde{\star}$-valuation overring of $D$ if and only if it is a
$*$-valuation overring of $\widetilde{D}$ (see Remark 4.20 for another
reasoning of this equality). ∎
###### Corollary 4.11.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$D$ is a $\widetilde{\star}$-quasi-Prüfer domain (e.g., if $D$ is a
P$\star$MD). Then
$\star[n]\text{-}\operatorname{dim}(D[n])=\star[n]\text{-}\operatorname{dim}_{v}(D[n])=n+\widetilde{\star}\text{-}\operatorname{dim}(D).$
Combining Corollary 4.11 with Theorem 3.5, we obtain the following corollary.
The special case of $\star=d_{D}$ is contained in [36].
###### Corollary 4.12.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}(D)=1$. The following statements are
equivalent:
* (1)
$\star[X]$-$\operatorname{dim}(D[X])=2$.
* (2)
$\star[m]\text{-}\operatorname{dim}(D[m])=m+1$ for any integer $m$.
In [1], to honor Jaffard, the authors defined a domain $D$ to be a _Jaffard
domain_ , in case $\operatorname{dim}(D)=\operatorname{dim}_{v}(D)<\infty$.
The class of Jaffard domains contains most of the well-known classes of finite
dimensional rings involved in dimension theory of commutative rings, such as
Noetherian domains, Prüfer domains, universally catenarian domains [4], and
stably strong S-domains [28, 24]. As the semistar analogue we define:
###### Definition 4.13.
Let $\star$ be a semistar operation on an integral domain $D$. The domain $D$
is said to be a $\widetilde{\star}$-Jaffard domain, if
$\widetilde{\star}$-$\operatorname{dim}(D)<\infty$ and
$\widetilde{\star}$-$\operatorname{dim}(D)=\widetilde{\star}$-$\operatorname{dim}_{v}(D)$.
Note that the notion of $d_{D}$-Jaffard domain coincides with the “classical”
notion of Jaffard domain. Note that $D$ is $\widetilde{\star}$-Jaffard domain
if and only if $\widetilde{\star}$-$\operatorname{dim}(D)<\infty$ and
$\star[r]$-$\operatorname{dim}(D[r])=r+\widetilde{\star}$-$\operatorname{dim}(D)$
for every $r\in\mathbb{N}$. Indeed let
$k=\widetilde{\star}$-$\operatorname{dim}_{v}(D)$. Then by Corollaries 4.8 and
4.7 respectively we have
$k+\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\star[k]\text{-}\operatorname{dim}_{v}(D[k])=\star[k]\text{-}\operatorname{dim}(D[k])=k+\widetilde{\star}\text{-}\operatorname{dim}(D).$
Hence
$\widetilde{\star}$-$\operatorname{dim}(D)=\widetilde{\star}$-$\operatorname{dim}_{v}(D)$.
The converse is true by Corollary 4.9.
Every $\widetilde{\star}$-Noetherian domain and every
$\widetilde{\star}$-quasi-Prüfer domain (e.g., every P$\star$MD) of finite
$\widetilde{\star}$-dimension is a $\widetilde{\star}$-Jaffard domain. As
Theorem 3.5 and Corollary 4.12 show, if $\widetilde{\star}$-dimension is one,
then $\widetilde{\star}$-quasi-Prüfer domains and $\widetilde{\star}$-Jaffard
domains coincide. For the general case, we have the following theorem. See
also [34, Theorem 4.3] for several other characterizations of
$\widetilde{\star}$-quasi-Prüfer domains. The special case of $\star=d_{D}$,
of the following theorem is contained in [3].
###### Theorem 4.14.
Let $\star$ be a semistar operation on an integral domain $D$. Suppose that
$\widetilde{\star}$-$\operatorname{dim}(D)$ is finite. Then the following
statements are equivalent:
* (1)
$D$ is a $\widetilde{\star}$-quasi-Prüfer domain.
* (2)
Each $(\star,\star^{\prime})$-linked overring $T$ of $D$ is a
$\widetilde{\star^{\prime}}$-quasi-Prüfer domain, where $\star^{\prime}$ is a
semistar operation on $T$.
* (3)
Each $(\star,\star^{\prime})$-linked overring $T$ of $D$ is a
$\widetilde{\star^{\prime}}$-Jaffard domain, where $\star^{\prime}$ is a
semistar operation on $T$.
* (4)
Each overring $T$ of $D$ is a $\widetilde{\star}_{\iota}$-Jaffard domain,
where $\iota$ is the canonical embedding of $D$ into $T$.
###### Proof.
$(1)\Rightarrow(2)$. Suppose that $D$ is a $\widetilde{\star}$-quasi-Prüfer
domain. Hence $\operatorname{Na}(D,\star)$ is a quasi-Prüfer domain by [7,
Theorem 2.16]. If $T$ is a $(\star,\star^{\prime})$-linked overring of $D$,
where $\star^{\prime}$ is a semistar operation on $T$, then by [9, Theorem
3.8], we have
$\operatorname{Na}(D,\star)\subseteq\operatorname{Na}(T,\star^{\prime})$.
Consequently $\operatorname{Na}(T,\star^{\prime})$ is a quasi-Prüfer domain by
[12, Corollary 6.5.14]. Therefore $T$ is a $\widetilde{\star^{\prime}}$-quasi-
Prüfer domain by [7, Theorem 2.16].
$(2)\Rightarrow(3)$ and $(3)\Rightarrow(4)$ are trivial.
$(4)\Rightarrow(1)$. In order to show that $D$ is a $\widetilde{\star}$-quasi-
Prüfer domain, it suffices by [7, Theorem 2.16], to show that $D_{P}$ is a
quasi-Prüfer domain for all $P\in\operatorname{QMax}^{\widetilde{\star}}(D)$.
And for this, it suffices to prove that each overring $T$ of $D_{P}$, is a
Jaffard domain by [12, Theorem 6.7.4]. To this end let $P$ be an arbitrary
quasi-$\widetilde{\star}$-maximal ideal of $D$, and $T$ be an overring of
$D_{P}$. Let $V$ be a valuation overring of $T$. Since $D_{P}\subseteq V$, and
$P$ is a quasi-$\widetilde{\star}$-maximal ideal of $D$, we have $V$ is a
$\widetilde{\star}$-valuation overring of $D$ by [14, Theorem 3.9]. Thus
$V^{\widetilde{\star}}=V$ by [10, Page 34]. This means that $V$ is a
$\widetilde{\star}_{\iota}$-valuation overring of $T$ ([10, Page 34]), where
$\iota$ is the canonical embedding of $D$ into $T$. So we obtain that
$\operatorname{dim}_{v}(T)=\widetilde{\star}_{\iota}$-$\operatorname{dim}_{v}(T)$.
Therefore by the hypothesis we have:
$\operatorname{dim}(T)\leq\operatorname{dim}_{v}(T)=\widetilde{\star}_{\iota}\text{-}\operatorname{dim}_{v}(T)=\widetilde{\star}_{\iota}\text{-}\operatorname{dim}(T)\leq\operatorname{dim}(T).$
Thus $\operatorname{dim}(T)=\operatorname{dim}_{v}(T)$, that is $T$ is a
Jaffard domain. Hence $D_{P}$ is a quasi-Prüfer domain for all
$P\in\operatorname{QMax}^{\widetilde{\star}}(D)$, that is $D$ is a
$\widetilde{\star}$-quasi-Prüfer domain. ∎
Recall that if $D$ is a Krull domain then it is a P$v$MD (c.f. [10, Remark
4.2]). Hence from the above theorem, it can be seen that a Krull domain is
$w$-Jaffard. There is an old question (see [5]) asking if is it possible to
find a UFD (or a Krull domain) which is not Jaffard. So, the natural question
is the following: is it possible to find a $w$-Jaffard non Jaffard domain?
Next, we wish to establish that, if $D$ is a $\widetilde{\star}$-Jaffard
domain, then $\operatorname{Na}(D,\star)$ is a Jaffard domain. First we
compute the Krull dimension of the $\star$-Nagata ring.
###### Theorem 4.15.
Let $\star$ be a semistar operation on an integral domain $D$. Then
$\operatorname{dim}(\operatorname{Na}(D,\star))=\star[X]\text{-}\operatorname{dim}(D[X])-1$.
In particular if $D$ is a $\widetilde{\star}$-Jaffard domain, then
$\operatorname{dim}(\operatorname{Na}(D,\star))=\widetilde{\star}\text{-}\operatorname{dim}(D)$.
###### Proof.
Note that if $Q$ is an upper to zero, then, $\operatorname{ht}(Q)\leq 1$. Also
if $Q\in\operatorname{Spec}(D[X])$, and $P:=Q\cap D$, such that
$P[X]\subsetneq Q$, then $\operatorname{ht}(Q)=\operatorname{ht}(P[X])+1$ by
[17, Lemma 30.17]. So we have:
$\displaystyle\star[X]\text{-}\operatorname{dim}(D[X])=$
$\displaystyle\sup\\{\operatorname{ht}(Q)|Q\in\operatorname{QMax}^{\star[X]}(D[X])\\}$
$\displaystyle=$ $\displaystyle\sup\\{\operatorname{ht}(Q)|Q\cap
D\in\operatorname{QMax}^{\widetilde{\star}}(D)\\}$ $\displaystyle=$
$\displaystyle\sup\\{\operatorname{ht}(P[X])+1|P\in\operatorname{QMax}^{\widetilde{\star}}(D)\\}$
$\displaystyle=$
$\displaystyle\sup\\{\operatorname{ht}(P[X])|P\in\operatorname{QMax}^{\widetilde{\star}}(D)\\}+1$
$\displaystyle=$
$\displaystyle\operatorname{dim}(\operatorname{Na}(D,\star))+1.$
For the third equality note that if
$Q\in\operatorname{QMax}^{\star[X]}(D[X])$, and $P:=Q\cap D$, then
$P[X]\subsetneq Q$. Otherwise $Q=P[X]$. Note that
$P\in\operatorname{QSpec}^{\widetilde{\star}}(D)$ (or equal to zero). Due to
the fact that $(P[X]+(X))\cap D=P$, we obtain by Remark 2.3 that
$P[X]+(X)\in\operatorname{QSpec}^{\star[X]}(D[X])$. Since
$P[X]\in\operatorname{QMax}^{\star[X]}(D[X])$ and is contained in $P[X]+(X)$,
we have $P[X]=P[X]+(X)$. Then $(X)\subseteq P[X]$ and therefore
$D=c_{D}((X))\subseteq c_{D}(P[X])\subseteq P$ which is a contradiction. For
the last equality note that
$\operatorname{Max}(\operatorname{Na}(D,\star))=\\{P\operatorname{Na}(D,\star)|P\in\operatorname{QMax}^{\widetilde{\star}}(D)\\}$
[14, Proposition 3.1 (3)]. ∎
Next we compute the valuative dimension of the $\star$-Nagata ring. Before
that, we need some observations and one lemma. Let $D$ be an integral domain
and $\star$ a semistar operation on $D$. One can consider the contraction map
$h:\operatorname{Spec}(\operatorname{Na}(D,\star))\to\operatorname{QSpec}^{\widetilde{\star}}(D)\cup\\{0\\}$.
Indeed if $N$ is a prime ideal of $\operatorname{Na}(D,\star)$, then there
exists a quasi-$\widetilde{\star}$-maximal ideal $M$ of $D$, such that
$N\subseteq M\operatorname{Na}(D,\star)$. So that
$h(N)=N\cap D\subseteq M\operatorname{Na}(D,\star)\cap
D=M\operatorname{Na}(D,\star)\cap K\cap D=M^{\widetilde{\star}}\cap D=M.$
The third equality holds by [14, Proposition 3.4 (3)]. So that
$h(N)\in\operatorname{QSpec}^{\widetilde{\star}}(D)\cup\\{0\\}$, since it is
contained in $M$ and [11, Lemma 4.1 and Remark 4.5]. Note that if
$P\in\operatorname{QSpec}^{\widetilde{\star}}(D)$, then
$h(P\operatorname{Na}(D,\star))=P\operatorname{Na}(D,\star)\cap
D=P\operatorname{Na}(D,\star)\cap K\cap D=P^{\widetilde{\star}}\cap D=P.$
Therefore
$h(\operatorname{Spec}(\operatorname{Na}(D,\star)))=\operatorname{QSpec}^{\widetilde{\star}}(D)\cup\\{0\\}$.
In fact using [7, Theorem 2.16], the map $h$ is bijective if and only if $D$
is a $\widetilde{\star}$-quasi-Prüfer domain.
###### Lemma 4.16.
Let $\star$ be a semistar operation on an integral domain $D$. Then each
valuation overring of $\operatorname{Na}(D,\star)$ is a $\star[X]$-valuation
overring of $D[X]$.
###### Proof.
Let $W$ be a valuation overring of $\operatorname{Na}(D,\star)$. Let $M$ be
the maximal ideal of $W$. Set $\mathfrak{Q}:=M\cap\operatorname{Na}(D,\star)$
and $Q:=M\cap D[X]$. Since
$\mathfrak{Q}\in\operatorname{Spec}(\operatorname{Na}(D,\star))$, we have
$h(\mathfrak{Q})=\mathfrak{Q}\cap D=Q\cap
D\in\operatorname{QSpec}^{\widetilde{\star}}(D)\cup\\{0\\}$. Thus by Remark
2.3, we obtain that $Q$ is a quasi-$\star[X]$-prime ideal of $D[X]$. Now
choose a quasi-$\star[X]$-maximal ideal $\mathcal{M}$ of $D[X]$ containing
$Q$. Thus we have $D[X]_{\mathcal{M}}\subseteq D[X]_{Q}\subseteq W$.
Consequently by [14, Theorem 3.9], we obtain that $W$ is a
$(\widetilde{\star[X]}=)\star[X]$-valuation overring of $D[X]$. ∎
Recall that for each domain $D$,
$\operatorname{dim}_{v}(D)=\sup\\{\operatorname{dim}_{v}(D_{M})|M\in\operatorname{Max}(D)\\}$.
In fact if $n=\operatorname{dim}_{v}(D)$, then there exists a valuation
overring $V$, with maximal ideal $N$, of $D$ such that
$\operatorname{dim}(V)=n$. Put $M:=N\cap D$. So that $V$ is a valuation
overring of $D_{M}$. Hence
$\operatorname{dim}_{v}(D)=n=\operatorname{dim}_{v}(V)\leq\operatorname{dim}_{v}(D_{M})\leq\operatorname{dim}_{v}(D)=n$.
Actually one can assume that $M$ is a maximal ideal of $D$.
Let $\star$ be a semistar operation on an integral domain $D$. Recall from
[16] that the _Kronecker function ring of $D$ with respect to the semistar
operation $\star$_ is defined by:
$\operatorname{Kr}(D,\star):=\left\\{f/g\bigg{|}\begin{array}[]{l}f,g\in
D[X],g\neq 0,\text{ and there exists }h\in D[X]\backslash\\{0\\}\\\ \text{
with }(c(f)c(h))^{\star}\subseteq(c(g)c(h))^{\star}\end{array}\right\\}.$
It is an overring of the $\star$-Nagata ring with quotient field $K(X)$, which
is a Bézout domain [16]. From [15, Theorem 3.5], we have $V$ is
$\star$-valuation overring of $D$ if and only if $V(X)$ is a valuation
overring of $\operatorname{Kr}(D,\star)$. Now we are ready to prove the
following theorem.
###### Theorem 4.17.
Let $\star$ be a semistar operation on an integral domain $D$. Then
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(\operatorname{Na}(D,\star)).$
###### Proof.
Consider the following inequalities:
$\displaystyle\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)\leq\operatorname{dim}_{v}(\operatorname{Kr}(D,\widetilde{\star}))\leq$
$\displaystyle\operatorname{dim}_{v}(\operatorname{Na}(D,\star))$
$\displaystyle\leq$
$\displaystyle\star[X]\text{-}\operatorname{dim}_{v}(D[X])=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)+1.$
The first inequality follows from the fact that if $V$ is a
$\widetilde{\star}$-valuation overring of $D$, then $V(X)$ is a valuation
overring of $\operatorname{Kr}(D,\widetilde{\star})$ and that
$\operatorname{dim}(V)=\operatorname{dim}(V(X))$; second inequality follows
from the fact that
$\operatorname{Na}(D,\star)\subseteq\operatorname{Kr}(D,\widetilde{\star})$,
while the third one uses the Lemma 4.16. So that we can assume that
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)$ and
$\operatorname{dim}_{v}(\operatorname{Na}(D,\star))$ are finite numbers. Now
by observation before the theorem, choose a quasi-$\widetilde{\star}$-maximal
ideal $P$ of $D$, such that the maximal ideal $M:=P\operatorname{Na}(D,\star)$
has the property that
$\operatorname{dim}_{v}(\operatorname{Na}(D,\star))=\operatorname{dim}_{v}(\operatorname{Na}(D,\star)_{M})=\operatorname{dim}_{v}(D_{P}(X)).$
But since $P\in\operatorname{QMax}^{\widetilde{\star}}(D)$, each valuation
overring of $D_{P}$, is a $\widetilde{\star}$-valuation overring of $D$ [14,
Theorem 3.9]. Hence we find the inequality
$\operatorname{dim}_{v}(D_{P})\leq\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)$.
Consequently we have
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)\leq\operatorname{dim}_{v}(\operatorname{Na}(D,\star))=\operatorname{dim}_{v}(D_{P}(X))=\operatorname{dim}_{v}(D_{P})\leq\widetilde{\star}\text{-}\operatorname{dim}_{v}(D),$
in which the second equality holds by [1, Proposition 1.22]. Thus we find the
desired equality
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(\operatorname{Na}(D,\star)).$
∎
As an immediate corollary we have:
###### Corollary 4.18.
Let $\star$ be a semistar operation on an integral domain $D$. Then:
* (a)
$D[X]$ is a $\star[X]$-Jaffard domain, if and only if,
$\operatorname{Na}(D,\star)$ is a Jaffard domain.
* (b)
$D$ is a $\widetilde{\star}$-Jaffard domain if and only if
$\operatorname{Na}(D,\star)$ is a Jaffard domain and
$\star[X]$-$\operatorname{dim}(D[X])=\widetilde{\star}$-$\operatorname{dim}(D)+1$.
###### Proof.
Both statements are easy consequences of Theorems 4.15 and 4.17, and for $(a)$
use also Theorem 4.8. ∎
###### Remark 4.19.
By the proof of the above theorem, we have
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(\operatorname{Kr}(D,\widetilde{\star}))$.
Since $\operatorname{Kr}(D,\widetilde{\star})$ is a Bézout, and hence a Prüfer
domain, we have
$\widetilde{\star}\text{-}\operatorname{dim}_{v}(D)=\operatorname{dim}_{v}(\operatorname{Kr}(D,\widetilde{\star}))=\operatorname{dim}(\operatorname{Kr}(D,\widetilde{\star})).$
###### Remark 4.20.
Let $D$, $\widetilde{D}$, $\star$, and $*$ be as in the Proposition 4.10. Note
that by the proof of part $(6_{\widetilde{\star}})\Rightarrow(10_{\star_{f}})$
of [7, Theorem 2.16] we have
$\operatorname{Na}(\widetilde{D},*)=\overline{\operatorname{Na}(D,\star)}$. So
that by Theorem 4.17 we have
$*\text{-}\operatorname{dim}_{v}(\widetilde{D})=\operatorname{dim}_{v}(\operatorname{Na}(\widetilde{D},*))=\operatorname{dim}_{v}(\overline{\operatorname{Na}(D,\star)})=\operatorname{dim}_{v}(\operatorname{Na}(D,\star))=\widetilde{\star}\text{-}\operatorname{dim}_{v}(D),$
which is another reason for the last equality in the proof of Proposition
4.10.
ACKNOWLEDGMENT
My sincere thanks goes to Muhammad Zafrullah for his useful comments on this
paper and to Marco Fontana for the valuable suggestions regarding Remark 3.9
and discussions on the notes after Theorem 4.14.
## References
* [1] D. F. Anderson, A. Bouvier, D. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domain, Expo. Math., 6, (1988), 145–175.
* [2] J. T. Arnold, On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc., 138, (1969), 313–326.
* [3] A. Ayache, P. Cahen and O. Echi, Anneaux quasi-Prüfériens et P-anneaux, Boll. Un. Mat. Ital., 10-B, (1996), 1–24.
* [4] A. Bouvier, D. E. Dobbs and M. Fontana, Universally catenarian integral domains, Advances in Math., 72 (1988), 211–238.
* [5] A. Bouvier and S. Kabbaj, Examples of Jaffard domains, J. Pure Appl. Algebra, 54, (2-3), (1988), 155–165.
* [6] G.W. Chang and M. Fontana, Uppers to zero and semistar operations in polynomial rings, J. Algebra, 3.18, (2007), 484–493.
* [7] G.W. Chang and M. Fontana, Uppers to zero in polynomial rings and Prüfer-like domains, Comm. Algebra 37 (2009), 164–192.
* [8] C. C. Chevalley, La notion d’anneau de décomposition, Nagoya Math. J. 7, (1954), 21–33.
* [9] S. El Baghdadi and M. Fontana, Semistar linkedness and flatness, Prüfer semistar multiplication domains, Comm. Algebra 32 (2004), 1101–1126.
* [10] S. El Baghdadi, M. Fontana and G. Picozza, Semistar Dedekind domains, J. Pure Appl. Algebra 193 (2004), 27–60.
* [11] M. Fontana and J. A. Huckaba, Localizing systems and semistar operations, in: S. Chapman and S. Glaz (Eds.), Non Noetherian Commutative Ring Theory, Kluwer, Dordrecht, 2000, 169–197.
* [12] M. Fontana, J. Huckaba, and I. Papick, Prüfer domains, Marcel Dekker, 1997.
* [13] M. Fontana, P. Jara and E. Santos, Prüfer $\star$-multiplication domains and semistar operations, J. Algebra Appl. 2 (2003), 21–50.
* [14] M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings and related semistar operations, Comm. Algebra 31 (2003), 4775–4801.
* [15] M. Fontana and K. A. Loper, A Krull-type theorem for semistar integral closure of an integral domain, ASJE Theme Issue “Commutative Algebra” 26 (2001), 89–95.
* [16] M. Fontana and K. A. Loper, Kronecker function rings: a general approach, in: D. D. Anderson and I. J. Papick (Eds.), Ideal Theoretic Methods in Commutative Algebra, Lecture Notes Pure Appl. Math. 220 (2001), Dekker, New York, 189–205.
* [17] R. Gilmer, Multiplicative ideal theory, New York, Dekker, 1972.
* [18] F. Halter-Koch, Generalized integral closures, in “Factorization in Integral Domains” (D.D. Anderson, Editor), Lecture Notes Pure Appl. Math., Marcel Dekker, 187 (1997), 349–358.
* [19] E. Houston, Prime $t$-ideals in $R[X]$, in: P.-J. Cahen, D.G. Costa, M. Fontana, S-E. Kabbaj (Eds.), Commutative Ring Theory, Dekker Lecture Notes 153, 1994, 163–170.
* [20] E. Houston, S. Malik and J. Mott Characterizations of $\star$-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52.
* [21] E. Houston and M. Zafrullah, On $t$-invertibility, II, Comm. Algebra 17 (1989), 1955–1969.
* [22] P. Jaffard, Les Systèmes d’Idéaux, Dunod, Paris, 1960.
* [23] P. Jaffard, Théorie de la dimension dans les anneaux de polynomes, Gauthier-Villars, Paris, 1960.
* [24] S. Kabbaj, Sur les S-domaines forts de Kaplansky, J. Algebra 137 (1991), 400–415.
* [25] F. Halter-Koch. Localizing systems, module systems, and semistar operations J. Algebra, 238, no.2 (2001), 723–761.
* [26] I. Kaplansky, Commutative rings, rev. ed., Univ. Chicago Press, Chicago, 1974.
* [27] W. Krull, Jacobsonsche Rings, Hilbertsche Nullstellensatz, Dimension Theorie, Math. Z. 54 (1951), 354–387.
* [28] S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra, 28 (1983), 249–264.
* [29] M. Nagata, Local rings, Wiley-Interscience, New York, 1962.
* [30] A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1–21.
* [31] A. Okabe and R. Matsuda, Star operations and generalized integral closures, Bull. Fac. Sci., Ibaraki Univ. 24 (1992), 7–13.
* [32] G. Picozza, A note on semistar noetherian domains, Houston J. Math. 33 (2007), 415–432.
* [33] G. Picozza, Star operations on overrings and semistar operations, Comm. Algebra, 33 (2005), 2051–2073.
* [34] P. Sahandi, Universally catenarian integral domains, strong S-domains and semistar operations, Comm. Algebra to appear.
* [35] A. Seidenberg, A note on the dimension theory of rings, Pasific J. Math. 3 (1953), 505–512.
* [36] A. Seidenberg, On the dimension theory of rings II, Pasific J. Math. 4 (1954), 603–614.
* [37] F. G. Wang, On $w$-dimension of domains, Comm. Algebra 27 (1999), 2267–2276.
* [38] F. G. Wang, On $w$-dimension of domains, II, Comm. Algebra 29 (2001), 2419–2428.
|
arxiv-papers
| 2008-09-08T10:42:53
|
2024-09-04T02:48:57.713777
|
{
"license": "Public Domain",
"authors": "Parviz Sahandi",
"submitter": "Parviz Sahandi Dr.",
"url": "https://arxiv.org/abs/0809.1305"
}
|
0809.1307
|
# Minimal prime ideals and semistar operations
Parviz Sahandi Department of Mathematics, University of Tabriz, Tabriz, Iran
and School of Mathematics, Institute for Research in Fundamental Sciences
(IPM), Tehran, Iran. sahandi@tabrizu.ac.ir, sahandi@ipm.ir
###### Abstract.
Let $R$ be a commutative integral domain and let $\star$ be a semistar
operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We
show that, if every minimal prime ideal of $I$ is the radical of a
$\star$-finite ideal, then the set $\operatorname{Min}(I)$ of minimal prime
ideals over $I$ is finite.
###### Key words and phrases:
Star operation, semistar operation, minimal prime ideal.
###### 2000 Mathematics Subject Classification:
13A15, 13G05
## 1\. Introduction
In [12, Theorem 88], Kaplansky proved that: Let $R$ be a commutative ring
satisfying the _ascending chain condition_ (a.c.c. for short) on radical
ideals, and let $I$ be an ideal of $R$. Then there are only a finite number of
prime ideals minimal over $I$.
This result was generalized in [9, Theorem 1.6] by showing that (see also
[1]): Let $R$ be a commutative ring with identity, and let $I\neq R$ be an
ideal of R . If every prime ideal minimal over $I$ is the radical of a
finitely generated ideal, then there are only finitely many prime ideals
minimal over $I$.
In 1994, Okabe and Matsuda [13] introduced the concept of _semistar operation_
to extend the notion of classical _star operations_ as described in [8,
Section 32]. Star operations have been proven to be an essential tool in
_multiplicative ideal theory_ , allowing one to study different classes of
integral domains. Semistar operations, thanks to a higher flexibility than
star operations, permit a finer study and new classifications of special
classes of integral domains.
Throughout this note let $R$ be a commutative integral domain, with identity
and let $K$ be its quotient field.
The purpose of this note is to prove the semistar analogue of Kaplansky’s [12,
Theorem 88] and Gilmer and Heinzer [9, Theorem 1.6] results. More precisely we
prove the following theorem.
Theorem. Let $\star$ be a semistar operation of finite type on the integral
domain $R$, and $I$ be a quasi-$\star$-ideal of $R$. If every minimal prime
ideal of $I$ is the radical of a $\star$-finite ideal, then $I$ has finitely
many minimal prime ideals.
Now we recall some definitions and properties related to semistar operations.
Let $\overline{\mathcal{F}}(R)$ denote the set of all nonzero $R$-submodules
of $K$ and let $\mathcal{F}(R)$ be the set of all nonzero _fractional_ ideals
of $R$, i.e. $E\in\mathcal{F}(R)$ if $E\in\overline{\mathcal{F}}(R)$ and there
exists a nonzero $r\in R$ with $rE\subseteq R$. Let $f(R)$ be the set of all
nonzero finitely generated fractional ideals of $R$. Then, obviously
$f(R)\subseteq\mathcal{F}(R)\subseteq\overline{\mathcal{F}}(R)$. A semistar
operation on $R$ is a map
$\star:\overline{\mathcal{F}}(R)\rightarrow\overline{\mathcal{F}}(R)$,
$E\rightarrow E^{\star}$, such that, for all $x\in K$, $x\neq 0$, and for all
$E,F\in\overline{\mathcal{F}}(R)$, the following properties hold:
* $\star_{1}$
$(xE)^{\star}=xE^{\star}$;
* $\star_{2}$
$E\subseteq F$ implies that $E^{\star}\subseteq F^{\star}$;
* $\star_{3}$
$E\subseteq E^{\star}$ and $E^{\star\star}:=(E^{\star})^{\star}=E^{\star}$,
cf. for instance [13]. Recall that, given a semistar operation $\star$ on $R$,
for all $E,F\in\overline{\mathcal{F}}(R)$, the following basic formulas follow
easily from the axioms:
* (1)
$(EF)^{\star}=(E^{\star}F)^{\star}=(EF^{\star})^{\star}=(E^{\star}F^{\star})^{\star}$;
* (2)
$(E+F)^{\star}=(E^{\star}+F)^{\star}=(E+F^{\star})^{\star}=(E^{\star}+F^{\star})^{\star}$;
* (3)
$(E\cap F)^{\star}\subseteq E^{\star}\cap F^{\star}=(E^{\star}\cap
F^{\star})^{\star}$, if $E\cap F\neq 0$.
cf. for instance [13, Proposition 5].
A _(semi)star operation_ is a semistar operation that, when restricted to
$\mathcal{F}(R)$, is a star operation (in the sense of [8, Section 32]). It is
easy to see that a semistar operation $\star$ on $R$ is a (semi)star operation
if and only if $R^{\star}=R$.
Let $\star$ be a semistar operation on the integral domain $R$. An ideal $I$
of $R$ is called a _quasi- $\star$-ideal_ of $R$ if $I^{\star}\cap R=I$. It is
easy to see that for any ideal $I$ of $R$, the ideal $I^{\star}\cap R$ is a
quasi-$\star$-ideal. An ideal is said to be a _quasi- $\star$-prime_, if it is
prime and a quasi-$\star$-ideal. Let $\star$ be a semistar operation, put
$E^{\star_{f}}=\bigcup F^{\star}$ where the union taken over all finitely
generated $F\subseteq E$, for every $E\in\overline{\mathcal{F}}(R)$. It is
easy to see that $\star_{f}$ defines a semistar operation on $R$ called _the
semistar operation of finite type associated to_ $\star$. Note that there is
the equality $(\star_{f})_{f}=\star_{f}$. A semistar operation $\star$ is said
to be _of finite type_ if $\star=\star_{f}$; in particular $\star_{f}$ is of
finite type. An element $E\in\overline{\mathcal{F}}(R)$ is said _$\star$
-finite_, if $E^{\star}=F^{\star}$ for some $F\in f(R)$. Note that if $I$ is
an ideal of $R$ then $I^{\star}$ is an ideal of the overring $R^{\star}$ of
$R$. Denote by $\star$-$\operatorname{Min}(I)$ the set of _minimal quasi-
$\star$-prime_ ideals over $I$. So that when $\star=d$ is the identity
semistar operation, then $d$-$\operatorname{Min}(I)=\operatorname{Min}(I)$. It
can be seen that if $I$ is an ideal of $R$, then
$\star$-$\operatorname{Min}(I)\subseteq\operatorname{Min}(I^{\star}\cap R)$.
Using [4, Lemma 2.3(d)] we have each minimal prime over a
quasi-$\star_{f}$-ideal is a quasi-$\star_{f}$-ideal. Therefore If $I$ is a
quasi-$\star_{f}$-ideal, we have
$\star_{f}$-$\operatorname{Min}(I)=\operatorname{Min}(I)$.
The most widely studied (semi)star operations on $R$ have been the identity
$d_{R}$, $v_{R}$, and $t_{R}:=(v_{R})_{f}$ operations, where
$E^{v_{R}}:=(E^{-1})^{-1}$, with $E^{-1}:=(R:E):=\\{x\in K|xE\subseteq R\\}$.
Our terminology and notation come from [8].
## 2\. Main result
Before proving the main result of this paper, we need a lemma.
###### Lemma 2.1.
Suppose that $\star$ is a semistar operation of finite type on the integral
domain $R$, and that $I$ is a quasi-$\star$-ideal. Then $\sqrt{I}^{\star}\cap
R=\sqrt{I}$, that is $\sqrt{I}$ is also a quasi-$\star$-ideal of $R$.
###### Proof.
Since $\sqrt{I}\subseteq\sqrt{I}^{\star}\cap R$, it is enough to show that
$\sqrt{I}^{\star}\cap R\subseteq\sqrt{I}$. Let $x\in\sqrt{I}^{\star}\cap R$.
Then for every $P\in\operatorname{Min}(I)$ we have
$xR^{\star}\subseteq\sqrt{I}^{\star}\subseteq P^{\star}$. Since $P$ is a
quasi-$\star$-prime ideal of $R$ by [4, Lemma 2.3(d)], we obtain that $x\in
P$. Hence $x\in\sqrt{I}$ as desired. ∎
###### Remark 2.2.
Suppose that $\star$ is a semistar operation of finite type on the integral
domain $R$, and that $I$ is a quasi-$\star$-ideal. Then $\sqrt{I^{\star}}\cap
R=\sqrt{I}$. Indeed suppose that $x\in\sqrt{I^{\star}}\cap R$, then there is a
positive integer $n$ such that $x^{n}\in I^{\star}$. Since $I$ is a
quasi-$\star$-ideal, and $x\in R$ we obtain that $x^{n}\in I$. Hence
$x\in\sqrt{I}$. Consequently we have $\sqrt{I^{\star}}\cap
R\subseteq\sqrt{I}\subseteq\sqrt{I^{\star}\cap R}=\sqrt{I^{\star}}\cap R$,
which gives us the desired equality.
We next give the main result of this paper.
###### Theorem 2.3.
Let $\star$ be a semistar operation of finite type on the integral domain $R$,
and $I$ be a quasi-$\star$-ideal of $R$. If every minimal prime ideal of $I$
is the radical of a $\star$-finite ideal, then $I$ has finitely many minimal
prime ideals.
###### Proof.
Note that $\sqrt{I}$ is a radical quasi-$\star$-ideal by Lemma 2.1. Hence
since we have $\operatorname{Min}(I)=\operatorname{Min}(\sqrt{I})$, it is
harmless to assume that $I$ is a radical quasi-$\star$-ideal.
Let $S=\\{P_{1}\cdots P_{n}|\text{ each }P_{i}\text{ is a prime ideal minimal
over }I\\}$. If for some $C=P_{1}\cdots P_{n}\in S$ we have
$C^{\star}\subseteq I^{\star}$, hence $C\subseteq C^{\star}\cap R\subseteq
I^{\star}\cap R=I$. Then any prime ideal minimal over $I$ contains some
$P_{i}$, so $\\{P_{1},\cdots,P_{n}\\}$ is the set of minimal prime ideals of
$I$. Hence we may assume that $C^{\star}\nsubseteq I^{\star}$ for each $C\in
S$. Consider the set $\mathcal{A}$ consisting of all radical
quasi-$\star$-ideals $J$ of $R$ containing $I$ such that $C^{\star}\nsubseteq
J^{\star}$ for each $C\in S$. Since $I\in\mathcal{A}$ we have
$\mathcal{A}\neq\emptyset$. The set $\mathcal{A}$ is partially ordered under
inclusion $\subseteq$, and we show that it is inductive under this ordering.
Let $\\{J_{\alpha}\\}_{\alpha\in\Gamma}$ be a chain in $\mathcal{A}$. Put
$J=\cup J_{\alpha}$. So that $J$ is a radical quasi-$\star$-ideal of $R$
containing $I$ such that $J^{\star}=(\cup J_{\alpha})^{\star}=\cup
J_{\alpha}^{\star}$, in which the second equality holds, since $\star$ is a
semistar operation of finite type. Now assume that $C^{\star}\subseteq
J^{\star}$ for some $C\in S$. Suppose that $C=P_{1}\cdots P_{n}$, and that
$P_{i}=\sqrt{L_{i}}$ for some $\star$-finite ideal $L_{i}$ of $R$, for
$i=1,\cdots,n$. Let $F_{i}$s be finitely generated ideals of $R$, such that
$L_{i}^{\star}=F_{i}^{\star}$, for $i=1,\cdots,n$. Now consider $(F_{1}\cdots
F_{n})^{\star}=(L_{1}\cdots L_{n})^{\star}\subseteq(P_{1}\cdots
P_{n})^{\star}\subseteq\cup J^{\star}_{\alpha}$, which implies that
$F_{1}\cdots F_{n}\subseteq\cup J_{\alpha}$. Therefore there exists an index
$\alpha\in\Gamma$ such that $F_{1}\cdots F_{n}\subseteq J_{\alpha}$ and hence
$(L_{1}\cdots L_{n})^{\star}=(F_{1}\cdots F_{n})^{\star}\subseteq
J^{\star}_{\alpha}$. Thus $L_{1}\cdots L_{n}\subseteq J_{\alpha}$, and so we
have $P_{1}\cdots P_{n}\subseteq\sqrt{L_{1}\cdots
L_{n}}\subseteq\sqrt{J_{\alpha}}=J_{\alpha}$. Consequently $(P_{1}\cdots
P_{n})^{\star}\subseteq J^{\star}_{\alpha}$ which is impossible. Now Zorn’s
Lemma gives us a maximal element $Q$ of $\mathcal{A}$. One can actually assume
that $Q\neq R$. We show that $Q$ is a prime ideal of $R$. To this end let $a$,
$b$ be two elements of $R$ such that $ab\in Q$ and assume that $a$, $b\notin
Q$. Since $Q\varsubsetneq(Q+aR)\subseteq\sqrt{(Q+aR)^{\star}\cap R}$, and
$\sqrt{(Q+aR)^{\star}\cap R}$ is a radical quasi-$\star$-ideal (by Lemma 2.1)
containing $I$, there exists an element $C_{1}\in S$ such that
$C_{1}^{\star}\subseteq\sqrt{(J+aR)^{\star}\cap R}^{\star}$. By the same
reason there exists again an element $C_{2}\in S$ such that
$C_{2}^{\star}\subseteq\sqrt{(Q+bR)^{\star}\cap R}^{\star}$. Therefore we have
$\displaystyle(C_{1}C_{2})^{\star}=(C_{1}^{\star}C_{2}^{\star})^{\star}\subseteq$
$\displaystyle(\sqrt{(Q+aR)^{\star}\cap R}^{\star}\sqrt{(Q+bR)^{\star}\cap
R}^{\star})^{\star}$ $\displaystyle=$ $\displaystyle(\sqrt{(Q+aR)^{\star}\cap
R}\sqrt{(Q+bR)^{\star}\cap R})^{\star}$ $\displaystyle=$
$\displaystyle(\sqrt{((Q+aR)^{\star}\cap R)((Q+bR)^{\star}\cap R)})^{\star}$
$\displaystyle\subseteq$
$\displaystyle(\sqrt{((Q+aR)^{\star}(Q+bR)^{\star})\cap R})^{\star}$
$\displaystyle=$
$\displaystyle(\sqrt{((Q^{\star})^{2}+aQ^{\star}+bQ^{\star}+abR^{\star})\cap
R})^{\star}$ $\displaystyle\subseteq$ $\displaystyle(\sqrt{Q^{\star}\cap
R})^{\star}=(\sqrt{Q})^{\star}=Q^{\star},$
which is a contradiction. Therefore $Q$ is a prime ideal of $R$. But since
$I\subseteq Q$, it contains a prime ideal $P$ minimal over $I$ by [12, Theorem
10]. Thus $P\in S$ and $P^{\star}\subseteq Q^{\star}$, a contradiction. ∎
Defining different semistar operations, we can derive different corollaries.
###### Corollary 2.4.
(Gilmer and Heinzer [9, Theorem 1.6] and Anderson [1]) Let $R$ be an integral
domain, and let $I$ be an ideal of $R$. If each minimal prime of the ideal $I$
is the radical of a finitely generated ideal, then $I$ has only finitely many
minimal primes.
The following result proved recently by El Baghdadi and Gabelli [6,
Proposition 1.4] over P$v$MDs. They used the lattice isomorphism between the
lattice of t-ideals of R and the lattice of ideals of the $t$-Nagata ring of
$R$ over P$v$MDs.
###### Corollary 2.5.
Let $I$ be a $t$-ideal of the integral domain $R$. If each minimal prime ideal
of $I$ is the radical of a $t$-finite ideal, then $\operatorname{Min}(I)$ is
finite.
###### Corollary 2.6.
Suppose that $R$ has a Noetherian overring $S$. If $I$ is an ideal of $R$ such
that $IS\cap R=I$, then $\operatorname{Min}(I)$ is finite.
###### Proof.
Define a semistar operation $\star$, by $E^{\star}=ES$, for each
$E\in\overline{\mathcal{F}}(R)$. Thus $I=IS\cap R=I^{\star}\cap R$ is a
quasi-$\star$-ideal of $R$. Note that $\star$ is a semistar operation of
finite type. Let $P\in\operatorname{Min}(I)$. Since $S$ is a Noetherian ring,
we have $P^{\star}=PS=(x_{1},\cdots,x_{n})S=(x_{1},\cdots,x_{n})^{\star}$, for
some elements $x_{1},\cdots,x_{n}$ of $P$. This means that $P$ is a
$\star$-finite ideal. Now the result is clear from Theorem 2.3. ∎
Recall that a semistar operations $\star$ on the integral domain $R$ is called
_stable_ , if $(E\cap F)^{\star}=E^{\star}\cap F^{\star}$ for every $E$, $F$
in $\overline{\mathcal{F}}(R)$. Again recall from the introduction that
$\star$-$\operatorname{Min}(I)$ is the set of quasi-$\star$-prime ideals
minimal over $I$.
###### Lemma 2.7.
Suppose that $\star$ is a semistar operation stable and of finite type on the
integral domain $R$, and that $I$ is a nonzero ideal of $R$, such that
$I^{\star}\cap R\neq R$. Then
$\star$-$\operatorname{Min}(I)=\operatorname{Min}(I^{\star}\cap R)$. In
particular
$\sqrt{I^{\star}}\cap R=\sqrt{I^{\star}\cap
R}=\cap_{P\in\star\text{-}\operatorname{Min}(I)}P.$
###### Proof.
One sees easily that
$\star$-$\operatorname{Min}(I)\subseteq\operatorname{Min}(I^{\star}\cap R)$.
For the reverse inclusion, let $P\in\operatorname{Min}(I^{\star}\cap R)$. So
that $I\subseteq I^{\star}\cap R\subseteq P$. Choose by [12, Theorem 10] a
prime ideal $Q$ minimal over $I$ contained in $P$. Note that $Q$ is a
quasi-$\star$-ideal, since it is contained in $P$ ([7, Corollary 3.9, Lemma
4.1, and Remark 4.5]). Then $I^{\star}\subseteq Q^{\star}\subseteq P^{\star}$
and so $I\subseteq I^{\star}\cap R\subseteq Q^{\star}\cap R=Q\subseteq
P^{\star}\cap R=P$. Thus $Q=P$, since $P$ is minimal over $I^{\star}\cap R$. ∎
###### Corollary 2.8.
Let $\star$ be a semistar operation stable and of finite type on the integral
domain $R$, and $I$ be a nonzero ideal of $R$, such that $I^{\star}\cap R\neq
R$. If every quasi-$\star$-prime ideal minimal over $I$ is the radical of a
$\star$-finite ideal, then $\star$-$\operatorname{Min}(I)$ is finite.
###### Proof.
By the above lemma we have
$\star$-$\operatorname{Min}(I)=\operatorname{Min}(I^{\star}\cap R)$. Thus
every prime ideal minimal over $I^{\star}\cap R$ is the radical of a
$\star$-finite ideal. Noting that $I^{\star}\cap R$ is a quasi-$\star$-ideal
of $R$, and using Theorem 2.3, we have $\star$-$\operatorname{Min}(I)$ is a
finite set. ∎
In [5, Section 3] the authors defined and studied the _semistar Noetherian
domains_ , that is, domains having the ascending chain condition on quasi
semistar ideals. In [14] Picozza generalize several of the classical results
that hold in Noetherian domains to the case of semistar operations stable and
of finite type, for instance, Cohen’s Theorem, primary decomposition,
principal ideal Theorem, Krull intersection Theorem, etc.
###### Corollary 2.9.
([14, Proposition 2.4(2)]) Suppose that $\star$ is a stable semistar operation
of finite type on the integral domain $R$, and that $R$ is a
$\star$-Noetherian domain. Then $\star$-$\operatorname{Min}(I)$ is finite for
every ideal $I$ of $D$.
Next we give equivalent conditions that every quasi-$\star$-prime of $R$ is
the radical of a $\star$-finite ideal.
###### Proposition 2.10.
Let $\star$ be a semistar operation of finite type on the integral domain $R$,
the following then are equivalent:
* (1)
Each quasi-$\star$-prime is the radical of a $\star$-finite ideal.
* (2)
Each radical quasi-$\star$-ideal is the radical of a $\star$-finite ideal.
* (3)
$R$ satisfies the a.c.c. on radical quasi-$\star$-ideals.
###### Proof.
$(1)\Rightarrow(2)$ Consider the following set.
$\mathcal{A}=\\{I|I=\sqrt{I},I=I^{\star}\cap R,\text{ which is not the radical
of a $\star$-finite ideal}\\}.$
If $\mathcal{A}\neq\emptyset$, let $\beta=\\{I_{\alpha}\\}$ be a chain of
elements of $\mathcal{A}$. Put $I=\cup I_{\alpha}$. Hence $I$ is a radical
quasi-$\star$-ideal of $R$ such that $I^{\star}=(\cup I_{\alpha})^{\star}=\cup
I_{\alpha}^{\star}$. Suppose that $I=\sqrt{L}$ for some $\star$-finite ideal
$L$. Let $L^{\star}=F^{\star}$ for some $F\in f(R)$. So that
$F^{\star}=L^{\star}\subseteq I^{\star}=\cup I_{\alpha}^{\star}$ which implies
that $F\subseteq\cup I_{\alpha}$. Therefore there is an index $\alpha$ such
that $F\subseteq I_{\alpha}$. Consequently $L^{\star}=F^{\star}\subseteq
I_{\alpha}^{\star}$ and hence $L\subseteq I_{\alpha}$. So we obtain that
$I_{\alpha}=\sqrt{L}$, which is impossible. Hence $I\in\mathcal{A}$. Thus by
Zorn’s Lemma $\mathcal{A}$ has a maximal element $P$. Let $a$, $b$ be two
elements of $R$ such that $ab\in P$ and suppose that $a$, $b\notin P$. Since
$P\varsubsetneq(P+aR)\subseteq\sqrt{(P+aR)^{\star}\cap R}$, and
$\sqrt{(P+aR)^{\star}\cap R}$ is a radical quasi-$\star$-ideal (by Lemma 2.1),
we have $\sqrt{(P+aR)^{\star}\cap R}=\sqrt{L}$, for some $\star$-finite ideal.
By the same reason $\sqrt{(P+bR)^{\star}\cap R}=\sqrt{N}$, where $N$ is a
$\star$-finite ideal. The same proof as Theorem 2.3 shows that $P=\sqrt{LN}$,
which is impossible, since $LN$ is a $\star$-finite ideal. Hence $P$ is a
quasi-$\star$-prime, a contradiction. Hence $\mathcal{A}=\emptyset$.
$(2)\Rightarrow(3)$ Suppose that $(I_{n})_{n\in\mathbb{N}}$ be an ascending
chain of radical quasi-$\star$-ideals, and set
$I=\bigcup_{n\in\mathbb{N}}I_{n}$. Then $I$ is a radical quasi-$\star$-ideal.
Hence $I=\sqrt{L}$ for some $\star$-finite ideal $L$. So, there is an integer
$n_{0}$ such that $I_{n_{0}}=\sqrt{L}=I$. Hence $(I_{n})_{n\in\mathbb{N}}$ is
stationary.
$(3)\Rightarrow(1)$ Suppose that $(1)$ is false. Then we can construct a chain
$(I_{n})_{n\in\mathbb{N}}$ of radical quasi-$\star$-ideals strictly ascending.
Indeed, let $P$ be a quasi-$\star$-prime ideal which is not the radical of a
$\star$-finite ideal. Set $I_{1}=\sqrt{(x)}$, where $0\neq x\in P$. Given
$I_{n}=\sqrt{(x_{1},\cdots,x_{n})^{\star}\cap R}$, where
$x_{1},\cdots,x_{n}\in P$, then
$I_{n+1}=\sqrt{(x_{1},\cdots,x_{n},x_{n+1})^{\star}\cap R}$, where $x_{n+1}\in
P\backslash I_{n}$. ∎
###### Corollary 2.11.
Suppose that $\star$ is a semistar operation of finite type on the integral
domain $R$. If $R$ satisfies the a.c.c. on radical quasi-$\star$-ideals, then
$\star$-$\operatorname{Min}(I)$ is finite for every ideal $I$ of $R$.
###### Proof.
Note $\star$-$\operatorname{Min}(I)\subseteq\operatorname{Min}(I^{\star}\cap
R)$ and use Theorem 2.3. ∎
###### Remark 2.12.
(1) One can prove Theorem 2.3 for arbitrary rings with zero divisors. Let $R$
be a commutative ring, with total quotient ring $T(R)$. Let $\mathcal{F}(R)$
denote the set of all $R$-submodules of $T(R)$. Suppose an operation
$*:\mathcal{F}(R)\rightarrow\mathcal{F}(R)$, $E\rightarrow E^{*}$, satisfies,
for all $E,F\in\mathcal{F}(R)$, and for all $x\in T(R)$, the following:
* $*_{1}$
$xE^{*}\subseteq(xE)^{*}$ and if $x$ is regular, then $xE^{*}=(xE)^{*}$;
* $*_{2}$
$E\subseteq F$ implies that $E^{*}\subseteq F^{*}$;
* $*_{3}$
$E\subseteq E^{*}$ and $E^{**}:=(E^{*})^{*}=E^{*}$.
Then from these axioms the following directly follow:
* (i)
$(EF)^{*}=(E^{*}F)^{*}=(EF^{*})^{*}=(E^{*}F^{*})^{*}$;
* (ii)
$(E+F)^{*}=(E^{*}+F)^{*}=(E+F^{*})^{*}=(E^{*}+F^{*})^{*}$;
* (iii)
$(E\cap F)^{*}\subseteq E^{*}\cap F^{*}=(E^{*}\cap F^{*})^{*}$.
It is clear that any semistar operation satisfies these axioms.
It is routine to see that [10, Lemma 3.3] the $v$-operation satisfies, these
axioms, where $E^{v}=(E^{-1})^{-1}$, in which $E^{-1}=(R:E)=\\{x\in
T(R)|xE\subseteq R\\}$, for $E\in\mathcal{F}(R)$.
By this operation Theorem 2.3, is true for rings with zero divisors.
(2) It is interesting to note that if we take $R$ to be the ring of all
sequences from $\mathbf{Z}/2\mathbf{Z}$ that are eventually constant, with
pointwise addition and multiplication, then $R$ is a zero-dimensional Boolean
ring with minimal prime ideals $P_{i}=\\{\\{a_{n}\\}\in R|a_{i}=0\\}$ and
$P_{\infty}=\\{\\{a_{n}\\}\in R|a_{n}=0\text{ for large }n\\}$ and each
$P_{i}$ is principal but $P_{\infty}$ is not finitely generated. Thus while
$R$ has infinitely many minimal prime ideals, only one is not the radical of a
finitely generated ideal.
In the rest of the paper we will define a class of rings, that satisfy the
a.c.c. on radical quasi-$\star$-ideals.
Let $R$ be a commutative ring. An ideal $I$ of $R$ is called an _ideal of
strong finite type_ (_SFT-ideal_ for short) if there exist a finitely
generated ideal $J\subseteq I$ and a positive integer $k$ such that $a^{k}\in
J$ for each $a\in I$. The ring $R$ is called an _SFT-ring_ if each ideal of
$R$ is an SFT-ideal. These concepts were introduced by J. T. Arnold in [2].
The condition that $R$ is an SFT-ring plays a key role in computing the Krull
dimension of the power series ring $R[[X]]$ over $R$. In [11], Kang and Park
defined and studied the $\star=t$ analogue of SFT-rings. Now we define the
more general semistar-SFT-rings.
Let $R$ be a domain and $\star$ a semistar operation on it. We define a
nonzero ideal $I$ of $R$ to be a _$\star$ -SFT-ideal_ if there exist a
finitely generated ideal $J\subseteq I$ and a positive integer $k$ such that
$a^{k}\in J^{\star}$ for each $a\in I^{\star_{f}}$. The ring $R$ is said to be
a _$\star$ -SFT-ring_ if each nonzero ideal of $R$ is a $\star$-SFT-ideal.
Obvious examples of a $\star$-SFT-ring is $\star$-Noetherian domains.
###### Proposition 2.13.
Suppose that $\star$ is a semistar operation of finite type on the integral
domain $R$. If $R$ is a $\star$-SFT-ring, then $R$ satisfies the a.c.c. on
radical quasi-$\star$-ideals.
###### Proof.
Let $P$ be a quasi-$\star$-prime ideal. Since $P$ is a $\star$-SFT-ideal there
is a finitely generated subideal $J\subseteq P$ such that
$\sqrt{P^{\star}}=\sqrt{J^{\star}}$. Now consider
$P=\sqrt{P}=\sqrt{P^{\star}\cap R}=\sqrt{P^{\star}}\cap R=\sqrt{J^{\star}}\cap
R=\sqrt{J^{\star}\cap R}.$
Since $J^{\star}\cap R$ is an $\star$-finite ideal, the result follows by
Proposition 2.10. ∎
###### Corollary 2.14.
Each quasi-$\star$-ideal of a $\star$-SFT-ring $R$, has only finitely many
minimal prime ideals.
We close the paper with the following characterization of $\star$-SFT-rings.
###### Proposition 2.15.
Suppose that $\star$ is a semistar operation of finite type on the integral
domain $R$. Then $R$ is a $\star$-SFT-ring if and only if each
quasi-$\star$-prime ideal of $R$ is a $\star$-SFT-ideal.
###### Proof.
Suppose that $R$ is not a $\star$-SFT-ring. Therefore the set
$\mathcal{A}=\\{I|I=I^{\star}\cap R,\text{ and is not a }\star\text{-SFT-
ideal}\\},$
is not an empty set. The set $\mathcal{A}$ is partially ordered under
inclusion, and is inductive under this ordering. By Zorn’s lemma,
$\mathcal{A}$ contains a maximal element $P$. Assume that $a_{1},a_{2}$ be two
elements of $R$ such that $a_{1}a_{2}\in P$ and $a_{1},a_{2}\notin P$. Since
$P\varsubsetneq(P+a_{i}R)^{\star}\cap R$, $(P+a_{i}R)^{\star}\cap R$ is a
$\star$-SFT-ideal of $R$. Consequently there exist a finitely generated ideal
$L_{i}\subseteq(P+a_{i}R)^{\star}\cap R$, and a positive integer $k_{i}$ such
that $c^{k_{i}}\in(L_{i})^{\star}$ for each $c\in(P+a_{i}R)^{\star}$. Let
$L=L_{1}L_{2}$ and $k=k_{1}+k_{2}$. Then $L$ is a finitely generated subideal
of $P$ such that
$c^{k}=c^{k_{1}}c^{k_{2}}\in(L_{1})^{\star}(L_{1})^{\star}\subseteq(L_{1}L_{2})^{\star}$,
for each $c\in P^{\star}$. Thus $P$ is a $\star$-SFT-ideal, a contradiction.
Therefore, $P$ is a quasi-$\star$-prime ideal which is not a $\star$-SFT-
ideal. ∎
ACKNOWLEDGMENT
I would like to thank Professor Marco Fontana for his useful comments on this
paper.
## References
* [1] D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc., 122, No. 1. (1994), 13–14.
* [2] J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc., 177, (1973), 299- 304.
* [3] J. T. Arnold, Power series rings over Prüfer domains, Pacific J. Math., 44, (1973), 1–11.
* [4] S. El Baghdadi, M. Fontana, Semistar linkedness and flatness, Prüfer semistar multiplication domains, Comm. Alg., 32, (2004), 1101- 1126.
* [5] S. El Baghdadi, M. Fontana, and G. Picozza, Semistar Dedekind domains, J. Pure Appl. Alg., 193, (2004), 27–60.
* [6] S. El Baghdadi, S. Gabelli, Ring-theoretic properties of PvMDs, Comm. Alg., 35, (2007), 1607–1625.
* [7] M. Fontana, J. A. Huckaba, Localizing systems and semistar operations, in: S. Chapman, S. Glaz (Eds.), Non Noetherian Commutative Ring Theory, Kluwer Academic Publishers, Dordrecht, 2000, pp. 169–197 (Chapter 8).
* [8] R. Gilmer, Multiplicative ideal theory, New York: M. Deker, 1972.
* [9] R. Gilmer, W. Heinzer, Primary ideals with finitely generated radicals in a commutative rings, manuscripta math., 78, (1993), 201–221.
* [10] I. Huckaba, I. Papick, Quotient rings of polynomial rings, Manuscripta Math., 31, (1980), 167–196.
* [11] B. G. Kang, M. H. Park, A note on $t$-SFT-rings, Comm. Alg., 34, (2006), 3153–3165.
* [12] I. Kaplansky, Commutative rings, revised edition, Univ. of Chicago Press, Chicago, 1974.
* [13] A. Okabe, R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ., 17, (1994), 1 -21.
* [14] G. Picozza, A note on semistar noetherian domains, Houston J. Math., 33, no. 2, (2007), 415–432.
|
arxiv-papers
| 2008-09-08T10:50:11
|
2024-09-04T02:48:57.721388
|
{
"license": "Public Domain",
"authors": "Parviz Sahandi",
"submitter": "Parviz Sahandi Dr.",
"url": "https://arxiv.org/abs/0809.1307"
}
|
0809.1444
|
001–008
#
(2009)
††volume: Volume XXVIIA††journal: Transactions IAU, Volume XXVIIA††editors:
Karel A. van der Hucht, ed.
COMMISSION 10 | SOLAR ACTIVITY
---|---
| ACTIVITÉ SOLAIRE
PRESIDENT | James A. Klimchuk
---|---
VICE-PRESIDENT | Lidia van Driel-Gesztelyi
SECRETARY | Carolus J. Schrijver
PAST PRESIDENT | Donald B. Melrose
ORGANIZING COMMITTEE | Lyndsay Fletcher, Nat Gopalswamy,
| Richard A. Harrison, Cristina H. Mandrini,
| Hardi Peter, Saku Tsuneta,
| Bojan Vršnak, Jing-Xiu Wang
TRIENNIAL REPORT 2006-2009
## 1 Introduction
Commission 10 deals with solar activity in all of its forms, ranging from the
smallest nanoflares to the largest coronal mass ejections. This report reviews
scientific progress over the roughly two-year period ending in the middle of
2008. This has been an exciting time in solar physics, highlighted by the
launches of the Hinode and STEREO missions late in 2006. The report is
reasonably comprehensive, though it is far from exhaustive. Limited space
prevents the inclusion of many significant results. The report is divided into
the following sections: Photosphere and Chromosphere; Transition Region;
Corona and Coronal Heating; Coronal Jets; Flares; Coronal Mass Ejection
Initiation; Global Coronal Waves and Shocks; Coronal Dimming; The Link Between
Low Coronal CME Signatures and Magnetic Clouds; Coronal Mass Ejections in the
Heliosphere; and Coronal Mass Ejections and Space Weather. Primary authorship
is indicated at the beginning of each section.
## 2 Photosphere and Chromosphere (C. J. Schrijver)
### 2.1 Quiet-Sun Field Within the Photosphere
The Solar Optical Telescope (SOT) on board the Hinode spacecraft provides an
unprecedented combination of spatial resolution and continuity of
observations. The SpectroPolarimeter focal-plane instrumentation exploits that
to measure the polarization signals from the photospheric plasma. Lites et al.
(2008) and Ishikawa et al. (2008) show direct evidence that much of the
magnetic field in the quiet-Sun photosphere is essentially horizontal to the
solar surface. This observation is the direct confirmation of the existence of
the weak field for which less direct evidence had been found by Harvey et al.
(2007), and contributes to Hanle de-polarization effects discussed by, e.g.,
Trujillo Bueno et al. (2004).
The nearly vertical component is found primarily in the downflow network of
the granular convection, corresponding to the well-known network field. The
horizontal field is mostly found in the interior of the convective cells.
Despite this significant preference for a separation by upflow and downflow
domains, flux has been observed to also emerge already largely vertical even
within the interior of the granular convective cells (Orozco-Suarez et al.
2008). And, perhaps not surprisingly, the conceptually expected evolutionary
pattern of emerging flux is also seen: Centeno et al. (2007) report on
observations in which field is seen to first surface nearly horizontally and
subsequently—as it is advected to the downflow lanes—rights itself to be
nearly vertical.
Lites et al. (2008) measure the mean flux density of the horizontal field to
be about five times higher than that associated with the nearly vertical field
component. Interestingly, radiative MHD simulations of near-surface stratified
convection by Schuessler and Voegler (2008) show a very similar orientation-
dependent ratio for the field. Steiner et al. (2008) reach a similar
conclusion based on their numerical experiments: they argue that the granular
upflows allow field to be stretched horizontally, being advected from over the
cell centers only slowly in the stagnating, overshooting, upper-photospheric
flows. Both studies support the conclusion that near-surface turbulent dynamo
action significantly contributes to the internetwork photospheric field. A
study by Abbett (2007) elucidates how such a turbulent-dynamo field would
connect sub-photospheric and coronal layers through a complex and dynamic
chromospheric layer in between; work by Isobe et al. (2008) explores
numerically the frequent reconnective interactions expected with the overlying
chromospheric canopy field, suggesting that this and the associated wave
generation may have significant consequences for atmospheric heating and
driving of the solar wind.
### 2.2 The Solar Dynamo(s): Global and Local Aspects
Ephemeral bipolar regions are at the small end of the active region spectrum.
Their properties over the solar cycle are an extension of those of their
larger counterparts: they follow the general butterfly pattern, and have the
proper preferential orientation of their dipole axes relative to the equator,
but with a spread about the mean that increases towards the smaller bipoles.
In this, they are a natural extension of the active region population. Where
they were known to differ from the large regions is in the fact that they are
the first to appear and last to fade for a given sunspot cycle.
Now, work by Hagenaar et al. (2008) uncovers another distinct property of
ephemeral regions: the emergence frequency decreases with increasing local
flux imbalance (consistent with findings by Abramenko et al. (2006) and Zhang
et al. (2006) who differentiated only coronal hole regions from other quiet-
Sun regions). Hagenaar et al. (2008) find that the rate of flux emergence is
lower within strongly unipolar network regions by at least a factor of 3
relative to flux-balanced quiet Sun. One consequence of this is that because
coronal holes overlie strong network regions, there are fewer ephemeral
regions, and therefore fewer EUV or X-ray bright points within coronal holes.
The ephemeral-region population thus takes an interesting position in the
study of solar magnetic activity: with the smallest-scale internetwork field
perhaps largely generated by a local turbulent dynamo (Schuessler & Voegler
2008), and with the active regions associated with a global dynamo action, the
ephemeral region population has signatures of both. Voegler & Schuessler
(2007) show that local dynamo action can lead to a mixed-polarity field
similar to the flux balanced very-quiet network field. It remains to be seen
what such experiments predict in case there is a net flux imbalance, i.e., a
background ‘guide field’: are fewer ephemeral regions generated, or does
reconnection with the background guide field cause fewer of them to survive
the rise to the surface (see discussion by Hagenaar et al. 2008).
### 2.3 Emerging Flux: Observations and Numerical Experiments
Observations made with the Hinode SOT show unambiguously that magnetic flux
bundles that form active regions do not emerge as simply curved arches, but
rather as fragmented collections of undulating flux bundles. Each bundle
likely crosses the photosphere one or more times between the extremes of the
emerging region (e.g., Lites 2008). This is likely the result of the coupling
to the near-surface convective motions, and the difficulty of relatively heavy
sub-photospheric material to drain from the dipped field segments.
Reconnection between neighboring supra-photospheric flux bundles could pinch
off the sub-photospheric mass pockets, thus allowing the field to rise into
the corona. Radiative MHD simulations of emerging flux by Cheung et al. (2008)
support this interpretation: they show the ‘serpentine’ nature of the emerging
flux, with characteristics that resemble the observed patterns of emerging
flux, flux cancellation with associated downflows, convective collapse into
strong-field flux concentrations, and photospheric bright points. Note that an
example of field dipping into sub-photospheric layers is also discussed by
Abbett (2007).
### 2.4 Upper-Chromospheric Dynamics: Spicules and Waves
Spectacularly sharp Ca II H narrow-band filter observations made both with
Hinode’s SOT and the Swedish Vacuum Solar Telescope reveal ubiquitous jet-like
features (called spicules or fibrils) above the solar limb. The relatively
long-lived, broad population among these (discussed by De Pontieu et al.
2007a) appear to be caused by acoustic shock waves propagating upward from the
photosphere. These shock waves cause the chromospheric material to undulate
with almost perfectly parabolic height-time profiles, and saw-tooth velocity
patterns. These shock-induced fibrils occur both in plages and in quiet Sun.
A more enigmatic phenomenon is formed by the much finer and more transient
population of hair-like high extension of the chromospheric plasma discussed
by De Pontieu et al. (2007b). Their origin remains subject to debate, but
their transverse displacements point to the ubiquitous existence of Alfven-
like waves propagating into the corona. This is the most direct observational
evidence for the existence of such waves reported to date. The estimated power
suffices to heat the quiet-Sun corona and power the solar wind: these waves
have amplitudes of 10-25 km s-1 for periods of 100-500 seconds. Alfven-like
waves with similar periods have also been observed for the first time in
coronal loops, using the CoMP instrument (Tomczyk et al. 2007).
## 3 Transition Region (H. Peter)
The transition region from the chromosphere to the corona, originally thought
of as a simple thin onion shell-like layer, is a spatially and temporally
highly complex part of the solar atmosphere. So far we are missing a unifying
picture combining the numerous phenomena observed in emission lines formed
from a couple of 10,000 K to several 100,000 K. Some of the aspects are re-
interpreted by Judge (2008), who attempts to explain the transition region as
being due to cross-field conduction of neutral atoms. The emission measure
increasing towards low temperatures and the persistent redshifts are two of
the major observational facts to be explained. A collection of one-dimensional
transient models (Spadaro et al. 2006) and a three-dimensional MHD model
(Peter et al. 2006) gave quantitative explanations for this. In agreement with
the latter model, Doschek (2006) could show that the bulk of the (low)
transition region emission originates from small cool loops. Using rocket
imaging data, Patsourakos et al. (2007) give further direct observational
evidence for the existence of such small loop-like structures dominating the
transition region emission.
The magnetic field has to connect the transition region structures down to the
chromosphere, and in case they are part of hot coronal elements also to the
corona (Peter 2007). However, correlations between the transition region and
the photosphere cannot be identified in a unique way (Sánchez Almeida et al.
2007). Larger features in the photosphere, such as moving magnetic features,
might well leave an imprint in the outer atmosphere (Lin et al. 2006). In
general, the connection from the chromosphere to the transition region is
quite subtle and hard to identify in observations (Hansteen et al. 2007). A
new way to investigate the relation between transient events in the transition
region and the chromosphere was presented by Innes (2008). She studied the
chromospheric emission of molecular hydrogen near 111.9 nm during microflaring
events and proposes that the (coronal) energy deposition in the microflare
also heats the chromosphere and thus affects the opacity for molecular
hydrogen lines.
The energy balance, one part being the heating process, is largely determining
the pressure of the transition region and thus implicitly also the mass
loading. Combining various models and observations, Aschwanden et al. (2007)
argue that the bulk part of the heating is located deep down, basically
reflecting an exponential decay of the heating rate with height, on average.
As speculated earlier, Tian et al. (2008a) could now show that the persistent
blueshifts in upper transition region lines are not due to the solar wind
outflow, but due to mass loading of loops. Recent Doppler shift observations
with the Extreme ultraviolet Imaging Spectrometer (EIS; Culhane et al. 2007a)
on board Hinode indicate that the redshifts are due to radiative cooling and
subsequent bulk downflows within the loops (Bradshaw 2008).
New investigations of coronal moss, i.e. the (upper) transition region
footpoint areas of large hot loops, show that in moss regions the temperature
is inversely related to the density (Tripathi et al. 2006). Using comparisons
with models, Warren et al. (2008b) show that in order to understand this moss
within the framework of a steady uniform heating model, one needs to assume
that the moss plasma is not fully filling the volume. However, it remains to
be seen if such a static model is applicable at all, because one might suspect
a spatially varying heating rate (see above), and if the assumption of static
moss is justified.
Motivated by direct magnetic field measurements in the corona indicating the
presence of Alfven waves (Tomczyk et al. 2007) and observations with SOT on
Hinode above the limb, McIntosh et al. (2008) re-interpreted the widths of
transition region lines across the solar disk. They conclude that the present
observations are consistent with a line-of-sight superposition of Alfvenic
disturbances in small-scale structures. How this relates to the new finding of
Doschek et al. (2007) that the (non-thermal) line widths are largest not in
the brightest parts of an active region but in dimmer regions adjacent to
bright loops remains to be seen. Doschek et al. (2007) find broad lines
related to potential outflow locations, so maybe this problem hints to
different acceleration and heating mechanisms in open and closed field
regions. Another new difference between (globally) open and closed field
regions, was proposed by Tian et al. (2008b), who find evidence that the
expansion of transition region structures is more rapid in the coronal holes
as compared to the quiet Sun. Dolla & Solomon (2008) analyzed line widths
above the limb in order to determine the (kinetic) temperatures of minor ions
in presumably open field regions. They find the smallest mass-to-charge-ratio
ions to be the hottest at a given height, but their analysis remains
inconclusive with regard to supporting or disproving the proposed heating by
ion-cyclotron resonances.
While being in orbit nearly 13 years now, the instruments on board SOHO, SUMER
in particular, still give numerous new valuable results on the transition
region. The EIS instrument on board Hinode covers wavelengths around 17-21 nm
and 24-29 nm. This mainly includes emission lines formed from 1 to several MK,
but also a small number of lines from the transition region, and allows good
density diagnostics (Feldman et al. 2008). Given the spectral range, the main
science topics are grouped around active region phenomena, while the
transition region can also be investigated (Young et al. 2007). Besides these
instruments, which will provide the main source for observations of transition
region lines in the coming years, rocket experiments complement these data.
## 4 Corona and Coronal Heating (J. A. Klimchuk)
The past two years have seen considerable progress in understanding the
magnetically-closed corona and how it is heated. This short report highlights
just some of the important contributions. Much effort has been devoted to
determining the properties of the heating—how it varies in time and space and
whether it depends on physical parameters such as the strength of the magnetic
field and the length of field lines. Some studies have concentrated on
individual coronal loops, while others have addressed active regions as a
whole. These efforts have both clarified some issues and raised new questions.
Let us first consider distinct, measurable loops. A short history is useful.
For many years after the Skylab soft X-ray observations, it was thought that
loops are static equilibrium structures maintained by steady heating. Then
came the EUV observations from EIT/SOHO and TRACE. These revealed that warm
($\sim$ 1 MK) loops are much too dense for static equilibrium and have super-
hydrostatic scale heights. Modeling efforts showed that the excess densities
and large scale heights could be explained by impulsive heating. Because of
their temperature response, EIT and TRACE are sensitive to the loops when they
are cooling by radiation, well after the heating has ceased. The problem is
that loops are observed to persist for longer than a cooling time, so a
monolithic model is not viable. This led to the suggestion that loops are
bundles of thin, unresolved strands. The observed high densities, large scale
heights, and long lifetimes can all be explained if the strands are heated at
different times by a storm of nanoflares. Since the strands are in different
stages of cooling, a range of temperatures should be present within the loop
bundle at any given time. In particular, there should be a small amount of
very hot ($>5$ MK) plasma. See Klimchuk (2006) for a discussion of these
points and original references.
Whether loops are isothermal or multi-thermal has been intensely debated over
the past several years. Double- and triple-filter observations from TRACE seem
to suggest that the most narrow loops are isothermal (Aschwanden 2008).
However, it has been demonstrated that many different thermal distributions,
including ones that are broad, can reproduce the observed intensities, even
with 3 filters (Schmelz et al. 2007a; Patsourakos & Klimchuk 2007; Noglik &
Walsh 2007). Spectrometer observations provide far superior plasma
diagnostics. The results here are mixed. Studies made with CDS/SOHO continue
to find evidence for both isothermal loops and highly multi-thermal loops
(Schmelz et al. 2007b; Cirtain et al. 2007), while studies made with the new
EIS instrument on Hinode find that loops tend to be mildly multi-thermal
(Ugarte-Urra et al. 2008; Warren et al. 2008a). Where temporal information is
available, there is clear evidence that the loops are evolving, but the
evolution is generally slower than expected for radiative cooling. Loop
lifetimes are extremely important and require further investigation. A loop
bundle will be only mildly multi-thermal if the storm of nanoflares is short-
lived; however, the observed lifetime of the loop will then be correspondingly
short. If a loop is observed to persist for much longer than a cooling time
(López Fuentes et al. 2007), then its thermal distribution is expected to be
broad. More work is needed on whether the lifetimes and thermal distributions
of loops are consistent. Finally, Landi & Feldman (2008) have found that one
particular active region is dominated by three distinct temperatures, which
would greatly challenge our understanding if correct.
Modeling the plasma properties of whole active regions is a relatively new
endeavor. In addition to providing valuable information on coronal heating,
these research models are the forerunners of eventual operational models for
nowcasting and forecasting the solar spectral irradiance. This is of great
practical value, since the irradiance controls the dynamics, chemistry, and
ionization state of the terrestrial upper atmosphere and thereby affects radio
signal propagation, satellite drag, etc. Active region models based on static
equilibrium are able to reproduce the observed soft X-ray emission reasonably
well, but they fail, often miserably, at reproducing the EUV emission. The
model corona is too faint in the EUV (there are no warm loops) and the moss at
the transition region footpoints of hot loops is too bright. Winebarger et al.
(2008) have demonstrated that better agreement can be obtained in the core of
an active region by using a combination of flux tube expansion and filling
factors near 10%. Filling factors of this magnitude have been measured in moss
with EIS (Warren et al. 2008b). Small filling factors are consistent with the
idea of unresolved loop strands. Reale et al. (2007) have developed a new
multi-filter technique using XRT/Hinode data that reveals considerable thermal
structure on small but resolvable scales.
Active region models based on impulsive heating are in much better agreement
with observations than are static models. In particular, the predicted coronal
EUV emission is greatly enhanced (Warren & Winebarger 2007; Patsourakos &
Klimchuk 2008a). The predicted moss emission is still too bright, but these
models assume constant cross section flux tubes, and expanding tubes will
improve the agreement, as they do for static models. It is also likely that
the brightness of the observed moss is diminished by spicules and possibly by
other cool absorbing material.
Nanoflare models predict that very hot plasma should be present throughout the
corona, albeit in very small quantities (Klimchuk et al. 2008; this paper
presents a highly efficient IDL code for modeling dynamic loops and is
available upon request). The intensities of very hot spectral lines are
expected to be extremely faint, due to the small emission measures and
possible also to ionization nonequilibrium effects (Bradshaw & Cargill 2006;
Reale & Orlando 2008). Measureable quantities of very hot ($\sim$ 10 MK)
plasma have been detected outside of flares by the CORONAS-F
spectroheliometers (Zhitnik et al. 2006), RHESSI (McTiernan 2008), and XRT
(Siarkowski et al. 2008; Reale et al. 2008). The derived differential emission
measure distributions, DEM($T$), are consistent with the predictions of
nanoflare models. The DEM($T$) derived from EIS spectra for $T\leq 5$ MK are
also consistent with the predictions (Patsourakos & Klimchuk 2008b). Other
tests of the nanoflare idea include emission line Doppler shifts, broadening,
and wing enhancements that are associated with evaporating and condensing
plasma (Patsourakos & Klimchuk 2006; Hara et al. 2008; Bradshaw 2008).
Information on the distribution of nanoflare energies can be inferred from the
intensity fluctuations of observed loops (Parenti et al. 2006; Pauluhn &
Solanki 2007; Parenti & Young 2008; Sarkar & Walsh 2008; Sakamoto et al. 2008;
Bazarghan et al. 2008). Proper flares are known to have a power law energy
distribution with an index $<2$. Extrapolating to smaller energies implies
that nanoflares cannot heat the corona, as first pointed out by Hudson (1991).
However, it is now believed that the power law index for small events is $>2$,
though with a large uncertainty (Benz 2004; Pauluhn & Solanki 2007; Bazarghan
et al. 2008). Furthermore, the subset of proper flares that are not associated
with CMEs also have a power law index $>2$ (Yashiro et al. 2006; see Section
6). Since the physics of eruptive events and noneruptive events (nanoflares
and confined flares) is likely to be much different, it is not surprising that
they obey different power laws.
Thermal nonequilibrium, a phenomenon thought to be important for prominence
formation (Karpen & Antiochos 2008), may also play an important role in
ordinary loops. Loop equilibria do not exist for steady heating if the heating
is highly concentrated low in the loop legs. Instead, cool condensations form
and fall to the surface in a cyclical pattern that repeats on a time scale of
hours. Resolvable condensations are indeed observed in active regions, but
only in a small fraction of loops (Schrijver 2001). As a possible explanation
for other EUV loops, Klimchuk & Karpen (2008) appeal to the multi-strand
concept. The individual tiny condensations that occur within each strand will
not be detected as long as the strands are out of phase. It is encouraging
that the models predict excess densities similar to those of observed EUV
loops. Mok et al. (2008) report thermal nonequilibrium behavior in their
active region simulations. Hot loops form and cool, but without producing
localized condensations.
We can summarize the state of understanding as follows. Much of the
magnetically-closed corona is certainly not in static equilibrium, but much of
it could be. A significant portion—perhaps the vast majority—is heated
impulsively or is in thermal nonequilibrium, or some combination thereof. Most
coronal heating mechanisms that have been proposed involve impulsive energy
release (Klimchuk 2006; Uzdensky 2007; Cassak et al. 2008; Dahlburg et al.
2008; Rappazzo et al. 2008; Ugai 2008). It should be noted, however, that
nanoflares that recur sufficiently frequently within the same flux strand (on
a time scale much shorter than the cooling time) will produce quasi-static
conditions. It is clear that more observational and theoretical work is
required before the coronal heating problem will be solved.
## 5 Coronal Jets (L. van Driel-Gesztelyi)
Coronal bright points are often observed to have jets—collimated transient
ejections of hot plasma. Hinode (Kosugi et al. 2007) can now study the fine
detail of jets which tend to occur preferentially inside coronal holes, which
is consistent with reconnection taking place between the open magnetic field
of the coronal hole and the closed loop field lines. Observations with the XRT
instrument (Golub et al. 2007) revealed that jets from polar coronal holes are
more numerous than previously thought (60 jets day-1, Savcheva et al. 2007;
and even 10 jets hour-1, Cirtain et al. 2007). The EIS instrument (Culhane et
al. 2007a) allows direct measurement of the velocity of jets in the corona for
the first time. The footpoints of the loops are seen to be red-shifted which
is consistent with downflowing cooling plasma following reconnection. The
(blue-shifted) jet is the dominant feature in velocity space but not in
intensity (Kamio et al. 2007). Another new feature of jets is post-jet
enhancement of cooler coronal lines observed by EIS. This can be explained by
the hot plasma in the jet not having sufficient velocity to leave the Sun and
then falling back some minutes later (Culhane et al 2007b).
XRT observations of jets at the poles have shown mean velocities for jets of
160 km s-1 (Savcheva et al. 2007). Multiple velocity components were found in
jets by Cirtain et al. (2007) in XRT polar coronal hole data: a spatio-
temporal average of about 200 km s-1 as well as a much higher velocity
measured at the beginning of each jet—with speeds reaching 800 km s-1. Cirtain
et al. (2007) interpret this early (and sometimes recurrent) fast flow as
being due to plasma ejected at the Alfven speed during the relaxation phase
following magnetic reconnection. The mass flux supplied by about 10 jets per
hour occurring in the two polar coronal holes was estimated to produce a net
flux of 1012 protons m-2 s-1 which is only a factor of 10 less than the
current estimates of the average solar wind flux. These small jets are
providing a substantial amount of mass that is being carried into
interplanetary space. A 3D numerical simulation has been carried out to
compare with these observations (Moreno-Insertis et al. 2008) and is found to
be consistent with several key observational aspects of polar jets such as
their speeds and temperatures.
A study of the 3D morphology of jets became possible for the first time with
stereoscopic observations by the EUVI/SECCHI imagers (Howard et al. 2008)
onboard the twin STEREO spacecraft. The most important geometrical feature of
the observed jets was found to be helical structures showing evidence of
untwisting (Patsourakos et al. 2008). This is in agreement with the 3D model
proposed by Pariat et al. (2008) with magnetic twist (untwisting) being the
jet’s driver.
## 6 Flares (L. Fletcher and J. Wang)
In this brief review we focus on progress in flare energy build-up and flare
prediction, flare photospheric effects, high energy coronal sources, non-
thermal particles, the flare-CME relationship, and recent advances with
Hinode.
Where is the magnetic free energy stored in a flaring active region? Using the
increasingly robust methods for extrapolating magnetic fields from vector
magnetic field measurements, Schrijver et al. (2008) find evidence for pre-
flare filamentary coronal currents located $<20$ Mm above the photosphere and
Regnier & Priest (2007) show that in a newly-emerged active region the free
energy is concentrated within the first 50 Mm (in an older, decaying region it
resides at higher altitudes). Horizontal shear flows close to the neutral line
prior to large flares (Deng et al. 2006) confirm the concentration of free
energy in a small spatial scale, and Schrijver (2007) finds that if the
unsigned flux within 15 Mm of the polarity inversion line exceeds
$2\\!\times\\!10^{21}$ Mx a major flare will occur within a day. Though Leka &
Barnes (2007) find that the probability of flaring has only a weak
relationship to the state of the photospheric magnetic field at any time,
single or synthesized magnetic parameters are being used with some success to
quantify flare probability and productivity. Georgoulis & Rust (2007)
introduce the effective connected magnetic field of an active region, finding
that this exceeds 1600 and 2100 G for M- and X-class flares, respectively, at
95% probability. LaBonte et al. (2007) surveyed the helicity injection prior
to X-class flares producing a CME, finding occurrence only if the peak
helicity flux exceeds $6\\!\times\\!10^{36}$ Mx s-1. Cui et al. (2007) find
that flare probability increases with active region complexity,
nonpotentiality, and length of polarity inversion line. Impulsive phase HXR
sources are concentrated where the magnetic field is strong, and where the
reconnection rate is high (Temmer et al. 2007; Jing et al. 2008; Liu et al.
2008).
The last three years have seen effort directed towards understanding the
magnetic and seismic effects of flares near the photosphere. It is clear that
the photospheric magnetic field changes abruptly and non-reversibly during the
flare impulsive phase (see e.g. Sudol & Harvey 2005 for a recent survey).
Rapid changes in sunspot structure have also been detected by Chen et al.
(2007) in 40% of X-class flares, 17% of M flares, and 10% C flares, while Wang
(2006) finds variations in magnetic gradient close to the polarity inversion
line consistent with a sudden release of magnetic shear. The obvious future
task is to analyze vector magnetograms to identify changes in the ‘twist’
component of the field.
Flare-generated seismic waves, discovered by Kosovichev & Zharkova (1998) and
amply confirmed in Cycle 23, also show the flare’s photospheric impact. Donea
et al. (2006), Kosovichev (2006, 2007) and Zharkova & Zharkov (2007) show that
flare HXR sources and seismic sources correlate in space and time. Seismic
sources are associated also with white light kernels, responsible for the
majority of the flare radiated energy and strongly correlated with HXR sources
(Fletcher et al. 2007) but the total acoustic energy is a small fraction of
total flare energy (Donea et al. 2006). Nonetheless, looking at the cyclic
variation of the total energy in the Sun’s acoustic spectrum, Karoff &
Kjeldsen (2008) propose that—analogous with earthquakes—flares may excite
long-duration global oscillations.
The RHESSI mission has discovered several new types of flare coronal HXR
sources, and we highlight here a hard-spectrum HXR source at least 150 Mm
above the photosphere, with a nonthermal electron fraction of about 10%
(Krucker et al. 2007b). Hard spectrum gamma-ray (200-800 keV) coronal sources
have also been found, suggesting coronal electron trapping (Krucker et al.
2008). A soft-hard-soft spectral variation with time is present in some
coronal sources as well as footpoints (Battaglia & Benz 2006), and this may be
explicable by a combination of coronal trapping and stochastic electron
acceleration (Grigis & Benz 2006). A curious observation with the Owens Valley
Solar Telescope of terahertz emission may come from a compact coronal source
of electrons at 800 keV, if the electrons are radiating in a volume with a
magnetic flux density of 4.5 kG (Silva et al. 2007).
Discovering the origin and properties of the flare electron distribution
continues to motivate advanced modeling and observations. Particle-in-cell
simulations, used for some years in magnetospheric physics, have been
harnessed to study acceleration in coronal magnetic islands produced by
magnetic reconnection (Drake et al. 2006), and wave-particle distributions in
current sheet and uniform magnetic field geometries (Karlicky & Barta 2007;
Sakai et al. 2006). Flare Vlasov simulations are also being developed (e.g.
Miteva et al. 2007; Lee et al. 2008). Detailed RHESSI HXR spectroscopy has led
to a new diagnostic for the flare electron angular distribution, based on
photospheric HXR albedo (Kontar et al. 2006a). This diagnostic suggests that
electron distributions might not be strongly downward-beamed in the
chromosphere (Kontar et al. 2006b; Kasparova et al. 2007; though see Zharkova
& Gordovskyy 2006 for an alternative explanation). Xu et al. (2008) studied
RHESSI flares having an extended coronal source, finding evidence for an
extended coronal accelerator. Looking at the larger coronal context, Temmer et
al. (2008) show that peaks in the flare electron acceleration rate and in the
CME acceleration rate are simultaneous within observational constraints
($\sim$ 5 minutes). RHESSI and WIND observations suggest that the spectral
indices of flare and interplanetary electrons are correlated, but in a way
that is inconsistent with existing models for flare X-ray emission, and the
number of escaping electrons is only about 1/500th of the number of electrons
required to produce the chromospheric HXR flux (Krucker et al. 2007a).
However, outstanding questions about the total electron number and supply in
solar flares has prompted various authors to suggest alternatives to the
‘monolithic’ coronal electron beam picture. For example Dauphin (2007),
Gontikakis et al. (2007) examine acceleration in multiple, distributed coronal
region sites, while Fletcher & Hudson (2008) investigate the transport of
flare energy to the chromosphere in the form of the Poynting flux of large-
scale Alfvenic pulses, in strong and low-lying coronal magnetic fields.
The relationship between flares and CMEs continues to be an important topic.
The general consensus regarding the spatial correspondence between CME
position angle and flare location in the pre-SOHO era was that the flare is
located anywhere under the span of the CME (e.g., Harrison 2006). However,
using 496 flare-CME pairs in the SOHO era, Yashiro et al. (2008) found that
the offset between the flare position and the central position angle (CPA) of
the associated CME has a Gaussian distribution centered on zero, meaning the
flare is typically located radially below the CME leading edge. This finding
suggests a closer flare-CME relationship as implied by the CSHKP eruption
model. Many flares are not associated with CMEs. Yashiro et al. (2006) studied
two sets of flares one with and the other without CMEs. The number of flares
as a function of peak X-ray flux, fluence, and duration in both sets followed
a power law. Interestingly, the power law index was $>2$ for flares without
CMEs, while $<2$ for flares with CMEs. In flares without CMEs, the released
energy seems to go entirely into heating, which suggests that nanoflares may
contribute significantly to coronal heating (see Section 4).
The launch of Hinode promises significant advances in flare physics in the
next cycle. Thus far there have only been a small number of well-observed
large flares, but observations of small flares start to show the combined
power of RHESSI and Hinode. For example, Hannah et al. (2008) find a
microflare not conforming to the usual relationship between flare thermal and
nonthermal emission, and Milligan et al. (2008) show evidence for hot
downflowing plasma in the flare corona, not explained in any existing flare
model. Observational evidence for a new kind of reconnection, called slip-
running reconnection, has been found by Aulanier et al. (2007), and sub-
arcsecond structure in the white light flare sources has been demonstrated by
Isobe et al. (2007). We look forward to the continued operation of these
instruments, and the theoretical advances that they will bring, in the rise of
Cycle 24.
## 7 Coronal Mass Ejection Initiation (L. van Driel-Gesztelyi)
In recent years our physical understanding of CMEs has evolved from cartoons
inspired by observations to full-scale numerical 3D MHD simulations
constrained by observed magnetic fields. Notably, there has been progress made
in simulating CME initiation by flux rope instabilities as inspired by
observed filament motions during eruption which frequently include helical
twisting and writhing (e.g. Rust & LaBonte 2005; Green et al. 2007). Several
of these simulations use the analytical model of a solar active region by
Titov & Démoulin (1999) as initial condition. The model contains a current-
carrying twisted flux rope that is held in equilibrium by an overlying
magnetic arcade. The two instabilities considered as eruption drivers are the
ideal MHD helical kink and torus instabilities. The helical kink instability
sets in if a certain threshold of (flux rope) twist ($\sim$ 2.5 turns for
line-tied flux ropes) is reached (e.g., Török & Kliem 2005). Above this
threshold, twist becomes converted into writhe during the eruption, deforming
the flux rope (or filament) into a helical kink shape. On the other hand, a
current-carrying ring (or flux rope) situated in an external poloidal magnetic
field ($B_{ex}$) is unstable against radial expansion when the Lorentz self-
force or hoop force decreases more slowly with increasing ring radius than the
stabilizing Lorentz force due to $B_{ex}$. Known as the torus instability, its
possible role in solar eruptions has been examined by Kliem & Török (2006) and
Isenberg & Forbes (2007). In solar eruptions, the torus instability does not
require a pre-eruptive, highly twisted flux rope, but (i) a sufficiently steep
poloidal field decrease with height above the photosphere and (ii) an
(approximately) semi-circular flux rope shape. Both the helical kink and torus
instabilities may be responsible for initiating and driving
prominence/filament eruptions and thus CMEs. The magnetic field decrease with
height above the filament was shown to be critical whether a confined eruption
or a full eruption occurs as well as for determining the acceleration profile,
corresponding to fast CMEs for rapid (field) decrease, as it is typical of
active regions, and to slow CMEs for gentle decrease, as is typical of the
quiet Sun (Török & Kliem 2005, 2007; Liu 2008). The latter means that CMEs
from complex active regions with steep field gradients in the corona are more
likely to give rise to fast CMEs—something that is indeed observed.
More complex CME initiation models involve multiple magnetic flux systems,
such as in the magnetic break-out model (e.g. Antiochos et al. 1999, DeVore &
Antiochos 2008). In this model, magnetic reconnection removes unstressed
magnetic flux that overlies the highly stressed core field and this way allows
the core field to erupt. The magnetic break-out model involves specific
nullpoints and separatrices. A multi-polar configuration was also included in
the updated catastrophe model (Lin & van Ballegooijen 2005), the flux
cancellation model (Amari et al. 2007), and the MHD instability models (Török
& Kliem 2007). In an attempt to test CME initiation models with special
attention to the breakout, Ugarte-Urra et al. (2007) analyzed the magnetic
topology of the source regions of 26 CME events using potential field
extrapolations and TRACE EUV observations. They found only seven events which
could be interpreted in terms of the breakout model, while a larger number of
events (12) could not be interpreted in those terms. The interpretation of the
rest remained uncertain. On the other hand, the CME event analyzed by Williams
et al. (2005) provided a good example to indicate that also a combination of
several mechanisms, e.g. magnetic break-out and kink instability, can be at
work in initiating CMEs.
## 8 Global Coronal Waves and Shocks (B. Vršnak)
The research on globally propagating coronal disturbances (large-amplitude
waves, shocks, and wave-like disturbances) continued to be very dynamic. Maybe
the most prominent characteristic of the past triennium was an enhanced effort
to combine detailed multi-wavelength observations with the theoretical
background. The empirical research resulted in a number of new findings,
leading to new ideas and interpretations, whereas the theoretical research
provided a better understanding of physical processes governing the formation
and propagation of global coronal disturbances.
For the first time the EUV signatures of a global coronal wave (‘EIT waves’)
were measured at high cadence by EUVI/STEREO, related to the eruption of 2007
May 19 (Long et al. 2008; Veronig et al. 2008). Long et al. (2008) reported
for the first time the wave signatures at 304 Å. Furthermore, they confirmed
the idea by Warmuth et al. (2001) that velocities of EIT waves measured by
EIT/SOHO are probably significantly underestimated due to the low cadence of
the EIT instrument. Veronig et al. (2008) revealed reflection of the wavefront
from the coronal hole boundary, indicating that the observed disturbance
represents a freely-propagating MHD wave.
The data from pre-STEREO instruments continued to be exploited fruitfully.
Mancuso & Avetta (2008) analyzed the UV-spectrum (UVCS/SOHO) of the 2002 July
23 coronal shock, and concluded that the plasma-to-magnetic presure ratio
$\beta$ could be an important parameter in determining the effect of ion
heating at collisionless shocks. Employing the extensive data on CMEs, solar
energetic particle (SEP) events, and type II radio bursts during the SOHO era,
Gopalswamy et al. (2008b) demonstrated that essentially all type II bursts in
the decameter-hectometric (DH) wavelength range are associated with SEP events
once the source location is taken into account. Shen et al. (2007) proposed a
method to determine the shock Mach number by employing the CME kinematics,
type II burst dynamic spectrum, and the extrapolated magnetic field. Analyzing
one Moreton wave that spanned over almost $360^{\rm o}$, Muhr et al. (2008)
revealed two separate radiant points at opposite ends of the two-ribbon flare,
indicating that the wave was driven by the CME expanding flanks. Veronig et
al. (2006) found out that the Moreton/EIT wave segments where the front
orientation is normal to the coronal hole boundary can intrude into the
coronal hole up to 60-100 Mm.
Regarding the nature of global coronal disturbances, some new ideas appeared.
For example, Attrill et al. (2007) attributed EIT waves to successive
reconnections of CME flanks with coronal loops, which could explain the
association of EIT ‘waves’ and shallow coronal dimmings which are formed
behind the bright front. Wills-Davey et al. (2007) proposed that slow EIT
waves are caused by MHD slow-mode soliton-like waves. Delannee et al. (2008)
performed a 3D MHD simulation to show that EIT ‘waves’ could be a signature of
a current shell formed around the erupting structure. Balasubramaniam et al.
(2007) demonstrated that the visibility of Moreton waves increases when
sweeping over filaments and filament channels, so they put forward the idea
that a significant contribution to the Moreton-wave Hα signature might be
coming from coronal material of enhanced density.
The question of the origin of coronal shocks and large amplitude waves
continues to be one of the central topics in this field. The published studies
showed a variety of results, some biased towards the CME-driven option, some
favoring the flare-ignited scenario, and some finding arguments for small
scale ejecta (e.g., Chen 2006; Pohjolainen & Lehtinen 2006; Shanmugaraju et
al. 2006a,b; Subramanian & Ebenezer 2006; Cho et al., 2007; Liu et al. 2007;
Reiner et al. 2007; White 2007; Grechnev et al. 2008; Muhr et al. 2008;
Veronig et al. 2008; Magdalenic et al. 2008; Mancuso & Avetta 2008;
Pohjolainen 2008). To illustrate the current level of ambiguity in such
studies, let us mention that for one well observed event two sets of authors
came to diametrally opposite conclusions: Vršnak et al. (2006) favored a flare
driver, whereas Dauphin et al. (2006) advocate a CME. The status of the
‘CME/flare controversy’ was reviewed recently by Vršnak & Cliver (2008).
Related to the formation and propagation of large-amplitude waves and shocks,
a number of important theoretical papers were published. Pagano et al. (2007)
investigated the role of magnetic fields and showed that a CME-driven wave
propagates to longer distances in the absence of magnetic field than in the
presence a weak open field. Ofman (2007) modeled the wave activity following a
flare by launching a velocity pulse into a model active region and
demonstrated that the resulting global oscillations are in good agreement with
observations. Employing the photospheric magnetic field measurements, Liu et
al. (2008) performed a 3D MHD simulation of a CME, and showed that the shock
segment at the nose of the CME remains quasi-parallel most of the time. In the
simulation of reconnection in a vertical current sheet, Barta et al. (2007)
revealed the formation of large-amplitude waves associated with changes of the
reconnection rate, which might explain flare-associated type II bursts in the
wake of CMEs. Zic et al. (2008) developed an analytical MHD model describing
the formation of large-amplitude waves by impulsively expanding 3D pistons.
The model provides an estimate of the time/distance at which the shock should
be formed, dependent on the source-surface acceleration, the terminal
velocity, the initial source size, the ambient Alfven speed, and plasma
$\beta$.
Finally, it should be noted that a comprehensive review on coronal waves and
shocks was published by Warmuth (2007). Gopalswamy (2006e) reviewed the
relationship between CMEs and type II bursts, while Mann & Vršnak (2007)
surveyed the relationship between CMEs, flares, coronal shocks, and particle
acceleration.
## 9 Coronal Dimming (R. Harrison and L. van Driel-Gesztelyi)
There is no strict definition of the phenomenon which we call coronal dimming.
Most authors consider coronal dimming to be a depletion of extreme-UV (EUV) or
X-ray emission from a large region of the corona, which is thought to be
closely associated with coronal mass ejection (CME) activity. Clearly,
understanding the onset phase of a CME is one of the key issues in solar
physics today, so the study of such dimming activity could well be of critical
importance. However, most of the literature deals with dimming in a rather
hand-waving manner, with the emphasis on phenomenological studies and
associations, no strict definitions of what constitutes a dimming event (e.g.,
the depth of the depletion in intensity, the size of the dimming region, etc.)
and little in terms of a physical interpretation of the plasma characteristics
of the dimming region. Having said that, some key studies are emerging which
do tackle such issues head on, and with the advent of the new STEREO and
Hinode spacecraft, along with the on-going SOHO and TRACE missions, as well as
the up-coming SDO mission, we have many tools to address this area of research
effectively.
Coronal dimming is not a newly discovered phenomenon; Rust and Hildner (1976)
reported such an event using Skylab observations. More recently, from the late
1990s, dimming was reported using X-ray and EUV, imaging and spectroscopic
data, from the SOHO and Yohkoh spacecraft (e.g. Sterling & Hudson 1997;
Harrison 1997; Gopalswamy & Hanaoka 1998; Zarro et al. 1999; Harrison & Lyons
2000), and dimming has taken center-stage in the study of mass ejection onset
in recent years (e.g., recent studies include Moore & Sterling 2007; Zhang et
al. 2007; Reinard & Biesecker 2008). In many ways coronal dimming has become a
well established phenomenon.
The majority of dimming reports involve EUV or X-ray imaging, and we have
excellent tools aboard SOHO, TRACE, STEREO and Hinode to identify and study
the topology and evolution of dimming regions. On the other hand, there are
spectroscopic studies of dimming which are providing key plasma information,
despite having limited fields of view or cadence. The combination of imaging
and spectroscopy is essential, but it is worth stressing some of the
spectroscopic studies because they stress the physical processes which are
involved in the dimming and, perhaps, the CME onset process.
EUV spectroscopy has been used to confirm that the dimming process represents
a loss of mass—i.e., it is a density depletion—rather than a change in
temperature (Harrison & Lyons 2000; Harrison et al. 2003). Indeed, these
studies have demonstrated the loss of between $4.3\\!\times\\!10^{10}$ and
$2.7\\!\times\\!10^{14}$ kg, in each case consistent with the mass of an
overlying, associated CME. If we are identifying the plasma which becomes
(part of) the CME, then this is an exciting phenomenon; studies focusing on
the properties of the dimming plasma, before, during and after the event, will
be essential for understanding the CME onset (Harrison & Bewsher 2007).
Hudson et al. (1996) showed that the timescale of the dimming formation
observed in Yohkoh/Soft X-ray Telescope (SXT; Tsuneta et al. 1991) data is
much faster than corresponding conductive and radiative cooling times. More
recently, data obtained by the Hinode/Extreme ultra-violet Imaging
Spectrometer (EIS; Culhane et al. 2007) have shown detection of Doppler
blueshifted plasma outflows of velocity $\approx 40$ km s-1 corresponding to a
coronal dimming (Harra et al. 2007). This result confirms a similar finding
(Harra & Sterling 2001) obtained with the SOHO/Coronal Diagnostic Spectrometer
(CDS; Harrison et al. 1995). In addition, SOHO/CDS limb observations have been
used to show the formation of a dimming region through the outward expansion
of pre-CME EUV loops (Harrison & Bewsher, 2007), which is consistent with such
blueshifts. Imada et al. (2007) find that Hinode/EIS data of a dimming shows a
dependence of the outflow velocity on temperature, with hotter lines showing a
stronger plasma outflow (up to almost 150 km s-1. These works collectively
support the primary interpretation of coronal dimmings as being due to plasma
evacuation.
Statistical studies are becoming important in truly establishing the
relationship with CMEs. Reinard & Biesecker (2008) have recently studied the
properties of 96 dimming events, using EUV imaging, associated with CME
activity. They confirmed earlier studies which showed that the dimming events
could be long-lasting, ranging from 1 to 19 hours, and compared the size of
the dimming regions to the associated CMEs. They also tracked the number of
dimming pixels through each event and showed that the ‘recovery’ after the
dimming often took the form of a two-part slope (plotted as dimming area vs.
time).
Bewsher et al. (2008) have produced the first statistical and probability
study of the dimming phenomenon using spectroscopy. They recognized that while
we have associated CMEs and dimming, there has not been a thorough statistical
study which can really identify the degree of that association, i.e., to put
that relationship on a firm footing. Using spectroscopy, they also recognized
the importance of studying this effect for different temperatures. They made
use of over 200 runs of a specific campaign using the SOHO spacecraft with an
automated procedure for identifying dimming.
Key results included the following: Up to 84% of the CMEs in the data period
can be back-projected to dimming events—and this appears to confirm the
association that we have been proposing. However, they also showed, as did
other spectral studies, that the degree of dimming varies between temperatures
from event to event. If different dimming events have different effects at
different temperatures then this is a problem for monitoring such events with
fixed-wavelength imagers.
Assuming that magnetic field lines of the CME are mostly rooted in the
dimmings, several properties derived from the study of dimmings can be used to
obtain information about the associated CME. Firstly, calculations of the
emission measure and estimates of the volume of dimmings can give a proxy for
the amount of plasma making up the CME mass (Sterling & Hudson 1997; Harrison
& Lyons 2000; Harrison et al. 2003; Zhukov & Auchère 2004). Secondly, the
spatial extent of coronal dimmings can give information regarding the angular
extent of the associated CME (Thompson et al. 2000; Harrison et al. 2003;
Attrill et al. 2007, van Driel-Gesztelyi et al. 2008). Thirdly, quantitative
measurement of the magnetic flux through dimmings can be compared to the
magnetic flux of modeled magnetic clouds (MC) at 1 AU (Webb et al. 2000;
Mandrini et al. 2005; Attrill et al. 2006; Qiu et al. 2007), see Démoulin
(2008) for a review. Fourth, studying the evolution of the dimmings,
particularly during their recovery phase can give information about the
evolution of the CME post-eruption (Attrill et al. 2006; Crooker & Webb 2006)
providing proof for e.g. magnetic interaction between the expanding CME and
open field lines of a neigboring coronal hole. Finally, study of the
distribution of the dimmings, their order of formation and measurement of
their magnetic flux contribution to the associated CME enabled Mandrini et al.
(2007) to derive an understanding of the CME interaction with its surroundings
in the low corona for the case of the complex 28 October 2003 event. They,
building on the model proposed by Attrill et al. (2007), demonstrated that
magnetic reconnection between field lines of the expanding CME with
surrounding magnetic structures ranging from small- to large-scale (magnetic
carpet, filament channel, active region) make some of the field lines of the
CME ‘step out’ from the flaring source region. Magnetic reconnection is driven
by the expansion of the CME core resulting from an over-pressure relative to
the pressure in the CME’s surroundings. This implies that the extent of the
lower coronal signatures match the final angular width of the CME. Through
this process, structures over a large-scale magnetic area become CME
constituents (for a review see van Driel-Gesztelyi et al. 2008). From the
wide-spread coronal dimming some additional mass is supplied to the CME.
Observations show that coronal dimmings recover whilst suprathermal uni- or
bi-directional electron heat fluxes are still observed at 1 AU in the related
ICME, indicating magnetic connection to the Sun. The questions why and how
coronal dimmings disappear whilst the magnetic connectivity is maintained was
investigated by Attrill et al. (2008) through the analysis of three CME-
related dimming events. They demonstrated that dimmings observed in SOHO/EIT
data recover not only by shrinking of their outer boundaries but also by
internal brightenings. They show that the model developed in Fisk & Schwadron
(2001) of interchange reconnections between ‘open’ magnetic field and small
coronal loops is applicable to observations of dimming recovery. Attrill et
al. (2008) demonstrate that this process disperses the concentration of ‘open’
magnetic field (forming the dimming) out into the surrounding quiet Sun, thus
recovering the intensity of the dimmings whilst still maintaining the magnetic
connectivity of the ejecta to the Sun.
Although this brief summary cannot report on all studies, it is clear that we
have made progress very recently in putting the dimming phenomenon on a firm
footing—the association is real—and we are making in-roads into studies of the
plasma activities leading to the dimming/CME onset process. With the
continuation of the SOHO mission, as well as TRACE, combined with the new
STEREO and Hinode missions and the up-coming SDO mission, this is a topic
which will receive much attention in the next few years.
## 10 The Link Between Low-Coronal CME Signatures and Magnetic Clouds (C.
Mandrini)
A major step to understanding the variability of the space environment is to
link the sources of coronal mass ejections (CMEs) to their interplanetary
counterparts, mainly magnetic clouds (MCs), a subset of interplanetary CMEs
characterized by enhanced magnetic field strength when compared to ambient
values, a coherent and large rotation of the magnetic field vector, and low
proton temperature (Burlaga 1995). Identifying the solar sources and comparing
qualitatively and quantitatively global characteristics and physical
parameters both in the Sun and the interplanetary medium provide useful tools
to constrain models in both environments.
Under the assumption that dimmings (see Section 9) at the Sun mark the
position of ejected flux rope footpoints (Webb et al. 2000), the magnetic flux
through these regions can be used as a proxy for the magnetic flux involved in
the ejection and, thus, be compared to the magnetic flux in the associated
interplanetary MC. Another proxy for the flux involved in an ejection is the
reconnected magnetic flux swept by flare ribbons, as they separate during the
evolution of two-ribbon flares. Using EUV dimmings as proxies and
reconstructing the MC structure from one spacecraft observations, Mandrini et
al. (2005) and Attrill et al. (2006) found that the magnetic flux in dimming
regions was comparable to the azimuthal MC flux, while the axial MC flux was
several times lower. Qui et al. (2007) analyzed and compared the reconnected
magnetic flux to the total MC flux, finding similar results (see also
Yurchyshyn et al. 2006; Longcope et al. 2007; Möstl et al. 2008, where MC data
from two spacecraft were used). These results led to the conclusion that the
ejected flux rope is formed by successive reconnections in a sheared arcade
during the eruption process, as opposed to the classical view of a previously
existing flux rope being ejected. However, in extreme events that occur in not
isolated magnetic configurations, it was found that the flux in dimmings did
not agree with the MC flux (Mandrini et al. 2007). This mismatch led these
authors to propose a scenario in which dimmings spread out to large distances
from the initial erupting region through a stepping reconnection process (in a
similar process to that proposed by Attrill et al., 2007, for the
interpretation of EIT waves). An overview of earlier works on quantitative
comparisons of solar and interplanetary global magnetohydrodynamic invariants,
such as magnetic flux and helicity, can be found in Démoulin (2008).
Qualitative comparisons are also useful tools to understand the eruption
process. Studying the temporal and spatial evolution of EUV dimmings, together
with soft X-ray coronal observations, in conjunction with interplanetary in
situ data of suprathermal electron fluxes, Attrill et al. (2006) and Crooker &
Webb (2006) derive an eruption scenario in which interchange magnetic
reconnection between the expanding CME loops and the open field lines of a
polar coronal hole led to the opening of one leg of the erupting flux rope.
Harra et al. (2007), combining EUV and Hα solar observations of eruptive
events with in situ magnetic field and suprathermal electron data, were able
to understand the sequence of events that produced two MCs with opposite
magnetic field orientations from the same magnetic field configuration.
The simple comparison of the magnetic field orientation in the erupting
configurations, which can be inferred from magnetograms, the directions of
filaments, coronal arcades or loops, with the axis of the associated MCs, can
give clues about the mechanism at the origin of solar eruptions. Green et al.
(2007) analyzed in detail associations of filament eruptions and corresponding
MCs, and they found that when the filament and MC axis differed by a large
angle, the direction of rotation was related to the magnetic helicity sign of
the erupting configuration (see also Harra et al. 2007). The rotation was
consistent with the conversion of twist into writhe, under the ideal MHD
constraint of helicity conservation, providing support for the assumption of a
flux rope topology where the kink instability sets in during the eruption (see
the review by Gibson et al. 2006).
## 11 Coronal Mass Ejections in the Heliosphere (R. Harrison)
In the 1970s the Helios spacecraft operated from solar orbits with perihelion
0.31 AU. Zodiacal light photometers were used to detect CMEs in the inner
heliosphere (see e.g. Richter et al. 1982; Jackson & Leinert 1985). CME images
were constructed from three photometers which scanned the sky using the
spacecraft rotation. More recently, a major advance was made with the launch,
in 2003, of the Solar Mass Ejection Imager (SMEI) aboard the Coriolis
spacecraft (Eyles et al. 2003). This instrument maps the entire sky with three
cameras each scanning $60^{\rm o}$ slices of the sky as the spacecraft moves
around the Earth, and thus, it has pioneered full-sky mapping aimed
specifically at the detection of CMEs propagating through the inner
heliosphere (see e.g. recent papers by Kahler & Webb 2007 and Jackson et al.
2007).
The combination of wide-angle heliospheric mapping from out of the Sun-Earth
line is now being satisfied by the Heliospheric Imagers (HI) (Harrison et al.
2008) aboard the NASA STEREO spacecraft. The development of these instruments
has come very much from the SMEI heritage and, with the unique opportunities
from the STEREO spacecraft locations, these instruments are able to image
those CME events directed towards the Earth. Indeed, for the first time, the
HI instruments provide a view of the passage of CMEs along virtually the
entire Sun-Earth line and such observations represent a major milestone in
investigations of the influence of solar activity on the Earth and human
systems.
Each HI instrument consists of two wide-angle telescopes mounted within a
baffle system enabling imaging of the heliosphere from the corona out to
Earth-like distances and beyond. The low scattered light levels and
sensitivity allow the detection of stars down to magnitudes of 12-13. This
performance is excellent for the detection of solar ejecta and solar wind
structure through the detection of Thomson scattered photospheric light off
free electrons in regions of density enhancement.
The STEREO spacecraft were launched in October 2006 with full scientific
operation of the HI instruments starting from April 2007. The spacecraft are
in near Earth-like solar orbits, with one ahead and one behind the Earth in
its orbit. They are drifting away at $22.5^{\rm o}$ per year (Earth-Sun-
spacecraft angle). The spacecraft are labelled STEREO A and STEREO B, for
ahead and behind.
The first HI observations of CMEs in the heliosphere, tracked out to Earth-
like distances, were reported by Harrison et al. (2008). The same instruments
are also reaping the benefits of wide-angle imaging of the heliosphere with
observations of comets (Fulle et al. 2007; Vourlidas et al. 2007), even the
imaging of co-rotating interaction regions (Sheeley et al. 2008a,b; Rouillard
et al. 2008a) and impacts of CMEs at other planets (Rouillard et al. 2008b).
With the HI instruments we now have a real opportunity to begin to relate the
coronal events that we call CMEs with their heliospheric counterparts,
commonly referred to as ICMEs - Interplanetary CMEs. Most ICME studies have
been performed utilizing in situ particle and field observations, and it is
clear that heliospheric imaging can provide a thorough test of the
interpretation of such in situ data on the topology and propagation of CMEs in
the heliosphere. Indeed, the uniqueness of this opportunity is well
illustrated by the fact that there are a number of extremely basic
observational tests which can be made with the new facility to underline our
current understanding of how CMEs travel out through the Solar System.
Crooker & Horbury (2006) have recently reviewed the propagation of ICMEs in
the heliosphere, utilizing in situ data. They note that cartoon sketches of
ICMEs commonly show magnetic field lines connected to the Sun at both ends.
Furthermore, reporting on the work of Gosling et al. (1987), Crooker et al.
(2002), and others, they note that it is widely accepted that counter-
streaming particle beams in ICMEs are a sign that both ends of the ICME are
indeed connected to the Sun. This is known as a ‘closed’ ICME. On the other
hand, uni-directional beams may signal connection at only one end—an ‘open’
ICME. Logically, then, the lack of beams would appear to signal disconnection
at both ends. In this case the ICME has become an isolated plasmoid.
Given this interpretation, in situ observations of ICMEs appear to show many
events which are apparently connected to the Sun at both footpoints, and
rather fewer events which appear to be connected at one end. Complete
disconnection of an ICME (a plasmoid) appears to be rare. In addition, the in
situ observations suggest that CMEs are connected to the Sun over extremely
long distances; Riley et al. (2004) looked for the degree of ‘openness’ of
ICMEs using observations of counter-streaming electrons from Ulysses data and
could detect no trend in the openness of ICMEs with distance out to Jupiter.
If ICME connectivity to the Sun is the same at 1 AU as it is at 5 AU then it
can be argued that an ascending CME could still be rooted at the Sun for a
week, or, indeed, much longer.
In reality, an ascending flux rope would most likely contain a mix of open and
closed field lines, driven by apparently random reconnection events (Crooker &
Horbury 2006; Gosling et al. 1995). Complete disconnection of the structure
appears to be unlikely.
With the new STEREO HI data we should be able to test this scenario, and this
has been reported by Harrison et al. (2008). The HI data appear to confirm the
in situ interpretation showing coherent structures, apparently still connected
to the Sun over long distances. There is no evidence for events pinching-off.
However, this in turn presents us with an anomaly. McComas (1995) has argued
that the heliospheric magnetic flux does not continually build up, so flux
must be shed through reconnection somehow during the ICME process. If we are
rejecting the plasmoid or disconnected ICME scenario then we must find another
way of limiting the flux build up over time.
In the absence of evidence for the pinching-off of CMEs, an interchange
reconnection process has been suggested as the mechanism by which CMEs
disconnect from the Sun (Gosling et al. 1995; Crooker et al. 2002). The basic
idea is that the ascending CME can travel a considerable distance, well beyond
the Earth, still connected to the Sun, and that perhaps days or even weeks
after the onset, the legs of the CME, still rooted in the Sun, will interact
with adjacent open field lines at low altitude in the corona; reconnection
results in the formation of low-lying loops as one CME leg reconnects with the
adjacent fields and an outward ascending kink-shaped structure ascends into
the heliosphere from the site of one of the original CME footpoints.
This approach has a few attractive points. For example, it seems logical that
the site of the greatest field density, magnetic complexity and field-line
motion would be the most likely site of any reconnection in the ascending CME.
However, assuming that such interchange reconnection is the ‘end game’ of a
CME, and that this low level reconnection results in the outward propagation
of a kinked field-line configuration, what might we expect to observe and,
indeed, have we seen such features? Harrison et al. (2008) indeed point to
observations of narrow V-shaped structures identified in the HI data that
could be candidates for such reconnection events.
It is early days for this work using STEREO but the indications are that there
is plenty to be gained from these studies. As the mission progresses we
anticipate more opportunities where we have the chance to combine both imaging
and in situ measurements of specific events, and their impacts, as well as to
model CMEs in the heliosphere in 3D as never before. Thus, this report should
be take as an early statement on the progress and direction of this work which
is opening a new chapter in solar, heliospheric and space weather physics.
## 12 Coronal Mass Ejections and Space Weather (N. Gopalswamy)
CMEs cause adverse space weather in two ways: (i) when they arrive at Earth’s
magnetosphere, they can couple to Earth’s magnetic field and cause major
geomagnetic storms (Gosling et al. 1990) and (ii) they can drive fast mode MHD
shocks that accelerate solar energetic particles (Reames 1999). Significant
progress has been made on both these aspects over the past few years. In the
case of geomagnetic storms, connecting the magnetic structure and kinematics
of ICMEs observed at 1 AU to the CME source region at the Sun has received
considerable attention. In the case of SEPs, assessing the contribution from
flare reconnection and shock to the observed SEP intensity has been the focus.
The importance of the variability in the Alfven speed profiles in the outer
corona is also under investigation because of its importance in deciding the
shock formation.
### 12.1 Geomagnetic Storms
High-Speed Solar Wind Streams (HSS) interacting with the slow solar wind
result in corotating interaction regions (CIRs), which also can produce
geomagnetic storms (Vršnak et al. 2007a), but they are generally weaker than
the CME-produced storms (Zhang et al. 2007). Occasionally, the CIR and ICME
structures combine to produce major storms (Dal Lago et al. 2006). Multiple
CMEs are often involved producing some super-intense storms (Gopalswamy et al.
2007; Zhang et al. 2007). There are numerous effects produced by the ICMEs in
the magnetosphere and various other layers down to the ground (see Borovsky et
al. 2006; Kataoka & Pulkkinen 2008).
The key element of ICMEs for the production of geomagnetic storms is the
southward magnetic field component. While the quite heliospheric field has no
out of the ecliptic field component (except for Alfvenic fluctuations in the
solar wind), a CME adds this component to the interplanetary (IP) magnetic
field. If an ICME has a flux rope structure, one can easily see that the
azimuthal component of the flux-rope field or its axial component forms the
out of the ecliptic component. In ICMEs with a flux rope structure (i.e.,
magnetic clouds), it is easy to locate the southward component from the
structure of the cloud (Gopalswamy 2006a; Wang et al. 2007; Gopalswamy et al.
2008a). In non-cloud ICMEs, it is not easy to infer the location of the
southward component. If the ICMEs are shock-driving, then the magnetosheath
between the shock and the driving ICME (Kaymaz & Siscoe 2006; Lepping et al.
2008) can contain southward field and hence cause geomagnetic storms
(Gopalswamy et al. 2008a). The cloud and sheath storms can be substantially
different (Pulkkinen et al. 2007).
Once an IP structure has a southward magnetic field, the efficiency with which
it causes geomagnetic storm depends on the strength of the magnetic field and
the speed with which it hits the magnetosphere (Gonzalez et al. 2007;
Gopalswamy 2008d). Statistical investigations have shown that the storm
intensity (measured e.g., by the Dst index) is best correlated with the speed-
magnetic field product in magnetic clouds and their sheaths. Interestingly, an
equally good correlation is obtained when the magnetic cloud/sheath speed is
replaced by the CME speed measured near the Sun (Gopalswamy et al. 2008a).
This suggests that if one can estimate the magnetic field in CMEs near the
Sun, the strength of the ensuing magnetic storm can be predicted. The ICME
speed can be predicted based on the CME speed by quantifying the interaction
between CMEs and the solar wind (Xie et al. 2006; Nakagawa et al. 2006; Jones
et al. 2007; Vršnak & Zic, 2007). Most of the storm-causing CMEs are halo
CMEs, which are subject to projection effects and hence space speeds cannot be
easily measured (Kim et al. 2007; Gopalswamy & Xie 2008; Howard et al. 2007;
Vršnak et al. 2007b). There have been several attempts to use the sky-plane
speed of CMEs to obtain their space speed (Xie et al. 2006; Michalek et al.
2008; Zhao 2008) with varying extents of success. The magnetic field strength
and kinetic energy of CMEs are somehow related to the free energy available in
the source region. Quantifying this free energy has been a difficult task
(Ugarte-Urra et al. 2007; Schrijver et al. 2008).
The solar sources of CMEs need to be close to the disk center for the CMEs to
make a direct impact on Earth and they have to be fast. In fact the solar
sources of magnetic clouds, storm-causing CMEs, and halo CMEs have been shown
to follow the butterfly diagram suggesting that only sunspot regions have the
ability to produce such energetic CMEs (Gopalswamy 2008d). The average near-
Sun speed of CMEs that cause intense geomagnetic storms is $\sim 1000$ km s-1
(Gopalswamy 2006b; Zhang et al. 2007), similar to the average speed of halo
CMEs (Gopalswamy et al. 2007) because many of the storm-producing CMEs are
halo CMEs. Halo CMEs are more energetic (Lara et al. 2006; Liu 2007;
Gopalswamy et al. 2007, 2008a) and end up being magnetic clouds at 1 AU. Most
halo CMEs ( 70%) are geoeffective. Non-geoeffective halos are generally
slower, originate far from the disk center, and originate predominantly in the
eastern hemisphere of the Sun. The geoeffectiveness rate of halo CMEs has been
reported to be anywhere from 40% to more than 80% (Yemolaev & Yermolaev 2006),
but the difference seems to be due to different definitions used for halo CMEs
(some authors have included all CMEs with width $>120^{\rm o}$ as halos) and
the sample size (Gopalswamy et al. 2007). The geoeffectiveness rate of CMEs
may be related to the fact that more ICMEs are observed as magnetic clouds
during solar minimum than during solar maximum (Riley et al. 2006). It is
possible that all ICMEs are magnetic clouds if viewed appropriately (Krall
2007). This suggestion is consistent with the ubiquitous nature of post
eruption arcades, which seem to indicate flux rope formation in the eruption
process (Kang et al. 2006; Qiu et al. 2007; Yurchyshyn 2008). While the
reconnection process certainly forms a flux rope, it is not clear if the
reconnection creates a new flux rope or fattens an existing one.
### 12.2 SEP Events
Energetic storm particle (ESP) events are the strongest evidence for SEP
acceleration by shocks, but this happens when the shocks arrive at the
observing spacecraft near Earth (Cohen et al. 2006). This means the shocks
must have been stronger near the Sun accelerating particles to much higher
energies. The strongest evidence for SEPs in flares is the gamma-ray lines,
which are now imaged by RHESSI (Lin 2007). All shock-producing CMEs are
associated with major flares (M- or X-class in soft X-rays), so both
mechanisms must operate in most SEP events. There has been an ongoing debate
as to which process is dominant based on SEP properties such as the spectral
and compositional variability at high energies (Tylka & Lee 2006; Cane et al.
2007).
The easiest way to identify shocks near the Sun are the type II radio bursts
especially at frequencies below 14 MHz, which correspond to the near-Sun IP
medium (Gopalswamy 2006c). Analyzing electrons and protons in SEP events,
Cliver & Ling (2007) have found evidence for a dominant shock process
including flatter SEP spectra, apparent widespread sources, and high
association with long wavelength type II bursts. A recent statistical study
finds the SEP association rate of CME steadily increases with CME speed and
width especially and there is one-to-one correspondence between SEP events and
CMEs from the western hemisphere with long wavelength type II bursts
(Gopalswamy et al. 2008b). Type II burst studies have also have concluded that
the variability in Alfven speed in the outer corona decides the formation and
strength of shocks (Shen et al. 2007; Gopalswamy et al. 2008c). For example, a
400 km s-1 CME can drive a shock, while a 1000 km s-1 CME may not drive a
shock, depending on the local Alfven speed.
James A. Klimchuk
President of the Commission
## References
* Abbett (2007) Abbett, W. P. 2007, ApJ, 665, 1469
* Abramenko et al. (2006) Abramenko, V. I., Fisk, L. A., & Yurchyshyn, V. B. 2006, ApJ (Lett), 641, L65
* (3) Amari, T., Aly, J. J., Mikic, Z., & Linker, J. 2007, ApJ (Lett) 671, L189
* (4) Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485
* (5) Aschwanden, M. J. 2008, ApJ (Lett), 672, L135
* (6) Aschwanden, M. J., Winebarger, A., Tsiklauri, D. & Peter, H. 2007, ApJ, 659, 1673
* (7) Attrill, G. D. R., Harra, L. K., van Driel-Gesztelyi, L., & Démoulin, P. 2007, ApJ 656, L101
* (8) Attrill, G. D. R., Nakwacki, M. S., Harra, L. K., van Driel-Gesztelyi, L., Mandrini, C. H., et al. 2006, Solar Phys. 238, 117
* (9) Attrill, G. D. R., van Driel-Gesztelyi, L., Démoulin, P., Zhukov, A. N., Steed, K., et al. 2008, Solar Phys. in press
* (10) Aulanier, G.; Golub, L.; DeLuca, E. E.; Cirtain, J. W.; Kano, R. et al. 2007, Science, 318, 1588
* Balasubramaniam et al. (2007) Balasubramaniam, K. S., Pevtsov, A. A., & Neidig, D. F. 2007, ApJ 658, 1372
* Barta et al. (2007) Barta, M., Karlicky, M., Vršnak, B., & Goossens, M. 2007, Cent. Eur. Astrophys. Bull. 31, 165
* (13) Battaglia, M.; Benz, A. O. 2006, A&A, 456, 751
* (14) Bazarghan, M., Safari, H., Innes, D. E., Karami, E., & Solanki, S. K. 2008, A&A, submitted
* (15) Benz, A. O. 2004, in: A. K. Dupree & A. O. Benz (eds.), Stars as Suns: Activity, Evolution, and Planets, Proc. IAU Symp. No. 219, p. 461
* (16) Borovsky, J. E., and Denton, M. H. 2006. JGR,111,A07S08.
* (17) Bradshaw, S. J. 2008, A&A, 486, L5
* (18) Bradshaw, S. J. & Cargill, P. J. 2006, A&A, 458, 987
* (19) Burlaga, L.F. 1995, Interplanetary Magnetohydrodynamics, New York: Oxford University Press
* (20) Cane, H. V., Richardson, I. G., and Rosenvinge, T. 2007, Space Sci. Rev, 130, 301, 2007
* (21) Cassak, P. A., Mullan, D. J., & Shay, M. A. 2008, ApJ (Lett), 676, L69
* Centeno et al. (2007) Centeno, R., Socas-Navarro, H., Lites, B., Kubo, M., Frank, Z. 2007, ApJL, 666, 137
* Chen (2006) Chen, P. F. 2006, ApJ 641, L153
* (24) Chen, W.-Z.; Liu, C.; Song, H.; Deng, N.; Tan, C.-Y.; Wang, H. 2007, Ch. J. A&A, 7, Issue 5, 733
* Cheung et al. (2008) Cheung, M. C. M., Schüssler, M., Tarbell, T. D., & Title, A. M. 2008, ApJ, in press
* Cho et al. (2007) Cho, K. S., Lee, J., Moon, Y. J., Dryer, M., et al. 2007, A&A 461, 1121
* (27) Cirtain, J. W., Golub, L., Lundquist, L., van Ballegooijen, A., Savcheva, A., Shimojo, M., et al. 2007, Science 318\. 1580
* (28) Cirtain, J. W. et al. 2007, ApJ, 655, 598
* (29) Cliver, E. W. & Ling, A. G. 2007, ApJ, 658, 1349
* (30) Cohen, C. M. S. 2006, in: N. Gopalswamy, R. Mewaldt, J. Torsti (eds.), Solar Eruptions and Energetic Particles, Geophysical Monograph Series, Vol. 165, p. 275
* (31) Crooker, N.U., Gosling, J.T., Kahler, S.W. 2002, JGR, 107 (A2)
* (32) Crooker, N.U., Horbury, T.S. 2006, Sp. Sci. Rev., 123, 93
* (33) Crooker, N. U., & Webb, D. F. 2006, JGR 111(A10), 8108
* (34) Cui, Y.; Li, R.; Wang, H.; He, H. 2007, Solar Phys., 242, 1
* (35) Culhane, J. L., Harra, L. K., James, A. M., Al-Janabi, K., Bradley, L. J., Chaudry, R. A., et al. 2007a, Solar Phys. 243, 19
* (36) Culhane, L., Harra, L. K., Baker, D., van Driel-Gesztelyi, L., Sun, J., Doschek, G. A., et al. 2007b, PASJ S751
* (37) Dahlburg, R. B., Liu, J.-H., Klimchuk, J. A., & Nigro, G. 2008, ApJ, submitted
* (38) Dal Lago, A. et al. 2006, JGR, 111, A07S14
* (39) Dauphin, C.; Vilmer, N.; Anastasiadis, A. 2007, A&A, 468, 273
* Dauphin et al. (2006) Dauphin, C., Vilmer, N., & Krucker, S. 2006, A&A 55, 339
* Delannée et al. (2008) Delannee, C., Torok, T., Aulanier, G., & Hochedez, J.-F. 2008, Solar Phys. 247, 123
* (42) Démoulin, P. 2008, Ann. Geophys. in press
* (43) Deng, N.; Xu, Y.; Yang, G.; Cao, W.; Liu, C. et al. 2006, ApJ, 644, 1278
* De Pontieu et al. (2007a) De Pontieu, B., Hansteen, V. H., Rouppe van der Voort, L., van Noort, M., & Carlsson, M. 2007a, ApJ, 655, 624
* De Pontieu et al. (2007b) De Pontieu, B., McIntosh, S. W., Carlsson, M., Hansteen, V. H., Tarbell, T. D. 2007b, Science, 318, 1574
* (46) DeVore, C. R., &Antiochos, S. K. 2008, ApJ 680, 740
* (47) Dolla, L. & Solomon, J. 2008, A&A 483, 271
* (48) Donea, A.-C.; Besliu-Ionescu, D.; Cally, P. S.; Lindsey, C.; Zharkova, V. V. 2006, Solar Phys., 239, Issue 1-2, 113
* (49) Doschek, G. A. 2006, ApJ 649, 515
* (50) Doschek, G. A., Mariska, J. T. & Warren, H. P. 2007, ApJ 667, L109
* (51) Drake, J. F.; Swisdak, M.; Che, H.; Shay, M. A. 2006, Nature, 443, 553
* (52) Eyles, C. J., Simnett, G. M., Cooke, M. P., Jackson, B. V., Buffington, A. 2003, Solar Phys., 217, 319
* (53) Feldman, U., Landi, E. & Doschek, G. A. 2008, ApJ 679, 843
* (54) Fisk, L. A., & Schwadron, N. A. 2001, ApJ 560, 425
* (55) Fletcher, L.; Hannah, I. G.; Hudson, H. S.; Metcalf, T. R. 2007, ApJ, 656, 1187
* (56) Fletcher, L.; Hudson, H. S., 2008 ApJ, 675, 1645
* (57) Fulle, M., Leblanc, F., Harrison, R. A., Davis, C. J., Eyles, C. J., Halain, J.-P. 2007, ApJ (Lett), 661, L93
* (58) Georgoulis, M. K.; Rust, D. M. 2007, ApJ (Lett), 661, L109
* (59) Gibson, S.E., Fan, Y., Török, T., Kliem, B. 2006, Space Scien. Rev. 124, 131
* (60) Golub, L., Deluca, E., Austin, G., Bookbinder, J., Caldwell, D., Cheimets, P., et al. 2007, Solar Phys. 243, 63
* (61) Gontikakis, C.; Anastasiadis, A.; Efthymiopoulos, C. 2007, MNRAS, 378, 1019
* (62) Gonzalez, W. D., Clua-Gonzalez, A. L., Echer, E. & Tsurutani, B. T. 2007, GRL, 34, L06101, doi: 10.1029/2006GL028879.
* (63) Gopalswamy, N. 2006a, Space Sci. Rev, 124, 145
* (64) Gopalswamy, N. 2006b, J. Astrophys. Astron., 27, 243
* (65) Gopalswamy, N. 2006c, in: N. Gopalswamy, R. Mewaldt, J. Torsti (eds.), Solar Eruptions and Energetic Particles, Geophysical Monograph Series, Vol. 165, p. 207
* (66) Gopalswamy, N. 2006d, J. Atm. Solar Terrestrial Phys., doi:10.1016/j.jastp.2008.06.010
* Gopalswamy (2006) Gopalswamy, N. 2006e, Geophys. Monogr. Ser., 165, 207
* (68) Gopalswamy, N., Akiyama, S., Yashiro, S., Michalek, G., & Lepping, R. P. 2008a., J. Atm. Solar Terrestrial Phys., 70, 245
* (69) Gopalswamy, N., & Xie, H. 2008, JGR, doi:10.1029/2008JA013030, in press
* (70) Gopalswamy, N., Yashiro, S., & Akiyama, S. 2007, JGR, 112, A06112
* (71) Gopalswamy, N., S. Yashiro, S. Akiyama, P. Makela, H. Xie, M., et al. 2008b, Ann. Geophysicae, 26, 1
* (72) Gopalswamy, N., Yashiro, S., Xie, H., Akiyama, S., Aguilar-Rodriguez, E., et al. 2008c, ApJ, 674, 560
* (73) Gosling, J. T., Baker, D. N., Bame, S. J., Feldman, W. C., Zwickl, R. D. 1987, JGR, 92 (11), 8519
* (74) Gosling, J. T., Bame, S. J. McComas, D. J., and Phillips, J. L. 1990, GRL, 127, 901
* (75) Gosling, J. T., Birn, J., Hesse, M. 1995, GRL, 22, 869
* (76) Green, L.M., Kliem, B., Török, T., van Driel-Gestelyi, L., Attrill, G.D.R., Solar Phys. 246, 365
* (77) Grigis, P. C.; Benz, A. O. 2006, A&A, 458, 641
* Grechnev et al. (2008) Grechnev, V. V., Uralov, A. M., Slemzin, V. A., Chertok, I. M., Kuzmenko, I. V., et al. 2008, Solar Phys. in press
* Hagenaar et al. (2008) Hagenaar, H. J., DeRosa, M. L., & Schrijver, C. J. 2008, ApJ, 678, 541
* (80) Hannah, I. G.; Krucker, S.; Hudson, H. S.; Christe, S.; Lin, R. P. 2008, A&A, 481, L45
* (81) Hansteen, V. H., de Pontieu, B. & Carlsson, M. 2007, PASJ 59, S699
* (82) Hara, H. et al. 2008, ApJ (Lett), 678, L67
* (83) Harra, L. K., Crooker, N. U., Mandrini, C. H., van Driel-Gesztelyi, L., Dasso, S. et al. 2007, Solar Phys. 244, 95
* (84) Harra, L. K., Hara, H., Imada, S., Young, P. R., Williams, D. R., et al. 2007, PASJ 59, S801
* (85) Harra, L.K., & Sterling, A.C. 2001, ApJ 561, L215
* (86) Harrison, R. A. 2006, in: N. Gopalswamy, R. Mewaldt, J. Torsti (eds.), Solar Eruptions and Energetic Particles, Geophysical Monograph Series 165, p. 73
* (87) Harrison, R. A., & Bewsher, D. 2007, A&A, 461, 1155
* (88) Harrison, R. A., Davis, C. J., Bewsher, D., Davies, J. A., Eyles, C. J. 2008, Adv. Space Res., submitted
* (89) Harrison, R. A., Davis, C. J., Eyles, C. J., Bewsher, D., Crothers, S. 2008, Solar Phys., 247, 171
* (90) Harrison, R. A., Sawyer, E. C., Carter, M. K., Cruise, A. M., Cutler, R. M., et al. 1995, Solar Phys. 162, 233
* (91) Howard, R. A., Moses, J. D., Vourlidas, A., Newmark, J. S., Socker, D. G., Plunkett, S. P., et al. 2008, SSRv 136, 67
* (92) Howard, T. A., D. Nandy, and A. C. Koepke 2008, JGR, 113, A01104
* (93) Hudson, H. S. 1991, Solar Phys. 133, 357
* (94) Hudson, H. S., Acton, L. W., & Freeland, S. L. 1996, ApJ 470, 629
* (95) Imada, S., Hara, H., Watanabe, T., Kamio, S., Asai, A., et al. 2007, PASJ 59, S793
* Ishikawa et al. (2008) Ishikawa, R., Tsuneta, S., Ichimoto, K., Isobe, H., Katsukawa, Y. 2008, A&A, 481, L25
* (97) Innes, D. 2008, A&A 481, L41
* (98) Isenberg, P.A., & Forbes, T.G. 2007, ApJ 670, 1453
* (99) Isobe, H.; Kubo, M.; Minoshima, T.; Ichimoto, K.; Katsukawa, Y et al. 2007, PASJ, 59, S807
* (100) Jackson, B. V., Hick, P. P., Buffington, A., Bisi, M. M. and Jensen, E. A. 2007, Proc. SPIE 6689
* (101) Jackson, B. V. & Leinert, C. 1985, JGR, 90, 10,759
* (102) Ji, H.; Huang, G.; Wang, H. 2007, ApJ, 660, Issue 1, 893
* (103) Jing, J.; Chae, J.; Wang, H. 2008, ApJ, 672, L73
* (104) Jones, R. A., A. R. Breen, R. A. Fallows, A. Canals, M. M. Bisi, et al. 2007, JGR, 112, A08107
* (105) Judge, P. J. 2008, ApJ 683, 87
* (106) Kahler, S. W., and D. F. Webb 2007, JGR, 112, A09103
* (107) Kamio, S., Hara, H., Watanabe, T., Matsuzaki, K., Shibata, K., et al. 2007, PASJ S757
* (108) Kang, S., Y.-J. Moon, K.-S. Cho, Y. Kim, Y. D. Park, et al. 2006, JGR, 111, A05102
* (109) Karlicky, M.; Barta, M. 2007, A&A, 464, 735
* (110) Karoff, C.; Kjeldsen, H. 2008, ApJ, 678, Issue 1, L73
* (111) Karpen, J. T. & Antiochos, S. K. 2008, ApJ, 676, 658
* (112) Kasparov , J.; Kontar, E. P.; Brown, J. C. 2007, A&A, 466, 705
* (113) Kataoka, R., and A. Pulkkinen 2008, JGR, 113, A03S12
* (114) Kim, K.-H., Y.-J. Moon, and K.-S. Cho 2007, JGR, 112, A05104
* (115) Kliem, B., & Török, T. 2007, Phys. Rev. L. 96(25), 255002.
* (116) Klimchuk, J. A. 2006, Solar Phys., 234, 41
* (117) Klimchuk, J. A. & Karpen, J. T. 2008, ApJ, in preparation
* (118) Klimchuk, J. A., Patsourakos, S., & Cargill, P. J. 2008, ApJ, 682, 1351
* (119) Kontar, Eduard P.; Brown, John C. 2006b, ApJ, 653, L149
* (120) Kontar, E. P.; MacKinnon, A. L.; Schwartz, R. A.; Brown, J. C. 2006a, A&A, 446, 1157
* (121) Kosovichev, A. G. 2006, Solar Phys., 238, 1
* (122) Kosovichev, A. G. 2007, ApJ, 670, L65
* (123) Kosovichev, A. G.; Zharkova, V. V. 1998, Nature, 393, 317
* (124) Kosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., et al. 2007, Solar Phys. 243, 3
* (125) Krall, J. 2007, ApJ, 657, 559
* (126) Krucker, S.; Hurford, G. J.; MacKinnon, A. L.; Shih, A. Y.; Lin, R. P. 2008, ApJ, 678, L63
* (127) Krucker, S.; Kontar, E. P.; Christe, S.; Lin, R. P. 2007a, ApJ, 663, L109
* (128) Krucker, S.; White, S. M.; Lin, R. P. 2007b, ApJ, 669, L49
* (129) LaBonte, B. J.; Georgoulis, M. K.; Rust, D. M. 2007, ApJ, 671, 955
* (130) Landi, R. & Feldman, U. 2008, ApJ, 672, 674
* (131) Lara, A., N. Gopalswamy, H. Xie, E. Mendoza-Torres, R. Perez-Er quez, et al. 2006, JGR, 111, A06107
* (132) Lee, K. W.; Buchner, J.; Elkina, N. 2008, A&A, 478, 889
* (133) Leka, K. D.; Barnes, G. 2007, ApJ, 656, 1173
* (134) Lepping, R. P.; Wu, C.-C.; Gopalswamy, N.; Berdichevsky, D. B. 2008, Solar Phys., 248, 125
* (135) Lin, C.-H., Banerjee, D., O’Shea, E. & Doyle, J. G. 2006, A&A 460, 597
* (136) Lin, J., & van Ballegooijen, A. A. 2005, ApJ 629, 582.
* Lites (2008) Lites, B. W. 2008, in A. Balogh (ed.), ISSI proceedings of a workshop on the solar dynamic magnetic field
* Lites et al. (2008) Lites, B. W., Kubo, M., Socas-Navarro, H., Berger, T., Frank, Z. 2008, ApJ, 672, 1237
* (139) Liu, C.; Lee, J.; Jing, J.; Gary, D. E.; Wang, H. 2008, ApJ, 672, L69
* Liu et al. (2007) Liu, C., Lee, J., Yurchyshyn, V., Deng, N., Cho, K-S., et al. 2007, ApJ 669, 1372
* (141) Liu, Y. 2007,ApJ, 654, L171
* (142) Liu, Y. 2008, ApJ 679, L151
* Liu et al. (2008) Liu, Y. C.-M., Opher, M., Cohen, O., Liewer, P. C., & Gombosi, T. I. 2008, ApJ 680, 757
* Long et al. (2008) Long, D. M., Gallager, P. T., McAteer, R. T. J., & Bloomfield, D. S. 2008, ApJ 680, L81
* (145) Longcope, D., Beveridge, C., Qiu, J., Ravindra, B., Barnes, G., et al. 2007, Solar Phys. 244, 45
* (146) López Fuentes, M. C., Klimchuk, J. A., & Mandrini, C. H. 2007, ApJ, 657, 1127
* Magdalenić et al. (2008) Magdalenić, J., Vršnak, B., Pohjolainen, S., Temmer, M., Aurass, H., et al. 2008, Solar Phys. in press
* Mancuso & Avetta (2008) Mancuso, S. & Avetta, D. 2008, ApJ 677, 683
* (149) Mandrini, C. H., Nakwacki, M. S., Attrill, G., van Driel-Gesztelyi, L., Démoulin, P., et al. 2007, Solar Phys. 244, 25
* (150) Mandrini, C. H., Pohjolainen, S., Dasso, S., Green, L. M., Démoulin, P., et al. 2005, A&A 434, 725
* (151) McIntosh, S. W., De Pontieu, B. & Tarbell, T. D. 2008, ApJ 673, L219
* (152) McTiernan, J. M. 2008, ApJ, submitted
* (153) Michalek, G.; Gopalswamy, N.; Yashiro, S. 2008, Solar Phys., 248, 113
* (154) Milligan, R. O. 2008, ApJ, 680, L157
* (155) Miteva, R.; Mann, G.; Vocks, C.; Aurass, H. 2007, A&A, 461, 1127
* (156) Mok, Y., Mikic, Z., Lionello, R., & Linker, J. A. 2008, ApJ (Lett), 679, L161
* (157) Moreno-Insertis, F.. Galsgaard, K., & Ugarte-Urra, I. 2008, ApJ 673, L211
* (158) Möstl, C., Miklenic, C., Farrugia, C.J., Temmer, M., Veronig, A., et al. 2008, Annales Geophys. in press
* Muhr et al. (2008) Muhr, N., Temmer, M., Veronig, A., Vrsnak, B., & Hanslmeier A. 2008, Cent. Eur. Astrophys. Bull. 32,79
* (160) Nakagawa, T., N. Gopalswamy, & S. Yashiro 2006, JGR, 111, A01108
* (161) Nitta, N. V.; Mason, G. M.; Wiedenbeck, M. E.; Cohen, C. M. S.; Krucker, S. et al. 2008, ApJ, 675, L125
* (162) Noglik, J. B. & Walsh, R. W. 2007, ApJ, 655, 1127
* Ofman (2007) Ofman, L. 2007, ApJ 655, 1134
* Orozco Suárez et al. (2007) Orozco Suárez, D., Bellot Rubio, L. R., del Toro Iniesta, J. C., Tsuneta, S., Lites, B. W. 2007, ApJ(Lett), 670, L61
* Pagano et al. (2007) Pagano, P., Reale, F., Orlando, S., & Peres, G. 2007, A&A 464, 753
* (166) Parenti, S., Buchlin, E., Cargill, P. J., Galtier, S., & Vial, J.-C. 2006, ApJ, 651, 1219
* (167) Parenti, S. & Young, P. R. 2008, A&A, submitted
* (168) Pariat, E., Antiochos, S.K., DeVore, C.R. 2008, ApJ in press
* (169) Patsourakos, S., Gouttebroze, P. & Vourlidas, A. 2007, ApJ 774, 1214
* (170) Patsourakos, S. & Klimchuk, J. A. 2006, ApJ, 647, 1452
* (171) Patsourakos, S. & Klimchuk, J. A. 2007, ApJ, 667, 591
* (172) Patsourakos, S. & Klimchuk, J. A. 2008a, ApJ, in press
* (173) Patsourakos, S. & Klimchuk, J. A. 2008b, ApJ, submitted
* (174) Patsourakos, S., Pariat, E., Vourlidas, A., Antiochos, S.K., Wuelser, J.P. 2008, ApJ 680, L73
* (175) Pauluhn, A., & Solanki, S. K. 2007, A&A, 462, 311
* (176) Peter, H. 2007, Adv. Space Res. 39, 1814
* (177) Peter, H., Gudiksen, B. & Nordlund, Å., 2006, ApJ 638, 1086
* Pohjolainen (2008) Pohjolainen, S. 2008, A&A 483, 297
* Pohjolainen & Lehtinen (2006) Pohjolainen, S. & Lehtinen, N. J. 2006, A&A 449, 359
* (180) Pulkkinen, T. I., Partamies, N., Huttunen, K. E. J., Reeves, G. D., & Koskinen, H. E. J. 2007, GRL34, L02105
* (181) Qiu, J., Hu, Q., Howard, T. A., & Yurchyshyn, V. B. 2007, ApJ 659, 758
* (182) Rappazzo, A. F., Velli, M., Einaudi, G., & Dahlburg, R. B. 2008, ApJ, 677, 1348
* (183) Reames, D. V. 1999, Space Sci. Rev, 90, 413
* (184) Reale, F. et al. 2007, Science, 318, 1582
* (185) Reale, F. & Orlando, S. 2008, ApJ, in press
* (186) Reale, F. et al. 2008, in preparation
* (187) Regnier, S.; Priest, E. R. 2007, A&A, 468, 701
* Reiner et al. (2007) Reiner, M. J., Krucker, S., Gary, D. E., Dougherty, B. L., Kaiser, M. L., et al. 2007, ApJ 657, 1107
* (189) Richter, I., Leinert, C., Planck, B. 1982, A&A, 110, 115
* (190) Riley, P., Gosling, J.T., Crooker, N.U. 2004, ApJ, 608, 1100
* (191) Riley, P., Schatzman, C., Cane, H. V., Richardson, I. G., Gopalswamy, N. 2006. ApJ, 647, 648
* (192) Rouillard, A. P., Davies, J. A., Forsyth, R. J., Davis, C. J., Harrison, R.A. 2008a, GRL, 35, L10110
* (193) Rouillard, A. P., Davies, J. A., Rees, A., Zhang, T., Forsyth, R. J. 2008b, JGR, in press
* (194) Rust, D. M., & LaBonte, B. J. 2005, ApJ 622, 69
* (195) Sakai, J. I.; Nagasugi, Y.; Saito, S.; Kaufmann, P. 2006, A&A, 457, 313
* (196) Sanchez Almeida, J., Teriaca, L. & Sütterlin, P, et al. 2007, A&A 475, 1101
* (197) Sakamoto, Y., Tsuneta, S., & Vekstein, G. 2008, ApJ, 689, in press
* (198) Sarkar, A. & Walsh, R. W. 2008, ApJ, 683, 516
* (199) Savcheva, A., Cirtain, J., Deluca, E. E., Lundquist, L. L., Golub, L., et al. 2007, PASJ 59, S771
* (200) Schmelz, J. T., Kashyap, V. L., & Weber, M. A. 2007a, ApJ (Lett), 660, L157
* (201) Schmelz, J. T. et al. 2007b, ApJ (Lett), 658, L119
* (202) Schrijver, C. J. 2001, Solar Phys., 198, 325
* (203) Schrijver, C. J. 2007, ApJ, 655, L117
* (204) Schrijver, C. J.; DeRosa, M. L.; Metcalf, T.; Barnes, G.; Lites, B. et al. 2008, ApJ, 675, 1637
* (205) Schrijver, C. J. et al. 2008, ApJ, 675, 1637
* Schüssler and Vögler (2008) Schüssler, M. & Vögler, A. 2008, A&A, 481, L5
* Shanmugaraju et al. (2006b) Shanmugaraju, A., Moon, Y.-J., Cho, K.-S., Dryer, M., & Umapathy, S. 2006b, Solar Phys. 233, 117
* Shanmugaraju et al. (2006a) Shanmugaraju, A., Moon, Y.-J., Kim, Y.-H., Cho, K.-S., Dryer, M. & Umapathy, S. 2006a, A&A 458, 653
* (209) Sheeley, N. R., Herbst, A. D., Palatchi, C. A., Wang, Y.-M., Howard, R. A. 2008, ApJ 674, L109
* (210) Sheeley, N. R., Herbst, A. D., Palatchi, C. A., Wang, Y.-M., Howard, R. A. 2008, ApJ 675, 853
* Shen et al. (2007) Shen,C., Wang, Y., Ye, P., Zhao, X. P., Gui, B., & Wang S. 2007, ApJ 670, 849
* (212) Siarkowski, M., Falewicz, R., Kepa, A., & Rudawy, P. 2008, Ann. Geophys., submitted
* (213) Silva, A. V. R.; Share, G. H.; Murphy, R. J.; Costa, J. E. R.; de Castro, C. G. et al. 2007, Solar Phys., 245, 311
* (214) Spadaro, D., Lanza, A. F., Karpen, J. T. & Antiochos, S. K. 2006, ApJ 642 579
* Subramanian & Ebenezer (2006) Subramanian, K. R. & Ebenezer, E. 2006, A&A 451, 683
* (216) Sudol, J. J.; Harvey, J. W. 2005, ApJ, 635, 647
* (217) Temmer, M.; Veronig, A. M.; Vrsnak, B.; Miklenic, C. 2007, ApJ, 654, 665
* (218) Temmer, M.; Veronig, A. M.; Vrsnak, B.; Rybak, J.; Gomory, P. et al. 2008, ApJ, 673, L95
* (219) Thompson, B. J., Cliver, E. W., Nitta, N., Delannée, C., & Delaboudiniére, J.P. 2000, GRL 27, 1431
* (220) Tian, H., Marsch, E., Tu, C.-Y. et al. 2008b, A&A 482, 267
* (221) Tian, H., Tu, C.-Y., Marsch, E. et al. 2008a, A&A 478, 915
* (222) Titov, V. S., Démoulin, P. 1999, A&A 351, 707
* (223) Tomczyk, S., McIntosh, S., Keil, S. W. et al. 2007, Science 317, 1192
* (224) Török, T., & Kliem, B. 2007, Astron. Nachr. 328, 743
* (225) Török, T., & Kliem, B. 2005, ApJ 630, L97
* (226) Tripathi, D., Mason, H. E., Young, P. R. & Del Zanna, G. 2008, A&A 481, L53
* (227) Tsuneta, S., Acton, L., Bruner, M., Lemen, J., Brown, W., et al. 1991, Solar Phys. 136, 37
* (228) Tylka, A. J. & Lee, M. A. 2006, ApJ, 646, 1319
* (229) Ugai, M. 2008, Phys. Plasmas, 15, 082306
* (230) Ugarte-Urra, I., Warren, H. P., & Brooks, D. H. 2008, ApJ, submitted
* (231) Ugarte-Urra, I., Warren, H. P., & Winebarger, A. R. 2007, ApJ 662, 1293
* (232) Uzdensky, D. A. 2007, ApJ, 671, 2139
* (233) van Driel-Gesztelyi, L., Attrill, G. D. R., Démoulin, P., Mandrini, C. H., & Harra, L.K. 2008, Ann. Geophys. in press
* (234) Veronig, A. M.; Karlicky, M.; Vrsnak, B.; Temmer, M.; Magdaleni, J. et al. 2006, A&A, 446, 675
* Veronig et al. (2008) Veronig, A. M., Temmer, M., & Vršnak, B. 2008, ApJ 681, L113
* Veronig et al. (2006) Veronig, A. M., Temmer, M., Vršnak, B. & Thalmann J. K. 2006, ApJ 647, 1466
* Vögler and Schüssler (2007) Vögler, A. & Schüssler, M. 2007, A&A Let., 465, 43
* (238) Vourlidas, A., Davis, C.J., Eyles, C.J., Crothers, S.R., Harrison, R.A. 2007, ApJ (Lett) 668, L79
* Vršnak, & Cliver (2008) Vršnak, B. & Cliver E. W. 2008, Solar Phys. in press
* (240) Vršnak, B.; Sudar, D.; Ru djak, D.; Zic, T. 2007b, A&A, 469, 339
* (241) Vršnak, B., Temmer, M., & Veronig, A. M. 2007a, Solar Phys., 440, 331
* Vršnak et al. (2006) Vršnak, B., Warmuth, A., Temmer, M., et al. 2006, A&A 448, 739
* (243) Vršnak, B. & Zic, T 2007, A&A, 472, 937
* (244) Wang, H. 2006, ApJ, 649, 490
* (245) Wang, Y., Ye, P., & Wang, S. 2007, Solar Phys., 240, 373
* Warmuth (2007) Warmuth, A. 2007, LNP 725, 107
* Warmuth et al. (2001) Warmuth, A., Vršnak, B., Aurass, H., & Hanslmeier, A. 2001, ApJ 560, L105
* (248) Warren, H. P. & Winebarger, A. R. 2007, ApJ, 666, 1245
* (249) Warren, H. P., Ugarte-Urra, I., Doschek, G. A., Brooks, D. H., & Williams, D. R. 2008a, ApJ, submitted
* (250) Warren, H. P., Winebarger, A. R., Mariska, J. T., Doschek, G. A., & Hara, H. 2008b, ApJ, 677, 1395
* (251) Webb, D. F., Lepping, R. P., Burlaga, L. F., DeForest, C. E., Larson, D. E., Martin, S. F., Plunkett, S. P., & Rust, D. M. 2000, JGR 105, 27251
* White (2007) White, S. M. 2007, Asian J. Phys. 16, 189
* (253) Williams, D. R., Török, T., Démoulin, P., van Driel-Gesztelyi, L., & Kliem, B. 2005, ApJ 628, L163
* Wills-Davey et al. (2007) Wills-Davey, M. J., DeForest, C. E., & Stenflo, J. O. 2007, ApJ 664, 556
* (255) Winebarger, A. R., Warren, H. P., & Falconer, D. A. 2008, ApJ, 676, 672
* (256) Xie, H.; Gopalswamy, N.; Manoharan, P. K.; Lara, A.; Yashiro, S.; et al. 2006, JGR, 111, A01103
* (257) Xu, Y.; Emslie, A. G.; Hurford, G. J. 2008, ApJ, 673, 576
* (258) Yashiro, S., Akiyama, S., Gopalswamy, N., Howard, R. A. 2006, ApJ (Lett), 650, L143
* (259) Yashiro, S., Michalek, G., Akiyama, S., Gopalswamy, N., & Howard, R. A. 2008, ApJ, 673, 1174
* (260) Yermolaev, Yu.I., & Yermolaev, M.Yu. 2006, Adv. Space Res., 37 (6), 1175
* (261) Young, P. R., Del Zanna, G. & Mason, H. E. 2007, PASJ 59, S727
* (262) Yurchyshyn, V. 2008, ApJ, 675, L49
* (263) Yurchyshyn, V.B., Liu, C., Abramenko, V., Krall, J. 2006, Solar Phys. 239, 317
* (264) Zhang, J., Ma, J., & Wang, H. 2006, ApJ 649, 464
* (265) Zhang, J., et al. 2007, JGR 112, A10102.
* (266) Zhang, Y., Wang, J., Attrill, G. D. R., Harra, L. K.; Yang, Z., et al. 2007, Solar Phys., 241, 329
* (267) Zhao, X. P. 2008, JGR, 113, A02101
* (268) Zharkova, V. V.; Gordovskyy, M. 2006, ApJ, 651, 553
* (269) Zharkova, V. V.; Zharkov, S. I. 2007, ApJ, 664, 573
* (270) Zhitnik, I. A. et al. 2006, Solar System Research, 40, 272
* (271) Zhukov, A.N., & Auchère, F. 2004, A&A 427, 705
* Žic et al. (2008) Žic, T., Vršnak, B., Temmer, M., & Jacobs, C. 2008, Solar Phys. in press
|
arxiv-papers
| 2008-09-08T20:23:25
|
2024-09-04T02:48:57.729693
|
{
"license": "Public Domain",
"authors": "J. A. Klimchuk, L. van Driel-Gesztelyi, C. J. Schrijver, D. B.\n Melrose, L. Fletcher, N. Gopalswamy, R. A. Harrison, C. H. Mandrini, H.\n Peter, S. Tsuneta, B. Vrsnak, J. Wang",
"submitter": "James Klimchuk",
"url": "https://arxiv.org/abs/0809.1444"
}
|
0809.1508
|
# Fermion Tunneling Beyond Semiclassical Approximation
Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
E-mail: bibhas@bose.res.in
Abstract:
Applying the Hamilton-Jacobi method beyond the semiclassical approximation
prescribed in [12] for the scalar particle, Hawking radiation as tunneling of
Dirac particle through an event horizon is analysed. We show that, as before,
all quantum corrections in the single particle action are proportional to the
usual semiclassical contribution. We also compute the modifications to the
Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black
hole. Finally, the coefficient of the logarithmic correction to entropy is
shown to be related with the trace anomaly.
Semiclassical methods of modeling Hawking radiation as a tunneling effect were
developed over the past decade and have generated a lot of interest [1, 2, 3,
4, 5, 6, 7, 8, 9, 10]. From this approach an alternative (intuitive) way of
understanding black hole radiation emerged. However, most of the calculations
in the literature [1, 2, 4, 5, 6, 7, 8, 9, 10] have been performed just for
scalar particles. Since a black hole can radiate all types of particles like a
black body, the emission spectrum should contain particles of all spins.
Therefore a detailed study of spin one-half particle emission is necessary.
Although there exist some computations [11] in this context, these are
confined to the semiclassical approximation and do not consider quantum
corrections.
In our previous work [12], we formulated the Hamilton-Jacobi method of
tunneling beyond semiclassical approximation by considering all the terms in
the expansion of the one particle action for a scalar particle. We showed that
the higher order terms are proportional to the semiclassical contribution.
This result, together with properties of conformal transformations, eventually
led to corrected expressions for thermodynamic variables of a black hole. It
is not obvious whether a similar analysis is valid for the case of spin-half
fermion tunneling. This issue is addressed here.
In this paper we will discuss the Dirac particle tunneling beyond
semiclassical approximation employing the Hamilton-Jacobi method suggested in
[12]. We will explicitly show that the higher order terms in the single
particle action are again proportional to the semiclassical contribution. By
dimensional argument the form of these proportionality constants, upto some
dimensionless parameters, are determined. In particular for Scwarzschild
spacetime, these are given by the inverse powers of the square of the mass of
the black hole, because in this case, the only macroscopic parameter is mass.
Using the principle of “detailed balance” [2, 9] the modified Hawking
temperature is identified. Then the corrections to the Bekenstein-Hawking area
law are derived by using the Gibbs form of first law of thermodynamics.
Interestingly, the leading order correction to the entropy is the logarithmic
of the semiclassical entropy which was found earlier in [13, 14, 15, 16, 17,
18, 19, 20, 21, 22]. Finally, using a constant scale transformation to the
metric, we show that the coefficient of the logarithmic correction is related
to trace anomaly.
Our method involves calculating the imaginary part of the action for the
(classically forbidden) process of s-wave emission across the horizon which in
turn is related to the Boltzmann factor for emission at the Hawking
temperature. We consider a massless Dirac particle in a general class of
static, spherically symmetric spacetime of the form
$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\Omega^{2}$ (1)
where the horizon $r=r_{H}$ is given by $f(r_{H})=g(r_{H})=0$. The massless
Dirac equation is given by
$\displaystyle i\gamma^{\mu}\nabla_{\mu}\psi=0$ (2)
where for this case the $\gamma$ matrices are defined as,
$\displaystyle\gamma^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{f(r)}}\left(\begin{array}[]{c c}i&0\\\
0&-i\end{array}\right);\,\,\ \gamma^{r}=\sqrt{g(r)}\left(\begin{array}[]{c
c}0&\sigma^{3}\\\ \sigma^{3}&0\end{array}\right)$ (7)
$\displaystyle\gamma^{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left(\begin{array}[]{c c}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right);\,\,\,\
\gamma^{\phi}=\frac{1}{r\textrm{sin}\theta}\left(\begin{array}[]{c
c}0&\sigma^{2}\\\ \sigma^{2}&0\end{array}\right).$ (12)
The covariant derivative is given by,
$\displaystyle\nabla_{\mu}=\partial_{\mu}+\frac{i}{2}\Gamma{{}^{\alpha}}{{}_{\mu}}{{}^{\beta}}\Sigma_{\alpha\beta};\,\,\
\Gamma{{}^{\alpha}}{{}_{\mu}}{{}^{\beta}}=g^{\beta\nu}\Gamma^{\alpha}_{\mu\nu};\,\,\
\Sigma_{\alpha\beta}=\frac{i}{4}\Big{[}\gamma_{\alpha},\gamma_{\beta}\Big{]}$
$\displaystyle\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$ (13)
Since for radial trajectories only the $(r-t)$ sector of the metric (1) is
important, (2) can be expressed as
$\displaystyle
i\gamma^{\mu}\partial_{\mu}\psi-\frac{1}{2}\Big{(}g^{tt}\gamma^{\mu}\Gamma^{r}_{\mu
t}-g^{rr}\gamma^{\mu}\Gamma^{t}_{\mu r}\Big{)}\Sigma_{rt}\psi=0.$ (14)
Here the required nonvanishing connections are
$\displaystyle\Gamma^{r}_{tt}=\frac{f^{\prime}g}{2};\,\,\
\Gamma^{t}_{tr}=\frac{f^{\prime}}{2f}.$ (15)
Therefore, under the metric (1), the Dirac equation (2) reduces to
$\displaystyle
i\gamma^{t}\partial_{t}\psi+i\gamma^{r}\partial_{r}\psi+\frac{f^{\prime}g}{2f}\gamma^{t}\Sigma_{rt}\psi=0$
(16)
and the matrix form of $\Sigma_{rt}$ from (12) and (13) is given by
$\displaystyle\Sigma_{rt}=\frac{i}{2}\left(\begin{array}[]{c c c
c}0&0&i\sqrt{\frac{f}{g}}&0\\\ 0&0&0&-i\sqrt{\frac{f}{g}}\\\
-i\sqrt{\frac{f}{g}}&0&0&0\\\ 0&i\sqrt{\frac{f}{g}}&0&0\end{array}\right).$
(21)
To solve (16) we employ the following ansatz for the spin up (i.e. +ve
$r$-direction) and spin down (i.e. -ve $r$-direction) $\psi$ as
$\displaystyle\psi_{\uparrow}(t,r)=\left(\begin{array}[]{c}A(t,r)\\\ 0\\\
B(t,r)\\\
0\end{array}\right){\textrm{exp}}\Big{[}\frac{i}{\hbar}I_{\uparrow}(t,r)\Big{]}$
(26) $\displaystyle\psi_{\downarrow}(t,r)=\left(\begin{array}[]{c}0\\\
C(t,r)\\\ 0\\\ D(t,r)\\\
\end{array}\right){\textrm{exp}}\Big{[}\frac{i}{\hbar}I_{\downarrow}(t,r)\Big{]}$
(31)
where $I(r,t)$ is the one particle action which will be expanded in powers of
$\hbar$. Here we will only solve the spin up case explicitly since the spin
down case is fully analogous. Substituting the ansatz (26) in (16), we obtain
the following two equations:
$\displaystyle\Big{(}\frac{iA}{\sqrt{f}}\partial_{t}I_{\uparrow}+B\sqrt{g}\partial_{r}I_{\uparrow}\Big{)}+\hbar\Big{(}\frac{1}{\sqrt{f}}\partial_{t}A-i\sqrt{g}\partial_{r}B+i\frac{f^{\prime}\sqrt{g}}{4f}B\Big{)}=0$
(32)
$\displaystyle\Big{(}-\frac{iB}{\sqrt{f}}\partial_{t}I_{\uparrow}+A\sqrt{g}\partial_{r}I_{\uparrow}\Big{)}+\hbar\Big{(}-\frac{1}{\sqrt{f}}\partial_{t}B-i\sqrt{g}\partial_{r}A+i\frac{f^{\prime}\sqrt{g}}{4f}A\Big{)}=0.$
(33)
Since the last terms within the first bracket of the above equations do not
involve the single particle action, they will not contribute to the
thermodynamic entities of the black hole. Therefore we will drop these two
terms. Now taking $I_{\uparrow}=I$ and expanding $I,A$ and $B$ in powers of
$\hbar$, we find,
$\displaystyle I(r,t)=I_{0}(r,t)+\displaystyle\sum_{i}\hbar^{i}I_{i}(r,t)$
$\displaystyle
A=A_{0}+\displaystyle\sum_{i}\hbar^{i}A_{i};\,\,\,B=B_{0}+\displaystyle\sum_{i}\hbar^{i}B_{i}.$
(34)
where $i=1,2,3,......$. In these expansions the terms from ${\cal{O}}(\hbar)$
onwards are treated as quantum corrections over the semiclassical value
$I_{0}$, $A_{0}$ and $B_{0}$ respectively. Substituting (34) in (32) and (33)
and then equating the different powers of $\hbar$ on both sides, we obtain the
following two sets of equations:
$\displaystyle{\textrm{Set I}}:~{}\hbar^{0}:$
$\displaystyle\frac{i}{\sqrt{f}}A_{0}\partial_{t}I_{0}+\sqrt{g}B_{0}\partial_{r}I_{0}=0$
(35) $\displaystyle\hbar^{1}:$
$\displaystyle\frac{i}{\sqrt{f}}A_{0}\partial_{t}I_{1}+\frac{i}{\sqrt{f}}A_{1}\partial_{t}I_{0}+\sqrt{g}B_{0}\partial_{r}I_{1}+\sqrt{g}B_{1}\partial_{r}I_{0}=0$
(36) $\displaystyle\hbar^{2}:$
$\displaystyle\frac{i}{\sqrt{f}}A_{0}\partial_{t}I_{2}+\frac{i}{\sqrt{f}}A_{1}\partial_{t}I_{1}+\frac{i}{\sqrt{f}}A_{2}\partial_{t}I_{0}$
$\displaystyle+$
$\displaystyle\sqrt{g}B_{0}\partial_{r}I_{2}+\sqrt{g}B_{1}\partial_{r}I_{1}+\sqrt{g}B_{2}\partial_{r}I_{0}=0$
. . . and so on. $\displaystyle{\textrm{Set II}}:~{}\hbar^{0}:$
$\displaystyle-\frac{i}{\sqrt{f}}B_{0}\partial_{t}I_{0}+\sqrt{g}A_{0}\partial_{r}I_{0}=0$
(38) $\displaystyle\hbar^{1}:$
$\displaystyle-\frac{i}{\sqrt{f}}B_{0}\partial_{t}I_{1}-\frac{i}{\sqrt{f}}B_{1}\partial_{t}I_{0}+\sqrt{g}A_{0}\partial_{r}I_{1}+\sqrt{g}A_{1}\partial_{r}I_{0}=0$
(39) $\displaystyle\hbar^{2}:$
$\displaystyle-\frac{i}{\sqrt{f}}B_{0}\partial_{t}I_{2}-\frac{i}{\sqrt{f}}B_{1}\partial_{t}I_{1}-\frac{i}{\sqrt{f}}B_{2}\partial_{t}I_{0}$
$\displaystyle+$
$\displaystyle\sqrt{g}A_{0}\partial_{r}I_{2}+\sqrt{g}A_{1}\partial_{r}I_{1}+\sqrt{g}A_{2}\partial_{r}I_{0}=0$
. . . and so on.
Equations (35) and (38) are collectively known as the semiclassical Hamilton-
Jacobi equations for a Dirac particle. Since the metric (1) is stationary it
has timelike Killing vectors. Thus we will look for solutions of (35) and (38)
which behave as
$\displaystyle I_{0}=\omega t+W(r),$ (41)
where $\omega$ is the energy of the particle. Substituting this in (35) and
(38) we obtain,
$\displaystyle\frac{iA_{0}}{\sqrt{f}}\omega+B_{0}\sqrt{g}W^{\prime}(r)=0$
$\displaystyle-\frac{iB_{0}}{\sqrt{f}}\omega+A_{0}\sqrt{g}W^{\prime}(r)=0.$
(42)
These two equations have two possible solutions:
$\displaystyle A_{0}=iB_{0};\,\,\
W_{+}(r)=\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$ $\displaystyle
A_{0}=-iB_{0};\,\,\,W_{-}(r)=-\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$
(43)
where $W_{+}$($W_{-}$) corresponds to ingoing (outgoing) solutions. The limits
of the integration are chosen such that the particle goes through the horizon
$r=r_{H}$. Therefore the solution for $I_{0}(r,t)$ is
$\displaystyle I_{0}(r,t)=\omega
t\pm\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}.$ (44)
Now, it is interesting to note that using (43) and (44) in the equations of
Set I and Set II simultaneously and then solving we get relations connecting
different orders in the expansion of $A$ with those of $B$:
$\displaystyle A_{a}=\pm iB_{a}$ (45)
where $a=0,1,2,3,....$. These lead to a simplified form of all the equations
in Set I and Set II as,
$\displaystyle\partial_{t}I_{a}=\pm\sqrt{fg}\partial_{r}I_{a}$ (46)
i.e. the functional form of the above individual linear differential equations
is same and is identical to the usual semiclassical Hamilton-Jacobi equations
(35) and (38). Therefore the solutions of these equations are not independent
and $I_{i}$’s are proportional to $I_{0}$. A similar situation happened for
scalar particle tunneling [12]. Since $I_{0}$ has the dimension of $\hbar$ the
proportionality constants should have the dimension of inverse of $\hbar^{i}$.
Again in the units $G=c=k_{B}=1$ the Planck constant $\hbar$ is of the order
of square of the Planck Mass $M_{P}$ and so from dimensional analysis the
proportionality constants have the dimension of $M^{-2i}$ where $M$ is the
mass of black hole. Specifically, for Schwarzschild type black holes having
mass as the only macroscopic parameter, these considerations show that the
most general expression for $I$, following from (34), valid for (46), is given
by,
$\displaystyle
I(r,t)=\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}I_{0}(r,t).$
(47)
where $\beta_{i}$’s are dimensionless constant parameters.
The above analysis shows that to obtain a solution for $I(r,t)$ it is
therefore enough to solve for $I_{0}(r,t)$ which has the solution of the form
(44). In fact the standard Hamilton-Jacobi solution determined by this
$I_{0}(r,t)$ is just modified by a prefactor to yield the complete solution
for $I(r,t)$. Substituting (44) in (47) we obtain
$\displaystyle
I(r,t)=\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}\Big{[}\omega
t\pm\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$ (48)
Therefore the ingoing and outgoing solutions of the Dirac equation (2) under
the background metric (1) is given by exploiting (26) and (48),
$\displaystyle\psi_{{\textrm{in}}}\sim{\textrm{exp}}\Big{[}-\frac{i}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega
t+\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$ (49)
and
$\displaystyle\psi_{{\textrm{out}}}\sim{\textrm{exp}}\Big{[}-\frac{i}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega
t-\omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}.$ (50)
Now for the tunneling of a particle across the horizon the nature of the
coordinates change. The sign of the metric coefficients in the $(r-t)$ sector
is altered. This indicates that ‘$t$’ coordinate has an imaginary part for the
crossing of the horizon of the black hole and correspondingly there will be a
temporal contribution to the probabilities for the ingoing and outgoing
particles. This has similarity with[23] where they show for the Schwarzschild
metric that two patches across the horizon are connected by a discrete
imaginary amount of time.
The ingoing and outgoing probabilities of the particle are, therefore, given
by,
$\displaystyle
P_{{\textrm{in}}}=|\psi_{{\textrm{in}}}|^{2}\sim{\textrm{exp}}\Big{[}\frac{2}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega{\textrm{Im}}~{}t+\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(51)
and
$\displaystyle
P_{{\textrm{out}}}=|\psi_{{\textrm{out}}}|^{2}\sim{\textrm{exp}}\Big{[}\frac{2}{\hbar}(1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}})\Big{(}\omega{\textrm{Im}}~{}t-\omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(52)
Now the ingoing probability $P_{\textrm{in}}$ has to be unity in the classical
limit (i.e. $\hbar\rightarrow 0$) - when there is no reflection and everything
is absorbed - instead of zero or infinity [12].Thus, in the classical limit,
(51) leads to,
$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}.$
(53)
From the above one can easily show that ${\textrm{Im}}~{}t=-2\pi M$ for the
Schwarzschild spacetime which is precisely the imaginary part of the
transformation $t\rightarrow t-2i\pi M$ when one connects the two regions
across the horizon as shown in [23]. Therefore the probability of the outgoing
particle is
$\displaystyle
P_{{\textrm{out}}}\sim{\textrm{exp}}\Big{[}-\frac{4}{\hbar}\omega\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$
(54)
Now using the principle of “detailed balance” [2, 9]
$\displaystyle
P_{{\textrm{out}}}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{h}}\Big{)}P_{\textrm{in}}={\textrm{exp}}\Big{(}-\frac{\omega}{T_{h}}\Big{)}$
(55)
we obtain the temperature of the black hole as
$\displaystyle
T_{h}=T_{H}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\Big{)}^{-1}$
(56)
where
$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{(}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}^{-1}$
(57)
is the standard semiclassical Hawking temperature of the black hole and other
terms are the corrections due to the quantum effect. Using this expression and
knowing the metric coefficients $f(r)$ and $g(r)$ one can easily find out the
temperature of the corresponding black hole. The same result was also obtained
in [12] for scalar particle tunneling.
For the Schwarzschild black hole the metric coefficients are
$\displaystyle f(r)=g(r)=(1-\frac{r_{H}}{r});\,\,\,r_{H}=2M.$ (58)
Therefore using (56) and (57) it is easy to write the corrected Hawking
temperature:
$\displaystyle T_{h}=\frac{\hbar}{8\pi
M}\Big{(}1+\sum_{i}\beta_{i}\frac{\hbar}{M^{2i}}\Big{)}^{-1}.$ (59)
Now use of the Gibbs form of first law of thermodynamics gives the corrected
form of the Bekenstein-Hawking entropy:
$\displaystyle S_{\textrm{bh}}$ $\displaystyle=$
$\displaystyle\int\frac{dM}{T_{h}}=\frac{4\pi M^{2}}{\hbar}+8\pi\beta_{1}\ln
M-\frac{4\pi\hbar\beta_{2}}{M^{2}}+{\textrm{higher order terms in $\hbar$}}$
(60) $\displaystyle=$ $\displaystyle\frac{\pi
r_{H}^{2}}{\hbar}+8\pi\beta_{1}\ln
r_{H}-\frac{16\pi\hbar\beta_{2}}{r_{H}^{2}}+{\textrm{higher order terms in
$\hbar$}}$
The area of the event horizon is
$\displaystyle A=4\pi r_{H}^{2}$ (61)
so that,
$\displaystyle S_{\textrm{bh}}$ $\displaystyle=$
$\displaystyle\frac{A}{4\hbar}+4\pi\beta_{1}\ln
A-\frac{64\pi^{2}\hbar\beta_{2}}{A}+......................$ (62)
It is noted that the first term is the usual semiclassical contribution to the
area law $S_{\textrm{BH}}=\frac{A}{4\hbar}$ [24, 25]. The other terms are the
quantum corrections. Now it is possible to express the quantum corrections in
terms of $S_{\textrm{BH}}$ by eliminating $A$:
$\displaystyle S_{\textrm{bh}}=S_{\textrm{BH}}+4\pi\beta_{1}\ln
S_{\textrm{BH}}-\frac{16\pi^{2}\beta_{2}}{S_{\textrm{BH}}}+.........$ (63)
Interestingly the leading order correction is logarithmic in $A$ or
$S_{\textrm{BH}}$ which was found earlier in [13, 14] by field theory
calculations and later in [15, 20] by quantum geometry method. The higher
order corrections involve inverse powers of $A$ or $S_{\textrm{BH}}$.
To determine the value of the coefficients $\beta_{1},\beta_{2}$ etc we will
adopt the following steps. The point is that nonzero values for these
coefficients are related to quantum corrections (loop effects). Such
corrections, in a field theoretical approach, are manifested by the presence
of anomalies. Now it is a well known fact that it is not possible to
simultaneously preserve general coordinate (diffeomorphism) invariance and
conformal invariance. Retaining general coordinate invariance, one finds the
breakdown of conformal invariance leading to the presence of nonvanishing
trace of the stress tensor. We now show that the coefficients appearing in
(63) are related to this trace anomaly.
We begin by studying the behaviour of the action (47) upto order $\hbar^{2}$
under an infinitesimal constant scale transformation, parametrised by $k$, of
the metric coefficients,
$\displaystyle\tilde{g}{{}_{\mu\nu}}=kg_{\mu\nu}\simeq(1+\delta k)g_{\mu\nu}.$
(64)
Under this the metric coefficients of (1) change as
$\tilde{f}=kf,\tilde{g}=k^{-1}g$. Also, in order to preserve the scale
invariance of the Dirac equation (2), the field $\psi$ should transform as
$\tilde{\psi}=k^{\frac{1}{2}}\psi$. On the other hand, $\psi$ has the
dimension of ${(\textrm{mass})}^{\frac{3}{2}}$ and since in our case the only
mass parameter is the black hole mass $M$, the infinitesimal change of it is
given by,
$\displaystyle\tilde{M}=k^{\frac{1}{3}}M\simeq(1+\frac{1}{3}\delta k)M.$ (65)
Now from (54) the imaginary part of the semiclassical contribution of the
single particle action is
$\displaystyle\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}=-2\omega{\textrm{Im}}~{}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$
(66)
where $\omega$ gets identified with the energy (i.e. mass $M$) of a stable
black hole [8]. Therefore $\omega$ and $\hbar$ transforms like (65) and
$M^{2}$ respectively under (64).
Considering only the $\hbar$ and $\hbar^{2}$ order terms in (47) and using
(65) we obtain, under the scale transformation,
$\displaystyle{\tilde{I}}_{(1+2)}$ $\displaystyle\equiv$
$\displaystyle\hbar\textrm{Im}\tilde{I}{{}_{1}}{{}_{(\textrm{out})}}+\hbar^{2}\textrm{Im}\tilde{I}{{}_{2}}{{}_{(\textrm{out})}}$
(67) $\displaystyle=$
$\displaystyle\Big{(}\frac{{\tilde{\hbar}}\beta_{1}}{\tilde{M}^{2}}+\frac{{\tilde{\hbar}}^{2}\beta_{2}}{\tilde{M}^{4}}\Big{)}\textrm{Im}\tilde{I}{{}_{0}}{{}_{(\textrm{out})}}$
$\displaystyle\simeq$
$\displaystyle\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}+\frac{{{\hbar}}^{2}\beta_{2}}{{M}^{4}}\Big{)}(1+\frac{1}{3}\delta
k)\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$ $\displaystyle=$ $\displaystyle
I_{(1+2)}+\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}+\frac{{{\hbar}}^{2}\beta_{2}}{{M}^{4}}\Big{)}\frac{1}{3}\delta
k\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$
Therefore
$\displaystyle\delta I_{(1+2)}$ $\displaystyle=$
$\displaystyle{\tilde{I}}_{(1+2)}-I_{(1+2)}$ (68) $\displaystyle\simeq$
$\displaystyle\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}+\frac{{{\hbar}}^{2}\beta_{2}}{{M}^{4}}\Big{)}\frac{1}{3}\delta
k\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$
leading to,
$\displaystyle\frac{\delta I_{(1+2)}}{\delta
k}=\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}+\frac{{{\hbar}}^{2}\beta_{2}}{{M}^{4}}\Big{)}\frac{1}{3}\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$
(69)
Now use of the definition of the energy-momentum tensor and (69) yields,
$\displaystyle\textrm{Im}\int d^{4}x\sqrt{-g}T_{\mu}^{\mu}=\frac{2\delta
I_{(1+2)}}{\delta
k}=\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}+\frac{{{\hbar}}^{2}\beta_{2}}{{M}^{4}}\Big{)}\frac{2}{3}\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$
(70)
Thus, in the presence of a trace anomaly, the action is not invariant under
the scale transformation. Since for the Schwarzschild black hole $f(r)$ and
$g(r)$ are given by (58), from (66) we obtain
$\textrm{Im}S_{0}^{(\textrm{out})}=-4\pi\omega M$. Substituting this in (70)
we obtain for $\omega=M$ as,
$\displaystyle\hbar\beta_{1}+\frac{\hbar^{2}\beta_{2}}{M^{2}}=-\frac{3}{8\pi}\textrm{Im}\int
d^{4}x\sqrt{-g}T_{\mu}^{\mu}$ (71)
where $T_{\mu}^{\mu}$ is calculated upto two loops. Starting from the action
(47) and following the identical steps as above, a similer relation among all
$\beta$’s with the right hand side of (71) can be established. In this case
$T_{\mu}^{\mu}$ is due to all loop expansions.
Since the higher loop calculations to get $T_{\mu\nu}$ (from which
$T_{\mu}^{\mu}$ is obtained) is very much complicated, usually in literature
[28] only one loop calculation for $T_{\mu\nu}$ is discussed. Thus, comparing
only the $\hbar^{1}$ order on both sides of (71), we obtain,
$\displaystyle\beta_{1}=-\frac{3}{8\pi}{\textrm{Im}}\int
d^{4}x\sqrt{-g}{T^{\mu}_{\mu}}^{(1)}$ (72)
This relation clearly shows that $\beta_{1}$ is connected to the trace
anomaly. A similar relation is given in [26] where it has been shown that the
coefficient $\beta_{1}$ is related to trace anomaly for the scalar particle
tunneling. The only difference is the factor before the integration. This
agrees well with the earlier conclusion [27, 13] where using conformal field
theory technique it was shown that this $\beta_{1}$ is related to trace
anomaly and is given by,
$\displaystyle\beta_{1}=-\frac{1}{360\pi}\Big{(}-N_{0}-\frac{7}{4}N_{\frac{1}{2}}+13N_{1}+\frac{233}{4}N_{\frac{3}{2}}-212N_{2}\Big{)}$
(73)
‘$N_{s}$’ denotes the number of fields with spin ‘$s$’. In our case
$N_{\frac{1}{2}}=1$ and $N_{0}=N_{1}=N_{\frac{3}{2}}=N_{2}=0$.
To conclude, we have successfully extended our approach [12] of scalar
particle tunneling beyond semiclassical approximation to the model of fermion
tunneling. We have considered all orders in the single particle action for
fermion tunneling through the event horizon of the black hole. We showed that
higher order correction terms of the action are proportional to the
semiclassical contribution. A similar result was shown earlier in [12] for the
scalar particle tunneling. By dimensional argument and principle of “detailed
balance” the same form of the modified Hawking temperature, as in the scalar
case, was recovered. The logarithmic and inverse powers of area corrections to
the Bekenstein-Hawking area law were also reproduced. Finally, we showed that
the coefficient of the logarithmic term of entropy is related to trace
anomaly. However, the prefactor appearing in this term is different from the
scalar particle example, a result that is supported by earlier works [27, 13].
Here we have only told about $\beta_{1}$. Discussions on other coefficients
can also be given from (71), but since no information about $T_{\mu\nu}$ due
to multi-loops is available in the literature, we cannot say anything about
them at this moment.
Acknowledgment:
I wish to thank Prof. Rabin Banerjee for suggesting this investigation and
constant encouragement.
## References
* [1] M.K.Parikh and F.Wilczek, Phys. Rev. Lett. 85, 5042 (2000) [arXiv:hep-th/9907001].
M.K.Parikh, Int. J. Mod. Phys. D 13, 2351 (2004) [arXiv:hep-th/0405160].
* [2] K.Srinivasan and T.Padmanabhan, Phys. Rev. D 60, 024007 (1999) [arXiv:gr-qc/9812028].
S. Shankaranarayanan, K. Srinivasan and T. Padmanabhan, Mod. Phys. Lett. A 16,
571 (2001) [arXiv:gr-qc/0007022].
S.Shankaranarayanan, T.Padmanabhan and K.Srinivasan, Class. Quantum Grav. 19,
2671 (2002) [arXiv:gr-qc/0010042].
S.Shankaranarayanan, Phys. Rev. D 67, 084026 (2003) [arXiv:gr-qc/0301090].
* [3] V.A.Berezin, A.M.Boyarsky and A.Yu.Neronov, Gravitation and Cosmology 5, 16 (1999) [arXiv:gr-qc/0605099].
* [4] M.Arzano, A.J.M.Medved and E.C.Vagenas, JHEP 0509, 037 (2005) [arXiv:hep-th/0505266].
A.J.M.Medved and E.C.Vagenas, Mod. Phys. Lett. A 20, 2449 (2005) [arXiv:gr-
qc/0504113].
* [5] Qing-Quan Jiang, Shuang-Qing Wu and Xu Cai, Phys. Rev. D 73 064003 (2006) [arXiv:hep-th/0512351].
Yapeng Hu, Jingyi Zhang and Zheng Zhao, Mod. Phys. Lett. A 21 2143 (2006)
[arXiv:gr-qc/0611026].
Zhibo Xu and Bin Chen, Phys. Rev. D 75 024041 (2007) [arXiv:hep-th/0612261].
Cheng-Zhou Liu and Jian-Yang Zhu, “Hawking radiation as tunneling from
Gravity’s rainbow”, [arXiv:gr-qc/0703055].
Qing-Quan Jiang and Shuang-Qing Wu, Phys. Lett. B 635, 151 (2006) [arXiv:hep-
th/0511123].
Yapeng Hu, Jingyi Zhang and Zheng Zhao, Int.J.Mod.Phys. D 16, 847 (2007)
[arXiv:gr-qc/0611085].
G.E.Volovik, “On de Sitter radiation via quantum tunneling”,
[arXiv:0803.3367].
Jingyi Zhang, “Black hole quantum tunnelling and black hole entropy
correction” [arXiv:0806.2441].
* [6] E.T.Akhmedov, V.Akhmedova and D.Singleton, Phys. Lett. B 642, 124 (2006) [arXiv:hep-th/0608098].
E.T.Akhmedov, V.Akhmedova, D.Singleton and T.Pilling, Int.J.Mod.Phys. A 22,
1705 (2007) [arXiv:hep-th/0605137].
T.Pilling, Phys. Lett. B 660, 402 (2008) [arXiv:0709.1624].
T.K.Nakamura, “Factor two discrepancy of Hawking radiation temperature”,
[arXiv:0706.2916].
B.D.Chowdhury, Pramana 70, 593 (2008) [arXiv:hep-th/0605197].
* [7] M.Angheben, M.Nadalini, L.Vanzo and S.Zerbini, JHEP 0505, 014 (2005) [arXiv:hep-th/0503081].
R.Kerner and R.B.Mann, Phys. Rev. D 73, 104010 (2006) [arXiv:gr-qc/0603019].
P.Mitra, Phys. Lett. B 648, 240 (2007) [arXiv:hep-th/0611265].
* [8] R.Banerjee and B.R.Majhi, Phys. Lett. B 662, 62 (2008) [arXiv:0801.0200].
* [9] R.Banerjee, B.R.Majhi and S.Samanta, Phys. Rev. D 77 124035 (2008) [arXiv:0801.3583].
* [10] R.Banerjee, B.R.Majhi and S.K.Modak, “Area Law in Noncommutative Schwarzschild Black Hole”, [arXiv:0802.2176].
* [11] R.Kerner and R.B.Mann, Class. Quant. Grav. 25, 095014 (2008) [arXiv:0710.0612].
Ran Li and Ji-Rong Re, Phys. Lett. B 661 370 (2008) [arXiv:0802.3954].
R.D.Criscienzo and L.Vanzo, “Fermion Tunneling from Dynamical Horizons”,
[arXiv:0803.0435].
Ran Li and Ji-Rong Ren, Class. Quant. Grav. 25 125016 (2008)
[arXiv:0803.1410].
R.Kerner and R.B.Mann, Phys. Lett. B 665 277 (2008) [arXiv:0803.2246].
De-You Chen, Qing-Quan Jiang, Shu-Zheng Yang and Xiao-Tao Zu, “Fermions
tunneling from the charged dilatonic black holes”, [arXiv:0803.3248].
D.Y.Chen, Q.Q.Jiang and X.T.Zu, Phys. Lett. B 665 106 (2008)
[arXiv:0804.0131].
Shiwei Zhou, Wenbiao Liu, Phys. Rev. D 77 104021 (2008).
Q.Q.Jiang, Phys.Rev. D 78 044009, (2008).
* [12] R.Banerjee and B.R.Majhi, JHEP 0806 095 (2008) [arXiv:0805.2220].
* [13] D.V.Fursaev, Phys. Rev. D 51, R5352 (1995) [arXiv:hep-th/9412161].
* [14] R.B.Mann and S.N.Solodukhin, Nucl. Phys. B 523, 293 (1998) [arXiv:hep-th/9709064].
* [15] R.K.Kaul and P.Majumdar, Phys. Rev. Lett. 84, 5255 (2000) [arXiv:gr-qc/0002040].
* [16] T.R.Govindarajan, R.K.Kaul and V.Suneeta, Class. Quantum Grav. 18, 2877 (2001) [arXiv:gr-qc/0104010].
* [17] S.Das, P.Majumdar and R.K.Bhaduri, Class. Quantum Grav. 19, 2355 (2002) [arXiv:hep-th/0111001].
* [18] S.S.More, Class. Quantum Grav. 22, 4129 (2005) [gr-qc/0410071].
* [19] S.Mukherji and S.S.Pal, JHEP 0205, 026 (2002) [arXiv:hep-th/0205164].
* [20] A.Ghosh and P.Mitra, Phys. Lett. B 616, 114 (2005) [arXiv:gr-qc/0411035].
* [21] For a review and a complete list of papers on logarithmic corrections, see D.N.Page, New Journal of Phys. 7, 203 (2005) [arXiv:hep-th/0409024].
* [22] S.K.Modak, “Corrected entropy of BTZ black hole in tunneling approach”, [arXiv:0807.0959].
* [23] E.T.Akhmedov, T.Pilling and D.Singleton, “Subtleties in the quasi-classical calculation of Hawking radiation”, [arXiv:0805.2653].
* [24] J.D.Bekenstein, PhD Thesis Princeton University, Princeton, NJ (1972).
J.D.Bekenstein, Lett. Nuovo Cimento 4 737 (1972).
J.D.Bekenstein, Phys. Rev. D 7, 2333 (1973).
J.D.Bekenstein, Phys. Rev. D 9 3292 (1974).
* [25] J.M.Bardeen, B.Carter and S.W.Hawking, Commun. Math. Phys. 31, 161 (1973).
* [26] R.Banerjee and B.R.Majhi, “Quantum Tunneling and Trace Anomaly”, [arXiv:0808.3688].
* [27] S.M.Christensen and M.J.Duff, Phys. Lett. B 76 571 (1978).
* [28] B.S.DeWitt, Phys. Rep. 19, 295 (1975).
|
arxiv-papers
| 2008-09-09T08:54:59
|
2024-09-04T02:48:57.739918
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/0809.1508"
}
|
0809.1633
|
# RF Breakdown with and without External Magnetic Fields
R. B. Palmer, R. C. Fernow, Juan C. Gallardo, Diktys Stratakis Brookhaven
National Laboratory, Upton NY 11973 Derun Li Lawrence Berkeley National
Laboratory, Berkeley, CA 94720
###### Abstract
Neutrino Factories and Muon Colliders’ cooling lattices require both high
gradient rf and strong focusing solenoids. Experiments have shown that there
may be serious problems operating rf in the required magnetic fields. The use
of high pressure gas to avoid these problems is discussed, including possible
loss problems from electron and ion production by the passage of an ionizing
beam. It is also noted that high pressure gas cannot be used in later stages
of cooling for a muon collider. Experimental observations using vacuum rf
cavities in magnetic fields are discussed, current published models of
breakdown with and without magnetic fields are summarized, and some of their
predictions compared with observations.
A new theory of magnetic field dependent breakdown is presented. It is
proposed that electrons emitted by field emission on asperities on one side of
a cavity are focused by the magnetic field to the other side where they melt
the cavity surface in small spots. Metal is then electrostatically drawn from
the molten spots, becomes vaporized and ionized by field emission from the
remaining damage and cause breakdown. The theory is fitted to existing 805 MHz
data and predictions are made for performance at 201 MHz. The model predicts
breakdown gradients significantly below those specified for either the
International Scoping Study (ISS) iss Neutrino Factory or a Muon Collider
collider .
Possible solutions to these problems are discussed, including designs for
‘magnetically insulated rf’ in which the cavity walls are designed to be
parallel to a chosen magnetic field contour line and consequently damage from
field emission is suppressed. An experimental program to study these problems
and their possible solution is outlined.
###### pacs:
29.20.-c, 29.25.-7, 29.27.-a,52.59.-f,79.70.+q
††preprint: BNL-81489-2008-JA
## I Introduction
### I.1 The use of rf in Magnetic Fields for Neutrino Factories and Muon
Colliders
Low frequency (330-200 MHz) rf is needed for phase rotation and early cooling
in the currently proposed Neutrino Factory iss and Muon Collider collider
designs. The magnitude of the required magnetic fields are of the order of
1.75 T for the phase rotation and around 3 T in the early cooling lattices.
The rf gradients specified are between 12 and 15 MV/m neuffer .
Experimental data exists on the operation of 805 MHz vacuum rf in magnetic
fields. These were obtained with two very different cavity types: 1) a multi-
cell cavity with open irises noremopen , and 2) a single ‘pillbox’ cavity with
irises closed with Cu plates or Be windows xxx ,norempill . Both showed
significant problems and will be discussed in this article.
There is very little data on operations of 201 MHz vacuum cavities in
significant magnetic fields, although experiments are underway with local
fields of about 1 T on a limited part of the cavity. Tests are planned for the
201 MHz cavity in fields of the order of 3 T and geometries that can be made
close to those in the machine designs.111These tests will be undertaken when
the first large superconducting ‘Coupling Coil’ is delivered from Harbin,
China, within the next 12 months.
There is also data on the DC operation of a test cavity in high pressure
hydrogen gas that showed no magnetic field dependence gas . However, there may
be other problems arising when an ionizing beam passes through such a gas
filled cavity; in addition, such gas filled cavities cannot be used in the
later stages of cooling for a Muon Collider because the Coulomb scattering in
the gas would cause too much emittance growth.
We first discuss the gas filled cavity experiments and the possible problems
with their use with ionizing beams; then we consider the experiments and
models of vacuum cavities ( no gas in them) in magnetic fields.
### I.2 805 MHz High Pressure Gas filled Test Cavity Experiments
Figure 1: (Color) Schematic of a test cavity.
A simple test cavity (Fig. 1) has been operated with hydrogen at different
pressures and with ‘buttons’ made of different materials that define the small
high gradient gap gas . It was found that at lower pressures, the breakdown
gradient follows the Paschen prediction, but at higher pressures the gradient
is limited to values (50 MV/m for Cu) close to those observed
($53~{}\frac{\text{MV}}{\text{m}}$ in the open multi-cell cavity discussed in
Sec. I.4) in vacuum cavities after conditioning. This similarity in the
observed maximum gradients suggests that the initiating mechanism for
breakdown in the high pressure gas, and vacuum cavities is the same. Models
for this phenomena will be discussed in Sec. II. No change in breakdown was
observed in the presence of an external magnetic field of 3 T. This is as
would be expected from the mechanism for vacuum breakdown with magnetic fields
that is described in Sec. I.5.
### I.3 Theoretical Expectation for the Effects of an Ionizing Beam passing
through a High Pressure Gas filled Cavity
Tollestrup alvin1 has studied the likely effects of a muon beam passing
through gas filled rf cavities. It is concluded that the lifetime of the
electrons and ions produced by the ionization of those beams are long compared
with the likely duration of the muon beams. It is further concluded that such
electrons in the cavity will be driven backwards and forwards as the rf
voltage oscillates, and that this will lead to heating of the gas and loss of
the rf energy. The final $Q$ of three different cavities exposed to $10^{11}$
muons are estimated and given in Table 1.
Table 1: Final $Q$ of three cavities. Frequency (MHz) | 400 | 800 | 1600
---|---|---|---
rf gradient (MV/m) | 16 | 16 | 16
Final Q | 647 | 325 | 163
The final $Q$ for other cases can be approximately estimated by scaling from
these numbers. This is done for three cases:
* •
The phase rotation and initial cooling for the ISS Neutrino Factory iss
* •
The phase rotation and initial cooling for the low emittance muon collider
parameters as presented at The Neutrino Factory & Muon Collider Collaboration
(NFMCC) 2008 meeting mainpage , alexahin1 .
* •
The phase rotation and initial cooling for the high emittance muon collider
alexahin1 .
Table 2: Parameters for a Neutrino Factory and low and high emittance Muon Collider. | $\nu$ Factory | Muon Collider
---|---|---
| | Low emittance | High emittance
Frequency (MHz) | 200 | 200 | 200
rf gradient (MV/m) | 16 | 16 | 16
Repetition rate (Hz) | 50 | 65 | 12
Wall power (GW) | 4 | 3.6 | 3.2
10 GeV protons on target ($10^{12}$) | 51 | (35) | (170)
Final muons per charge ($10^{12}$) | | 10 $\times$ 0.1 | 2
Cooling transmission (%) | | 30 | 7
Muons per proton | 0.33 | (0.17) | (0.33)
Muons before cooling, $N$ ($10^{12}$) | 17 | 6 | 57
Final Q | 7.6 | 22 | 2.3
Table 2 gives the needed parameters of the three cases. For the two collider
cases, the numbers of initial muons are estimated from the final number of
muons, multiplied by two for the 2 charges, and divided by the quoted
transmission. This is a somewhat optimistic assumption since the phase
rotation and initial cooling will also see unwanted kaons, very high energy
pions, protons and knock-ons from neutrons, that will add to the losses. For
the neutrino factory case the total number of muons is derived from the number
of protons per pulse at 10 GeV together with MARS mars simulated total pion
production at this energy.
We assume that the final $Q$ of the cavity is given by
$Q=Q_{o}\times\frac{f_{o}}{f}\times\frac{N_{o}}{N}$ (1)
where $f$ is the cavity frequency, $N$ is the number of muons per pulse and
the subscripted values are from the 400 MHz case in Ref. alvin1, .
Since $Q$ tells us the energy loss per period of the rf, the $Q=22$ for the
low emittance case means that little rf is left after only three cycles. Yet
for the phase rotation schemes under study, the muons are spread over 10 or
more cycles prior to bunching and rotation. Further down the cooling channel
the numbers of muons are less, but until and unless the bunches are merged,
the rf must remain effective over 10 or more cycles, and the required $Q$ must
be well above 60 222The introduction of other gasses to rapidly capture the
electrons may avoid this problem, but there remain concerns that the presence
of accumulating numbers of ions will cause trouble.. In any case, it must be
noted that high pressure gas cannot be used in the later cooling stages where
the emittance is very small. In these cases the Courant Snyder $\beta_{\perp}$
must be very small where any material is introduced. This can be achieved in
local areas, or inside very high field solenoids, but not over the lengthy rf
systems. Final cooling for a muon collider will thus inevitably require vacuum
acceleration near strong fields.
### I.4 Experimental Operations of a Multi-cell 805 MHz Open Cavity in
approximately Axial Magnetic Fields
Figure 2: (Color) Schematic of a multi-cell cavity first tested in the lab G
magnet at Fermilab. The lines shown are the magnetic field lines extending
from two irises to the end window.
An ionization cooling lattice had been designed that employed liquid hydrogen
absorbers inside high field solenoids separated by lower field matching
solenoids surrounding 6 cell, open iris, rf cavities. Such lattices used early
in a cooling sequence, would use 201 MHz, but higher frequency and high
magnetic fields would be used for later cooling.
To test the system a 6 cell cavity (Fig. 2) was designed with iris apertures
tailored to fit the beam profile in the matching fields noremopen . The cavity
was then tested with and without magnetic fields. The available fields were
not those in the matching design, and the cavity could not be positioned
symmetrically in the magnet, so the fields used in the test were asymmetrical.
The maximum achieved surface gradients were quite high ($\approx 53$ MV/m),
comparable with those in the gas filled cavity, and were not strongly
dependent on the magnetic fields (Fig. 3a). However, both X-rays and dark
currents were greatly increased with the magnetic field on. After some time,
vacuum was lost, and it was found that the end vacuum window was so damaged as
to cause the vacuum leak.
It was found that the location of the window damage corresponded to a focused
dark current coming from one of the high field irises. In addition radiation
damage patterns were observed showing that many beamlets were focused by the
magnetic fields onto the end window. There was no indication that dark
currents from one high gradient iris were focused on to another, which may
explain why the maximum achieved gradients remained high (see Sec. III.2).
### I.5 Pillbox Cavity Breakdown in Magnetic Fields
In a linac with open irises, the peak surface fields are typically a factor of
two higher than the average accelerating gradient. Subsequent to the design of
the multi-cell open iris cavity, it was realized that with muons one could
introduce thin Be windows at each iris and obtain accelerating gradients much
closer to the maximum surface fields. In addition, the resulting ‘pillbox’
cavities give more acceleration for a given rf power. A test ‘pillbox’ cavity
was thus designed (Fig. 3b) and tested in a magnetic field. This time the
cavity was mounted in the center of the magnet and could thus be tested with
symmetrical fields. It was also tested with one of the magnets coils unpowered
to give asymmetrical fields and with the coils powered in opposite directions
to study the effects with these different field geometries.
Figure 3: (Color) Left: a) breakdown gradients vs axial magnetic field. Right:
b) schematic of a pillbox cavity. (Data from Ref. xxx, )
Without field the maximum achieved gradients norempill were somewhat lower
than for the multi-cell cavity, but this was, at least to some extent, the
result of more conservative operation after the severe damage seen in the
earlier cavity. With magnetic fields, the maximum gradients were found to be
strongly dependent on field (Fig. 3a). It was also found that over time its
performance deteriorated. Examination of the inside of the cavity showed
severe pitting on the irises. The Be windows themselves showed no visible
damage, but there was a spray of Cu over their surface and Cu powder in the
bottom splash .
## II Breakdown models without magnetic fields
It is assumed in all models that breakdown is initiated at ‘asperities’, where
the local electric fields is higher, by a factor, $\beta_{\rm FN}$ introduced
by Fowler-Nordheim refFN . The average values of these factors can be
determined by observing the electron currents (dark current) emitted by the
sum of many asperities, each of which has a specific value of $\beta_{\rm
FN}$. The field emitted average electron current density
$J_{F}(\frac{\text{A}}{\text{m}^{2}})$ for a surface field
$\bm{E}(\frac{\text{V}}{\text{m}})$, and local field
$\bm{E}_{\text{local}}=\beta_{\rm FN}\bm{E}$ is given by ohmic
$J_{F}=6\times 10^{-12}\times
10^{4.52\phi^{-0.5}}\frac{\bm{E}_{\text{local}}^{2.5}}{\phi^{1.75}}\exp{\left[-\frac{\zeta\phi^{1.5}}{\bm{E}_{\text{local}}}\right]}$
(2)
where $\phi$ is the material work function in (eV) ($\phi=4.5$ eV for Cu) and
$\zeta=6.53\times 10^{9}(\text{eV})^{-1.5}(\frac{\text{V}}{\text{m}}).$ (3)
In vacuum cavities with a thin window, one can measure some fraction of all
field emitted electrons and observe its field dependence. From this
dependence, given an assumed work function $\phi$, one can then extract an
average value of $\left<\beta_{\rm FN}\right>\left<\bm{E}\right>$.
It is reasonable to assume that breakdown occurs where the local field, and
thus $\beta_{\rm FN}\bm{E}$ is maximum, and thus higher by a factor $\alpha$
than the average value determined from the gradient dependence of the dark
current from many asperities, so
$\frac{\bm{E}_{\rm
local}}{\alpha}=\left<\beta_{\text{FN}}\right>\left<\bm{E}\right>$ (4)
where $\alpha\gtrsim 1$ depends on the probability distribution of $\beta_{\rm
FN}$.
Observed breakdown gradients are found to depend on frequency ref:freq
($\propto\sqrt{f}$), rf pulse length, and cavity dimensions, but it has been
found noremfreq that, over a range of frequencies from DC to a few GHz, and
for differing pulse lengths, cavity dimensions and in waveguides, the values
of $\frac{\bm{E}_{\text{local}}}{\alpha}$ fall in a relatively narrow range
around 7 GV/m. Breakdown thus appears to be related to the local electric
fields at asperities (points, cracks or other causes of the field
enhancement), or to the field emitted currents that are strongly dependent on
these fields. Several plausible but speculative mechanisms for the initiation
of rf breakdown have been proposed, which will be discussed next.
### II.1 Breakdown Models
#### II.1.1 Mechanical fracture model
The mechanism, as discussed in Ref. noremmodel, , assumes the following
sequence of events:
1) The surface contains asperities at the top of which the local field is
given by the average field multiplied by a Fowler-Nordheim field enhancement
factor $\beta_{\text{FN}}$. 2) The outward electrostatic tension
$F_{s}=\frac{\epsilon_{o}}{2}(\beta_{\text{FN}}\bm{E})^{2}$ is equal in
magnitude to the energy density of the field, hence it is proportional to the
square of the field (7 GV/m would induce a tension of 300 MPa), breaks off the
tip and the small piece now moves away from the remainder of the asperity. 3)
The piece is bombarded by field emitted electrons from the remaining asperity
and becomes vaporized and ionized. 4) Following this formation of a local
plasma other mechanisms cause the plasma to spread, or other mechanisms short
out the cavity leading to breakdown.
Within this model the breakdown occurs when the electrostatic outward tension
at the asperity equals the tensile strength of the material. For the local
field
$\bm{E}_{\text{local}}=\alpha\left<\beta_{\text{FN}}\right>\left<\bm{E}\right>$,
the average field at the surface of the asperity is given by
$\left<\bm{E}\right>\propto\frac{\sqrt{\textbf{T}}}{\left<\beta_{\text{FN}}\right>\alpha}$
(5)
where T is the tensile strength of the material. There is relatively little
data on breakdown, including $\beta_{\rm FN},$ of materials of significantly
different tensile strength. Figure 4a shows the observed dependencies in an
experiment wg at 11 GHz in a special tapered rf waveguide. The error bars on
tensile strengths indicate the range of quoted values, depending on material
treatment, with the highest strengths given for fine drawn wire. It is
reasonable to assume that on the nanometer scale of an asperity, the material
strength is of the order of that the highest strengths observed. The plotted
curve represents the theoretic predictions assuming $\alpha=1$. However,
independently of the value of $\alpha$, the theoretical expectation is not
observed in the experimental dependency; indeed, the higher strength stainless
steel has a lower minimum local breakdown field than expected.
Figure 4: (Color) Top: a) minimum local rf gradient at an asperity over
$\alpha$ vs tensile strength of materials of an 11 GHz waveguide; the error
bars on tensile strengths indicate the range of quoted values, depending on
material treatment. Bottom: b) local rf gradients vs the product of the
melting temperature $T_{\text{m}}$ times the thermal conductivity $K$ divided
by the electrical resistivity $\rho$ for an 11 GHz vacuum waveguide. (Data
from Refs. gas, , wg, )
#### II.1.2 Ohmic heating model
It has been suggested ohmic that, when the field emission current density is
sufficiently high, breakdown is initiated by ohmic heating that melts the tip
of an asperity. Once liquefied electrostatic forces would pull the molten
material away, just as the broken piece of the asperity was pulled away in the
first model. This molten material, as it lifts from the remains of the
asperity, will be exposed to field emission from the remaining asperity left
behind and it will be further heated, vaporized and ionized to form a plasma.
For submicron asperities, the time constant for achieving a steady thermal
state is only of the order of a nanosecond, so the temperatures reached depend
only on the geometry, electrical resistivity, thermal conductivity, and
current densities at the tip.
Figure 5: (Color) Schematic for asperity heating calculation
Assuming the asperity to have a conical shape (see Fig. 5), with solid angle
$\Omega$, and emitting area $A=h_{o}^{2}\Omega,$ then, given the electrical
resistivity $\rho$ and current density $j_{o}$, the heat $Q$ flowing back to
the base of the asperity is
$Q(h)=\int_{{h_{0}}}^{h}\,\frac{I^{2}~{}\rho}{\Omega
h^{2}}\,dh\approx\frac{I^{2}~{}\rho}{\Omega}\left(\frac{1}{h_{0}}-\frac{1}{h}\right)$
(6)
The temperature difference between the tip and the base $\Delta T$, as a
function of the thermal conductivity $K$, assuming $h_{1}\gg h_{o}$ is
$\displaystyle\Delta T=\int_{{h_{0}}}^{{h_{1}}}\,\frac{Q}{Kh^{2}\Omega}\,dh$
$\displaystyle\approx$
$\displaystyle\left(\frac{I^{2}\rho}{2h_{0}^{2}K\Omega^{2}}\right)$ (7)
$\displaystyle=$ $\displaystyle\left(\frac{j_{o}^{2}A\rho}{2K\Omega}\right)$
The field emission current density is approximately proportional to the local
field $\bm{E}_{\text{local}}$ to the tenth power noremopen , so for a fixed
emission area $A$ and cone angle $\Omega$, the field needed to melt the
material is proportional to
$\bm{E}_{\text{local}}\propto\left(\frac{K~{}T_{m}}{\rho}\right)^{1/20}.$ This
expression contains a number of approximations: the temperature dependence of
the parameters is ignored, the asperity shape is assumed to be a cone of fixed
angle, and the field emission is approximated by a power law, but the result
should be qualitatively correct, and is plotted in Fig. 4b. It is seen that
the material dependencies agree with the prediction within the errors, and, in
particular, the lower gradients achieved with stainless steel are as
predicted.
#### II.1.3 Thermal runaway model
Figure 6: (Color) Rates of heating from electrical resistivity $\rho$, and
cooling from thermal conductivity $K$, vs the effective temperature at which
$\rho$ and $K$ are determined.
The above calculation of ohmic heating was carried out under the assumption
that the resistivity and thermal conductivity are independent of temperature,
which they are not. Assuming that these expressions are approximately valid
using resistivities and thermal conductivities at an intermediate effective
temperature $T_{\text{effective}}$, then one can look at the relative rates of
heating and cooling as a function of that effective temperature. Figure 6
shows the heating and cooling vs the effective temperature for pure Cu kandl .
For a low current density $j_{0}$ giving a temperature less than a critical
value $(T_{\text{effective}}<T_{\text{critical}}),$ then the cooling which is
$\propto\Delta T\times K$ rises more rapidly than the heating
$(\propto\rho(T))$, and a stable temperature is possible. On the other hand at
higher current densities, which leads to temperatures above
$T_{\text{critical}}$, the rate of heating vs temperature rises faster than
the cooling and the temperature will ‘runaway’. For the data used, the
critical effective temperature is about $300^{\circ}.$ The actual critical
temperature at the tip will be somewhat higher. It is worth noting that
repeated heating to temperatures of this order may induce fatigue, leading to
damage and an increased probability of breakdown after many rf pulses.
#### II.1.4 Reverse bombardment model
A fourth model wilson assumes that some initial mechanism generates a local
plasma (called a ‘plasma spot’), that by itself does not directly cause
breakdown. Plasma spots, tiny sources of light, have been observed in DC and
pulsed gaps without breakdown. In this model, breakdown occurs when electrons
emitted by the local plasma are returned to their source spots by the rf
electric field. The energy given to the source by these returning electrons is
required to cause the plasma to grow and cause the actual breakdown. But as we
will see later (Fig. 10), at least for 805 and 201 MHz in the absence of an
axial magnetic field, electrons emitted at the highest field location never
come back to their source, no matter what their initial phase. This appears to
be true for cavities in general, with the exception of emission on the axis in
pillbox cavities. Yet breakdown on the axis of pillbox cavities is rarely
observed.
#### II.1.5 Surface damage by heating
As we have noted, breakdown gradients rise approximately as the root of the rf
frequencies; but this dependency does not continue at frequencies above 10
GHz. Fatigue damage from cyclical surface heating appears then to limit the
gradients. This damage is worse at locations with maximum surface currents,
where surface electric fields are usually low. But the damage, which causes
cracks to form at grain boundaries can be so severe that breakdown is
initiated in regions with both electric fields and surface current. Since this
phenomena is only seen at such high frequencies, it is not expected to be a
problem for neutrino factories or muon colliders.
### II.2 Observed Dependencies
Figure 7: (Color) Top: a) breakdown gradients vs the approximate energy to
melt a given volume of the material. Bottom: b) breakdown gradients vs the rf
pulse length. (Data from Refs. gas, , wg, )
Although the above discussion suggest that ohmic heating initiates breakdown,
this conclusion depends on a limited number of experiments; therefore one has
to conclude that the initial mechanism that starts a breakdown is not fully
understood yet. But there are some dependencies that appear fairly
consistently:
* •
Over a wide range of frequencies (0.2 to 3 GHz), breakdown gradients are
approximately proportional to the square root of the rf frequency, and this
dependency arises because of changes in the observed field enhancement factor
$\beta_{\text{FN}}$.
* •
Breakdown appears dependent on the required energy to melt a given volume of
the electrode material (see Fig. 7a). In the vacuum waveguide experiment case,
this dependence is again caused by changes in $\beta_{\text{FN}}$, rather than
in the local fields.
* •
Breakdown occurs at lower gradients for long rf pulses than for short pulses
(Fig. 7b), and there is some indication that this too arises from the pulse
lengths influence on $\beta_{\text{FN}}$.
### II.3 Conditioning
In all cases discussed here, the cavities or waveguides were ‘conditioned’
prior to achieving the quoted gradients. After one or more breakdowns at one
gradient, the cavity will subsequently withstand a somewhat higher gradient.
Typically, many hundred successive breakdowns are induced prior to the cavity
reaching its ‘final gradient’. When dark current distributions are studied as
the gradient is increased by this conditioning, then it is again found
conditioning that it is the enhancement $\beta_{\text{FN}}$ that is
decreasing and not that the local field is increasing.
A reasonable assumption is that, whatever the initial cause, the effect of a
breakdown is to remove asperities and consequently lowering
$\beta_{\text{FN}}.$ At the same time, however, the breakdowns, depending on
the energy available, will create new asperities thus incrasing
$\beta_{\text{FN}}.$ In a ‘conditioned’ cavity, these competing processes have
reached an equilibrium. The rf energy available (stored or in a longer pulse),
divided by the energy required to melt a given volume of metal, gives the
total amount of melted metal. Assuming that more molten metal will be more
likely to create bad asperities, leads directly to the observed dependencies
with material properties and rf pulse length.
In the case of the frequency dependency, since lower frequency cavities are
usually larger, the observed dependency once again agrees with the
expectation.
## III Breakdown Models with External Magnetic Fields
### III.1 Published Breakdown Model with Magnetic Fields
It has been proposed in the twist model noremopen that the magnetic field
dependence on breakdown arises from the torque forces on an asperity due to
the inflow of current feeding the field emission reacting to the external
magnetic field; that is $F~{}\propto~{}I~{}\times~{}B$ where $\bm{E}$ is the
cavity electric field gradient and approximately $I~{}\propto~{}\bm{E}^{10}$
noremopen . Thus for breakdown at a fixed force $F$, we expect
$\bm{E}_{\text{breakdown}}~{}\propto B^{-1/10}$ (8)
Figure 8: (Color) Breakdown gradients vs axial magnetic fields. Black line is
dependency predicted by asperity twist model. Red lines are fit to the plotted
Lab G norempill breakdown data. Blue lines are the calculated values for a
201 MHz cavity. Dotted lines are for Be surfaces.
In Fig. 8 the observed pillbox cavity breakdowns are plotted as a function of
the external average magnetic field. The points plotted are those where both
superconducting coils were powered so that the magnetic fields were relatively
uniform over the cavity. The dependency predicted by this mechanism is shown
by the solid black line; it is a poor fit at higher fields where the breakdown
gradient falls much faster than predicted.
### III.2 Introduction to the Proposed Mechanism
Figure 9: (Color) Proposed mechanism for breakdown with an external magnetic
field.
We propose a new model for breakdown with a magnetic field that is independent
of the breakdown mechanism in the absence of magnetic fields (see Fig. 9).
Breakdown occurs by this mechanism only if its breakdown gradient is lower
than that from the case without a magnetic field. Its elements are:
* •
‘Dark Current’ electrons are field emitted from an asperity, accelerated by
the rf fields, and impact another location in the cavity. In the absence of a
magnetic field these impacts are spread over large areas and do no harm.
* •
With sufficient magnetic field they are focused to small spots, where they can
melt the surface producing local damage. If such damage is at a low gradient
location there is no immediate breakdown, but the damage can accumulate until,
for instance, a hole is made in a window.
* •
If the electrons are focused onto a location with high surface rf gradient,
then electrostatic forces will pull the molten metal out and away from the
surface. This metal, as it leaves the now damaged location, will be exposed to
field emitted electrons from the damaged area and will be vaporized and
ionized, leading to a local plasma and subsequent breakdown.
* •
For higher energy electrons, the melting will start deeper in the material
where the ionization loss is greater, and expand to the surface. Thus, when
the melting reaches the surface, significant quantities of molten metal can be
sprayed onto other surfaces in the cavity splash .
Breakdown will be dependent on a) the Fowler-Nordheim field enhancement
$\beta_{\text{FN}}$ that determines the strength of the field emitted current,
b) the local geometry of the asperity that will determine the initial particle
distributions and effects of space charge, and c) on the geometry and magnetic
fields that focus the electrons onto other locations.
### III.3 Electron Motion in a Cavity
Figure 10: (Color) Trajectories of electrons field emitted at different phases
from the highest surface field location in an 805 MHz pillbox cavity with a)
no external magnetic field, b) an axial field of 0.1 T, and c) an axial field
of 1 T. The axial electric field is 25 MV/m. Phases are in degrees relative to
the maximum.
A program CAVEL cavel tracks particles from arbitrary positions on the walls
of a cavity until they end on some other surface. The program uses SUPERFISH
superfish to determine the rf electric and magnetic fields, and uses a map of
external magnetic fields calculated for arbitrary coil dimensions and
currents. Fig. 10 shows trajectories, for differing initial rf phases,
starting from the highest field location on an iris. Without an external
magnetic field, none of the trajectories from the high field location come
back to their common origin. Tracks emitted at a phase of $20^{\circ}$ do hit
the opposing iris at a high gradient location, but they are not focused there
and are spread out over a significant distance. But with a sufficient external
axial magnetic field, the tracks are either focused to the high gradient
location on the opposite iris or returned to their source.
There remains a small dependency of the arrival positions with phase, which
arises from the combined effects of the perpendicular external and rf magnetic
fields, but this dependency is small. If there were no other mechanism to
spread out the electrons, then damage would appear as lines, which are not
observed. In addition, damage would be much worse on the axis than at larger
radii, also not observed. From this, one can assume that the damaging emission
is dominated by field emission which is restricted to a limited range of
phases ($\pm~{}\approx 20^{\circ}),$ and that another mechanisms, such as
space charge, spreads the beamlets out to a greater extent than this phase
dependent effect. Figure 11 shows the energies of the electrons on impact and
indicates which phases are returned and which arrive on the next iris. Field
emitted electrons $(\pm 25^{\circ})$ do not come back, but are focused to a
local area on the opposite iris. Experimental observations noremopen of the
time structure of dark current show that this dark current is concentrated at
phases close to zero. It seems unlikely then that, even with an external axial
magnetic field, significant currents of electrons will be returned to their
source.
The electron energies for the 805 MHz cavity with axial rf fields of 25 MV/m
are approximately 1 MeV. For a 201 MHz cavity, and the same gradient, they are
of the order of 4 MeV. At these energies, the electrons penetrate to
significant depths in the Cu cavity walls. If Be is used, the penetration is
even deeper. The relative surface heating and thus probability of melting and
damage depends on the fraction of energy deposited in a surface layer, taking
into account thermal conduction away from the deposition. A full calculation
would include the time dependence of both deposition and conduction. As an
approximation, the temperature rise is estimated from the deposition in a
depth corresponding to the thermal diffusion depth at the surface.
Figure 11: (Color) Energies of electrons on impact vs. their phase of
emission. Red indicates electrons that returned, blue those that arrive on the
next iris. Axial gradient is 17 MV/m.
### III.4 The Effects of Space Charge on the Transverse Distribution of Field
Emitted Current
Without an asperity and emission from a small area, the space charge forces
give transverse momenta to emitted electrons causing the beamlet’s radius to
increase. As the beamlet increases in radius and the electrons are
accelerated, the space charge forces drop and it can be shown that the induced
rms transverse momentum is $\sigma_{p\perp}\propto\sqrt{I}.$ But if the
electrons are emitted from the tip of an asperity then they will first be
spread by the approximately spherically symmetric local electric fields, and
the effect of the space charge is consequently modified.
A simple simulation was performed (see Fig. 12). The initial electric fields
were assumed to have strength $\beta_{\text{FN}}\bm{E}$, and exact spherical
symmetry out to a distance $X$. Beyond this distance, the fields were assumed
to be perpendicular to the average surface with strength $\bm{E}$. No space
charge effects are included in the spherically symmetric part. Radial space
charge forces, inversely proportional to an average radius, are assumed beyond
the distance $X$.
Figure 12: (Color) Schematic of approximate simulation of space charge effects
on electrons emitted by an asperity.
We found that with $X=0.2~{}\mu$m, the transverse momenta could be
approximated by $\sigma_{p\perp}\propto{I^{j}},$ but with the power $j=0.3$
instead of 0.5 as in the simple case without asperity.
Figure 13: (Color) The simulated final electron energy ${\cal E}_{e}$ as a
function of axial rf gradient for (red) a 805 MHz pillbox cavity, and (blue) a
201 MHz cavity.
If $X$ was smaller, then the best fit exponent $j$ was found to increase in
value. Since neither the asperity height, nor its shape, are known, it is
reasonable to treat the exponent $j$ as an unknown that is fitted to the
experimental data. In a better simulation, this fit would give us information
on the asperity dimensions, but this model is too simple for this at this
stage. At distances from the source large compared with $X$, space charge
becomes negligible, but the transverse momentum is focused by the axial
magnetic field, giving a beamlet with an rms radial size
$\sigma_{r}\propto\frac{I^{j}}{B}$ where $B$ is the axial magnetic field. The
power per unit area $W$ of the electron beamlet hitting the opposing surface
is given by
$W=\frac{I{\cal E}_{e}}{\pi\sigma_{r}^{2}}~{}\propto\frac{I^{(1-2j)}~{}{\cal
E}_{e}~{}B^{2}}{\pi}$ (9)
where ${\cal E}_{e}$ is the final electron energy in $(\text{MeV})$ determined
from CAVEL cavel simulations. ${\cal E}_{e}$ is plotted in Fig. 13 as a
function of the axial rf gradient, for 1) 805 MHz pillbox cavity norempill ,
and 2) 201 MHz cavity designed for the MICE cooling experiment mice .
### III.5 Fraction of Energy Deposited in the Thermal Diffusion Depth
The thermal diffusion length $\delta$ corresponding to the rf field pulse
duration $\tau(s)$ is, $\delta=10^{-2}\sqrt{D~{}\tau}$ where $D=\frac{K}{\rho
C_{s}}$ is the thermal diffusion constant in (m), $K$ is the thermal
conductivity and $C_{s}$ the specific heat. It is assumed that all the energy
deposited in this diffusion depth is spread uniformly within that depth
$\delta.$ The penetration depth $d$ in $(\mu m)$ of low energy electrons is
approximately given by penetration
$d=0.0267\frac{A}{\rho Z^{0.89}}{\cal E}_{e}^{1.67}$ (10)
where ${\cal E}_{e}$ is the electron incident energy in (keV), $\rho$ is the
material density in $(\frac{g}{cm^{3}}),$ and $Z$ and $A$ are the atomic
number and atomic weight respectively.
If ${\cal E}(x)$ is the energy of an electron with penetration depth $x$ and
we define $Q$ as the fraction of the electron energy deposited in the thermal
diffusion length $\delta$, then:
* •
For electrons whose penetration depth is less than the diffusion depth
($d<\delta$): $\quad Q=1$.
* •
For electrons whose penetration depth is greater than the diffusion depth, but
not much greater ($\delta<d<10\times\delta$) then: $\quad Q=\frac{{\cal
E}_{e}-{\cal E}_{e}(d-\delta)}{{\cal E}_{e}}.$
* •
For electrons with penetration depth $d>10\times\delta$ then:
$Q=\frac{\frac{d{\cal E}_{e}}{dx}\delta}{{\cal E}_{e}}.$
### III.6 Dependency of Local Temperature Rise
The surface temperature rises as a fraction of the melting temperature is then
$\frac{\Delta T}{T_{\text{m}}}~{}\propto W\left(\frac{\tau
Q}{\delta~{}\rho~{}C_{s}~{}T_{\text{m}}}\right)$ (11)
where $\rho$ is the density, $C_{s}$ is the specific heat, $T_{m}$ is the
melting temperature, and $\tau$ is the rf pulse length, taken to be 20 and
$160~{}\mu s$ at 805 and 201 MHz respectively ($\tau\propto\lambda^{3/2}$).
Parameters used for Cu and Be are given in Table 3.
Table 3: Atomic and Nuclear Properties of Cu and Be | Z | A | K | $\rho$ | $C_{s}$ | D | $\delta$(805) | $\delta$(201) | $T_{\text{m}}$
---|---|---|---|---|---|---|---|---|---
| | | (W/cm$-^{o}$C) | (g/cm3) | (J/g$-^{o}$C) | (cm2/s) | ($\mu$m) | ($\mu$m) | (oC)
Cu | 29 | 63.5 | 4.01 | 8.96 | 0.385 | 1.16 | 48 | 186 | 1085
Be | 4 | 9.0 | 2.18 | 1.85 | 1.825 | 0.81 | 40 | 155 | 1287
### III.7 Fit to Field Dependence Data from Lab-G Pillbox and Predictions
for 201 MHz
The above is a very approximate analysis. A full simulation of the problem is
being pursued. The simulations were done with uniform magnetic field; tracks
were only simulated from the single maximum gradient location; electron
impacts were assumed at $90^{\circ}$ to the surface; the thermal diffusion
calculation ignored the rise time shape and used an approximate calculation;
both current scale and space charge strength were normalized to fit data.
However, if this simple approach can qualitatively fit the data, it should
allow a qualitative extrapolation to 201 MHz and other materials. A fuller
simulation should provide more quantitative results. The red curved line in
Fig. 8 shows the fit to the 805 MHz Cu cavity in Lab G breakdown data
norempill ; the fitted value of the current exponent was $j=0.35.$ More data
july has been taken since the cavity had suffered significant damage. These
data (not shown) lie at somewhat lower breakdown gradients and presumably
correspond to a now higher value of the Fowler-Nordheim $\beta_{\text{FN}}$.
But since $\beta_{\text{FN}}$, has not been redetermined from dark current
measurements of the damaged cavity, this analysis with that data is not
possible. The black line in the plot shows the dependence predicted by the
twist model noremopen , which does not fit the data well. The dashed lines
show the calculated breakdown limits for Be. The horizontal red line indicates
the gradient limit from the assumed model without magnetic fields, assuming
the local field limit to be 7 GV/m.
### III.8 Extrapolation to 201 MHz
Without an external magnetic field, breakdown gradients have been observed to
follow approximately a $\sqrt{f}$ behavior (see. Sec. II). Under the
assumptions used here noremmodel , the local field at breakdown is independent
of the frequency; this implies that $\beta_{\text{FN}}$ decreases
approximately as $\frac{1}{\sqrt{f}}.$ Using this assumption, and using the
appropriately modified diffusion depth, the predicted breakdown limits at 201
MHz, for Cu (solid blue line) and Be (dashed blue line) are shown in Fig. 8.
The predicted 201 MHz breakdown gradients are seen to be a factor of 2 to 2.5
below those for 805 MHz. This factor comes primarily from the higher expected
$\beta_{\text{FN}}$, but also from the longer rf pulse duration and thus
longer time to heat and melt the surfaces. At the higher gradients specified
for Neutrino Factory phase rotation and cooling (15 MV/m at B$\leq$ 3 T) are
well above this prediction.
### III.9 Other Experimental Results Consistent with this Analysis
The pillbox cavity used in the above experimental study has, more recently,
been tested button with one of its two Be window replaced by a flat Cu plate
with an easily replaceable central ‘button’. The button had dimensions such
that the local field on the tip was 1.7 times that on the outer Cu ‘iris’. The
intent was to allow a study of magnetic field dependent breakdown as a
function of materials, the assumption being that breakdown would occur at the
high gradient on the tip of the button. The cavity was found to operate with
gradients on the button significantly higher (by approximately a factor of
1.7) than had been observed without the button. It was noted however that this
breakdown occurred with gradients on the iris, and on the flat support plate,
that were essentially the same as those present without the button. This
suggested that breakdown was not occurring on the button, but rather at other
locations that did not have the 1.7 times field enhancement: on the irises
and/or the flat button support plate. When the cavity was later disassembled
it was indeed found that was there little damage on the button, where the
field was maximum. But there was significant damage in a distinct band from 3
to 6 cm on the TiN coated Cu support plate (see Fig. 14a).
These observations are consistent with the breakdown mechanism described in
this paper. SUPERFISH superfish calculations showed that the fields on the Be
window opposite the button and support plate were maximal just in the band 3
to 6 cm, where the damage was observed (see Fig. 14b). In this model, emission
from the Cu button, falling on the Be is less liable to cause damage and
breakdown, whereas the emission from the Be focused onto the Cu plate should
cause damage just where the gradients were maximal on the Be.
Figure 14: (Color) Top: a) Button support plate showing damage band between 3
and 6 cm radius. Bottom: b) Surface fields on Be (red) and Cu (blue) vs
radius, with bands showing where damage was concentrated. (Data from Ref.
button, )
## IV Possible Solutions, or Reductions, of these Problems
There are several possible approaches to these breakdown problems in a Muon
Collider or Neutrino Factory.
* •
Redesign the phase rotation and cooling channels to use lower rf fields. This
approach would clearly hurt performance, and, in addition, risks a slow
deterioration of performance as occasional breakdown events continue to spray
Cu around the cavity - as observed in the pillbox tests.
* •
Use cooling lattices with high pressure hydrogen gas in the rf cavities. No
degradation of rf performance has been observed in a small 805 MHz test cavity
with axial magnetic fields and rf gradients similar to those in vacuum
cavities. If the loading from beam induced electrons is not a problem, or is
slowed by introducing gas impurities, then this solution should offer no loss
of performance in early cooling stages. But to cool to very low emittances, it
will probably require lattices with lower Courant Snyder $\beta$ at the
absorber than can be achieved in the rf. In this case, the addition of
hydrogen gas at the higher $\beta$s in the rf would cause unacceptable
emittance growth. So for later cooling, high pressure hydrogen gas is probably
not a solution.
* •
Build cavities with exceptionally good surfaces so that the
$\beta_{\text{FN}}$ is sufficiently low initially that no breakdown occurs.
With Atomic Layer Deposition (ALD) this may be a realistic option ald . The
fear would be that a single breakdown spoils the surface in such a way that
there will be a cause of further breakdown and a conditioning that approaches
the same gradient limits seen in a conventional cavity.
* •
Design lattices with magnetic field shielded from the rf. The above prediction
suggests that so long as the field is less than about 0.2 T, no adverse
effects will be observed. Attempts to design such lattices have,
unfortunately, shown significantly worse performance.
* •
Design lattices using multi-cell open cavities, with alternating current coils
in their irises (Fig. 15). In this case, as in the original multi-cell open
cavity tests, the focused electrons would be directed to low field regions in
the cavity and would thus not initiate breakdowns. Nevertheless damage done to
those locations and molten Cu ejected from such damage could cause eventual
deterioration of performance. In addition, the use of open, instead of pill-
box cavities implies lower acceleration for a given surface field.
Figure 15: (Color) Open cavity with alternated solenoid coils in irises.
* •
Damage might be eliminated if cavities were designed such that all high
electric gradient surfaces were parallel to the magnetic fields (Fig. 16).
This could provide ‘magnetic insulation’ maginsul . Dark current electrons
would be constrained to move within short distances of the surfaces, would
gain little energy, would cause no X-rays, and do no damage. Possible
difficulties might be: a) cavities so designed will not give optimum
acceleration for given surface fields, and b) multipactoring might occur, now
that the energies with which electrons do return to the surfaces are in the
few hundred volt range where secondary emission is maximal. In addition, the
use of open, instead of pill-box, cavities implies lower acceleration for
given surface fields.
Figure 16: (Color) Top: a) Magnetic field lines from coils in a cavity
lattice, together with cavity shape that follows these field lines. Bottom: b)
Energies of returning electrons as a function of their initial phase.
## V Experiments needed to study these problems
Two critical experiments are already planned:
1) The testing of the existing 201 MHz cavity in magnetic fields similar to
those in current MICE and cooling designs. These experiments would use a MICE
‘Coupling Coil’ when it is available.
2) Operating a high pressure hydrogen filled test cavity in a proton beam to
study the possible breakdown and rf losses due to the ionization of the gas by
the beam.
Whatever the results of these tests, further experiments will likely be needed
to study the observed problems at 805 MHz and test possible solutions. If the
tests of the 201 MHz cavity in magnetic fields show problems also at that
frequency, then further 805 MHz tests would be needed to explore solutions for
these lower frequencies. Eventually, however, tests of any proposed solutions
would have to be done at 201 MHz.
If with the proton beam the rf losses in the 805 MHz test cavity are
sufficiently low, then further tests of high pressure gas filled cavities will
be needed. In particular, cavities must be tested with more stored energy and
with a similar shape to those needed. Thin windows must be designed, safety
problems must be addressed and beams with time structure and intensity nearer
to those in the applications should be be employed.
### V.1 Experiments with a Simple Vacuum Pillbox Cavity
Figure 17: (Color) Simple pillbox cavity with mounting in two orientations
within the lab G solenoid.
The ‘pillbox’ cavities that have so far been tested have relatively complex
shapes, and the simulations of their performance with fields in differing
directions are complex. It would thus be desirable to test a simple pillbox
shaped cavity whose performance would be far easier to simulate. The cavity
should be designed so that it can be mounted in the center of the of the Lab G
magnet in at least two orientations: with its axis parallel to the solenoids
and perpendicular to that angle. Ideally it should be possible to test it at
other angles as well. A square cavity moretti1 (Fig. 17) would meet this
requirements, would be easy to build and easy to arrange with coupling ports
to the waveguide in either of two locations to meet the two angle requirement.
Such a simple cavity design would also be a good test vehicle for testing
surface treatments including Atomic Layer Deposition (ALD). Since the design
is so simple, several versions could be made to compare their performances.
As illustrated in Fig. 18, CAVEL simulations of a cavity with its axis
perpendicular to the solenoid axis shows that the electrons are constrained to
lie within a very small distance from the surface and gain little energy. Such
a cavity would test whether multipactoring is a serious problem.
Figure 18: (Color) A schematic illustration of an emitted electron’s orbit
when the magnetic field is parallel with the emitting surface.
In addition, if such a cavity were tested with the emission surface at a small
angle to the field then the electrons could gain some energy, but would end up
on the cylindrical outer surfaces where there would be no electric field. This
would test the configuration of the coil-in-iris solution without the shape
modified to achieve magnetic insulation. Surface damage might be expected, but
it should not cause breakdown.
### V.2 A Single Cell Experiment with ‘Magnetic Insulation’
Figure 19: (Color) Single cell with magnetic insulation.
In this experiment (Fig. 19) superconducting coils would be mounted on either
side of a cavity whose shape is such that the magneric fields are strictly
parallel with the high gradient surfaces. The two coils close to the open pipe
are powered in opposite directions. A second pair of coils is mounted outside
the inner coils and powered with currents in the opposite direction to those
they surround. The coils would be separately powered so that the sensitivity
to deviations from the insulated condition can be studied.
The rf cavity would be operated at liquid nitrogen temperatures in order to
minimize the radiation falling on the superconducting coils that will be very
close to its surfaces. There would be a vacuum both outside the rf cavity (for
thermal insulation) and inside the cavity, although the quality of the latter
should be higher, and would be separately monitored. Super-insulation and an
outer liquid nitrogen shield are needed but not shown. Tuning of the cavity
would be provided by axially squeezing or stretching the cavity. A variant of
this experiment would have Be windows, making it into more of a pillbox design
and raising the accelerating field. This would not provide such complete
magnetic insulation but, because of the special properties of the Be might
still have acceptable performance.
### V.3 A Multi-purpose Test Stand for these Experiments
It is proposed that this and the following experiments would be carried out on
a multi-experiment test stand. To allow ease of assembly, mounting of
instrumentation, and making changes, all connections (rf, cryogens, magnet and
instrumentation leads etc.) would be brought in through a single support plate
(on the right side in the illustration). The vacuum container would be in the
form of a dome that joins to the support plate with a single flange, and can
thus be easily removed without disturbing the connections.
### V.4 A Multi-cell Experiment with ‘Magnetic Insulation’
Figure 20: (Color) Multi-cell cavity with magnetic insulation.
The next experiment would be of a multi-cell magnetically insulated cavity
(Fig. 20). This would test the magnetic insulation in a geometry similar to
that required in the 201 MHz acceleration for phase rotation or early cooling.
In addition, it would address the problem of joining cavities inside the bore
of the solenoids. This joint does not need to carry much rf current, nor does
it need to hold high vacuum (since there is vacuum on both sides). The method
requiring simple axial pressure is shown in the figure, but other ideas may be
explored.
It should be noted that this experiment, and the previous one, test a model of
the 201 MHz lattices that would be used in the phase rotation, Neutrino
Factory cooling, and the early cooling in a Muon Collider.
### V.5 Testing of Components for later 6D Cooling
Later 6D cooling will require compact higher field (10-15 T) solenoids in
order to focus the beams to lower $\beta$’s and thus cool to lower emittances.
Test coils could conveniently be tested on the proposed test stand, since it
would allow for easy mounting, cooling and testing with easily accessible
instrumentation. Later, these coils would be used in conjunction with 805 MHz
cavities (see Fig. 21).
Figure 21: (Color) Lattice for later cooling to lower emittances.
The first stage of the 10-20 T solenoid development could appropriately be
tested in the same test stand discussed above. Subsequently, the rf cavities
would be included. The final step would be to add the required hydrogen
absorber. The details of the lattice, whether it is a snake alexahin1 or
Guggenheim Gug configuration for instance, would not change the nature of the
components needing test. Figure 21 is thus not meant as a definitive design of
the required lattice, but rather as a test of the kind of components needed;
the cavity shapes shown are not those with true magnetic insulation. The true
shapes with their coils are yet to be determined. Figure 21 is thus intended
only as an illustration of the direction of the required R& D.
### V.6 Testing of Magnetic Field Solutions at 201 MHz
The experiments already planned with the 201 MHz cavity in several Tesla
fields could show satisfactory performance. If however this is not the case,
then whatever solutions are proposed on the basis of the 805 MHz tests will
need to be demonstrated at 201 MHz.
## VI Conclusions
The main results and predictions of the model presented in this paper can be
summarized as:
* •
A review of models for breakdown without external magnetic fields suggests
that the initiation of breakdown is best explaned by ohmic heating due to
field emission current at asperities. The dependence of breakdown on
frequency, pulse length and cavity materials can be explained by the competing
processes of asperity destruction and the creation of new asperities.
* •
We have proposed a model for damage in rf cavities operated in significant
axial magnetic fields. The model fits the existing data reasonably well.
* •
The model also fits some otherwise surprising results from a pillbox cavity
with a Be window facing a Cu plate with a central button.
* •
The model predicts relatively low gradient breakdowns at 201 MHz in magnetic
fields. These predicted breakdowns occur at significantly lower gradients than
the operating gradients specified for the phase rotation and initial cooling
in the ISS iss Neutrino Factory and in a Muon Collider.
* •
Methods to address these problems are discussed, including the use of
‘magnetically insulated rf’. An experimental program to study this concept is
outlined.
###### Acknowledgements.
We would like to thank J. Norem, A. Moretti and A. Bross for many discussions
and sharing the experimental data. This work has been supported by U.S.
Department of Energy under contracts AC02-98CH10886 and DE-AC02-76CH03000.
## References
* (1) International Scoping Study (ISS); http://www.cap.bnl.gov/mumu/project/ISS/.
* (2) R. B. Palmer et al., in Proceedings of the 2007 Particle Accelerator Conference, ed. C. Petit-Jean-Genaz, p. 3193.
* (3) D. Neuffer, Presentation to the 2008 NFMCC Collaboration Meeting,Fermilab 2008, http://www.cap.bnl.gov/mumu/conf/MC-080317/talks/DNeuffer1-080317.pdf.
* (4) J. Norem et al., Dark current, breakdown, and magnetic field effects in a multicell, 805 MHz cavity, Phys. Rev. ST Accel. Beams 6, 072001 (2003).
* (5) J. Norem et al., private communication.
* (6) A. Moretti et al., Effects of high solenoidal fields on rf accelerating cavities, Phys. Rev. ST Accel. Beams 8, 072001 (2005).
* (7) P. Hanlet, et al., High pressure rf cavities in magnetic fields, Proc EPAC 2006, Edinburgh; TUPCH147; and R. Johnson; private communication.
* (8) A. Tollestrup presentation to the 3rd LEMC workshop, FNAL, 2008, http://www.muonsinc.com/lemc2008/presentations/alvin_BeamGasCavityLEworkshopFNAL4_22_08.ppt.
* (9) NFMCC web page http://www.cap.bnl.gov/mumu.
* (10) Y. Alexahin, Presentation to NFMCC Collaboration meeting, Fermilab 2008, http://www.cap.bnl.gov/mumu/conf/MC-080317/talks/
YAlexahin1-080317.pdf.
* (11) MARS, http://www-ap.fnal.gov/MARS/.
* (12) See Fig. 17 in ref. conditioning, .
* (13) R. H. Fowler and L. Nordheim, Proc. R. Soc. London, Ser. A119, 173 (1928).
* (14) G.A. Loew, J. W. Wang, rf breakdown studies in room temperature electron linac structures, SLAC-PUB-4647 (1988); and again proposed for high pressure gas filled cavities, R. Johnson (private communication).
* (15) W. D. Kilpatrick, Rev. Sci. Instr. 28, 824 (1957); G.A. Loew, J. W. Wang, Handbook of Accelerator Physics and Engineering, edited by A.W. Chao and M. Tigner, p. 390 (1999).
* (16) A. Hassanein, et al., The effects of surface damage on rf cavity operation, Phys. Rev. ST Accel. Beams 9, 062001 (2006).
* (17) J. Norem et al., Mechanisms limiting high gradient rf cavities, in Proceedings of the 2003 Particle Accelerator Conference, ed. J. Chew, P. Lucas, and S. Webber (IEEE, Piscataway, NJ, 2003), p. 1246.
* (18) V. A. Dolgashev, S. G. Tantawi, RF breakdown in X-band waveguides, Proc EPAC 2002, Paris, France.
* (19) G W C Kaye & T H Laby, Tables of Physical and Chemical Constants 14th edition, Longman (1978).
* (20) P. Wilson, Gradient limitation in accelerating structures imposed by surface melting, SLAC-PUB-9953 (2003); and Proc Workshop on High Gradient RF; ANL (2003); P. Wilson et al., in Proceedings of LINAC 2004, Lubeck, Germany, p. 189 (2004).
* (21) See Figs. 1 & 8 in ref. noremfreq, .
* (22) R. Fernow, CAVEL, http://pubweb.bnl.gov/users/fernow/www/cavel/.
* (23) J. H. Billen and L. M. Young, SUPERFISH, Los Alamos National Laboratory Report No. LA-UR-96-1834, 1996.
* (24) MICE, http://www.mice.iit.edu/.
* (25) K. Kanaya and S. Okayama, J. Phys. D, Vol. 5, 43 (1972); J. Wittke, electron microprobe Notes, http://www4.nau.edu/microanalysis/Microprobe/Probe.html.
* (26) D. Huang, Fermilab MTA 805 MHz program, MUTAC presentation, April 8, 2008; http://www.cap.bnl.gov/mumu/conf/MUTAC-080408/talks/09AM/DHuang1-080408.pdf
* (27) D. Huang, Recent update of 805 MHz cavity material test, NFMCC Friday Meeting 5/13/2008; http://www.fnal.gov/projects/muon_collider/FridayMeetings/23-MAY-2008/Huang.ppt; D. Huang et al., Report No MUC-525, 2008.
* (28) J. Norem; NFMCC Friday meeting 1/11/08; Recent results of high gradient studies at Argonne; http://www.fnal.gov/projects/muon_collider/FridayMeetings/11-JAN-2008/Norem.pdf.
* (29) F. Winterberg, Rev. Scient. Instrum., 41, 1756 (1970); Rev. Scient. Instrum., 43, 814 (1972).
* (30) A. Moretti, Presentation to the 2008 NFMCC Collaboration Meeting, Fermilab 2008, http://www.cap.bnl.gov/mumu/conf/MC-080317/talks/
AMoretti1-080317.pdf.
* (31) Amit Klier, presentation at the Low Emittance Muon Collider Workshop, Fermilab (6-10 Feb 2006), http://www.muonsinc.com/mcwfeb06/; P. Snopok, presentation at the 2008 NFMCC Collaboration Meeting, Fermilab, March 2008; http://www.cap.bnl.gov/mumu/conf/MC-080317/talks/PSnopok1-080317.pdf.
|
arxiv-papers
| 2008-09-09T18:57:28
|
2024-09-04T02:48:57.748510
|
{
"license": "Public Domain",
"authors": "Robert B. Palmer, Richard C. Fernow, Juan C. Gallardo, Diktys\n Stratakis, Derun Li",
"submitter": "Juan C. Gallardo",
"url": "https://arxiv.org/abs/0809.1633"
}
|
0809.1810
|
In particle simulations, often the dynamics results from the summation of all
pair-wise forces in the ensemble of particles. Such situations arise in
astrophysics, molecular dynamics, plasma physics, and certain formulations of
fluid dynamics problems, for example. The total field of interest
(gravitational, electrostatic, etc.) at one evaluation point requires adding
the contribution of all source points or particles, and so if both evaluation
points and particles are numbered at $N$, a total of $N^{2}$ operations is
needed. This fact was for a long time an impediment to the wider use of
particle simulations, as the computational effort becomes prohibitive for
large numbers of particles.
The above scenario changed dramatically with the introduction of tree-codes
and the fast multipole method (FMM), which appear in the late 1980s for
accelerated evaluation of $N$-body problems. Tree-codes
Appel1985,BarnesHut1986 are generally perceived to be easier to grasp and
program, and provide a complexity of $\mathcal{O}(N\log N)$. The FMM was
introduced as an algorithm for the rapid evaluation of gravitational or
Coulombic interactions greengard+rokhlin1987 and promises a reduction in
computational complexity to $\mathcal{O}(N)$. It has, since its dissemination,
been adapted for many applications: for fast evaluation of boundary elements
Gaspar1998, for vortex sheet problems with desingularized equations
HamiltonMajda1995, for long-range electrostatics in DNA simulations
FenleyETal1996, and many others. The impact of the FMM has been undeniably
huge, resulting in it being chosen as one of the Top 10 Algorithms of the 20th
Century DongarraSulli2000.
Despite the great volume of work using and adapting the FMM in many
application areas, there remains some lack of insight regarding how the
algorithm can be efficiently used to obtain an accurate representation of the
field of interest. The error of the FMM approximation is estimated by
theoretical bounds, which as could be expected reveal a trade-off between
accuracy and efficiency of the computation. However, there is not much
literature providing measurements of the accuracy of the approximation, in
practice. One may often find such assertions in published works as “only the
first three moments of the expansion were used”, or something to that effect.
But just as often there is no information provided about the actual errors
which are observed. Of course, it is not easy to provide such measures of
observed error, as this would require additional computations using the direct
$\mathcal{O}(N^{2})$ method, for comparison purposes. Nevertheless, it is
important for users of the algorithm to know what to expect in terms of
accuracy and efficiency, depending on the choice of algorithm parameters.
We aim to contribute to this gap in understanding by presenting a methodical
investigation into the errors of the approximation used by the FMM, when the
underlying ‘client’ application is the calculation of the velocity field
induced by $N$ regularized particles of vorticity. This application is rather
more demanding than the Newtonian force calculation, because in the latter
case the gravitational interaction is dominated by the first moment —due to
the fact that all mass is positive. Therefore, keeping only the first three
moments could easily give the desired accuracy. On the other hand, as in
Coulomb electrostatic calculations, the vortex particles can be both positive
and negative, and thus an acceptable accuracy may require that more terms in
the expansion be kept.
For the purposes of this study, a prototype code of the FMM computation of the
velocity induced by $N$ vortex particles was implemented using the
Python††http://www.python.org/ language. The nice features of Python —such as
dynamic typing, extensive numerical libraries, and high programmer
productivity— helped us produce a piece of software which is easy to use and
easy to understand. We are currently using this Python code as a starting
point for a parallel version of the code which, in collaboration with members
of the developer team, will be incorporated to the PETSc library for
scientific computingpetsc-manual. This project will be reported elsewhere, but
preliminary results are being presented in the upcoming Parallel CFD
meetingCruzBarbaKnepley2008a. Our final aim is to contribute to the community
of particle simulations with an open source FMM implementation which is
parallel and portable. For the time being, the Python code is being made
available publicly and we welcome correspondence from interested
researchers††http://www.maths.bris.ac.uk/ aelab/research/pyFMM.html .
Using the Python FMM code, more than 900 calculations were performed, varying
the numerical parameters: $N$, the number of particles, $l$, the number of
levels in the tree, and $p$, the truncation level of the multipole expansion.
We looked not only at the maximum error in the domain, which would be the
conventional approach; we also present results showing how the error varies in
space, revealing some interesting features of the method. Through this
presentation of the results, we believe a clear characterization of the nature
of the FMM approximation is obtained.
The paper is organized as follows. The next section presents an outline of the
vortex particle method, for completeness. In §fmm, we offer an overview of the
FMM, with some details of our implementation. Following, in §errors, we
discuss briefly the sources of errors in the FMM algorithm. And finally,
§results reports the detailed experiments using the FMM for evaluation of the
velocity of $N$ vortex particles; the behavior of the method will be
illustrated for varying parameters, as well as the impact on the efficiency of
the calculation, for different problem sizes.
|
arxiv-papers
| 2008-09-10T15:06:08
|
2024-09-04T02:48:57.757280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Felipe A. Cruz, L. A. Barba",
"submitter": "Felipe Cruz",
"url": "https://arxiv.org/abs/0809.1810"
}
|
0809.1949
|
# Protocol Channels
Steffen Wendzel
Kempten University of Applied Science swendzel (at) ploetner-it (dot) de,
www.wendzel.de
(2009-07-26)
###### Abstract
Covert channel techniques are used by attackers to transfer data in a way
prohibited by the security policy. There are two main categories of covert
channels: timing channels and storage channels. This paper introduces a new
storage channel technique called _protocol channels_. A protocol channel
switches one of at least two protocols to send a bit combination to a
destination. The main goal of a protocol channel is that packets containing
covert information look equal to all other packets of the system what makes a
protocol channel hard to detect.
Keywords: protocol channel, covert channel, data hiding
## Protocol Channels
For attackers, it is usual to transfer different kinds of hidden information
trough hacked or public networks. The solution for this task can be to use a
network covert channel technique like they are well known since many years.
There are currently two different main types of covert channels, so called
_storage channels_ (which include hidden information in attributes of
transfered network packets) and _timing channels_ (which make use of the
timings of sent packets to transfer hidden information) [Owens02].
A new storage channel technique called a “protocol channel” includes hidden
information only in the header part of protocols that specify an incapsulated
protocol (e.g. the field “Ether Type” in Ethernet, the “Protocol” value in
PPP, the “Next Header” value in IPv6 or the source/destination port of TCP and
UDP). For instance, if a protocol channel would use the two protocols ICMP and
ARP, while ICMP means that a 0 bit was transfered and ARP means that a 1 bit
was transfered, then the packet combination sent to transfer the bit
combination “0011” would be ICMP, ICMP, ARP, ARP. A protocol channel must not
contain any other information that identifies the channel. It is also
important that a protocol channel only uses usual protocols of the given
network. An algorithm to identify such usual protocols for adaptive covert
channels (protocol hopping covert channels) was introduced by [YADALI08].
The higher the number of available protocols for a protocol channel is, the
higher amount of information can be transfered within one packet since more
states are available. Given the above example, 2 different states are
available, what represents 1 bit. If the attacker could use 4 different
protocols, a packet would represent 2 bits.
This does not allow high covert channel transfer rates but is enough to
transfer sniffed passwords or other tiny information. Specially if the
attacker uses some compressing algorithm (like modifing ASCII texts he
converts to a 6 bit representation of the most printable characters), the need
for a high transfer rate decreases. The proof of concept code “pct” uses a
minimalized 5 bit ASCII encoding and a 6th bit as a parity bit.
## Problems
Since a protocol channel only contains one or two (usualy not more) bits of
hidden information per packet, it is not possibly to include reliability
information (like an ACK flag or a sequence number). If a normal packet that
not belongs to the protocol channel would be accepted by the receiver of a
protocol channel, the whole channel would become desyncronized. It is not
possibly to identify packets which (not) belong to the protocol channel if
they use one of the protocols the protocol channel uses.
Another problem is the defragmentation as well as the loss of packets. If a
packet was defragmented, the receiver would receive it two times what means
that the bit combination would be used two times and the receiver-side bit
combination would be destroyed. The channel would end up desyncronized in this
case too. The receiver could check for packets that include the “More
Fragments” flag of IPv4 as a solution for this problem. Lost packets create a
hole in the bit combination what results in the same desyncronization problem.
## Conclusion
Protocol channels provide attackers a new way to send hidden information
through networks. Even if a detection by network security monitoring systems
is possible – e.g. because of unusual protocols used by the attacker – a
regeneration of the hidden data is as good as impossible since it would need
information about the transfered data type, the way the sent protocol
combinations are interpreted (e.g. big-endian or little-endian) and a
recording of all sent packets to make a regeneration possible.
## References
* [OWENS02] M. Owens: A Discussion of Covert Channels and Steganography, SANS Institute, 2002.
* [YADALI08] F. Yarochkin, S.-Y. Dai, C.-H. Lin, Y. Huang, S.-Y. Kuo: Towards Adaptive Covert Communication System, Dep. of Electrical Engineering, National Taiwan University, 2008.
|
arxiv-papers
| 2008-09-11T10:57:19
|
2024-09-04T02:48:57.761917
|
{
"license": "Public Domain",
"authors": "Steffen Wendzel",
"submitter": "Steffen Wendzel",
"url": "https://arxiv.org/abs/0809.1949"
}
|
0809.2293
|
# High Degree Diophantine Equation $\mathbf{c^{q}=a^{p}+b^{p}}$
WU Sheng-Ping Wuhan University, Wuhan, Hubei Province, The People’s Republic
of China. hiyaho@126.com
(Date: May 4, 2010)
###### Abstract.
The main idea of this article is simply calculating integer functions in
module (Modulated Function and digital function). This article studies power
and exponent functions, logarithm function between integer modules, module of
complex number, the analytic method of digit by digit, and modular integration
and differentials in discrete subspace. Finally prove a condition of non-
solution of Diophantine Equation $a^{p}+b^{p}=c^{q}$:
$a,b>0,(a,b)=(b,c)=1,p,q\geq 41$, $p$ is prime.
###### Key words and phrases:
High degree Diophantine equation, Modulated function, Modulated logarithm,
Digital Analytic, Discrete geometry, Fermat’s Last Theorem
###### 2000 Mathematics Subject Classification:
Primary 11D41, Secondary 11T06, 11C08, 11C20, 13F20
###### Contents
1. 1 Introduction
2. 2 Modulated Function
3. 3 Some Definitions
4. 4 The Modulus Of Prime $p=4n-1$ On Complex Numbers
5. 5 Digital Analytic
6. 6 Modular Integration
7. 7 Discreet Geometry and Subspace
8. 8 Diophantine Equation $a^{p}+b^{p}=c^{q}$
Truth is ordinary.
## 1\. Introduction
When talking about high degree diophantine equation the most famous result is
Fermat’s last theorem. This article applies purely algebraic method to discuss
unequal logarithms of finite integers under module, and get a nice result on
equation $c^{q}=a^{p}+b^{p}$.
## 2\. Modulated Function
In this section $p$ is a prime greater than $2$ unless further indication.
###### Definition 2.1.
Function of $x\in\mathbf{Z}$: $c+\sum_{i=1}^{m}c_{i}x^{i}$ is power-analytic
(i.e power series). Function of $x$: $c+\sum_{i=1}^{m}c_{i}e^{ix}$ is linear
exponent-analytic of bottom $e$. $e,c,c_{i},i$ are constant integers. $m$ is
finite positive integer.
###### Theorem 2.2.
Power-analytic functions modulo $p$ are all the function from mod $p$ to mod
$p$, if $p$ is a prime. And $(1,x^{i}),(0<i\leq p-1)$ are linear independent
group. (for convenience always write $1$ as $x^{0}$, and $x^{p-1}$ is
different from $x^{0}$)
###### Proof.
Make n-th order matrix $X$:
$X_{i,1}=1,Xij=i^{j-1}\quad(1\leq i\leq p,2\leq j\leq p)$
The columnar vector of this matrix is the values of $x^{i}$. This matrix is
Vandermonde’s matrix and its determinant is not zero modulo $p$. The number of
functions in mod $p$ and the number of the linear combinations of the columnar
vectors are the same $p^{p}$. So the theorem is valid. ∎
A proportion of the row vector is values of exponent function modulo $p$.
###### Theorem 2.3.
Exponent-analytic functions modulo $p$ and of a certain bottom are all the
functions from mod $p-1$ to mod $p$, if $p$ is a prime.
###### Proof.
From theorem 2.2, $p-1$ is the least positive number $a$ for:
$\forall x\neq 0\textrm{ mod $p$}(x^{a}=1\textrm{ mod $p$})$
or exists two unequal number $c,b$ mod $p-1$ such that functions
$x^{c},x^{b}:x^{c}=x^{b}$ mod $p$. Hence exists $e$ whose exponent can be any
member in mod $p$ except $0$. Because the part of row vector in matrix $X$ (as
in the previous theorems) is values of exponent function, so this theorem is
valid. ∎
###### Theorem 2.4.
$p$ is a prime. The members except zero factors in mod $p^{n}$ is a multiple
group that is generated by single element $e$ (here called generating element
of mod $p^{n}$).
Think about $p+1$ which is the generating element of all the subgroups of rank
$p^{i}$.
###### Definition 2.5.
(Modulated Logarithm modulo $p^{m}$) $p$ is a prime, $e$ is the generating
element as in the last theorem:
$lm_{e}(x):\quad{x\in\mathbf{Z}((x,p)=1)}\to\textrm{mod }p^{m-1}(p-1):\quad
e^{lm_{e}(x)}=x\textrm{ mod }p^{m}$
Similarly it’s written as that $y=lm_{b}(x)$ mod $p^{m-1}$, $b=e^{p-1}$ mod
$p$. Because for $x$ such that $x=1$ mod $p$ there is only one $y$ mod
$p^{m-1}$ meeting $b^{y}=x$ mod $p^{m}$. $p$ is prime.
###### Lemma 2.6.
$lm_{e}(-1)=p^{m-1}(p-1)/2\textrm{ mod $p^{m-1}(p-1)$}$
$p$ is a prime. $e$ is defined in mod $p^{m}$.
###### Lemma 2.7.
The power series expansions of $log(1+x),(|x|<1)$ (real natural logarithm),
$exp(x)$ (real natural exponent), and the series for $exp(log(1+x)),(|x|<1)$
with the previous two substituted into are absolutely convergent.
###### Definition 2.8.
Because:
$\frac{a}{p^{m}}=kp^{n}\leftrightarrow a=0\textrm{ mod $p^{m+n}$}$
$a,k\in\mathbf{Z}$, it’s valid to make the rational number set modulo integers
if it applies to equations (written as $a/p^{m}=0$ mod $p^{n}$).
###### Definition 2.9.
$p^{i}||a$ means $p^{i}|a$ and not $p^{j}|a,j>i$.
###### Theorem 2.10.
$p$ is a prime greater than $2$. Defining
$E=\sum_{i=0}^{n}\frac{p^{i}}{i!}\textrm{ mod $p^{m}$}$
$n$ is sufficiently great and dependent on $m$. $e^{1-p^{m}}=E$ mod $p^{m}$,
$e$ is the generating element (Here the logarithm: $lm_{e}(x)$ is written as
$lm(x)$). Then for $x\in\mathbf{Z}$:
$E^{x}=\sum_{i=0}^{n}\frac{p^{i}}{i!}x^{i}\textrm{ mod $p^{m}$}$
$lm_{E}(px+1)=\sum_{i=1}^{n}\frac{(-1)^{i+1}p^{i-1}}{i}x^{i}=:f(x)\textrm{ mod
$p^{m-1}$}$
$lm_{E}(x^{1-p^{m}})=lm(x^{1-p^{m}})/lm(E)=lm(x^{1-p^{m}})=lm(x)$ mod
$p^{m-1}$. In fact $m$ is free to choose. And $E$ is nearly $exp(p)$.
If $2|x$ this theorem is also valid for $p=2$.
###### Proof.
To prove the theorem, contrast the coefficients of $E^{x}$ and $E^{f(x)}$ to
those of $exp(px)$ and $exp(log(px+1))$. ∎
###### Theorem 2.11.
Set $d_{m}:p^{d_{m}}||{p^{m}}/{m!}$. It’s valid $d_{m(>p^{n})}>d_{p^{n}}$.
###### Theorem 2.12.
(Modulated Derivation) $p$ is a prime greater than $2$. $f(x)$ is a certain
power-analytic form mod $p^{m}$, $f^{(i)}(x)$ is the $i$-th order real
derivation (hence called modulated derivation relative to the special
difference by $zp$ as this theorem): ($n$ is sufficiently great)
$f(x+zp)=\sum_{i=0}^{n}\frac{p^{i}}{i!}z^{i}f^{(i)}(x)\textrm{ mod $p^{m}$}$
If $2|z$ this theorem is also valid for $p=2$.
###### Definition 2.13.
(Example of Modulated Function) Besides taking functions as integer function,
some functions can be defined (for increasingly positive integer $m$) by
equations _modulo $p^{m}$_, even though with irrational value as real function
in form sometimes. This kinds of function is called Modulated Function. For
example:
$(1+p^{2}x)^{\frac{1}{p}}\textrm{ mod $p^{m}$}$
is as the unique solution of the equation for $y$:
$1+p^{2}x=y^{p}\textrm{ mod $p^{m+1}$}$
By calculation to verify:
$plm_{E}(y)=lm_{E}(1+p^{2}x)=\sum_{i=1}(-1)^{i+1}p^{i-1}\frac{(px)^{i}}{i}\textrm{
mod $p^{m+1}$}$
###### Lemma 2.14.
$(E^{x})^{\prime}=pE^{x}\textrm{ mod $p^{m}$}$
This modulated derivation is not necessary to relate to difference by $zp$,
it’s valid for difference by $1$.
###### Lemma 2.15.
The derivation of $(1+x)^{1/p}$ mod $p^{m+2}$ at the points $x:p^{2}|x$ is:
$\displaystyle((1+x)^{1/p})^{\prime}=(E^{\frac{1}{p}lm_{E}(1+x)})^{\prime}\textrm{
mod $p^{m}$}$ $\displaystyle=$ $\displaystyle
pE^{\frac{1}{p}lm_{E}(1+x)}(\frac{1}{p}lm_{E}(1+x))^{\prime}=\frac{1}{p}(1+x)^{1/p}\frac{1}{1+x}\textrm{
mod $p^{m}$}$
###### Theorem 2.16.
Because
$1-x^{p^{n-1}(p-1)}=\left\\{\begin{array}[]{l}0,(x\neq 0\textrm{ mod $p$})\\\
1,(x=0\textrm{ mod $p$})\end{array}\right.\textrm{ mod $p^{n}$}$
and in $x=0$ mod $p$, any power-analytic function is of the form:
$\sum_{i=0}^{n-1}a_{i}x^{i}$
hence the power-analytic function is of the form:
$\sum_{i=0}^{p-1}(1-(x-i)^{p^{n-1}(p-1)})(\sum_{k=0}^{n-1}a_{ki}(x-i)^{k})\textrm{
mod $p^{n}$}$
###### Theorem 2.17.
Modulated Derivation of power-analytic and modulated function $f(x)$ mod
$p^{2m}$ can be calculated as
$f^{\prime}(x)=(f(x+p^{m})-f(x))/p^{m}\text{ mod ${p^{m}}$}$
The modulated derivations of equal power analytic functions mod $p^{2m}$ are
equal mod $p^{m}$.
###### Theorem 2.18.
Modulated $plm(x)$ is power-analytic modulo $p^{m}$.
## 3\. Some Definitions
In this section $p,p_{i}$ is prime. $m,m^{\prime}$ are sufficiently great.
###### Definition 3.1.
$x\to a$ means the variable $x$ is set value $a$.
###### Definition 3.2.
$a,b,c,d,k,p,q$ are integers,$(p,q)=1$:
$[a]_{p}=[a+kp]_{p}$ $[a]_{p}+[b]_{p}=[a+b]_{p}$
$[a=b]_{p}$ means $[a]_{p}=[b]_{p}$.
$[a]_{p}[b]_{q}=[x:[x=b]_{p},[x=b]_{q}]_{pq}$ $[a]_{p}\cdot[b]_{p}=[ab]_{p}$
Easy to verify:
$[a+c]_{p}[b+d]_{q}=[a]_{p}[b]_{q}+[c]_{p}[d]_{q}$
$[ka]_{p}[kb]_{q}=k[a]_{p}[b]_{q}$
$[a^{k}]_{p}[b^{k}]_{q}=([a]_{p}[b]_{q})^{k}$
###### Definition 3.3.
$\sigma(x)$ is the Euler’s character number as the least positive integer $s$
meeting
$\forall y((y,x)=1\to[y^{s}=1]_{x})$
###### Definition 3.4.
The complete logarithm on composite modules is complicated. But it can be
easily defined
$[lm(x)]_{p_{1}^{n_{1}}p_{2}^{n_{2}}\cdot\cdot\cdot
p_{m}^{n_{m}}}:=[lm(x)]_{p_{1}^{n_{1}}}[lm(x)]_{p_{2}^{n_{2}}}\cdot\cdot\cdot[lm(x)]_{p_{m}^{n_{m}}}$
$p_{i}$ is distinct primes. This definition will be used without detailed
indication.
###### Definition 3.5.
$x={{}_{q}}[a]:[a=x]_{q},0\leq x<q$
###### Definition 3.6.
For module $p^{i}$: $F_{p^{i}}(x)(:=p^{n})$ means $F_{p^{i}}(x)||x$; For
composite module $Q_{1}Q_{2}$ meeting $(Q_{1},Q_{2})=1$:
$F_{Q_{1}Q_{2}}(x):=F_{Q_{1}}(x)F_{Q_{2}}(x)$.
###### Definition 3.7.
$P(q)$ is the multiple of all the distinct prime factors of $q$.
###### Definition 3.8.
$Q(x)=\prod_{i}[p_{i}]_{p_{i}^{m}}$, $p_{i}$ is all the prime factors of $x$.
$m$ is sufficiently great.
###### Theorem 3.9.
$2|q\to 2|x$:
$[Q(q)lm(1+xq)=\sum_{i=1}(xq)^{i}(-1)^{i+1}/i]_{q^{m}}$
The method of proof is getting result in powered prime module and synthesizing
them in composite module.
###### Definition 3.10.
$[a^{1/2}:=e^{{{}_{p-1}}[lm(a)]/2}]_{p}$
It can be can proven that
$[a^{1/2}(1/a)^{1/2}=-1]_{p}$
###### Definition 3.11.
$[lm(pk)=plm(k)]_{p^{m}}$
$p$ is a prime.
###### Theorem 3.12.
$[plm(x)=(x^{p^{m}(1-p^{m})}-1)/p^{m}]_{p^{m}},[x\neq 0]_{p}$
$plm(x)=\sum_{1}^{m^{\prime}}(-1)^{i+1}((x-x^{p^{m}})/x^{p^{m}})^{i}/i,[x\neq
0]_{p}$
Hence
$[plm^{\prime}(x)=1/x]_{p^{m}},[x\neq 0]_{p}$
###### Definition 3.13.
$[y=x^{1/a}]_{p}$ is the solutions of equation $[y^{a}=x]_{p}$. When
$(a,p-1)\neq 1$, $x^{1/a}$ is multi-valued function or empty at all.
## 4\. The Modulus Of Prime $p=4n-1$ On Complex Numbers
In this section $p$ is prime other than $2$.
For prime $p=4n-1$ the equation $[i^{2}=-1]_{p}$ has no solution, then it’s
suitable to extend the module to complex numbers.
###### Definition 4.1.
$\mathbf{PZ}=\\{x+yi:x,y\in\mathbf{Z}\\}$
For $p=4n-1$, define
$[x:x\in\mathbf{PZ}]_{p^{n}}=[x+tp^{n}:t\in\mathbf{PZ}]_{p^{n}}$
This definition is sound and good because there is no zero factor other than
$p^{j}$.
###### Definition 4.2.
For prime $p=4n-1$. $a,b\in\mathbf{Z}$. Define $e^{i}$ in $\mathbf{PZ}$, for
any $j\in\mathbf{Z}$ and some $a,b$:
$[e^{j\cdot i}=\frac{2ab}{a^{2}+b^{2}}+i\frac{a^{2}-b^{2}}{a^{2}+b^{2}}]_{p}$
$[e^{(1-p^{2m})i}=\sum_{j=0}^{n}\frac{p^{j}i^{j}}{j!}=:E^{i}]_{p^{m}}$
($n$ is sufficiently great and dependent on $m$). Analyzing the group formed
by the all solutions of $[z^{*}z=1]_{p}$ in mod $p$ (count $p+1$ and
$[z^{p}=z^{*}]_{p}$) can find this definition is all right.
Define $[e^{a+bi}=e^{a}e^{bi}]_{p^{m}}$.
It can be found the results on exponent’s and logarithm’s expansion are valid
in $\mathbf{PZ}$ similar to the form as in $\mathbf{Z}$.
###### Definition 4.3.
$(q_{1},q_{2})=1,a,b,a^{\prime},b^{\prime}\in\mathbf{Z}$:
$[a+bi]_{q_{1}}[a^{\prime}+b^{\prime}i]_{q_{2}}=[a]_{q_{1}}[a^{\prime}]_{q_{2}}+[i[b]_{q_{1}}[b^{\prime}]_{q_{2}}]_{q_{1}q_{2}}$
Also define the triangular functions by $e^{z}$.
###### Definition 4.4.
For mod $p^{m},p=4n+1$, $\omega:[\omega^{2}=-1]_{p^{m}}$ was chosen as
_pseudo-imaginary_ unit $i$ (and treat $\omega,i$ differently), define for all
equations that is formed by functions of arguments
$z_{1}i+z_{2},z_{1},z_{2}\in\mathbf{Z}$, with property _pseudo-conjugation_ :
$[z=a+bi=0]_{p^{m}}\to[z^{*}=a-bi=0]_{p^{m}}$
$a,b\in\mathbf{Z}$. Then the $i$ has the similar property like the true
imaginary unit because from the above condition, it’s implied:
$[a+bi=0]_{p^{m}}\to[a=0,b=0]_{p^{m}}$
Elements like $a,b$ are called pseudo-real.
Notice that the original value of $i$: $\omega$ should be treated as pseudo-
real.
For $i$ (pseudo-imaginary unit for $p=4n+1$) actually real, Strengthen $i$ as
$[1]_{p-1}[i]_{p^{m}}$ (but there will be trouble in composite modulus
regarding exponent) and set pseudo-conjugation to all the equations involved.
Defining $e^{i}$ for pseudo-real $a,b$ meeting
$[e^{i}=a+bi]_{p^{m}},[a,b\neq 0]_{p}$ $[e^{i(1-p^{m})}=E^{i}]_{p^{m}}$
$[(e^{i})^{*}=e^{-i}=a-bi]_{p^{m}}$
it means to meet the harmony in real and complex senses. Think about the group
formed by $[z:zz^{*}=1]_{p}$, whose elements count $p-1$ ie. all non-zero
$[n]_{p}$, it can be found the above identities are possible.
Complete logarithm is complicated, but logarithm mod $p^{n}$ is easy, it will
be used without detailed indication.
Pseudo-conjugation will be used without detailed indication.
## 5\. Digital Analytic
In this section $p$ is prime unless further indication. $m,m^{\prime}$ are
sufficiently great.
###### Definition 5.1.
$T(q^{\prime},x)=y:[x=y]_{q^{\prime}},2|q^{\prime}:-q^{\prime}/2<y<q^{\prime}/2+1\textrm{
otherwise }-q^{\prime}/2<y<q^{\prime}/2$
_Digit_ can be express as
$D_{q^{n}}(x):=(T(q^{n},x)-T(q^{n-1},x))/q^{n-1}$
_Digital function_ is digit in digits’s power analytic function. Also define
$D_{(q)p}(x):=D_{p}((x-T(q,x))/q)$
###### Definition 5.2.
_Independent Digital variables (Functions)_ is the digits can not be
constrained in root set of a nonzero digital function.
###### Theorem 5.3.
_Resolve function digit by digit._ For an integer function $f(x)$ mod $p^{m}$,
whose value can be express by its digits. The digit of the function is
determined by its arguments’s digits, then the digit can be express by
_Digital function_
$D_{p^{k}}(f(x))=\sum_{j}a_{j}\prod_{i=1}^{n}D^{j_{i}}_{p^{i}}(x)$
$0\leq j_{i}\leq p-1$. With this method of _Digit by Digit_ the whole function
can be resolved in the similar form for each digits of the function.
Digital functional resolution has some important properties, it can express
arbitrary map $f(x)$ between the same module.
###### Definition 5.4.
_Digital functions group_
$[s_{i}=f_{i}(,x_{j},)]_{p},i,j=0\cdot,\cdot,\cdot,n$
called _square group_ or _square function_.
###### Theorem 5.5.
Functional independent square group is invertible.
###### Proof.
Independence means the function value travel all, it’s one to one map and
invertible:
$[x_{i}=g_{i}(,s_{j},)]_{p},i,j=0\cdot,\cdot,\cdot,n$
∎
## 6\. Modular Integration
In this section $p$ is prime unless further indication. $m,m^{\prime}$ are
sufficiently great.
###### Definition 6.1.
The size of a set is called the freedom of the set.
###### Definition 6.2.
With consideration of mod $p$:
$\delta(,x_{i}-C_{i},):=\left\\{\begin{array}[]{ll}1&[(,x_{i},)=(,C_{i},)]_{p}\\\
0&\textrm{otherwise}\end{array}\right.$
###### Definition 6.3.
The algebraic derivation of the shortest expression (clean expression) of
digital function is called clean derivation. The clean derivation of
$[f(x)]_{p}$ is denoted as $f^{D}(x),Df(x)/Dx$ formally. This definition will
be used without detailed indication. The real algebraic derivation, or
modulated derivation is denoted as $f^{\prime}(x),df(x)/dx$ for $f(x)$ or
$f(x)^{p^{n}}$.
###### Theorem 6.4.
The clean derivation expressed in algebraic derivation is
$[f^{D}(x)=d_{k}f(x)]_{p}$
with $k$ sufficiently great, and $d_{k}$ is:
$d_{0}:=d/dx,d_{1}:=d_{0}+(d^{p}/d^{p}x)/p!$
$d_{n}:=d_{n-1}+(d^{n(p-1)+1}/d^{n(p-1)+1}x)/(n(p-1)+1)!$
###### Definition 6.5.
For convenience it’s taken that
$[1/x:=x^{p(p-1)-1}]_{p^{2}}$
when calculate digital functions.
###### Definition 6.6.
$[f^{D}(x)=-\sum_{t=0}^{p-1}f(t)(t-x)^{p-2}]_{p}$
or concisely
$[=-\sum_{t}f(t)(t-x)^{p-2}]_{p}$
Prove this by the formula of power sum and bernoulli number.
###### Definition 6.7.
The reduced function is clean and without a term that has a factor of the
highest degree on single argument.
###### Theorem 6.8.
$[I^{t}(x):=-\sum_{i=0}^{p-2}x^{p-1-i}t^{i+1}/(i+1)]_{p}$
$[I^{t}_{t_{0}}(x):=I^{t}(x)-I^{t_{0}}(x)]_{p}$
then
$[\int_{0}^{t}f(x)dx=\sum_{x}f(x)I^{t}(x)]_{p}$
$f$ is reduced and clean.
###### Theorem 6.9.
The $I^{t}(C)$ has $p-1$ distinct values and two zero values if $[C\neq
0]_{p}$.
###### Proof.
Take the clean function as vector with units $x^{n}$. If the equation
$I^{t}(x)=C$ has roots, observe the freedom of the set generated by transform
$f(x)\to\sum_{x}f(x)(I^{t}(x)-C)$
∎
###### Theorem 6.10.
$[I^{t}(x)\neq-t]_{p},[t\neq 0]_{p}$ $[I^{t}(x)=-I^{x}(t)]_{p}$
###### Definition 6.11.
$[f(x)\cdot I^{t}(x):=\sum_{x}f(x)I^{t}(x)]_{p}$
$[f(,x_{i},)\cdot\prod_{i}I^{t_{i}}(x_{i}):=\sum_{,x_{i},}f(,x_{i},)\prod_{i}I^{t_{i}}(x_{i})]_{p}$
###### Definition 6.12.
$[Dt:=I^{t}(t)-I^{t-1}(t)-I^{t}(1)]_{p}$ $[f(t)\cdot Dt:=f(t)\cdot
I^{t}(t)-f(t)\cdot I^{t-1}(t)-f(t)\cdot I^{t}(1)]_{p}$ $[f(t)\cdot
I^{t}(1):=\sum_{x}f(x)I^{t}(1)]_{p}$
Define the _modular integration_ in an area $A$:
$[\int_{A}f(,x_{i},)\prod_{i=0}^{n}Dx_{i}:=\sum_{(,x_{i},)}\delta_{A}(,x_{i},)(f(,x_{i},)\cdot\prod_{i=0}^{n}Dx_{i})]_{p}$
$[\delta_{A}(,x_{i},)=\sum_{(,x_{i},)\in A}\delta(,x_{i},)]_{p}$
$[\int_{a}^{b}f(x)Dx:=\sum_{x\in(a,b]}f(x)\cdot Dx]_{p}$
Obviously
$[\int^{x}_{0}\delta(x)Dx=\int^{x}_{0}\delta(x)D(x+C)]_{p}$
$[\int^{x}_{0}f(x)Dx=\int^{x}_{0}f(x)D(x+C)]_{p}$ $[\delta(x)\cdot
Dx=-I^{x}(1)]_{p}$ $\forall x[f(x)\cdot Dx=0]_{p}\leftrightarrow\forall
x[f(x)=0]_{p}$
###### Definition 6.13.
Define
$[f^{I}(x):=f(x)\cdot Dx]_{p}$ $[f^{\Sigma}(t):=\sum_{x=1}^{t}f^{I}(x)]_{p}$
$[f^{\Delta}(x):f^{\Delta}(x)-f^{\Delta}(x-1)=f^{I}(x),f^{\Delta}(0)=0]_{p}$
$\int f(x)Dx:=f^{\Delta}(x)+C$
$f^{\Delta}(x)$ (is called _original function_) is defined by $f^{I}(x)$
uniquely except for a Constant difference. For example
$[\int\delta(x)Dx=-(x^{p}-x)/p+C=xlm(x)|_{[x\neq
0]_{p}}+C=\sum_{z=0}^{x}I^{1}(z)+C]_{p}$
Note that the function $xlm(x)$ is not digital function, in fact it’s defined
in mod $p^{2}$, this means the integration is depend on integral track,
especially as the track $(a,b]$ crosses zero mod $p$.
It’s obvious that
$[Df^{I}(x)/Dx=f(x)-f(x-1)]_{p}$
The definition is extended to multi-arguments function as
$[f^{I}(,x_{i},):=f(,x_{i},)\cdot\prod_{i}Dx_{i}]_{p}$
$[f^{\Sigma}(,t_{i},):=(\prod_{i}\sum_{x_{i}=0}^{t})f^{I}(,x_{i},)]_{p}$
$[f^{\Delta}(,x_{i},):f^{\Delta}(x_{0},,x_{i},)-f^{\Delta}(x_{0}-1,,x_{i}-1,)=f^{I}(,x_{i},),f^{\Delta}(,x_{i-1},0,x_{i+1},)=0]_{p}$
$(\prod_{i}\int)f(,x_{i},)\prod_{i}Dx_{i}:=f^{\Delta}(,x_{i},)+C(,x_{i},)$
###### Definition 6.14.
Define _modular derivation_ of digital function formally as
$[f^{D}(x):=\sum_{t}-f(x^{p}+t^{p})/t^{p}]_{p}$
It’s the inverse of the modular integration.
###### Theorem 6.15.
$[(\prod_{i}\int_{x_{i}=a_{i}}^{b_{i}}f(,x_{i},)\prod_{i}Dx_{i}=(\prod_{i}\Delta_{x_{i}=a_{i}}^{b_{i}}f^{\Delta}(,x_{i},)]_{p}$
$a_{i},b_{i}$ are constants.
## 7\. Discreet Geometry and Subspace
In this section $m$ is sufficiently great. $p$ is prime.
###### Definition 7.1.
Define _modular differential_ as the inversion of _square (modular) linear
integration_ :
$[\int_{l}f_{i}(X)Dx_{i}=\sum_{i}\int_{l_{i}}f_{i}(X)Dx_{i}]_{p}$
$X=(,x_{i},)$ $l=\sum_{i}l_{i}$
$l_{i}=(,x_{i-1,1},x_{i},x_{i+1,0},x_{i+2,0},)$ $x_{i}=(x_{i,0},x_{i,1}]$
And define
$[DF(X):=\sum_{i}\frac{DF(X)}{D{x_{i}}}Dx_{i}]_{p}$
$\frac{DF(X)}{D{x_{i}}}$ is clean partial derivation. $X,Dx$ are called
original relatively to $F(X),DF(X)$.
###### Definition 7.2.
The so-called _discreet geometry_ always discuss the square box
$(x_{1},x_{2})=([a,b],[c,d]),(,x_{i},)=(,[a_{i},b_{i}],)$
And take these square box of different dimensions into consideration of real
geometry. Obviously it can be find the result is similar to that in real
differential geometry
$[\int_{D}D^{\wedge}F=\int_{\partial D}F]_{p}$
$F$ is _antisymmetrical modular differential tensor_.
###### Definition 7.3.
$D(G+G^{\prime})=DG+DG^{\prime},D^{\wedge}(G+G^{\prime})=D^{\wedge}G+D^{\wedge}G^{\prime}$
$G,G^{\prime}$ is differential tensor.
$[D(K(X)\bigotimes_{i}Dx_{\sigma(i)})=DK(X)\bigotimes_{i}Dx_{\sigma(i)}]_{p}$
$[D^{\wedge}(K(X)\bigwedge_{i}Dx_{\sigma(i)})=DK(X)\bigwedge_{i}Dx_{\sigma(i)}]_{p}$
$\sigma$ is a map in $\mathbf{N}$.
###### Definition 7.4.
Use $\Delta_{k}x$ and $D_{k}x$ to express difference and differential, and for
example, $D_{k}^{2}x$ means $D_{k}x\cdot D_{k}x$, $D_{2}^{2}xD_{1}x$ means
$(D_{2}x\cdot D_{2}x)\otimes D_{1}x$.
###### Definition 7.5.
For the discrete space
$[F(X)=(f_{0}(,x_{i}),,f_{i}(,x_{j},),,f_{n-1}(,x_{j},))]_{p}$
($F(X)$ is square invertible), the _subspace_ in digital functions set
sub $f_{k\in A}(X)$
is the module of the ideal generated from $f_{k\in A}(X)$.
###### Definition 7.6.
_Span function_ of arguments $(,x_{i},)$ is digital function
$[F(,x_{i},,\Delta_{k}x_{i},)]_{p}$
###### Definition 7.7.
The difference of span function is defined by
$[\Delta^{\prime}\Delta X_{i}=0]_{p}$
For all $i$.
###### Theorem 7.8.
The difference can be calculate by operator
$[\Delta=\sum_{n=1}^{m}(\sum_{i}\Delta x_{i}\frac{D}{Dx_{i}})^{n}/n!]_{p}$
###### Theorem 7.9.
$\Delta(f(x)g(x))=g(x)\Delta f(x)+f(x)\Delta g(x)+\Delta f(x)\Delta g(x)$
###### Definition 7.10.
The _Correspondence_ between clean span function $S$ and tensor $T$ is
substitution:
$S\to T=TC(S):\Delta_{k}x_{i}\to D_{k}x_{i},T\to
S=SC(T):D_{k}x_{i}\to\Delta_{k}x_{i}$
###### Definition 7.11.
The sum of all terms of the lowest degree of the difference of original
arguments in the span function $f$ is denoted as $LD(f)$
###### Definition 7.12.
The differential tensor in subspace sub $f_{i\in A}(,x_{j},)$ is defined as
module of all
$[\prod_{i}D_{\sigma(i)}(\prod_{k\in A}f_{k}^{j_{k}}(X))]_{p}$
$\sigma$ is arbitrary map in $\mathbf{N}$.
###### Definition 7.13.
The span function in subspace sub $f_{i\in A}(,x_{j},)$ is defined as module
of all
$[(\prod_{i=1}^{n}\Delta_{\sigma(i)})(\prod_{k\in
A}f_{k}^{j_{k}}(X))]_{p},n\geq 0$
###### Theorem 7.14.
$[F=0\to\Delta F=0]_{p}\textrm{ sub $f_{i\in A}(,x_{i},)$}$
$F$ is a span function.
###### Theorem 7.15.
$[G=0\leftrightarrow SC(G)=0]_{p}\textrm{ sub $f_{i\in A}(,x_{i},)$}$
$G$ is is differential tensor.
###### Theorem 7.16.
$[T=0\to DT=0]_{p}\textrm{ sub $f_{i\in A}(,x_{i},)$}$
$T$ is a differential tensor.
###### Theorem 7.17.
If a function in subspace
$[Dg(,x_{j},)=0]_{p}\textrm{ sub $f_{i\in A}(,x_{j},)$}$
Then
$[g(,x_{i},)=C]_{p}\textrm{ sub $f_{i\in A}(,x_{j},)$}$
###### Proof.
The span function in sub $f_{i\in A}(,x_{j},)$ is in fact the substitution
$f_{i\in A}(,x_{j},)\to 0$
and taking the arguments of $f_{i}(,x_{j},)$ to express the functions. ∎
###### Definition 7.18.
Derivation of the group of functions $(,f_{i}(,x_{i},),),0\leq i<n$
$G_{(,x_{i},)}^{(,f_{i},)}:=|\frac{\partial(,f_{i}(,x_{j},),)}{\partial(,x_{j},)}|$
is called _geometry derivation_. If the derivation is operated on clean
functions group in a prime module then it’s called clean geometry derivation.
###### Definition 7.19.
For convenience for the latter define
$\delta(x):=1-x^{p-1}$
###### Definition 7.20.
Express a square digital function as sum of delta branches
$[f_{j}(x_{0},,x_{i},,x_{n})=a_{(j)k_{0},,k_{i},,k_{n}}\delta(x_{0}-k_{0},,x_{i}-k_{i},,x_{n}-k_{n})]_{p}$
The clean geometry derivation in $(,x_{i},)=(,k_{i},)$ is only depend on the
delta branches of
$[(a_{i}\delta_{(,k_{i},)(k_{i}/b_{i})})_{i}]_{p}$
$\delta(,x_{i}-k_{i},):=\delta_{(,k_{i},)},(,k_{i-1},k_{i},k_{i+1},)(k_{i}/k):=(,k_{i-1},k,k_{i+1},)$
The points as in the supporting set of these delta branches are called
_relative chain_ of the point $(,k_{i},)$. The points in relative chain of
point $P$ is denoted as $RC(P)$.
###### Theorem 7.21.
For a square digital function, the relative chain of the point of $P$ are $R$,
and the functional values of members of $R$ are distinct. Then Construct like
this: Alter the function’s value by adding on delta branches but do not alter
that of the chain $R$, hence to form a invertible function with the clean
geometry derivations are unchanged at the point $P$, and the determinant of
the square sub-matrix of the partial derivation matrix (of dimensions at least
2) of the square digital functions, is also unchanged.
## 8\. Diophantine Equation $a^{p}+b^{p}=c^{q}$
$m$ are sufficiently great.
###### Definition 8.1.
For real number $a$
$[a]=\max(x\in\mathbf{Z}:x\leq a)$
###### Theorem 8.2.
$0<b<a<q/P^{3}(q),P^{11}(q)|q$, $(a,q)=(b,q)=1$. For any prime $p|q$ meeting
$[a^{2}\neq b^{2}]_{p}$. Then
$[lm(a)\neq lm(b)]_{q^{2}}$
###### Proof.
$r=P(q)=\prod_{i}p_{i}$. If $[lm(a)=lm(b)]_{q^{2}}$.
Considering module $q^{2}r$. Make
$f(x,y)=\prod_{i}[T((q^{2},p_{i}^{m}),(a+rx)^{p_{i}-1}+(b+ry)^{p_{i}-1}-a^{p_{i}-1}-b^{p_{i}-1})+(q^{2},p_{i}^{m})s]_{p_{i}^{m}})$
$x=\sum_{i}q_{i}D_{i},q_{i}=q/(q,p_{i}^{m})$ $y=\sum_{i}q_{i}D^{\prime}_{i}$
set
$D_{i}=T((q,p_{i}^{m}),D_{i}),D^{\prime}_{i}=T((q,p_{i}^{m}),D^{\prime}_{i}),[x=D_{i}]_{(q^{2},p_{i}^{m})},[y=D^{\prime}_{i}]_{(q^{2},p_{i}^{m})}$
The function $f(x,y)$ can be expressed digit by digit in the square function
$[F_{ji}(,D_{li}(x),D_{l+1,i}(x),,D_{li}(y),D_{l+1,i}(y),)=D_{p_{i}^{j}}(f(x,y))]_{p_{i}^{m}},p_{i}|p_{i}^{j}|q^{2}r$
$D_{li}(x):=D_{p_{i}^{l}}(D_{i}),p_{i}^{l}|q$
$F_{ji}$ depends on $D_{k},D^{\prime}_{k}$. The derivation on
$D_{li}(x),D_{li}(y)$ of the square group $F_{ji}$ is derived from modulated
derivation by noticing the special power of the function $f(x,y)$:
$[df(x,y)=\sum_{j}p_{i}^{j}dF_{ji}]_{(p_{i}^{m},q^{2}r)}$
$dF_{ji}$ is in the form of the real algebraical differential on variable
forms of $D_{li}(x),D_{li}(y)$. It can be observed that replace the
differential unit $dD_{li}(x),dD_{li}(y)$ with free digits, then the according
replacements of $df(x,y)$ is generated at sub-module of mod
$(q^{2}r,p_{i}^{m})$. If the derivation on $D_{li}(x),D_{li}(y)$ between
several $[F_{ji}]_{p_{i}}$ are independent of each others then the according
digits of the replacements of these $df(x,y)$ are free of each others.
Calculate the case of the derivation at $(x,y)\to(0,0),s\to 0$ and that
$\Delta x,\Delta y$ replace $dx,dy$. The distinct generated function values
mod $q^{2}r$ count less than
$rab(q/a+q/b)$
Make
$rab(q/a+q/b)<q^{2}/r$
to formulate for some $p_{i}$
$[\bigwedge_{l>1,p_{i}^{l}|q^{2}}DF_{li}|_{(x,y)=(0,0)}=0]_{p_{i}}$
Further more for some $k$ at $(x,y)=(0,0)$
$[DF_{ki}=0]_{p_{i}}\textrm{ mod $[DF_{ji}]_{p_{i}},p_{i}^{j}|q^{2},j\geq k$}$
This results is proved by reasoning in that: the replacements of $df$ is
$CDr:=Cr\sum_{i}q_{i}(D_{i}/a+D^{\prime}_{i}/b)=:Cr(q_{i}A(=D_{i}/a+D^{\prime}_{i}/b))+(p_{i}^{m},q)B)$
Then varying $A$ (or $B$) is to draw a straight line in 2-dimensional space
$(\textrm{mod $(p_{i}^{m},q^{2})$, mod $q^{2}/(p_{i}^{m},q^{2})$})$
like
$([D]_{(p_{i}^{m},q^{2})},[D]_{q^{2}/(p_{i}^{m},q^{2})})$
Presume doesn’t exist $D_{1},D_{2}:0<|D_{1}|,|D_{2}|<p_{i}$:
$[D_{1}/a=D_{2}/b]_{(q/r,p_{i}^{m})}$
Consider points $(x,y)=(0,0)$ and its relative chain in mod
$(q^{2}r,p_{i}^{m})$. Operate on the domain of $x,y$ to set, except the first
digit invariable, $f(x,y)$ all distinct mod $(p_{i}^{m},q^{2}r)$, like the
theorem 7.21, with functional values of the point $(x,y)=(0,0)$ and its
relative chain unchanged, and also set
$[f(RC(0,q_{i}s))\to
f(RC(0,0))+p_{i}^{k-1}s^{\prime}]_{(q^{2},p_{i}^{m})},s=T(p_{i},s)$
$[s=s^{\prime}]_{p_{i}}$, $s^{\prime}$ is choose to not disturb the digits of
$f(RC(0,0))$ except the $k$-th digit and the $RC(0,q_{i}s)$ and $RC(0,0)$ are
created by adding the same modular value on $(0,q_{i}s),(0,0)$ respectively.
After these it’s valid that for the new digits $F_{ji}$
$[0=\delta(F_{ki}-F_{vi})DF_{ki}=Ds=DF_{ki}]_{p}\textrm{ sub
$F_{ji},F_{vi}-F_{ki},p_{i}^{j}|q^{2},p_{i}^{v}=(p_{i}^{m},q^{2}r),j\neq k$}$
$[F_{ki}=C]_{p}\textrm{ sub
$F_{ji},F_{vi}-F_{ki},p_{i}^{j}|q^{2},p_{i}^{v}=(p_{i}^{m},q^{2}r),j\neq k$}$
it’s obviously not valid.
If exists $D_{1},D_{2}:0<|D_{1}|,|D_{2}|<p_{i}$:
$[D_{1}/a=D_{2}/b]_{(q/r,p_{i}^{m})},(D_{1},D_{2})=1$
discussion on
$[f(x,y)=T(p_{i}^{10},(d_{1}+p_{i}x)^{p_{i}-1}+(d_{2}+p_{i}y)^{p_{i}-1}-d_{1}^{p_{i}-1}-d_{2}^{p_{i}-1})+p_{i}^{10}s]_{p_{i}^{1}1})$
$p_{i}<\max(|d_{1}|,|d_{2}|)<p_{i}^{2},d_{1}=D_{1}^{2^{n}},d_{2}=D_{2}^{2^{n}}$
This case can be proved impossible unless $[a=\pm b]_{(q^{2}/r,p_{i}^{m})}$. ∎
###### Theorem 8.3.
$0<b<a<q/P^{3}(q)$,$(a,q)=(b,q)=1$. $P^{11}(q)|q$. Then
$[lm(a)\neq lm(b)]_{q^{4}/P^{5}(q)}$
###### Proof.
If not, then:
If the product of all prime sub-modules in which
$a^{2}=b^{2}$
is greater than $q^{2}/P^{5}(q)$, then
$a^{2}=b^{2}$
∎
###### Theorem 8.4.
For prime $p$ and positive integer $q$ the equation
$a^{p}+b^{p}=c^{q}$
has no integer solution $(a,b,c)$ such that $a,b>0,(a,b)=(b,c)=(a,c)=1$ if
$p,q\geq 41$.
###### Proof.
The method is to make logarithm in mod $c^{q}$. It’s a condition sufficient
for controversy:
$\frac{q}{p}\leq 6[(q-2)/39]$
∎
|
arxiv-papers
| 2008-09-12T21:40:29
|
2024-09-04T02:48:57.768807
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Sheng-Ping Wu",
"submitter": "Sheng-Ping Wu",
"url": "https://arxiv.org/abs/0809.2293"
}
|
0809.2306
|
This submission has been withdrawn by arXiv administrators because of
inappropriate authorship claims.
|
arxiv-papers
| 2008-09-13T00:36:34
|
2024-09-04T02:48:57.773243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhixiang Wu",
"submitter": "Zhixiang Wu",
"url": "https://arxiv.org/abs/0809.2306"
}
|
0809.2354
|
# Generalized Mean Field Approach to a Resonant Bose-Fermi Mixture
D.C.E. Bortolotti JILA, NIST and Department of Physics, University of
Colorado, Boulder, CO 80309-0440 LENS and Dipartimento di Fisica, Universitá
di Firenze, and INFM, Sesto Fiorentino, Italy A.V. Avdeenkov Institute of
Nuclear Physics, Moscow State University, Vorob’evy Gory, 119992, Moscow,
Russia J.L. Bohn JILA, NIST and Department of Physics, University of
Colorado, Boulder, CO 80309-0440
###### Abstract
We formulate a generalized mean-field theory of a mixture of fermionic and
bosonic atoms, in which the fermion-boson interaction can be controlled by a
Feshbach resonance. The theory correctly accounts for molecular binding
energies of the molecules in the two-body limit, in contrast to the most
straightforward mean-field theory. Using this theory, we discuss the
equilibrium properties of fermionic molecules created from atom pairs in the
gas. We also address the formation of molecules when the magnetic field is
ramped across the resonance, and present a simple Landau-Zener result for this
process.
## I introduction
The use of magnetic Feshbach resonances to manipulate the interactions in
ultracold quantum gases has greatly enriched the study of many-body physics.
Notable examples include the crossover between BCS (Bardeen-Cooper-Schrieffer
BCS1957 ) and BEC (Bose-Einstein Condensate bose1924 ; einstein1924 )
superfluidity in ultracold Fermi gases regal2003nature_sup ;
bartenstein2004cmb ; zwierlein2004cpf , and the “Bose Nova” collapse in Bose
gases donley2002nature . Recent experimental developments, olsen2004prl_rbk ;
zaccanti2006pra_rbk ; stan2004ofr ; ospelkaus:120402 ; jin2008 ; ketterle2008
have enabled the creation of an ultracold mixture of bosons and fermions,
where an interspecies Feshbach resonance may introduce a rich source of new
phenomena. From the theoretical point of view, studies of Bose-Fermi mixtures
to date have been mostly limited to non resonant physics, focusing mainly on
mean field effects in trapped systems roth2002mfi ; roth2002sas ;
modugno2002cdf ; hu2003ttb ; liu2003fte ; salasnich2007sbd ; pedri2005tdb ;
adhikari:043617 ; buchler2004psa ; kanamoto2006psf ; adhikari2005fbs , phases
in optical lattices albus2003mba ; lewenstein2004abf ; roth2004qpa ;
sanpera2004afb ; gunter2006bfm ; tit2008 ,or equilibrium studies of
homogeneous gases, focusing mainly on phonon induced superfluidity or beyond-
mean-field effects bijlsma2000ped ; heiselberg2000iii ; efremov2002pwc ;
viverit2002bis ; matera2003fpb ; albus2002qft ; wang2006sct . Pioneering
theoretical work on the resonant gas include Ref. powell2005prb , in which a
mean-field equilibrium study of the gas is supplemented with a beyond-mean-
field analysis of the bosonic depletion; and Ref. storozhenko2005pra , where
an equilibrium theory is developed using a separable-potentials model.
The aim of this article is to develop and solve a mean-field theory describing
an ultracold atomic Bose-Fermi mixture in the presence of an interspecies
Feshbach resonance. This goal appears innocuous enough at first sight, since
mean-field theories for resonant Bose-Bose Timmermans1999prl ;
holland2002prl_ramsey and Fermi-Fermi milstein2002pra ; griffin2002prl ;
stoof2003joptb gases exist, and have been studied extensively. In both of
these theories, the mean-field approximation consists of considering the
bosonic Feshbach molecules as being fully condensed, and this greatly
simplifies the treatment, since the Hamiltonian reduces to a standard
Bogoliubov-like integrable form landau9 .
The fundamental difference between these examples and the Bose-Fermi mixtures
is that in the latter the Feshbach molecules are fermions, and therefore their
center of mass-momentum must be included explicitly. The most obvious mean-
field approach consists in considering the atomic Bose gas to be fully
condensed. However, as we will show below, resonant molecules are really
composed of two bound atoms, which spend their time together vibrating around
their center of mass. It follows that outright omission of the bosonic
fluctuations of the atoms, disallows the bosonic constituents to oscillate
(i.e. fluctuate) at all, and therefore this leads to an improper description
of the physics of atom pairs.
This article is organized as follows. In section II we introduce the field
theory model used to study Feshbach resonances, describing briefly the
parametrization used, and outlining the exact solution of this model in the
two-body limit. The section ends with a test of this two-body theory, by
comparing the binding and resonance energies predicted by the model and the
virtually-exact analogues obtained from two-body close coupling calculations.
Section III introduces the simplest mean-field many-body theory of the gas,
obtained by disregarding all bosonic fluctuations. The solution of the theory
is outlined, and its limitations highlighted. In spite of these limitations,
mean-field theory provides a useful language for dealing with the problem, the
utility of which will persist even beyond the limits of applicability of the
theory itself.
Finally in section IV we introduce our generalized mean-field theory, which
is, in short, similar to the mean-field theory described in section III, but
with the notable improvement of using properly renormalized molecules as
building blocks, instead of their bare counterparts. This approach is not
trivially described in the Hamiltonian formalism, where substituting dressed
molecules for free ones would lead to double counting of diagrams. In this
section we therefore shift to the Green-function/path-integral language, where
this double-counting can be avoided quite easily. Finally we proceed to the
numerical solution of this theory, and note that for narrow resonances the
results are consistent with their mean-field equivalents. This encourages us
to develop a simple theory to study the molecular formation via magnetic field
ramps, and, using an approach based on the Landau-Zener formalism lz1 ; lz2 ,
we derive analytic expressions in section V.
Throughout this article, we work with zero temperature gases, in the free
space thermodynamic limit. These are limitations which render the results
obtained here hard to directly compare with experimental results. One of the
main possible future directions of this work should include solving the same
problem in a trap, and generalization to higher temperatures.
## II The Model
We are interested primarily in the effects of resonant behavior on the
otherwise reasonably understood properties of the system. To this end we use a
model which has become standard in the last few years. This model has been
useful in studying the effect of resonant scattering in Bose
holland2002prl_ramsey ; stoof2004pr ; burnett2003pra and Fermi holland2002pra
; stoof2004prl ; gurarie2004prl ; griffin2003pra ; sasha2004jpb ; sasha2005pra
gases. In the case of the bose-fermi mixture, this model has been used in
Ref.powell2005prb ,and bortolotti2006jpb ; bortolotti2006sfm . We refer to
these works for further details about the origin and justification of the
Hamiltonian we use here, and for details on the solution in the two body
regime.
The Hamiltonian for the system reads:
$H=H_{0}+H_{I},$ (1)
where
$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle\sum_{p}\epsilon_{p}^{F}\
\hat{a}_{p}^{\dagger}\hat{a}_{p}+\sum_{p}\epsilon_{p}^{B}\
\hat{b}_{p}^{\dagger}\hat{b}_{p}+\sum_{p}\left(\epsilon_{p}^{M}+\nu\right)\
\hat{c}_{p}^{\dagger}\hat{c}_{p}$ $\displaystyle+$ $\displaystyle{\gamma\over
2V}\sum_{p,p^{\prime},q}\hat{b}_{p-q}^{\dagger}\hat{b}_{p^{\prime}+q}^{\dagger}\hat{b}_{p}\hat{b}_{p^{\prime}}$
$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle{V_{bg}\over
V}\sum_{p,p^{\prime},q}\hat{a}_{p-q}^{\dagger}\hat{b}_{p^{\prime}+q}^{\dagger}\hat{a}_{p}\hat{b}_{p^{\prime}}$
$\displaystyle+$
$\displaystyle{g\over\sqrt{V}}\sum_{q,p}(\hat{c}_{q}^{\dagger}\hat{a}_{-p+q/2}\hat{b}_{p+q/2}+h.c.).$
Here $\hat{a}_{p},\hat{b}_{p}$, are the annihilator operators for,
respectively, fermions and bosons, $\hat{c}_{p}$ is the annihilator operator
for the molecular field holland2002pra ; sasha2004jpb ; sasha2005pra ;
$\gamma=4\pi a_{b}/m_{b}$ is the interaction term for bosons, where $a_{b}$ is
the boson-boson scattering length; $V_{bg},\nu$, and $g$ are parameters
related to the Bose Fermi interaction, yet to be determined; the single
particle energies are $\epsilon^{\alpha}=p^{2}/2m_{\alpha}$, where
$m_{\alpha}$ indicates the mass of bosons, fermions, or pairs; and $V$ is the
volume of a quantization box with periodic boundary conditions.
The first step is to find the values for $V_{bg},\nu,g$, in terms of
measurable parameters. We will, for this purpose, calculate the 2-body
T-matrix resulting from the Hamiltonian in Eq. LABEL:act-bfm. Integrating the
molecular field out of the real time path integral, leads to the following
Bose-Fermi interaction Hamiltonian:
$H_{I}^{2body}={1\over V}\left(V_{bg}+{g^{2}\over
E-\nu}\right)\sum_{p}\hat{a}_{p}^{\dagger}\hat{b}_{-p}^{\dagger}\hat{a}_{p}\hat{b}_{-p}.$
(3)
This expression is represented in center of mass coordinates, and $E$ is the
collision energy of the system. From the above equation we read trivially the
zero energy scattering amplitude in the saddle point approximation:
$T=(V_{bg}-{g^{2}\over\nu}),$ (4)
which corresponds to the Born approximation (This is akin to identifying the
scattering amplitude $f=a_{sc}$ in the Gross-Pitaevskii equation, where the
interaction term would be ${2\pi\over m_{bf}}a_{b}$ ). We emphasize that this
approximation is only valid in the zero energy limit, and it does not,
therefore, describe the correct binding energy as a function of detuning.
However, with this approach we obtain an adequate description of the behavior
of scattering length as a function of detuning, which allows us to relate the
parameters of our theory to experimental observables via the conventional
parametrization holland2002pra ; stoof2004pr
$T(B)={2\pi\over m_{bf}}a_{bg}\left(1-{\Delta_{B}\over(B-B_{0})}\right),$ (5)
where $a_{bg}$ is the value of the scattering length far from resonance,
$\Delta_{B}$ is the width, in magnetic field, of the resonance, $m_{bf}$ is
the reduced mass, and $B_{0}$ is the field at which the resonance is centered.
The identification of parameters between Eqns., (4) and (5) proceeds as
follows: far from resonance, $|B-B_{0}|>>\Delta_{B}$, the interaction is
defined by a background scattering length, via $V_{bg}={2\pi a_{bg}\over
m_{bf}}$. To relate magnetic field dependent quantity $B-B_{0}$ to its energy
dependent analog $\nu$, requires defining a parameter
$\delta_{B}=\partial\nu/\partial B$, which may be thought of as a kind of
magnetic moment for the molecules. It is worth noting that $\nu$ does not
represent the position of the resonance nor the binding energy of the
molecules, and that, in general $\delta_{B}$ is a field-dependent quantity,
since the thresholds move quadratically with field, because of nonlinear
corrections to the Zeeman effect. For current purposes we identify
$\delta_{B}$ by its behavior far from resonance, where it is approximately
constant. Careful calculations of scattering properties using the model in Eq.
(LABEL:act-bfm), however, leads to the correct Breit-Wigner behavior of the
2-body T-matrix newton .
Finally we get the following identifications:
$\displaystyle V_{bg}={2\pi a_{bg}\over m_{bf}}$ $\displaystyle
g=\sqrt{V_{bg}\delta_{B}\Delta_{B}}$ $\displaystyle\nu=\delta_{B}(B-B_{0}).$
Synthesizing the approach descibed in bortolotti2006jpb , and diagrammatically
represented in Fig. 1, we can obtain the exact two-body T-matrix of the system
by solving the Dyson equation
$\displaystyle T$ $\displaystyle=$ $\displaystyle gD^{0}g+gD^{0}g\ \Pi\
gD^{0}g+gD^{0}g\ \Pi\ gD^{0}g\ \Pi\ gD^{0}g+\cdots$ (7) $\displaystyle=$
$\displaystyle gDg$
where $T$ is the T-matrix for the collision, and which has formal solution
$T=gDg={g^{2}\over(D^{0})^{-1}-g^{2}\Pi}.$ (8)
These quantities take the explicit form
$\displaystyle D^{0}(E)$ $\displaystyle=\left({V_{bg}\over g^{2}}+{1\over
E-\nu}\right)$ (10) $\displaystyle\Pi(E)$ $\displaystyle=-i\int\ {d\omega\over
2\pi}{d{\bf p}\over(2\pi)^{3}}{1\over(\hbar\omega-{p^{2}\over
2m_{b}}+i0^{+})}\times$ $\displaystyle{1\over(E-\hbar\omega-{p^{2}\over
2m_{f}}+i0^{+})}$ $\displaystyle\approx
i{m_{bf}^{3/2}\over\sqrt{2}\pi}\sqrt{E}+{m_{bf}\Lambda\over\pi^{2}},$
where $m_{bf}$ is the boson-fermion reduced mass, and $\Lambda$ is an
ultraviolet momentum cutoff needed to hide the unphysical nature of the
contact interactions. Note that $D^{0}$ represents an effective molecular
field, accounting for the fermion-boson background interaction, and, as
described in detail in bortolotti2006jpb , it is obtained by integrating the
original molecular field, and performing a Hubbard-Stratonovich transformation
negele to eliminate the direct boson-fermion interaction in favour of the
effective molecular field $D^{0}$.
Figure 1: Feynman diagrams representing the resonant collision of a fermion
and a boson. Solid lines represent fermions, dashed lines bosons, and double
solid-dashed lines represent the effective composite fermions.
Regularization of the theory bortolotti2006jpb ; milstein2002pra is obtained
by the substitutions
$\displaystyle\bar{V_{bg}}=V_{bg}\left(1\over 1-{m_{bf}\Lambda
V_{bg}\over\pi^{2}}\right)$ $\displaystyle\bar{g}=g\left(1\over
1-{m_{bf}\Lambda V_{bg}\over\pi^{2}}\right)$
$\displaystyle\bar{\nu}=\nu+\bar{g}g{m_{bf}\Lambda V_{bg}\over\pi^{2}}.$ (11)
Finally the two-body T-matrix takes the form
$T(E)=\left[{1\over V_{bg}+{g^{2}\over
E-\nu}}+i{m_{bf}^{3/2}\over\sqrt{2}\pi}\sqrt{E}\right]^{-1}.$ (12)
### II.1 Poles of the T-Matrix: Testing the Model
Bound states and resonances of the two-body system are identified in the
structure of poles of the T-matrix (Eq.12). This is illustrated in Figure 2,
where real and imaginary parts of the poles’ energies are plotted as a
function of magnetic field. The resonance portrayed in the figure is the
$544.7G$ resonance present in the $|9/2,-9/2\rangle|1,1\rangle$ state of
40K-87Rb. For $B<544.7G$ (corresponding to detunings $\nu<0$) the two-body
system possesses a true bound state, whose binding energy is denoted by the
solid line. In this case, the pole occurs for real energies. This bound state
vanishes as the detuning goes to zero, where the resonance occurs.
For positive detunings, $\nu>0$, on the other hand, the poles are complex, and
the inverse of the imaginary part is proportional to the lifetime of the
metastable resonant state.
Figure 2: The top panel shows the scattering length versus magnetic field for
the 544.7 G resonance, present in the $|9/2,-9/2\rangle|1,1\rangle$ states of
the 40K-87Rb collision. The bottom panel shows the poles of the model two-body
T-matrix (Eq. 12) parameterized for the same resonance, as a function of
magnetic field. Thick solid and dashed lines denote the real parts of relevant
poles, representing bound and resonance states respectively. The thin dashed
lines are the real parts of unphysical poles. The empty circles and squares,
represent the position of the resonance and the bound state, obtained via a
virtually exact close coupling calculation, and are presented to show the
level of accuracy of the model.
In this regime, there is no longer a true bound state, but there may be a
scattering resonance, indicated in Fig. 2 by a thick dashed line. This
resonance appears for magnetic fields $B>544.7$ for this particular resonance,
well before the disappearance of the bound state. This value is highly
dependent upon the value of the background potential. We will see in section
IV that for $V_{bg}=0$, the resonance actually appears at positive detunings.
In the case of 40K-87Rb, $V_{bg}<0$, implying that there is a weak potential
resonance in the open channel which interferes with the closed-channel
resonance, and causes it to cross the axis at negative detunings. For
$V_{BG}>0$ (unpublished ) the positive background scattering length is set by
a bound state in the open channel, which does not affect the resonance states,
but which interferes with the bound state at negative detunings.
The thin dashed lines in Fig. 2 are physically meaningless solutions to the
Schrödinger equation, in which the amplitude in the resonant state would grow
exponentially in time, rather than decay. These poles do not therefore
identify any particular features in the energy-dependent cross section of the
atoms, and will not modify the physics of the system. Finally Fig. 2 contains
data obtained from virtually exact close-coupling calculations which show the
extent of validity of the model. For the purposes at hand this agreement is
sufficient.
It should be noted that the agreement is not as good for positive background
scattering length systems, since the open-channel bound state determining this
scattering length is not adequately described by the model, which treats the
background physics as an essentially zero range interaction. This implies that
the relation between the background scattering length and open channel bound-
state energy is exactly $E_{b}=1/2\mu a_{bg}^{2}$, while in the physical
system this relation depends on the details of the interaction potential. This
problem has been addressed in the literature kokkelmans2004pra , but no
treatable field theory has yet been proposed.
$\hskip 14.22636ptB_{0}(G)\hskip 14.22636pt$ | $\hskip 14.22636pt\Delta_{B}(G)$ | $\delta_{B}(K/G)$ | $a_{bg}(a.u.)$
---|---|---|---
$492.49$ | $0.134$ | $3.624\times 10^{-5}$ | $-176.5$
$544.7$ | $3.13$ | $1.576\times 10^{-4}$ | $-176.5$
$659.2$ | $1.0$ | $2.017\times 10^{-4}$ | $-176.5$
Table 1: Parametrization of the three main Feshbach resonances used in this
thesis. All Three resonances are in the $|9/2,-9/2\rangle|1,1\rangle$ states
of the ${}^{40}K-^{87}Rb$ collision
## III Mean-Field Theory
In this section we introduce the many-body physics of the system, by first
analyzing it in a mean-field approach. Because of the statistical properties
of the system, we will see right away that mean field theory does not recover
the correct two body physics in the low density limits. In spite of this
substantial weakness, however, the approach has several qualitative features
which persist even in the improved theory that we introduce below.
Furthermore, since the model is exactly solvable, it will allow us to develop
a language which will help us to understand the problem in simpler terms, and
to identify some small physical effects, which, when ignored, can greatly
simplify the beyond mean-field approach presented in the next section.
### III.1 The Formalism
Starting with the Hamiltonian described by equation (1), we obtain the mean-
field Hamiltonian by substituting the boson annihilator $\hat{b}$ by its
expectation value $\phi=\langle\hat{b}\rangle$, a complex number. The number
operator $\hat{b}_{p}^{\dagger}\hat{b}_{p}$ therefore becomes
$|\phi|^{2}=N_{b}$, where $N_{b}$ is the number of condensed bosons. The grand
canonical Hamiltonian therefore becomes
$\displaystyle H$ $\displaystyle=$ $\displaystyle
E_{b}+\sum_{p}\left(\epsilon_{p}^{F}-\mu_{f}+V_{bg}n_{b}\right)\hat{a}_{p}^{\dagger}\hat{a}_{p}$
$\displaystyle+$
$\displaystyle\sum_{p}\left(\epsilon_{p}^{M}+\nu-\mu_{m}\right)\
\hat{c}_{p}^{\dagger}\hat{c}_{p}$ $\displaystyle+$ $\displaystyle
g\sqrt{n_{b}}\sum_{p}\left(\hat{c}_{p}^{\dagger}\hat{a}_{p}+h.c.\right),$
where $n_{b}$ is the density of condensed bosons, $E_{b}/V=\gamma
n_{b}^{2}-\mu_{b}n_{b}$ is the energy per unit volume of the (free) condensed
bosons, a constant contribution to the total energy of the system, and
$\mu_{(b,f,m)}$ are the chemical potentials. These are Lagrange multipliers
that serve to keep the densities constant as we minimize the energy to find
the ground state. In the following we will drop the volume term, absorbing it
in the definition of the creator/annihilator operators, such that the expected
value of the number operator represents a density, instead of a number.
Before proceeding with the analysis of this Hamiltonian, we introduce the set
of self-consistent equations we wish to solve. To this end we define the
quantities $n_{b(f)}^{0}$, representing the total density of bosons (fermions)
in the system, at detuning $\nu\rightarrow\infty$. At finite detunings some of
these atoms will combine into molecules, and the densities will be denoted as
$n_{(b,f,m)}$ for bosons, fermions and molecules respectively.
The system, therefore, is described by six quantities, namely three densities
and three chemical potentials, which require six equations to determine. These
equations, which can be derived by number-conservation constraints and energy
minimization arguments, are:
$\displaystyle n_{f}+n_{m}-n_{f}^{0}=0$ (14.a) $\displaystyle
n_{b}+n_{m}-n_{b}^{0}=0$ (14.b) $\displaystyle n_{f}={d\ \Omega\over
d\mu_{f}}$ (14.c) $\displaystyle n_{m}={d\ \Omega\over d\mu_{m}}$ (14.d)
$\displaystyle{d\Omega\over d\phi}=0$ (14.e)
$\displaystyle\mu_{b}+\mu_{f}=\mu_{m},$ (14.f)
where $\Omega=\langle H\rangle/V$ is the Gibbs free energy. Equations (14.a)
and (14.b) follow from the simple counting argument that for every molecule
created, there is one less free boson and one less free fermion in the gas.
Equations 14.c, and 14.d are simply the Lagrange multiplier constraint
equations, equation (14.e) follows from the mean-field approximation, whereby
the bosonic field is simply a complex number, and minimization of the energy
can therefore be done directly. Finally equation (14.f) is the law of mass
action, which follows from the fact that to make a molecule it takes one free
atom of each kind.
The next step is to write down $\Omega$ for the system, by taking the
expectation value of the Hamiltonian in equation( III.1), obtaining
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
E_{b}/V+\sum_{p}\left(\epsilon_{p}^{F}-\mu_{f}+V_{bg}n_{b}\right)\eta_{f}(p)$
$\displaystyle+$
$\displaystyle\sum_{p}\left(\epsilon_{p}^{M}+\nu-\mu_{m}\right)\eta_{m}(p)$
$\displaystyle+$ $\displaystyle 2g\sqrt{n_{b}}\sum_{p}\eta_{mf}(p),$
where $\eta_{f}(p)=\langle\hat{a}_{p}^{\dagger}\hat{a}_{p}\rangle$ and
$\eta_{m}(p)=\langle\hat{c}_{p}^{\dagger}\hat{c}_{p}\rangle$ are the fermionic
and molecular momentum distributions,
$\eta_{mf}(p)=\langle\hat{c}_{p}^{\dagger}\hat{a}_{p}\rangle$ is an off-
diagonal correlation term arising from the interactions in the system, and the
densities are given by $n_{f,m,mf}=\int{dp\over 2\pi^{2}}\eta_{f,m,mf}(p)$.
Equations 14.a-14.f then read
$\displaystyle n_{f}+n_{m}-n_{f}^{0}=0$ (16.a) $\displaystyle
n_{b}+n_{m}-n_{b}^{0}=0$ (16.b) $\displaystyle n_{f,m,mf}=\int{dp\over
2\pi^{2}}\eta_{f,m,mf}(p)$ (16.c) $\displaystyle g\
n_{mf}-\mu_{b}\sqrt{n_{b}}+\gamma n_{b}^{3/2}=0$ (16.d)
$\displaystyle\mu_{b}+\mu_{f}=\mu_{m}$ (16.e)
The remaining task is now to find expressions to calculate the expected values
$\eta_{f,m,mf}(p)$. To this end we follow a Bogoliubov-like approach, similar
to that described in powell2005prb . The mean-field Hamiltonian is bilinear in
all creation/annihilation operators, which means that it can be diagonalized
via a change of basis, whereby introducing the operators
$\displaystyle\hat{\alpha}_{p}=A_{\alpha}\hat{a}_{p}+C_{\alpha}\hat{c}_{p}$
(17) $\displaystyle\hat{\beta}_{p}=A_{\beta}\hat{a}_{p}+C_{\beta}\hat{c}_{p},$
for some appropriately chosen coefficients $A_{\alpha,\beta}$ and
$B_{\alpha,\beta}$, the Hamiltonian will read
$H^{\prime}=E_{0}+\sum_{p}\lambda_{\alpha}(p)\hat{\alpha}_{p}^{\dagger}\hat{\alpha}_{p}+\sum_{p}\lambda_{\beta}(p)\hat{\beta}_{p}^{\dagger}\hat{\beta}_{p}.$
(18)
At this point we note that the Hamiltonian is just a separable sum of free-
particle Hamiltonians, where the free particles are fermions, with dispersion
relations $\lambda_{\alpha,\beta}(p)$. We can readily write down the
distribution
$\eta_{\alpha,\beta}(p)=\Theta(-\lambda_{\beta,\alpha}(p))$ (19)
where $\Theta$ is the step function, and calculate the densities
$n_{\alpha,\beta}$. The step function could be replaced by the free Fermi
distribution for non-zero temperatures, but the mean-field assumption that all
bosons are condensed would no longer hold. If these were ordinary free
fermions with dispersion $p^{2}/2m-\mu$, equation (19) would reduce to the
standard zero-temperature Fermi distribution. We will see below that
$\lambda_{\alpha,\beta}(p)$ are dispersion relations of quasi-particles that
are a mixture of atoms and molecules.
Below we show how these ideas, together with equation (14.a) - 14.f, give us
the tools we require to calculate the observable atomic and molecular
densities as a function of the chemical potentials.
To illustrate more explicitly the diagonalization procedure we define the
vectors
$A=\left(\begin{array}[]{r}\hat{a}_{p}\\\ \hat{c}_{p}\end{array}\right)\ \ \
A^{\dagger}=(\ \hat{a}_{p}^{\dagger}\ \ \hat{c}_{p}^{\dagger}\ ),$ (20)
and
$B=\left(\begin{array}[]{r}\hat{\alpha}_{p}\\\
\hat{\beta}_{p}\end{array}\right)\ \ \ B^{\dagger}=(\
\hat{\alpha}_{p}^{\dagger}\ \ \hat{\beta}_{p}^{\dagger}\ ),$ (21)
whereby the Hamiltonian can be written as $A^{\dagger}\hat{H}A$, and
$B^{\dagger}\hat{H^{\prime}}B$, where
$\hat{H}=\left(\begin{array}[]{cc}\left(\epsilon_{p}^{F}-\mu_{f}+V_{bg}n_{b}\right)&g\sqrt{n_{b}}\\\
g\sqrt{n_{b}}&\left(\epsilon_{p}^{M}+\nu-\mu_{m}\right)\end{array}\right),$
(22)
and
$\hat{H^{\prime}}=\left(\begin{array}[]{cc}\lambda_{\alpha}(p)&0\\\
0&\lambda_{\beta}(p)\end{array}\right).$ (23)
Diagonalizing $\hat{H}$, we get the two eigenvalues
$\displaystyle\lambda_{\alpha,\beta}(p)$ $\displaystyle=$
$\displaystyle{h_{f}(p)+h_{m}(p)\over 2}$ $\displaystyle\pm$
$\displaystyle{1\over 2}\sqrt{4g^{2}n_{b}+(h_{m}(p)-h_{f}(p))^{2}},$
where we have defined
$h_{f}(p)=\left(\epsilon_{p}^{F}-\mu_{f}+V_{bg}n_{b}\right)$, and
$h_{m}(p)=\left(\epsilon_{p}^{M}+\nu-\mu_{m}\right)$, and the unitary
eigenvector matrix
$U=\left(\begin{array}[]{cc}A_{\alpha}&B_{\alpha}\\\
A_{\beta}&B_{\beta}\end{array}\right).$ (25)
The transformation in eq. 17 can then be written as $A=U^{\dagger}B$, and its
inverse $B=UA$.
Our goal now is to write the densities $\eta_{m,f,mf}(p)$ in terms of the
known densities $\eta_{\alpha,\beta}(p)$. In component notation, (where
$A_{i}=\hat{a}_{p}$, etc.), we can write
$\langle A_{i}^{\dagger}A_{l}\rangle=\langle
B_{j}^{\dagger}U_{ji}(U^{\dagger})_{lj}B_{k}\rangle=U_{lj}U^{*}_{ij}\langle
B_{j}^{\dagger}B_{j}\rangle,$ (26)
where we have used the fact that since the Hamiltonian is diagonal in the $B$
basis, then $\langle B_{j}^{\dagger}B_{k}\rangle=\langle
B_{j}^{\dagger}B_{j}\rangle\delta_{jk}$.
Using this formalism we obtain the relations:
$\displaystyle\eta_{f}(p)$ $\displaystyle=$
$\displaystyle|A_{\alpha}|^{2}\eta_{\alpha}(p)+|B_{\alpha}|^{2}\eta_{\beta}(p)$
(27) $\displaystyle\eta_{f}(p)$ $\displaystyle=$
$\displaystyle|A_{\beta}|^{2}\eta_{\alpha}(p)+|B_{\beta}|^{2}\eta_{\beta}(p)$
$\displaystyle\eta_{fm}(p)$ $\displaystyle=$ $\displaystyle
A_{\alpha}^{*}A_{\beta}\eta_{\alpha}(p)+B_{\alpha}^{*}B_{\beta}\eta_{\beta}(p).$
Using these expressions in conjunction with eqs. 16.a-16.e will then allow us
to compute the equilibrium properties of the system.
This simplified, mean-field version of the solution can only approximately
reproduce the energies of atomic and molecular states, as is shown in figure
3. In this example, we have assumed a uniform mixture of 40K and 87Rb atoms
with densities $8.2\times 10^{14}$ cm-3 and $4.9\times 10^{15}$ cm-1,
respectively. These densities correspond to the central density of each
species assuming it is confined to a $100$ Hz spherical trap. Far from the
resonance, these energies asymptote to zero (representing the atomic state)
and to the detuning (representing the bare molecular state that went into the
theory). Near zero detuning, these levels cross owing to the coupling term
$g\sqrt{n_{b}}$ in Eqn. (22). The size of this crossing is therefore larger
the larger the bosonic density is. In the crossing region the eigenstates do
not clearly represent either atoms or molecules, but linear combinations of
the two.
Figure 3: The thick solid lines represent the “renormalized” mean-field
energy levels $\lambda_{\alpha,\beta}(k_{f})$ , while the thin dashed lines
represent their bare counterparts.
### III.2 Mean field: noninteracting case
To better understand the structure of the mean-field theory, in this section
we detail its results for a noninteracting gas, by setting
$g=V_{bg}=\gamma=0$. We contrast two different physical regimes, based on the
ratio of bosons to that of fermions, $r_{bf}\equiv n_{b}/n_{f}$.
Figure 4: Equilibrium chemical potentials (top panel) and populations (bottom
panel) as a function of detuning for a non interacting gas with $r_{bf}=.6$.
The solid lines represent fermions, dashed lines molecules, and dashed-dotted
lines bosons. The dotted lines in the top panel represent the bare molecular
and fermionic internal energies, respectively $\nu$ and $0$. The vertical
lines labeled a) and b) are discussed in the text, and represent the detuning
at which molecular formation begins and ends, respectively
Figure 5: Equilibrium chemical potentials (top panel) and populations (bottom
panel) as a function of detuning for a non interacting gas with $r_{bf}=6$.
The solid lines represent fermions, dashed lines molecules, and dashed-dotted
lines bosons. The dotted lines in the top panel represent the bare molecular
and fermionic internal energies, respectively $\nu$ and $0$.
In the case of high fermion density, we set $r_{bf}=0.6$, and plot chemical
potentials and populations of the various states in Figure 4. Consider what
happens in an infinitely slow ramp from positive detuning (no molecules) to
negative detuning (introducing bound molecular states). For large positive
detuing, the fermionic and molecular chemical potentials are the same, since
the chemical potential of the condensed bosons vanishes. For large enough
detuning, the molecular chemical potential remains below the detuning, so it
is energetically unfavorable to make molecules. When the detuning dips below
the chemical potential (detuning a in the figure), fermions begin to pair with
bosons to make molecules (see populations in lower panel). This process
continues until all the bosons are consumed (detuning b), at which point the
populations stabilize. For detunings less that this, there remain both
fermions and fermionic molecules in the gas, and there are two Fermi surfaces
present. Because the internal energy of the molecules continues to diminish at
lower detuning, so does the molecular chemical potential. The two Fermi
surfaces therefore split from one another, although the relative population of
the two fermions is fixed.
By contrast, the case where the bosons outnumber the fermions is shown in
Figure 5, where we have set the density ratio to $r_{bf}=6$. As in the
previous case, no molecules are generated until the detuning drops lower than
the chemical potential of the atomic Fermi gas. Since there are enough bosons
to turn all the fermions into molecules, there there are no fermionic atoms at
sufficiently negative detuning, and the gas possesses only a single Fermi
surface. The chemical potential of the remaining bosons is still zero, since
these bosons are condensed. Formally, then, the chemical potential for atomic
fermions is negative, meaning that their formation at negative detuning is
energetically forbidden.
### III.3 Mean field: interacting case
In most experimental circumstances, the density of bosons is larger than that
of fermions, since condensed bosons cluster to the center of the trap, whereas
fermions are kept away by Pauli blocking. We therefore focus on this case
hereafter, setting the bose and fermi densities to $n_{b}=4.9\times 10^{15}$
cm-3 and $n_{f}=8.2\times 10^{14}$ cm-3. The coupling term $g\sqrt{n_{b}}$ in
equation (22) is the perturbative expansion parameter for the problem, and
since it has units of energy, it must be compared with the characteristic non-
perturbed energy of the gas, which in this case is $E_{f}$. Also since in the
perturbative expansions it always appears squared (see eq. 7), we can define
the unitless small parameter for the system as
$\epsilon_{SM}=g^{2}n_{b}/E_{f}^{2}=g^{2}n_{f}/E_{f}^{2}r_{bf}$, For the
$492G$ resonance in table 1, we have $\epsilon_{SM}=6.35\times 10^{-2}r_{bf}$,
and since $r_{bf}=6$, the small parameter is of order $0.1$, appropriate for
perturbative treatment.
Figure 6 shows the equilibrium chemical potentials for the system obtained via
a self-consistent solution of equations (III.1) and 16.a-16.e. This figure is
qualitatively similar to the corresponding non-interacting result in Fig. 5,
but contains important differences. First, the nonzero boson-boson interaction
generates a nonzero bosonic chemical potential $\mu_{b}$ that breaks the
degeneracy between the molecular and fermionic chemical potentials. In
physical terms, this means that there is an energy cost in maintaining bosons
unpaired, and therefore we need to take this into account in the kinematic
analysis. Namely, to make molecules energetically favorable no longer requires
a detuning $\nu$ such that $\nu=\mu_{f}$, but now requires
$\nu=\mu_{f}+\mu_{b}$. The net result is to shift the chemical potential up
and to the left by an amount $\mu_{b}$, and to shift the molecular population
curve to the left by this amount.
Figure 6: Mean-field equilibrium chemical potentials (top panel) and
populations (bottom panel) as a function of detuning. The solid lines
represent fermions, dashed lines molecules, and dashed-dotted lines bosons.
A second difference is that molecule creation takes place more gradually as a
function of detuning in the interacting case. This is simply the result of the
avoided crossing smearing out the molecular energy.
Finally, we exploit the simplicity of the mean field approach to test some
approximations that will simplify the beyond-mean-field approach in the next
section. These approximations have been tested numerically, and they give
corrections of the order of $.1\%$ or less in calculated molecular populations
for all regimes of interest here. The approximations are: i) incorporate the
boson-boson interaction $\gamma N_{b}^{3}$ by shifting the detuning and
chemical potential as discussed above; ii) disregard the background scattering
between bosons and fermions, i.e., set $V_{bg}=0$, since this interaction is
dominated by its resonance part; and iii) disregard the correlation function
$\langle\eta_{b}f(p)\rangle$ ((analogous to the boson polarization operator in
the Green function formalism), since its contribution to $\mu_{b}$ is much
smaller than that of $\gamma n_{b}^{3}$. It is difficult to directly verify
the validity of these approximations in the beyond-mean-field approach.
Nevertheless, we expect that these approximations remain valid, since the
generalized mean field theory is, after all, a mean-field theory at heart.
## IV Generalized Mean-Field Theory
In section III we reached the conclusion that the mean-field approach to the
resonant Bose-Fermi system does not properly account for the correct two body
physics of the system. In this section we wish to improve on this, by
introducing a generalization to mean-field theory, via an appropriate
renormalization of the molecular propagator, which is able to reproduce the
correct two-body physics in the low-density limit. To accomplish this, we will
have to abandon the Hamiltonian treatment of the previous section, in favor of
a perturbative approach based on the Green’s function formalism, much as was
done for two bodies in II. Throughout this section we use the approximations
made above, namely, $\gamma n_{b}^{3}=V_{bg}=\langle\eta_{b}f(p)\rangle=0$.
We begin by recasting the rn field result from Sec. III in the language of
Green functions. The self-consistent Dyson equations that describe this system
are
$\displaystyle G^{MF}_{F}(E,P)$ $\displaystyle=$ $\displaystyle
G^{0}_{F}(E,P)+g^{2}n_{b}\ D^{0}(E,P)\ G^{MF}_{F}(E,P)$ $\displaystyle
D^{MF}(E,P)$ $\displaystyle=$ $\displaystyle D^{0}(E,P)+g^{2}n_{b}\
G^{0}_{F}(E,P)\ D^{MF}(E,P),$ (28)
where the free propagators are simply
$\displaystyle D^{0}={1\over\omega-\xi^{M}(p)+i\eta\ {\rm sign}(\xi(p))}$
$\displaystyle G^{0}_{F}={1\over\omega-\xi^{F}(p)+i\eta\ {\rm sign}(\xi(p))},$
(29)
and $\xi^{M,F}(p)=(\epsilon_{p}^{M,F}-\mu_{m,f})$ These propagators are
described diagrammatically in Fig. 7. They represent the fact that a free
fermion may encounter a condensed boson and associate with it, temporarily
creating a molecule; or that a free molecule may temporarily split into a
fermion and a condensed boson. Self-consistency ensures that these processes
may be repeated coherently an infinite number of times. We neglect the bosonic
renormalization equation $\phi^{MF}=n_{b}^{0}+g^{2}n_{b}\ G^{0}_{F}(E,P)\
D^{0}(E,P)$, whereby a condensed boson may pick-up a fermion to create a
molecule; this is equivalent to the condition $\langle\eta_{b}f(p)\rangle=0$.
Figure 7: Feynman diagrams included in the mean-field theory. Thin (thick)
solid lines represent free (renormalized) fermions, thin double dashed-solid
lines represent free molecules, and thick double dashed lines represent
renormalized molecules. The little lightning bolts represent condensed bosons,
whereby the arrow indicates whether they are taken from or released into the
condensate.
Solutions to these equations take the form
$\displaystyle G^{MF}_{F}(E,P)$ $\displaystyle=$ $\displaystyle{1\over
G^{0}_{F}(E,P)^{-1}-g^{2}n_{b}\ D^{0}(E,P)}$ $\displaystyle D^{MF}(E,P)$
$\displaystyle=$ $\displaystyle{1\over D^{0}(E,P)^{-1}-g^{2}n_{b}\
G^{0}(E,P)}.$ (30)
Using the definitions of $G^{0}_{F}(E,P)$ and $D^{0}(E,P)$ from Eq. 29, we can
find the poles corresponding to many-body bound states. In this case, it can
be shown that the poles are exactly the mean-field eigenvalues
$\lambda_{\alpha,\beta}(p)$ from section II. Moreover, the equations (IV) are
symmetric with respect to interchange of $G_{F}$ and $D$, which implies that
both renormalized green functions have the same poles, and the same residues.
We can therefore study the properties of the fermions by only looking at the
molecules. This is not completely surprising, since, given that the condensed
bosons are relatively inert, every molecule corresponds exactly to a missing
fermion, and vice-versa.
The most important deficiency of this mean-field approach is that it only
allows molecules to decay into a free fermion and a condensed boson,
disregarding the possibility that the bosonic byproduct may be noncondensed.
We must allow noncondensed bosons somehow, and yet these bosons make a
perturbation to the result, as seen by the following argument. The fundamental
mean field assumption is that the gas is at zero temperature, and therefore
the noncondensed population should be negligible at equilibrium. Furthermore,
if a molecule is composed of a zero-momentum boson and a fermion from the
Fermi sea, dissociating into a noncondensed boson implies that the outgoing
fermion would have momentum lower than the Fermi momentum, an event which
Pauli blocking makes quite unlikely. Therefore, if a molecule does indeed
decay yielding a non-condensed boson, it should immediately recapture the
boson in a virtual process such as that described in Fig. 1. It is only
convenient that these events are exactly the kind of events which will
correctly renormalize the binding energy of the molecules, leading to a theory
which will reproduce the exact two-body resonant physics.
The Dyson equation describing this generalized mean-field theory are:
$\displaystyle G^{GMF}_{F}(E,P)$ $\displaystyle=$ $\displaystyle
G^{0}_{F}(E,P)+g^{2}n_{b}\ D(E,P)\ G^{GMF}_{F}(E,P)$ $\displaystyle
D^{GMF}(E,P)$ $\displaystyle=$ $\displaystyle D(E,P)+g^{2}n_{b}\
G^{0}_{F}(E,P)\ D^{GMF}(E,P).$
Here we have replaced the free propagator $D^{0}$ by the renormalized
molecular propagator $D$ from equation (8). A diagrammatic representation of
this theory appears in Fig. 8. By analogy with the mean field version, the
solution to these equations are:
$\displaystyle G^{GMF}_{F}(E,P)$ $\displaystyle=$ $\displaystyle{1\over
G^{0}_{F}(E,P)^{-1}-g^{2}n_{b}\ D(E,P)}$ $\displaystyle D^{GMF}(E,P)$
$\displaystyle=$ $\displaystyle{1\over D(E,P)^{-1}-g^{2}n_{b}\ G^{0}(E,P)}.$
(32)
These equations preserve the symmetrical nature of the mean-field theory
described above, and also the avoided crossing of atomic and molecular levels.
This is demonstrated in Figure 9, where we reproduce the $P=k_{f}$ pole of $D$
from Fig. 2 as dashed lines. We also present in this figure the corresponding
poles for the generalized mean-field theory, as solid lines. As in figure 3 we
note the splitting in two energy levels, avoiding each-other around $\nu=0$.
The fundamental difference in this case is that the molecular curve does not
asymptote to the bare detuning, but rather to the correct molecular binding
and resonance energies.
Figure 8: Feynman diagrams included in the generalized mean-field theory.
Like in the mean field case (Fig. 7), thin (thick) solid lines represent free
(renormalized) fermions, thin double dashed-solid lines represent free
molecules, and thick double dashed lines represent renormalized molecules. The
little lightning bolts represent condensed bosons, whereby the arrow indicates
whether they are taken from or released into the condensate. The novelty here
in the inclusion of the 2-body dressed molecules from Fig. 1.
Figure 9: The thick lines represent the poles of $D^{GMF}(k_{f})$, and
correspond to molecular and fermionic energies. The dashed lines represent the
molecular two body poles, corresponding to the molecular binding and resonant
energies obtained disregarding the background interaction bortolotti2006sfm .
As in the mean-field theory case, the effect of the interaction with the
condensate is to create an avoided crossing between the atomic and molecular
states. The “bulge” in the upper solid curve is a consequence of Pauli
blocking which has the consequence of favouring molecular stability. See
bortolotti2006sfm for more details.
Studying the equilibrium properties of the system is now a matter solving the
self-consistent set of equations (16.a-16.e), while setting $\eta_{mf}$ and
$\lambda$ equal to zero. To do this we first need to extract the distributions
$\eta_{f,m}$ from the Green functions $D^{GMF}$ and $G^{GMF}_{F}$. To avoid
taking a distracting detour here, we refer the reader to appendix A for
details. As in the previous section, we will consider a mixture composed of a
free gas of fermionic ${}^{40}K$ atoms, with a density of $8.2\times 10^{14}$
cm-3, and a gas of condensed ${}^{87}Rb$ bosons with density $4.9\times
10^{15}$ cm-3 (corresponding to the respective Thomas-Fermi densities of
$10^{6}$ atoms of either species in the center of a $100Hz$ spherical trap).
Figure 10 shows the equilibrium molecular population as a function of
detuning, for the $492.5G$ resonance. For the densities assumed, the mean-
field parameter $\epsilon_{SM}=g^{2}n_{b}/E_{f}^{2}\approx 0.4$ is indeed
perturbative. For this narrow resonance, the agreement between mean-field and
generalized mean-field is quite good, and we could as easily have used the
bare molecular positions to calculate this quantity. However, the situation is
completely different for the wide resonance at 544.7G, for which the
equilibrium molecular populations are shown in Fig. 11. Here the
“perturbative” parameter has the value $\epsilon_{SM}=38.7$, and is not
perturbative at all. For a given small detuning, the simple mean-field
approximation would greatly overestimate the number of molecules in the gas at
equilibrium. The more realistic generalized mean field theory accounts for the
fact that the actual molecular bound-state energy is higher than the bare
detuning. This fact in turn hinders molecular formation, according to chemical
potential arguments analogous to those in Sec. III.2.
Figure 10: Equilibrium molecular population as a function of detuning for the
narrow $492.49G$ resonance. The solid line represents results obtained via the
generalized mean field theory presented in the text, while the dashed-dotted
line represents the mean field results.
Figure 11: Equilibrium molecular population as a function of detuning for the
wider $544.7G$ resonance. The solid line represents results obtained via the
generalized mean field theory presented in the text, while the dashed-dotted
line represents the mean field results.
## V Molecule Formation
Moving beyond equilibrium properties, we are also interested in the prospects
for molecule creation upon ramping a magnetic field across the resonance. In
reference bortolotti2006jpb , the mean-field equations of motion for the
system at hand were derived, as follows:
$\displaystyle i\hbar{\partial\over\partial t}\phi$ $\displaystyle=$
$\displaystyle(V_{bg}\rho_{F}+\gamma|\phi|^{2})\phi+g\rho_{MF}^{*}$ (33.a)
$\displaystyle\hbar{\partial\over\partial t}\eta_{F}(p)$ $\displaystyle=$
$\displaystyle-2g\ \Im m(\phi\eta_{MF}(p))$ (33.b)
$\displaystyle\hbar{\partial\over\partial t}\eta_{M}(p)$ $\displaystyle=$
$\displaystyle 2g\ \Im m(\phi\eta_{MF}(p))$ (33.c) $\displaystyle
i\hbar{\partial\over\partial t}\eta_{MF}(p)$ $\displaystyle=$
$\displaystyle\left[\epsilon_{p}^{F}-\epsilon_{p}^{M}-\nu+V_{bg}|\phi|^{2}\right]\eta_{MF}(p)-$
(33.d) $\displaystyle\ \ \ \ g\phi^{*}\left(\eta_{F}(p)-\eta_{M}(p)\right),$
where $\eta_{F}(p)=\langle\hat{a}_{p}^{\dagger}\hat{a}_{p}\rangle$ is the the
fermionic distribution, $\eta_{M}(p)$ its molecular counterpart, and
$\rho_{M,F}=\int{dp\over 2\pi^{2}}p^{2}\eta_{M,F}(p)$ the fermionic and
molecular densities. Similarly
$\eta_{MF}(p)=\langle\hat{c}_{p}^{\dagger}\hat{a}_{p}\rangle$ and is the
distribution for molecule-fermion correlation, with the associated density
$\rho_{MF}$.
Reference bortolotti2006jpb outlined the limitations of the non-equilibrium
theory, by claiming that to obtain the correct two-body physics in the low-
density limit it would be necessary to include three point and possibly higher
correlations. While this fact is indeed true, we have amended it in the
previous section. For experimentally reasonable parameters, the mean-field
theory can be complemented by the correct renormalized propagator to
accurately describe the equilibrium properties of the gas. Encouraged by this
argument, we now apply it to the problem of a field ramp as well.
In the following we wish to study molecular formation via a time dependent
ramp of the magnetic field across the resonance. To this end we use two
approaches: the first consists of propagating equations (33.a-33.d), ramping
the detuning linearly in time from a large positive value to a large negative
one, and plotting the final molecular population as a function of detuning
ramping rate $R$. The second approach consists in noticing that if $\nu(t)$ is
a linear function of time, then the mean-field Hamiltonian (eq. 22) is ideally
suited to a Landau-Zener treatment, whereby the final molecular population as
a function of detuning can be readily written as
$n_{m}/min(n_{b},n_{f})=1-e^{-R\over\tau}$ (34)
Here $n_{m}/max(n_{b},n_{f})$ is the fraction of possible molecules formed,
and $R=1/{\partial B\over\partial t}$ is the inverse ramp rate, and the
exponential time constant is given by $\tau={\hbar\delta_{B}\over
g^{2}n_{b}}={m_{bf}\over ha_{bg}n_{b}\Delta_{B}}$, where $a_{bg}$ is the
background scattering length, $h$ is Plank’s constant, and $\Delta_{B}$ is the
magnetic field width of the resonance.
Remarkably, the characteristic sweep rate $\tau$ does not depend on the
fermionic density. This arises from the fact that in mean-field theory the
momentum-states of the fermionic gas are uncoupled, except via the depletion
of the condensate. Since in the Landau-Zener approach the depletion is assumed
small, it follows that the various fermionic momentum states are considered
independently, and thus the probability of transition of the gas is equal to
the probability of transition of each individual momentum state. This
approximation is only valid for narrow resonances, such as the $492.5G$
resonance in table 1.
Molecular formation rate versus ramp rate is shown in Figure 12, for the cases
$r_{bf}=6$ (more fermions than bosons) and $r_{bf}=0.6$ (more fermions than
bosons. In both cases the Landau-Zener result agrees nearly perfectly with
direct numerical integration. This agreement is surprising in the case of more
fermions, since the width of the crossing is proportional to density of
leftover bosons, and we expect that this number will change substantially as
the bosonic population is depleted via the formation of molecules. This type
of time dependent crossing should not be properly described by the Landau-
Zener formula. However, monitoring the time evolution of the molecular
population as a function of time shows that the majority of the transfer takes
place quite abruptly somewhat after crossing the zero-detuning region, whereby
the change in bosonic density does not modify the energy levels substantially.
Figure 12: Transition probability into molecular state via a magnetic field
ramp across the $492.5G$ Feshbach resonance. The solid lines are obtained via
numerical solutions of equations (33.a-33.d), while the dashed lines represent
the Landau-Zener equivalent. In the top panel the gas is composed of more
bosons than fermions $r_{bf}=6$, while in the bottom panel the opposite is
true $r_{bf}=.6$
## VI Conclusion
In this article, we developed and solved a generalized mean-field theory
describing an ultracold atomic Bose-Fermi mixture in the presence of an
interspecies Feshbach resonance. The theory is “generalized” in the sense that
it correctly incorporates bosonic fluctuations, at least to the level that it
reproduces the correct two-body physics in the extreme dilute limit. This
theory, like any mean-field theory, presents undeniable limitations.
Nevertheless, any useful many-body treatment must start from a well conceived
mean-field theory. Future directions of this work should include the
generalization to finite temperature, and the inclusion of a trap, initially
in a local-density approximation. These advances would be essential to check
for empirical confirmation of the theory.
###### Acknowledgements.
DCEB and JLB acknowledge support from the DOE and the Keck Foundation.
## Appendix A Green Function Methods for Fermions
In this Appendix we briefly introduce some of the Green function techniques
that we found useful in our calculations.
### A.1 Free Green functions
We start from the Green function for a gas of free fermions, which is given,
in the frequency-momentum representation by
$G^{0}(w,{\bf q})={1\over\omega-\xi({\bf p})+i\eta\ {\rm sign}(\xi({\bf
p}))},$ (A-35)
where $\xi({\bf p})={\bf p}^{2}/2m-\mu$. The momentum distribution, at
equilibrium, is given by
$n({\bf p})=-i\lim_{\eta\to 0^{+}}\int{{d\omega\over
2\pi}e^{i\omega\eta}G^{0}(w,{\bf q})},$ (A-36)
Here the limit comes from the equilibrium condition; the frequency, in the
Green function definition, is the fourier space equivalent of time, whereby
the real time green function represents the evolution of the system from from
time $t$ to $t^{\prime}$, and the observables obtained this way, represent
expected values of the kind
$\langle\psi(t)|\mathcal{O}|\psi(t^{\prime})\rangle$. However, since we want
equilibrium conditions, we need to take the limit $t\to t^{\prime}$, which is
non trivial, since $G^{0}$ is defined by a green function equation of the form
$\mathcal{L}\ G^{0}(t-t^{\prime})\propto\delta(t-t^{\prime})$, where
$\mathcal{L}$ is some linear operator, and which highlights a peculiar
behavior in the limit we desire. However, since we know on physical grounds
that observables, such as the momentum distribution, must be defined and well
behaved at equilibrium, then by first taking the expectation value integral,
and then the limit, we can circumvent the problem. In eq. A-36 this implies we
cannot quite get rid of the fourier transform exponent
$e^{i\omega(t-t^{\prime})}$ until after the $\omega$ integral.
To perform the integral in A-36, we exploit the fourier exponent, by noting
that since $\eta$ is positive, $e^{i\omega\eta}\to 0$ as $\omega\to+i\infty$,
so that the integral is identical to a contour integral over the path defined
by the real $\omega$ axis, closed in the upper complex $\omega$ plain by an
infinite radius semicircle, which, as we have just seen, gives no contribution
to the integral. We can now integrate using the residue theorem.
We note that the integrand in A-36 has a simple pole at $\omega=\xi({\bf
p})-i\eta\ sign(\xi({\bf p}))$. Thus, if $\xi({\bf p})>0$, then the pole is in
the lower complex plane, and the integral vanishes, and if $\xi({\bf p})>0$,
then the pole is in the upper complex plane, with residue 1. Using the residue
theorem, and summarizing these results we finally get
$n({\bf p})=\Theta(-\xi({\bf p})),$ (A-37)
which we recognize as the zero temperature fermi distribution.
### A.2 Interacting Green functions
According to Dyson’s equation, the green function for an interacting system
has the form
$G(w,{\bf q})={1\over\omega-\xi({\bf p})-\Sigma(\omega,{\bf p})},$ (A-38)
where $\Sigma(\omega,{\bf p})$ is an arbitrarily complicated function
summarizing all the interactions in the system, which is known as self energy.
The prescription to find $\Sigma$ is quite straightforward, and it consists of
adding all amputated connected feynman diagrams for the system. The fact that,
in general, the number of such diagrams is infinite, makes this task virtually
impossible. Nonetheless, eq. A-38 is very powerful, since it allows one to
include the effect of infinite subsets of the total number of diagrams in the
system, by only having to explicitly calculate a few representative ones.
An alternative standard approach, leads to the exact result (NOTE: Abrikosov
measures energy from $\mu$, here we measure from 0, which is more standard.)
$G(\omega,{\bf p})=\int_{0}^{\infty}{d\omega^{\prime}\left[{A(\omega,{\bf
p})\over\omega-\omega^{\prime}+i\eta}+{B(\omega,{\bf
p})\over\omega+\omega^{\prime}-i\eta}\right]},$ (A-39)
Where $A$ and $B$ are, again, arbitrary complicated functions, though they are
known to be finite.
To understand $A$ and $B$ more closely, we need to introduce the following
well known identity:
$\lim_{\nu\to 0}{1\over x\pm i\nu}=\mathcal{P}{1\over x}\mp i\pi\delta(x),$
(A-40)
where $\mathcal{P}$ is a Cauchy principal value, which represents the
contribution due to a discontinuity in a Riemann sheet (branch cut), and the
delta function represents the contribution due to the pole.
Applying A-40 to A-39 we get
$\displaystyle Re\ G(\omega,{\bf
p})=\mathcal{P}\int_{0}^{\infty}{d\omega^{\prime}\left[{A(\omega,{\bf
p})\over\omega-\omega^{\prime}+i\eta}+{B(\omega,{\bf
p})\over\omega+\omega^{\prime}-i\eta}\right]}$ (A-41) $\displaystyle Im\
G(\omega,{\bf p})=\left\\{\begin{array}[]{r c c}-\pi A(\omega,{\bf
p})&if&\omega>0\\\ \pi B(-\omega,{\bf p})&if&\omega<0\end{array}\right.$
(A-44)
Finally, eq. A-36 represents a fundamental property of green functions, and it
can be generalized to interacting systems simply substituting $G^{0}$ with
$G$. Applying it to eq. A-39, and performing the $\omega$ integral first, we
get
$n({\bf p})=\int_{0}^{\infty}{d\omega^{\prime}B(\omega,{\bf
p})}=\int_{-\infty}^{0}{d\omega{-1\over\pi}Im\ G(\omega,{\bf p}).}$ (A-45)
Introducing the function $\rho(\omega,{\bf p})=-2ImG(\omega,{\bf p})$,
generally called spectral function, the above equation can be written as
$n({\bf p})=\int{{d\omega\over 2\pi}\rho(\omega,{\bf p})\Theta(-\omega)}.$
(A-46)
An important property of the spectral function is that for all ${\bf p}$,
$\int{{d\omega\over 2\pi}\rho(\omega,{\bf p})}=1.$ (A-47)
This can be understood as a sum rule in the following sense: if we wish to
calculate the number of holes in the system, we would take the $\eta\to 0^{-}$
limit in equation A-36. The distribution would then have been $n_{holes}({\bf
p})=1-n({\bf p})=\int_{0}^{\infty}{d\omega^{\prime}A(\omega^{\prime},{\bf
p})}$, so that
$1=\int_{0}^{\infty}{d\omega^{\prime}\left[A(\omega^{\prime},{\bf
p})+B(\omega^{\prime},{\bf p})\right]}=\int{{d\omega\over
2\pi}\rho(\omega,{\bf p})}$.
Using eq.A-38, together with the definition of $\rho$, we can write
$\rho(\omega,{\bf p})={-2Im\Sigma(\omega,{\bf p})\over\left[\omega-\xi({\bf
p})-Re\Sigma(\omega,{\bf p})\right]^{2}+\left[Im\Sigma(\omega,{\bf
p})\right]^{2}}.$ (A-48)
Furthermore, if $\Sigma$ were to be real, or if, equivalently, the pole of the
green function were to be real, for some momentum ${\bf p}$, then taking the
limit $Im\Sigma\to 0$ of A-38, and using eq,A-40, we get
$\rho(\omega,{\bf p})=2\pi\delta(\omega-\xi({\bf p})-Re\Sigma(\omega,{\bf
p})),$ (A-49)
which can be simplified, using the properties of the delta function, to
$\rho(\omega,{\bf p})=2\pi Z({\bf p})\delta(\omega-\omega_{0}({\bf p})),$
(A-50)
where Z, known as spectral weight is given by
$Z({\bf p})={1\over\left|1-{\partial\over\partial\omega}Re\Sigma(\omega,{\bf
p})\right|_{w=\omega_{0}({\bf p})}},$ (A-51)
and $\omega_{0}({\bf p})$ is the pole of the green function, defined by
$\omega_{0}({\bf p})-\xi({\bf p})-\Sigma(\omega_{0}({\bf p}),{\bf p})=0.$
(A-52)
The momentum distribution in this case is thus given by
$n({\bf p})=Z({\bf p})\int{\delta(\omega-\omega_{0}({\bf
p}))\Theta(-\omega)}=Z({\bf p})\Theta(-\omega_{0}({\bf p})).$ (A-53)
## References
* [1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Phys. Rev., 108(5):1175–1204, Dec 1957.
* [2] S.N. Bose. Z. Phys, 26:178–181, 1924.
* [3] A. Einstein. Physik- Mathematik, pages 261–267, 1924.
* [4] M. Greiner, C.A. Regal, and D.S. Jin. Nature, 426(6966):537–540, 2003.
* [5] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.H. Denschlag, and R. Grimm. Physical Review Letters, 92(12):120401, 2004.
* [6] MW Zwierlein, CA Stan, CH Schunck, SMF Raupach, AJ Kerman, and W. Ketterle. Physical Review Letters, 92(12):120403, 2004.
* [7] E.A. Donley, N.R. Claussen, S.T. Thompson, and C.E. Wieman. Arxiv preprint cond-mat/0204436, 2002.
* [8] S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, and D. S. Jin. Phys. Rev. Lett., 93(18):183201, Oct 2004.
* [9] Francesca Ferlaino, Chiara D’Errico, Giacomo Roati, Matteo Zaccanti, Massimo Inguscio, Giovanni Modugno, and Andrea Simoni. Phys. Rev. A, 73(4):040702, Apr 2006.
* [10] CA Stan, MW Zwierlein, CH Schunck, SMF Raupach, and W. Ketterle. Physical Review Letters, 93(14):143001, 2004.
* [11] J.J.Zirbel, K.-K.Ni, S.Ospelkaus, J.P D Incao, C.E.Wieman, J.Ye, and D.S.Jin. Physical Review Letters, 100(11):143201, 2008.
* [12] Yong-il Shin, A.Schirotzek, C.H.Schunck, and W.Ketterle Physical Review Letters, 101(15):070404, 2008.
* [13] C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock, and K. Bongs. Physical Review Letters, 97(12):120402, 2006.
* [14] R. Roth and H. Feldmeier. Physical Review A, 65(2):21603, 2002.
* [15] R. Roth. Physical Review A, 66(1):13614, 2002.
* [16] G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R.J. Brecha, and M. Inguscio. 2002\.
* [17] H. Hu and X.J. Liu. Physical Review A, 68(2):23608, 2003.
* [18] X.J. Liu, M. Modugno, and H. Hu. Physical Review A, 68(5):53605, 2003.
* [19] L. Salasnich, S.K. Adhikari, and F. Toigo. Physical Review A, 75(2):23616, 2007.
* [20] P. Pedri and L. Santos. Two-Dimensional Bright Solitons in Dipolar Bose-Einstein Condensates. Physical Review Letters, 95(20):200404, 2005.
* [21] Sadhan K. Adhikari. Physical Review A (Atomic, Molecular, and Optical Physics), 70(4):043617, 2004.
* [22] HP Büchler and G. Blatter. Physical Review A, 69(6):63603, 2004.
* [23] R. Kanamoto and M. Tsubota. Physical Review Letters, 96(20):200405, 2006.
* [24] S.K. Adhikari. Physical Review A, 72(5):53608, 2005.
* [25] A. Albus, F. Illuminati, and J. Eisert. Physical Review A, 68(2):23606, 2003.
* [26] M. Lewenstein, L. Santos, MA Baranov, and H. Fehrmann. Physical Review Letters, 92(5):50401, 2004.
* [27] R. Roth and K. Burnett. Physical Review A, 69(2):21601, 2004.
* [28] A. Sanpera, A. Kantian, L. Sanchez-Palencia, J. Zakrzewski, and M. Lewenstein. Physical Review Letters, 93(4):40401, 2004.
* [29] K. Günter, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger. Physical Review Letters, 96(18):180402, 2006.
* [30] I.Titvinidze, M.Snoek, and W.Hofstetter. Physical Review Letters, 100(14):100401, 2008.
* [31] MJ Bijlsma, BA Heringa, and HTC Stoof. Physical Review A, 61(5):53601, 2000.
* [32] H. Heiselberg, CJ Pethick, H. Smith, and L. Viverit. Physical Review Letters, 85(12):2418–2421, 2000.
* [33] DV Efremov and L. Viverit. Physical Review B, 65(13):134519, 2002.
* [34] L. Viverit. Physical Review A, 66(2):23605, 2002.
* [35] F. Matera. Physical Review A, 68(4):43624, 2003.
* [36] AP Albus, SA Gardiner, F. Illuminati, and M. Wilkens. Physical Review A, 65(5):53607, 2002.
* [37] D.W. Wang. Physical Review Letters, 96(14):140404, 2006.
* [38] S. Powell, S. Sachdev, and H.P. Büchler. Physical Review B, 72(2), 2005.
* [39] A. Storozhenko, P. Schuck, T. Suzuki, H. Yabu, and J. Dukelsky. Physical Review A (Atomic, Molecular, and Optical Physics), 71(6):063617, 2005.
* [40] Eddy Timmermans, Paolo Tommasini, Mahir Hussein, and Arthur Kerman. Phys. Rep., 315:199, 1999.
* [41] S. J. J. M. F. Kokkelmans and M. J. Holland. Phys. Rev. Lett., 89(18):180401, Oct 2002.
* [42] S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo, R. Walser, and M. J. Holland. Phys. Rev. A, 65(5):053617, May 2002.
* [43] Y. Ohashi and A. Griffin. Phys. Rev. Lett., 89(13):130402, Sep 2002.
* [44] RA Duine and HTC Stoof. Journal of Optics B: Quantum and Semiclassical Optics, 5(2):S212–S218, 2003.
* [45] E.M. Lifshitz and L.P. Pitaevskii. Statistical Physics part 2. Butterworth-Heinemann, 1996.
* [46] C Zener. C. Proc. R. Soc. London A, 137(696), 1932.
* [47] L.D. Landau. Phys. Z. Sowjetunion, 1(89), 1932.
* [48] RA Duine and HTC Stoof. Physics Reports, 396(3):115–195, 2004.
* [49] Thorsten Köhler, Thomas Gasenzer, and Keith Burnett. Phys. Rev. A, 67(1):013601, Jan 2003.
* [50] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser. Phys. Rev. Lett., 87(12):120406, Aug 2001.
* [51] GM Falco and HTC Stoof. Physical Review Letters, 92(13):130401, 2004.
* [52] AV Andreev, V. Gurarie, and L. Radzihovsky. Physical Review Letters, 93(13):130402, 2004.
* [53] Y. Ohashi and A. Griffin. Physical Review A, 67(3):33603, 2003.
* [54] A.V. Avdeenkov. Journal of Physics B Atomic Molecular and Optical Physics, 37(1):237–246, 2004.
* [55] AV Avdeenkov and JL Bohn. Physical Review A, 71(2):23609, 2005.
* [56] D.C.E. Bortolotti, A.V. Avdeenkov, C. Ticknor, and J.L. Bohn. Journal of Physics B: Atomic, Molecular, and Optical Physics, 39(1):189–203, 2006.
* [57] A.V. Avdeenkov, D.C.E. Bortolotti, and J.L. Bohn. Physical Review A, 74(1):12709, 2006.
* [58] R.G. Newton. Scattering Theory of Waves and Particles. Courier Dover Publications, 2002.
* [59] J.W. Negele and H. Orland. Quantum Many-Particle Systems. Addison-Wesley, 1988.
* [60] D.C.E. Bortolotti. unpublished, 2006.
* [61] B. Marcelis, E. G. M. van Kempen, B. J. Verhaar, and S. J. J. M. F. Kokkelmans. Phys. Rev. A, 70(1):012701, Jul 2004.
|
arxiv-papers
| 2008-09-13T17:06:39
|
2024-09-04T02:48:57.777273
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D.C.E. Bortolotti, A.V.Avdeenkov, J.L.Bohn",
"submitter": "Alexander Avdeenkov",
"url": "https://arxiv.org/abs/0809.2354"
}
|
0809.2616
|
††thanks: This work was supported by the U.S. Department of Energy, Office of
Basic Energy Sciences under contract DE-AC02-06CH11357.
# Precise Orbital Tracking of an Asteroid with a Phased Array of Radio
Transponders
Bernhard W. Adams Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL
60439, USA adams@aps.anl.gov
###### Abstract
Deflecting an asteroid from an Earth impact trajectory requires only small
velocity changes, typically of the order of microns per second, if done many
years ahead of time. For this, a highly precise method of determining the
need, magnitude, and direction of a deflection is required. Although the
required precision can be achieved by much less accurate extended
observations, an intrinsic resolution of $\mu$m/s permits the live monitoring
of nongravitational orbital perturbations (Yarkovsky effect), and of a
deflection effort itself. Here, it is proposed to deploy on the asteroid’s
surface multiple radio units to form a phased array capable of measuring
radial velocities relative to Earth to about 1 $\mu$m/s and ranges to 5 m. The
same technology can also be used for scientific applications such as very-long
baseline radio astronomy, milli-Hertz gravitational wave detection, or mapping
of the solar wind.
## I Introduction
Asteroid impacts on Earth are infrequent, but potentially devastating events.
One example is (99942) Apophis, a 300-meter rock, which will miss Earth in
2029 by less than the orbital height of geostationary satellites. If, in this
encounter, its center of gravity passes through the so-called keyhole – an
imaginary, 600-meter wide region in space near Earth – then gravitational
deflection will set it up for an impact in 2036 with an energy release similar
to the explosion of Krakatoa in 1883. Making the asteroid miss the keyhole
requires much less effort than avoiding the entire planet once the keyhole has
been traversed: Up to three years before 2029, a velocity change $\Delta
v<1\mu$m/s tangential to the orbit is sufficient to avoid the keyhole san ;
Schweickart et al. (2006), but a $\Delta v$ of the order of cm/s is necessary
for steering away from the entire planet after a keyhole passage. Therefore, a
determination whether a deflection will be necessary should be done well
before 2029. To obviate the need for an unnecessarily large deflection, the
orbit should be measured to a precision commensurate with the required
velocity change, i.e., 1 $\mu$m/s. Passive radar can resolve velocities to a
few cm/s and ranges to tens of meters, and only at distances much less than
the orbital diameter of Earth and Apophis. A much higher accuracy can be
achieved by use of a radio transponder traveling with or on the asteroid.
Placement and operation of a radio transponder on an asteroid is challenging
in several ways, including (i) deliverable payload, (ii) power supply, (iii)
unknown surface topography and composition, (iv) landing without bouncing off
faster than the escape velocity of a few cm/s, (v) maintaining a directional
link to Earth while the asteroid is rotating, (vi) degradation of radio
signals due to interplanetary-plasma scintillations, and (vii) the space
environment. The problem of landing would be irrelevant, and the power supply
would be simpler with a transponder aboard a spacecraft orbiting/accompanying
the asteroid, but transponders fixed to the asteroid surface can reach a much
higher accuracy (see below). Therefore, the present proposal will concentrate
on the latter approach.
## II Design Goals
First, to the accuracy: Doppler tracking of planetary probes has shown
performance at the stipulated level, most recently 0.5 $\mu$m/s for the
gravitational-wave experiment Asmar et al. (2005) on Cassini currently
orbiting Saturn. Orbital refinement techniques achieve this with much less
precise individual measurements over an orbital period, but a high-precision
technology permits live monitoring of a deflection maneuver, as well as of the
Yarkovsky Chesley et al. (2003) effect (orbital perturbations due to radiation
pressure). This would also be useful for using the Yarkovsky itself for
deflection Spitale (2002). In addition to Doppler tracking, this proposal also
aims for ranging at 5 m resolution, corresponding to 1 $\mu$m/s over two
months, about 20% of Apophis’ orbital period. These Doppler and ranging
accuracies are stipulated for most of the asteroid’s orbit, except where the
line of sight passes within 100 solar radii from the sun, and radio-signal
degradation due to scintillations (see below) becomes too severe. This makes
the maximum tracking distance 1.82 astronomical units (A.U.), i.e., 2.72
$\cdot 10^{8}$ km.
A radio transponder could be placed on the surface or aboard a companion
spacecraft orbiting or tracking the asteroid. The latter option can be
realized with proven spacecraft technology, but it cannot reach $\mu$m/s
accuracy for the asteroid’s center of mass in a single measurement: due to the
irregular shape and rotation of the asteroid, orbits are generally nonperiodic
and unstable. Therefore, long averaging is necessary without orbit correction
manouvers, but these would be required to prevent crashing or ejection. Active
locking of a companion spacecraft to surface features of the asteroid using
laser or radar ranging is also unlikely to be accurate to the $\mu$m/s level.
Therefore, instantaneous high-precision measurements for live monitoring of
deflections, or of the Yarkovsky effect, require a transponder on the
asteroid’s surface.
## III Details of the Design
For reasonable power consumption, a directional antenna is required that
compensates for the asteroid’s rotation. Using a tracking dish antenna on the
low-gravity surface seems challenging. A much better approach is found in
phased-array technology, where the phases of received and transmitted signals
in a multitude of dipole antennae are controlled electronically without any
mechanical motion for directionality through constructive interference. The
phased array discussed here Adams (2007) consists of 475 radio units grouped
in 19 nodes of 25 each (Fig. 1). These numbers are somewhat arbitrary, and are
meant only as an example.
Figure 1: Left: The large web connecting 19 nodes by optical fibers and thin
electrical wires. Right: One node containing 25 transponders, connected to
each other by optical fibers and thin metal wires attached to a thin plastic
foil. As shown, the total length of wire connecting the nodes is 11.4 km.
Lower right: cross section of a wire with carbon fibers, optical fibers, and
metal filaments.
The nodes and the radio units within each node are linked together by a net of
thin threads containing carbon fibers for mechanical strength, glass fibers
for the distribution of a precise timing signal (for phasing) and digital
communication, and metal wires for electric-power sharing. Each radio unit has
24 antennae for a total of 11400 for each of the frequencies used (see below).
Each node contains a highly stable oscillator, one of which generates a
reference frequency that is distributed over the glass-fiber network. Shortly
before arrival, the transfer spacecraft aims for a collision with the asteroid
at a velocity of about 0.1 m/s (roughly the escape velocity from the
asteroid’s surface). As shown in Fig. 2, it then sequentially ejects 18
containers
Figure 2: Deployment of the coarse web. A space vehicle (1) carries the array
to the asteroid. It is comprised of three sections, each containing six
slices. These are ejected in sequence (2-3-4) to deploy the web. After
ejection of all slices, the entire web is made to rotate slowly to keep it
stretched out (see text). Finally, each of the slices deploys a sub-web, as
shown in fig. 3.
for the nodes, which are connected by carbon/metal/glass-fiber threads. To
keep the fully deployed net spread out over about 800 meters, it is then spun
up by gas jets to about one revolution per hour. Next, each container
(including one in the remainder of the spacecraft) releases a thin plastic
foil with attached radio units and inflates it to 50 m diameter by injecting
gas into thin tubes (Fig. 3).
Figure 3: Ejection and unfolding of the sub-webs: 1: container opens, 2:
folded plastic foil is ejected, 3: foil unfolds as gas is blown into tubes
formed between two layers of the foil.
Continuing on its collision course, the assembly wraps itself around the
asteroid. After several slow-motion bounces that are restrained by the net,
everything settles down in fixed random locations. Some damage is tolerable as
it only leads to a slightly degraded performance (see below).
Next, Earth sends a radio signal sufficiently strong for reception without
phasing (see below). As the asteroid rotates, the array compares the phases of
this signal in each antenna to the local reference, and thus acquires the
information necessary for phased-array operation at each of the frequencies
used. This information is continually refined as the asteroid and Earth
progress along their orbits.
The Doppler-tracking resolution is degraded by scintillations due to
fluctuations of the radio refractive index n of the turbulent interplanetary
plasma (solar wind). Extrapolating spacecraft-communication data Woo (1978)
for a frequency of $\nu=2.3$ GHz from 20 to 100 solar radii, and applying
scaling Ho et al. (2002) with $\nu^{-1.2}$, the spectral broadening due to
scintillations of an 8-GHz signal at 100 solar radii is about 0.01 Hz. Ways to
mitigate them are discussed in the literature on spacecraft-based
gravitational-wave experiments Armstrong et al. (2003); Armstrong (2006),
using the dispersion $n-1\sim n^{-2}$ of $n$ with multiple frequencies $\nu$
to measure, and then correct for the fluctuations. Here, frequencies of about
8 GHz and 16 GHz will be used (slightly offset in up- and downlink).
Doppler shifts are detected through phase shifts between ground and remote
clocks, and a radial-velocity resolution of $\Delta v_{r}=1\mu$m/s requires
the reference clocks to be stable to $\Delta v_{r}/c=3\cdot 10^{-15}$, where
$c$ is the speed of light. With a radio frequency of $\nu=8\mbox{ GHz}$ and a
phase sensitivity of $s=2\pi/100$ (3.5 degrees), this stability must be
maintained over the time $\tau=(c/\Delta v_{r})s/(2\pi\nu)=375$ s. This
exceeds the state of the art of 1 part in $10^{-13}$ over 1000 s in spaceborne
ultra-stable oscillators (USOs), but can be achieved with a coherent link
Howard et al. (1974); Woo et al. (1976), where the uplink signal is used to
phase-lock the space-borne oscillator from which the return signal is derived.
Recently, the modified technique of noncoherent Doppler tracking DeBoy et al.
(2005) was used, where the spacecraft has a freely-running oscillator and
transmits its phase slippage relative to the signal received from Earth
through a digital communication channel. This latter technique is proposed
here Adams (2007) to make it easier to accommodate radio signals at two
frequencies that fluctuate differently as they traverse the solar wind. With
the above oscillator stability of $10^{-13}$, the 1-$\sigma$ walk-off of an
8-GHz signal is $2\pi$ in 1250 s. Thus, to resolve the phase to $s=2\pi/100$,
about 8 bits of digital data need to be transmitted every 1250 s for
1-$\sigma$ certainty, and 10 bits for 4-$\sigma$.
For ranging, Earth and the array transmit radio pulses to each other with a
pseudo-random number (PRN) phase modulation similarly to GPS, and cross-
correlate the received signals with the local ones. The array transmits the
correlation result through the digital communication channel. With Gaussian
Minimum-Shift Keying (GMSK) and a bit-time-bandwidth product of BT = 0.3
(standard in cell phonesAgilent Technologies (2005)), the signal bandwidth for
a 5-meter resolution is about 20 MHz Adams (2007). Here, $\nu=2.25$ GHz is
chosen for the uplink, and $\nu=2.1$ GHz for downlink ranging. The pulse
duration must cover the initial ranging uncertainty of about 1 km, that is 3
$\mu$s, but shorter PRN sequences may be used once the range is known better.
The radio system thus uses four frequencies in each direction: two for Doppler
tracking, one for ranging, and one for digital communication. Up- and downlink
frequencies are slightly offset from each other to isolate the receivers at
each end from the transmitters. Each of the up/down frequency pairs has 11400
antennae in the array.
The transmitter powers of the array and the ground station are determined by
their mutual distance $d$, the ground and array antenna gains $G_{g}$ and
$G_{a}$, and the receiver sensitivities. On average, half of the 11400
antennae in the array are in view of the Earth at a given time, and their
random orientations introduce Adams (2007) another factor of $1/4$, so
$G_{a}=10\log(11400/8)=31.5\mbox{ dB}$ for all wavelengths. Signal diminuition
with distance is described by the free-space loss $L=(4\pi d/l)^{2}$, i.e.,
the reciprocal solid angle subtended by a dipole antenna at one end of the
radio link, as seen from the other. For a maximum operating distance of 1.82
AU at a frequency of 8 GHz, ${\cal L}=10\log_{10}L=-279\mbox{ dB}$. Finally,
the receiver on Earth is assumed to use a 70-meter dish antenna (gain
$G_{g}=65\mbox{ dBi}$ at 8 GHz) of the deep-space network Miller (2006). The
sum $G_{a}+G_{g}+{\cal L}$ gives a total transmission loss of -182.5 dB for
all radio frequencies (${\cal L}$ and $G_{g}$ go opposite in frequency,
$G_{a}$ is frequency-independent). The receiver system noise temperature of
the ground station is estimated Manshadi (2000) at 17 K, corresponding to a
spectral power density of -186.3 dBm/Hz. For downlink Doppler tracking, this
noise power needs to be matched within a scintillation-broadened bandwidth of
0.01 Hz. With a 20-dB noise margin and a transmission loss of -182.5 dB, the
array thus needs to output -3.8 dBm (0.4 mW) for each of the two radio
frequencies. The design should foresee a slightly higher power to allow for a
reduction in array gain due to a potential loss of radio units. For downlink
ranging at 20-MHz bandwidth and 0-dB noise margin, the transmitter pulse power
is 69 dBm (8.3 kW), to be shared among – on average – 5472 antennae. Each
antenna driver thus outputs a peak power of about 1.5 W, i.e., about as much
as a cell phone. With 3-$\mu$s-long ranging pulses transmitted once in 30 ms,
the average power for ranging is 800mW for the array, 3.4 mW per radio unit
(assuming 235 of 475 being in view of Earth and active), and 140 $\mu$W per
antenna. The lack of noise margin is compensated by repeated ranging pulses:
10000 pulses for a 20dB margin taking 300 s define the time for one ranging
measurement. Once the range is known to tens of meters, shorter PRNs can be
used for lower average power or improved noise margin. Loss of some radio
units does not disable ranging, but leads to an increase in the measurement
time.
For the uplink, a higher transmitter power is available, but the receiver
noise temperature is much higher (assumed here 1000 K , which is typical for
cell phones). Thus, the ground station has to transmit at 60 times the
downlink power, i.e., bursts of about 0.5 MW for ranging and a continuous
signal of 24 mW for Doppler tracking. The initial-phasing signal has to be
received without array gain ($G_{a}=0\mbox{ dBi}$), and it needs to have a
bandwidth given by the Doppler-shift uncertainty $\Delta\nu$ due to an initial
velocity uncertainty $\Delta v$. For $\Delta v=1m/s$ and $\nu=8$ GHz,
$\Delta\nu=27Hz$, and thus the ground station has to transmit at 1.6 kW for a
noise margin of 20 dB.
Power is generated by radiation-tolerant, thin-film, amorphous-silicon solar
cells deposited on the plastic foils of the nodes (see Fig. 4). This
technology achieves a power of 2 kW per kg of solar-cell/substrate material
United Solar Ovonic (2007) for operation at 1 A.U. distance from the sun.
Outside of the solar-cell areas, the foils have a broadband dielectric mirror
coating to reflect visible and near-infrared sunlight, but let far-infrared
escape. This reduces the temperature under the coating to well below the 321 K
in black-body radiation equilibrium at 1 A.U. from the sun, so the radio unit
can shed excess heat.
Figure 4: Left: The plastic foil of one node with 25 transponders sitting on
it. Lower right: Closeup of the foil surrounding one transponder with
microwave strip lines going out to the antennae, and a section of the plastic
foil coated with a dielectric-mirror multilayer that reflects visible and
near-infrared sunlight, but lets far infrared through. Upper right: Side view
of the same. Sunlight is reflected off the coating, but far infrared heat
radiation can escape into space. The transponder can thus radiate heat off to
the ground surrounding it.
At any given time, about half the radio units are in the sunlight and have to
power themselves and the other half through the metal wires between and within
the nodes, which are switched into rapidly varying configurations of closed
circuit loops containing power sources and sinks. At an efficiency of 10 %,
the radio transmitters will consume an average power of 50 mW (10 times (3.4
mW for ranging, 2 $\mu$W for Doppler, 1.6 mW for digital communincation), 50
mW are needed for the onboard digital electronics, and 900 mW for cooling
radio units in the sunlight, so a radio unit in the sunlight needs to harvest
1 W for itself. Radio units in the shade need 100 mW for the electronics, 100
mW power reserve for heating, and 100 mW for internal power conversion losses.
As estimated below, a sunlit radio unit then needs to harvest another 2.4 W to
power up to two units in the shade. With wires made of a high-strength
aluminum alloy with a specific resistance of $2.9\cdot 10^{-8}$ $\Omega$m
Adams (2007) and a cross section of $2.0\cdot 10^{-8}$ m2 (250 wires of 10
$\mu$m diameter, each), the wires have a resistance of 1.48 $\Omega$/m. For
example, in a 1250-meter-long current loop with two sunlit nodes powering four
in the shade (each with 25 radio units), a current of 100 mA needs to run 50 %
of the time to supply 30W at a voltage drop of 600 V (150 V per node). There
is an additional voltage drop of 185 V in the wires. Therefore, each of the
two sunlit nodes need to generate a voltage of 393 V to power the loop. At 66
% power-conversion efficiency each sunlit node has to harvest about 60 W for
those in the shade, i.e., 2.4 W per radio unit. For more detail, see ref.
Adams (2007).
To determine the payload to be delivered to the asteroid, the following masses
are estimated Adams (2007): 47.5 kg for the 475 radio, 5 kg for the plastic
foil of the 19 nodes (excluding the solar cells), 0.81 kg of solar cells
(475*3.4 W / (2 kW/kg)), 1.1 kg for 12 km of carbon/glass fiber/metal wires, 6
kg for the power-conversion electronics, master clocks and other
infrastructure in the nodes, and 9.5 kg for the containers in which the nodes
are stowed until they are deployed (Fig. 2). This brings the total array mass
to about 70 kg. Another 80 kg can be estimated for the delivery spacecraft
(structure, engines, tanks, etc.) for a mass $m_{s}=150$ kg to be delivered to
the asteroid. An optimal transfer trajectory from low-earth orbit (LEO) to
Apophis requires a total velocity change of $\Delta v=7.71$ km/s Adams (2007).
Applying the Tsiolkovsky rocket equation $m_{t}=m_{s}\exp(\Delta v/v_{p})$ for
the total mass $m_{t}$ at LEO, where $v_{p}$ is the rocket exhaust velocity
(3.07 km/s for hydrazine thrusters), the mass to launch to LEO is about 1900
kg, which is well within the capabilities of commercial satellite launchers.
## IV Other Possible Applications
Space-based phased arrays – based on asteroids, or floating freely – could
also be used for other scientific applications, especially so if multiple ones
are used. These applications make use of the capability of a phased array to
transmit and receive simultaneously in different directions. One example is
very-long-baseline (VLBI) radio astronomy Adams (2007) with baselines the size
of Earth’s orbit. The arrays would maintain radio links with each other to
distribute a master clock signal, and would simultaneously receive
directionally from the radio source of interest. As the asteroids progress
along their orbits, different Fourier components of the source’s angular
distribution on the sky can be resolved to assemble an extremely high-
resolution image, corresponding to the spacing between the asteroid-based
receivers. Another possible application is in gravitational-wave detection at
milli-Hertz frequencies, as is currently being done with the Cassini
spacecraft Asmar et al. (2005). Because this application requires a very high
velocity accuracy, the arrays have to be based on asteroids to minimize their
susceptibility to orbital perturbations due to the solar wind, the Yarkovsky
effect, etc. With phased arrays based on several asteroids, multiple coherent
links can be maintained similtaneously in different directions between
asteroids to precisely locate a source Adams (2008). Yet another possible
application would be in the mapping of the solar wind by measuring the
scintillations in radio links between several arrays. The transmitter power
for the latter two applications may be estimated as before for the downlink in
Doppler tracking, with the bandwidth again given by scintillations. However,
the receiver temperature is now 1000K (instead of 17 K), and the receiver
antenna gain is 31.5 dBi (instead of 65 dBi). This requires an increase of the
transmitter power by a factor of $1.1\cdot 10^{5}$ over the value of 0.4 mW
estimated before. Thus, for each link the array must transmit at 45 W, and
each of the about 200 to 250 radio units in view of the respective receiver
must transmit about 200 mW. These scientific applications are not subject to
the time constraints of a mission to track an asteroid before an impact or a
keyhole passage. Therefore, lengthier but more energy-efficient transfer
trajectories Marsden and Ross (2006) are admissible, which will drastically
reduce the mass to launch to LEO, and thus the mission cost.
## V Summary
In summary, a way is proposed to precisely track the orbit of an asteroid by
deploying a radio transponder on its surface, which establishes a two-
frequency noncoherent link for Doppler tracking at 1 $\mu$m/s accuracy,
returns broadband-modulated pulses for ranging to 5 m, and communicates
digital data representing correlations between uplink and downlink Doppler and
ranging signals. The transponder is a phased array of 11400 antennae for each
of the four frequencies used, driven by 475 radio units. These are linked with
each other by a net that serves the multiple purposes of enabling the landing
maneuver in the low gravity of the asteroid, phasing the array, and sharing
solar-generated power. Due to a high degree of redundancy, the array can
tolerate some damage, which will only lead to a slight reduction in antenna
gain. The same phased-array technology can also be used for very-long-baseline
radio astronomy or for gravitational-wave detection.
## References
* (1) _The order of magnitude can be estimated by converting Kepler’s 3rd law to express the orbital period dependent on the velocity, and then taking the derivative._
* Schweickart et al. (2006) R. Schweickart, C. Chapman, D. Durda, P. Hut, B. Bottke, and D. Nesvorny, White paper no. 39 at NASA NEO Workshop held in Vail, Colorado, 26-28 June, 2006 (2006), www.b612foundation.org/papers/wpdynamics.pdf.
* Asmar et al. (2005) S. Asmar, J. Armstrong, L. Iess, and P. Tortora, Radio Science 40, 1 (2005).
* Chesley et al. (2003) S. Chesley, S. Ostro, D. Vokrouhlicky, D. Capek, J. Giorgini, M. Nolan, J. Margot, A. Hine, L. Benner, and A. Chamberlin, Science 302, 1739 (2003).
* Spitale (2002) J. N. Spitale, Science 296, 77 (2002).
* Adams (2007) B. W. Adams, contribution to the Apophis mission design competition of the Planetary Society (2007), supplementary material.
* Woo (1978) R. Woo, Astrophys. J. 219, 727 (1978).
* Ho et al. (2002) C. Ho, M. Sue, A. Bedrossian, and R. Sniffin, Proc. GA02 1278-1287 14, 1277 (2002).
* Armstrong (2006) J. Armstrong, Living Reviews in Relativity 9, 1 (2006).
* Armstrong et al. (2003) J. Armstrong, L. Iess, P. Tortora, and B. Bertotti, Astrophys. J. 599, 806 (2003).
* Howard et al. (1974) H. Howard, G. Tyler, G. Fjeldbo, A. Kliore, G. Levy, D. Brunn, R. Dickinson, R. Edelson, W. Martin, R. Postal, et al., Science 183, 1297 (1974).
* Woo et al. (1976) R. Woo, F. Yang, K. Yip, and W. Kendall, Astrophys. J. 210, 568 (1976).
* DeBoy et al. (2005) C. C. DeBoy, C. Haskins, D. Duven, R. Schulze, J. R. Jensen, M. Bernacik, and W. Millard, Acta Astronautica 57, 540 (2005).
* Agilent Technologies (2005) Agilent Technologies, web document (2005), http://eesof.tm.agilent.com/docs/adsdoc2005A/gsm/gm013.html.
* Miller (2006) R. Miller, web document (2006), www.ioag.org/ioag6_nasajpl_dsn_status.pdf.
* Manshadi (2000) F. Manshadi, IEEE Aerospace Conf. Proc. 5, 73 (2000).
* United Solar Ovonic (2007) L. United Solar Ovonic, web site (2007), http://www.uni-solar.com.
* Adams (2008) B. W. Adams, manuscript in preparation (2008).
* Marsden and Ross (2006) J. Marsden and S. Ross, Bull. Am. Amth. Soc. 43, 43 (2006).
|
arxiv-papers
| 2008-09-15T20:39:32
|
2024-09-04T02:48:57.786638
|
{
"license": "Public Domain",
"authors": "Bernhard W. Adams",
"submitter": "Bernhard Adams",
"url": "https://arxiv.org/abs/0809.2616"
}
|
0809.2631
|
# Effect of edge removal on topological and functional robustness of complex
networks
Shan He Sheng Li lisheng@sjtu.edu.cn Hongru Ma Department Of Physics,
Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
###### Abstract
We study the robustness of complex networks subject to edge removal. Several
network models and removing strategies are simulated. Rather than the
existence of the giant component, we use total connectedness as the criterion
of breakdown. The network topologies are introduced a simple traffic dynamics
and the total connectedness is interpreted not only in the sense of topology
but also in the sense of function. We define the topological robustness and
the functional robustness, investigate their combined effect and compare their
relative importance to each other. The results of our study provide an
alternative view of the overall robustness and highlight efficient ways to
improve the robustness of the network models.
###### pacs:
89.75.Fb, 89.75.Hc
## I introduction
Complex networks are ubiquitous in our world. They exhibit not only diverse
structural characteristics ref1 ; ref24 , such as the power-law tail of degree
distribution and the small-world phenomenon of average path length, but also
different levels of robustness, e.g., scale-free networks display higher
tolerance to error but more vulnerability to attack than exponential networks
ref5 ; ref6 ; ref7 ; ref16 . Robustness evaluates the ability of a network to
maintain its original attributes and functions when constituent loss or other
kinds of damage are present. Once the robustness is identified, the weakness
of a network is pointed out for optimizations or countermeasures, e.g., the
weakness of a communication network can be overcome to increase reliability,
while the weakness of an epidemic network can be utilized for efficient
destruction. For these practical applications, the study of the robustness
have received a lot of interests.
Several measures of the robustness have been proposed. A frequently used one
is the existence of the giant component ref5 ; ref6 ; ref7 ; ref8 ; ref16 .
The giant component is the only component in a network whose size scales
linearly with the number of vertices. It was found that the damage to a
network, such as random removal of vertices or edges, can be exactly mapped to
a standard percolation process. The network percolates if the giant component
exists, indicating that the general connectedness of the network is
maintained. There are also other measures based on quantities such as
efficiency ref8 ; ref15 and average path length ref5 ; ref8 . The measure we
adopt in the present work is the preservation of total connectedness ref9 .
The total connectedness of a network is preserved only when communication is
effective between every pair of vertices in the network. The more damage
needed to destroy the total connectedness, the more robust the network is.
This measure can be useful to describe the robustness of networks that have no
vertex redundancy. In such networks, each vertex contributes to the whole in a
way that cannot be replaced by the others and even the unavailability of a
single vertex affects the overall functionality, e.g., in a scenario that many
computers collaborate through a network to accomplish some calculation-
intensive task, if one or more members lose communication, the performance of
the collaboration may be degraded or the task may even fail. Note that the
total connectedness used in the measure is not limited within the sense of
topology, i.e., physical connection is not the only factor that influences the
effectiveness of the communication; there are other factors that can hinder
the communication even though all the vertices are connected. These factors
are often related to the function of a network, e.g., it was reported that in
a transmission network, a path connecting two vertices may become so long
after damage that this path is unusable ref17 . As both topology and function
related factors can prevent the communication between vertices, two types of
robustness are involved: the topological robustness and the functional
robustness. Studying their combined effect and the dominance of one over the
other can provide additional insights into the network robustness. However,
such studies have not yet been carried out extensively.
Besides the measures, the damage of constituent loss has also been widely
studied in the form of vertex removal. Removing a vertex is an appropriate
abstraction of several real-world events, e.g., a user leaves a P2P network, a
website goes offline. There are other events that should be abstracted more
appropriately by removing an edge, e.g., a network cable is unplugged, a
flight between two cities is canceled. However, only a few works cover edge
removal. In fact, there are remarkable differences between the two types of
removal. In the sense of topology, vertex removal inflicts more damage, as
each time a vertex is removed, all of its edges are removed as well. Thus to
achieve the same effect, smaller amount of vertices is removed than edges. If
the damage of removal is assessed in terms of edges rather than vertices, the
vulnerability of a network could be moderated ref13 . In the sense of
function, the removal of vertices reduces the total amount of quantity
transmitting on a network, as vertices usually not only deliver but also
generate the quantity. If the reduction of the total amount can compensate the
damage, the robustness of the network is enhanced ref9 . On the other hand,
the enhancement may not be expected for edge removal because there is no
change in the total amount. These differences indicate that edge removal needs
separate studies from vertex removal.
In this paper, we propose to study the effect of edge removal on the
topological robustness and the functional robustness, using the total
connectedness as the criterion of breakdown. Several network topologies and
removing strategies are simulated. We combine a simple traffic dynamics with
the network topologies and successively remove edges until the total
connectedness of the networks is destroyed. The fractions of removed edges
characterize the robustness of the networks. Moreover, as the destroy of the
total connectedness is related to either the topological factors or the
functional factors, we see which type of the factors is dominant. The purpose
of our study is to explore the relationship between the topological robustness
and the functional robustness, which is different from the purposes of
previous works that also deal with edge removal ref11 ; ref19 ; ref20 ; ref21
; ref30 . Particularly, we only concern the times when the networks lose the
total connectedness, and do not discuss the evolutions thereafter, such as the
occurrence of cascading failure ref14 . The results of our study provide an
alternative view of the network robustness and highlight efficient ways for
improvements.
The paper is organized as follows: In Sec. II, the model we studied is
described in detail. In Sec. III and IV, the simulation results are shown
according to the removing strategies. In Sec. V, we give the conclusion.
## II network topologies, robustness and removing strategies
The network models of topology we studied are the Erdös-Rényi (ER) model of
random network ref1 , the Barabási-Albert (BA) model of scale-free network
ref2 and the Newman-Watts (NW) model ref4 , which is a variant of the Watts-
Strogatz (WS) model of small-world network ref3 . For all the network models,
we set vertex number $N=1024$ and average degree $\langle k\rangle=8$. The
edges are undirected and have no weight. Multiple edges are not allowed. In
order to study the robustness, we ensure that the topologies are totally
connected when they are intact. The ER model yields exponential degree
distribution. Each vertex pair is linked by an edge with probability $\langle
k\rangle/(N-1)$. Since $\langle k\rangle>\ln(N)$, the ER networks are almost
surely connected ref1 . The BA model features growth and preferential
attachment. At any time step of a growth, the network from the previous step
is connected. A new vertex is added with $\langle k\rangle/2$ preferentially
linked edges in the current step and the connectedness is preserved. If a BA
network grows infinitely large, the probability of a randomly selected vertex
having degree $k$ is proportional to $k^{-3}$. The NW model starts with a
regular structure, which is identical to the WS model. But unlike the WS
model, the NW model builds shortcuts by randomly inserting $pN$ edges, where
$p$ is the rate of the shortcuts. This way of building shortcuts eliminates
the possible network fragment during the process of rewiring in the WS model
ref4 . As total connectedness is the special concern, we adopted the NW model
instead of the WS model. With $p$ changes from small to large, a NW network
can undergo a transition from “large world” to “small world”. We choose
$p=0.02$ in the small-world phase without loss of generality. The $pN$
shortcuts are inserted into a periodic square lattice with side length $L=32$
and coordination number $z=8$. As $p$ is a small quantity, the average degree
is $8+p\approx 8$. Among the three models, the BA model produces scale-free
degree distribution which is observed in many real networks ref1 . The NW
model is based on a square lattice, which is the topology that can reproduce
some of the observed real Internet features ref31 . The randomly inserted
shortcuts are for the small-world property, though high clustering is absent.
The ER model does not match real networks in nearly all aspects. Nevertheless,
it has the significance that many of its properties can be obtained through
probabilistic approaches. Note that the average path lengths of all the three
models scale logarithmically with the number of vertices. It is interesting to
compare the robustness of them.
The robustness of the network models refers to the ability of maintaining
total connectedness when a portion of edges are removed. The total
connectedness is defined as the effective communication between every pair of
vertices, which is affected by the twofold effects of the edge removal. One is
that the physical connection between a pair of vertices may be cut off. If the
number of edges in a network is decimated to be less than $N-1$, the network
is certainly not totally connected. The other effect is related to the fact
that a network usually performs some function, e.g., transmits some quantity.
The edge removal may redistribute the quantity transmitting on the network and
the vertices may lose communication due to congestion ref10 , e.g., the
quantity from a source vertex is congested on some intermediate vertices and
never reaches the target. The former effect is topological, while the latter
is functional. Either of the two effects can destroy the total connectedness,
and cause the breakdown of the networks.
The topological breakdown of the networks occurs when the networks are just
fragmented from one component into two during the edge removal, i.e., at least
one of the vertices is isolated physically from the others ref9 . This
condition is more restrictive on the connectivity than the existence of the
giant component, which was usually used in previous works ref5 ; ref6 ; ref7 ;
ref8 ; ref13 ; ref14 ; ref16 . We adopt this condition in favor of the
emphasis on the completeness of a network and that the functional breakdown
can be defined in a similar manner.
We incorporate a simple traffic dynamics to define the functional breakdown of
the networks. At each time step, a certain amount $\rho$ of quantity, which
can be data, energy, etc., is transmitted between every pair of vertices,
along the shortest paths. Transmitting along the shortest paths minimizes the
delay of distributing the quantity. For the logarithmic average path length of
the network models, the transmission scales well. The number of shortest paths
passing through a vertex $j$, which defines the vertex betweenness $B_{j}$ of
$j$ ref26 , determines the load $L_{j}$ of the transmitting quantity on $j$,
$L_{j}=\rho B_{j}$ ref10 . The load is handled by the vertex within its
capacity, i.e., the amount of quantity that can pass through $j$ is at most
the capacity of the vertex. It would be efficient to set the capacity of $j$
proportional to $B_{j}$, so as to meet the load. However, the calculation of
betweenness needs global information and takes time $O\left(\langle k\rangle
N^{2}\right)$ ref22 , which is resource consuming. For simplicity, we set
uniform capacity $C$ for each vertex. As a result, the vertex with the largest
betweenness $B_{\max}$ is the most likely to encounter overload. The largest
generation rate $\rho_{c}$ of the transmitting quantity is $C/B_{\max}$. The
rate $\rho_{c}$ marks the capacities of the networks. It was found that
homogeneous networks have higher capacities than heterogeneous ones ref10 . As
the purpose of the study is to compare the different aspects of the robustness
rather than the absolute capacities, we normalize $\rho$ with respect to
$\rho_{c}$ for each individual network, i.e., the load of $j$ changes to
$L_{j}=\rho B_{j}C/B_{\max}$, where $\rho$ is the normalized generation rate
of the quantity. The load $L_{j}$ is initially less than $C$ with $0<\rho<1$,
but $L_{j}$ could be larger than $C$ as the vertex betweenness redistributes
in the process of edge removal. If removing an edge leads to $L_{j}>C$ for
some vertex $j$, congestion occurs ref10 . The excessive load accumulates
continuously on $j$ and the vertex is unable to communicate with other
vertices. Similar to the topological breakdown, the total connectedness is
destroyed. This situation defines the functional breakdown of the networks.
One can see that the functional breakdown actually depends on the evolution of
the ratio $B_{j}/B_{\max}$; the study of the functional robustness is
essentially a test of the response of vertex betweenness to edge
perturbations. Though the dynamics is simple, some essential functional
properties of the networks are reflected and we are able to study the relative
importance of the functional robustness to the topological robustness under
different values of the only parameter $\rho$.
To study the two types of the robustness, we start from an undamaged network
generated by one of the network models and remove the edges one at a time
until the network breaks down for either the topological or the functional
reasons. The robustness are studied in the contexts of different removing
strategies.
Two removing strategies are used: random failure and attack. The random
failure strategy removes edges with uniform probability, which can be seen as
a simple abstraction of the successive error in a communication network. The
attack strategy removes edges in the descending order of their importance,
which tries to model a sophisticated attacker who knows the global information
of a network and always targets the most important link. There are several
quantities that can define the importance of an edge, such as edge betweenness
and edge degree ref30 ; ref8 . We choose edge betweenness as the definition of
the importance, because betweenness directly measures the load of the
transmitting quantity in our traffic dynamics. Analogous to vertex
betweenness, edge betweenness is defined as the number of shortest paths
passing through an edge ref30 . Every time the most important edge is removed,
we recalculate the betweenness to find the most important one in the remaining
edges. If there are multiple edges having the same largest betweenness, we
randomly select one for removal. Random failure and attack are usually studied
together to obtain a collective characterization of network robustness ref5 .
In the next two sections, we show the results according to the two strategies.
## III the random failure strategy
In this section, edges are randomly removed from the networks. We first study
the total breakdown, which is the combined effect of the topological breakdown
and the functional breakdown. Figure 1 shows the cumulative distribution
function $CDF_{total}$ of the total breakdown as a function of the fraction
$f$ of removed edges for the network models. The results are shown for
different values of $\rho$.
Figure 1: The cumulative distribution function $CDF_{total}$ of the total
breakdown under the random failure strategy as a function of the fraction $f$
of removed edges for (a) the ER model, (b) the BA model and (c) the NW model.
When $\rho=0$, the total breakdown is in fact the topological breakdown; when
$\rho>0$, the total breakdown is the combined effect of the topological
breakdown and the functional breakdown. All the data is averaged over $10240$
realizations.
When $\rho=0$, there is no quantity transmitting on the networks. The total
breakdown is in fact the topological breakdown, which is characterized by a
critical fraction $f_{c}^{topo}$ of removed edges. When $f<f_{c}^{topo}$, each
of the networks is totally connected; when $f\geq f_{c}^{topo}$, each of the
networks is split into two or more components, where one of the components
could be the giant component.
For the ER model, almost every model realization is connected if $\langle
k\rangle>\ln(N)$ ref1 . Thus $f_{c}^{topo}=\left[\langle
k\rangle-\ln(N)\right]/\langle k\rangle$.
For the BA model, a simple estimate of $f_{c}$ is sought under the framework
of random graph with prescribed degree sequence ref29 . Seeing that the BA
model yields no assortative mixing ref28 , i.e., no degree correlation, we
equate the BA networks to random scale-free networks. Although the BA model
features network assembly and evolution instead of complete randomness, some
qualitative result can be obtained by the estimate and confirmed through
simulation. For a random network with arbitrary degree distribution $P(k)$,
the probability $\pi_{s}$ of a randomly selected vertex being in a finite
component of size $s$ has been obtained in Ref. ref27 ,
$\pi_{s}=\frac{\langle
k\rangle}{(s-1)!}\left[\frac{d^{s-2}}{dz^{s-2}}\left[g_{1}(z)\right]^{s}\right]_{z=0},$
(1)
where $g_{1}(z)=\sum_{k=0}^{\infty}\frac{(k+1)P(k+1)}{\langle k\rangle}z^{k}$
and $s>1$. Particularly, $\pi_{1}=P(0)$. We now consider a scale-free random
network, which has the same $N$, $\langle k\rangle$ and $P(k)$ as the BA
networks. With the minimum degree larger than $2$, $g_{1}(z)$ has no constant
terms and $\pi_{s}=0$. There are no finite components, all the vertices are
connected in the giant component. It was reported that the giant component
always exists for a random network with $P(k)\sim k^{-3}$ when vertices are
randomly removed ref6 . The result is similar if edges are removed ref13 .
Thus the network is always split into the giant component and a finite
component. Denote $\pi_{s}(f)$ the distribution of the size of the finite
component after a fraction $f$ of edges are removed. When $f\to 0$, the
network is nearly not affected and $\pi_{s}(f)\to 0$. Though $\pi_{s}(f)$ is
small, various sizes of the finite components are probable. With the existence
of the giant component, $\pi_{s}(f)\approx\pi_{s}(0)$ decays exponentially
ref27 . We then neglect the higher order components and focus only on the
components of size one and two. The new degree distribution and the new
average degree after edge removal are
$P_{f}(k)=\sum_{k_{0}=k}^{\infty}P\left(k_{0}\right){k_{0}\choose
k}(1-f)^{k}f^{k_{0}-k}$ and $\langle k\rangle_{f}=(1-f)\langle k\rangle$,
respectively ref6 . We calculate the ratio
$\displaystyle r(f)$ $\displaystyle=$
$\displaystyle\frac{\pi_{2}(f)}{\pi_{1}(f)}=\frac{\left[P_{f}(1)\right]^{2}}{\langle
k\rangle_{f}P_{f}(0)}$ (2) $\displaystyle=$
$\displaystyle\frac{(1-f)\left[\sum_{k=1}^{\infty}kP(k)f^{k}\right]^{2}}{f^{2}\langle
k\rangle\sum_{k=1}^{\infty}P(k)f^{k}}.$
This ratio is a monotonically decreasing function of $f$, which implies that
in addition to the exponential decay of $\pi_{s}(f)$ when $f\to 0$, the
components of size two become even less probable than the components of size
one as $f$ becomes larger; in most cases, the network is split into the giant
component and a finite component consisting of an isolated vertex. In the
simulation of the BA model, we observed that in more than 99.8% realizations,
the finite component is of size one. This fact allows us to estimate the
critical fraction $f_{c}^{topo}$ by the emergence of an isolated vertex, which
has empty degree. If one such vertex can be sampled, the total connectedness
is destroyed, i.e., $f_{c}^{topo}$ satisfies
$P_{f_{c}{}^{topo}}(0)=\frac{1}{N}$.
For the NW model, the critical fraction $f_{c}^{topo}$ is obtained in a
similar manner. Employing the expression of $\pi_{s}(f)$ ref27 , we get
$\displaystyle\pi_{s}(f)$ $\displaystyle=$
$\displaystyle\frac{(1-f)^{s-1}\langle
k\rangle}{(s-1)!}\left[\frac{d^{s-2}}{dz^{s-2}}\left[z^{7}\right]^{s}\right]_{z=f}$
(3) $\displaystyle=$ $\displaystyle\frac{(7s)!\langle k\rangle
f^{6s+2}(1-f)^{s-1}}{(s-1)!(6s+2)!},$
where the shortcuts in the model are neglected for convenience and the
original degree distribution reads $P(k_{0})=\delta(k_{0}-8)$. When $s$ is
large,
$\frac{\pi_{s}(f)}{\pi_{s-1}(f)}\approx\frac{7^{7}}{6^{6}}(1-f)f^{6}.$ (4)
This ratio tends to zero when $f\to 0$ and increases monotonically until it
reaches the maximum value $1$ at $f=f_{c}=\frac{6}{7}$, where $f_{c}$ is the
critical percolation fraction ref6 . As the total connectedness is more
restrictive than the existence of the giant component, we claim that
$0<f_{c}^{topo}<f_{c}$. In this region, $\pi_{s}(f_{c}^{topo})$ drops
exponentially, and again, each of the NW networks is split into the giant
component and an isolated vertex, $f_{c}^{topo}$ satisfies
$P_{f_{c}{}^{\text{topo}}}(0)=\frac{1}{N}$. Particularly, we get
$f_{c}^{topo}\sim N^{-\frac{1}{8}}$ for the NW model. The result is consistent
that if $N\to\infty$, then $f_{c}^{topo}\to 0$ and $\pi_{s}(f_{c}^{topo})$
decays very quickly with increasing $s$. In Fig. 2, we plotted the scaling
relation between $P_{f_{c}^{topo}}(0)$ and $1/N$ obtained in the simulation
for the BA model and the NW model. As the figure shows, the agreement between
the theoretical estimate and the simulation is reasonable. The above
calculations show that $f_{c}^{topo}$ is strongly correlated to the local
connectivity of vertices. In the BA model, the majority of the vertices have
degree four, while in the NW model, all the vertices have degree at least
eight. Thus as Fig. 1 shows, the NW model is more topologically robust than
the BA model and they both are more topologically robust than the ER model.
Figure 2: The scaling relation between $P_{f}(0)$ and $1/N$ at
$f=f_{c}^{topo}$ for the BA model and the NW model when edges are randomly
removed. Symbols are the results of simulation, each of which is an average
over $102400$ networks. The two solid lines (They nearly overlap.) are the
linear fits of the simulation results. The slopes of the lines are $0.991$ and
$0.980$.
When $\rho>0$, the functional breakdown can also happen. Intuitively, networks
with larger $\rho$ are more functionally vulnerable. We showed in Fig. 1 the
cumulative distribution function of the total breakdown only for near capacity
generation rates. When $\rho=0.95$, the ER networks are very likely to break
down when only a small fraction of edges are removed. For the same $\rho$, the
NW model has slightly larger $CDF_{total}$ than the BA model, which suggests
that while the NW model is more topologically robust than the BA model, it is
less functionally robust. Note in the figure that while $CDF_{total}$ for
large $\rho$ increases fast in the region $0<f<0.1$ for all the network
models, $CDF_{total}$ for $\rho=0$ increases much slower for the ER model or
remains nearly zero for the BA model and the NW model. Thus in this region of
$f$, the total breakdown is mainly determined by the functional breakdown.
With $\rho$ close to $1$, the functional breakdown reflects the change in the
network capacity during the edge removal. In Fig. 3, we plotted the average
network capacity $\langle\rho_{c}\rangle_{f}$ as a function of $f$ for the
network models. ($\langle\rho_{c}\rangle_{f}$ is normalized to the initial
value $\langle\rho_{c}\rangle_{0}$.) The capacity of all the network models
decreases with the edge removal, because the shortest paths concentrate on the
remaining edges and the betweenness of vertices becomes more heterogeneous.
For the ER networks, the degree distribution remains exponential when edges
are randomly removed, i.e., the ER networks are still ER networks after the
edge removal. The vertex betweenness $B$ of ER networks has a power-law
relation $B\sim k^{\alpha}$ with degree $k$ ref23 , hence
$\displaystyle\left\langle\rho_{c}\right\rangle_{f}$ $\displaystyle=$
$\displaystyle\frac{C}{B_{\max}(f)}=\frac{C}{\sum_{j}B_{j}(f)\frac{\left[k_{\max}(f)\right]^{\alpha}}{\sum_{k}k^{\alpha}}}$
(5) $\displaystyle=$
$\displaystyle\frac{C}{\sum_{j}B_{j}(f)\frac{(1+\alpha)\left[k_{\max}(f)\right]{}^{\alpha}}{\left[k_{\max}(f)\right]{}^{\alpha+1}-1}}$
$\displaystyle\approx$
$\displaystyle\frac{Ck_{\max}(f)}{\sum_{j}B_{j}(f)(1+\alpha)},$
where $B_{\max}(f)$ is the maximum vertex betweenness and $k_{\max}(f)$ is the
maximum degree when a fraction $f$ of edges are removed. The vertex
betweenness also satisfies $\sum_{j}B_{j}(f)=DN(N-1)$, where
$D\sim\ln(N)/\ln(\langle k\rangle_{f})$ is the average path length ref1 . Then
we get
$\left\langle\rho_{c}\right\rangle_{f}\sim\frac{Ck_{\max}(f)\ln\left(\langle
k\rangle_{f}\right)}{(1+\alpha)N(N-1)\ln(N)}$ (6)
and the normalized average capacity
$\frac{\left\langle\rho_{c}\right\rangle_{f}}{\left\langle\rho_{c}\right\rangle_{0}}\sim
k_{\max}(f)\ln[\langle k\rangle(1-f)]=F(f),$ (7)
where $k_{\max}(f)$ and $f$ have an implicit relation ref24 through the
regularized gamma function $P(\alpha,x)$ that $P\left[\left\lfloor
k_{\max}(f)+1\right\rfloor,(1-f)\langle k\rangle\right]=\frac{1}{N}$. The
linear relation between
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
and $F(f)$ is also shown in Fig. 3. The average network capacity of the ER
model decreases much more quickly than those of the other two network models,
thus the ER model has the worst functional robustness.
For the BA model, though there is a similar power-law relation $B\sim
k^{\eta}$ in scale-free networks ref25 , scale-free degree distribution does
not remain scale-free when edges are randomly removed ref20 . Instead, we
compare the trend of the the load distribution for the BA model and the NW
model. The two network models both have heterogeneous load distributions. Hub
vertices or shortcuts carry large load. The BA model also has heterogeneous
connectivity while the NW model has homogeneous connectivity. When each edge
is removed with equal probability, the hub vertices in the BA networks are
more likely to be diminished than the non-hub vertices, while the shortcuts in
the NW model are not biased. The BA networks tend to have more homogeneous
load distribution than the NW networks. As shown in Fig. 3, the average
capacity of the BA networks decreases slightly slower than that of the NW
networks. The BA networks are a little more functionally robust.
Figure 3: The normalized average capacity
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
under the random failure strategy as a function of the fraction $f$ of removed
edges for the network models. The inset shows the linear relation between
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
and $F(f)$ for the ER model, where $F(f)=k_{\max}(f)\ln[\left\langle
k\right\rangle(1-f)]$. All the data is averaged over $10240$ realizations. The
line in the inset is the linear fit.
After studying the combined effect, we compare the dominance of the
topological breakdown and the functional breakdown. During the simulation, we
measure the occurrence probability $P_{topo}$ of the topological breakdown. If
this probability is larger than 50%, the topological breakdown is dominant,
otherwise the functional breakdown is dominant. As a supplement to the
absolute robustness studied above, the dominance is a sign of the relative
robustness, i.e., which breakdown is less likely to happen than the other. The
result is shown in Fig. 4 as a function of $\rho$. For all the network models,
the topological breakdown is dominant when $\rho$ is small, and the functional
breakdown is dominant when $\rho$ is close to $1$. There is a shift in the
dominance, which corresponds to a particular value of $\rho$. The smaller this
value is, the better relative topological robustness a network model has. As
the figure shows, the NW model has the best relative topological robustness as
well as the best absolute topological robustness, and the BA model has the
best relative functional robustness as well as the best absolute functional
robustness. Though the ER model has both the worst absolute robustnesses, its
relative robustness is intermediate.
Figure 4: The occurrence probability $P_{topo}$ of the topological breakdown
under the random failure strategy as a function of the normalized generation
rate $\rho$ for the network models. The data is averaged over $10240$
realizations. The dashed line is for $P_{topo}=50\%$. The network model that
corresponds to the smallest $\rho$ at the point of intersection between
$P_{topo}$ and this line has the best relative topological robustness.
## IV the attack strategy
In this section, edges are removed in the descending order of their
betweennesses. For the sake of clarity, we first show the results for the ER
model and the BA model, and then for the NW model. We study the same
quantities as in the previous section.
The cumulative distribution function $CDF_{total}$ of the total breakdown was
plotted as a function of $f$ in Fig. 5 for the ER model and the BA model. We
first discuss the topological robustness, which corresponds to $\rho=0$. The
ER networks are very vulnerable to attack, the removal of only $0.5\%$ edges
almost surely destroys the total connectedness. The vulnerability is due to
the lack of loops. With nearly no loops ref1 , the ER networks can be roughly
seen as trees, i.e., there is only one self-avoiding path connecting a pair of
vertices. The only path is at the same time the shortest path. Many shortest
paths concentrate on the high betweenness edges, which are removed by the
attacker with high priority. The loss of the edges prevents communication
between vertex pairs easily, as there are no alternative paths. On the other
hand, the BA networks are much more robust, they can afford the removal of
nearly half of the edges before almost surely break down. There is a strong
correlation ref8 in BA networks that $C_{B}(e)\sim k_{e}$, where $C_{B}(e)$
is the betweenness of an edge $e$ and $k_{e}$ is the product of the degrees of
the two vertices connected by the edge. This correlation implies that at the
early stage of the attack, when the characteristics of the networks are not
affected too much, only the edges between vertices with large degree are
targeted. The removal of these edges generally does not destroy the total
connectedness and the BA networks can be reduced up to tree-like. The
topological robustness of the ER model and the BA model has been studied by
examining the size of the largest component in Ref. ref8 . The authors focused
on the effects of various attack strategies. Here we examine the total
connectedness and provide explanations for the effects of the specific
strategy. Moreover, the topological robustness is compared with the functional
robustness. As shown in the figure, plots corresponding to empty load and high
load almost overlap for both of the network models, indicating that the
functional breakdown hardly occurs. We then investigate the average network
capacity.
Figure 5: The cumulative distribution function $CDF_{total}$ of the total
breakdown under the attack strategy as a function of the fraction $f$ of
removed edges for (a) the ER model and (b) the BA model. All the data is
averaged over $10240$ realizations. The plots corresponding to empty load and
high load almost overlap for both of the network models.
The normalized average capacity
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
was plotted as a function of $f$ in Fig. 6 for the ER model and the BA model.
When the edge with the highest betweenness is attacked, the shortest paths
that originally pass through the edge take detours. The highest edge
betweenness is dispersed and the network capacity is increased. This is the
case for both of the network models. The capacity of the ER networks
monotonically increases during the whole process, indicating that the networks
are free of the functional breakdown. The capacity of the BA networks keeps
increasing until a certain value of $f$, after which the shortest paths revert
to collect on the remaining edges and the capacity is decreased. As shown in
the figure, the capacity is boosted up to nearly 8 times as the initial value
at $f=0.35$ and then drops. We observed in the simulation that only in few
realizations does the capacity drop below the initial value, thus the BA
networks are almost free of the functional breakdown. Note that the growth and
decay of the capacity was studied in Ref. ref11 with the purpose of finding
the optimum transmission efficiency. At each time step, the author chose to
remove the edge with the largest weight, which is the product of the
betweennesses of the two vertices an edge connects, and intentionally avoided
the disintegration of network. Though there are some variations in the models,
our result is qualitatively no different.
Figure 6: The normalized average capacity
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
under the attack strategy as a function of the fraction $f$ of removed edges
for (a) the ER model and (b) the BA model. All the data is averaged over
$10240$ realizations.
An interesting point worthy of note is that the attack strategy does not
fulfill its name for the BA model. By comparing Fig. 5(b) with Fig. 1(b), we
find that the BA model is more robust under the attack strategy than under the
random failure strategy for any value of $\rho$. This result coincides with
the moderated vulnerability of networks in terms of edge ref13 and is in
contrast to the higher vulnerability of the BA model under vertex attack than
error when the existence of the giant component is used as the criterion of
breakdown ref5 .
The cumulative distribution function $CDF_{total}$ of the total breakdown and
the normalized average capacity
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
are shown as functions of $f$ for the NW model in Fig. 7. The figure can be
interpreted as follows: Starting from a complete NW network, the attacker
first targets the shortcuts, because they collect the most shortest paths.
After all the shortcuts are removed, the rest of the network is a two
dimensional $L\times L$ periodic lattice with each vertex connected to its
neighbors and next neighbors. The structure becomes completely homogeneous and
the network capacity is boosted. Thus we see the first peak of the network
capacity at $f=f_{1}=2p/\langle k\rangle=0.005$ in Fig. 7(b). The attacker
continues by randomly targeting an edge. We show a diagram in Fig. 8(a) to
help the interpretation. The lattice is put on a $x$-$y$ plane and Fig. 8(a)
shows a slice along the $x$ direction. For every edge in the figure, there are
$L-1$ parallel counterparts in the $y$ direction. The $L$ edges form a group
that if any edge in the group is removed, the others are removed in the next
few steps as well, because shortest paths that originally pass through the
whole group concentrate on the remaining members, raising the betweennesses of
them. Suppose that edge $e_{1}$ in the figure is removed, the network capacity
drops until all the counterparts of the edge are removed. The heterogeneity of
the raised betweenness is alleviated and we see the second peak of the network
capacity, which is $L$ edges away from the first peak. However, the capacity
is not fully restored, as edge $e_{2}$, $e_{3}$ and their counterparts collect
larger number of shortest paths than before. Thus two peaks follow, each at an
interval of $L$ edges. With the $3L$ edges gone, the lattice is no longer
periodic and the center area in the $x$ direction collects the highest
betweenness. Then the following attack removes another $3L$ edges in this area
and the network is split. For the most of the model realizations, six peaks of
the network capacity are observed, which conforms to the arguments above.
There is a special case that allows the observation of one more peak. As
depicted in Fig. 8(b), it is possible that a shortcut $e_{4}$ is placed
between vertex $a$ and $b$. This kind of shortcut spans less lattice distance
than the abundant next-neighbor links, thus collects less shortest paths and
is removed very late in the attack. The existence of such shortcut brings
higher betweenness to a group of $L$ edges in the $y$ direction, e.g., all the
shortest paths passing through $a$ and $c$ collect on the edge $e_{6}$ in the
figure. Removing this group of edges induces the seventh peak. The seven peaks
are pointed out in Fig. 7(b) by arrows. We see in the figure that the network
capacities on the peaks are larger than the initial value, thus only when the
capacities in the valleys drop below the initial value does the network
functionally break down, i.e., $CDF_{total}$ shown in Fig. 7(a) resembles
stairs for $\rho\neq 0$. When $\rho=0.95$, nearly $90\%$ model realizations
functionally break down before $f=0.005$. The situation is only a little
better than the ER model and much worse than the BA model. For the topological
case $\rho=0$, there is a plateau in the figure which corresponds to the
emergence of the seventh peak. We estimate the emergence probability as
${pN\choose 1}P_{sc}\left(1-P_{sc}\right)^{pN-1}\approx 7.3\%$, where
$P_{sc}=\frac{4}{N-8}$ is the probability of the occurrence of the special
shortcut and we only consider the first order of this probability. The result
is supported by Fig. 7 that the topological breakdown takes place just before
the seventh peak of the network capacity, at $f=f_{7}=f_{1}+6L/(\langle
k\rangle N/2)=0.0519$, for about $92.7\%$ model realizations. The topological
breakdown for the rest of the model realizations takes place after $L$ more
edges are removed. The NW model has intermediate topological robustness
between the ER model and the BA model.
Figure 7: The simulation results of the attack strategy for the NW model: (a)
The cumulative distribution function $CDF_{total}$ of the total breakdown as a
function of the fraction $f$ of removed edges for different values of the
normalized generation rate $\rho$; (b) The normalized average capacity
$\left\langle\rho_{c}\right\rangle_{f}/\left\langle\rho_{c}\right\rangle_{0}$
as a function of $f$. All the data is averaged over $10240$ realizations. The
arrows point out the seven peaks of the network capacity. The dashed lines are
guides to the eye: the topological breakdown ($\rho=0$) takes place just
before the seventh peak for about $92.7\%$ model realizations. Figure 8: A NW
network after all the shortcuts are removed, which is a two dimensional
periodic lattice with coordination number $8$, is put on a $x$-$y$ plane. Both
the arcs and lines represent edges, and the circles represent vertices. (a) A
slice of the lattice in the $x$ direction is shown. Every edge in the complete
lattice has equal betweenness. If $e_{1}$ is removed, the shortest paths going
from the left-hand side to the right-hand side have fewer choices of edge than
before. Thus $e_{2}$ and $e_{3}$ collect larger betweenness than the others
and are the immediate targets in the following attack. Note that $e_{1}$,
$e_{2}$ and $e_{3}$ each represent a group of $L$ edges in the $y$ direction.
(b) The edge $e_{4}$ is such a special shortcut that it spans less lattice
distance than the next-neighbor links and collects fewer shortest paths.
Though the betweenness of the shortcut is low, the betweennesses of the
neighboring edges are increased, e.g., the shortest paths connecting $a$, $c$
and $a$, $d$ both pass through $b$, the edges $e_{5}$ and $e_{6}$ are biased.
Different from $e_{6}$, the edge $e_{5}$ is in the $y$ direction. In addition
to the six peaks in the $x$ direction, one more peak of the network capacity
could be seen. In fact the effect of the shortcut $e_{4}$ is weak that the
counterpart of $e_{5}$ which connects to $a$ in the $y$ direction is not
removed before the lattice is split. For clarity, some next-neighbor links are
omitted.
The relative robustness of the network models for the attack strategy is
studied in the same way as in the previous section for the random failure
strategy. The result is shown in Fig. 9. The ER model has the worst absolute
topological robustness, and also the worst relative topological robustness, as
for any value of $\rho$, no functional breakdown takes place. While the BA
model has the best absolute topological robustness, the relative topological
robustness is still bad, nearly the same as the ER model. For the NW model, we
know that the topological breakdown only takes place when at least $6L$ edges
are removed, but the functional breakdown could happen if network capacity
drops below the initial value. The figure shows that for $\rho>0.2$, the
topological breakdown hardly takes place. Thus the NW model has the best
relative topological robustness.
Figure 9: The occurrence probability $P_{topo}$ of the topological breakdown
under the attack strategy as a function of the normalized generation rate
$\rho$ for the network models. The data is averaged over $10240$ realizations.
The dashed line is for $P_{topo}=50\%$. The network model that corresponds to
the smallest $\rho$ at the point of intersection between $P_{topo}$ and this
line has the best relative topological robustness.
## V conclusion
In this paper, we have studied the topological robustness and the functional
robustness of several network models under two strategies of edge removal,
using the total connectedness as the measure. For each removing strategy, we
have examined the combined effect and the relative importance of the two types
of robustness. Through the study of the combined effect, we have found out the
network topology which is the most robust in a specific environment, e.g., the
NW model is the most robust in the environment of the random edge removal,
while the BA model is the most robust in the environment of the edge attack.
Through the study of the relative importance, we have known with evidence how
to strengthen a network efficiently, e.g., as the NW model has the best
relative topological robustness, improvements on the functional robustness
should be emphasized, such as increasing the capacity of vertices; the BA
model has bad relative topological robustness under the edge attack, thus
enhancements on the connectivity such as building redundant links are
appropriate. These results can have applications in designing and optimizing
artificial networks, such as implementing a robust P2P network where a
connection between peers has a constant probability to fail. There are also
some extensions for further studies. The traffic dynamics in our model is far
from realistic, more elements abstracted from real-world traffic can be
incorporated. Moreover, networks are not limited to carrying traffic. It is an
open question how the interaction between the topologies and different
dynamics influences the network robustness.
###### Acknowledgements.
The work was supported by the National Natural Science Foundation of China
under Grant No. 10334020.
## References
* (1) R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
* (2) M. E. J. Newman, SIAM Rev. 45, 167 (2003).
* (3) A.-L. Barabási and R. Albert, Science 286, 509 (1999).
* (4) D.J. Watts and S.H. Strogatz, Nature (London) 393, 440 (1998).
* (5) M. E. J. Newman and D. J. Watts, Phys. Rev. E 60, 7332 (1999).
* (6) R. Albert, H. Jeong, and A.-L. Barabási, Nature (London) 406, 378 (2000).
* (7) R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000).
* (8) R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 86, 3682 (2001).
* (9) D.S. Callaway, M.E.J. Newman, S.H. Strogatz, and D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000).
* (10) P. Holme, B. Kim, C. Yoon, and S. Han, Phys. Rev. E 65, 056109 (2002).
* (11) A. E. Motter and Y.-C. Lai, Phys. Rev. E 66, 065102 (2002).
* (12) J.-L. Guillame, M. Latapy, and C. Magnien, Lect. Notes Comput. Sci. 3544, 186 C196 (2005).
* (13) J. Duch and A. Aranas, Proc. SPIE 6601, 66010O (2007).
* (14) R. Guimerà, A. Díaz-Guilera, F. Vega-Redondo, A. Cabrales, and A. Arenas, Phys. Rev. Lett. 89, 248701 (2002).
* (15) P. Crucitti, V. Latora, M. Marchiori, A. Rapisarda, Physica A 320, 622 (2003).
* (16) E. López, R. Parshani, R. Cohen, S. Carmi, and S. Havlin, Phys. Rev. Lett. 99, 188701 (2007).
* (17) J.-H. Kim, K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 058701 (2003).
* (18) G.-Q. Zhang, D. Wang, G.-J. Li, Phys. Rev. E 76, 017101 (2007).
* (19) A. E. Motter, T.Nishikawa, and Y. C. Lai, Phys. Rev. E 66, 065013 (2002).
* (20) S. Martin, R. D. Carr, and J.-L. Faulon, Physica A 371, 870 (2006).
* (21) P. Holme, Phys. Rev. E 66, 036119 (2002).
* (22) M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002).
* (23) U. Brandes, J. Math. Sociol. 25, 163 (2001).
* (24) M. Barthélemy, Eur. Phys. J. B 38, 163 (2003).
* (25) K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 87, 278701 (2001).
* (26) M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).
* (27) M. E. J. Newman, Phys. Rev. E 76, 045101 (2007).
* (28) M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002).
* (29) M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, 026118 (2001).
* (30) R. V. Solé and S. Valverde, Physica A 289, 595 (2001).
|
arxiv-papers
| 2008-09-16T02:19:49
|
2024-09-04T02:48:57.791948
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shan He, Sheng Li, Hongru Ma",
"submitter": "Sheng Li",
"url": "https://arxiv.org/abs/0809.2631"
}
|
0809.2683
|
# Apply current exponential de Finetti theorem to realistic quantum key
distribution
Yi-Bo Zhao zhaoyibo@mail.ustc.edu.cn Key Lab of Quantum Information,
University of Science and Technology of China, (CAS), Hefei, Anhui 230026,
China Zheng-Fu Han Key Lab of Quantum Information, University of Science and
Technology of China, (CAS), Hefei, Anhui 230026, China Guang-Can Guo Key Lab
of Quantum Information, University of Science and Technology of China, (CAS),
Hefei, Anhui 230026, China
###### Abstract
In the realistic quantum key distribution (QKD), Alice and Bob respectively
get a quantum state from an unknown channel, whose dimension may be unknown.
However, while discussing the security, sometime we need to know exact
dimension, since current exponential de Finetti theorem, crucial to the
information-theoretical security proof, is deeply related with the dimension
and can only be applied to finite dimensional case. Here we address this
problem in detail. We show that if POVM elements corresponding to Alice and
Bob’s measured results can be well described in a finite dimensional subspace
with sufficiently small error, then dimensions of Alice and Bob’s states can
be almost regarded as finite. Since the security is well defined by the smooth
entropy, which is continuous with the density matrix, the small error of state
actually means small change of security. Then the security of unknown-
dimensional system can be solved. Finally we prove that for heterodyne
detection continuous variable QKD and differential phase shift QKD, the
collective attack is optimal under the infinite key size case.
###### pacs:
03.67.Dd,03.67.Hk
## I Introduction:
Information-theoretical security proof Renner thesis is a powerful and
general way to prove the security for quantum key distribution (QKD). In this
method, to give the amount of unconditional secret keys, we only need to
discuss upper or lower bounds of some entropies. The exponential de Finetti
theorem is crucial to this method, which support that as the key size goes to
infinite, Eve cannot get more information from the coherent attack than from
the collective attack Renner thesis . Since in the collective attack Eve
attacks each signal independently with the same method, it is much easy for us
to discuss the security. However, current exponential de Finetti theorem
relying on the dimension and even diverges if the dimension is infinite, while
in practice the dimension is often unknown or infinite.
There are also some other kind of quantum de Finetti theorems. In Ref.
Matthias ; Matthias2 ; finite de fi , several de Finetti theorems for
different conditions are given. These de Finetti theorems can be independent
with the dimension. Even under the infinite dimensional case, they still
converge. However, These de Finetti theorems are polynomial and not
exponential. As the key size goes to infinite, they can not exponentially
converge to zero. Whether such polynomial de Finetti theorems can be applied
to QKD requires further discussion.
We can think about a more general case. Alice and Bob respectively get a
quantum state from a channel and do measurement and thus hold classical data
finally. Realistically, they only know the classical data and do not know
anything about the dimension of quantum state beforehand. Therefore, it is not
realistic for us to assume the dimension before discussing the security. If
the dimension of quantum state is unknown, current exponential de Finetti
theorem may not be directly applied and the security against the most general
attack is difficult to given by the information-theoretical method. In Ref.
Renner nature , Renner gave some concrete examples to show the de Finetti
theorem. From these examples we can see that if the dimension of individual
quantum state is higher than the block size, the whole state may be far away
from an almost i.i.d. state. For some QKDs, the dimension problem can be
solved by introducing the squashing model squashingmode ; squashingmode1 .
However for some other protocols we do not know whether there exist a
squashing model, i.e. continuous variable (CV) QKD binaryCVQKD ; zhaoc and
differential phase shift (DPS) QKD DPSQKD . Then, it is necessary for Alice
and Bob to get some information about their dimensions.
Here we give a general way to estimate the effective dimension (In the
following we will see that Alice and Bob’s measurement data are obtained
almost only from a finite dimensional subspace. Here we call the dimension of
this subspace as effective dimension.) of a system, and a general method to
apply current information theoretical security proof to practical QKD. Finally
we prove that if POVM elements corresponding to Alice and Bob’s measured
results can be well described in a finite dimensional subspace with
sufficiently small error, the security of unknown dimensional system is very
close to that of a finite dimensional system, where Alice and Bob put finite
dimensional filters before their detectors. The security of this finite
dimensional system is covered by current information theoretical security
proof method. Then the security of unknown-dimensional system can be solved.
Our solution is based on the estimation of the effective dimension of a
system. In Ref. min dimen , Wehner et al. gave an estimation to the lower
bound of the dimension of a system. We hope future works can shrink the gap
between these two results. Up to now, some efforts have been done for the
finite key size case ZhaoIEEE ; scar . The security under finite key size case
may be much different from that under infinite key size case. To give a better
result for finite key size case, it is necessary to give a tight estimation to
the effective dimension.
We may think that the world is always finite, so regard it as guaranteed that
current exponential de Finetti theorem can be directly applied to practical
system. It is not necessary the case. Firstly, finite measurement result does
not always mean finite dimensional quantum state. A finite measurement result
can also be generated from an infinite quantum state. Secondly, to know the
upper bound of the dimension of quantum state is required if we consider the
finite key size case. From the Ref. Renner thesis we know that the amount of
secret key rate under the finite key size case is deeply related with the
dimension. Our estimation of effective dimension is expected to be favorable
to finite key size situation.
We noted two parallel works shown in Ref. de Finetti ; lev . In these two
works, the unconditional security of CVQKD is addressed. In Ref. de Finetti ,
Renner et al. modified previous exponential de Finetti theorem and this new
theorem can be directly applied to CVQKD. From this new de Finetti theorem we
can see that in CVQKD if the variance of Bob’s measurement result is finite,
the state Alice, Bob and Eve share can still be approximated by an almost
i.i.d. state. Our result only works for heterodyne CVQKD and requires the
maximum value of Alice and Bob’s heterodyne detection to be finite. Under the
infinite key size case, our result can give the same approximation that the
state describes the whole infinite communications can be approximated by an
almost product state with arbitrarily small error. In Ref. lev , Leverrier et
al. directly addressed the unconditional security of CVQKD without the de
Finetti theorem. Their work is based on the Gaussian optimality. In this paper
we approximate the CVQKD by a finite dimension protocol. The security of
finite dimension protocol can be covered by current information theoretical
security proof. Then the unconditional security of CVQKD is possible to prove.
Compared with these two works, one advantage of our work is its application to
photon number detection protocols, e.g. another coherent state protocol,
DPSQKD. In the following, we will demonstrate how to apply our result to
DPSQKD.
The basic idea of our approach is as following. Although the dimension of
quantum state Alice, Bob and Eve initially share is totally unknown, after
obtaining measurement results, Alice and Bob collapse Eve’s state into a less
complex state and can know some information about the effective dimension of
their state. Then we can construct another finite dimensional protocol, where
Alice and Bob put a finite dimensional filter right before their detection
equipments that can filter out high dimensional components. We prove that
final state of this new finite dimensional protocol is only slightly different
from the original one. Then the security of that unknown-dimensional protocol
can be approximated by this new protocol. The security of this new finite
dimensional protocol is covered by Ref. Renner thesis , then the security of
that unknown-dimensional protocol can be solved.
In the following we will introduce a general QKD protocol at first and then
discuss unknown-dimensional problem. Latter we will introduce a finite
dimensional protocol and prove that if components of POVM elements
corresponding to Alice and Bob’s measured results on high dimensional bases
are small enough then the security of original unknown-dimensional protocol
can be well approximated by this finite dimensional protocol. While discussing
their difference, we will introduce an entanglement version measurement to
describe Alice and Bob’s detection. Finally some application examples will be
given. In application examples, we will only discuss the infinite key size
case, while our result is also useful under the finite key size case.
## II Protocol:
Here we limit our analysis to the following protocol.
Alice and Bob take $N$ quantum states from a channel respectively. Then they
permute their subsystem according to a commonly chosen random permutation.
They separate $N$ states into $l$ blocks and perform POVM measurement to each
state. Without loss of generality, we assume Alice and Bob respectively hold
several POVMs, $M^{Ai}=\\{M_{x^{i}}^{Ai}\\}$ and $M^{Bi}=\\{M_{y^{i}}^{Bi}\\}$
($i=1,...,l$), where $x^{i}$ and $y^{i}$ denote corresponding measurement
results, and they perform the POVM $M^{Ai}=\\{M_{x^{i}}^{Ai}\\}$ and
$M^{Bi}=\\{M_{y^{i}}^{Bi}\\}$ to the $i$-th blocks (we assume the choice of
POVMs is publicly known). Then they publish measurement results from the first
block to estimate the channel. Before the classical procedure they estimate
the dimension of their quantum state according to the region of their
measurement results. Then they give up partial of their measurement results
(required by the information-theoretical security proof Renner thesis ) and
finally obtain classical strings. After performing data processing,
information reconciliation and privacy amplification, they finally generate
secret keys. Here, we allow Alice and Bob to hold several POVMs, mainly
because in many QKD protocols, Alice and Bob need to randomly change their
measurement bases. One POVM corresponds to one choice of bases.
From current de Finetti theorem we know that if the dimension of the channel
is finite, the state Alice, Bob and Eve share after many communications is
close to an almost product state. It has been shown that such almost product
state almost has the same property with the product state. The product state
corresponds to the collective attack. Then we only need to consider collective
attack Renner thesis . However, if the dimension is infinite, the de Finetti
theorem may diverge. Then we cannot know the difference between collective
attack and coherent attack.
We assume after getting quantum state, Alice, Bob and Eve share the state
$\rho_{A^{N}B^{N}E^{N}}$. Since discarding subsystem never increases mutual
information, we can safely assume that Eve holds the purification of
$\rho_{A^{N}B^{N}E^{N}}$, so that $\rho_{A^{N}B^{N}E^{N}}$ is pure Neilson ;
zhaoc ; Renner thesis . After measuring all $N$ states, Alice and Bob know the
region of their measurement results. For example, in DPSQKD DPSQKD , if they
use photon number resolving detector, they can know the maximum photon number
they received from one pulse. In CVQKD binaryCVQKD with heterodyne detection,
they can know the maximum amplitude they get. Here, we will show that such
information is enough for Alice and Bob to know whether their system can be
approximated by a finite dimensional system.
Alice and Bob can make an initial estimation to their state according to
measurement results. After Alice and Bob knows the region of their measurement
results, they can only consider such $\rho_{A^{N}B^{N}E^{N}}$ that can
generate their measured results with probability higher than certain small
parameter $\varepsilon$. Then the collection of states they need to consider
is largely reduced. The insecure probability introduced by such method is no
larger than $\varepsilon$ and the strength of security will be reduced by
$\varepsilon$ e-secure . This procedure is required by our proof.
More precisely, we assume Alice and Bob’s measurement results from a single
state of $i$-th block belong to the region $\Xi(X^{i})$ and $\Xi(Y^{i})$
respectively. We let
$D^{Ai}=\sum_{x^{i}\in\Xi(X^{i})}M_{x^{i}}^{Ai}$
$D^{Bi}=\sum_{y^{i}\in\Xi(Y^{i})}M_{y^{i}}^{Bi}$
Then $D^{Ai}$ and $D^{Bi}$ actually are POVM elements that correspond to Alice
and Bob’s measurement results belonging to the region $\Xi(X^{i})$ and
$\Xi(Y^{i})$ respectively. $D^{Ai}$ ($D^{Bi}$) may be different for different
blocks. To avoid distinguishing different $D^{Ai}$s ($D^{Bi}$s), here we
define POVM elements, $\tilde{D}^{A}$ and $\tilde{D}^{B}$ satisfying that for
arbitrary state $\rho$ and $i$, we always have
$\begin{array}[]{c}tr(\tilde{D}^{A}\rho)\geq tr(D^{Ai}\rho)\\\
tr(\tilde{D}^{B}\rho)\geq tr(D^{Bi}\rho)\end{array}$ (1)
To know the requirement given in Eq. (1) well, we can see some examples. It
can be seen that $\tilde{D}^{A}=I$ is one trivial element that always satisfy
Eq. (1). Also, if all $D^{Ai}$s are just the same, $\tilde{D}^{A}=D^{Ai}$ is
the one satisfying Eq. (1). Furthermore, since for any $i$ and arbitrary
$\rho$, the expectation value of $\tilde{D}^{A}-D^{Ai}$ and
$\tilde{D}^{B}-D^{Bi}$ are non-negative, $\tilde{D}^{A}-D^{Ai}$ and
$\tilde{D}^{B}-D^{Bi}$ are non-negative operators. Therefore, if
$\tilde{D}^{A}-D^{Ai}$ and $\tilde{D}^{B}-D^{Bi}$ are not zero, they are also
valid POVM elements. Then $\\{D^{Ai},\tilde{D}^{A}-D^{Ai},I-\tilde{D}^{A}\\}$
and $\\{D^{Bi},\tilde{D}^{B}-D^{Bi},I-\tilde{D}^{B}\\}$ constitute a POVM
respectively (It should be noted that $I$ may be an operation of an infinite
dimensional space.). Here we define the POVM elements $\tilde{D}^{A}$ and
$\tilde{D}^{B}$ mainly because the maximum value of measurement result of
different blocks may be different and then $D^{Ai}$s are not the same.
Nevertheless, for most of current protocols, it is not difficult to find a
tight $\tilde{D}^{A}$ and $\tilde{D}^{B}$. For example, in the heterodyne
detection CVQKD, Alice and Bob do not change the basis, so there are only two
blocks, one used for parameter estimation, one used to generate secret keys.
We assume at Alice’s side the maximum value of one block is $V_{A}^{\max 1}$,
and that of the other is $V_{A}^{\max 2}$. Then $D^{A1}=\sum_{x^{i}\leq
V_{A}^{\max 1}}M_{x^{i}}^{A1}$ and $D^{A2}=\sum_{x^{i}\leq V_{A}^{\max
2}}M_{x^{i}}^{A2}$. Since Alice uses the same POVM for these two blocks, we
have $M_{x^{i}}^{A1}=M_{x^{i}}^{A2}$. If $V_{A}^{\max 1}>V_{A}^{\max 2}$, we
can choose $\tilde{D}^{A}=\sum_{x^{i}\leq V_{A}^{\max 1}}M_{x^{i}}^{A1}$,
which satisfies Eq. (1). Then $\tilde{D}^{A}-$ $D^{A1}=0$ and $\tilde{D}^{A}-$
$D^{A2}=\sum_{V_{A}^{\max 2}<x^{i}\leq V_{A}^{\max 1}}M_{x^{i}}^{A1}$, which
is a POVM element. In the DPSQKD, Bob also does not change his bases, so a
similar result can be obtained.
To analysis the security we can only consider the state
$\rho_{A^{N}B^{N}E^{N}}$ that satisfies
$tr[(\tilde{D}^{A}\tilde{D}^{B})^{\otimes
N}\rho_{A^{N}B^{N}E^{N}}]\geq\varepsilon$ (2)
while the final strength of security will be reduced by $\varepsilon$, where
$\tilde{D}^{A}$ and $\tilde{D}^{B}$ are properly chosen elements that satisfy
Eq. (1). Then the collection of $\rho_{A^{N}B^{N}E^{N}}$ we need to consider
is largely reduced. It can be seen that to shrink the collection of
$\rho_{A^{N}B^{N}E^{N}}$, we need to find tight $\tilde{D}^{A}$ and
$\tilde{D}^{B}$. In the following, we will see that this technique is required
by our argument.
After Alice and Bob’s measurement, the state Alice, Bob and Eve hold becomes
$\rho_{X^{N}Y^{N}E^{N}}$, where $X$ and $Y$ are classical variable that can
take the value $x^{i}$ and $y^{i}$ and can be expressed by orthogonal quantum
state Renner thesis . Actually, the security of QKD system directly related to
the state $\rho_{X^{N}Y^{N}E^{N}}$, rather than original state
$\rho_{A^{N}B^{N}E^{N}}$. Therefore, if we can find a finite dimensional
system that generates another state $\tilde{\rho}_{X^{N}Y^{N}E^{N}}$ very
close to $\rho_{X^{N}Y^{N}E^{N}}$, then the security of the original unknown
system can be approximated by this finite dimensional system.
Now we can compare two schemes as illustrated in Fig. 1. One is the original
unknown-dimensional scheme, and the other is a modified scheme, in which Alice
and Bob respectively put filters before their detectors. We assume these two
filters can totally filter out high dimensional component of received state
and dimensions of output states of these two filters are $d_{A}$ and $d_{B}$
respectively. For convenience, we will call the original protocol as protocol
1 and the modified one as protocol 2. Then in the protocol 2 dimensions of
Alice and Bob’s received states are $d_{A}$ and $d_{B}$ respectively. In the
following, we will see that if we properly set the filter and choose high
enough $d_{A}$ and $d_{B}$, then the security of protocol 1 can be
approximated by that of protocol 2.
Figure 1: Illustration of protocol 1 and protocol 2, where in protocol 2
finite dimensional filters are put before the detectors. If the filters are
properly chosen, the security of protocol 1 can be well approximated by that
of protocol 2, while the security of protocol 2 is covered by current
information theoretical security proof method. The exact difference between
protocol 1 and protocol 2 becomes significant while we consider the the finite
key size case.
To simplify our discussion, it is necessary to avoid distinguishing different
blocks. We know that $\tilde{D}^{A}-D^{Ai}$ and $\tilde{D}^{B}-D^{Bi}$ are
also POVM elements. Here we introduce other two classical data $x^{\prime i}$
and $y^{\prime i}$ that correspond to POVM elements $\tilde{D}^{A}-D^{Ai}$ and
$\tilde{D}^{B}-D^{Bi}$ respectively. Then in protocol 1 Alice and Bob’s
measurement results of $i$-th block are within the region
$\Xi(X^{i})\cup\\{x^{\prime i}\\}$ and $\Xi(Y^{i})\cup\\{y^{\prime i}\\}$
respectively. Therefore, this protocol does not change if it runs as follows.
While getting measurement results from a state of $i$-th block, Alice and Bob
accept them only when they belong to region $\Xi(X^{i})\cup\\{x^{\prime i}\\}$
and $\Xi(Y^{i})\cup\\{y^{\prime i}\\}$ respectively. Otherwise, they discard
them. Now we can calculate the difference between the protocol 1 and the
protocol 2.
## III Estimation of $L_{1}$-distance based on observations:
Before calculating the difference, here we introduce a entanglement version
measurement. There are several interpretations for the quantum measurement,
e.g. von Neumann measurement scheme and Many-worlds interpretation quantum
measure . Here we are not to give a new philosophical interpretation, but to
construct a physical model that can effectively perform POVM measurement. This
physical model allows us to easily find the difference between protocol 1 and
protocol 2. For briefness, here we only take Alice’s measurement as an
example. Alice’s POVM measurement can be performed by the equipment shown in
Fig. 2. The measurement procedure is realized by an interaction among her
received state, detector and the environment. After the interaction, Alice
gives up received state and the environment and thus only holds the detector,
which directly gives her the classical data. After reading out the classical
data, Alice set the detector and environment to the initial state to do the
next measurement. In this model we require initial states of Alice’s detector
and environment are pure and respectively to be $|de\rangle_{A}$ and
$|$Env$\rangle_{A}$. For convenience, here we let $|ini\rangle_{A}$ denote
$|de\rangle_{A}|$Env$\rangle_{A}$. Then the interaction among the received
state, detector and environment for $i$-th block can be given by
$U_{A}^{i}=\sum_{x^{i}}|x^{i}\rangle|Q_{x^{i}}\rangle_{A}\langle
ini|\sqrt{M_{x^{i}}^{Ai}}$ (3)
where $|x^{i}\rangle$s are orthogonal states of detector, $|Q_{x^{i}}\rangle$
describes orthogonal state of the environment, $\langle ini|$ denotes the
initial pure state of Alice’s detector and environment and
$\sqrt{M_{x^{i}}^{Ai}}$ is the POVM operators corresponding to POVM element
$M_{x^{i}}^{Ai}$ Neilson . To check the validity of this measurement, we can
apply it to a two parties system $\rho_{AB}$. After the interaction described
by $U_{A}^{i}$, the state of whole system becomes
$\displaystyle\rho_{ABXQ_{X}}$ $\displaystyle=$ $\displaystyle
U_{A}^{i}\rho_{AB}\otimes|ini\rangle_{A}\langle ini|U_{A}^{i+}$
$\displaystyle=$
$\displaystyle\sum_{x^{i}}|x^{i}\rangle|Q_{x^{i}}\rangle\sqrt{M_{x^{i}}^{Ai+}}\rho_{AB}\sum_{x^{j}}\langle
x^{j}|\langle Q_{x^{j}}|\sqrt{M_{x^{j}}^{Aj}}$
where $Q_{X}$ denotes the environment and all of $|Q_{x^{i}}\rangle$s are
orthogonal with each other. After we trace out the system A and environment we
immediately obtain the state
$\rho_{XB}=\sum_{x^{i}}|x^{i}\rangle\langle x^{i}|\otimes
tr_{A}(\sqrt{M_{x^{i}}^{Ai+}}\rho_{AB}\sqrt{M_{x^{i}}^{Ai}})$
We see that
$tr_{A}(\sqrt{M_{x^{i}}^{Ai+}}\rho_{AB}\sqrt{M_{x^{i}}^{Ai}})=P(x^{i})\rho_{B}^{x^{i}}$
where $P(x^{i})$ is the probability of the out come $x^{i}$ and
$\rho_{B}^{x^{i}}$ denotes Bob’s conditional state while Alice’s measurement
result is $x^{i}$. Then $\rho_{XB}$ becomes
$\rho_{XB}=\sum_{x^{i}}P(x^{i})|x^{i}\rangle\langle
x^{i}|\otimes\rho_{B}^{x^{i}}$
which consists with the POVM measurement. Since Alice only accept the data
within the collection $\Xi(X^{i})\cup\\{x^{\prime i}\\}$, we can reduce the
unitary transformation given in Eq. 3 to a general quantum operation
$\hat{O}_{A}^{i}$ to describe Alice’s effective measurement, which is given by
$\hat{O}_{A}^{i}=\sum_{x^{i}\in\Xi(X^{i})\cup\\{x^{\prime
i}\\}}|x^{i}\rangle|Q_{x^{i}}\rangle_{A}\langle ini|\sqrt{M_{x^{i}}^{Ai}}$ (4)
Here we can see that $\hat{O}_{A}^{i}$ may not be a unitary transformation.
The quantum operation that describes Alice’s total $N$ detection is
$\hat{O}_{A^{N}}=\bigotimes\limits_{i=1}^{l}(\hat{O}_{A}^{i})^{\otimes n_{i}}$
(5)
where $n_{i}$ denotes the length of $i$-th block. By the same way we can give
the operator describing Bob’s whole $N$ detections (By substituting the
notation $A$ by $B$.).
Figure 2: Illustration of entanglement version measurement, where
$|$Env$\rangle$ and $|de\rangle$ respectively denote the initial state of
environment and the detector. The measurement can be realized by the unitary
operation among received state, detector and the environment. After the
operation, the received state and the environment are given up and the
measurement result can be directly given by the state of detector, denoted by
$\rho_{X}$.
Since we only need to consider the case that $\rho_{A^{N}B^{N}E^{N}}$ is pure,
here we assume the initial state Alice and Bob receive is
$|\Psi_{A^{N}B^{N}E^{N}}\rangle$. The initial state of Alice and Bob’s
detector and environment is
$|\Psi_{X^{N}Y^{N}Q_{X}^{N}Q_{Y}^{N}}\rangle=|ini\rangle_{A}|ini\rangle_{B}$.
Then for the protocol-1, after Alice and Bob’s measurement (general quantum
operation Neilson ) the state describing Alice, Bob, Eve, detectors and
environment becomes
$\displaystyle|\Psi_{P-1}\rangle$ $\displaystyle=$
$\displaystyle|\Psi_{X^{N}Y^{N}A^{N}B^{N}Q_{X}^{N}Q_{Y}^{N}E^{N}}\rangle$ (6)
$\displaystyle=$
$\displaystyle\frac{\hat{O}_{A^{N}}\hat{O}_{B^{N}}|\Psi_{A^{N}B^{N}E^{N}}\rangle|\Psi_{X^{N}Y^{N}Q_{X}^{N}Q_{Y}^{N}}\rangle}{\sqrt{\langle\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}\rangle}}$
where $X$ and $Y$ denotes Alice and Bob’s detectors, $Q_{X}$ and $Q_{Y}$
denote the environment around Alice and Bob, $\hat{O}_{B^{N}}$ is the operator
describing Bob’s whole $N$ detections and
$\langle\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}\rangle$
describes expectation value of
$\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}$.
In protocol-2 Alice and Bob respectively put a $d_{A}$ and $d_{B}$ dimensional
filter before their detectors. The filter can be described by a projection
into a subspace. Here we let the projector $P_{A}^{d_{A}}$ and $P_{B}^{d_{B}}$
denotes Alice and Bob’s filters. For convenience we let $P_{AB}^{d_{A}d_{B}}$
denotes $P_{A}^{d_{A}}\otimes P_{B}^{d_{B}}$. Then for the protocol-2 after
Alice and Bob’s measurement the whole state becomes
$\displaystyle|\Psi_{P-2}\rangle=|\tilde{\Psi}_{X^{N}Y^{N}A^{N}B^{N}Q_{X}^{N}Q_{Y}^{N}E^{N}}\rangle$
(7)
$\displaystyle=\frac{\hat{O}_{A^{N}}\hat{O}_{B^{N}}(P_{AB}^{d_{A}d_{B}})^{\otimes
N}|\Psi_{A^{N}B^{N}E^{N}}\rangle|\Psi_{X^{N}Y^{N}Q_{X}^{N}Q_{Y}^{N}}\rangle}{\sqrt{\langle(P_{AB}^{d_{A}d_{B}})^{\otimes
N}\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}(P_{AB}^{d_{A}d_{B}})^{\otimes
N}\rangle}}$
After tracing out received quantum state $A^{N}$ and $B^{N}$, and the
environment $Q^{N}_{X}$ and $Q^{N}_{Y}$ we obtain the state Alice, Bob and Eve
finally hold. For the protocol-1, they finally hold the state
$\rho_{X^{N}Y^{N}E^{N}}$ and for the protocol-2, they finally share
$\tilde{\rho}_{X^{N}Y^{N}E^{N}}$. If we know the $L_{1}$ distance between
$\rho_{X^{N}Y^{N}E^{N}}$ and $\tilde{\rho}_{X^{N}Y^{N}E^{N}}$ we can know the
difference between securities of protocol-1 and protocol-2 Renner thesis .
Since tracing out the subsystem never increases the $L_{1}$ distance Renner
thesis , the $L_{1}$ distance between $\rho_{X^{N}Y^{N}E^{N}}$ and
$\tilde{\rho}_{X^{N}Y^{N}E^{N}}$ is no larger than that between
$|\Psi_{P-1}\rangle$ and $|\Psi_{P-2}\rangle$. We know that
$\displaystyle|\Psi_{A^{N}B^{N}E^{N}}\rangle=$ (8)
$\displaystyle(P_{AB}^{d_{A}d_{B}})^{\otimes
N}|\Psi_{A^{N}B^{N}E^{N}}\rangle+\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}|\Psi_{A^{N}B^{N}E^{N}}\rangle$
where $\overline{P}$ denotes the orthogonal complement space of $P$. By
putting the Eq. (8) into Eq. (6) we quickly know
$|\Psi_{P-1}\rangle=\alpha|\Psi_{P-2}\rangle+\beta|\Psi^{\prime}\rangle$
where
$|\beta|=\frac{\sqrt{\langle\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\rangle}}{\sqrt{\langle\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}\rangle}}.$
(9)
and
$|\Psi^{\prime}\rangle=\frac{\hat{O}_{A^{N}}\hat{O}_{B^{N}}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}|\Psi_{A^{N}B^{N}E^{N}}\rangle|\Psi_{X^{N}Y^{N}Q_{X}^{N}Q_{Y}^{N}}\rangle}{\sqrt{\langle\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\hat{O}_{A^{N}}^{+}\hat{O}_{B^{N}}^{+}\hat{O}_{A^{N}}\hat{O}_{B^{N}}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\rangle}}$
is a state obtained from the complimentary space
$\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$, which may not be orthogonal
with $|\Psi_{P-2}\rangle$. From the Appendix-A of Ref. Renner thesis we know
that the $L_{1}$ distance of two pure state $|\Psi_{1}\rangle$ and
$|\Psi_{2}\rangle$ can be given by
$|||\Psi_{1}\rangle-|\Psi_{2}\rangle||=2\sqrt{1-|\langle\Psi_{1}|\Psi_{2}\rangle|^{2}}$
where $||\cdot||$ denotes the $L_{1}$ distance. Then the $L_{1}$ distance
between $|\Psi_{P-1}\rangle$ and $|\Psi_{P-2}\rangle$ is no larger than
$2|\beta|$, which yields
$||\rho_{X^{N}Y^{N}E^{N}}-\tilde{\rho}_{X^{N}Y^{N}E^{N}}||\leq 2|\beta|$
For convenience here we let $\tilde{D}=\tilde{D}^{A}\tilde{D}^{B}$. If we put
Eqs. (4) and (5) into Eq. (9) and apply the fact that operators of different
detectors are commutate, we can know that
$|\beta|=\sqrt{\frac{tr_{ABE}[\tilde{D}^{\otimes
N}\overline{(P_{AB}^{d_{A},d_{B}})^{\otimes
N}}\rho_{A^{N}B^{N}E^{N}}\overline{(P_{AB}^{d_{A},d_{B}})^{\otimes
N}}]}{tr_{ABE}[\tilde{D}^{\otimes N}\rho_{A^{N}B^{N}E^{N}}]}}$ (10)
Now we can see that if all of pure state $\rho_{A^{N}B^{N}E^{N}}$ satisfying
Eq. (2) make $|\beta|$ small enough, then protocol-1 can be well approximated
by protocol-2.
To estimate $|\beta|$ here we give a very useful theorem.
###### Theorem 1
Let $|1\rangle_{A}$, $|2\rangle_{A}$, …, and $|1\rangle_{B}$, $|2\rangle_{B}$,
…, be bases of Alice and Bob’s Hilbert spaces respectively, by which
projectors $P_{A}^{d_{A}}$ and $P_{B}^{d_{B}}$ can be respectively given by
$P_{A}^{d_{A}}=|1\rangle_{A}\langle 1|+...+|d_{A}\rangle_{A}\langle d_{A}|$
and $P_{B}^{d_{B}}=|1\rangle_{B}\langle 1|+...+|d_{B}\rangle_{B}\langle
d_{B}|$. Then if we have $\sum_{i=1,j=d_{A}}^{\infty,\infty}|_{A}\langle
i|\tilde{D}^{A}|j\rangle_{A}|+\sum_{i=1,j=d_{B}}^{\infty,\infty}|_{B}\langle
i|\tilde{D}^{B}|j\rangle_{B}|\leq$ $\frac{\varepsilon^{3}}{N}$, for arbitrary
$\rho_{A^{N}B^{N}E^{N}}$ it is always satisfied that
$tr_{AB}[\tilde{D}^{\otimes N}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\rho_{A^{N}B^{N}}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}]:=L\leq\varepsilon^{3}$.
Proof: We can see that $L$ is no larger than
$\max_{|\Psi^{N}\rangle}\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$, where
$|\Psi^{N}\rangle\in\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$. If we
expand $\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle=\langle\Psi^{N}|\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\tilde{D}^{\otimes N}|\Psi^{N}\rangle$ into product spaces
$\overline{P_{A}^{d_{A}}}(P_{A}^{d_{A}})^{\otimes N-1}(P_{B}^{d_{B}})^{\otimes
N},...,$ and do straightforward calculation, we can immediately find that
$L\leq N[\sum_{i=1,j=d_{A}}^{\infty,\infty}|_{A}\langle
i|\tilde{D}^{A}|j\rangle_{A}|+\sum_{i=1,j=d_{B}}^{\infty,\infty}|_{B}\langle
i|\tilde{D}^{B}|j\rangle_{B}|]\leq$ $\varepsilon^{3}$. (The straightforward
calculation is too bothering to show here. Detailed one can be seen in the
appendix.) $\square$
Since $tr_{ABE}[\tilde{D}^{\otimes N}\rho_{A^{N}B^{N}E^{N}}]\geq\varepsilon$,
from Eq. (10), we can see that Theorem 1 actually gives a sufficient condition
for $|\beta|\leq\varepsilon$.
Now, we can know the distance between protocol 1 and protocol 2 from the
measurement results. The only remained problem is to give the difference
between securities of protocol-1 and protocol-2 if the state difference of
them is known.
###### Theorem 2
If $||\rho_{X^{N}Y^{N}E^{N}}-\tilde{\rho}_{X^{N}Y^{N}E^{N}}||\leq 2\delta$ for
all $\rho_{A^{N}B^{N}E^{N}}$ satisfying
$tr[(\tilde{D}^{A}\tilde{D}^{B})^{\otimes
N}\rho_{A^{N}B^{N}E^{N}}]\geq\delta$, then the $5\delta+\epsilon$-secure
secret key rate of protocol-1 is no less than the $2\delta+\epsilon$-secure
secret key rate of protocol-2, while Alice and Bob take results from
protocol-1 as that from protocol-2 to estimate the secret key rate of
protocol-2 by the information theoretical method.
Proof: The $L_{1}$ distance cannot be increased by quantum operations and thus
classical bit-wise processing Renner thesis . If
$||\rho_{X^{N}Y^{N}E^{N}}-\tilde{\rho}_{X^{N}Y^{N}E^{N}}||\leq 2\delta$, then
we have $||\rho_{\bar{X}^{k}\bar{E}}-\tilde{\rho}_{\bar{X}^{k}\bar{E}}||\leq
2\delta$ and $||\rho_{X^{N}Y^{N}}-\tilde{\rho}_{X^{N}Y^{N}}||\leq 2\delta$,
where $\bar{X}^{k}$, $\bar{Y}^{k}$ and $\bar{E}$ denote Alice and Bob’s
classical data and Eve’s state after the data processing respectively, during
which some information maybe announced. The security is well defined by smooth
min- and max-entropies. The amount of $\epsilon$-secure secret keys can be
given by
$H_{\min}^{\epsilon^{\prime}}(\rho_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}$
expl , while the strength of parameter estimation is
$\epsilon^{\prime\prime\prime}$, where
$H_{\min}^{\epsilon^{\prime}}(\cdot|\cdot)$ denotes the smooth min-entropy,
$leak_{IR}$ denotes the amount of information published during the
$\epsilon^{\prime\prime}$-secure reconciliation and
$\epsilon^{\prime}+\epsilon^{\prime\prime}+\epsilon^{\prime\prime\prime}=\epsilon$
Renner thesis . Since
$||\rho_{\bar{X}^{k}\bar{E}}-\tilde{\rho}_{\bar{X}^{k}\bar{E}}||\leq 2\delta$,
the smooth min-entropy satisfies
$H_{\min}^{2\delta+\epsilon}(\rho_{\bar{X}^{k}\bar{E}}|\bar{E})\geq
H_{\min}^{\epsilon}(\tilde{\rho}_{\bar{X}^{k}\bar{E}}|\bar{E})$ Renner thesis
. Also, if Alice and Bob use the data from the protocol-1 as that from the
protocol-2 to estimate the state of protocol-2, the security of the parameter
estimation Renner thesis will be reduced by $2\delta$, because
$||\rho_{X^{N}Y^{N}}-\tilde{\rho}_{X^{N}Y^{N}}||\leq 2\delta$. Furthermore, if
we only consider the $\rho_{A^{N}B^{N}E^{N}}$ satisfying
$tr[(\tilde{D}^{A}\tilde{D}^{B})^{\otimes
N}\rho_{A^{N}B^{N}E^{N}}]\geq\delta$, the strength of security will also be
reduced by $\delta$. In all, the
$\epsilon^{\prime}+\epsilon^{\prime\prime}+\epsilon^{\prime\prime\prime}$
secure security of protocol 2 is given by
$H_{\min}^{\epsilon^{\prime}}(\tilde{\rho}_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}$,
while the strength of parameter estimation is $\epsilon^{\prime\prime\prime}$
and the $\tilde{\rho}_{\bar{X}^{k}\bar{E}}$ is estimated by the data obtained
from protocol 2. The
$\epsilon^{\prime}+\epsilon^{\prime\prime}+\epsilon^{\prime\prime\prime}+2\delta$
secure security of protocol 2 is given by
$H_{\min}^{\epsilon^{\prime}}(\tilde{\rho}_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}$,
while the strength of parameter estimation is
$\epsilon^{\prime\prime\prime}+2\delta$, where the
$\tilde{\rho}_{\bar{X}^{k}\bar{E}}$ is estimated by data obtained from
protocol 1. Finally the
$\epsilon^{\prime}+\epsilon^{\prime\prime}+\epsilon^{\prime\prime\prime}+5\delta$
secure secret key rate of protocol 1 can be given by
$H_{\min}^{2\delta+\epsilon^{\prime}}(\rho_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}\geq
H_{\min}^{\epsilon^{\prime}}(\tilde{\rho}_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}$,
while the strength of parameter estimation is
$\epsilon^{\prime\prime\prime}+2\delta$, the state
$\tilde{\rho}_{\bar{X}^{k}\bar{E}}$ is estimated by the data obtained from
protocol 1 and only the $\rho_{A^{N}B^{N}E^{N}}$ satisfying
$tr[(\tilde{D}^{A}\tilde{D}^{B})^{\otimes N}\rho_{A^{N}B^{N}E^{N}}]\geq\delta$
is considered. Here the term
$H_{\min}^{\epsilon^{\prime}}(\tilde{\rho}_{\bar{X}^{k}\bar{E}}|\bar{E})-leak_{IR}$
is amount of the
$\epsilon^{\prime}+\epsilon^{\prime\prime}+\epsilon^{\prime\prime\prime}+2\delta$
secure secrete keys of protocol 2, while the strength of parameter estimation
is $\epsilon^{\prime\prime\prime}+2\delta$, where $2\delta$ comes from the
fact that the state $\tilde{\rho}_{\bar{X}^{k}\bar{E}}$ is estimated by the
data obtained from protocol 1. $\square$
The state distance
$||\rho_{X^{N}Y^{N}E^{N}}-\tilde{\rho}_{X^{N}Y^{N}E^{N}}||\leq 2\delta$ can be
evaluated from the measurement results through theorem 1. The security of
protocol 2 is covered by current information theoretical security proof
method. Then the the security of protocol 1 can be solved.
## IV Security of Protocol 2.
If Alice and Bob’s received initial state in protocol 1 is
$\rho_{A^{N}B^{N}}$, the state they received in protocol 2 is
$\tilde{\rho}_{A^{N}B^{N}}=\frac{1}{p}(P_{AB}^{d_{A}d_{B}})^{\otimes
N}\rho_{A^{N}B^{N}}(P_{AB}^{d_{A}d_{B}})^{\otimes N}$, where $\frac{1}{p}$ is
introduced for normalization. After quantum communication, Alice and Bob will
permute their state, then $\rho_{A^{N}B^{N}}$ is permutation invariant. The
projection operator $(P_{AB}^{d_{A}d_{B}})^{\otimes N}$ commutates with the
permutation operator, so $\tilde{\rho}_{A^{N}B^{N}}$ is also a permutation
invariant state. The dimension of individual state of
$\tilde{\rho}_{A^{N}B^{N}}$ is $d_{A}d_{B}$. Then there is a symmetric
purification for $\tilde{\rho}_{A^{N}B^{N}}$ in a Hilbert space of dimension
$(d_{A}d_{B})^{2N}$, which actually is $\tilde{\rho}_{A^{N}B^{N}E^{N}}$ Renner
thesis ; expl4 . Then the dimension of the individual state of
$\tilde{\rho}_{A^{N}B^{N}E^{N}}$ is $(d_{A}d_{B})^{2}$. According to current
exponential de Finetti theorem, the state $\tilde{\rho}_{A^{N}B^{N}E^{N}}$ is
close to an almost product state expl5 . Then we can only consider the
collective attack. Since under the collective attack, Eve attacks all of
signals independently by the same method, here we let $\tilde{\rho}_{ABE}$
denotes the state Alice, Bob and Eve share after a single communication.
Before calculating the secret key rate, we need to estimate possible
$\tilde{\rho}_{ABE}$ from measurement results. It should be noted that
although $\tilde{\rho}_{ABE}$ belongs to a Hilbert space of dimension
$(d_{A}d_{B})^{2}$, we do not really need to estimate it only in a
$(d_{A}d_{B})^{2}$ dimensional subspace. We can still construct it in an
infinite dimensional space, because a state belonging to a $(d_{A}d_{B})^{2}$
dimensional Hilbert space also belongs to a infinite dimensional Hilbert space
expl2 . This point shows that while we discuss the collective attack for
protocol 2, we do not need to take the filter in to account. If we give up
filters in protocol 2, the protocol 2 becomes the same as protocol 1. Then if
we do not take the filter into account, the security against collective attack
of protocol 2 is actually equivalent to that of protocol 1. Finally, our
conclusion is as follows. The security of protocol 1 can be approximated by
that of protocol 2. For the protocol 2 we only need to consider the collective
attack. While the Hilbert space of protocol 2 is only a subspace of protocol
1, then the secrete key rate of protocol 2 against collective attack is no
less than that of protocol 1 against collective attack. Finally, we actually
give the difference between coherent attack and collective attack for protocol
1. We introduce the filter only to apply current de Finetti theorem and to
give the difference between coherent attack and collective attack for protocol
1.
The $5\delta+\epsilon$-secure unconditional secret key rate of protocol-1 is
no less than the $2\delta+\epsilon$-secure secret key rate of protocol-2.
Under the infinite key size case, the unconditional secrete key rate of
protocol 2 is given by the secret key rate against collective attacks Renner
thesis . The secret key rate under collective attack of protocol 2 is no less
than that of protocol 1. Also under the infinite key size case, the parameter
$\epsilon$ can approach to zero. Then the $5\delta$-secure unconditional
secret key rate of protocol-1 is no less than the $2\delta$ secure
unconditional secret key rate of protocol-2 and no less than its secret key
rate against collective attacks, where $2\delta$ comes from the fact that
Alice and Bob use the data of protocol 1 to estimate the state of protocol 2.
In addition, under the infinite key size case, we may choose large enough
$d_{A}$ and $d_{B}$ so as to make $\delta$approach to zero. Then we can
directly say that for protocol 1 if the POVM elements corresponding to the
measured results can be arbitrarily well described in a finite dimensional
space, the collective attack is optimal under the infinite key size case.
For many practical QKDs, the projection of POVM elements of measured results
on high dimensional basis is extremely small. For example, the POVM element
for heterodyne detection corresponding to measured result $(p,q)$ is
$M_{p,q}=\frac{1}{\pi}|p+iq\rangle\langle p+iq|$, whose component on the
photon number basis $|m\rangle$ exponentially goes to zero as $m$ increase.
The POVM of inefficient photon number resolving detector POVM also has
similar property. Then if a QKD protocol utilize such detectors, Alice and Bob
can announce the maximum $p^{2}+q^{2}$ or maximum photon number received from
one pulse. Then Alice and Bob can construct the big POVM $\tilde{D}$ and for a
given $\frac{\varepsilon^{3}}{N}$ they can find a big enough $d$ (smaller than
$N$) that in photon number picture satisfies
$\sum_{i=1,j=d}^{\infty,\infty}|\langle i|\tilde{D}|j\rangle|\leq$
$\frac{\varepsilon^{3}}{N}$. Then the difference between states of protocol 1
and protocol 2 can be smaller than $2\varepsilon$. The
$5\varepsilon+\epsilon$-secure secret key rate can be given by
$2\varepsilon+\epsilon$-secure secret key rate of protocol 2, which is covered
by Ref. Renner thesis .
## V Applications:
In the realistic case, the measured result is always finite. In heterodyne
detection protocols, the maximum value of measured result is limited. In
photon number detection protocol, the maximum received photon number is
finite. Such realistic cases allow us readily apply our results.
Here we give two application examples. We will see that our result can be
readily used for heterodyne detection and photon number detection case. It
should be noted that in the following we only proved that for CVQKD and DPSQKD
the collective attack is optimal under infinite key size case. How to prove
their security against collective attack has not been solved in this paper.
For short, we only take the infinite key size case for examples. It seems that
our estimation of effective dimension is meaningless under this case. However,
we should note that under the finite key size case, the estimation of
effective dimension will be useful.
### V.1 Unconditional security of CVQKD
Now we apply our results to the heterodyne detection CVQKD and prove that as
the key size goes to infinite the collective is optimal. In the prepare &
measurement CVQKD, Alice prepare a continuous variable EPR pair, and sends one
part to Bob. Alice and Bob respectively do heterodyne detection to their held
states. The security of such scheme against collective attack is discussed in
Ref. binaryCVQKD . Here, we prove that for this protocol the collective attack
is optimal under the infinite key size case. We denote Alice and Bob’s
measurement result by $(p_{A},q_{A})$ and $(p_{B},q_{B})$ respectively. The
corresponding POVM elements are respectively
$M_{p_{A},q_{A}}=\frac{1}{\pi}|p_{A}+iq_{A}\rangle\langle p_{A}+iq_{A}|$ and
$M_{p_{B},q_{B}}=\frac{1}{\pi}|p_{B}+iq_{B}\rangle\langle p_{B}+iq_{B}|$. In a
realistic system, the maximum value of Alice and Bob’s measurement results is
finite (or Alice and Bob can give up some extremely larger measurement
results). Then their final shared data is within certain region. We assume
$V_{A}^{\max}$ and $V_{B}^{\max}$ are large enough, so that for all possible
$(p_{A},q_{A})$s and $(p_{B},q_{B})$s Alice and Bob hold satisfy
$p_{A}^{2}+q_{A}^{2}\leq V_{A}^{\max}$ and $p_{B}^{2}+q_{B}^{2}\leq
V_{B}^{\max}$ (or Alice and Bob only accept the data with amplitude no larger
than $V_{A}^{\max}$ and $V_{B}^{\max}$). Then we can construct $\tilde{D}^{A}$
and $\tilde{D}^{B}$ respectively to be
$\displaystyle\tilde{D}^{A}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int\limits_{p_{A}^{2}+q_{A}^{2}\leq
V_{A}^{\max}}|p_{A}+iq_{A}\rangle\langle p_{A}+iq_{A}|dp_{A}dq_{A}$
$\displaystyle\tilde{D}^{B}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int\limits_{p_{B}^{2}+q_{B}^{2}\leq
V_{A}^{\max}}|p_{B}+iq_{B}\rangle\langle p_{B}+iq_{B}|dp_{B}dq_{B}$
The filter $P_{A}^{d_{A}}$ and $P_{B}^{d_{B}}$ can be chosen in photon number
space. We let
$\displaystyle P_{A}^{d_{A}}$ $\displaystyle=$
$\displaystyle|0\rangle_{A}\langle 0|+|1\rangle_{A}\langle
1|+...+|d_{A}-1\rangle_{A}\langle d_{A}-1|$ $\displaystyle P_{B}^{d_{B}}$
$\displaystyle=$ $\displaystyle|0\rangle_{B}\langle 0|+|1\rangle_{B}\langle
1|+...+|d_{B}-1\rangle_{B}\langle d_{B}-1|$
where $|i\rangle_{A}\langle i|$ and $|j\rangle_{B}\langle j|$ denote the
photon number state. Now we utilize theorem 1 to discuss the difference
between protocol 1 and protocol 2. We see that
$\displaystyle\sum_{i=0,j=d_{A}-1}^{\infty,\infty}|_{A}\langle
i|\tilde{D}^{A}|j\rangle_{A}|$ (11) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int\limits_{p_{A}^{2}+q_{A}^{2}\leq
V_{A}^{\max}}\sum_{i=0,j=d_{A}-1}^{\infty,\infty}|_{A}\langle
i|p_{A}+iq_{A}\rangle$ $\displaystyle\langle
p_{A}+iq_{A}|j\rangle_{A}|dp_{A}dq_{A}$ $\displaystyle=$ $\displaystyle
2\int\limits_{r_{A}^{2}\leq
V_{A}^{\max}}\sum_{i=0,j=d_{A}-1}^{\infty,\infty}\frac{r_{A}^{i}r_{A}^{j}\exp[-r_{A}^{2}]}{\sqrt{i!j!}}dr_{A}$
$\displaystyle=$ $\displaystyle 2\int\limits_{r_{A}^{2}\leq
V_{A}^{\max}}\sum_{i=0}^{\infty}\frac{r_{A}^{i}\exp[-r_{A}^{2}]}{\sqrt{i!}}\sum_{j=d_{A}-1}^{\infty}\frac{r_{A}^{j}}{\sqrt{j!}}dr_{A}$
where in the forth line we used the result that $|_{A}\langle
i|p_{A}+iq_{A}\rangle|=\frac{r_{A}^{i}\exp[-r_{A}^{2}/2]}{\sqrt{i!}}$ and let
$r_{A}^{2}=$ $p_{A}^{2}+q_{A}^{2}$. Under the case that $d_{A}\gg
V_{A}^{\max}$, we can use the Stirling formula to approximate $\sqrt{j!}$.
Then we have
$\sum_{j=d_{A}}^{\infty}\frac{r_{A}^{j}}{\sqrt{j!}}\varpropto(r_{A}/\sqrt{d_{A}})^{d_{A}}$,
which exponentially goes to zero as $d_{A}$ increases. Then the whole term
$\sum_{i=0,j=d_{A}}^{\infty,\infty}|_{A}\langle i|\tilde{D}^{A}|j\rangle_{A}|$
will exponentially go to zero with the increase of $d_{A}$. By the same way we
can prove that the term $\sum_{i=1,j=d_{B}}^{\infty,\infty}|_{B}\langle
i|\tilde{D}^{B}|j\rangle_{B}|$ will also exponentially goes to zero with the
increase of $d_{B}$. Finally, for a given $\varepsilon^{3}$ and large enough
$N$, we can find a $d_{A}\ll N$ and $d_{B}\ll N$, that satisfy
$\sum_{i=1,j=d_{A}}^{\infty,\infty}|_{A}\langle
i|\tilde{D}^{A}|j\rangle_{A}|+\sum_{i=1,j=d_{B}}^{\infty,\infty}|_{B}\langle
i|\tilde{D}^{B}|j\rangle_{B}|:=Err\leq\ \frac{\varepsilon^{3}}{N}$
Then from the theorem 1 and 2 we know that the security of this CVQKD scheme
can be approximated by the security of a scheme of dimension
$(d_{A}d_{B})^{2}\ll N$ with errors no larger than $5\varepsilon$ ($Err$
exponentially approach to zero with the increase of $d_{A}$ and $d_{B}$, so
that $d_{A}$ and $d_{B}$ are proportional with $\log(N/\varepsilon^{3})$. Then
for large enough $N$, we can have $(d_{A}d_{B})^{2}\ll N$). Then
$5\varepsilon+\epsilon$-secure secret key rate of heterodyne detection CVQKD
can be given by $2\varepsilon+\epsilon$-secure secret key rate of protocol-2,
where Alice and Bob respectively put filters $P_{A}^{d_{A}}$ and
$P_{B}^{d_{B}}$ before their detectors. As $N\rightarrow\infty$, we can find
large enough $(d_{A}d_{B})^{2}\ll N$, that allow $\varepsilon\rightarrow 0$,
and the security parameter $\epsilon$ can goes to zero From the Ref. Renner
thesis we know that, under the case that $(d_{A}d_{B})^{2}\ll
N\rightarrow\infty$, the collective attack is optimal for protocol 2 and its
secret key rate can be given by that under collective attack. Since the
secrete key rate against collective attack of protocol 2 is no larger than
that of protocol 1, under the infinite key size case the unconditional secret
key rate of heterodyne detection CVQKD equal to its secret key rate under
collective attacks and the collective is optimal. Here, we require Alice and
Bob give up such data whose amplitude is larger than $V_{A}^{\max}$ and
$V_{B}^{\max}$. We can expect that for large $V_{A}^{\max}$ and
$V_{B}^{\max}$, the proportion of given up data is extremely small. Such
procedure only causes extremely small change of state, and thus only cause
extremely small change of security. On the other hand, a realistic security
proof for CVQKD should take such cut off procedure into account. After all, in
a realistic situation, the maximum value of measurement results is always
finite.
### V.2 Unconditional security of DPSQKD
Now we apply our result to coherent state DPSQKD, whose dimension is infinite
in principle. Up to now, the security against collective attack for DPSQKD
under noiseless case is proved DPSQKD . Here we show that that proof actually
is unconditional security proof. To allow Alice and Bob do random permutation,
in Ref. DPSQKD Zhao et al. cut the long sequence of coherent states into
blocks and regarded one block as one big state. Then Alice and Bob can permute
these big states. In the DPSQKD Alice sends Bob a big state
$|\Psi_{\vec{x}}^{N_{b}}\rangle={\bigotimes\limits_{i=1}^{N_{b}}}|(-1)^{x_{i}+1}\alpha\rangle$
(denotes the state of a block), according to her binary string
$\vec{x}=(x_{1},x_{2},...,x_{N_{b}})$, where $|(-1)^{x_{i}+1}\alpha\rangle$ is
a coherent state. Then Bob measures the phase difference between each two
individual state. The collective attack means Eve attack these big states
(blocks) independently with the same method. Here we require Bob use the
photon number resolving detector. After many rounds of quantum communications,
Bob announces the maximum photon number received from one big state (one
block). Then if Bob put a filter that filters out all the state whose photon
number is larger than certain criteria, the measured results should not change
too much.
We see that if the efficiency of photon number resolving detector is 100%,
then we can definitely know the actual dimension of Bob’s received state.
However, if that efficiency is not 100%, we cannot determine the exact
dimension of Bob’s state from the measured photon numbers.
Here we discuss the imperfect detector case. In Ref. POVM2 , the POVM element
of ineffective photon number resolving detector is given. In that reference,
the spacial mode of received photon state has not been considered. If we take
the spacial mode and other components into account, we can extend that POVM
element corresponding to $n$ photons to be
$\Pi_{n}=\sum\limits_{m=n}^{\infty}C^{n}_{m}\gamma^{n}(1-\gamma)^{m-n}P_{m}$
(12)
where $\gamma$ denotes detector efficiency and $P_{m}$ denotes the projector
to $m$ photon number subspace. We assume the dimension of $m$ photon number
subspace is $f_{m}$, and $P_{m}$ to be
$P_{m}=\sum_{k=1}^{f_{m}}|\varphi_{k}^{m}\rangle\langle\varphi_{k}^{m}|$ (13)
where $|\varphi_{k}^{m}\rangle$ denotes the orthogonal state of $m$ photon
number subspace. It can be prove that $f_{m}\leq l(m+l-1)!/m!$, where $l$
denotes the block size.
If Bob’s maximum received photon number is $n_{0}$, then the POVM element
corresponding to this event can be given by
$\tilde{D}^{B}=\sum_{n=0}^{n=n_{0}}\Pi_{n}$ (14)
In DPSQKD, if the block size is $l$, the dimension of Alice’s modulation is
$2^{l}$, which is finite. Therefore we only need to discuss Bob’s state. We
can construct Bob’s filter to be
$P_{B}^{d_{B}}=\sum_{m=0}^{m=m_{0}}P_{m}$
where $P_{m}$ is given by Eq. (13). Now we can use theorem 1 to estimate the
difference between protocol 1 and protocol 2. We enumerate the basis of the
filter by $|\varphi_{k}^{m}\rangle$. Then we have
$\displaystyle
Diff:=\sum_{m=0,m^{\prime}=m_{0},k,k^{\prime}}\langle\varphi_{k}^{m}|\tilde{D}^{B}|\varphi_{k^{\prime}}^{m^{\prime}}\rangle$
(15)
$\displaystyle\leq\sum_{n=0}^{n_{0}}\gamma^{n}\sum_{m=m_{0}}C_{m}^{n}(1-\gamma)^{m-n}l(m+l-1)!/m!$
$\displaystyle\leq\sum_{n=0}^{n_{0}}\gamma^{n}\sum_{m=m_{0}}(1-\gamma)^{m-n}l/n!(m+l-1)^{l+n-1}$
where in the second line we have used Eqs. (12), (13) and (14) and the fact
that $f_{m}\leq l(m+l-1)!/m!$ and in the third line we used the fact that
$m(m-1)...(m-n)\leq m^{n}$. It can be seen that $Diff$ exponentially goes to
zero as $m_{0}$ increases. Then for a given security parameter we can find a
large enough key size $N$ that gives the required security.
It also can be seen that if Bob use the perfect photon number resolving
detector or a detector that can given the upper bound of the number of
received photons (e.g. bourn up if received photon number is too high), then
they can find a protocol 2 that is exactly same as protocol 1. Then we can
immediately get a conclusion that the collective attack is optimal under the
infinite key size case.
## VI Conclusion:
In the above we give a method to apply current exponential de Finetti theorem
to realistic QKD. In realistic QKD, the number of Alice and Bob received
photons is always finite and their measurement results always belong to a
finite region. This property allow us effectively describe the QKD protocol in
a finite dimensional subspace with sufficiently small error. In this paper, we
introduce another finite dimensional protocol by putting finite dimensional
filters before the detectors, and shown the security difference between the
original unknown-dimensional protocol and this finite dimensional protocol
based on measurement results. Since the security of that finite dimensional
protocol is covered by current information theoretical security proof method,
the security of a realistic unknown dimensional system can be solved. Our
result can be used to prove the unconditional security of heterodyne detection
CVQKD and DPSQKD. Finally, we prove that for heterodyne detection CVQKD and
DPSQKD collective attack is optimal under the infinite key size case. The
difference between protocol 1 and protocol 2 will be meaningful if we consider
the finite key size case.
Acknowledgement: Special thanks are given to R. Renner for fruitful
discussions. This work is supported by National Natural Science Foundation of
China under Grants No. 60537020 and 60621064.
## Appendix A Detailed proof for Theorem 1
At first we can see that $tr_{AB}[\tilde{D}^{\otimes
N}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes
N}}\rho_{A^{N}B^{N}}\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}]$ is no
larger than $\max_{|\Psi^{N}\rangle}\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$ where
$|\Psi^{N}\rangle\in\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$. To find
$\max_{|\Psi^{N}\rangle}\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$, we need to expand the space
$\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$ by product spaces
$\overline{P_{A}^{d_{A}}}(P_{A}^{d_{A}})^{\otimes N-1}(P_{B}^{d_{B}})^{\otimes
N},...$. Here, we let $P_{A_{k}}^{d_{A}}$ ($P_{B_{k}}^{d_{B}}$) denote the
projector to Alice’s (Bob’s) $k$-th state. Also we distinguish bases of $k$-th
state of Alice (Bob) as $|1\rangle_{A_{k}},|2\rangle_{A_{k}},...$,
($|1\rangle_{B_{k}},|2\rangle_{B_{k}},...$). We know that
$I_{A_{k}}=P_{A_{k}}^{d_{A}}+\overline{P_{A_{k}}^{d_{A}}}$ and
$I_{B_{k}}=P_{B_{k}}^{d_{A}}+\overline{P_{B_{k}}^{d_{A}}}$, where $I_{A_{k}}$
and $I_{B_{k}}$ are the identity matrixes corresponding to Alice and Bob’s
$k$-th states. Then we have
$\displaystyle\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle=\langle\Psi^{N}|(P_{A_{1}}^{d_{A}}+\overline{P_{A_{1}}^{d_{A}}})\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$
$\displaystyle=\langle\Psi^{N}|\overline{P_{A_{1}}^{d_{A}}}\tilde{D}^{\otimes
N}|\Psi^{N}\rangle+\langle\Psi^{N}|P_{A_{1}}^{d_{A}}\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$ $\displaystyle=C_{F}^{1}+C_{L}^{1}$ (16)
where $C_{F}^{1}$ and $C_{L}^{1}$ respectively denote the first and second
term in the second line. Since
$\begin{array}[]{c}\overline{P_{A_{1}}^{d_{A}}}=|d_{A}+1\rangle_{A_{1}}\langle
d_{A}+1|+|d_{A}+2\rangle_{A_{1}}\langle d_{A}+2|+...\\\
I_{A_{1}}=|1\rangle_{A_{1}}\langle 1|+|2\rangle_{A_{1}}\langle
2|+...\end{array}$
the $C_{F}^{1}$ can be given by
$\displaystyle C_{F}^{1}$ $\displaystyle=$
$\displaystyle\langle\Psi^{N}|\overline{P_{A_{1}}^{d_{A}}}\tilde{D}^{\otimes
N}I_{A_{1}}|\Psi^{N}\rangle$ $\displaystyle=$
$\displaystyle\sum_{m_{1}=d_{A}+1,m_{1}^{\prime}=1}^{\infty,\infty}\langle
m_{1}|\tilde{D}^{A}|m_{1}^{\prime}\rangle\cdot$
$\displaystyle\langle\Psi^{N}|m_{1}\rangle(\tilde{D}^{A})^{\otimes
N-1}(\tilde{D}^{B})^{\otimes N}\langle m_{1}^{\prime}|\Psi^{N}\rangle$
where we have used the fact that $P_{A_{k}}^{d_{A}}$ and $\tilde{D}_{j}^{A}$
and $\tilde{D}_{j}^{B}$ are commutate if $k\neq j$ and $\tilde{D}_{j}^{A}$ and
$\tilde{D}_{j}^{B}$ denote POVM elements corresponding to $j$-th state. We
know there exist two pure states $|\Phi_{1}^{m_{1}}\rangle$ and
$|\tilde{\Phi}_{1}^{m_{1}^{\prime}}\rangle$ that can let $\langle
m_{1}^{\prime}|\Psi^{N}\rangle$ and $\langle m_{1}|\Psi^{N}\rangle$ be written
as $\langle m_{1}|\Psi^{N}\rangle=\lambda_{1}|\Phi_{1}^{m_{1}}\rangle$ and
$\langle
m_{1}^{\prime}|\Psi^{N}\rangle=\lambda_{1}^{\prime}|\tilde{\Phi}_{1}^{m_{1}^{\prime}}\rangle$,
where $|\lambda_{1}|\leq 1$ and $|\lambda_{1}^{\prime}|\leq 1$. Then
$C_{F}^{1}$ can be given by
$\displaystyle C_{F}^{1}$ $\displaystyle=$
$\displaystyle\langle\Psi^{N}|\overline{P_{A_{1}}^{d_{A}}}\tilde{D}^{\otimes
N}I_{A_{1}}|\Psi^{N}\rangle$ $\displaystyle=$
$\displaystyle\lambda_{1}\lambda_{1}^{\prime}\sum_{m_{1}=d_{A}+1,m_{1}^{\prime}=1}^{\infty,\infty}\langle
m_{1}|\tilde{D}^{A}|m_{1}^{\prime}\rangle\cdot$
$\displaystyle\langle\Phi_{1}^{m_{1}}|(\tilde{D}^{A})^{\otimes
N-1}(\tilde{D}^{B})^{\otimes N}|\tilde{\Phi}_{1}^{m_{1}^{\prime}}\rangle$
Before giving the upper bound to $C_{F}^{1}$, we will discuss the upper bound
of $|\langle\Phi_{1}^{m_{1}}|(\tilde{D}^{A})^{\otimes
N-1}(\tilde{D}^{B})^{\otimes N}|\tilde{\Phi}_{1}^{m_{1}^{\prime}}\rangle|$. It
is known that arbitrary POVM element $M$ can be written into a diagonal form.
We assume an arbitrary $M$ can be written as
$M=a_{1}|\varphi_{1}\rangle\langle\varphi_{1}|+a_{2}|\varphi_{2}\rangle\langle\varphi_{2}|+...$
where $|\varphi_{1}\rangle,|\varphi_{2}\rangle,...$are orthogonal bases, and
$a_{1},a_{2},...$ are positive real numbers and satisfy $a_{i}\leq 1$. We let
$|\psi\rangle$ and $|\psi^{\prime}\rangle$ are two arbitrary states. Now we
consider the following value for $|\psi\rangle$ and $|\psi^{\prime}\rangle$.
$|\langle\psi|M|\psi^{\prime}\rangle|=|a_{1}\langle\psi|\varphi_{1}\rangle\langle\varphi_{1}|\psi^{\prime}\rangle+a_{2}\langle\psi|\varphi_{2}\rangle\langle\varphi_{2}|\psi^{\prime}\rangle+...|$
From the fact that
$\displaystyle|\langle\psi|\varphi_{1}\rangle|^{2}+|\langle\psi|\varphi_{2}\rangle|^{2}+...$
$\displaystyle\leq$ $\displaystyle 1$ (19)
$\displaystyle|\langle\psi^{\prime}|\varphi_{1}\rangle|^{2}+|\langle\psi^{\prime}|\varphi_{2}\rangle|^{2}+...$
$\displaystyle\leq$ $\displaystyle 1$
we know that for arbitrary states $|\psi\rangle$ and $|\psi^{\prime}\rangle$
and POVM element $M$, it is always satisfied that
$\displaystyle|\langle\psi|M|\psi^{\prime}\rangle|$ $\displaystyle\leq$
$\displaystyle\sqrt{|a_{1}\langle\psi|\varphi_{1}\rangle|^{2}+|a_{2}\langle\psi|\varphi_{2}\rangle|^{2}+...}$
$\displaystyle\leq$
$\displaystyle\sqrt{|\langle\psi|\varphi_{1}\rangle|^{2}+|\langle\psi|\varphi_{2}\rangle|^{2}+...}\leq
1$
where in the first line we applied the Cauchy-Schwartz inequality which says
that
$\displaystyle|a_{1}b_{1}+a_{2}b_{2}+...|\leq$
$\displaystyle\sqrt{|a_{1}|^{2}+|a_{2}|^{2}+...}\sqrt{|b_{1}|^{2}+|b_{2}|^{2}+...}$
and in the second line we applied Eq. (19) and the fact that $a_{i}\leq 1$.
Now we put Eq. (A) into Eq. (A) and obtain
$|C_{F}^{1}|\leq\sum_{m_{1}=d_{A}+1,m_{1}^{\prime}=1}^{\infty,\infty}|\langle
m_{1}|\tilde{D}^{A}|m_{1}^{\prime}\rangle|$ (21)
By the same way $C_{L}^{1}$ can be given by
$\displaystyle C_{L}^{1}$ $\displaystyle=$
$\displaystyle\langle\Psi^{N}|P_{A_{1}}^{d_{A}}\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$ $\displaystyle=$
$\displaystyle\langle\Psi^{N}|P_{A_{1}}^{d_{A}}\overline{P_{A_{2}}^{d_{A}}}\tilde{D}^{\otimes
N}|\Psi^{N}\rangle+\langle\Psi^{N}|P_{A_{1}}^{d_{A}}P_{A_{2}}^{d_{A}}\tilde{D}^{\otimes
N}|\Psi^{N}\rangle$ $\displaystyle=$ $\displaystyle C_{F}^{2}+C_{L}^{2}$
where $C_{F}^{2}$ and $C_{L}^{2}$ respectively denote the first and the second
term in the second line.
As the $C_{F}^{1}$, the $C_{F}^{2}$ can be rewritten as
$\displaystyle C_{F}^{2}$ $\displaystyle=$
$\displaystyle\langle\Psi^{N}|P_{A_{1}}^{d_{A}}\overline{P_{A_{2}}^{d_{A}}}\tilde{D}^{\otimes
N}I_{A_{2}}|\Psi^{N}\rangle$ (22) $\displaystyle=$
$\displaystyle\sum_{m_{2}=d_{A}+1,m_{2}^{\prime}=1}^{\infty,\infty}\langle
m_{2}|\tilde{D}^{A}|m_{2}^{\prime}\rangle\cdot$
$\displaystyle\langle\Psi^{N}|m_{2}\rangle
P_{A_{1}}^{d_{A}}(\tilde{D}^{A})^{\otimes N-1}(\tilde{D}^{B})^{\otimes
N}\langle m_{2}^{\prime}|\Psi^{N}\rangle$
Also there exist a pure state $|\Phi_{2}^{m_{2}}\rangle$ by which
$\langle\Psi^{N}|m_{2}\rangle P_{A_{1}}^{d_{A}}$ can be written as
$\lambda_{2}\langle\Phi_{2}^{m_{2}}|$ and a pure state
$|\tilde{\Phi}_{2}^{m_{2}^{\prime}}\rangle$ by which $\langle
m_{2}^{\prime}|\Psi^{N}\rangle$ can be given by
$\lambda_{2}^{\prime}|\tilde{\Phi}_{2}^{m_{2}^{\prime}}\rangle$. Since
$|\lambda_{1}|\leq 1$ and $|\lambda_{2}|\leq 1$, from the Eqs. (A) and (22) we
know that
$|C_{F}^{2}|\leq\sum_{m_{2}=d_{A}+1,m_{2}^{\prime}=1}^{\infty,\infty}|\langle
m_{2}|\tilde{D}^{A}|m_{2}^{\prime}\rangle|$ (23)
If we continuously do such procedure, we will find that
$\langle\Psi^{N}|(\tilde{D}^{A}\tilde{D}^{B})^{\otimes
N}|\Psi^{N}\rangle=\sum_{i=1}^{2N}C_{F}^{i}+C_{L}^{2N}$ (24)
and
$|C_{F}^{i}|\leq\sum_{m_{i}=d_{A}+1,m_{i}^{\prime}=1}^{\infty,\infty}|\langle
m_{i}|\tilde{D}^{A}|m_{i}^{\prime}\rangle|$ (25)
for $i\leq N$, and
$|C_{F}^{i}|\leq\sum_{m_{i}=d_{B}+1,m_{i}^{\prime}=1}^{\infty,\infty}|\langle
m_{i}|\tilde{D}^{B}|m_{i}^{\prime}\rangle|$ (26)
for $i>N$, where
$C_{L}^{2N}=\langle\Psi^{N}|(P_{AB}^{d_{A}d_{B}})^{\otimes
N}\tilde{D}^{\otimes N}|\Psi^{N}\rangle=0$ (27)
and we have applied the fact that $|\Psi^{N}\rangle\in$
$\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$. Finally from Eqs. (24), (25)
(26) and (27) we can see that
$\displaystyle|\langle\Psi^{N}|\tilde{D}^{\otimes
N}|\Psi^{N}\rangle|\leq\sum_{i=1}^{2N}|C_{F}^{i}|$ (28) $\displaystyle\leq
N[\sum_{i=1,j=d_{A}}^{\infty,\infty}|_{A}\langle
i|\tilde{D}^{A}|j\rangle_{A}|+\sum_{i=1,j=d_{B}}^{\infty,\infty}|_{B}\langle
i|\tilde{D}^{B}|j\rangle_{B}|]$
Since Eq. (28) holds for arbitrary
$|\Psi^{N}\rangle\in\overline{(P_{AB}^{d_{A}d_{B}})^{\otimes N}}$, the Theorem
1 is proved.
## References
* (1) R. Renner, aXiv: quant-ph/0512258 (2005).
* (2) M. Christandl, R. Koenig, G. Mitchison, R. Renner, Comm. Math. Phys. 273, 473 (2007).
* (3) M. Christandl and B. Toner, arXiv:0712.0916 (2007).
* (4) C. D́Cruz, T. J. Osborne, R. Schack, Phys. Rev. Lett. 98, 160406 (2007).
* (5) R. Renner, Nature Physics 3, 645 (2007).
* (6) T. Tsurumaru, K. Tamaki, Phys. Rev. A 78, 032302 (2008).
* (7) N. J. Beaudry, T. Moroder and N. Lütkenhaus, Phys. Rev. Lett. 101, 093601 (2008).
* (8) Y.-B. Zhao, M. Heid, J. Rigas, N. Lütkenhaus, Phys. Rev. A 79, 012307 (2009).
* (9) R. Garcia-Patron and N. J. Cerf, Phys. Rev. Lett. 97, 190503 (2006).
* (10) Y.-B. Zhao, C.-H. F. Fung, Z.-F. Han and G.-C. Guo, Phys. Rev. A 78, 042330 (2008).
* (11) S. Wehner, M. Christandl, A. C. Doherty, arXiv:0808.3960 (2008).
* (12) Y.-B. Zhao, Y.-Z. Gui, J.-J. Chen, Z.-F. Han, G.-C. Guo IEEE Trans. Inform. Theory, 54, 2803 (2008).
* (13) V. Scarani and R. Renner, Phys. Rev. Lett. 100, 200501 (2008).
* (14) R. Renner and J. I. Cirac, arXiv: 0809.2243 (2008).
* (15) A. Leverrier, E. Karpov, P. Grangier and N. J. Cerf, arXiv:0809.2252 (2008).
* (16) M. A. Nielsen and I. L. Chuang, Quantum Computing and Quantum Information, (Cambridge University Press,Cambridge, UK, 2000).
* (17) From the definition of the universal security, we know $\varepsilon$-secure key can be considered indentical to an ideal key, except with probability $\varepsilon$ Renner thesis .
* (18) V. B. Braginsky and F. Y. Khalili, Quantum Measurement, Camebridge University Press, (1992).
* (19) Here we omit a small term proportional to $O[\text{log}(1/\epsilon^{\prime})]$.
* (20) There may be many different purifications, but all of them are different by local unitary transformations at Eve’s side. The local unitary transformation does not change the smooth min-entropy. Therefore, all purifications are equivalent actually.
* (21) Here we assume $(d_{A}d_{B})^{2}\ll N$
* (22) Under the collective attack, the secret key rate can be given by $\min_{\tilde{\rho}_{ABE}\in\Xi(\tilde{\rho}_{ABE})}H(X|E)-leak_{IR}$, where $\Xi(\tilde{\rho}_{ABE})$ is the collection of all $\tilde{\rho}_{ABE}$s (in a finite dimensional space) that consist with the observation. If we define a collection $\Xi(\rho_{ABE})$ to denote the collection of $\rho_{ABE}$s (belong to a infinite dimensional space) that consist with the observation, we can find that $\Xi(\tilde{\rho}_{ABE})\in\Xi(\rho_{ABE})$. Therefore we have $\min_{\tilde{\rho}_{ABE}\in\Xi(\tilde{\rho}_{ABE})}H(X|E)\geq\min_{\rho_{ABE}\in\Xi(\rho_{ABE})}H(X|E)$. This inequality means that the secret key rate against the collective attack of protocol 2 is no less than that of protocol 1.
* (23) A. M. Branczyk, T. J. Osborne, A. Gilchrist and T. C. Ralph. Phys. Rev. A 68, 043821 (2003).
* (24) S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, arXiv:quant-ph/0204073 (2002).
|
arxiv-papers
| 2008-09-16T14:51:44
|
2024-09-04T02:48:57.799162
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi-Bo Zhao, Zheng-Fu Han, Guang-Can Guo",
"submitter": "Yibo Zhao",
"url": "https://arxiv.org/abs/0809.2683"
}
|
0809.2756
|
# Stochastic Electron Acceleration in Shell-Type Supernova Remnants II
Siming Liu Zhong-Hui Fan Christopher L. Fryer
###### Abstract
We discuss the generic characteristics of stochastic particle acceleration by
a fully developed turbulence spectrum and show that resonant interactions of
particles with high speed waves dominate the acceleration process. To produce
the relativistic electrons inferred from the broadband spectrum of a few well-
observed shell-type supernova remnants in the leptonic scenario for the TeV
emission, fast mode waves must be excited effectively in the downstream and
dominate the turbulence in the subsonic phase. Strong collisionless non-
relativistic astrophysical shocks are studied with the assumption of a
constant Aflvén speed. The energy density of non-thermal electrons is found to
be comparable to that of the magnetic field. With reasonable parameters, the
model explains observations of shell-type supernova remnants. More detailed
studies are warranted to better understand the nature of supernova shocks.
###### Keywords:
acceleration of particles — MHD — plasmas — shock waves — turbulence
###### :
98.38.Mz
## 1 Turbulence Cascade and Stochastic Particle Acceleration
In the Kolmogorov phenomenology, the free energy dissipation rate is given by
$Q\equiv C_{1}\rho u^{3}/L\,,$ where $C_{1}$ is a dimensionless constant, and
$u$ and $L$ are the eddy speed and the turbulence generation scale,
respectively. The eddy turnover speed and time at smaller scales are given
respectively by $v^{2}_{edd}(k)\equiv 4\pi W(k)k^{3}\propto k^{-2/3}$ and
$\tau_{edd}(k)\equiv 2\pi/kv_{edd}=\pi^{1/2}(Wk^{5})^{-1/2}\propto
k^{-2/3}\,,$ where $W(k)=(u^{2}/4\pi)(2\pi/L)^{2/3}k^{-11/3}=(4\pi)^{-1}(2\pi
Q/C_{1}\rho)^{2/3}k^{-11/3}\propto k^{-11/3}\,$ is the isotropic turbulence
power spectrum, $k=2\pi/l$ is the wave number and $l$ is the eddy size. From
the three-dimensional Kolmogorov constant $C\simeq 1.62$ (Yeung & Zhou, 1997),
we obtain $C_{1}=2\pi/C^{3/2}=3.05\,.$ At the turbulence generation scale
$k_{m}=2\pi/L$, $v_{edd}=u$, $Q=2C_{1}\rho[4\pi W^{3}k^{11}]^{1/2}=C_{1}\rho
v^{2}_{edd}(k)/\tau_{edd}(k)\,,$ and the total turbulence energy is given by
$\int W(k)4\pi k^{2}{\rm d}k=(3/2)u^{2}\,.$ The turbulence decay time is
therefore given by $\tau_{d}=3\tau_{edd}(k_{m})/C_{1}\,,$ i.e., eddies decay
after making $3C_{1}^{-1}\sim 1$ turns.
We are interested in the acceleration of particles through scattering randomly
with heavy scattering centers with the corresponding acceleration time
$\tau_{ac}=\tau_{sc}[3v^{2}/v^{2}_{edd}(k)]\,,$ where $\tau_{sc}=2\pi/kv=l/v$
is the scattering time, $v$ is the particle speed, and we have assumed that
the scattering mean free-path is equal to $l$. For the above isotropic
Kolmogorov turbulence spectrum, $\tau_{ac}(k)=3v/2Wk^{4}\propto k^{-1/3}\,.$
To have significant stochastic particle acceleration (SA), the acceleration
time $\tau_{ac}(k)$ should be shorter than the turbulence decay time
$\tau_{d}$, which implies $u^{2}>C_{1}vv_{edd}(k)\,.$ So, in general, the SA
is more efficient at smaller scales. The onset scale of the SA is given by
$k_{c}=(C_{1}v/u)^{3}k_{m}\,.$ Therefore, to produce energetic particles with
a speed of $v$ by a Kolmogorov spectrum of scattering centers, the turbulence
must have a dynamical range greater than $D_{k}=(C_{1}v/u)^{3}\,.$
In the Kraichnan phenomenology, the turbulence decay is suppressed by the wave
propagating effect with $Q=C_{1}\rho u^{4}/Lv_{F},$
$W(k)=(u^{2}/4\pi)k_{m}^{1/2}k^{-7/2},$ $\tau_{edd}\propto
k^{-3/4}\,,v_{edd}\propto k^{-1/4}\,,\tau_{ac}\propto k^{-1/2}\,,$ and the
turbulence decay time $\tau_{d}=3\tau_{edd}(k_{m})v_{F}/C_{1}u\,,$ where the
wave speed $v_{F}\gg u$. To have significant acceleration through scattering
with the eddies, the dynamical range of the turbulence must be greater than
$D_{w}=(C_{1}v/v_{F})^{2}\,,$ which is much less than $D_{k}=(C_{1}v/u)^{3}$.
The resonant interactions of particles with waves can be much more efficient
in accelerating particles in this case. For a wave speed $v_{F}$ independent
of $k$, the acceleration time is given by
$\tau_{ac}=3\tau_{sc}v^{2}/v_{F}^{2}\propto k^{-1}\,.$ To have significant
acceleration, the scattering mean free path of the particles must be shorter
than $(v_{F}^{3}/C_{1}vu^{2})L=D_{k}^{2/3}D_{w}^{-3/2}L\,.$
Several Shell-Type Supernova Remnants (STSNRs) have been observed extensively
in the radio, X-ray, and TeV bands. X-ray observations with Chandra, XMM-
Newton, and Suzaku and TeV observations with HESS have made several surprising
discoveries that challenge the classical diffusive shock particle acceleration
model (Liu et al., 2006; Takaaki et al., 2008). The SNR RX J1713.7-3946 is
about $t=1600$ years old (Wang et al., 1997) with a radius of $R\simeq 10$ pc
and a distance of $D\simeq 1$ kpc. By fitting its broadband spectrum with an
electron distribution of $f\propto\gamma^{-p}\exp-(\gamma/\gamma_{c})^{1/2}$,
we find that $p=1.85$, $B=12.0\;\mu$G, $\gamma_{c}m_{e}c^{2}=3.68$ TeV, and
the total energy of relativistic electrons with the Lorentz factor
$\gamma>1800$ $E_{e}=3.92\times 10^{47}$ erg (Fig. 6).
The X-ray emitting electrons have a gyro-radius of $r_{g}\simeq 10^{15}$ cm,
which shouldn’t be shorter than the scattering mean free path. To produce
these electrons through the SA, the turbulence must be generated on scales
greater than $D_{k}r_{g}$, $D_{w}r_{g}$, and $D_{w}^{3/2}D_{k}^{-2/3}r_{g}$
for the non-resonance Kolmogorov, Kraichnan phenomenology, and the resonant
interactions, respectively. For STSNRs, $u\sim v_{F}\sim 0.01c$,
$D_{k}r_{g}\sim 10$ kpc, which is much larger than the radii of the remnants.
The SA by eddies with a Kolmogorov spectrum is therefore insignificant.
$D_{w}r_{g}\sim 30$ pc, which is also too thick.
$D_{w}^{3/2}D_{k}^{-2/3}r_{g}\sim 0.1$ pc, which is much greater than the
particle inertial length and may be generated through the Kelvin-Helmholtz
instabilities or cosmic ray drifting upstream (Micono et al., 1999; Niemiec et
al., 2008). Therefore if relativistic electrons from the STSNRs are
accelerated through the SA, they must be energized through resonant
interactions with high speed plasma waves. Low speed waves also require a
large turbulence dynamical range to accelerate particles.
## 2 Shock Structure, Wave Damping, and Stochastic Electron Acceleration by
Fast Mode Waves in the Downstream
We next study the SA in the shock downstream by weakly magnetized turbulence
with the Alfvén speed $v_{A}=(B^{2}/4\pi\rho)^{1/2}\ll u$, where $B$, and
$\rho$ are the magnetic field, and mass density, respectively. For strong non-
relativistic shocks with the shock frame upstream speed $U$ much higher than
the speed of the parallel propagating fast mode waves in the upstream
$v_{F}=(v_{A}^{2}+5v_{S}^{2}/3)^{1/2}$, where $v_{S}^{2}=P/\rho$ is the
isothermal sound speed and $P$ is the gas pressure, mass, momentum, and energy
conservation across the shock front require
$U^{2}=5v_{S}^{2}+5u^{2}+2v_{A}^{2}+U^{2}/16\,,$ (1)
where we have assumed that the turbulence behaves as an ideal gas and ignored
the wave propagation effects. The shock structure can be complicated due to
the present of turbulence. We assume that the turbulence is isotropic and has
a generation scale of $L$, which does not change in the downstream. The speeds
$v_{S}$, $v_{A}$, and $u$ therefore should be considered as averaged
quantities on the scale $L$. $v_{A}$ depends on the upstream conditions and/or
the dynamo process of magnetic field amplification(Cho & Vishniac, 2000;
Niemiec et al., 2008). Here we assume it a constant in the downstream. One can
then quantify the evolution of other speeds in the downstream.
For the Kolmogorov phenomenology,
${3{\rm d}\rho u^{2}\over 2{\rm d}t}=-Q\ \ \ {\rm i.e.,}\ \ \ {3U{\rm
d}u(x)^{2}\over 8{\rm d}x}=-{C_{1}u(x)^{3}\over L}\,.$ (2)
Near the shock front, we denote the isothermal sound speed and Aflvén speed by
$v_{S0}$ and $v_{A0}$, respectively. Then the eddy speed at the shock front is
given by $a^{1/2}U/4$ with $a=3-16v_{S0}^{2}/U^{2}-32v_{A0}^{2}/5U^{2}$.
Integrate equation (2) from the shock front $(x=0)$ to downstream ($x>0$), we
then have
$\displaystyle{u(x)\over U}$ $\displaystyle=$ $\displaystyle{1\over
4C_{1}x/3L+4/a^{1/2}}\,,$ (3) $\displaystyle{v_{S}(x)\over U}$
$\displaystyle=$ $\displaystyle\left[{3\over 16}-{1\over
16\left(C_{1}x/3L+a^{-1/2}\right)^{2}}-{2v_{A}^{2}\over
5U^{2}}\right]^{1/2}\,,$ (4) $\displaystyle{v_{F}(x)\over U}$ $\displaystyle=$
$\displaystyle\left[{5\over 16}-{5\over
48\left(C_{1}x/3L+a^{-1/2}\right)^{2}}+{v_{A}^{2}\over
3U^{2}}\right]^{1/2}\,.$ (5)
Figure 1: Evolution of the eddy speed $u$ and speed of parallel propagating
fast mode waves $v_{F}$ in the downstream for $v_{A}=0.0633U$.
As mentioned in the previous section, to produce the observed X-ray emitting
electrons in the STSNRs through the SA processes, fast mode waves must be
excited efficiently. The MHD wave period is given by
$\tau_{F}(k)=2\pi/v_{F}k$. Then the transition scale from the Kolmogorov to
Kraichnan phenomenology occurs at $\tau_{F}(k_{t})=\tau_{edd}(k_{t})$ or
$v_{F}=v_{edd}(k_{t})$(Jiang et al., 2008). We then have
$k_{t}=(u/v_{F})^{3}k_{m}\,.$ (6)
For $k>k_{t}>k_{m}$, the turbulence spectrum in the inertial range is given by
$W(k)=u^{2}(4\pi)^{-1}k_{m}^{2/3}k_{t}^{-1/6}k^{-7/2}=(4\pi)^{-1}v_{F}^{1/2}u^{3/2}k_{m}^{1/2}k^{-7/2}\,.$
Although the turbulence energy exceeds $(3/2)u^{2}$ when the wave propagation
effect is considered, we still assume that the enthalpy of the turbulence is
given by $(5/2)u^{2}$ for $v_{F}<u$ so that equation (1) and the above
solutions for the speed profiles remain valid.
In the subsonic phase with $v_{F}>u$, we assume that fast mode waves can still
be excited efficiently to maximize the SA efficiency. Then the Kraichnan
phenomenology prevails and
$\displaystyle W(k)$ $\displaystyle=$ $\displaystyle
u^{2}(4\pi)^{-1}k_{m}^{1/2}k^{-7/2}\,,$ (7) $\displaystyle{3U{\rm
d}u(x)^{2}\over 8{\rm d}x}$ $\displaystyle=$
$\displaystyle-{C_{1}u(x)^{4}\over Lv_{F}}$ (8)
where from equation (1) one has
$v_{F}=\left[{5U^{2}/16}+{v_{A}^{2}/3-{5u^{2}(x)/3}}\right]^{1/2}.$ These
equations can be solved numerically to get the speed profiles in the subsonic
phase. Figure 1 shows the $v_{F}$ and $u$ profiles with $v_{A}=v_{A0}=0.0633U$
in the downstream and $v_{S0}=v_{A0}\ll U$.
In summary,
$W(k)=u^{3/2}(4\pi)^{-1}\min({v_{F}^{1/2},u^{1/2}})k_{m}^{1/2}k^{-7/2}\,$ (9)
in the Kraichnan regime. The collisionless damping starts at the coherent
length of the magnetic field $l_{d}=2\pi/k_{d}\,,$ where the period of Alfvén
waves $2\pi/kv_{A}$ is comparable to the eddy turnover time, i.e.,
$v_{A}^{2}=4\pi
W(k_{d})k_{d}^{3}=\min({v_{F}^{1/2},u^{1/2}})u^{3/2}k_{m}^{1/2}k_{d}^{-1/2}\,.$
Then we have
$k_{d}=[u^{3}\min(v_{F},u)/v_{A}^{4}]k_{m}\,.$ (10)
For a fully ionized hydrogen plasma with isotropic particle distributions,
which is reasonable in the absence of strong large scale magnetic fields, the
transit-time damping (TTD) rate is given by (Stix, 1962; Petrosian et al.,
2006)
$\displaystyle\Lambda_{T}(\theta,k)={(2\pi k_{\rm
B})^{1/2}k\sin^{2}\theta\over 2(m_{e}+m_{p})\cos\theta}\times$
$\displaystyle\left[\left(T_{e}m_{e}\right)^{1/2}e^{-{m_{e}\omega^{2}\over
2k_{\rm B}T_{e}k_{||}^{2}}}+(T_{p}m_{p})^{1/2}e^{-{m_{p}\omega^{2}\over
2k_{\rm B}T_{p}k_{||}^{2}}}\right]$ (11)
where $k_{\rm B}$, $T_{e}$, $T_{p}$, $m_{e}$, $m_{p}$, $\theta$, $\omega$, and
$k_{||}=k\cos\theta$ are the Boltzmann constant, electron and proton
temperatures, masses, angle between the wave propagation direction and mean
magnetic field, wave frequency, and parallel component of the wave vector,
respectively. The first and second terms in the brackets on the right hand
side correspond to damping by electrons and protons, respectively. For weakly
magnetized plasma with $v_{A}<v_{S}$, proton heating always dominates the TTD
for $\omega^{2}/k_{||}^{2}\sim v_{S}^{2}\sim k_{\rm B}T_{p}/m_{p}$. If $v_{A}$
does not change dramatically in the downstream, the continuous heating of
background particles through the TTD processes makes
$T_{p}\rightarrow(m_{p}/m_{e})T_{e}$ since the heating rates are proportional
to $(mT)^{1/2}$, where $m$ and $T$ represent the mass and temperature of the
particles, respectively. We see that parallel propagating waves (with
$\sin\theta=0$) are not subject to the TTD processes and can accelerate some
particles to very high energy through cyclotron resonances. Obliquely
propagating waves are damped efficiently by the background particles. Although
the damping rates for waves propagating nearly perpendicular to the magnetic
field ($\cos\theta\simeq 0$) are also low, these waves are subject to damping
by magnetic field wandering (Petrosian et al., 2006). The turbulence power
spectrum cuts off sharply when the damping rate becomes comparable to the
turbulence cascade rate
$\Gamma=\tau_{edd}^{-2}/(\tau^{-1}_{F}+\tau^{-1}_{edd})\simeq\tau_{edd}^{-2}\tau_{F}$
(Jiang et al., 2008). One can define a critical propagation angle
$\theta_{c}(k)$, where $\Lambda_{T}(\theta_{c},k)=\Gamma(k)$. Equations (9)
and (11) give
${v_{A}^{2}k_{d}^{1/2}\over
2^{1/2}\pi^{3/2}v_{S}v_{F}k^{1/2}}\simeq{\sin^{2}\theta_{c}\over\cos\theta_{c}}\exp\left(-{v_{F}^{2}\over
2v_{S}^{2}\cos^{2}\theta_{c}}\right)\,,$ (12)
where the electron damping term has been ignored. The turbulence spectra at
several locations in the downstream are shown in Figure 2.
Figure 2: The turbulence spectra $v_{edd}^{2}(k)$ at several locations in the
downstream as indicated. The Kolmogorov, Kraichnan, and damping ranges are
indicated for the supersonic phase spectrum with $x=0.010L$. At the other
locations, the turbulence is subsonic and there are only Kraichnan and damping
ranges. The sharp drops of the turbulence spectra in the damping range are due
to the onset of thermal damping at the coherent length of the magnetic field
$2\pi/k_{d}$. Figure 3: Evolution of the acceleration efficiency $\eta$
(dotted), cutoff Lorentz factor $\gamma_{c}$ (dotted-dashed), spectral index
$p$ (dashed), and $\tau=\tau_{\rm ac}/\tau_{\rm d}$ (solid) in the downstream
for $v_{A}=0.0633U$ and $U=0.01c$. The particle acceleration is significant
for $\tau<1$, i.e., between the two vertical dashed lines indicating $x_{1}$
and $x_{2}$. For $\gamma_{c}$, we have assumed that $B=12\mu$G and $L=0.15$pc.
For $\eta$, the two thin lines are for $U=0.015c$ (higher) and $0.0067c$
(lower). See the following section for details
The escape time of relativistic particles with $v\simeq c$, where $c$ is the
speed of light, from the particle acceleration region is given by
$\tau_{esc}=(L^{2}/4c^{2})\tau_{sc}$ and the spectral index of the accelerated
particles in the steady state is given by
$\displaystyle p$ $\displaystyle=$ $\displaystyle\left({9\over
4}+{\tau_{ac}\over\tau_{esc}}\right)^{1/2}-{1\over 2}=\left({9\over
4}+{12c^{2}k_{m}^{2}\over v_{F}^{2}k_{d}^{2}}\right)^{1/2}-{1\over 2}$ (13)
$\displaystyle=$ $\displaystyle\left[{9\over 4}+{12c^{2}v_{A}^{8}\over
u^{6}v_{F}^{2}\min(v_{F}^{2},u^{2})}\right]^{1/2}-{1\over 2}\,.$
We note that for $v_{A}$ independent of $x$ in the downstream, $p$ reaches its
minimum at the transonic point $x_{0}$, where $v_{F}=u$. The maximum energy
that particles can reach though resonant interactions with these parallel
propagating waves is given by
$\gamma_{c}={2\pi qB\over m_{e}c^{2}k_{d}}={qBLv_{A}^{4}\over
m_{e}c^{2}u^{3}\min(v_{F},u)}\,,$ (14)
where $q$ is the elementary charge units. The ratio of the dissipated energy
carried by non-thermal particles to that of the thermal particles should be
greater than
$\eta={\theta_{c}^{2}(k_{d})\over 4}={e^{5/6}v_{A}^{2}\over
2(2\pi)^{3/2}v_{S}v_{F}}={e^{5/6}v_{A}^{2}\over 2(2\pi)^{3/2}v_{S}v_{F}}\,,$
(15)
where $e=2.72$, since the isotropic turbulence with $k<k_{d}$ can also
accelerate particles with the Lorentz factor $\gamma\geq\gamma_{c}$. To have
efficient acceleration of relativistic particles, the turbulence decay time
$\tau_{d}=3\max(u,v_{F})L/C_{1}u^{2}$ should be longer than the acceleration
time $\tau_{ac}=({3c^{2}/v_{F}^{2}})\tau_{sc}=6\pi
c/v_{F}^{2}k_{d}=3cv_{A}^{4}L/v_{F}^{2}u^{3}\min(v_{F},u)\,,$ which implies
$\max(u,v_{F})L/C_{1}u^{2}>cv_{A}^{4}L/v_{F}^{2}u^{3}\min(v_{F},u)$, i.e.,
$C_{1}<u^{2}v_{F}^{3}/cv_{A}^{4}\,.$ There are at most two locations
$x_{1}<x_{2}$ in the down stream, where $\tau=\tau_{\rm ac}/\tau_{\rm d}=1$
and $C_{1}=u^{2}v_{F}^{3}/cv_{A}^{4}$. In combination with equation (13),
significant particle acceleration occurs for
$p<\left[{9/4}+{12v_{F}^{2}\max(u^{2},v_{F}^{2})/u^{4}C_{1}^{2}}\right]^{1/2}-{1/2}\,.$
The particle acceleration in the supersonic phase, i.e., $v_{F}<u$, therefore
produces very hard electron distributions with $p<1.39$ for $C_{1}=3$. Softer
electron distributions have to be produced in the subsonic phase. Figure 3
shows the evolution of $\eta$, $\gamma_{c}$, $p$, and $\tau=\tau_{\rm
ac}/\tau_{\rm d}$ in the downstream for $U=0.01c$. The profiles of $v_{F}/U$
and $u/U$ only depend on $v_{A}/U$. So is the profile of $\eta$. $\tau$ and
$p$ also depend on the absolute value of $U$. To obtain $\gamma_{c}$, one
needs to know $L$ and $B$ as well. In the extremely supersonic phase with
$v_{F}\ll u$, the SA is negligible. The SA is significant only after the
plasma is already heated up so that $v_{F}\sim u$. In the late subsonic phase,
$u\ll v_{F}$, the SA is also insignificant since most of the free energy of
the system has been converted into heat.
Figure 4: Normalized nonthermal electron distribution $f$ produced at several
locations in the downstream. Figure 5: The distributions of nonthermal
electrons $F$ in the downstream.
The particle distribution may be approximated reasonably well with
$f(x,\gamma)\propto\gamma^{-p(x)}\exp-[\gamma/\gamma_{c}(x)]^{1/2}$(Liu et
al., 2006). Then the distribution of non-thermal particle in the downstream
$F(x,\gamma)=\int_{x_{1}}^{x}f(x^{\prime},\gamma)\eta(x^{\prime})(4Q/m_{e}c^{2}U){\rm
d}x^{\prime}$ (16)
where $\int_{m_{p}/m_{e}}^{\infty}\gamma f(x^{\prime},\gamma){\rm d}\gamma=1$,
and $\int_{m_{p}/m_{e}}^{\infty}\gamma m_{e}c^{2}F(x,\gamma){\rm d}\gamma$
gives the energy density of non-thermal particles at $x$. If
$u^{5}/cv_{A}^{4}<C_{1}$ at the sonic point $x_{0}$, then $x_{0}<x_{1}$ and
there will be no particle acceleration in the supersonic phase. Figures 4 and
5 show the normalized electron distribution $f$ and $F$ at several locations
in the downstream, respectively.
## 3 Results
Here, we use the SNR RX J1713.7-3946 as an example to demonstrate how the SA
by fast mode waves accounts for the observed broadband spectrum. Figure 6
shows the best fit with $v_{\rm A}/U=0.0633$, $L=4.71\times 10^{17}$ cm,
$B=12$ $\mu$G and $U=0.01c$. Comparing to the thin dashed line, which is
derived by assuming an electron distribution
$\propto\gamma^{-p}\exp-(\gamma/\gamma_{c})^{1/2}$, there is a radio spectral
bump due to electron acceleration relatively far from the shock front (see
Figs. 4 and 5). In our model, there are five parameters: $B$, $U$, $v_{A}$,
$L$, and the equivalent volume of a uniform emission range. The last is 4
times bigger than the volume of the SNR, suggesting higher nonthermal electron
densities in the interior of the remnant than near the shock front. The
observed radio to X-ray spectral index, X-ray to TeV flux ratio, location of
the X-ray cutoff, and bolometric luminosity of the source give four
constraints, which leads to one more degree of freedom. Our model fit to the
spectrum therefore is not unique. However, $B$ is uniquely determined by the
ratio of the X-ray to TeV flux. To reproduce the observed spectral shape, the
profiles of $p$, $\gamma_{c}$, and $\eta$ should not change, which implies
that $v_{A}^{8}c^{2}\propto u^{10}$ and $L\propto u^{4}/v_{A}^{4}$ at the
transonic point $x_{0}$. For $v_{A}\ll U$, $u$ is proportional to $U$. We
therefore obtain $v_{A}$ and $L$ as functions of $U$ as indicated in Figure 7.
The density can be derived from $B$ and $v_{A}$, and the overall acceleration
efficiency is defined as $\eta_{eq}=\int_{x_{1}}^{x_{2}}\eta Q{\rm
d}x/\int_{0}^{\infty}Q{\rm d}x.$ Nearly identical spectrum can be obtained for
parameters on these lines. We note $\eta\propto v_{A}^{2}/v_{S}v_{F}\propto
v_{A}^{2}/U^{2}\propto U^{1/2}$. The acceleration is more efficient in the
earlier phase of the remnant evolution. The two thin dotted lines in Figure 3
show the dependence of $\eta$ on $U$.
Figure 6: Best fits to the observed spectrum. The dashed line is for a simple
power-law model with a gradual high energy cutoff. The solid line is for the
fiducial model. The low and high energy spectral peaks are produced through
the synchrotron and inverse Compon scattering of the background photos,
respectively. Figure 7: Nearly identical fit to the observations are obtained
for parameters on these lines.
## References
* Cho & Vishniac (2000) Cho, J., & Vishniac, E. T. 2000, ApJ, 538, 217
* Jiang et al. (2008) Jiang, Y. W., Liu, S. M., & Petrosian, V. 2008, astro-ph/0802.0910
* Liu et al. (2006) Liu, S. M., Fan, Z. H., Fryer, C. L., Wang, J. M., & Li, H., 2008, ApJ, 683, L163
* Micono et al. (1999) Micono, M., Zurlo, N., Massaglia, S., Ferrari, A., & Melrose, D. B. 1999, AA, 349, 323
* Niemiec et al. (2008) Niemiec, J., Pohl, M., Stroman, T., & Nishikawa, K. I. 2008, ApJ, 684, 1171
* Petrosian et al. (2006) Petrosian, V., Yan, H. R., & Lazarian, A. 2006, ApJ, 644, 603
* Stix (1962) Stix, T. H. 1962, The Theory of Plasma Waves (McGraw-Hill Book Company, inc.)
* Takaaki et al. (2008) Takaaki, T., et al. 2008, ApJ, in press, astro-ph/0806.1490
* Wang et al. (1997) Wang, Z. R., Qu, Q.-Y., & Chen, Y. 1997, A&A, 318L, 59
* Yeung & Zhou (1997) Yeung, P. K., & Zhou, Y. 1997,PhRvE, 56, 1746
|
arxiv-papers
| 2008-09-16T16:50:35
|
2024-09-04T02:48:57.805709
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Siming Liu, Zhong-Hui Fan, Christopher L. Fryer",
"submitter": "Zhonghui Fan",
"url": "https://arxiv.org/abs/0809.2756"
}
|
0809.2966
|
# Nuclear matter and neutron matter for improved quark mass density- dependent
model with $\rho$ mesons
Chen Wu1 and Ru-Keng Su1,2,3111rksu@fudan.ac.cn 1\. Department of Physics,
Fudan University,Shanghai 200433, P.R. China
2\. CCAST(World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China
3\. Center of Theoretical Nuclear Physics,
National Laboratory of Heavy Ion Collisions, Lanzhou 730000, P.R. China
###### Abstract
A new improved quark mass density- dependent model including u, d quarks,
$\sigma$ mesons, $\omega$ mesons and $\rho$ mesons is presented. Employing
this model, the properties of nuclear matter, neutron matter and neutron star
are studied. We find that it can describe above properties successfully. The
results given by the new improved quark mass density- dependent model and by
the quark meson coupling model are compared.
###### pacs:
12.39.-x, 14.20.-c, 05.45.Yv, 26.60.+c
## I Introduction
In our previous papers [1-6], a new quark meson coupling model bases on quark
mass density- dependent(QMDD) model is presented. The QMDD model suggested by
Fowler, Raha and Weiner [7] firstly assuming that the masses of u, d and s
quarks(and the corresponding antiquarks) satisfy:
$\displaystyle m_{q}=\frac{B}{3n_{B}}(i=u,d,\bar{u},\bar{d})$ (1)
$\displaystyle m_{s,\bar{s}}=m_{s0}+\frac{B}{3n_{B}}$ (2)
where $n_{B}$ is the baryon number density, $m_{s0}$ is the current mass of
the strange quark, and $B$ is the bag constant. As was explained in Refs.[1,
2, 5, 8], the ansatz Eqs. (1) and (2) corresponds to a quark confinement
hypothesis because when $V\rightarrow\infty,n_{B}\rightarrow 0$ and
$m_{q}\rightarrow\infty$, it prevents the quark goes to infinity. It is shown
that the properties of strange quark matter in the QMDD model are nearly the
same as those obtained in the MIT bag model [9, 10]. But the basic difference
is that instead of the MIT bag boundary condition, we have the density-
dependent masses of quarks in QMDD model according to Eqs. (1) and (2). It
means that the ansatz Eqs. (1) and (2) can replace MIT bag boundary condition
and get the nearly the same results.
Quark- meson coupling(QMC) model suggested by Guichon [11] firstly is a famous
hybrid quark meson model which can describe many physical properties of
nuclear matter and nuclei successfully [12]. In this model, the nuclear system
was suggested as a collection of MIT bag and mesons. The interactions between
quarks and mesons are limited within the MIT bag regions. As was pointed in
Refs. [1, 2, 6], this model has two major shortcomings: (1) It is a permanent
quark confinement model because the MIT bag boundary condition cannot be
destroyed by temperature and density. Therefore, it cannot describe the quark
deconfinement phase transition. (2) It is difficult to do nuclear many-body
calculation beyond mean field approximation(MFA) by means of QMC model,
because we cannot find the free propagators of quarks and mesons easily. The
reason is that the interactions between quarks and mesons are limited within
the bag regions, the multireflection of quarks and mesons by MIT bag boundary
must be taken into account for getting the free propagators. These two
shortcomings come from MIT bag constrain all.
To overcome these two shortcomings, we suggested an improved quark mass
density- dependent(QMDD) model in Refs. [1-6]. We added the $\sigma$\- meson
and $\omega$\- meson to improve the QMDD model. Instead of the MIT bag, after
introducing the nonlinear interaction of $\sigma$-mesons and qq$\sigma$
coupling, we construct a Friedberg- Lee soliton bag in nuclear system. The
quark masses are still density- dependent. The interactions between quarks and
mesons are extended to the whole system. Since the MIT bag constraint is given
up, our improved QMDD(IQMDD) model can describe the quark deconfinement phase
transition [6] and do the nuclear many- body calculations beyond MFA in
principle. We have proved that our model can successfully describe the
saturation properties, the equation of state, the compressibility and the
effective nucleon mass of symmetric nuclear matter and give a reasonable
critical temperature of quark deconfinement [1-6].
The motivation of this paper is to extend our study to asymmetric nuclear
matter, especially to the neutron matter and the neutron star. It means that
we must consider the isosipn dependence and distinguish the u-quark and
d-quark. We will add the isospin vector $\rho$ mesons to improve the IQMDD
model in this paper. We hope to compare the results of IQMDD model with those
obtained by QMC model and QHD-II model for neutron matter and neutron star. In
order to find their differences and similarity explicitly, we will use the
same approximation as that of the QMC model [13] in our calculations. Though
the study of neutron star employing QMC model in Ref. [14] is too simple, but
in order to exhibit the basic differences between the IQMDD model and the QMC
model, we still consider the neutron star by using the same approximation as
Ref. [14].
The organization of this paper is as follows. In the next section, we give the
main formulas of the IQMDD model under the mean field approximation. The main
formulas of neutron stars in also included. In the third section, some
numerical results are presented. The last section contains a summary and
discussions.
## II Formulas of the IQMDD model with $\rho$ meson
### II.1 IQMDD model for nuclear matter
The Lagrangian density of IQMDD model with $\sigma,\omega,$ and $\rho$ mesons
is :
$\displaystyle\mathcal{L}=\bar{\psi}[i\gamma^{\mu}\partial_{\mu}-m_{q}+g^{q}_{\sigma}\sigma-g^{q}_{\omega}\gamma^{\mu}\omega_{\mu}-g^{q}_{\rho}\gamma^{\mu}\vec{\tau}\cdot\vec{\rho}^{\mu}]\psi+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma\hskip
0.0pt$
$\displaystyle-U(\sigma)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}+\frac{1}{2}m_{\rho}^{2}\vec{\rho_{\mu}}\cdot\vec{\rho^{\mu}}-\frac{1}{4}\vec{G_{\mu\nu}}\vec{G^{\mu\nu}}$
(3)
where
$\displaystyle
U(\sigma)=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}b\sigma^{3}+\frac{1}{4}c\sigma^{4}+B,$
(4)
$\displaystyle-B=\frac{m_{\sigma}^{2}}{2}\sigma_{v}^{2}+\frac{b}{3}\sigma_{v}^{3}+\frac{c}{4}\sigma_{v}^{4},$
(5)
$\displaystyle\sigma_{v}=\frac{-b}{2c}[1+\sqrt{1-4m_{\sigma}^{2}c/b^{2}}],$
(6)
and the quark mass $m_{q}$(q = u, d) is given by Eq. (1). $m_{\sigma}$ and
$m_{\omega}$ are the masses of $\sigma$ and $\omega$ mesons,
$F_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$,
$\overrightarrow{G}_{\mu\nu}=\partial_{\mu}\overrightarrow{\rho}_{\nu}-\partial_{\nu}\overrightarrow{\rho}_{\mu}$,
$g_{\sigma}^{q}$, $g_{\omega}^{q}$ and $g_{\rho}^{q}$ are the coupling
constants between quark and $\sigma$ meson, quark and $\omega$ meson and quark
and $\rho$ meson respectively.
The equation of motion for quark field under MFA in the whole space is
$\displaystyle[i\gamma\cdot\partial-(m_{q}-g_{\sigma}^{q}\bar{\sigma})-\gamma^{0}(g_{\omega}^{q}\bar{\omega}+\frac{1}{2}g_{\rho}^{q}\tau_{z}\bar{\rho})]\psi=0$
(7)
where $\bar{\sigma},\bar{\omega}$ are the mean field values of the $\sigma$
field and the corresponding time component of $\omega$ field respectively,
$\bar{\rho}$ is the mean field value of the time component in the third
direction of isospin for $\rho$ field, $\tau_{z}$ is the third component of
the Pauli matrix. The effective quark mass $m_{q}^{*}$ is given by:
$\displaystyle m_{q}^{*}=m_{q}-g_{\sigma}^{q}\bar{\sigma}$ (8)
In nuclear matter, three quarks constitute a Freidberg-Lee soliton bag [15],
and the effective nucleon mass is obtained from the bag energy and reads:
$\displaystyle M_{N}^{*}=\Sigma_{q}E_{q}=\Sigma_{q}\frac{4}{3}\pi
R^{3}\frac{\gamma_{q}}{(2\pi)^{3}}\int_{0}^{K_{F}^{q}}\sqrt{{m^{*}_{q}}^{2}+k^{2}}(\frac{dN_{q}}{dk})dk$
(9)
where $\gamma_{q}$ is the quark degeneracy, $K_{F}^{q}$ is Fermi energy of
quarks. $dN_{q}/dk$ is the density of states for various quarks in a spherical
cavity. The expression of $dN_{q}/dk$ adopted in this paper con be found in
Ref. [16].
The Fermi energy $K_{F}^{q}$ of quarks is given by
$\displaystyle 3=\frac{4}{3}\pi R^{3}n_{B}$ (10)
where $n_{B}$ satisfies
$\displaystyle
n_{B}=\Sigma_{q}\frac{\gamma_{q}}{(2\pi)^{3}}\int_{0}^{K_{F}^{q}}(\frac{dN_{q}}{dk})dk$
(11)
The bag radius $R$ is determined by the equilibrium condition for the nucleon
bag:
$\displaystyle\frac{\delta M^{*}_{N}}{\delta R}=0$ (12)
In nuclear matter, the total energy density and pressure density read
$\displaystyle\varepsilon_{matter}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{N}}{(2\pi)^{3}}(\int_{0}^{K_{F}^{p}}+\int_{0}^{K_{F}^{n}})\sqrt{{M_{N}^{*}}^{2}+p^{2}}dp^{3}+\frac{g_{\omega}^{2}}{2m_{\omega}^{2}}\rho_{B}^{2}$
(13)
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\bar{\sigma}^{2}+\frac{1}{3}b\bar{\sigma}^{3}+\frac{1}{4}c\bar{\sigma}^{4}+\frac{g_{\rho}^{2}}{8m_{\rho}^{2}}{\rho_{3}^{2}}$
and
$\displaystyle p_{matter}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\frac{\gamma_{N}}{(2\pi)^{3}}(\int_{0}^{K_{F}^{p}}+\int_{0}^{K_{F}^{n}})\frac{p^{2}}{\sqrt{{M_{N}^{*}}^{2}+p^{2}}}dp^{3}+\frac{g_{\omega}^{2}}{2m_{\omega}^{2}}\rho_{B}^{2}$
(14)
$\displaystyle-\frac{1}{2}m_{\sigma}^{2}\bar{\sigma}^{2}-\frac{1}{3}b\bar{\sigma}^{3}-\frac{1}{4}c\bar{\sigma}^{4}+\frac{g_{\rho}^{2}}{8m_{\rho}^{2}}{\rho_{3}^{2}}$
where $\gamma_{N}=2$ is degeneracy of proton or neutron, $K_{F}^{p}$ and
$K_{F}^{n}$ is Fermi momenta of proton and neutron, and $\rho_{3}$ is the
difference between the proton and neutron densities, respectively. Therefore
$\displaystyle\rho_{p}=\frac{1}{3\pi^{2}}{K_{F}^{p}}^{3},\rho_{n}=\frac{1}{3\pi^{2}}{K_{F}^{n}}^{3},$
(15)
Where $\rho_{p}$ and $\rho_{n}$ is the density of proton and neutron
respectively, and the density of nuclear matter $\rho_{B}$ resds
$\displaystyle\rho_{B}=\rho_{p}+\rho_{n}$ (16)
In Eqs. (13, 14), $g_{\omega}$ and $g_{\rho}$ are the coupling constants
between nucleon and $\omega$ meson, nucleon and $\rho$ meson respectively.
They satisfies $g_{\omega}=3g_{\omega}^{q}$ and $g_{\rho}=g_{\rho}^{q}$ [13].
In MFA, the $\bar{\omega}$ is determied by baryon number conservation
$\displaystyle\bar{\omega}=\frac{g_{\omega}\rho_{B}}{m_{\omega}^{2}}$ (17)
As that of the QMC model [13], $\bar{\sigma}$ and $\bar{\rho}$ are given by
the thermodynamics conditions:
$\displaystyle(\frac{\partial\varepsilon_{matter}}{\partial\bar{\sigma}})_{R,\rho_{B}}=0,\
\ \ \ and\ \ \ \
(\frac{\partial\varepsilon_{matter}}{\partial\bar{\rho}})_{R,\rho_{B}}=0$ (18)
respectively. Therefore, $\bar{\rho}$ is expressed by
$\displaystyle\bar{\rho}=\frac{g_{\rho}}{2m_{\rho}^{2}}\rho_{3}$ (19)
and $\bar{\sigma}$ is given by
$\displaystyle
m_{\sigma}^{2}\bar{\sigma}+b\bar{\sigma}^{2}+c\bar{\sigma}^{3}=-\frac{\gamma_{N}}{(2\pi)^{3}}(\int_{0}^{K_{F}^{p}}+\int_{0}^{K_{F}^{n}})\frac{M_{N}^{*}}{\sqrt{{M_{N}^{*}}^{2}+p^{2}}}d^{3}p(\frac{\partial
M_{N}^{*}}{\partial\bar{\sigma}})_{R}$ (20)
Eqs. (13-20) form a complete set of equations and we can solve numerically.
Our numerical results will be shown in the next section.
Noting that the left hand side of Eq. (20) is a cubic order function of
$\bar{\sigma}$ , except one solution $\bar{\sigma}=0$, there are still two
solutions. This is a general character for adding a nonlinear scalar $\sigma$
field and using MFA to a physical model [17]. But as was pointed in Ref. [1],
we can prove that one of these solutions corresponds to unstable and
unphysical branch, and the other corresponds a stable soliton solution.
Hereafter we give up the unphysical solution and consider the physical
solution only.
We note that the expression for the total energy density, Eq. (13), is very
similar to that of QHD-II model and QMC model. The differences comes from the
effective nucleon mass Eqs. (9) and (1), and the self-consistency condition
for $\sigma$ field, Eq. (20). Let us consider the self-consistency condition
and $(\frac{\partial M_{N}^{*}}{\partial\bar{\sigma}})_{R}$ further. Using the
same argument as that of Ref. [13], we find that the $(\frac{\partial
M_{N}^{*}}{\partial\bar{\sigma}})_{R}$ can be expressed as
$\displaystyle(\frac{\partial
M_{N}^{*}}{\partial\bar{\sigma}})_{R}=-g_{\sigma}\times\left(\begin{array}[]{c}1\\\
C_{1}(\bar{\sigma})\\\
C_{2}(\bar{\sigma})\end{array}\right)\,\,\,\,for\left(\begin{array}[]{c}QHD-
II\\\ QMC\\\ IQMDD\end{array}\right)model$ (27)
where the expression of scalar density factor $C_{1}(\bar{\sigma})$ for QMC
model can be found in Ref. [13]. For IQMDD model, $C_{2}(\bar{\sigma})$ can be
obtained numerically. The curves of $C_{1}(\bar{\sigma})$ and
$C_{2}(\bar{\sigma})$ will be shown in Sec. 3.
### II.2 IQMDD model for neutron star
We now turn to investigate the neutron matter and neutron star for the IQMDD
model. Since the aim of this paper is to compare the IQMDD model and the QMC
model, we use the same approximation to study the neutron star as that of the
QMC model [14]. More detail treatment of neutron star such as the phase
transition for the quark matter and the neutron matter, the contribution of
hyperon and etc, has been neglected. Two basic assumptions: the neutron star
matter is charge neutrality and reaches to the $\beta$-equilibrium, are
adopted [14]. Since we assume that the nucleons and light leptons exist in the
neutron star only, charge neutrality is expressed as
$\displaystyle\rho_{p}=\sum_{l=e,\mu}\rho_{l},$ (28)
where $\rho_{i}$ is the number density of particle $i(=p,e,\mu)$. Under
$\beta$-equilibrium, the processes
$\displaystyle n\rightarrow p+e^{-}+\bar{\nu}_{e},p+e^{-}\rightarrow
n+\nu_{e}$ (29)
occur at the same rate. This condition can be satisfied when the chemical
potentials before and after the decay are same. The chemical potential of each
particle reads
$\displaystyle\mu_{n}$ $\displaystyle=$
$\displaystyle\sqrt{{K_{F}^{n}}^{2}+{m_{N}^{*}}^{2}}+g_{\omega}\bar{\omega}-\frac{1}{2}g_{\rho}\bar{\rho}$
(30) $\displaystyle\mu_{p}$ $\displaystyle=$
$\displaystyle\sqrt{{K_{F}^{p}}^{2}+{m_{N}^{*}}^{2}}+g_{\omega}\bar{\omega}+\frac{1}{2}g_{\rho}\bar{\rho}$
(31) $\displaystyle\mu_{l}$ $\displaystyle=$
$\displaystyle\sqrt{K_{l}^{2}+m_{l}^{2}}$ (32)
where $K_{l}$ is the Fermi momentum of the lepton $l(e,\mu)$. The chemical
equilibrium condition is expressed as
$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e},$ (33)
$\displaystyle\mu_{e}$ $\displaystyle=$ $\displaystyle\mu_{\mu}$ (34)
Once the solution has been found, the equation of state(EoS) can be calculated
from
$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\frac{\gamma_{N}}{(2\pi)^{3}}(\int_{0}^{K_{F}^{p}}+\int_{0}^{K_{F}^{n}})\sqrt{{M_{N}^{*}}^{2}+p^{2}}dp^{3}+\frac{g_{\omega}^{2}}{2m_{\omega}^{2}}\rho_{B}^{2}+\frac{1}{2}m_{\sigma}^{2}\bar{\sigma}^{2}$
(35)
$\displaystyle+\frac{1}{3}b\bar{\sigma}^{3}+\frac{1}{4}c\bar{\sigma}^{4}+\frac{g_{\rho}^{2}}{8m_{\rho}^{2}}{\rho_{3}}^{2}+\frac{1}{\pi^{2}}\sum_{l}\int_{0}^{k_{l}}\sqrt{k^{2}+m_{l}^{2}}k^{2}dk,$
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{3}\frac{\gamma_{N}}{(2\pi)^{3}}(\int_{0}^{K_{F}^{p}}+\int_{0}^{K_{F}^{n}})\frac{p^{2}}{\sqrt{{M_{N}^{*}}^{2}+p^{2}}}dp^{3}+\frac{g_{\omega}^{2}}{2m_{\omega}^{2}}\rho_{B}^{2}-\frac{1}{2}m_{\sigma}^{2}\bar{\sigma}^{2}$
(36)
$\displaystyle-\frac{1}{3}b\bar{\sigma}^{3}-\frac{1}{4}c\bar{\sigma}^{4}+\frac{g_{\rho}^{2}}{8m_{\rho}^{2}}{\rho_{3}}^{2}+\frac{1}{3\pi^{2}}\sum_{l}\int_{0}^{k_{l}}\frac{k^{4}}{\sqrt{k^{2}+m_{l}^{2}}}dk,$
Using the Oppenheimer and Volkoff equation
$\displaystyle\frac{dp(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{Gm(r)\varepsilon}{r^{2}}(1+\frac{p}{\varepsilon
C^{2}})(1+\frac{4\pi r^{3}p}{m(r)C^{2}})(1-\frac{2Gm(r)}{rC^{2}})^{-1}$ (37)
$\displaystyle dM(r)=4\pi r^{2}\varepsilon(r)dr$ (38)
where G is gravitational constant and C is the velocity of light, and the
equation of state for neutron matter given by Eqs. (29), (30) and (9), we can
study the physical behavior of neutron star for IQMDD model.
## III numerical result
Before numerical calculation, let us discuss the parameters in IQMDD model.
First, we choose $m_{\omega}=783$ MeV, $m_{\rho}=770$ MeV and $m_{\sigma}=509$
MeV as that of Ref. [18]. Fixing the nucleon mass $M_{N}=939$ MeV, we get
$B=174$ MeV fm-3. Obviously, the behaviors at the saturation point must be
explained for a successful model. It reveals that nuclear matter saturates at
a density $\rho_{0}=0.15$ fm-3 with a binding energy per particle $E/A=-15$
MeV at zero temperature, and the compression constant to be about
$K(\rho_{0})=210$ MeV. Therefore we fixed
$g_{\omega}^{q}=2.44,g^{q}_{\sigma}=4.67,b=-1460$ MeV to explain above data.
In addition, the symmetry energy coefficient $a_{sym}$ satisfies
$\displaystyle
a_{sym}=\frac{1}{2}(\frac{\partial^{2}(\varepsilon/\rho)}{\partial\alpha^{2}})_{\alpha=0}=(\frac{g_{\rho}}{m_{\rho}})^{2}\frac{k_{0}^{3}}{12\pi^{2}}+\frac{k_{0}^{2}}{6\sqrt{k_{0}^{2}+{M^{\star}_{N}}^{2}}}$
(39)
where
$\displaystyle\alpha=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}},k_{0}=1.42fm^{-1}$
(40)
Using the data $a_{sys}$=33.2 MeV we fix $g_{\rho}=9.07$. The parameters used
to calculate and the results of $K$ and $M_{N}^{*}$ for IQMDD model are shown
in Table 1. For comparison, we also show the corresponding parameters for QMC
model in Table 1. The data and results for QMC model are taken from Ref. [13].
We see in Table 1 all parameters and results are very similar for these two
models. Their differences are not remarkable.
TABLE 1. Comparison of calculated quantities in IQMDD and QMC model |
---|---
| ${g_{\sigma}^{q}}$ | ${g_{\omega}^{q}}$ | ${g_{\rho}^{q}}$ | R(fm) | K(MeV) | $M^{*}_{N}(\rho_{0})(MeV)$
QMC | 5.53 | 1.26 | 8.44 | 0.80 | 200 | 851
IQMDD | 4.67 | 2.44 | 9.07 | 0.85 | 210 | 775
Our results for symmetric nuclear matter and neutron matter are shown in Fig.
1-3. The scalar density factor $C(\bar{\sigma})$ as a function of
$\bar{\sigma}$ is shown in Fig. 1 where the dashed curve refers to
$C_{1}(\bar{\sigma})$ and solid curve to $C_{2}(\bar{\sigma})$ respectively.
This factor plays an essential role to demonstrate the main character of quark
structure for different models. We see from Fig. 1 that the scalar density
factors $C_{1}(\bar{\sigma})$ and $C_{2}(\bar{\sigma})$ are both smaller than
unity(QHD-II model) and decrease when $\bar{\sigma}$ increases. In particular,
$C_{2}(\bar{\sigma})$ is located between the line of unity and the curve of
$C_{1}(\bar{\sigma})$. It means that the values of main physical quantities
given by IQMDD model will almost located between the values given by QHD-II
model and by QMC model. Our results confirm this conclusion.
In Fig. 2, we draw the curves of energy per baryon vs. baryon number density
for both symmetric nuclear matter and for neutron matter respectively. We see
that ignoring $\rho$ meson coupling yields a smaller bound state around
$\rho_{B}\sim 0.10fm^{-3}\sim 0.66\rho_{0}$ in the dotted curve, but it
becomes unbound solid line when the $\rho$ meson contribution is introduced.
The saturation curve for symmetric nuclear matter is shown by dash-dotted
curve in Fig. 2. In fact, the behavior of curves in Fig. 2 is very similar to
that of QMC model, but the corresponding value of $\rho_{B}$ is 0.60$\rho_{0}$
for QMC model and 0.66$\rho_{0}$ for IQMDD model. (See Fig. 2 of Ref. [13])
The equation of state for neutron matter is shown in Fig. 3 where the dashed
curve presents the result when the $\rho$ meson contribution is ignored and
the solid curve corresponds to the full calculation. We see the contribution
of $\rho$ meson is important for the EoS. After comparing with the results of
QHD-II model and QMC model, we come to a conclusion that the shape of equation
of state for neutron matter in IQMDD model is qualitatively similar to that of
QHD-II model and QMC model, it is softer than that of QHD-II model but harder
than that of the QMC model, as is indicated in Fig. 1 by the behavior of
scalar density factor.
Having shown the IQMDD model can provide an successful description for nuclear
and neutron matter, we would like to study the structure and composition of
neutron stars for IQMDD model. We will show that it can successfully describe
the neutron star.
The EoS is given by Eqs. (9), (29) and (30) for IQMDD model when the neutron
star matter reaches to $\beta$ equilibrium. The curve of EoS is shown in Fig.
4. In Fig. 5, we show the particle population including $n,p,e,\mu$ for
different density by solid($n$), dashed($p$), dotted($e$), dash-dotted($\mu$)
curves respectively. The mass of neutron star in units of sun mass
$M/M_{\bigodot}$ as a function of central density $\varepsilon_{c}$ is plotted
in Fig. 6. In Fig. 7 we show the mass radius relation of the neutron star. The
maximum mass of neutron star $M_{max}$ found in Fig. 7 is 1.73 $M_{\bigodot}$.
It is smaller than the value of 2.2$M_{\bigodot}$ given by QMC model [14]. We
would like to emphasize that the above treatment for neutron star is too
rough. Therefore, the value of $M_{max}$ is not important. Our aim is to
demonstrate that all curves shown in Fig. 4-7 are in agreement with those
given by QMC model qualitatively [14]. We come to a conclusion that perhaps
the IQMDD model is a good candidate to replace the QMC model.
## IV Summary and discussion
In summary, we have added the $\rho$ meson to the IQMDD model to study the
asymmetric nuclear matter, especially, the neutron matter and neutron star.
The $u,d,$ quarks, nonlinear scalar $\sigma$ meson field, $\omega$ meson
field, $\rho$ meson field and the corresponding quark mesons couplings are
including in the new IQMDD model. The isospin effect has been considered by
introducing isovector $\rho$ mesons in this model. After fixing the parameters
by the experimental values such as the massed of nucleon, $\sigma$ meson,
$\omega$ meson, $\rho$ meson; saturation point, compression constant and the
symmetry energy, under MFA, we have investigate the physical properties of
nuclear matter and neutron matter. We found that the results given by IQMDD
model are similar to that of QMC model. The values of the main physical
quantities for neutron matter and nuclear matter given by IQMDD model locate
in the regions between the values given by the QHD-II model and by the QMC
model. Employing the IQMDD model, we have studied the neutron star and also
found its properties almost agree with these given by QMC model. We conclude
that the new IQMDD model with $\rho$ meson is successful for describing the
nuclear matter and neutron matter. Perhaps it can play the role to replace the
QMC model.
###### Acknowledgements.
The author C.W. wish to thank Prof. Ren-Xin Xu discussions and correspondence.
This work is supported in part by the National Natural Science Foundation of
People’s Republic of China.
## References
* (1) C. Wu and R.K. Su, Phys. Rev. C 77, 015203 (2008).
* (2) C. Wu, W.L. Qian and R.K. Su, Phys. Rev. C 72, 035205 (2005).
* (3) H. Mao, R.K. Su and W.Q. Zhao, Phys. Rev. C 74, 055204 (2006).
* (4) W.L. Qian and R.K. Su, Int. J. Mod. Phys. A 20, 1931 (2005).
* (5) Y. Zhang and R.K. Su, Phys. Rev. C65, 035202 (2002); Phys. Rev. C 67, 015202 (2003); J. Phys. G 30, 811 (2004).
* (6) C. Wu and R.K. Su, J. Phys. G(to be published)
* (7) G.N. Fowler, S. Raha, and R.M. Weiner, Z. Phys. C 9, 271 (1981)
* (8) S. Chakrabarty, S. Raha and B. Sinha, Phys. Lett. B 229, 112 (1989).
* (9) O.G. Benrenuto and G. Lugones, Phys. Rev. D 51, 1989 (1995); G. Lugones and O.G. Benrenuto, Phys. Rev. D 52, 1276 (1995).
* (10) S. Yin and R.K. Su, Phys. Rev. C77, 055204 (2008).
* (11) P.A.M. Guichon, Phys. Lett. B 200, 235 (1988).
* (12) K. Saito, K. Tsushima and A.W. Thomas, Prog. in. part. Nucl. Phys 58, 1 (2007) and Refs therein.
* (13) K. Saito and A.W. Thomas, Phys. Lett. B 327, 9 (1994).
* (14) D.P. Menezes, P.K. Panda and C. Providencia,, Phys. Phys. C 72, 035802 (2005).
* (15) T.D. Lee, Particle physics and introduction to field theory (Harwood Academic , New York, 1981).
* (16) Y. Zhang, W.L. Qian, S.Q. Ying and R.K. Su, J. Phys. G 27, 2241 (2001).
* (17) B.M. Waldhauser, J.A. Marahu, H. Stöcker and W. Greiner, Phys. Phys. C 38, 1003 (1988).
* (18) R.J. Furnstahl, B.D. Serot and H.B. Tang, Nucl. Phys. A 615, 441 (1997).
Figure 1: Scalar density factors $C_{2}(\bar{\sigma})$ and
$C_{1}(\bar{\sigma})$ as a function of $\bar{\sigma}$ for IQMDD model and QMC
model respectively. Figure 2: Energy per nucleon for symmetric nuclear matter
and for neutron matter. The dash-dotted curve is the saturation curve for
nuclear matter. The solid curve(with $\rho$ meson) and the dotted curve(omit
$\rho$ meson) show the results for neutron matter. Figure 3: The equation of
state for neutron matter. The dashed line show the result for omitting $\rho$
meson, while the solid line corresponds to full consideration. Figure 4:
Equation of state for $\beta$\- equilibrium neutron star matter. Figure 5:
Populations in neutron star matter as a function of density. Figure 6: Neutron
star mass as a function of the central density. Figure 7: Masses of neutron
stars vs. their radii.
|
arxiv-papers
| 2008-09-17T17:21:48
|
2024-09-04T02:48:57.813579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Wu and Ru-Keng Su",
"submitter": "Chen Wu",
"url": "https://arxiv.org/abs/0809.2966"
}
|
0809.3028
|
# DESY 08–150 ISSN 0418-9833
September 2008
New relationships between Feynman integrals
O. V. Tarasov
II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany On leave of absence from Joint
Institute for Nuclear Research, 141980 Dubna (Moscow Region), Russia.
###### Abstract
New types of relationships between Feynman integrals are presented. It is
shown that Feynman integrals satisfy functional equations connecting integrals
with different values of scalar invariants and masses. A method is proposed
for obtaining such relations. The derivation of functional equations for one-
loop propagator- and vertex - type integrals is given. It is shown that a
propagator - type integral can be written as a sum of two integrals with
modified scalar invariants and one propagator massless. The vertex - type
integral can be written as a sum over vertex integrals with all but one
propagator massless and one external momentum squared equal to zero. It is
demonstrated that the functional equations can be used for the analytic
continuation of Feynman integrals to different kinematic domains.
PACS numbers: 02.30.Gp, 02.30.Ks, 12.20.Ds, 12.38.Bx
Keywords: Feynman integrals, functional equations, Appell hypergeometric
function
## 1 A method for deriving functional equations
Feynman integrals play an important role in making precise perturbative
predictions in quantum field theory. As is well-known, these integrals satisfy
recurrence relations [1]\- [4]. In general such relations connect several
integrals $I_{1,n},...,I_{N,n}$ with $n$ internal lines and integrals with a
lesser number of internal lines. They may be written in the following form
$\sum_{i=1}^{N}Q_{i}(\\{m_{j}\\},\\{s_{q}\\},\nu_{l},d)~{}I_{i,~{}n}=\sum_{\begin{subarray}{c}r<n\\\
k\end{subarray}}R_{k,r}(\\{m_{j}\\},\\{s_{m}\\},\nu_{l},d)~{}I_{k,~{}r},$
(1.1)
where $I_{k,r}$ stands for integrals with $r$ internal lines, arbitrary powers
of propagators $\nu_{j}$ and arbitrary shifts of the space-time dimension $d$
; $\\{s_{q}\\}$ is a set of independent scalar invariants that may be formed
from the external momenta. $Q_{i},~{}R_{k}$ are ratios of polynomials
depending on $\\{s_{r}\\}$, masses $m_{j}$ , $\nu_{l}$ and $d$. On the left-
hand side of (1.1) we combined integrals with $n$ internal lines and on the
right hand side integrals with a lesser number of lines.
The key idea in the derivation of the functional equations is to remove
integrals with the maximal number of lines from the relations (1.1) by an
appropriate choice of scalar invariants $\\{s_{q}\\}$, masses $m_{j}^{2}$,
powers of propagators $\nu_{j}$ and space - time dimension $d$.
In order to obtain functional equations from a given equation (1.1) one should
first solve the polynomial system of equations
$Q_{i}(\\{m_{j}\\},\\{s_{q}\\},\nu_{l},d)=0,~{}~{}~{}~{}~{}~{}~{}~{}i=1,{\ldots},N$
(1.2)
with respect to $\\{s_{q}\\},\\{m_{i}\\},\nu_{l},d$ and then take those
solutions for which not all coefficients in front of integrals on the right-
hand side of Eq.(1.1) are vanishing. In many cases functional equations can be
obtained from (1.1) by choosing only kinematic variables $\\{s_{q}\\}$ and
masses $\\{m_{i}\\}$.
We illustrate the method by considering the derivation of the functional
equations for a one-loop integral depending on $n-1$ independent external
momenta:
$I_{n}^{(d)}(\\{m_{l}^{2}\\};~{}\\{p_{ir}\\})=\int\frac{d^{d}q}{i\pi^{{d}/{2}}}\prod_{j=1}^{n}\frac{1}{[(q-p_{j})^{2}-m_{j}^{2}]^{\nu_{j}}},$
(1.3)
where
$p_{ir}=(p_{i}-p_{r})^{2}.$ (1.4)
Here and below, the usual causal prescription for the propagators is
understood, i.e. $1/[q^{2}-m^{2}]\leftrightarrow 1/[q^{2}-m^{2}+i0]$. Any
relation of the form (1.1) can be used for deriving functional equations. In
this paper we will use the following relation [4], [5]:
$\displaystyle G_{n-1}\nu_{j}{\bf
j^{+}}I^{(d+2)}_{n}(\\{m_{l}^{2}\\};\\{p_{ir}\\})-(\partial_{j}\Delta_{n})I^{(d)}_{n}(\\{m_{l}^{2}\\};\\{p_{ir}\\})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\sum_{k=1}^{n}(\partial_{j}\partial_{k}\Delta_{n}){\bf
k^{-}}I^{(d)}_{n}(\\{m_{l}^{2}\\};\\{p_{ir}\\}),$ (1.5)
where the operators ${\bf j^{\pm}}$ etc. shift the indices
$\nu_{j}\to\nu_{j}\pm 1$, $G_{n-1}$ is the Gram determinant
$G_{n-1}=-2^{n}\left|\begin{array}[]{cccc}(p_{1}-p_{n})(p_{1}-p_{n})&(p_{1}-p_{n})(p_{2}-p_{n})&\ldots&(p_{1}-p_{n})(p_{n-1}-p_{n})\\\
(p_{1}-p_{n})(p_{2}-p_{n})&(p_{2}-p_{n})(p_{2}-p_{n})&\ldots&(p_{2}-p_{n})(p_{n-1}-p_{n})\\\
\vdots&\vdots&\ddots&\vdots\\\
(p_{1}-p_{n})(p_{n-1}-p_{n})&(p_{2}-p_{n})(p_{n-1}-p_{n})&\ldots&(p_{n-1}-p_{n})(p_{n-1}-p_{n})\end{array}\right|,$
(1.6)
and $\Delta_{n}$ is the modified Cayley determinant defined as:
$\Delta_{n}=\left|\begin{array}[]{cccc}2m_{1}^{2}{}{}{}{}&~{}~{}~{}~{}m_{1}^{2}+m_{2}^{2}-p_{12}&~{}~{}\ldots{}{}&~{}~{}~{}~{}m_{1}^{2}+m_{n}^{2}-p_{1n}\\\
m_{1}^{2}+m_{2}^{2}-p_{12}{}{}{}&~{}~{}2m_{2}^{2}&~{}~{}\ldots{}{}&~{}~{}~{}~{}m_{2}^{2}+m_{n}^{2}-p_{2n}\\\
\vdots&\vdots&~{}~{}\ddots&\vdots\\\
m_{1}^{2}+m_{n}^{2}-p_{1n}{}{}{}&~{}~{}~{}~{}m_{2}^{2}+m_{n}^{2}-p_{2n}&~{}~{}\ldots{}{}&2m_{n}^{2}\end{array}\right|,$
$\partial_{j}\equiv\frac{\partial}{\partial m_{j}^{2}}.$ (1.7)
We assume that the external momenta are not restricted to some specific
integer dimension and therefore $G_{n-1}$ and $\Delta_{n}$ do not satisfy any
condition specific to a particular value of the space-time dimension.
In the present paper we will consider functional equations only for integrals
$I_{n}^{(d)}$ with the first powers of propagators. Setting all $\nu_{k}=1$ in
Eq. (1.5) yields an equation of the form (1.1) connecting integrals with $n$
and $n-1$ lines. Functional equations for the integral $I_{n-1}^{(d)}$ can be
obtained for each particular $j$ by imposing two conditions:
$G_{n-1}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{j}\Delta_{n}=0,$
(1.8)
and solving them by an appropriate choice of scalar invariants $p_{ij}$ and
masses. There are only $n-1$ independent systems of relations of the type
(1.8), because
$\sum_{k=1}^{n}\partial_{k}~{}\Delta_{n}=-G_{n-1}.$ (1.9)
Since $G_{n-1}$ and $\partial_{j}\Delta_{n}$ are nonlinear in $p_{ij}$ and
masses, each system of equations may have several solutions. The number of
functional equations is less than the number of possible solutions. This is
firstly because coefficients in front of integrals on both sides of Eq. (1.1)
are simultaneously zero for some solutions, and secondly, because not all
functional equations are independent.
## 2 Functional equations for the one-loop propagator - type integral
In accordance with our method described in the previous section, functional
equations for the integral $I^{(d)}_{2}$ can be obtained from equation (1.5)
taken at $n=3$, $~{}\nu_{1}=\nu_{2}=\nu_{3}=1$. We will not derive all
possible functional equations, restricting ourselves only to the case $j=1$ in
(1.5):
$\displaystyle G_{2}{\bf
1^{+}}I_{3}^{(d+2)}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12})$
$\displaystyle-$
$\displaystyle(\partial_{1}\Delta_{3})I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12})$
(2.10) $\displaystyle=$ $\displaystyle
2(p_{12}+p_{23}-p_{13})I_{2}^{(d)}(m_{1}^{2},m_{2}^{2};~{}p_{12})$
$\displaystyle+$ $\displaystyle
2(p_{13}+p_{23}-p_{12})I_{2}^{(d)}(m_{1}^{2},m_{3}^{2};~{}p_{13})$
$\displaystyle-$ $\displaystyle
4p_{23}I_{2}^{(d)}(m_{2}^{2},m_{3}^{2};~{}p_{23}),$
where
$\displaystyle
I_{3}^{(d)}(m_{j}^{2},m_{k}^{2},m_{l}^{2};p_{kl},p_{jl},p_{jk})\\!=\\!\int\frac{d^{d}q}{i\pi^{{d}/{2}}}\frac{1}{[(q-p_{j})^{2}\\!-\\!m_{j}^{2}][(q-p_{k})^{2}\\!-\\!m_{k}^{2}][(q-p_{l})^{2}\\!-\\!m_{l}^{2}]},$
$\displaystyle
I_{2}^{(d)}(m_{j}^{2},m_{k}^{2};~{}p_{jk})=\int\frac{d^{d}q}{i\pi^{{d}/{2}}}\frac{1}{[(q-p_{j})^{2}-m_{j}^{2}][(q-p_{k})^{2}-m_{k}^{2}]}.$
(2.11)
In order to remove integrals $I_{3}^{(d)},I_{3}^{(d+2)}$ from this relation
two conditions must be fulfilled:
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
2p_{12}^{2}+2p_{13}^{2}+2p_{23}^{2}-4p_{12}p_{13}-4p_{12}p_{23}-4p_{13}p_{23}=0,$
$\displaystyle\partial_{1}\Delta_{3}$ $\displaystyle=$ $\displaystyle
2p_{23}(p_{13}+p_{12}-p_{23})-4m_{1}^{2}p_{23}$ (2.12)
$\displaystyle~{}~{}+2m_{2}^{2}(p_{23}+p_{13}-p_{12})+2m_{3}^{2}(p_{23}+p_{12}-p_{13})=0.$
One can solve this system of equations with respect to $p_{13}$ and $p_{23}$.
The nontrivial solutions of the system (2.12) are:
$\displaystyle
p_{13}=s_{13}(m_{1}^{2},m_{2}^{2},m_{3}^{2},p_{12})=\frac{\Delta_{12}+2p_{12}(m_{1}^{2}+m_{3}^{2})-(p_{12}+m_{1}^{2}-m_{2}^{2})\lambda}{2p_{12}},$
$\displaystyle
p_{23}=s_{23}(m_{1}^{2},m_{2}^{2},m_{3}^{2},p_{12})=\frac{\Delta_{12}+2p_{12}(m_{2}^{2}+m_{3}^{2})+(p_{12}-m_{1}^{2}+m_{2}^{2})\lambda}{2p_{12}},$
(2.13)
where
$\lambda=\pm\sigma(p_{12}-m_{1}^{2}+m_{2}^{2})~{}\sqrt{\Delta_{12}+4p_{12}m_{3}^{2}},$
(2.14)
$\sigma(x)=\left\\{\begin{array}[]{ll}+1&~{}~{}~{}~{}\textrm{if}~{}~{}~{}~{}x\geq
0\textrm{,}\\\
-1&~{}~{}~{}~{}\textrm{if}~{}~{}~{}~{}x<0\textrm{,}\end{array}\right.$ (2.15)
$\Delta_{ij}=p_{ij}^{2}+m_{i}^{4}+m_{j}^{4}-2p_{ij}m_{i}^{2}-2p_{ij}m_{j}^{2}-2m_{i}^{2}m_{j}^{2}.$
(2.16)
Substituting Eq. (2.13) into Eq. (2.10) yields the following relation:
$\displaystyle I_{2}^{(d)}(m_{1}^{2},m_{2}^{2};~{}p_{12})=$
$\displaystyle\frac{p_{12}+m_{1}^{2}-m_{2}^{2}-\lambda}{2p_{12}}~{}I_{2}^{(d)}(m_{1}^{2},m_{3}^{2};~{}s_{13}(m_{1}^{2},m_{2}^{2},m_{3}^{2},p_{12}))$
$\displaystyle+$
$\displaystyle\frac{p_{12}-m_{1}^{2}+m_{2}^{2}+\lambda}{2p_{12}}~{}I_{2}^{(d)}(m_{2}^{2},m_{3}^{2};~{}s_{23}(m_{1}^{2},m_{2}^{2},m_{3}^{2},p_{12})).$
(2.17)
All arguments of the integral $I_{2}^{(d)}$ on the left - hand side of (2.17)
are arbitrary. At the same time, the last argument in integrals on the right -
hand side satisfy conditions (2.12). The mass $m_{3}$ in the equation is an
arbitrary parameter and can be chosen at will. Setting $m_{3}=0$ in Eqs.
(2.13)-(2.17), yields
$\displaystyle I_{2}^{(d)}(m_{1}^{2},m_{2}^{2};~{}p_{12})$ $\displaystyle=$
$\displaystyle\frac{p_{12}+m^{2}_{1}-m^{2}_{2}-\alpha_{12}}{2p_{12}}~{}I_{2}^{(d)}(m_{1}^{2},0;~{}s_{13})$
(2.18) $\displaystyle+$
$\displaystyle\frac{p_{12}-m^{2}_{1}+m^{2}_{2}+\alpha_{12}}{2p_{12}}~{}I_{2}^{(d)}(0,m_{2}^{2};~{}s_{23}),$
where
$\displaystyle
s_{13}=\frac{\Delta_{12}+2p_{12}m_{1}^{2}-(p_{12}+m_{1}^{2}-m_{2}^{2})\alpha_{12}}{2p_{12}},$
$\displaystyle
s_{23}=\frac{\Delta_{12}+2p_{12}m_{2}^{2}+(p_{12}-m_{1}^{2}+m_{2}^{2})\alpha_{12}}{2p_{12}},$
(2.19)
$\alpha_{12}=\pm\sigma(p_{12}-m_{1}^{2}+m_{2}^{2})~{}\sqrt{\Delta_{12}}~{}.$
(2.20)
The analytic expression for the integral $I_{2}^{(d)}(0,m^{2};~{}p^{2})$ is
[6], [7]:
$I_{2}^{(d)}(0,m^{2};~{}p^{2})=I_{2}^{(d)}(0,m^{2};~{}0)\,{}_{2}F_{1}\\!\\!\left[\begin{array}[]{c}1,2-\frac{d}{2}\,;\\\
\frac{d}{2}\,;\end{array}\frac{p^{2}}{m^{2}}\right],$ (2.21)
where
$I_{2}^{(d)}(0,m^{2};~{}0)=-\Gamma\left(1-\frac{d}{2}\right)m^{d-4}.$ (2.22)
Thus, relations (2.18) and (2.21) give us the analytic result for the integral
$I_{2}^{(d)}$ with arbitrary masses and external momentum squared. Our result
is in agreement with that presented in [7].
Setting $m_{2}^{2}=0$ in equation (2.18), assuming $|p_{12}|>m_{1}^{2}$ and
taking solution (2.19) corresponding to the $+$ sign in formula (2.20) yields
$I_{2}^{(d)}(m_{1}^{2},0;~{}p_{12})=\frac{m_{1}^{2}}{p_{12}}I_{2}^{(d)}\left(m_{1}^{2},0;~{}\frac{m_{1}^{4}}{p_{12}}\right)+\frac{(p_{12}-m_{1}^{2})}{p_{12}}I_{2}^{(d)}\left(0,0;~{}\frac{(p_{12}-m_{1}^{2})^{2}}{p_{12}}\right).$
(2.23)
The first term on the right - hand side is the same integral $I_{2}^{(d)}$ as
on the left - hand side, but with the last argument inverted. The second term
corresponds to the simple integral $I_{2}^{(d)}$ with both propagators
massless:
$I_{2}^{(d)}(0,0;~{}p^{2})=\frac{1}{i\pi^{d/2}}\int\frac{d^{d}k_{1}}{k_{1}^{2}(k_{1}-p)^{2}}=\frac{-\pi^{\frac{3}{2}}~{}(-p^{2})^{\frac{d}{2}-2}}{2^{d-3}\Gamma\left(\frac{d-1}{2}\right)\sin\frac{\pi
d}{2}}.$ (2.24)
Formula (2.23) can be applied to the analytic continuation of the integral
$I_{2}^{(d)}(m_{1}^{2},0;~{}p_{12})$ into the region of large momenta
$|p_{12}|>m_{1}^{2}$. It can also be used for the analytic continuation of the
integrals $I_{2}^{(d)}$ on the right - hand side of (2.18). Therefore, the
relations (2.18) and (2.23) describe the integral $I_{2}^{(d)}$ with arbitrary
masses and momenta in the whole kinematic region.
It is interesting to note that equation (2.23) corresponds to the well-known
formula for the analytic continuation of Gauss’s hypergeometric function
(2.21) (see, for example, Ref.[8]) :
$\,{}_{2}F_{1}\\!\\!\left[\begin{array}[]{c}1,2-\frac{d}{2}\,;\\\
\frac{d}{2}\,;\end{array}z\right]=\frac{1}{z}~{}\,{}_{2}F_{1}\\!\\!\left[\begin{array}[]{c}1,2-\frac{d}{2}\,;\\\
\frac{d}{2}\,;\end{array}\frac{1}{z}\right]+\frac{\Gamma\left(\frac{d}{2}\right)\Gamma\left(\frac{d}{2}-1\right)}{\Gamma(d-2)}(-z)^{\frac{d}{2}-2}\left(1-\frac{1}{z}\right)^{d-3}.$
(2.25)
Indeed, substituting the explicit expressions (2.21), (2.24) into (2.23) and
canceling common factors we obtain relation (2.25) with $z=p_{12}/m_{1}^{2}$.
## 3 Functional equations for the one-loop vertex - type integral
Functional equations for the vertex - type integral $I_{3}^{(d)}$ will be
derived in the same fashion as for the propagator - type integral. Setting
$n=4$, $\nu_{1}={\ldots}=\nu_{4}=1$ and $j=1$ in Eq.(1.5) yields:
$\displaystyle G_{3}~{}{\bf
1^{+}}I_{4}^{(d+2)}(\\{m_{l}^{2}\\};~{}\\{p_{ij}\\})-(\partial_{1}\Delta_{4})I_{4}^{(d)}(\\{m_{l}^{2}\\};~{}\\{p_{ij}\\})=$
(3.26)
$\displaystyle~{}~{}(\partial_{1}^{2}\Delta_{4})~{}I_{3}^{(d)}(m_{2}^{2},m_{3}^{2},m_{4}^{2};~{}p_{34},p_{24},p_{23})$
$\displaystyle+$
$\displaystyle(\partial_{1}\partial_{2}\Delta_{4})~{}I_{3}^{(d)}(m_{1}^{2},m_{3}^{2},m_{4}^{2};~{}p_{34},p_{14},p_{13})$
$\displaystyle+$
$\displaystyle(\partial_{1}\partial_{3}\Delta_{4})~{}I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},m_{4}^{2};~{}p_{24},p_{14},p_{12})$
$\displaystyle+$
$\displaystyle(\partial_{1}\partial_{4}\Delta_{4})~{}I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12}).$
One can obtain a functional equation for
$I_{3}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12})$ with arbitrary
arguments by appropriately choosing the four variables:
$p_{14},p_{24},p_{34}$, $m_{4}^{2}$. To remove the integrals
$I_{4}^{(d)},I_{4}^{(d+2)}$ from (3.26), two conditions should be satisfied
$G_{3}=0,~{}~{}~{}~{}\partial_{1}\Delta_{4}=0.$ (3.27)
This system of equations depends on 10 variables
$p_{12},p_{13},p_{14},p_{23},p_{24},p_{34},$
$m_{1}^{2},m_{2}^{2},m_{3}^{2},m_{4}^{2}$ and it can be solved by excluding,
for example, $p_{14}$ and $p_{34}$. There are four solutions of the system
(3.27) but appropriate expressions are rather long and for this reason they
will not be presented here. Instead we consider simplified situation, namely
we set in Eq. (3.26) from the very beginning $m_{4}^{2}=0$, and impose the
following conditions
$G_{3}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{1}\Delta_{4}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{1}\partial_{2}\Delta_{4}=0.$
(3.28)
This system can be solved by an appropriate choice of $p_{14},p_{34},p_{24}$.
There are several solutions of (3.28), but only for two of them are
coefficients in front of integrals on the right hand side of (3.26) different
from zero. These solutions are
$\displaystyle p_{14}$ $\displaystyle=$ $\displaystyle s_{14}^{(13)},$
$\displaystyle p_{34}$ $\displaystyle=$ $\displaystyle s_{34}^{(13)},$
$\displaystyle p_{24}$ $\displaystyle=$ $\displaystyle
s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12})$ (3.29) $\displaystyle=$
$\displaystyle\frac{(p_{12}+p_{23}-m_{1}^{2}-m_{3}^{2})p_{13}+(p_{12}-p_{23}-m_{1}^{2}+m_{3}^{2})(m_{3}^{2}-m_{1}^{2}+\alpha_{13})}{2p_{13}},$
where
$\displaystyle s_{14}^{(ij)}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{ij}+2m_{i}^{2}p_{ij}-(p_{ij}+m_{i}^{2}-m_{j}^{2})\alpha_{ij}}{2p_{ij}},$
$\displaystyle s_{34}^{(ij)}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{ij}+2m_{j}^{2}p_{ij}+(p_{ij}+m_{j}^{2}-m_{i}^{2})\alpha_{ij}}{2p_{ij}},$
$\displaystyle\alpha_{ij}$ $\displaystyle=$
$\displaystyle\pm\sigma(p_{ij}-m_{i}^{2}+m_{j}^{2})\sqrt{\Delta_{ij}}.$ (3.30)
Substituting (3) into (3.26) leads to the following functional equation:
$\displaystyle
I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12})=$
$\displaystyle~{}~{}\frac{p_{13}+m_{3}^{2}-m_{1}^{2}+\alpha_{13}}{2p_{13}}~{}I_{3}^{(d)}(m_{2}^{2},m_{3}^{2},0;~{}s_{34}^{(13)},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),p_{23})$
$\displaystyle~{}+\frac{p_{13}-m_{3}^{2}+m_{1}^{2}-\alpha_{13}}{2p_{13}}~{}I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},0;~{}s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),s_{14}^{(13)},p_{12}).$
(3.31)
Relation (3.31) means that the integral $I_{3}^{(d)}$ with arbitrary arguments
can always be expressed in terms of integrals with at least one massless
propagator. The only exceptional case, when $p_{12}=p_{13}=p_{23}=0$, is
trivial. In turn, integrals $I_{3}^{(d)}$ with one massless propagator can be
represented as a sum over integrals with two massless propagators. Indeed,
setting $m_{2}^{2}=0$ in Eq. (3.31) yields:
$\displaystyle I_{3}^{(d)}(m_{1}^{2},0,m_{3}^{2};~{}p_{23},p_{13},p_{12})=$
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{p_{13}-m_{1}^{2}+m_{3}^{2}+\alpha_{13}}{2p_{13}}~{}I_{3}^{(d)}(0,m_{3}^{2},0;~{}s_{34}^{(13)},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),p_{23})$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{p_{13}+m_{1}^{2}-m_{3}^{2}-\alpha_{13}}{2p_{13}}~{}I_{3}^{(d)}(m_{1}^{2},0,0;~{}s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),s_{14}^{(13)},p_{12}).$
(3.32)
Taking into account the symmetry of the integral $I_{3}^{(d)}$ with respect to
its arguments one can use Eq. (3.32) to express integrals on the right hand
side of (3.31) in terms of integrals with two propagators massless. Thus in
case when external momenta squared are different from zero the following
relation holds:
$\displaystyle
I_{3}^{(d)}(m_{1}^{2},m_{2}^{2},m_{3}^{2};~{}p_{23},p_{13},p_{12})=$
$\displaystyle\frac{(p_{13}+m_{3}^{2}-m_{1}^{2}+\alpha_{13})(p_{23}+m_{3}^{2}-m_{2}^{2}+\alpha_{23})}{4p_{13}p_{23}}$
$\displaystyle\times
I_{3}^{(d)}(m_{3}^{2},0,0;~{}s_{24}(m_{2}^{2},m_{3}^{2},s_{34}^{(13)},p_{23},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12})),s_{34}^{(23)},s_{34}^{(13)})$
$\displaystyle+\frac{(p_{13}+m_{3}^{2}-m_{1}^{2}+\alpha_{13})(p_{23}-m_{3}^{2}+m_{2}^{2}-\alpha_{23})}{4p_{13}p_{23}}$
$\displaystyle\times
I_{3}^{(d)}(m_{2}^{2},0,0;~{}s_{24}(m_{2}^{2},m_{3}^{2},s_{34}^{(13)},p_{23},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12})),s_{14}^{(23)},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}))$
$\displaystyle+\frac{(p_{13}-m_{3}^{2}+m_{1}^{2}-\alpha_{13})(p_{12}+m_{2}^{2}-m_{1}^{2}+\alpha_{12})}{4p_{13}p_{12}}$
$\displaystyle\times
I_{3}^{(d)}(m_{2}^{2},0,0;~{}s_{24}(m_{1}^{2},m_{2}^{2},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),p_{12},s_{14}^{(13)}),s_{34}^{(12)},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}))$
$\displaystyle+\frac{(p_{13}-m_{3}^{2}+m_{1}^{2}-\alpha_{13})(p_{12}-m_{2}^{2}+m_{1}^{2}-\alpha_{12})}{4p_{12}p_{13}}$
$\displaystyle\times
I_{3}^{(d)}(m_{1}^{2},0,0;~{}s_{24}(m_{1}^{2},m_{2}^{2},s_{24}(m_{1}^{2},m_{3}^{2},p_{23},p_{13},p_{12}),p_{12},s_{14}^{(13)}),s_{14}^{(12)},s_{14}^{(13)}).$
(3.33)
There is one further simplification of note. Setting
$m_{1}^{2}=m_{3}^{2}=p_{24}=0$ in Eq. (3.26) from the very beginning and
solving system of equations
$G_{3}=0,~{}~{}~{}~{}~{}~{}\partial_{1}\Delta_{4}=0,$ (3.34)
with respect to $p_{14}$ and $p_{34}$ yields a nontrivial relationship:
$\displaystyle I_{3}^{(d)}(0,m^{2},0;~{}p_{23},p_{13},p_{12})=$
$\displaystyle-\frac{[b(p_{23},p_{12})-(p_{12}+m^{2})\alpha_{123}]m^{2}}{2\Lambda_{3}}~{}I_{3}^{(d)}(m^{2},0,0;~{}\kappa_{14},0,p_{12})$
$\displaystyle-\frac{[b(p_{12},p_{23})+(p_{23}+m^{2})\alpha_{123}]m^{2}}{2\Lambda_{3}}~{}I_{3}^{(d)}(m^{2},0,0;~{}\kappa_{34},0,p_{23})$
$\displaystyle+\frac{\Lambda_{3}-p_{13}(p_{12}p_{23}-m^{4})-m^{2}(p_{12}-p_{23})\alpha_{123}}{2\Lambda_{3}}~{}I_{3}^{(d)}(0,0,0;~{}\kappa_{34},\kappa_{14},p_{13}),$
(3.35)
where
$\displaystyle\kappa_{14}=\frac{a(p_{23},p_{12})+m^{2}b(p_{23},p_{12})\alpha_{123}}{2\Lambda_{3}},$
$\displaystyle\kappa_{34}=\frac{a(p_{12},p_{23})-m^{2}b(p_{12},p_{23})\alpha_{123}}{2\Lambda_{3}},$
(3.36)
$\displaystyle\Lambda_{3}=(m^{4}-p_{12}p_{23})(p_{23}-p_{12})-(p_{12}+m^{2})b(p_{12},p_{23}),$
$\displaystyle
a(p_{12},p_{23})=m^{2}(p_{12}-p_{23})^{2}(p_{23}-m^{2})+[2(m^{2}-p_{23})p_{12}-m^{2}p_{13}](p_{23}+m^{2})p_{13},$
$\displaystyle
b(p_{12},p_{23})=(m^{2}+p_{13}+p_{12}-p_{23})p_{23}+m^{2}(p_{13}-p_{12}),$
$\displaystyle\alpha_{123}=\sigma(b(p_{12},p_{23}))\sqrt{\Delta_{123}},$
$\displaystyle\Delta_{123}=p_{12}^{2}+p_{13}^{2}+p_{23}^{2}-2p_{12}p_{13}-2p_{12}p_{23}-2p_{13}p_{23}.$
(3.37)
Therefore, by using Eq.(3.35) we can represent the integral $I_{3}^{(d)}$ with
arbitrary masses and nonzero kinematic variables as a combination of integrals
with two massless propagators, one momentum squared equal to zero and
integrals with all propagators massless. An analytic result for the integral
$I_{3}^{(d)}$ with all propagators massless is known in terms of ${}_{2}F_{1}$
functions (see Ref. [9]).
Integrals $I_{3}^{(d)}$ with two massless propagators on the right - hand side
of (3.35) can be evaluated analytically. In the kinematic region $|p_{12}|\leq
m^{2}$ and $|p_{13}|\leq m^{2}$ we find
$\displaystyle
I_{3}^{(d)}(0,m^{2},0;~{}0,p_{13},p_{12})=-\frac{I_{2}^{(d)}(0,0;~{}p_{13})}{m^{2}}\,{}_{2}F_{1}\\!\\!\left[\begin{array}[]{c}1,\frac{d-2}{2}\,;\\\
d-2\,;\end{array}\frac{p_{12}-p_{13}}{m^{2}}\right]$ (3.40)
$\displaystyle~{}~{}~{}~{}~{}+\frac{1}{m^{2}}~{}I_{2}^{(d)}(0,m^{2};~{}0)~{}F_{1}\left(1,1,2-\frac{d}{2},\frac{d}{2};\frac{p_{12}-p_{13}}{m^{2}},\frac{p_{12}}{m^{2}}\right),$
(3.41)
where $F_{1}$ is the Appell hypergeometric function [10] which admits a simple
one-fold integral representation:
$F_{1}\left(1,1,2-\frac{d}{2},\frac{d}{2};x,y\right)=\frac{(d-2)}{2}\int_{0}^{1}du\frac{[(1-u)(1-yu)]^{\frac{d}{2}-2}}{(1-xu)}.$
(3.42)
Thus by using (3.41) one can obtain the result for the integral $I_{3}^{(d)}$
in terms of the Appell function $F_{1}$ and Gauss’s hypergeometric function
${}_{2}F_{1}$. This result is in agreement with the result obtained in Ref.
[11] and later in Ref. [12]. At $d=4$ the result for $I_{3}^{(4)}$ in terms of
Appell function $F_{3}$ was obtained in Ref. [13].
The Appell function $F_{1}$ in formula (3.41) has branch points if
$|p_{12}|\geq m^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
or}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|p_{12}-p_{13}|\geq m^{2}.$ (3.43)
To analytically continue the integral
$I_{3}^{(d)}(0,m^{2},0;~{}0,p_{13},p_{12})$ into regions (3.43) one can use
appropriate functional equations. When $|p_{12}|\geq m^{2}$, the following
functional equation can be applied:
$\displaystyle
I_{3}^{(d)}(0,m^{2},0;~{}0,p_{13},p_{12})=\frac{m^{2}}{p_{12}}~{}I_{3}^{(d)}\left(0,m^{2},0;~{}0,\frac{m^{2}(p_{13}-p_{12}+m^{2})}{p_{12}},\frac{m^{4}}{p_{12}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{(p_{12}-m^{2})}{p_{12}}~{}I_{3}^{(d)}\left(0,0,0;~{}\frac{m^{2}(p_{13}-p_{12}+m^{2})}{p_{12}},\frac{(p_{12}-m^{2})^{2}}{p_{12}},p_{13}\right).$
(3.44)
This relation can be derived from Eq. (3.26) with
$m_{1}^{2}=m_{3}^{2}=m_{4}^{2}=p_{23}=0$ and $m_{2}^{2}=m^{2}$ by imposing the
following conditions:
$G_{4}=0,~{}~{}~{}~{}\partial_{1}\Delta_{4}=0,~{}~{}~{}~{}\partial_{1}\partial_{3}\Delta_{4}=0.$
(3.45)
On the right - hand side of the relation (3.44) the last two arguments of the
integral $I_{3}^{(d)}$ in the first term are finite for large $|p_{12}|$.
If $|p_{12}-p_{13}|\geq m^{2}$ and $|p_{12}|\leq m^{2}$, the following
relation can be applied
$\displaystyle
I_{3}^{(d)}(0,m^{2},0;~{}0,p_{13},p_{12})=\frac{p_{12}m^{2}}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}}I_{3}^{(d)}\left(0,m^{2},0;~{}0,\frac{p_{12}^{2}(p_{13}-p_{12}+m^{2})}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}},p_{12}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{p_{12}(p_{13}-p_{12})}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}}~{}I_{3}^{(d)}\left(0,0,0;~{}\frac{m^{2}(p_{13}-p_{12})^{2}}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}},\frac{p_{12}^{2}(p_{13}-p_{12}+m^{2})}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}},p_{13}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{m^{2}(p_{13}-p_{12})}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}}I_{3}^{(d)}\left(0,m^{2},0;~{}0,\frac{m^{2}(p_{13}-p_{12})^{2}}{m^{2}p_{13}+p_{12}p_{13}-p_{12}^{2}},0\right).$
(3.46)
This relation can be derived from Eq. (3.26) with
$m_{1}^{2}=m_{3}^{2}=m_{4}^{2}=p_{23}=p_{24}=0$ and $m_{2}^{2}=m^{2}$ by
imposing the following conditions:
$G_{4}=0,~{}~{}~{}~{}\partial_{1}\Delta_{4}=0.$ (3.47)
The penultimate argument of the first integral on the right hand side of
(3.46) is finite for large values of $|p_{13}|\geq m^{2}$ if
$p_{12}\neq-m^{2}$. The second and the third integrals on the right - hand
side of this relation can be expressed in terms of the hypergeometric function
${}_{2}F_{1}$ and their analytic continuation causes no problems.
If both conditions (3.43) hold then the analytic continuation can be done by
applying both (3.44) and (3.46).
## 4 Conclusions
Finally, we summarize what we have accomplished in this paper.
First of all, we formulated the general method for deriving functional
equations for Feynman integrals.
Second, it was shown that integrals with many kinematic arguments can be
reduced to a combination of integrals with simpler kinematics.
Third, we demonstrated that our functional equations can be used for the
analytic continuation of Feynman integrals to all kinematic domains.
In the present paper we considered rather particular cases of functional
equations. The systematic investigation and classification of the proposed
functional equations requires application of the methods of algebraic geometry
and group theory.
A detailed consideration of our functional equations and their application to
the one-loop integrals with four, five and six external legs as well as to
some two- and three- loop Feynman integrals will be presented in future
publications.
## 5 Acknowledgment
I am very thankful to Ronald Reid-Edwards for carefully reading the manuscript
and useful remarks. This work was supported in part from DFG grants DFG
KN365/3 and BMBF 05HT6GUA. Part of this investigation was done during my stay
at the Institut für Theoretische Physik E, RWTH Aachen where I was supported
from the DFG grant Sonderforschungsbereich Transregio 9-03.
## References
* [1] B. Petersson, J. Math. Phys. 6, (1965) 1955.
* [2] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44, (1972) 189.
* [3] F. V. Tkachov, Phys. Lett. B 100 (1981) 65;
K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192 (1981) 159.
* [4] O. V. Tarasov, Phys. Rev. D 54 (1996) 6479 [arXiv:hep-th/9606018].
* [5] J. Fleischer, F. Jegerlehner and O. V. Tarasov, Nucl. Phys. B 566 (2000) 423 [arXiv:hep-ph/9907327].
* [6] C. G. Bollini and J. J. Giambiagi, Phys. Lett. B 40 (1972) 566.
* [7] E. E. Boos and A. I. Davydychev, Theor. Math. Phys. 89, (1991) 1052; Teor. Mat. Fiz. 89, (1991) 56 .
* [8] A. Erdèly et. al., Higher Transcendental Functions Vol.1, McGraw-Hill, New York, 1953.
* [9] A. I. Davydychev, Phys. Rev. D 61 (2000) 087701 [arXiv:hep-ph/9910224].
* [10] P. Appell and J. Kampé de Fériet Fonctions hypergeometriques et hyperspériques, Gauthier Villars, Paris, 1926.
* [11] J. Fleischer, F. Jegerlehner and O. V. Tarasov, Nucl. Phys. B 672, (2003) 303, [arXiv:hep-ph/0307113].
* [12] A. I. Davydychev, Nucl. Instrum. Meth. A 559 (2006) 293 [arXiv:hep-th/0509233].
* [13] L. G. Cabral-Rosetti and M. A. Sanchis-Lozano, J. Comput. Appl. Math. 115 (2000) 93 [arXiv:hep-ph/9809213].
|
arxiv-papers
| 2008-09-17T22:56:29
|
2024-09-04T02:48:57.821191
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O. V. Tarasov",
"submitter": "Oleg Tarasov",
"url": "https://arxiv.org/abs/0809.3028"
}
|
0809.3144
|
Gravitational Uncertainty and Black Hole Remnants
Kourosh Nozari*** knozari@umz.ac.ir and S. Hamid
Mehdipour†††h.mehdipour@umz.ac.ir
Department of Physics, Faculty of Basic Sciences, University of Mazandaran,
P. O. Box 47416-1467, Babolsar, IRAN
October 2005
###### Abstract
Possible existence of black holes remnants provides a suitable candidates for
dark matter. In this paper we study the possibility of existence for such
remnants. We consider quantum gravitational induced corrections of black
hole’s entropy and temperature to investigate the possibility of such relics.
Observational scheme for detection of these remnants and their cosmological
constraints are discussed.
PACS: 04.60.-m , 04.70.-s, 04.70.Dy, 95.35.+d
Key Words: Quantum Gravity, Generalized Uncertainty Principle, Varying Speed
of Light Models, Black Holes Thermodynamics, Dark Matter
## 1 Introduction
It is by now widely accepted that dark matter (DM) constitutes a substantial
fraction of the present critical energy density in the Universe. However, the
nature of DM remains an open problem. There exist many DM candidates, most of
them are non-baryonic weakly interacting massive particles (WIMPs), or WIMP-
like particles [1]. By far the DM candidates that have been more intensively
studied are the lightest supersymmetric (SUSY) particles such as neutralinos
or gravitinos, and the axions (as well as the axinos). There are additional
particle physics inspired dark matter candidates [1]. A candidate which is not
as closely related to particle physics is the relics of primordial black holes
(Micro Black Holes) [2,3]. Certain inflation models naturally induce a large
number of such a black holes. As a specific example, hybrid inflation can in
principle yield the necessary abundance of primordial black hole remnants for
them to be the primary source of dark matter [4,5]. In recent years it has
been suggested that measurements in quantum gravity should be governed by
generalized uncertainty principle (GUP). In fact, some evidences from string
theory, quantum geometry and black hole physics, have led some authors to re-
examine usual uncertainty principle of Heisenberg [6-13]. These evidences have
origin on the quantum fluctuations of the background space-time metric.
Existence of a minimal length scale on the order of Planck length is an
immediate consequence of GUP. Introduction of this idea has attract
considerable attention and many authors considered various problems in the
framework of generalized uncertainty principle [14-28]. The issue of black
holes remnants has been considered by some authors. Adler and his coworkers
have argued that contrary to standard viewpoint, GUP may prevent small black
holes total evaporation in exactly the same manner that the uncertainty
principle prevents the Hydrogen atom from total collapse [29]. Chen
considering inflation induced primordial black holes, have investigated the
issue of stability of such relics [30]. Recently, Varying Speed of Light
(VSL), as a new conjecture, which has been proposed to solve the problems of
standard cosmology, has attract some attentions. After introduction of this
conjecture, several alternative VSL theories have been proposed and some of
their novel implications have been examined extensively [31-35].
In this paper the issue of black hole remnants will be considered in the
framework of both GUP and VSL. The main consequence of this combination is
related to the stability problem of possible remnants. In forthcoming sections
first we show that GUP provides a reasonable framework for VSL. Then using a
simple VSL model we will show that if one consider both the effect of GUP and
VSL on the thermodynamics of black holes, the results of Bekenstein-Hawking
concerning total evaporation of black holes should be re-examined. In
Bekenstein-Hawking approach the total evaporation of micro black hole is
possible. Here we will see that it is possible to have relics of evaporating
black holes which can be considered as a possible candidate for dark matter.
The structure of the paper is as follows: in section 2 we show that VSL can be
considered as an immediate consequence of GUP. Section 3 is devoted to the
quantum gravitational corrected black hole thermodynamics. Some numerical
calculations have been down and their physical results are discussed. The
paper follows by conclusions and discussion regarding observational scheme for
detection of such relics and their cosmological constraints in section 4.
## 2 Preliminaries
As has been revealed in introduction, usual uncertainty principle of quantum
mechanics, the so-called Heisenberg uncertainty principle, should be re-
formulated due to noncommutative nature of spacetime at Planck scale. As a
consequence, it has been indicated that in quantum gravity there exists a
minimal observable distance on the order of the Planck length which governs on
all measurements in extreme quantum gravity limit. In the context of string
theories, this observable distance is referred to GUP. A generalized
uncertainty principle can be formulated as
$\Delta x\geq\frac{\hbar}{\Delta p}+const.G\Delta p,$ (1)
which, using the minimal nature of $l_{P}$ can be written as,
$\Delta x\geq\frac{\hbar}{\Delta p}+\frac{\alpha^{\prime}l_{p}^{2}\Delta
p}{\hbar}.$ (2)
The main consequence of GUP is that measurement of the position is possible
only up to Planck length, $l_{P}$. So one can not setup a measurement to find
more accurate particle position than Planck length, and this means that the
notion of locality breaks down. It is important to note that there are more
generalization which contain further terms in right hand side of equation (2)
(see [36]), but in some sense regarding dynamics, equation (2) has more
powerful physical grounds. Suppose that
$\Delta x\sim x,\quad\quad\Delta p\sim p,\quad\quad p=\hbar k,\quad\quad
x=\bar{\lambda}=\frac{\lambda}{2\pi}.$
Therefore one can write,
$\bar{\lambda}=\frac{1}{k}+\alpha^{\prime}l_{p}^{2}\,k\qquad
and\qquad\omega=\frac{c}{\bar{\lambda}}.$ (3)
In this situation the dispersion relation becomes,
$\omega=\omega(k)=\frac{kc}{1+\alpha^{\prime}l_{p}^{2}\,k^{2}}.$ (4)
This relation can be described in another viewpoint. By expansion of
$\Big{(}1+\alpha^{\prime}l_{p}^{2}\,k^{2}\Big{)}^{-1}$ and neglecting second
and higher order terms of $\alpha^{\prime}$, we find that
$\omega=kc\big{(}1-\alpha^{\prime}l_{p}^{2}\,k^{2}\big{)}$. This can be
considered as $\omega=kc^{\prime}$ where
$c^{\prime}=c\big{(}1-\alpha^{\prime}l_{p}^{2}\,k^{2}\big{)}$. This relation
indicates the possibility of variation in $c$. Accepting the possibility of
variation in $c$, one can consider its time variation also. So we consider the
time dependence of light speed.
Actually, there are some evidences indicating that fine structure constant,
$\alpha=\frac{e^{2}}{\hbar c}$ is not constant [11]. The question then arises
that which of the quantities: $e$, $c$ or $\hbar$ are variable? A possible
situation is the variation in $c$. This is referred as varying speed of light
(VSL) theories in literatures. After introduction of this idea several varying
speed of light models have been proposed to solve the problems of standard
cosmology. One of the simplest of these models is the model proposed by Barrow
[35]. Barrow has considered the speed of light as
$c(t)=c_{0}a^{n}(t),$ (5)
where $c_{0}$ and $n$ are constant. Using this form of $c(t)$ to solve
problems of standard cosmology, some constraint will be imposed on the value
of $n$, depending on the nature of the problems. For example if we consider
the equation of state for matter content of the Universe as $p=(\gamma-1)\rho
c^{2}(t)$, then exact solutions with varying $c(t)$ and $G(t)$, restrict $n$
to the following limit
$n\leq-1\,\,\,\,for\,\,\,\gamma=4/3\quad\quad\quad
Radiation\,\,Dominated\,\,Era$ (6) $n\leq-1/2\,\,\,for\,\,\,\gamma=1\quad\quad
Matter\,\,Dominated\,\,Era,\,Dust.$ (7)
## 3 Black Holes Thermodynamics
In the current standard viewpoint, small black holes emit black body radiation
at the Hawking temperature,
$T_{H}\approx\frac{\hbar c^{3}}{8\pi GM}=\frac{M^{2}_{P}c^{2}}{8\pi M},$ (8)
where $M_{P}=\sqrt{\frac{\hbar c}{G}}$ is the Planck mass and we have set
$k_{B}=1$. The related entropy is obtained by integration of
$dS=c^{2}T^{-1}dM$ which is the standard Bekenstein entropy,
$S_{B}=\frac{4\pi GM^{2}}{\hbar c}=4\pi\frac{M^{2}}{M^{2}_{P}}.$ (9)
If one consider the GUP as equation (2), the last two equations become
respectively,
$T_{GUP}=\frac{Mc^{2}}{4\pi}\Bigg{[}1\mp\sqrt{1-\frac{M^{2}_{P}}{M^{2}}}\,\Bigg{]},$
(10)
and
$S_{GUP}=2\pi\Bigg{[}\frac{M^{2}}{M^{2}_{P}}\Bigg{(}1-\frac{M^{2}_{P}}{M^{2}}+\sqrt{1-\frac{M^{2}_{P}}{M^{2}}}\,\Bigg{)}-\ln\Bigg{(}\frac{M+\sqrt{M^{2}-M^{2}_{P}}}{M_{P}}\,\Bigg{)}\,\Bigg{]}.$
(11)
In equation (10), to recover the corresponding result in the limit of large
mass ($T_{H}$), one should consider the minus sign. These equations strongly
suggest the existence of black holes remnants. As it is evident from figure 2,
in the framework of GUP black hole can evaporate until when it reachs the
Planck mass. In this view point black hole remnants are stable. Now consider
the case of VSL. For simplicity we assume that only $c$ is varying and $G$ and
$\hbar$ are constant which we set $G=\hbar=1$. In this situation
$M_{P}=\sqrt{c(t)}$. This is a novel concept: a time-varying Planck mass!. It
means that Planck scales are varying with time and are actually cosmological
models dependent via dependence of $c(t)$ to cosmological scale factor. The
corresponding equations both in Hawking-Bekenstein and GUP viewpoint will
become as follows respectively,
$T^{(VSL)}_{H}(t)=\frac{c^{3}(t)}{8\pi M},$ (12) $S^{(VSL)}_{B}(t)=\frac{4\pi
M^{2}}{c(t)},$ (13)
$T^{(VSL)}_{GUP}(t)=\frac{Mc^{2}(t)}{4\pi}\Bigg{[}1-\sqrt{1-\frac{c(t)}{M^{2}}}\,\Bigg{]},$
(14)
and
$S^{(VSL)}_{GUP}(t)=2\pi\Bigg{[}\frac{M^{2}}{c(t)}\Bigg{(}1-\frac{c(t)}{M^{2}}+\sqrt{1-\frac{c(t)}{M^{2}}}\,\Bigg{)}-\ln{\Bigg{(}\frac{M+\sqrt{M^{2}-c(t)}}{\sqrt{c(t)}}\,\Bigg{)}}\,\Bigg{]}.$
(15)
Now one should specify the time dependence of $c(t)$. Using equation (5),
since there exists several possibilities for $a(t)$ and
$c(t)=M^{2}_{P}(t)=a^{n}(t)$, we can consider de Sitter Universe as an
example,
$c(t)=\Big{[}a_{0}\cosh(\frac{t}{a_{0}})\Big{]}^{n}.$ (16)
Now, equations (12)-(15) for de Sitter Universe become respectively,
$T^{(VSL)}_{H}(t)=\frac{\cosh^{3n}(t)}{8\pi M},$ (17)
$S^{(VSL)}_{B}(t)=\frac{4\pi M^{2}}{\cosh^{n}(t)},$ (18)
$T^{(VSL)}_{GUP}(t)=\frac{M\cosh^{2n}(t)}{4\pi}\Bigg{[}1-\sqrt{1-\frac{\cosh^{n}(t)}{M^{2}}}\,\Bigg{]},$
(19)
and
$S^{(VSL)}_{GUP}(t)=2\pi\Bigg{[}\frac{M^{2}}{\cosh^{n}(t)}\Bigg{(}1-\frac{\cosh^{n}(t)}{M^{2}}+\sqrt{1-\frac{\cosh^{n}(t)}{M^{2}}}\,\Bigg{)}-\ln{\Bigg{(}\frac{M+\sqrt{M^{2}-\cosh^{n}(t)}}{\sqrt{\cosh^{n}(t)}}\,\Bigg{)}}\,\Bigg{]}.$
(20)
Where we have set $a_{0}=1$. The results of numerical calculations are shown
in figures. In these figures we have considered $n=-1$ and the results are
shown for de Sitter Universe. Note that for different $n$, the overall
behavior of the solutions do not change considerably and other model Universes
give similar results.
## 4 Conclusions and Discussion
Based on our model and numerical calculations, the following results are
obtained
1. 1-
When one considers the time variation of speed of light alone, total
evaporation of black hole is possible in principle. Thus in the framework of
VSL micro black hole can evaporate completely. This is in agreement with the
Hawking-Bekenstein and in contrast with the results of GUP.
2. 2-
Application of generalized uncertainty principle to black holes thermodynamics
strongly suggests the possible existence of black holes remnant (figure 1 and
2).
3. 3-
When one considers thermodynamics of black holes in the framework of both GUP
and VSL, some novel results are obtained. The figure for temperature of black
hole versus the mass and time in a combination of GUP and VSL (figure 5) shows
that where the mass of black hole is zero, its temperature is zero also. This
is in contrast to Hawking result and seems completely reasonable since in the
absence of matter there is no meaning for temperature. When the mass
increases, the temperature is increases until the mass becomes equal to the
Planck mass. After that, increasing of mass is corresponding to decreasing of
temperature in complete agreement with the results of GUP (Adler et al [29]).
But the situation for mass less than Planck mass is completely different from
GUP results.
4. 4-
The figure for entropy of black hole versus the mass and time in a combination
of GUP and VSL (figure 6) shows that when one approaches the Planck mass,
entropy do not vanishes. This is physically reasonable but rules out the
result of Adler et al since they have zero entropy for remnants. Increasing
the time will increase the entropy which is natural. The possibility of having
black hole remnant at Planck mass is evident from this figure. In our model
there is a remnant entropy for black remnant. This can be at least related to
background spacetime metric fluctuation.
5. 5-
Adler and his coworkers have constructed their formulation based on analogy
between hydrogen atom and black holes. They have argued that since uncertainty
principle prevents the hydrogen atom from total collapse, generalized
uncertainty principle may prevent black holes total evaporation in the same
manner. In our opinion, the basic mistake of Adler et al is that they have not
considered hydrogen atom in GUP. Our calculation shows that in GUP hydrogen
atom is not stable. Since,
$\Delta r\Delta p\geq\frac{\hbar}{2}(1+\beta(\Delta p)^{2}).$ (21)
Suppose that $\Delta p\sim p$ and $\Delta r\sim r$, then one finds
$pr=\frac{\hbar}{2}(1+\beta p^{2})\Rightarrow\hbar\beta p^{2}-2pr+\hbar=0.$
(22)
So one obtains,
$p=\frac{r\pm\sqrt{r^{2}-\beta\hbar^{2}}}{\beta\hbar}.$ (23)
As has been argued we should consider the minus sign in (23). Since
$E=\frac{p^{2}}{2m}-\frac{e^{2}}{r},$ (24)
one find,
$E=\frac{1}{2m}\bigg{(}\frac{r-\sqrt{r^{2}-\beta\hbar^{2}}}{\beta\hbar}\,\bigg{)}^{2}-\frac{e^{2}}{r}\,.$
(25)
Now we set, $\frac{dE}{dr}=0$ and find $r_{min}=\hbar\sqrt{\frac{\beta}{2}}$.
The extremum value of energy becomes,
$E_{min}=-\bigg{[}\Big{(}\frac{e^{2}}{\hbar}\,\sqrt{\frac{2}{\beta}}\,\Big{)}+i\frac{1}{2m\beta}\bigg{]},$
(26)
where its real part is,
$E_{min}^{real}=-\Big{(}\frac{e^{2}}{\hbar}\,\sqrt{\frac{2}{\beta}}\,\Big{)}.$
(27)
Since $r_{min}$ is very small length, the radius of stability for hydrogen
atom is very small. Therefore in GUP scale, the hydrogen atom is not stable
and will collapse completely. If this is the case, one can not construct
analogy between hydrogen atom in Heisenberg uncertainty principle viewpoint
and black hole in GUP viewpoint. If we consider hydrogen atom in GUP, as we
have shown, this atom will collapse totally. But in the framework of GUP black
holes evaporate until they reach Planck mass. The issue of stability for
remnant can be considered in the framework of symmetry principle in the
system. In this regard supersymmetry, in particular supergravity, stands a
very good framework of providing such black hole remnants [30].
Note that our arguments for the existence of black hole remnants based on GUP
is heuristic. The search for its deeper theoretical foundation is currently
underway. As interactions with black hole remnants are purely gravitational,
the cross section is extremely small, and direct observation of these remnants
seems unlikely. One possible indirect signature may be associated with the
cosmic gravitational wave background. Unlike photons, the gravitons radiated
during evaporation would be instantly frozen. Since, according to our notion,
the black hole evaporation would terminate when it reduces to a remnants, the
graviton spectrum should have a cutoff at Planck mass. Such a cutoff would
have by now been red-shifted to $\sim 10^{14}GeV$. Another possible
gravitational wave-related signature may be the gravitational wave released
during the gravitational collapse. The frequencies of such gravitational waves
would by now be in the range of $\sim 10^{7}-10^{8}Hz$. It would be
interesting to investigate whether these signals are in principle observable.
Another possible signature may be some imprints on the CMB fluctuations due to
the thermodynamics of black hole remnants-CMB interactions. Possible
production of such remnants in LHC (Large Hadron Collider) and also in cosmic
ray showers are under investigation. If we consider hybrid inflation as our
primary cosmological model, there will be some observational constraints on
hybrid inflation parameters. For example a simple calculation based on hybrid
inflation suggests that the time it took for black holes to reduce to remnants
is about $10^{-10}Sec$. Thus primordial black holes have been produced before
baryogenesis and subsequent epochs in the standard cosmology [30].
Acknowledgment
This Work has been supported partially by Research Institute for Astronomy and
Astrophysics of Maragha, Iran. Also we would like to appreciate an unknown
referee of MPLA for his (her) valuable comments on original version of the
paper.
## References
* [1] P. Gondolo, Lectures delivered at the NATO Advanced Study Institute ”Frontiers of the Universe”, 8-20 Sept 2003, Cargese, France; astro-ph/0403064.
* [2] Ya. B. Zeldovich and I. D. Novikov, Sov. Astron. 10 (1966) 602.
* [3] S. W. Hawking, Mon. Not. R. Astron. Soc. 152 (1971) 75.
* [4] E. J. Copeland et al, Phys. Rev. D 49 (1994) 6410.
* [5] D. H. Lyth and A. Riotto, Phys. Rep. 314 (1999) 1; A. Linde, Phys. Rep. 575 (2000) 333.
* [6] G. Veneziano, Europhys. Lett. 2 (1986) 199; Proc. of Texas Superstring Workshop (1989).
* [7] D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216 (1989) 41.
* [8] D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 197 (1987) 81; Int. J. Mod. Phys. A 3 (1988) 1615; Nucl. Phys. B 347 (1990) 530.
* [9] D. J. Gross and P. F. Mende, Phys. Lett. B 197 (1987) 129; Nucl. Phys. B 303 (1988) 407.
* [10] K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B 234 (1990) 276; R. Guida, K. Konishi and P. Provero, Mod. Phys. Lett. A 6 (1991) 1487.
* [11] M. Kato, Phys. Lett. B 245 (1990) 43.
* [12] L. J. Garay, Int. J. Mod. Phys. A 10 (1995) 145.
* [13] S. Capozziello, G. Lambiase and G. Scarpetta, Int. J. Theor. Phys. 39 (2000) 15.
* [14] M. Maggiore, Phys. Lett. B 304 (1993) 65.
* [15] C. Castro, Found. Phys. Lett. 10 (1997) 273.
* [16] A. Camacho, Gen. Rel. Grav. 34 (2002) 1839.
* [17] M. Maggiore, Phys. Rev. D 49 (1994) 5182.
* [18] M. Maggiore, Phys. Lett. B 319 (1993) 83.
* [19] S. Kalyana Rama, Phys. Lett. B 519 (2001) 103.
* [20] A. Camacho, Gen. Rel. Grav. 35 (2003) 1153.
* [21] F. Scardigli and R. Casadio, Class. Quant. Grav. 20 (2003) 3915.
* [22] S. Hossenfelder, Mod. Phys. Lett. A 19 (2004) 2727.
* [23] S. Hossenfelder, Phys. Rev. D 70 (2004) 105003.
* [24] S. Hossenfelder, Mod. Phys. Lett. A 19 (2004) 2727.
* [25] A. J. M. Medved and E. C. Vagenas, Phys. Rev. D 70 (2004) 124021\.
* [26] A. J. M. Medved, Class. Quant. Grav. 22 (2005) 133.
* [27] B. Bolen and M. Cavaglia, Gen. Rel. Grav. 37 (2005) 1255.
* [28] M. R. Setare, Phys. Rev. D 70 (2004) 087501.
* [29] P. Chen and R. J. Adler, Nucl. Phys. B (Proc.Suppl.) 124 (2003) 103; R. J. Adler, P. Chen and D. I. Santiago, Gen. Rel. Grav. 33 (2001) 2101.
* [30] P. Chen, New Astron. Rev. 49 (2005) 233.
* [31] A. Albrecht and J. Magueijo, Phys. Rev. D 59 (1999) 043516\.
* [32] J. Magueijo, Phys. Rev. D 62 (2000) 103521.
* [33] J. D. Barrow and J. Magueijo, Phys. Lett. B 443 (1998) 104; Phys. Lett. B 447 (1999) 246; Class. Quant. Grav. 16 (1999) 1435.
* [34] J. Magueijo, Phys. Rev. D 63 (2001) 043502.
* [35] J. D. Barrow, Phys. Rev. D 59 (1999) 043515.
* [36] A. Kempf et al., Phys. Rev. D 52 (1995) 1108.
Figure 1: Temperature of a Black Hole Versus the Mass. Mass is in units of the
Planck mass and temperature is in units of the Planck energy. The lower curve
is the Hawking result, and the upper curve is the result of GUP. Figure 2:
Entropy of a Black Hole Versus the Mass. Entropy is dimensionless and mass is
in units of the Planck mass. The upper curve is the Hawking result, and the
lower curve is the result of GUP. Figure 3: Temperature of Black Hole Versus
the Mass and Time in VSL. The units are as previous and the result is shown
for De Sitter model. As figure shows, the result of Hawking is recovered.
Decreasing of temperature with increasing of time is natural. Figure 4:
Entropy of Black Hole Versus the Mass and Time in VSL. The units are as
previous and the result is shown for De Sitter model. As figure shows, the
result of Hawking is recovered. Increasing of Entropy with increasing of time
is natural. Figure 5: Temperature of Black Hole Versus the Mass and Time in a
combination of GUP and VSL. The units are as previous and the result is shown
for De Sitter model. As figure shows, where the mass is zero the temperature
is zero also. This is in contrast to Hawking result. When the mass increases,
the temperature increases until the mass becomes equal to the Planck mass.
After that, increasing of mass is corresponding to decreasing of temperature.
This is a novel result of GUP+VSL Scenario. Figure 6: Entropy of Black Hole
Versus the Mass and Time in a Combination of GUP and VSL. The units are as
previous and the result is shown for De Sitter model. As figure shows, When
one approaches the Planck mass, entropy do not vanishes. Increasing entropy
with time is natural. This figure shows the possibility of having black holes
relics.
|
arxiv-papers
| 2008-09-18T16:15:52
|
2024-09-04T02:48:57.827062
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kourosh Nozari and S. Hamid Mehdipour",
"submitter": "Seyed Hamid Mehdipour",
"url": "https://arxiv.org/abs/0809.3144"
}
|
0809.3414
|
# Round Table Discussion at the Final Session of FPCP 2008:
The Future of Flavor Physics and CP
Jeffrey A.Appel Fermilab, Batavia, IL 60510 USA Jen-Feng Hsu Hsiang-nan Li
Institute of Physics, Academia Sinica, Taipei, Taiwan
###### Abstract
The final session of FPCP 2008 consisted of a round-table discussion among
panelists and audience. The panelists included Jeffrey Appel(moderator),
Martin Beneke, George W.S. Hou, David Kirkby, Dmitri Tsybychev, Matt Wingate,
and Taku Yamanaka. What follows is an edited transcript of the session.
## I Question: What are the big questions in flavor physics at FPCP08?
Jeff Appel
Many of us from many places have to write trip reports when we get back. And
perhaps when writing the trip reports we could start with the big questions in
flavor physics that came up here. This is meant to help to you write the trip
report as well as to focus the discussions to come. A number of topics were
suggested by people who sent an email. So you can read them here.
* •
$CP$ violation in charged vs neutral $B$ decays?
* •
Mixing induced $CP$ violation in the $B_{s}$ system?
* •
$D-\bar{D}$ mixing: How soon can we measure mixing parameter $x$?
* •
Spectroscopy: What are the $XYZ$ states in the charm sector (counterparts in
the bottom sector?)?
I don’t need to go through them one by one, but I will ask our panel members
to begin with what among these topics they found most important; what they
think missing from the list. Martin why don’t we begin with you?
Martin Beneke
The list includes most of the hot topics discussed at this conference. The
first two items refer to phenomena connected with $b\to s$ transitions, where
the window to new physics is still open widest. However, we have learned in
the past few years that the standard flavor theory is working quite well. The
much discussed hints in the $b\to s$ sector are either not conclusive (second
item) or possess alternative hadronic standard-model interpretations (first
item). The actual observation of $D-\bar{D}$ mixing is exciting as a
phenomenon, but because of theoretical uncertainties, does not tell us much
that we did not know before about new physics.
Matt Wingate
From the lattice QCD perspective, the most interesting thing discussed here
was the discrepancy between the HPQCD calculation of $f_{D_{s}}$ and the
experimental measurement. The lattice result is quite sound: the non-strange
decay constant $f_{D}$ is the one which requires more work, namely
extrapolating lattice data to the physical up/down quark mass. The fact that
$f_{D}$ agrees with experiment while $f_{D_{s}}$ does not is an interesting
puzzle. The precision quoted for the lattice result is very impressive, and
further details from the authors will allow other lattice experts to judge the
quality of the fits involved. It doesn’t seem plausible to me that the source
of the discrepancy could be blamed on the fourth-root hypothesis used in
staggered-quark calculations.
One thing which I am investigating is: What more can be done on the lattice in
studying $b\to s$ decays? There are difficulties for the lattice here which
are not present in $b\to u$ decays or neutral $B$ meson mixing. Nevertheless,
the $b\to s$ decays are of such great interest that all approaches, including
lattice QCD, should be pushed as far as possible. I think there are
calculations we can do which will add to the picture.
Dmitri Tsybychev
I just want to add that whether there is mixing induced $CP$ violation in the
$B_{s}$ system will remain a hot topic for next couple of years, and hopefully
both D0 and CDF experiments will have updates on their results; if not in the
summer 2008, then in the fall. There is room for improvement on the precision
of measurements of $\phi_{s}$ for both experiments. With continuing successful
running of the Tevatron, both experiments plan to collect up to 8 $fb^{-1}$ of
data. CDF already has a sample of 3 $fb^{-1}$. Their current result is based
on a data sample of only 1.3 $fb^{-1}$. The D0 experiment has already used the
full sample of 2.8 $fb^{-1}$ available to date. Therefore it will be able to
increase its sample only when new data are collected. However, D0 plans to
improve the selection of $B_{s}$ mesons decaying into $J/\psi\phi$. As was
already mentioned, D0 can increase the statistical significance of its sample
by 20% through a better selection. This will directly translate to an
improvement of the measurement.
Additionally, a question still remains involving SU(3) or U-symmetry. D0
constrains the strong phases involved in the $B_{s}$ angular analysis to the
similar phases that appear in $B_{d}\rightarrow J/\psi K^{*}$ decays, and are
measured at B-factories. The constraint is rather weak, and allows for SU(3)
symmetry breaking, which may be as big as 10%. Polarization amplitudes,
measured in $B_{s}$ and $B_{d}$ decays, are compatible within measured
uncertainties. This may indicate that such symmetry exists. The result on
$phi_{s}$ does not change significantly if the phase constraints are removed.
However there is no consensus whether such a constraint should be applied, and
one can benefit from a stronger theoretical motivation.
Jeff Appel
You think that the systematic errors are not coming soon for how well you can
do on this?
Dmitri Tsybychev
The fit result for the case of free strong phases is provided in the D0
article in PRL, and agrees very well within statistical uncertainty with
result of the constrained fit.
Jeff Appel
Anybody else? Does anybody in the audience want to add to this list?
David Kirkby
I think the main question in flavor physics is where the new physics is going
to show up, if anywhere. We should remember also that there are certainly
topics in flavor physics that have intrinsic interest: spectroscopy, for
example. But how likely is it for new physics to show up there? To get the
audience more involved, how about a show of hands? Where do you think that the
new physics is likely to come from? Raise your hand once at which one of these
four you think is the most promising. So, how about the first one?
Jeff Appel
You have to leave your hands long enough for count. 1, 2, 3, 4..
David Kirkby
So how about the second one, the $B_{s}$? Which one of the four is the new
physics most likely to show up?
Rahul Sinha
They are connected. If you find $\Delta_{S}$ not equal to zero in $B$ mixing
you are likely to find other signals of new physics such as a deviation in the
small $B_{s}$ mixing phase among other things. They are connected, since
$\Delta_{S}$ can be written in terms of the small $B_{s}-\bar{B}_{s}$ mixing
phase.
David Kirkby
The second one. What is generated from the $B_{s}$ system? Can we find new
physics there? How about $D-\bar{D}$ mixing? Well you don’t know, but what’s
your intuition? What’s your gut feeling?
Choong Sun Kim
I thought of the story of the $D-\bar{D}$ mixing. There is no standard model
prediction. How can you find new physics?
David Kirkby
How likely do you think you are going to find something there?
Rahul Sinha
Yes, you can measure the $D-\bar{D}$ mixing phase with a precision of about
1degree at Super-$B$, but we need 50 inverse attobarns.
David Kirkby
How about spectroscopy? Beyond the standard model? QCD is not new physics.
Unidentified voice
It’s kind of obvious that new physics will show up there, and new particles
can contribute to the amplitudes like penguin decays or $B\to\tau$ decays. In
my opinion, this will be the best place to look for new physics.
Jose Ocariz
I agree with the previous comment that it’s a necessary condition; but it’s
not sufficient. For example, if we think of item 1, I have a feeling that this
is more or less motivated by the measurement of the different $CP$ asymmetry
in $B\to K\pi$ decay. This is a non-controversial measurement, but the
interpretation is not uncontroversial. There is no way of falsifying the
standard model by this kind of measurement despite the fact there is potential
sensitivity to contributions from non-standard physics.
[Comment by Tom Browder added in preparing this report: The discussion seemed
to imply that there is no possible future resolution of this issue. However,
the isospin sum rule proposed by Gronau and Rosner is a model-independent test
for new physics. It requires much more data ($>$ a factor of ten) and much
more precise measurements of $A_{CP}(B\rightarrow K^{0}\pi^{0})$.]
Gerald Eigen
Martin, you brought up the $b\to s$ transitions. I agree with you that these
are important. Since point 1 is rather general, wouldn’t you rather split them
into subtopics that are associated with different points than including them
all under point 1?
Martin Beneke
I was thinking of mixing-induced $CP$ violation. Would you like to include
$b\to s\ell\ell$?
Gerald Eigen
Yes, and also the $b$ transitions involving $s\bar{s}$, like $\phi K_{s}$,
$\eta^{\prime}K_{s}$, etc. The leptonic penguins clearly belong under point 1,
while the gluonic penguins fit better under point 2.
(George) Wei-Shu Hou
I know I am viewed as a fanatic, saying that fourth generation this and that.,
fourth generation for everything. I actually quite agree with what Martin and
Jose said, and that this kind of discussion can be endless, and we are not
going to go very far. But the converse is not true; that if you show that in
some new physics model you can generate an observed effect, it would still be
of interest. So [going into a short presentation] this result here is
published in 2005, and I give you the diagram. I have said many times during
the conference, that having a t’ would bring in large Yukawa couplings and new
$CP$ violating CKM elements. Our study was re-done at next to leading order in
PQCD, and the effect on DCPV difference was not diluted. And there is another
thing, that it does push down $\Delta_{S}$.
It’s not sufficient to generate the central value of the experiment, but to me
that is very interesting. The two things mentioned in this conference are of
note to me. One thing.Maybe I pull this slide [from Derek Strom’s talk] back.
If you look at this, here is the Standard Model expectation for $\phi_{s}$,
and here are all four different, related measurements. All the measured values
fall to the left. And here is the actual published prediction from 4th
generation (which is smack in the middle of the experiments). I already stated
something like this, large $\sin(2\Phi_{B_{s}})$, in 2005. This is on the
record. At the moment, I am not a UT-fitter fan, and nobody here is. I am not
on the IAC. I would have voted for them to be here, just for the debate. At
the moment, you know, experimentally one can not yet say too much. It’s not
inconsistent with the Standard Model. I do point out that these numbers
normally would be scattered (if the SM is correct), but they are not. The
error bars will get reduced, say in next two years, from 1.35 inverse
femtobarns of data, to 3 to 5 to 8. In the last year or two, I used to say
that if the central value stays, I would then be willing to bet a good bottle
of red wine that the 4th generation is real. Starting a year ago at FPCP in
Slovenia, the data seem to be heading in this direction. Now here [another
slide on $A_{FB}$ from Eigen’s talk] is one thing that Gerald brought up but
didn’t really go through. The green line in the Belle plot is marked ”C9, C10
sign-flipped”, which is equivalent to C7 sign flip. The blue line in the BaBar
plot is for the Standard Model, almost zero, but slightly negative. Now the
upper figure was actually shown by Dmitri [Tsybychev] in his talk. The blue
dashed curve, is the fourth generation differential $A_{FB}$, and the marked
red line gives roughly the lower $q^{2}$ bin here. So you can understand why
the Standard Model is slightly negative and close to zero; because below the
zero is negative. Sorry that the sign convention is opposite to the B-factory
experiment. And above the zero is positive but there is a bit more negative
than positive so that you get the blue zero, or close to zero, of the SM in
the BaBar plot. But in our fourth generation analysis the line moves down. So
the zero moves further down, and there is not much negative part but large
positive part; so it’s more consistent with Belle/BaBar results. And I think
it was Uli who raised this issue, you know, complaining what is still called
by experimentalists the C7 sign flip. This is basically a way that
experimentalists say that there is a deviation. And this is why I stressed
that I want to treat these things more generally, to allow complex Wilson
coefficients. This gives the shaded area. I don’t want to go into any further
details. Let me change tone and say — I am willing to bet a good bottle of
Champagne now, if you want to take up the order. Why? Now this [yet another
slide] is the standard folklore that Standard Model $CP$ violation is
$10^{-20}$. Here is the Jarlskog invariant, and A here, the invariant CPV
area, is like $10^{-}5$. But the real suppression is coming from these small
masses. So if you put in numbers, when you normalize properly with, say, the
electroweak phase transition temperature, you get this $10^{-20}$. Now you see
the fourth generation does miraculous stuff here because it naturally has
large Yukawa couplings. So if you shift by one generation, this
$m_{c}^{2}-\mu^{2}$ becomes $m_{t}^{2}-m_{c}^{2}$, etc. This gives rise to a
very large enhancement. Well, it is still a suppression factor, but the
$m_{b}^{2}-m_{s}^{2}$ alone is the only suppression. So this gives a $10^{15}$
gain, where about a factor of 30 is from the $b\rightarrow s$ $CP$ violating
analysis. OK, but the factor of 30 compared to $10^{15}$ is nothing, so long
that this factor of 30 is not $10^{10}$, or something. So we have a very large
enhancement factor compared to the Standard Model three generation Jarlskog
invariant. I think this is another proof that Nature is more ingenious than
anyone of us here. But for me, to be able to jump back to put the $CP$
violation within Yukawa sector to be relevant for baryogenesis, that’s why I
say I am willing to bet a good bottle of Champagne now, … but only for ten
people, OK?
Choong Sun Kim
I do not know all the details of fourth generation, but I have some simple
questions. First, as you know, and as everyone knows, this fourth generation
neutrino mass is quite heavy, $>$ 45 GeV. So why do we have such a heavy
neutrino, much different from the first three generations? That’s very strange
to me. This kind of thing comes out more naturally if we have something like a
string-inspired E(6) model, which predicts rather heavy vector-like quarks,
unlike the fourth generation.
(George) Wei-Shu Hou
Well, there is a very simple answer to that. Vector like quarks will not have
this enhancement. These are not masses, these are Yukawa couplings. Dirac
masses go into the denominators, propagators and decoupling. So we can not
have enhancement.
Choong Sun Kim
Something like Kaluza-Klein or some other excited states. I think probably a
similar result will come out generally without a so-heavy neutrino problem.
(George) Wei-Shu Hou
Yea, OK. I can not argue with Kaluza-Klein. They are all legitimate, but this
one (4th generation) is within Standard Model dynamics! Now for neutrinos, we
firmly know there are only three light ones. But since 1998, as compared to
1989, we also learned that neutrinos have mass. So it’s a much richer sector
than we knew of. Furthermore, you didn’t mention electroweak precision tests,
right? There is a recent paper by Kribs et al (Plehn, Spannowsky and Tait),
which refutes the very stringent application of precision test against the
fourth generation in the PDG. list. So it’s not ruled out. But whatever you
say, I am just saying this more than ten-order-of-magnitude gain is so
enormous. I use this to argue that, despite electroweak precision tests, even
the neutrino stuff, the 4th generation is fairly legitimate. The thing is,
when you have high scale $CP$ violation for baryogenesis, such as
leptogenesis, you tend not to have a laboratory test. It’s a matter of physics
in the usual sense.
Taku Yamanaka
I didn’t vote for any of the four items up there. Since I am an
experimentalist, and since I work on kaons, I will vote for kaon physics
experiments. The sensitivity of a $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
experiment will first go down by three orders-of-magnitude, from O(1E-8) to
O(1E-11). Even beyond the Grossman-Nir limit, there is a two-orders-of-
magnitude parameter space for new physics to appear. So, do you want to vote
for a 10 percent effect, or do you want to vote for a large parameter space
with two orders-of-magnitude? I would vote for a two-orders-of-magnitude
effect.
## II Question: What are the big flavor-physics questions to come?
Jeff Appel
There is another way to continue this discussion which is the second question.
That is, what are likely to be the big flavor-physics questions after the
first Tevatron or LHC signal beyond the standard model? And a corollary
question is what would be the flavor-physics questions if we don’t see a new
signal at LHC?
The answer given for the first part is that the interesting flavor-physics
question will depend on what you see. However, almost anything you see will
have multiple possible answers, multiple models which can explain it. This may
mean that there are sensitivities to flavor physics across the board. In fact,
I don’t think a signal in a particular channel will lead to only one flavor-
physics parameter that you want to look at. That’s how I guess I would put it.
Tom Browder
If a signal really shows up early at the LHC, I think the big question will be
how any new particles at LHC do not produce flavor changing neutral currents.
The theorists will have to find brilliant ways for cancellations to not
produce flavor changing neutral currents, not just produce a new model.
Jeff Appel
So you don’t think there will be big signals from LHC? I didn’t mean to put
too many words into your mouth. Anybody on the panel want to respond to this
more ambiguous question?
David Kirkby
I think it is easy to imagine new physics at LHC where you wouldn’t really
know what to do at the Super-B-factory. So maybe the challenge to the audience
is ”Can you think of something we may find at the LHC where it would be
unclear what to do in flavor physics?” Are there other scenarios? Let’s talk
about that.
Rahul Sinha
If you see a signal of something at the LHC, you want to make sure that the
theoretical parameters corresponding to your favorite model/scenario, and that
are consistent with the signal, are not actually ruled out by precision tests;
and $B$ physics would provide a precision constraint, through loop
contributions. Therefore, you want to make sure that $B$ data is consistent
with the scenario and the observed signal. That is one way again of using
flavor physics.
David Kirkby
There are strong constraints from the data we already have.
Rahul Sinha
This is not enough. As to whether the current flavor constraints are good
enough - let me say we need to improve; we need as much improvement as
possible. With the LHC alone, we may see a signal of new physics, but we may
not be able to figure out what kink of new physics it corresponds to. Here is
where flavor physics comes in, ruling out or finding consistency among
different models given a particular signal. The better the precision, the
better the constraints. One requires flavor physics to enable pinning down
what is the new physics.
Martin Beneke
We discuss flavor physics in the context of the TeV scale. In doing that, we
almost always implicitly assume that electroweak symmetry breaking is caused
by some weak-coupling phenomena. That’s not guaranteed. An entirely different
way of seeing things would be needed if it turns out that electroweak symmetry
breaking happens through some QCD-like strong-coupling mechanism. Then the
flavor-physics puzzle is more severe, because if there is no weak coupling at
the TeV scale, we would know that flavor physics is probing much higher scales
which are disconnected from TeV scale. So, indirectly, one of the big flavor-
physics questions to come and be answered is what causes electroweak symmetry
breaking.
Keh-Fei Liu
I wonder if one of you could comment on neutron electric dipole moment in
terms of its discovery potential, and if there can be some effect found in the
next couple of years. Will the new physics be orthogonal or complimentary to
this flavor physics?
Jeff Appel
The coupling to the neutron electric dipole moment for any of these questions.
George?
(George) Wei-Shu Hou
You mean something specific. I actually asked Junji Hisano, the expert.
Basically 4th generation effects can enter through loops. I think that’s what
you are referring to. So that should be studied, yes.
Keh-Fei Liu
In the program of looking for new physics, I want to see whether there is a
discovery in a channel that would be complementary to the study here in flavor
physics. Or, is there some orthogonal result?
Dmitri Tsybychev
I think it’s a general problem. If you see something at the LHC, how do you
reconstruct the underlying physics? Say that 200 models can give you the same
signal. It will take more than just one significant deviation in one channel
to really understand the nature of the new physics.
Rahul Sinha
Typically, one talks about the missing $E_{T}$ signal at the LHC. I would like
to get an opinion as to whether flavor physics can help in pinning down the
nature of the new physics. With flavor-physics constraints included, it would
be interesting to come up with signals that can help to say whether it is SUSY
or not SUSY.
Dmitri Tsybychev
If you have missing $E_{T}$, it could be anything. It could be supersymmetry.
It could be a leptoquark. It could be extra dimensions. There are a number of
scenarios that will result in large missing $E_{T}$.
Rahul Sinha
Sure. But, what is it that should be really watched out for, say, for SUSY or
other new physics, and what kind of measurements in flavor physics can
actually help distinguish between the kinds of scenarios. Is it possible to do
that? Anybody?
Jeff Appel
I think the point is that too many things have missing-energy signals to say
that this or that is the specific answer.
Tom Browder
There is a sort of a worldwide effort, at CERN and other places. People are
writing very thick yellow books about the connection between flavor physics
and the physics at LHC. They do consider lots of different scenarios in the
possible impact of all the observables in B physics. You may find reading
these articles boring now because we don’t have a new physics signal at the
LHC to look at. But there have been pretty substantial efforts and a lot of
papers on this.
Enrico Lunghi
I have a general comment on the first two questions. ATLAS and CMS are mostly
”flavor-diagonal” experiments. On the one hand, they will tell us the mass
scales and the tree-level structure of whatever new physics model is realized
in nature. On the other hand, the quantum structure of the theory (e.g. loop
effects) will be hardly accessible. The latter task is perfectly suited for
flavor-physics experiments, that will act as a tie-breaker among the several
equivalent new physics models that will emerge from the first analyses of LHC
data. Of course, these kinds of studies require inputs from ATLAS and CMS.
Once a few masses and processes are known, one can construct complete models
and predict which flavor observables are expected to deviate from the SM
predictions. It is also possible that ATLAS and CMS will not find any new
physics. In this case, flavor physics (including lepton flavor violation) will
allow us to access to much higher scales (e.g. hundreds of TeV). There are two
scenarios. If ATLAS and CMS find TeV-scale new physics, flavor physics will
help to find out the detailed structure of the theory. If, on the other hand,
new physics turns out to be beyond the reach of direct production at the LHC,
we can still explore it via super-rare processes (e.g. lepton flavor
violation).
Choong Sun Kim
I have some unrelated questions for Hsiang-nan and Martin about the previous
discussions. Everyone knows that we, within the standard model, can not
calculate the $B$ to $\pi^{0}\pi^{0}$ branching fractions. Is that new
physics?
Martin Beneke
No.
Choong Sun Kim
Because the error is quite small. The experiment error is small.
Martin Beneke
But the theoretical error is not so small.
Choong Sun Kim
But you can explain all others except for $\pi^{0}\pi^{0}$. Even $B$ to
$\rho^{0}\rho^{0}$, which has exactly same quark diagrams as $\pi^{0}\pi^{0}$,
can be predicted rather well. When the measurements began, it was quite
different - theory predicted only 1/3 of the experimentally measured branching
fraction. So I think today’s value is kind of a post-diction. You just changed
the input parameters. Therefore, even though we think it is rather trivial,
like the color-suppressed tree, it can be something else - like beyond the
standard model.
Martin Beneke
We have learned that the dynamics behind the color-suppressed tree amplitude
is very different from the naive factorization picture, and also understand
why the theoretical uncertainties are large for this amplitude.
Rahul Sinha
I just want to ask something since you raised the question about factorization
and naive factorization. Naive factorization works so well in $D$ decays. We
all remember the classic paper of Bauer, Stech and Wirbel. Factorization,
however, does not work so well in $B$ decays as is evident from data. Is there
a good explanation for that? Why does factorization work better for $D$ decays
and not that well for $B$ decays?
Martin Beneke
I wouldn’t say that this is true. In $B$ decays, we discuss many more
challenging observables than just branching fractions of tree-dominated
decays; such as penguin-dominated decays, $CP$ asymmetries, and strong phases.
Rahul Sinha
Let us just go back to branching ratios for modes like $K\pi$, $\pi\pi$ and …
These things work so well in $D$ decays, but not that well in $B$ decays.
Hsiang-nan Li
But I think this question does not belong to this category. I think it’s still
too early to have any concrete conclusion because currently the theoretical
precision is just up to next-to-leading order, right? So there is next-to-
next-to-leading order, next-to-next-to-next to leading order. There is a long
way to go.
## III Question: What are the connections between observations in the quark
and lepton sectors?
Jeff Appel
This is pretty technical for the round-table level of discussion. I guess I’d
like to move on to our next question. I don’t have a lot of questions. Don’t
get too scared. I wonder about the connection between the flavor observations
in the quark and lepton sectors. Do we understand these? Or, do we have to
wait to get to Plank scale to figure it out.
Choong Sun Kim
The $\sin(\theta_{12})$ in neutrino-sector mixing and $\sin(\theta_{12})$ in
the quark sector, now adding up those 2 mixing angles comes up to about 45
degrees. It could be an accident. Or maybe there is some kind of connection
between the quark sector and the lepton sector. People say it’s
complementarity, something like that. Quark-lepton complementarity. Maybe
there is some reason behind it, or is it an accident?
## IV Question: Is there a flavor-physics community, and if so, has it
articulated its case well enough?
Jeff Appel
One reason why I put this question in here is to address the nature of this
conference and our community. I use the singular form, our community, the
flavor-physics community which covers quarks and leptons. This is the physics
we have discussed at FPCP 2008. Have quarks and leptons been brought together
at this meeting more strongly than in the past because of $CP$ violation only,
or there is something more fundamental that makes them part of the same
community? And if so, has this community articulated the case for support of
both axes strongly enough? I am thinking of the priorities that have been
expressed in the United Kingdom and in the United States. We also have heard
about the delay in kaon physics at J-PARC, and so on.
Taku Yamanaka
Well, let me first speak about the situation in Japan. The High Energy Physics
Committee in Japan, of which I am also a member, wrote up a report on what to
do in the future. In that report, we stated two things. One is, approach the
high energy frontier, including LHC and ILC etc. We also stated that the
intensity frontier, especially flavor physics, is important. This is
especially true because in Japan we have Belle and the neutrino program. The
experiments are very popular and are being supported. J-PARC is the key
facility for neutrino and kaon experiments. Even the people pushing for the
ILC are supporting the J-PARC program, because if J-PARC fails, then there is
no linear collider. From the viewpoint of the funding agency, that’s very
clear.
If the question is, is there a flavor-physics community in Japan, the answer
is yes. The people working on kaons, $B$ physics, and neutrino physics,
experimentalists and theorists, have joined forces and won a ”Grant-in-Aid for
Scientific Research on Priority Areas”, titled ”New Developments of Flavor
Physics”. The project is supported for 6 years, and the fund is being used for
building the T2K and Opera experiments, the J-PARC kaon experiment, $B$
physics at CDF, and Belle. We get together every year to have a small workshop
to present all the new findings. This is really making a close community of
people ranging from young students to older professors working on various
experiments and theories, all on flavor physics.
Martin Beneke
It may be unpopular to say this, but talking to people outside and even within
the flavour physics community, one may get an impression that flavor
physicists had their chance to find new physics. They did not, so it is time
to move on to the next thing - LHC physics. If something shows up there, then
we can go back to flavor physics to try to sort things out. We may be blamed
ourselves for that because we have been talking too much about new physics and
obscure 2$\sigma$ effects, and didn’t succeed to create interest in the
intrinsic physics itself, in the phenomena.
I am fascinated and mystified how neutrino physics is succeeding in this
respect – measuring a mass matrix in the lepton sector, which is after all not
so different from measuring the CKM matrix. And there is even less prospect of
discovering new physics by determining $\theta_{13}$ than there is in
$V_{ub}$!
Jeff Appel
There is an interesting corollary to the way you put it. In terms of selling
the physics these days, one tries to sell physics as ”paradigm-changing”
discoveries. What is the argument you would make to sell our physics, whatever
it is? The first thing one looks for is what people call paradigm-changing
discoveries, right? How would you sell the physics that you are talking about
in the world?
Martin Beneke
Once neutrinos have masses, there is no paradigm change in measuring mixing
angles or even CP violation. Nevertheless, there is some intrinsic interest in
investigating neutrino properties, because neutrinos are considered
mysterious, while quarks are not. In any case, returning to your question,
advertising paradigm change is dangerous, since paradigms usually change by
observations that come unexpectedly, not because of a systematic search.
Jeff Appel
Martin, that’s in fact exactly right. The neutrinos are interesting because we
were surprised. And it’s more a matter of surprise which sells newspapers,
rather than continuing to observe the things we expected to observe, the so
called standard model. That doesn’t sell newspapers. The articles we read
about 600 physicists ”failing” to find this, ”failing” to find that. We have a
problem selling the more precision measurement and standard things.
Bruce Yabsley
You asked how do you sell something more subtle when someone else is pushing
the paradigm-changing. The answer is: it’s damn hard! If you send your
children to their grandparents, and the grandparents feed them candy, they are
attacking kids’ weakness. I am sorry. We always push that this would change
the world, whatever. Think of all that has been happening in spectroscopy,
some of the most interesting stuff to come up in the $B$ factory. I don’t
believe that it’s nuclear physics.
Jeff Appel
And it’s interesting because it was a surprise? Or interesting for another
reason?
Bruce Yabsley
Again, it’s interesting because it’s a surprise. Now, if we get into a
position where we discover something that is both surprising and interesting!
Maybe we have to spend a few years in training on how to talk to guys from the
newspapers. Maybe we just do.
David Kirkby
Maybe one way to answer your question is to look at the nuclear physics
community because they are, at least in the US, well funded; and what they are
doing is not so different from spectroscopy in heavy quarks.
Rahul Sinha
The fact is that we initially set up the $B$ factories to test the CKM
hypothesis. We have succeeded; we have done that. We have not only succeeded
in doing that, but we have learned a lot more. We have new resonances and many
puzzles about them. This is at the very least ”surprising”. So in that sense,
there is no way to say that we have not actually had very good physics output.
Somehow, $B$ physics efforts have become the victim of various constraints
dictating the directions in physics, e.g. our desire to find a way probe the
Plank scale as fast as possible.
Eli Rosenberg
Let me say something that has already been said. The first slide you put up
there. It all had to do with where new physics is going to be found. You
already brought in the concept that to sell anything, it has to be something
new. And on your second slide, the reaction to what happens in the Tevatron
and LHC. God help this field if nothing is found in those places. This
conversation becomes entirely irrelevant simply because we have oversold the
idea that we have to find something new. Now I have a feeling that if we went
back to 30, 40 or 50 years ago, when particle physics was a virgin, people
were working on precision measurements of electromagnetic interactions. We
must have felt exactly the same way you are feeling in this room now - that
somehow we were undervalued by looking at things where you could make precise
measurements. And the real argument is, we are working in the area where you
can make precise measurements, where you can look for new things like lepton
flavor violation. We’d like to measure $D$ mixing because we didn’t expect to
see much of it. It’s interesting, and it has intrinsic interest of its own,
period. Whether it’s going to be something new or not, that is a different
issue. Now, how you sell that to our funding agencies is where the problem
seems to come in. The same thing happens in the $K$ meson sector. The $K$
meson sector had a resurgence at one point after being pushed down for a long
time. So this has been a continuing problem. But I think part of the problem
is that we have gotten so big and so expensive that we oversell everything.
The field as a whole has oversold everything. This is what you have to do.
That’s why you read the headline about 600 physicists failing to find this or
that; because we said we were going to find it. You know, we sold the SSC as
if we could do everything except cure baldness. So I think we just have a PR-
reality problem about what science is about.
(George) Wei-Shu Hou
I would like to make several remarks touching on all that has been said. I
think that on neutrino physics, I held back on one question that I used to
ask. If I take $V_{ub}$, it’s very hard to extract, correct? But if we take
the $V_{ub}$ analogy, because neutrino people have had ten spectacular years,
this is in part because of the very large mixing angles. They could not ordain
that, right? So if I take $V_{ub}$ or even $V_{cb}$, our $\theta_{13}$ or
$\theta_{23}$ for neutrinos, I don’t see a program yet to measure something of
that strength for $\theta_{13}$. They are entering a hard time. Without that
(a large $\theta_{13})$, forget about $CP$ violation in the lepton sector. OK,
Majorana neutrinos, (neutrinoless) double beta decay, there is always some
discovery potential. But they are not really doing better than we are. I don’t
know how, in the last ten minutes, we entered such a very gloomy mood. I think
we actually have a good situation. The LHC is starting. The Tevatron is still
working very hard. We are seeing things here and there. We are, of course,
used to seeing things disappear from before our eyes. But that’s how it is;
right? Seeing something emerge and then disappear; hoping that one of them is
true. And I think this mixing-dependent $CP$ violation phase (in $B_{s}$) is
of course the way to go. But, maybe we are overselling it. It may be a PR
problem, but we do have genuine indications, not just challenges. So I like
what Enrico said, I think at LHC, ATLAS and CMS analyses will find the scale.
The New Particles will likely be extrinsic to flavor. However there is also
LHCb, right? I guess we go back to Question 2. What if we see nothing beyond
the Standard Model? Maybe we see the Higgs, maybe we don’t. But if we see
nothing, and LHCb will measure $\sin(2\phi_{s})$ to plus or minus 4 percent,
no charged Higgs, no SUSY. Then no matter what PR we do, you can not get the
next big machine.
But I am optimistic about both LHC proper, the high energy frontier, and LHCb
also. I am also a full supporter of the Super KEKB or Super $B$-factory.
Because it’s really a PR question again. A Super $B$-factory is a multi-
purpose facility. And speaking from Asia, I am even more supportive of this.
Asia is rising. It has the population, etc. I would fully support it even just
only on that account. That it is a project to work on, to go forward. And if
not a discovery at this stage, then there will be a discovery at the next
stage. To me, Super KEKB is a regional cooperation concern.
Jeff Appel
In order to move to a more positive direction, perhaps there are other
questions people would like to address to the panel, or to each other before I
get to my last question?
Bruce Yabsley
Just inspired by the previous discussion, I would note that when the LHC turns
on, the field is going to undergo a kind of basic change. The kind of
information we are using to decide what studies to do, and how to do it, is
going to change. Now, the moment someone puts a preprint on hep-ph, everyone
at the $B$-factories, drops everything they are doing to pursue the suggested
analysis, or something like this. What’s going to happen now is that we are
going to get some sign of a particular mass scale. And that’s going to have an
influence on the things we should be studying. But the influence is going to
be a hundred percent, because it’s going to determine what the new physics is,
presumably, and more than one model will be possible. Here my question is: do
we have a mechanism so that we have a while to think about how we are going to
be influenced by the new information. Or, are we going to be driven by some
prejudice that what we see, is what we know, and so suddenly everyone rushes
in particular directions like ten-year-olds playing soccer. The situation
really is going to change. I am wondering if we thought forward to what
happens when we see data.
Jeff Appel
I think there is plenty of evidence that the community is a very good at
rushing together in singular directions, in effect, ten-year-olds playing
soccer as you called it. We also did this when the $J/\psi$ was discovered.
Every experiment asked if they could see evidence of that signal.
Enrico Lunghi
I would like to make a comment on the impossibility of a null result at the
LHC. In fact, unitarity tells us that either we’ll find at least one Higgs
particle. Otherwise, some other phenomena have to happen (e.g. strongly-
interacting vector bosons, new strong resonances, …). In any case, even if
just a SM-like Higgs is found, we still need a linear collider to study its
properties. I don’t think that we can really go there and see nothing.
Eli Rosenberg
Because you are so convinced, and we have been told that we will. And if we
see nothing, it is the most fascinating physics result of all - except that it
will kill the field. Aside from that, … I rest my case. Then why do we have to
build it? Because we are going to see something.
Choong Sun Kim
I have one question not related to politics or anything like that. We know
that George proposes a fourth generation. But if there is a fourth generation,
it is supposed to violate unitarity in three generations because, effectively,
the 3 by 3 part of the larger 4 by 4 matrix is non-unitary. OK. What I want to
ask is another thing, about gamma in the unitary triangle. The gamma or alpha
measurement is not actually measuring gamma or alpha. It’s beta plus gamma,
for example. So my question is, is it possible that LHCb, or Super-B, or any
future B-factory can find the 3 by 3 CKM matrix non-unitary? Is it possible to
find non-unitarity or not?
Jeff Appel
Yes, anybody working in $B$ factories would say yes, you can find the triangle
does not close. It is not unitary and there is going to be something else,
some new physics.
Chung-Sun Kim
The measurement of gamma or alpha is from the $\pi-(\alpha+\beta)$. So by
definition, you are just taking the angles as from a triangle.
Jeff Appel
The sides have to work too, right?
Eli Rosenberg
Gamma, perhaps from a Dalitz analysis, maybe from the $\Upsilon(5S)$. Does
that beta plus gamma match the beta plus gamma you get when they interfere?
That’s the test. And that’s equivalent to test unitarity; whether the standard
model is working. That’s what is, in short hand, called alpha. I agree with
you. Nobody measures alpha, but measures $\pi-(\beta+\gamma)$.
Rahul Sinha
The one thing to note is that $\gamma$, or one of the angles measured, has to
be from outside of the $B_{d}$, to see a breakdown of unitarity. It is well
known that if you measure the three phases using just $B_{d}$, the effect of
new physics will cancel out. In addition, the other thing one can do is to
measure both sides and the angle and then check if the triangle closes.
Jeff Appel
The same triangle.
Choong Sun Kim
The measurement of $\gamma$ or $\alpha$ is from $\pi-(\alpha+\beta)$ or
$\pi-(\beta+\gamma)$. So, by definition, you are just forcing a triangle if
you do not measure the 3 angles independently. Also, beta from $B\rightarrow
J/\psi K_{s}$ can effectively include new physics, too.
Rahul Sinha
Yes. But, you can also measure gamma, outside the $B_{d}$ system. There is a
method to measure it using $B_{s}\rightarrow DK$. If you do that, then there
is no problem. You can detect the breakdown of unitarity without measuring the
side.
Jose Ocariz
Another way of saying it is that you are measuring 4 parameters with 10
observables. If you have no consistency, you have no unitarity.
## V A final question.
Jeff Appel
We are reaching the end of our scheduled time. I do want end on one new
question, our last question, which is ”How can we thank our hosts enough for
their hospitality, the careful and caring organization of this meeting, their
holding back the worst of the rains for the excursion, and clearing the sky as
well for the highest level of FPCP banquet ever held? So, I think we should
close this session and thank our hosts very much.
(applause)
And I personally want to thank the panelists and all of you in the audience
for the very stimulating discussion. Good luck on your trips home, whether
near or far. And, again, thank you all.
|
arxiv-papers
| 2008-09-19T16:33:32
|
2024-09-04T02:48:57.834170
|
{
"license": "Public Domain",
"authors": "Jeffrey A. Appel (Fermilab), Jen-Feng Hsu, Hsiang-nan Li (Institute of\n Physics, Academia Sinica)",
"submitter": "Jeffrey A. Appel",
"url": "https://arxiv.org/abs/0809.3414"
}
|
0809.3426
|
# Superfluid and Fermi liquid phases of Bose-Fermi mixtures in optical
lattices
Kaushik Mitra, C. J. Williams and C. A. R. Sá de Melo Joint Quantum
Institute, University of Maryland, College Park, Maryland 20742,
and National Institute of Standards and Technology, Gaithersburg, Maryland
20899
###### Abstract
We describe interacting mixtures of ultracold bosonic and fermionic atoms in
harmonically confined optical lattices. For a suitable choice of parameters we
study the emergence of superfluid and Fermi liquid (non-insulating) regions
out of Bose-Mott and Fermi-band insulators, due to finite Boson and Fermion
hopping. We obtain the shell structure for the system and show that angular
momentum can be transferred to the non-insulating regions from Laguerre-
Gaussian beams, which combined with Bragg spectroscopy can reveal all
superfluid and Fermi liquid shells.
###### pacs:
03.75.Hh, 03.75.Kk, 03.75 Lm
Fermi and Bose degenerate quantum gases and liquids are amazing systems, which
have revealed individually several macroscopic quantum phenomena. For
instance, superfluidity is known to exist in neutral liquids such as 4He
(boson) and in 3He (fermion), as well as in a variety of electronic materials
studied in standard condensed matter physics. The role of quantum statistics
and interactions is of fundamental importance to understand the phases
emerging from purely bosonic or purely fermionic systems, and a substantial
amount of understanding of these individual Bose or Fermi systems can be found
in the atomic and condensed matter physics literature. However, new frontiers
can be explored when mixtures of bosons and fermions are produced in harmonic
traps or optical lattices. An important example of the richness of quantum
degenerate Bose-Fermi mixtures was revealed in standard condensed matter
systems, where for fixed 4He density and increasing amounts of 3He, the
critical temperature for superfluidity is reduced, and below a tricritical
point phase separation appears reppy-1967 . In this Bose-Fermi mixture of
standard condensed matter physics, essentially the only control parameter is
the ratio between the densities of 4He and 3He.
In atomic physics, a spectacular degree of control has been achieved in Bose-
Fermi mixtures, where not only the ratio between densities of bosons and
fermions can be adjusted, but also the interactions between fermions and
bosons can be controlled through the use of Feshbach resonances bongs-2006 ,
as demonstrated in mixtures of harmonically trapped Bose-Fermi polarized
Fermion mixtures of 40K and 87Rb. Furthermore, these same atoms have been
succesfully loaded into optical lattices esslinger-2006 and have produced a
system that has no counterpart in standard condensed matter systems. By
controlling the depths of optical lattices we can change not only the
interactions between boson and fermions, but also their hopping from site to
site, thus allowing the exploration of a very rich phase space, where
supersolid and phase separated states have been suggested blatter-2003 . The
list of Bose-Fermi mixtures in atomic physics is growing, and include systems
where the masses are close like 6Li and 7Li, 39K and 40K, or 172Yb and 173Yb;
or systems where the masses are quite different like 6Li and 40K, 7Li and 39K,
6Li and 23Na, 6Li and 87Rb, 40K and 23Na, or 40K and 87Rb. This suggests a
wide possibility of regimes that can be reached by tuning interactions,
density and geometry, which is not is possible in ordinary condensed matter
physics.
A few studies of quantum phases of Bose-Fermi mixtures have focused on
homogeneous three dimensional systems with lewenstein-2004 ; batrouni-2007 ,
and without optical lattices viverit-2000 . However, only one effort focused
on harmonically confined optical lattices illuminati-2004 . Most of the
descriptions of Bose-Fermi mixtures in optical lattices have relied on
numerical methods using either Gutzwiller projection lewenstein-2004 ;
illuminati-2004 or quantum Monte Carlo batrouni-2007 techniques. In this
paper, we present a fully analytical theory of boson and spin polarized
fermion mixtures in harmonically confined optical lattices by using degenerate
perturbation theory for finite hopping in conjunction with the local density
approximation. This work provides insight into the phase diagram of Bose-Fermi
mixtures, and into the detection of superfluid and Fermi liquid shells at low
temperatures.
This paper we analyses in detail the regime where the hopping parameters of
bosons and fermions are comparable and the repulsion between bosons and
fermions is a substantial fraction of the boson-boson repulsion. In this case,
the system presents regions of (I) coexisting Bose-Mott and Fermi-band
insulator, (II) coexisting Bose-Mott insulator and Fermi liquid, (III) Bose-
Mott insulator, and (IV) Bose superfluid, as shown in Fig. 1. We compute
analytically the boundaries between various phases, and obtain the spatially
dependent boson and fermions filling fractions in each region. Although one
can envisage other situations where, for example, one can have coexistence of
superfluid and Fermi band insulator, we confine our discussion to the
situation above for the sake of simplicity. Finally, we propose a detection
method of the shell strucuture of Bose-Fermi mixtures by using Laguerre-
Gaussian beams helmerson-2007 and Bragg spectroscopy raman-2006 , where
angular momentum is transferred only to regions with extended states such as
the superfluid and Fermi-liquid shells.
To describe Bose-Fermi mixtures in harmonically confined square (2D) or cubic
(3D) optical lattices we start with the Hamiltonian
$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle K_{B}+K_{F}-\sum_{\bf
r}\mu_{B}({\bf r}){\hat{n}}_{B}({\bf r})-\sum_{\bf r}\mu_{F}({\bf
r}){\hat{n}}_{F}({\bf r})$ $\displaystyle+$
$\displaystyle\frac{U_{BB}}{2}\sum_{\bf r}{\hat{n}}_{B}({\bf
r})\left[{\hat{n}}_{B}({\bf r})-1\right]+U_{BF}\sum_{\bf r}{\hat{n}}_{B}({\bf
r}){\hat{n}}_{F}({\bf r}),$
where $K_{B}=-t_{B}\sum_{\langle{\bf r},{\bf r}^{\prime}\rangle}b_{\bf
r}^{\dagger}b_{{\bf r}^{\prime}}$ and $K_{F}=-t_{F}\sum_{\langle{\bf r},{\bf
r}^{\prime}\rangle}f_{\bf r}^{\dagger}f_{{\bf r}^{\prime}}$ are the kinetic
energies of boson and fermions with nearest-neighbor hoppings $t_{B}$ and
$t_{F}$, and $b^{\dagger}_{\bf r}$ and $f^{\dagger}_{\bf r}$ are the bosonic
and fermionic creation operators at site ${\bf r}$. Here, the lattice sites
for bosons and fermions are assumed to be the same, but the hopping parameters
can be different. The number operators are ${\hat{n}}_{B}({\bf r})=b_{\bf
r}^{\dagger}b_{\bf r}$ and $\hat{n}_{F}({\bf r})=f_{\bf r}^{\dagger}f_{\bf
r}$, and the corresponding local chemical potentials are $\mu_{F}({\bf
r})=\mu_{F}-V_{F}({\bf r})$ and $\mu_{B}({\bf r})=\mu_{B}-V_{B}({\bf r})$,
where $V_{F}({\bf r})=\Omega_{F}(r/a)^{2}/2$ and $V_{B}({\bf
r})=\Omega_{B}(r/a)^{2}/2$ are the harmonically confining potentials and
$\mu_{F}$ and $\mu_{B}$ are the chemical potentials for fermions and bosons.
The origin of the lattice with spacing $a$ is chosen to be at the minimum of
the harmonically confining potential. The terms containing $U_{BB}$ $(U_{BF})$
represent the boson-boson (boson-fermion) interaction.
When $t_{B}=t_{F}=0$, the Hamiltonian ${\hat{H}}={\hat{H}}_{0}$ is a sum of
single-site contributions, and the eigenstates are tensor products of number
states with state vectors
$|\psi\rangle=|n_{B,0},n_{B,1},\cdots\rangle|n_{F,0},n_{F,1},\cdots\rangle$,
with $n_{B,{\bf r}}=0,1,2,...$ and $n_{F,{\bf r}}=0,1$ representing the
occupation number of bosons and fermions at site ${\bf r}$, respectively. At
site ${\bf r}$ the local energy is $E_{n_{B},n_{F}}({\bf
r})=U_{BB}n_{B}(n_{B}-1)/2+U_{BF}n_{B}n_{F}-\mu_{B}({\bf r})n_{B}-\mu_{F}({\bf
r})n_{F}$. For the ground state wavefunction the number of bosons at site
${\bf r}$ is determined by $max(0,\lfloor(U_{BB}+\mu_{B}({\bf
r}))/2U_{BB}\rfloor)$ if $E_{n_{B},0}({\bf r})<E_{n_{B},1}({\bf r})$ and
$max(0,\lfloor(U_{BB}+\mu_{B}({\bf r})-U_{BF})/2U_{BB}\rfloor)$ otherwise.
Similarly, the number of fermions per site is zero if $E_{n_{B},0}({\bf
r})<E_{n_{B},1}({\bf r})$ and one otherwise. The symbol $\lfloor\cdot\rfloor$
is the floor function. In the ground state solution shells $(n_{B},n_{F})$
with $n_{B}$ bosons and $n_{F}$ fermions are formed by those lattice sites
${\bf r}$ for which the local energy is the same. For our harmonic traps these
shells are nearly spherically symmetric. The boundary between shells
$(n_{B},n_{F})$ and $(n_{B}\\!+\\!1,n_{F})$ is determined by
$E_{n_{B},n_{F}}({\bf r})=E_{n_{B}\\!+\\!1,n_{F}}({\bf r})$, leading to the
radius $R_{B,n_{B},n_{F}}=a\sqrt{\Omega_{n_{B},n_{F}}/\Omega_{B}}$, where
$\Omega_{n_{B},n_{F}}=2(\mu_{B}-n_{B}U_{BB}-n_{F}U_{BF})$. Similarly, the
boundary between shells with occupation numbers $(n_{B},0)$ and $(n_{B},1)$ is
determined by equating the local energies $E_{n_{B},n_{F}}({\bf r})$ and
$E_{n_{B},n_{F}+1}({\bf r})$, leading to the radius
$R_{F,n_{B}}=a\sqrt{2(\mu_{F}-n_{B}U_{BF})/\Omega_{F}}$. We consider the
number of particles to be sufficiently large such that the radii of the
boundaries are much larger than $a$.
Next, we begin our discussion of finite hoppings by taking first $t_{B}\neq
0$, with $t_{F}=0$. The Bose superfluid region emerges due to kinetic
fluctuations at the boundaries between the $(n_{B},n_{F})$ and
$(n_{B}+1,n_{F})$ shells. At this boundary the local energy
$E_{n_{B}+1,n_{F}}({\bf r})$ is degenerate with $E_{n_{B},n_{F}}({\bf r})$. To
describe the emergence of superfluid regions, we introduce the order parameter
for superfluidity $\psi_{B,j}$ via the transformation
$b_{i}^{\dagger}b_{j}\to\psi_{B,i}^{*}b_{j}+b_{i}^{\dagger}\psi_{B,j}-{\psi_{B,i}^{*}\psi_{B,j}}$,
and then for analytical convenience make the continuum approximation
$\psi({\bf r}+{\bf\it a})=\psi({\bf r})+a_{i}\partial_{i}\psi({\bf
r})+(1/2)a_{i}a_{j}\partial_{i}\partial_{j}\psi({\bf r})$.
In the limit of $U_{BB}\gg t_{B}$, we can restrict our Hilbert space to the
number basis states $|n_{B},n_{F}\rangle$ and $|n_{B}+1,n_{F}\rangle$, as any
contribution from other basis states to the local energy is of order
$t_{B}^{2}/U_{BB}$. The hopping term $t_{B}$ affects the energies
$E_{n_{B}+1,n_{F}}({\bf r})$ and $E_{n_{B},n_{F}}({\bf r})$ by removing their
degeneracy, thus creating finite-width superfluid regions between shells
$(n_{B}+1,n_{F})$ and $(n_{B},n_{F})$. The effective local Hamiltonian then
becomes,
$H_{\mathbf{r}}^{\textrm{eff}}=\left(\begin{array}[]{cc}E_{n_{B},n_{F}}({\bf
r})+\Lambda({\bf r})&-\sqrt{n_{B}+1}\Delta({\bf r})\\\
-\sqrt{n_{B}+1}\Delta^{*}({\bf r})&E_{n_{B}+1,n_{F}}({\bf r})+\Lambda({\bf
r})\end{array}\right),$ (1)
where $\Lambda({\bf r})=\frac{1}{2}(\Delta({\bf r})\psi^{*}({\bf r})+cc)$ and
$\Delta({\bf r})=t_{B}(z\psi({\bf r})+{\it a}^{2}\nabla^{2}\psi({\bf r}))$.
Here, $z$ is the coordination number which depends on the lattice dimension
$d$.
The eigenvalues of Eq. (1) are given by,
$E_{\pm}(\mathbf{r})=E_{s}(\mathbf{r})\pm\sqrt{\left[E_{d}(\mathbf{r})\right]^{2}+(n_{B}+1)\left|\Delta(\mathbf{r})\right|^{2}},$
where $E_{s}(\mathbf{r})=\left[E_{n_{B}+1,n_{F})}({\bf
r})+E_{n_{B},n_{F}}({\bf r})\right]/2+\Lambda({\bf r})$ is proportional to the
sum of the diagonal terms, and $E_{d}(\mathbf{r})=\left[E_{n_{B}+1,n_{F}}({\bf
r})-E_{n_{B},n_{F}}({\bf r})\right]/2=(n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf
r})/2$ is proportional to their difference. Notice that $E_{-}({\bf r})$ is
the lowest local energy leading to the total ground state energy ${\bf
E}=\frac{1}{L^{d}}\int d{\bf r}E_{-}({\bf r}).$
The order parameter equation (OPE) is determined by minimization of ${\bf E}$
with respect to $\psi^{*}(\mathbf{r})$ leading to
$\Delta(\mathbf{r})-\frac{(n_{B}+1)t_{B}(z+a^{2}\nabla^{2})\Delta(\mathbf{r})}{2\sqrt{\left|E_{d}(\mathbf{r})\right|^{2}+(n_{B}+1)\left|\Delta(\mathbf{r})\right|^{2}}}=0.$
(2)
Notice that the OPE is not of the Gross-Pitaeviskii (GP) type, since the
superfluid regions emerge from local fluctuations between neighboring Mott
shells. Ignoring the spatial derivatives of $\psi$ in Eq. 2 leads to the
spatially dependent order parameter
$\left|\psi(\mathbf{r})\right|^{2}=\frac{n_{B}+1}{4}-\frac{\left(n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf
r})\right)^{2}}{4z^{2}t_{B}^{2}(n_{B}+1)}.$ (3)
Since $\left|\psi(\mathbf{r})\right|^{2}\geq 0$, hence
$|n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B,\bf{r}}|\leq(n_{B}+1)zt_{B}$, and the inner
$R_{n_{B},n_{F},-}$ and outer $R_{n_{B},n_{F},+}$ radii for the superfluid
shell between the $(n_{B},n_{F})$ and $(n_{B}+1,n_{F})$ Mott regions are
obtained by setting $\left|\psi(\mathbf{r})\right|^{2}=0$ leading to
$R_{n_{B},n_{F},\pm}=R_{B,n_{B},n_{F}}\sqrt{1\pm\frac{2zt_{B}(n_{B}+1)}{\Omega_{B}}\frac{a^{2}}{R_{B,n_{B},n_{F}}^{2}}}.$
This relation shows explicitly that $t_{B}$ splits the spatial degeneracy of
the $(n_{B},n_{F})$ and $(n_{B}+1,n_{F})$ insulating shells at
$r=R_{c,n_{B},n_{F}}$ or $\mu_{B}({\bf r})=n_{B}U_{B}B+n_{F}U_{BF}$ by
introducing a superfluid region of width $\Delta
R_{n_{B},n_{F}}=R_{n_{B},n_{F},+}-R_{n_{B},n_{F},-}$. (See Fig. 1 for
characteristic widths).
In addition, the local bosonic filling fraction
$n_{B}(\mathbf{r})=-\frac{\partial
E_{-}(\mathbf{r})}{\partial\mu_{B}}=n_{B}+\frac{1}{2}-\frac{n_{B}U_{BB}+n_{F}U_{BF}-\mu_{B}({\bf{r}})}{2zt_{B}(n_{B}+1)}$
in the same region interpolates between $n_{B}+1$ for $r\lesssim
R_{n_{B},n_{F},-}$ and $n$ for $r\gtrsim R_{n_{B},n_{F},+}$, while the
chemical potential $\mu_{B}$ is fixed by the total number of bosons
$N_{B}=\int d{\bf r}n({\bf r})$. The local bosonic compressibility
$\kappa_{B}({\bf r})=\partial n_{B}({\bf
r})/\partial\mu_{B}=1/2zt_{B}(n_{B}+1)$ of the superfluid shells is non-zero,
in contrast to the incompressible ($\kappa_{B}=0$) $(n_{B},n_{F})$ and
$(n_{B}+1,n_{F})$ insulating shells for $r<R_{n_{B},n_{F},-}$ and
$r>R_{n_{B},n_{F},+}$, respectively.
Figure 1: (color online) (a) Shell structure of Bose-Fermi mixtures in
harmonically confined optical lattices showing a coexisting Bose-Mott and
Fermi-band insulator region at the center $(n_{B}=1,n_{F}=1)$, a coexisting
Bose-Mott insulator and Fermi liquid region (in blue), a Bose-Mott insulator
region $(n_{B}=1,n_{F}=0)$, and Bose-Superfluid region at the edge (in red).
(b) filling factors for fermions shown as the solid (dark blue) curve and for
bosons shown at the dashed (red) curve. The solid parabolic curve (light blue)
shows the order parameter in the superfluid region. The parameters used are
$t_{F}=t_{B}=0.0325U_{BB}$, $U_{BF}=0.1U_{BB}$, $\mu_{B}=0.8U_{BB}$,
$\mu_{F}=0.4U_{BB}$ and $\Omega_{F}=\Omega_{B}=8\times 10^{-6}U_{BB}$, which
are representative of Bose-Fermi mixtures with nearly the same mass such as
6Li and 7Li, 39K and 40K, or 172Yb and 173Yb. For the parameters chosen, the
widths of the superfluid and FL shells are several times larger than the
lattice spacing $a$.
Now, we consider finite $t_{F}$. In order to have a tractable theory we assume
that the shell boundaries of the bosons and fermions are well separated. This
allows us to investigate the Fermi liquid near the shell boundary of the
fermions in the presence of a Bose-Mott insulator with $n_{B}$ bosons per
site. Furthermore, if we assume that the local density of the Fermi gas is
smoothly varying then the local number of fermions is
$n_{F}({\bf r})=\int_{\epsilon_{\rm min}}^{\epsilon_{\rm max}}d\epsilon{\cal
D}(\epsilon)f[\epsilon-\mu^{\rm eff}_{F}({\bf r})]$ (4)
where $f[x]$ is the Fermi function at temperature $T$, and ${\cal
D}(\epsilon)=\sum_{{\bf k}}\delta(\epsilon-\epsilon_{{\bf k}})$ is the density
of fermion states with energy dispersion $\epsilon_{{\bf
k}}=-2t_{F}\sum_{\ell}^{d}\cos(k_{\ell}a)$. The effective chemical potential
$\mu^{\rm eff}_{F}({\bf r})=\mu_{F}({\bf r})-n_{B}U_{BF}$ accounts for the
effect of the bosons. The band minimum and maximum of $\epsilon_{{\bf k}}$ are
$\epsilon_{\rm min}=-2dt_{F}$ and $\epsilon_{\rm max}=2dt_{F}$, respectively.
Thus, the Fermi liquid region is limited by the boundaries $\epsilon_{\rm
min}\leq\mu_{F}({\bf r})\leq\epsilon_{\rm max}$, leading to
$R_{F,\pm}=R_{F,n_{B}}\sqrt{1\pm 2zt_{F}a^{2}/\Omega_{F}R_{F,n_{B}}^{2}}$ for
the inner $R_{F,-}$ and outer $R_{F,+}$ radius of the FL shell. The width of
the FL region is $\Delta R_{F,n_{B}}=R_{F,+}-R_{F,-}$. (See Fig. 1 for
characteristic widths). The isothermal compressibility of the FL region is
$\kappa_{F}({\bf r})=\partial n_{F}({\bf r})/\partial\mu_{F}$, which leads at
zero temperature to $\kappa_{F}({\bf r})=0$ outside the FL shell, indicating
the presence of insulating regions and $\kappa_{F}({\bf r})={\cal
D}[\mu_{F}({\bf r})]$ inside the FL shell, indicating the presence of
conducting regions. The superfluid and FL shells for finite $t_{B}$ and
$t_{F}$, and their density profiles are shown in Fig. 1 for the two-
dimensional case.
Next, we propose an experiment to detect superfluid and Fermi liquid shells in
Bose-Fermi mixtures using a combination of Gaussian and Laguerre-Gaussian
beams followed by Bragg spectroscopy. To illustrate the idea, we discuss the
simpler case of a nearly two-dimensional configuration, where the harmonic
trap is very tight along the z-direction, loose along the x- and y-
directions. Upon application of Gaussian and Laguerre-Gaussian beams along the
z-direction, only angular momentum is transferred to the atoms in the
conducting phases (superfluid or Fermi liquid), imposing a rotating current
with a well defined velocity profile, while the insulating regions do not
absorb angular momentum due to their large gap in the excitation spectrum.
Figure 2: (Color online) Schematic plot for the detection of outer (red)
superfluid and inner (blue) Fermi liquid shells using Bragg spectroscopy. The
angles $\theta_{1}$ and $\theta_{2}$ indicate the locations of strongest
momentum transfer from the Bragg beams (large green arrows) to the rotating
superfluid and Fermi liquid shells of radii $R_{1}$ and $R_{2}$. The gray
arrows indicate the sense of rotation of the conducting shells.
To probe the rotating superfluid and Fermi liquid phases we propose the use of
two counter-propagating Bragg beams applied in the xy plane along the x
direction, as indicated in Fig. 2.
The Bragg beams transfer a net linear momentum $\hbar(k+k^{\prime}){\bf x}$ to
the atoms of mass $m$ which satisfy the energy conservation condition
$\hbar(\omega-\omega^{\prime})=\epsilon_{f}-\epsilon_{i}-v_{x}\hbar(k+k^{\prime})+\frac{\hbar^{2}(k+k^{\prime})^{2}}{2m},$
(5)
where $v_{x}$ is the component of the velocity ${\bf v}({\bf r})={\bf p}({\bf
r})/m$ along the x direction and $\epsilon_{i}$ and $\epsilon_{f}$ are the
experimentally accessible energies of the initial and final internal states
atom. For an atom carrying one unit of angular momentum, the velocity is ${\bf
v}_{i}=\hbar\hat{\theta}/mr$. Therefore, within a conducting shell with radius
$r=R$ atoms get a linear momentum kick of $\hbar(k+k^{\prime})\hat{\bf x}$
when the velocity $v_{x}=\hbar\sin\theta/mR$ satisfies the condition given in
Eq. 5. This leads to two Bragg angles $\theta=-\sin^{-1}(mRv_{x}/\hbar)$, and
$\pi-\theta$ for each conducting shell. As can be seen in Fig. 2, the Bragg
angles are $\theta_{1}$ and $\pi-\theta_{1}$ for the outer superfluid shell
labelled by $R_{1}$, and are $\theta_{2}$ and $\pi-\theta_{2}$ for the fermi
liquid shell labelled by $R_{2}$. Once these atoms are kicked out of the
conducting shells, they form two small expanding clouds, which can be detected
by direct absorption imaging.
Next, we discuss the time scales over which the rotation in the conducting
regions persist and can be detected experimentally. In the case of the
superfluid region we use the Landau criterion to show that the velocity
imposed to the superfluid through the angular momentum transfer is much
smaller than the local sound velocity $c({\mathbf{r}})=\sqrt{\rho_{s}({\bf
r})/\kappa}$, where $\rho_{s}({\bf r})=2t_{B}a^{2}|\psi({\bf r})|^{2}$ is the
local superfluid density, and $\kappa$ is the compressibility. Thus,
$c({\mathbf{r}})=2\sqrt{(n_{B}+1)z}ta|\psi(\mathbf{r})|$ vanishes at the
insulator boundaries where $|\psi({\bf r})|=0$, and only close to the edge of
the superfluid regions the local rotational speed $v({\bf r})=\hbar/mr$
exceeds $c({\bf r})$, which means that essentially all the superfluid region
can be detected and the angular momentum transferred does not decay over time
scales of at least seconds, limited by the lifetime of the trapped system.
In the case of the Fermi liquid region, the time scale over which the flow of
the fermions persist in presence of the Bose-Mott insulator background can be
calculated from the imaginary part of the fermionic self-energy
$\Sigma_{F}(k)=U_{BF}^{2}T^{2}\sum_{q_{1},q_{2}}G_{B}(q_{1})G_{B}(q_{2})G_{F}(k+q_{1}-q_{2}),$
where $k=({\bf k},i\omega)$ and $q_{i}=({\bf q}_{i},i\nu_{i})$, with $\omega$
$(\nu_{i})$ are fermionic (bosonic) Matsubara frequencies and $T$ is
temperature. The bare inverse bosonic propagator in the Bose-Mott phase is
$G_{B}^{-1}(q,{\bf r})=\epsilon_{{\bf q}}\left[1+\epsilon_{\bf
q}\left(\frac{n_{B}+1}{i\hbar\omega-E_{1}({\bf r})}-\frac{n_{B}}{i\hbar\omega-
E_{2}({\bf r})}\right)\right],$
where $E_{1}({\bf r})=(n_{B}-1)U_{BB}-\mu_{B}({\bf r})$, $E_{2}({\bf
r})=(n_{B}-2)U_{BB}-\mu_{B}({\bf r})$ and $\epsilon_{\bf
q}=-2t_{B}\sum_{\ell}^{d}\cos(k_{\ell}a)$. The bare inverse fermionic
propagator in the Fermi liquid phase is $G_{F}^{-1}(k,{\bf
r})=i\hbar\omega-\epsilon_{F}({\bf k},{\bf r}),$ where $\epsilon_{F}({\bf
k},{\bf r})=\epsilon_{F}({\bf k})-\mu_{F}({\bf r})$. For $n_{B}=1$ and $T=0$,
the imaginary part of the fermionic self-energy is
${\rm Im}\Sigma_{F}(k,{\bf
r})=-\pi{U^{2}_{BF}}\left[F(\hbar\omega)+F(-\hbar\omega)\right],$ (6)
where $F(\hbar\omega)=\Theta(\hbar\omega)\Theta(-\hbar\omega+\mu_{B}({\bf
r})){\cal D}(\hbar\omega+\mu_{F}({\bf r}))$, $\Theta$ is the Heaviside step
function, and ${\cal D}(\epsilon)$ is the Fermion density of states. For a
two-dimensional Fermi liquid shell, there is a Van Hove singularity in ${\cal
D}(\epsilon)$ at half filling. The expression in Eq. (6) is independent of
momentum, since the dominant excitations in the Bose-Mott region are number-
conserving and low-momentum particle-hole excitations, but strongly dependent
on position through $\mu_{B}({\bf r})$ and $\mu_{F}({\bf r})$, leading to a
characteristic decay time $\tau_{\bf r}=-h/{\rm Im}\Sigma(k,{\bf r})$. For the
parameters used in Fig. 1, (with $U_{BF}/h=1~{}{\rm kHz}$) the time scale for
the persistence of the flow near the edges (away from the Van Hove
singularity) is $\tau_{\bf r}\approx 13~{}{\rm ms}$. However, near the center
of the Fermi liquid region (close to the Van Hove singularity) $\tau_{\bf r}$
is extremely short, indicating that it is much easier to detect fermions at
the edge than at the center of Fermi liquid shells.
We have discussed the phase diagram of Bose-Fermi mixtures in harmonically
confined optical lattices in the regime where the hopping parameters of bosons
and fermions are comparable and the repulsion between bosons and fermions is a
substantial fraction of the boson-boson repulsion. We showed that the system
exhibits regions of (I) coexisting Bose-Mott and Fermi-band insulator, (II)
coexisting Bose-Mott insulator and Fermi liquid, (III) Bose-Mott insulator,
and (IV) Bose-superfluid. We have calculated analytically the boundaries
between these phases and obtained the spatially dependent filling fraction for
each region. Finally, we proposed a detection method of the superfluid and
Fermi liquid shells of Bose-Fermi mixtures by using Gaussian and Laguerre-
Gaussian beams followed by Bragg spectroscopy.
## References
* (1) E. H. Graf, D. M. Lee, and J. D. Reppy, Phys. Rev. Lett. 19, 417 (1967).
* (2) S. Ospelkaus et al., Phys. Rev. Lett. 97, 120403 (2006).
* (3) K. Günter et al., Phys. Rev. Lett. 96 180402 (2006).
* (4) H. P. Büchler and G. Blatter, Phys. Rev. Lett. 91, 130404 (2003).
* (5) L. Viverit, C. J. Pethick, and H. Smith, Phys. Rev. A A, 053605 (2000).
* (6) M. Lewenstein et al., Phys. Rev. Lett. 92, 050401 (2004).
* (7) F. Hébert et al., Phys. Rev. A 76, 043619 (2007).
* (8) M. Cramer, J. Eisert, and F. Illuminati, Phys. Rev. Lett. 93, 190405 (2004).
* (9) C. Ryu et al., Phys. Rev. Lett. 99, 260401 (2007).
* (10) S. R. Muniz, D. S. Naik, and C. Raman, Phys. Rev. A 73 041605(R) (2006).
|
arxiv-papers
| 2008-09-19T17:30:08
|
2024-09-04T02:48:57.840633
|
{
"license": "Public Domain",
"authors": "Kaushik Mitra, C. J. Williams and C. A. R. S\\'a de Melo",
"submitter": "Kaushik Mitra",
"url": "https://arxiv.org/abs/0809.3426"
}
|
0809.3458
|
# A note on gaps
Hisanobu Shinya
###### Abstract
Let $p_{k}$ denote the $k$-th prime and $d(p_{k})=p_{k}-p_{k-1}$, the
difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the
number of primes $\leq x$ which satisfy the inequality $d(p_{k})\leq(\log
p_{k})^{2+\epsilon}$, where $\epsilon>0$ is arbitrary and fixed, and by
$\pi(x)$ the number of primes less than or equal to $x$. In this paper, we
first prove a theorem that $\lim_{x\to\infty}N_{\epsilon}(x)/\pi(x)=1$. A
corollary to the proof of the theorem concerning gaps between consecutive
squarefree numbers is stated.
Email address: shinyah18@yahoo.co.jp
2000 MSC: 11B05.
## 1 Introduction
Let $p_{k}$ denote the $k$-th prime and for $k>1$,
$d(p_{k}):=p_{k}-p_{k-1}.$
Concerning $d(p_{k})$, Harald Cramér conjectured that there exists a positive
real number $M$ such that
$d(p_{k+1})=p_{k+1}-p_{k}\leq M(\log p_{k})^{2}$
for all $k\geq 1$.
Cramér himself showed [2] that
$p_{k+1}-p_{k}=o((\log p_{k})^{3})$
for all but at most $o\left(x/(\log x)^{4}\right)$ primes $\leq x$.
We denote the number of primes less than or equal to a positive real number
$x$ with $\pi(x)$. In this paper, we prove the following theorem, which
supports Cramér’s conjecture.
###### Theorem 1.
Let $x$ be any positive real number and $N_{\epsilon}(x)$ the number of primes
$\leq x$ which satisfy the inequality
$d(p_{k})\leq(\log p_{k})^{2+\epsilon},$
where $\epsilon>0$ is arbitrary and fixed. Then we have
$\lim_{x\to\infty}\frac{N_{\epsilon}(x)}{\pi(x)}=1.$
Note that the function for the upper bound of $d(p_{k})$ in Theorem 1 is
$(\log p_{k})^{2}$, while that in Cramér’s conjecture is $(\log p_{k-1})^{2}$;
the function is evaluated at the smaller prime of the gap in Cramér’s
conjecture. Nevertheless, the prime number theorem implies that
$\lim_{k\to\infty}\frac{p_{k}}{p_{k+1}}=1,$
so replacing the function for the upper bound in Cramér’s conjecture by $(\log
p_{k})^{2}$ does not change the statement of the conjecture essentially.
In proving Theorem 1, the following lemmas are used.
###### Lemma 1.
[1, pp. 77] For any arithmetical function $a(n)$ let
$A(x)=\sum_{n\leq x}a(n),$
where $A(x)=0$ if $x<1$. Assume that $f$ has a continuous derivative on the
interval $[y,x]$, where $0<y<x$. Then we have
$\sum_{y<n\leq x}a(n)f(n)=A(x)f(x)-A(y)f(y)-\int_{y}^{x}A(t)f^{\prime}(t)dt.$
###### Lemma 2.
[2] For all $k\geq 1$, we have
$d(p_{k+1})=O(p_{k}^{\frac{1}{2}+\frac{7}{200}}).$
## 2 Proof of Theorem 1
In this section, we give a proof of Theorem 1.
Define
$d(p_{1}):=p_{1}.$
We begin by an elementary analysis of the partial sums
$Q(t):=\sum_{p_{k}\leq t}d(p_{k})-\sum_{n\leq t}1.$
###### Lemma 3.
For all positive real numbers $t\geq 2$, if $p_{k-1}\leq t\leq p_{k}$ ($k\geq
2$) then we have
$-(p_{k}-p_{k-1})\leq Q(t)\leq 0.$
###### Proof.
It is easy to see that for $N=2,3,\ldots$, we have
$\begin{split}\sum_{p_{k}\leq
p_{N}}d(p_{k})&=d(p_{1})+d(p_{2})+d(p_{3})+\cdots+d(p_{N})\\\
&=p_{1}+p_{2}-p_{1}+p_{3}-p_{2}+\cdots+p_{N}-p_{N-1}\\\ &=p_{N}.\end{split}$
Hence, for $N\geq 2$ we have
$Q(p_{N})=0.$ (1)
If $t$ satisfies $p_{N-1}<t<p_{N}$ ($N\geq 2$), then the first partial sums of
$Q$ is constant, and the second ones decrease by $1$ as $t$ increases by $1$.
The lemma now follows from (1).
∎
Using all the preliminary lemmas above, Theorem 1 is proved as follows. Let
$\epsilon>0$ be arbitrary.
We define a function $\delta_{\epsilon}(x)$ such that
$N_{\epsilon}(x)=(1-\delta_{\epsilon}(x))\pi(x).$ (2)
It is plain that $0<\delta_{\epsilon}(x)<1$. We show that
$\lim_{x\to\infty}\delta_{\epsilon}(x)=0,$ (3)
thereby proving Theorem 1.
By the definition of $N_{\epsilon}(x)$, there are
$(1-\delta_{\epsilon}(x))\pi(x)$ primes which are $\leq x$ and satisfy the
inequality
$d(p_{k})\leq(\log p_{k})^{2+\epsilon}.$
This is equivalent to stating the following:
$\begin{split}&\text{ There are $\delta_{\epsilon}(x)\pi(x)$ primes which are
$\leq x$ and satisfy}\\\ &d(p_{k})>(\log p_{k})^{2+\epsilon}.\end{split}$ (4)
At this point, we pay our attention to the partial sums
$\sum_{p\leq x}\frac{d(p)}{p}-\sum_{n\leq x}\frac{1}{n}.$
Define
$p(n):=\begin{cases}1&:\text{$n$ is prime}\\\ 0&:\text{otherwise}.\end{cases}$
In Lemma 1, we choose
$f(t)=\frac{1}{t},\quad a(n)=d(n)p(n)-1,\quad\text{and}\quad y=1/2,$
and obtain
$\sum_{p\leq x}\frac{d(p)}{p}-\sum_{n\leq
x}\frac{1}{n}=\frac{Q(x)}{x}+\int_{1}^{x}\frac{Q(t)dt}{t^{2}}.$ (5)
By Lemmas 2 and 3, it is plain that the limit of the right side of (5) as
$x\to\infty$ exists. Since
$\sum_{n\leq x}\frac{1}{n}\sim\log x,\quad\text{as $x\to\infty$},$
it follows from (5) that
$\sum_{p\leq x}\frac{d(p)}{p}\sim\log x,\quad\text{as $x\to\infty$}.$ (6)
Now, given $\epsilon$ and $x>0$, let $S_{\epsilon}(x)$ be the set of all
primes $\leq x$ which satisfy the last inequality in (4), $|S_{\epsilon}(x)|$
the number of elements in $S_{\epsilon}(x)$, and
$\chi_{\epsilon,x}(n):=\begin{cases}1&:n\in S_{\epsilon}(x)\\\
0&:\text{otherwise}.\end{cases}$
Choosing
$f(t)=q_{\epsilon}(t):=\frac{(\log t)^{2+\epsilon}}{t},\quad
a(n)=\chi_{\epsilon,x}(n),\quad\text{and}\quad y=1/2$
in Lemma 1, we have for each $w\leq x$
$\begin{split}\sum_{p\in S_{\epsilon}(x),p\leq
w}\frac{d(p)}{p}&\geq\sum_{n\leq w}\frac{(\log
n)^{2+\epsilon}\chi_{\epsilon,x}(n)}{n}\\\ &=\frac{\xi_{\epsilon,x}(w)(\log
w)^{2+\epsilon}}{w}-\int_{2}^{w}\xi_{\epsilon,x}(t)q^{\prime}_{\epsilon}(t)dt,\end{split}$
where
$\xi_{\epsilon,x}(t):=\sum_{n\leq t}\chi_{\epsilon,x}(n)$
and $0\leq\xi_{\epsilon,x}(w)\leq|S_{\epsilon}(x)|$ for $w\leq x$. In
particular, when $w=x$, we have
$\sum_{p\in S_{\epsilon}(x),p\leq
x}\frac{d(p)}{p}\geq\frac{|S_{\epsilon}(x)|(\log
x)^{2+\epsilon}}{x}-\int_{2}^{x}\xi_{\epsilon,x}(t)q^{\prime}_{\epsilon}(t)dt.$
(7)
But since
$q^{\prime}_{\epsilon}(t)=\frac{(2+\epsilon)(\log t)^{1+\epsilon}-(\log
t)^{2+\epsilon}}{t^{2}},$
it is plain that for each arbitrary $\epsilon>0$, there exists
$t_{\epsilon}>0$ such that $q^{\prime}(\epsilon)(t)<0$ for all $t\geq
t_{\epsilon}$. This in turn implies that for each arbitrary $\epsilon>0$, with
$\xi_{\epsilon,x}(t)\geq 0\quad\text{for all $t\geq 0$},$
there exists a positive real number $M_{\epsilon}$, which depends only on
$\epsilon$, such that the integral in (7) satisfies
$-\int_{2}^{x}\xi_{\epsilon,x}(t)q^{\prime}_{\epsilon}(t)dt>-M_{\epsilon}$ (8)
for all $x\geq 2$.
With (8), the inequality (7) becomes
$\begin{split}\sum_{p\in S_{\epsilon}(x),p\leq
x}\frac{d(p)}{p}&=\frac{|S_{\epsilon}(x)|(\log
x)^{2+\epsilon}}{x}-\int_{2}^{x}\xi_{\epsilon,x}(t)q^{\prime}_{\epsilon}(t)dt\\\
&\geq\frac{|S_{\epsilon}(x)|(\log
x)^{2+\epsilon}}{x}-M_{\epsilon}.\end{split}$ (9)
Furthermore, by (4), we have $|S_{\epsilon}(x)|=\delta_{\epsilon}(x)\pi(x)$,
and so (9) gives
$\begin{split}\sum_{p\in S_{\epsilon}(x),p\leq
x}\frac{d(p)}{p}&\geq\frac{|S_{\epsilon}(x)|(\log
x)^{2+\epsilon}}{x}-M_{\epsilon}\\\ &=\frac{\delta_{\epsilon}(x)\pi(x)(\log
x)^{2+\epsilon}}{x}-M_{\epsilon}.\end{split}$ (10)
Finally, by (6) and the prime number theorem
$\pi(x)\sim\frac{x}{\log x},\quad\text{as $x\to\infty$},$
letting $x\to\infty$, the inequality (10) becomes
$\begin{split}\log x\sim\sum_{p\leq x}\frac{d(p)}{p}\geq\sum_{p\in
S_{\epsilon}(x),p\leq
x}\frac{d(p)}{p}\geq\frac{\delta_{\epsilon}(x)\pi(x)(\log
x)^{2+\epsilon}}{x}-M_{\epsilon},\end{split}$
by which (3) follows immediately. This completes the proof of Theorem 1.
## 3 Distribution of squarefree numbers
We note the following theorem, which is a corollary to the proof of Theorem 1
in the previous section.
###### Theorem 2.
Let $\\{a_{k}\\}$ be the sequence of squarefree numbers,
$s(n):=\begin{cases}1&:n=a_{k}\\\ 0&:\text{otherwise},\end{cases}$
$d(a_{k}):=a_{k}-a_{k-1},$
and
$S(x):=\sum_{n\leq x}s(n).$
Let $\epsilon>0$ be arbitrary and $N_{\epsilon}(x)$ the number of squarefree
numbers $\leq x$ which satisfy the inequality
$d(a_{k})\leq M(\log a_{k})^{1+\epsilon}$
for some positive constant $M$. Then we have
$\lim_{x\to\infty}\frac{N_{\epsilon}(x)}{S(x)}=1.$
###### Proof.
We recall that [3]
$S(x)=Ax+O(x^{\frac{1}{2}}),$ (11)
for some constant $A$. By (11), it is easy to see that
$d(a_{k})=O(a_{k}^{\frac{1}{2}}).$
If we recall how Lemma 3 was derived with Lemma 2, it is plain that
$R(t):=\sum_{a_{k}\leq t}d(a_{k})-\sum_{n\leq t}1=O(t^{\frac{1}{2}}).$
The rest of the proof follows a similar pattern. In particular, we note the
following analogues:
1. 1.
the analogue of (6) is
$\sum_{a_{k}\leq x}\frac{d(a_{k})}{a_{k}}\sim\log x,\quad\text{as
$x\to\infty$};$
2. 2.
the analogue of $Q(t)$ is $R(t)$;
3. 3.
the analogue of $q_{\epsilon}(t)$ is the function $\frac{(\log
t)^{1+\epsilon}}{t}$.
∎
From Theorem 2, one may wonder if the exact order of $d(a_{k})$ is $O(\log
a_{k})$.
## References
* [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976.
* [2] A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial J. 1 (1995), 12-28
* [3] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford, New York, 1960.
|
arxiv-papers
| 2008-09-19T20:14:30
|
2024-09-04T02:48:57.845315
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hisanobu Shinya",
"submitter": "Hisanobu Shinya",
"url": "https://arxiv.org/abs/0809.3458"
}
|
0809.3497
|
# Polarized Neutron Matter:
A Lowest Order Constrained Variational Approach
G.H. Bordbar 111Corresponding author 222E-mail : bordbar@physics.susc.ac.ir
and M. Bigdeli
Department of Physics, Shiraz University, Shiraz 71454, Iran333Permanent
address
and
Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
In this paper, we calculate some of the polarized neutron matter properties,
using the lowest order constrained variational method with the $AV_{18}$
potential and employing a microscopic point of view. A comparison is also made
between our results and those of other many-body techniques.
21.65.+f, 26.60.+c, 64.70.-p
## 1 Introduction
Pulsars are rapidly rotating neutron stars with strong surface magnetic fields
in the range of $10^{12}-10^{13}$ Gauss [1, 2, 3]. The physical origin of this
magnetic field remains an open problem and there is still no general consensus
regarding the mechanism to generate such strong magnetic fields in a neutron
star. There exist several possibilities of the generation of the magnetic
field in a neutron star, from the nuclear physics point of view, however, one
of the most interesting and stimulating mechanisms which have been suggested
is the possible existence of a phase transition to a ferromagnetic state at
densities corresponding to the theoretically stable neutron stars and,
therefore, of a ferromagnetic core in the liquid interior of such compact
objects. Such a possibility has been studied by several authors using
different theoretical approaches [4-25], but the results are still
contradictory. Whereas some calculations, like for instance the ones based on
Skyrmelike interactions predict the transition to occur at densities in the
range $(1-4)\rho_{0}$ ($\rho_{0}=0.16fm^{-3}$), others, like recent Monte
Carlo [20] and Brueckner-Hartree-Fock calculations [21-23] using modern two-
and three-body realistic interactions exclude such a transition, at least up
to densities around five times $\rho_{0}$. This transition could have
important consequences for the evolution of a protoneutron star, in particular
for the spin correlations in the medium which do strongly affect the neutrino
cross section and the neutrino mean free path inside the star [26].
In recent years, we have computed the equation of state of symmetrical and
asymmetrical nuclear matter and some of their properties such as symmetry
energy, pressure, etc. [27-30] and properties of spin polarized liquid
${}^{3}He$ [31] using the lowest order constrained variational (LOCV)
approach. The LOCV method which was developed several years ago is a useful
tool for the determination of the properties of neutron, nuclear and
asymmetric nuclear matter at zero and finite temperature [27-39]. The LOCV
method is a fully self-consistent formalism and it does not bring any free
parameters into calculation. It employs a normalization constraint to keep the
higher order term as small as possible [34, 27]. The functional minimization
procedure represents an enormous computational simplification over
unconstrained methods that attempt to go beyond lowest order.
In the present work, we compute the polarized neutron matter properties using
the LOCV method with the $AV_{18}$ potential [40] employing microscopic
calculations where we treat explicitly the spin projection in the many-body
wave functions.
## 2 Basic Theory
### 2.1 LOCV Formalism
We consider a trial many-body wave function of the form
$\displaystyle\psi=F\phi,$ (1)
where $\phi$ is the uncorrelated ground state wave function (simply the Slater
determinant of plane waves) of $N$ independent neutron and $F=F(1\cdots N)$ is
an appropriate N-body correlation operator which can be replaced by a Jastrow
form i.e.,
$\displaystyle F=S\prod_{i>j}f(ij),$ (2)
in which S is a symmetrizing operator. We consider a cluster expansion of the
energy functional up to the two-body term,
$\displaystyle
E([f])=\frac{1}{N}\frac{\langle\psi|H\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(3)
The one-body term $E_{1}$ for a polarized neutron matter can be written as
Fermi momentum functional ($k_{F}^{(i)}=(6\pi^{2}\rho^{(i)})^{\frac{1}{3}})$:
$\displaystyle
E_{1}=\sum_{i=1,2}\frac{3}{5}\frac{\hbar^{2}{k_{F}^{(i)}}^{2}}{2m}\frac{\rho^{(i)}}{\rho}\cdot$
(4)
Labels 1 and 2 are used instead of spin up and spin down neutrons,
respectively, and $\rho=\rho^{(1)}+\rho^{(2)}$ is the total neutron matter
density. The two-body energy $E_{2}$ is
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2A}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (5)
where
$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12)$,
$f(12)$ and $V(12)$ are the two-body correlation and potential. For the two-
body correlation function, $f(12)$, we consider the following form [27, 28]:
$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(12)O^{(k)}(12),$ (6)
where, the operators $O^{(k)}(12)$ are given by
$\displaystyle O^{(k=1-3)}(12)$ $\displaystyle=$ $\displaystyle 1,\
(\frac{2}{3}+\frac{1}{6}S_{12}),\ (\frac{1}{3}-\frac{1}{6}S_{12}),$ (7)
and $S_{12}$ is the tensor operator.
After doing some algebra we find the following equation for the two-body
energy:
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\left(\frac{h^{2}}{2m}\right)\sum_{JLSS_{z}}\frac{(2J+1)}{2(2S+1)}[1-(-1)^{L+S+1}]\left|\left\langle\frac{1}{2}\sigma_{z1}\frac{1}{2}\sigma_{z2}\mid
SS_{z}\right\rangle\right|^{2}\int
dr\left\\{\left[{f_{\alpha}^{(1)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(1)}}^{2}(k_{f}r)\right.\right.$
(8)
$\displaystyle\left.\left.+\frac{2m}{h^{2}}(\\{V_{c}-3V_{\sigma}+V_{\tau}-3V_{\sigma\tau}+2(V_{T}-3V_{\sigma\tau})+2V_{\tau
z}\\}{a_{\alpha}^{(1)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}-3V_{l2\sigma}+V_{l2\tau}-3V_{l2\sigma\tau}]{c_{\alpha}^{(1)}}^{2}(k_{f}r))(f_{\alpha}^{(1)})^{2}\right]+\sum_{k=2,3}\left[{f_{\alpha}^{(k)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{h^{2}}(\\{V_{c}+V_{\sigma}+V_{\tau}+V_{\sigma\tau}+(-6k+14)(V_{tz}+V_{t})-(k-1)(V_{ls\tau}+V_{ls})\right.\right.$
$\displaystyle\left.\left.+[V_{T}+V_{\sigma\tau}+(-6k+14)V_{tT}][2+2V_{\tau
z}]\\}{a_{\alpha}^{(k)}}^{2}(k_{f}r)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}+V_{l2\sigma}+V_{l2\tau}+V_{l2\sigma\tau}]{c_{\alpha}^{(k)}}^{2}(k_{f}r)+[V_{ls2}+V_{ls2\tau}]{d_{\alpha}^{(k)}}^{2}(k_{f}r)){f_{\alpha}^{(k)}}^{2}\right]\right.$
$\displaystyle\left.+\frac{2m}{h^{2}}\\{V_{ls}+V_{ls\tau}-2(V_{l2}+V_{l2\sigma}+V_{l2\sigma\tau}+V_{l2\tau})-3(V_{ls2}+V_{ls2\tau})\\}b_{\alpha}^{2}(k_{f}r)f_{\alpha}^{(2)}f_{\alpha}^{(3)}\right.$
$\displaystyle\left.+\frac{1}{r^{2}}(f_{\alpha}^{(2)}-f_{\alpha}^{(3)})^{2}b_{\alpha}^{2}(k_{f}r)\right\\},$
where $\alpha=\\{J,L,S,S_{z}\\}$ and the coefficient ${a_{\alpha}^{(1)}}^{2}$,
etc., are defined as
$\displaystyle{a_{\alpha}^{(1)}}^{2}(x)=x^{2}I_{L,S_{z}}(x),$ (9)
$\displaystyle{a_{\alpha}^{(2)}}^{2}(x)=x^{2}[\beta I_{J-1,S_{z}}(x)+\gamma
I_{J+1,S_{z}}(x)],$ (10) $\displaystyle{a_{\alpha}^{(3)}}^{2}(x)=x^{2}[\gamma
I_{J-1,S_{z}}(x)+\beta I_{J+1,S_{z}}(x)],$ (11) $\displaystyle
b_{\alpha}^{(2)}(x)=x^{2}[\beta_{23}I_{J-1,S_{z}}(x)-\beta_{23}I_{J+1,S_{z}}(x)],$
(12) $\displaystyle{c_{\alpha}^{(1)}}^{2}(x)=x^{2}\nu_{1}I_{L,S_{z}}(x),$ (13)
$\displaystyle{c_{\alpha}^{(2)}}^{2}(x)=x^{2}[\eta_{2}I_{J-1,S_{z}}(x)+\nu_{2}I_{J+1,S_{z}}(x)],$
(14)
$\displaystyle{c_{\alpha}^{(3)}}^{2}(x)=x^{2}[\eta_{3}I_{J-1,S_{z}}(x)+\nu_{3}I_{J+1,S_{z}}(x)],$
(15)
$\displaystyle{d_{\alpha}^{(2)}}^{2}(x)=x^{2}[\xi_{2}I_{J-1,S_{z}}(x)+\lambda_{2}I_{J+1,S_{z}}(x)],$
(16)
$\displaystyle{d_{\alpha}^{(3)}}^{2}(x)=x^{2}[\xi_{3}I_{J-1,S_{z}}(x)+\lambda_{3}I_{J+1,S_{z}}(x)],$
(17)
with
$\displaystyle\beta=\frac{J+1}{2J+1},\ \gamma=\frac{J}{2J+1},\
\beta_{23}=\frac{2J(J+1)}{2J+1},$ (18) $\displaystyle\nu_{1}=L(L+1),\
\nu_{2}=\frac{J^{2}(J+1)}{2J+1},\ \nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1},$
(19) $\displaystyle\eta_{2}=\frac{J(J^{2}+2J+1)}{2J+1},\
\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1},$ (20)
$\displaystyle\xi_{2}=\frac{J^{3}+2J^{2}+2J+1}{2J+1},\
\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1},$ (21)
$\displaystyle\lambda_{2}=\frac{J(J^{2}+J+1)}{2J+1},\
\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1},$ (22)
and
$\displaystyle I_{J,S_{z}}(x)=\int dqP_{S_{z}}(q)J_{J}^{2}(xq)\cdot$ (23)
In the above equation $J_{J}(x)$ is the Bessel’s function and, $P_{S_{z}}(q)$
is defined as follows,
$\displaystyle P_{S_{z}}(q)$ $\displaystyle=$
$\displaystyle\frac{2}{3}\pi[(k_{F}^{\sigma_{z1}})^{3}+(k_{F}^{\sigma_{z2}})^{3}-\frac{3}{2}((k_{F}^{\sigma_{z1}})^{2}+(k_{F}^{\sigma_{z2}})^{2})q$
(24)
$\displaystyle-\frac{3}{16}((k_{F}^{\sigma_{z1}})^{2}-(k_{F}^{\sigma_{z2}})^{2})^{2}q^{-1}+q^{3}],$
for
$\frac{1}{2}\left|k_{F}^{\sigma_{z1}}-k_{F}^{\sigma_{z2}}\right|<q<\frac{1}{2}\left|k_{F}^{\sigma_{z1}}+k_{F}^{\sigma_{z2}}\right|$,
$\displaystyle P_{S_{z}}(q)=\frac{4}{3}\pi
min(k_{F}^{\sigma_{z1}},k_{F}^{\sigma_{z2}}),$ (25)
for $q<\frac{1}{2}\left|k_{F}^{\sigma_{z1}}-k_{F}^{\sigma_{z2}}\right|$ and
$\displaystyle P_{S_{z}}(q)=0,$ (26)
for $q>\frac{1}{2}\left|k_{F}^{\sigma_{z1}}+k_{F}^{\sigma_{z2}}\right|$, where
$\sigma_{z1}$ or $\sigma_{z2}=\frac{1}{2},-\frac{1}{2}$ for spin up and spin
down, respectively.
Now, we can minimize the two-body energy Eq.(8), with respect to the
variations in the function ${f_{\alpha}}^{(i)}$ but subject to the
normalization constraint [28],
$\displaystyle\frac{1}{A}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0,$ (27)
where in the case of spin polarized neutron matter the function $h_{S_{z}}(r)$
is defined as
$\displaystyle h_{S_{z}}(r)$ $\displaystyle=$
$\displaystyle\left[1-9\left(\frac{J_{J}^{2}(k_{F}^{i})}{k_{F}^{i}}\right)^{2}\right]^{-1/2};\
S_{z}=\pm 1$ (28) $\displaystyle=$ $\displaystyle 1\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ S_{z}=0$
From the minimization of the two-body cluster energy, we get a set of coupled
and uncoupled differential equations which are the same as presented in
Ref.[28].
### 2.2 Magnetic Susceptibility
The magnetic susceptibility, $\chi$, which characterizes the response of a
system to the magnetic field, is defined by
$\displaystyle\chi=\left(\frac{\partial M}{\partial H}\right)_{H=0},$ (29)
where $M$ is the magnetization of system per unit volume and $H$ is the
magnetic field. By some simplification, the magnetic susceptibility can be
written as
$\displaystyle\chi=\frac{\mu^{2}\rho}{\left(\frac{\partial^{2}E}{\partial\delta^{2}}\right)_{\delta=0}},$
(30)
where $\mu$ is the magnetic moment of the neutron and $\delta$ is the spin
polarization parameter which is defined as
$\displaystyle\delta=\frac{\rho^{(1)}-\rho^{(2)}}{\rho}\cdot$ (31)
Usually, one is interested in calculating the ratio of $\chi$ to the magnetic
susceptibility for a degenerate free Fermi gas ($\chi_{F}$). $\chi_{F}$ can be
straightforwardly obtained from Eq. (30), using the total energy per particle
of free Fermi gas,
$\displaystyle\chi_{F}=\frac{\mu^{2}m}{\hbar^{2}\pi^{2}}k_{F},$ (32)
where $k_{F}=(3\pi^{2}\rho)^{1/3}$ is Fermi momentum. After a little algebra
one finds
$\displaystyle\frac{\chi}{\chi_{F}}=\frac{2}{3}\frac{E_{F}}{\left(\frac{\partial^{2}E}{\partial\delta^{2}}\right)_{\delta=0}},$
(33)
where $E_{F}={\hbar^{2}k_{F}^{2}}/{2m}$ is the Fermi energy.
## 3 Results
In Fig. 1, we have shown the energy per particle for various values of spin
polarization of the neutron matter as a function of density. As can be seen
from this figure, the energy of neutron matter becomes repulsive by increasing
the polarization for all relevant densities. According to this result, the
spontaneous phase transition to a ferromagnetic state in the neutron matter
does not occur. If such a transition existed a crossing of the energies of
different polarizations would be observed at some density, indicating that the
ground state of the system would be ferromagnetic from that density on. As is
shown in Fig. 1, there is no crossing point. On the contrary, it becomes less
favorable as the density increases. For the energy of neutron matter, we have
also made a comparison between our results and the results of other many-body
methods with the $AV_{18}$ potential as shown in Fig. 2. The BGLS calculations
are based on the Brueckner-Hartree-Fock approximation both for continuous
choice (BHFC) and standard choice (BHFG) [41]. The APR results have been
obtained using the variational chain summation (VCS) method [42] and the EHMMP
calculations have been carried out using the lowest order Brueckner (LOB)
technique [43]. We see that our results are in agreement with those of others,
specially with the APR and EHMMP calculations.
For the neutron matter, we have also considered the dependence of energy to
the spin polarization $\delta$. Let us examine this dependency in quadratic
spin polarization form for different densities as shown in Fig. 3. As can be
seen from this figure, the energy per particle increases as the polarization
increases and the minimum value of energy occurs at $\delta=0$ for all
densities. This indicates that the ground state of neutron matter is
paramagnetic. In Fig. 3, the results of ZLS calculations using the Brueckner-
Hartree-Fock theory with the $AV_{18}$ potential [25] are also given for
comparison. There is an agreement between our results and those of ZLS,
specially at low densities. From Fig. 3, it is also seen that the variation of
the energy of neutron matter versus $\delta^{2}$ is nearly linear. Therefore,
one can characterize this dependency in the following analytical form
$\displaystyle E(\rho,\delta)$ $\displaystyle=$ $\displaystyle
E(\rho,0)+a(\rho)\delta^{2}\cdot$ (34)
The density dependent parameter $a(\rho)$ can be interpreted as the measure of
the energy required to produce a net spin alignment in the direction of the
magnetic field, and its value can be determined as the slope of each line in
Fig. 3, for the corresponding density,
$\displaystyle a(\rho)=\frac{\partial
E(\rho,\delta)}{\partial\delta^{2}}\cdot$ (35)
In Fig. 4, the parameter $a(\rho)$ is shown as a function of the density and
as can be seen the value of this parameter increase by increasing the density.
In turn this indicate the energy which require to align spin at same direction
increases. An conclusion can be inferred again from this result is that a
phase transition to a ferromagnetic state is not to be expected from our
calculation. The parameter $a(\rho)$ obtained by ZLS [25] is also shown in
Fig. 4, for comparison.
In Fig. 5, we have plotted the ratio ${{\chi}/{\chi_{F}}}$ versus density. As
can be seen from Fig. 5, this ratio changes continuously for all densities.
Therefore, the ferromagnetic phase transition does not occur. For comparison,
we have also shown the results of ZLS [25] in this figure.
The equation of state of polarized neutron matter, $P(\rho,\delta)$, can be
simply obtained using
$\displaystyle P(\rho,\delta)=\rho^{2}\frac{\partial
E(\rho,\delta)}{\partial\rho}$ (36)
In Fig. 6, we have presented the pressure of neutron matter as a function of
density $\rho$ at different polarizations. We see that the equation of state
becomes stiffer by increasing the polarization. We also see that with
increasing density, the difference between the equations of state at different
polarization becomes more appreciable. In order to check the causality
condition for our equations of state, we have calculated the velocity of
sound, $v_{s}$, as shown in Fig. 7. It is seen that the velocity of sound
increases with both increasing polarization and density, but it is always less
than the velocity of light in the vacuum $(c)$. Therefore, all calculated
equations of state obey the causality condition.
As it is known, the Landau parameter, $G_{0}$, describes the spin density
fluctuation in the effective interaction. $G_{0}$ is simply related to the
magnetic susceptibility by the relation
$\displaystyle\frac{\chi}{\chi_{F}}=\frac{m^{*}}{1+G_{0}}$ (37)
where $m^{*}$ is the effective mass. A magnetic instability would require
$G_{0}<-1$. Our results for the Landau parameter have been presented in the
Fig. 8. It is seen that the value of $G_{0}$ is always positive and
monotonically increasing up to highest density and does not show any magnetic
instability for the neutron matter. In Fig. 8, the results of ZLS calculations
[25] are also given for comparison.
## 4 Summary and Conclusions
The properties of neutron matter is of primary importance in the study of
neutron star, and in particular, strongly magnetized ones (i.e. pulsars). It
is therefore important to calculate the properties of polarized neutron
matter, using an efficient and sufficiently accurate method. We have recently
computed various properties of the neutron matter using the lowest order
constrained variational (LOCV) scheme. In order to make our results more
general, we used this method for the polarized neutron matter. Energy per
particle for various values of spin polarization of the neutron matter was
computed as function of density, and shown to become repulsive as a result of
increasing the polarization. In addition, we considered the dependence of
energy of neutron matter to the spin polarization, and found it to increase
with the spin polarization for all densities. This dependence was represented
by a quadratic formula with the coefficient of the quadratic term $a(\rho)$
determined as a function of the density. This parameter, too, was shown to
increase monotonically with density. Magnetic susceptibility, which
characterizes the response of the system to the magnetic field was calculated
for the system under consideration. We have also computed the equation of
state of neutron matter at different polarizations. Our results for higher
values of polarization show a stiff equation of state. The velocity of sound
was computed to check the causality condition of equation of state and it was
shown that it is always lower than the velocity of light in vacuum. We have
also investigated the Landau parameter $G_{0}$ which shows that the value of
$G_{0}$ is always positive and monotonically increasing up to high densities.
Finally, our results showed no phase transition to ferromagnetic state. We
have also compared the results of our calculations for the properties of
neutron matter with the other calculations.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, and Shiraz University Research Council.
## References
* [1] S. Shapiro and S. Teukolsky, Blak Holes, White Dwarfs and Neutron Stars, (Wiley-New york,1983).
* [2] F. Pacini, _Nature_ (London) 216 (1967) 567.
* [3] T. Gold, _Nature_ (London) 218 (1968) 731.
* [4] D. H. Brownell and J. Callaway, _Nuovo Cimento_ B 60 (1969) 169.
* [5] M. J. Rice, _Phys. Lett_. A 29 (1969) 637.
* [6] J. W. Clark and N. C. Chao, _Lettere Nuovo Cimento_ 2 (1969) 185.
* [7] J. W. Clark, _Phys. Rev. Lett_. 23 (1969) 1463.
* [8] S. D. Silverstein, _Phys. Rev_. Lett. 23 (1969) 139.
* [9] E. østgaard, _Nucl. Phys_. A 154 (1970) 202.
* [10] J. M. Pearson and G. Saunier, _Phys. Rev. Lett_. 24 (1970) 325.
* [11] V. R. Pandharipande, V. K. Garde and J. K. Srivastava, _Phys. Lett_. B 38 (1972) 485.
* [12] S. O. Backman and C. G. Kallman, _Phys. Lett_. B 43 (1973) 263.
* [13] P. Haensel, _Phys. Rev_. C 11 (1975) 1822.
* [14] A. D. Jackson, E. Krotscheck, D. E. Meltzer and R. A. Smith, _Nucl. Phys_. A 386 (1982)125.
* [15] M. Kutschera and W. W ojcik, _Phys. Lett_. B 223 (1989) 11.
* [16] S. Marcos, R. Niembro, M. L. Quelle and J. Navarro, _Phys. Lett_. B 271 (1991) 277.
* [17] P. Bernardos, S. Marcos, R. Niembro, M. L. Quelle, _Phys. Lett_ , B 356 (1995) 175.
* [18] A. Vidaurre, J. Navarro and J. Bernabeu, _Astron. Astrophys_. 135 (1984) 361.
* [19] M. Kutschera and W. W ojcik, _Phys. Lett_. B 325 (1994) 271.
* [20] S. Fantoni, A. Sarsa and K. E. Schmidt, _Phys. Rev. Lett_. 87 (2001) 181101.
* [21] I. Vida na, A. Polls and A. Ramos, _Phys. Rev_. C 65 (2002) 035804.
* [22] I. Vida na and I. Bombaci, _Phys. Rev_. C 66 (2002) 045801.
* [23] W. Zuo, U. Lombardo and C.W. Shen, in Quark-Gluon Plasma and Heavy Ion Collisions, Ed. W.M. Alberico, M. Nardi and M.P. Lombardo, World Scientific, p. 192 (2002).
* [24] A. A. Isayev and J. Yang, _Phys. Rev_. C 69 (2004) 025801.
* [25] W. Zuo, U. Lombardo and C. W. Shen, nucl-th/0204056.
W. Zuo, C. W. Shen and U. Lombardo, Phys. Rev C 67 (2003) 037301\.
* [26] J. Navarro, E. S. Hern andez and D. Vautherin, _Phys. Rev_. C 60 (1999) 045801.
* [27] G. H. Bordbar, M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 23 (1997) 1631.
* [28] G. H. Bordbar and M. Modarres, _Phys. Rev._ C 57 (1998) 714.
* [29] M. Modarres and G. H. Bordbar, _Phys. Rev._ C 58 (1998) 2781.
* [30] G. H. Bordbar, _Int. J. M. Phys._ A 18 (2003)3629.
* [31] G. H. Bordbar and S. M. Zebarjad, M. R. Vahdani, M. Bigdeli, _Int. J. M. Phys._ C 23 (2005) 3379
* [32] M. Modarres and J. M. Irvine, _J. Phys. G: Nucl. Part. Phys_. 5 (1979) 511; 5 (1979) 7.
* [33] C. Howes, R. F. Bishop, and J. M. Irvine, _J. Phys. G: Nucl. Part. Phys_. 4 (1978) 89; 4 (1979) 11.
* [34] J. C. Owen, R. F. Bishop, and J. M. Irvine, _Nucl. Phys._ A 277 (1977) 45.
* [35] C. Howes, R. F. Bishop, and J. M. Irvine, _J. Phys. G: Nucl. Part. Phys_. 4 (1978) 123.
* [36] R. F. Bishop, C. Howes, J. M. Irvine, and M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 4 (1978) 1709.
* [37] M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 19 (1993) 1349.
* [38] M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 21 (1995) 351.
* [39] M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 23 (1997) 923.
* [40] R. B. Wiringa, V. Stoks and R. Schiavilla, _Phys. Rev._ C 51 (1995) 38.
* [41] M. Baldo, G. Giansiracusa, U. Lombardo and H. Q. Song, Phys. Lett. B 473 (2000) 1.
* [42] A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58 (1998) 1804.
* [43] L. Engvik et al., Nucl. Phys. A 627 (1997) 85\.
Figure 1: The energy per particle versus density ($\rho$) for different values
of the spin polarization ($\delta$) of the neutron matter.
Figure 2: Comparison between our results for the energy per particle of
neutron matter and those of BGLS [41], APR [42] and EHMMP [43] calculations
with the $AV_{18}$ potential.
Figure 3: Our results (full curves) for the energy difference of polarized and
unpolarized cases versus quadratic spin polarization ($\delta$) for different
values of the density($\rho$) of the neutron matter. The results of ZLS [25]
(dashed curves) are also presented for comparison.
Figure 4: Our results (full curve) for the parameter $a(\rho)$ as a function
of the density($\rho$). The results of ZLS [25] (dashed curve) are also given
for comparison.
Figure 5: As Fig. 4, but for the magnetic susceptibility
(${{\chi}/{\chi_{F}}}$).
Figure 6: The equation of state of neutron matter for different values of the
spin polarization ($\delta$).
Figure 7: The velocity of sound in the unit of $c$ versus density ($\rho$) for
different values of the spin polarization ($\delta$).
Figure 8: As Fig. 4, but for the Landau parameter, $G_{0}$.
|
arxiv-papers
| 2008-09-21T06:25:59
|
2024-09-04T02:48:57.849368
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar and M. Bigdeli",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0809.3497"
}
|
0809.3498
|
# Spin Polarized Asymmetric Nuclear Matter and Neutron Star Matter Within the
Lowest Order Constrained Variational Method
G.H. Bordbara,b 111Corresponding author 222E-mail : bordbar@physics.susc.ac.ir
and M. Bigdelia,c aDepartment of Physics, Shiraz University, Shiraz 71454,
Iran333Permanent address,
bResearch Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
and
cDepartment of Physics, Zanjan University, Zanjan, Iran
###### Abstract
In this paper, we calculate properties of the spin polarized asymmetrical
nuclear matter and neutron star matter, using the lowest order constrained
variational (LOCV) method with the $AV_{18}$, $Reid93$, $UV_{14}$ and
$AV_{14}$ potentials. According to our results, the spontaneous phase
transition to a ferromagnetic state in the asymmetrical nuclear matter as well
as neutron star matter do not occur.
###### pacs:
21.65.-f, 26.60.-c, 64.70.-p
## I Introduction
The magnetic properties of polarized neutron matter, nuclear matter and
neutron star matter have crucial role for studying the possible onset of
ferromagnetic transition in the neutron star core. The most interesting and
stimulating mechanisms that have been suggested for magnetic source of
pulsars, is the possible existence of a phase transition to a ferromagnetic
state at densities corresponding to the theoretically stable neutron stars
and, therefore, of a ferromagnetic core in the liquid interior of such compact
objects. Pulsars are rapidly rotating neutron stars with strong surface
magnetic fields in the range of $10^{12}-10^{13}$ Gauss shap ; paci ; gold .
Such a possibility has been studied by several authors using different
theoretical approaches [4-29], but the results are still contradictory. In
most calculations, neutron star matter is approximated by pure neutron matter,
as proposed just after the discovery of pulsars. Vidana et al. vida and
Vidana and Bombaci vidab have considered properties of spin polarized neutron
matter and polarized asymmetrical nuclear matter using the Brueckner-Hartree-
Fock (BHF) approximation by employing three realistic nucleon- nucleon
interactions, Nijmegen II, Reid93 and NSC97e respectively. Zuo et al. zls
have also obtained properties of spin polarized neutron and symmetric nuclear
matter using same method with $AV_{18}$ potential. The results of those
calculations show no indication of ferromagnetic transition at any density for
neutron and asymmetrical nuclear matter. Fantoni et al. fanto have calculated
spin susceptibility of neutron matter using the Auxiliary Field Diffusion
Monte Carlo (AFDMC) method employing the AU6 + UIX three-body potential, and
have found that the magnetic susceptibility of neutron matter shows a strong
reduction of about a factor 3 with respect to its Fermi gas value. Baldo et
al. bgls , Akmal et al. apr and Engvik et al. ehmmp have considered
properties of neutron matter with $AV_{18}$ potential using BHF approximation
both for continuous choice (BHFC) and standard choice (BHFG), variational
chain summation (VCS) method and lowest order Brueckner (LOB) respectively. On
the other hand some calculations, like those of Brownell and Callaway brown
and Rice rice considered a hard sphere gas model and showed that neutron
matter becomes ferromagnetic at $k_{F}\approx 2.3fm^{-1}$. Silverstein silv
and Østgaard ostga found that the inclusion of long range attraction
significantly increased the ferromagnetic transition density (e.g., Østgaard
predicted the transition to occur at $k_{F}\approx 4.1fm^{-1}$ using a simple
central potential with hard core only for the singlet spin states). Clark
clark and Pearson and Saunier pear calculated the magnetic susceptibility
for low densities $(k_{F}\leq 2fm^{-1})$ using more realistic interactions.
Pandharipande et al. pandh , using the Reid soft-core potential, performed a
variational calculation arriving to the conclusion that such a transition was
not to be expected for $k_{F}\leq 5fm^{-1}$. Early calculations of the
magnetic susceptibility within the Brueckner theory were performed by Bäckmann
and Källman backm employing the Reid soft-core potential, and results from a
correlated basis function calculation were obtained by Jackson et al. jack
with the Reid v6 interaction. A different point of view was followed by
Vidaurre et al. vida , who employed neutron-neutron effective interactions of
the Skyrme type, finding the ferromagnetic transition at $k_{F}\approx
1.73-1.97fm^{-1}$. Marcos et al. marcos have also studied the spin stability
of dense neutron matter within the relativistic Dirac-Hartree-Fock
approximation with an effective nucleon-meson Lagrangian, predicting the
ferromagnetic transition at several times nuclear matter saturation density.
This transition could have important consequences for the evolution of a
protoneutron star, in particular for the spin correlations in the medium which
do strongly affect the neutrino cross section and the neutrino mean free path
inside the star navarro .
In our pervious works, the properties of unpolarized asymmetrical nuclear
matter have been considered by us using the lowest order constrained
variational (LOCV) method bordb with the $AV_{18}$ potential wiring . We have
found that the energy per particle of nuclear matter alter monotonically and
linearly by quadratic asymmetrical parameter.
Recently, we have computed the properties of polarized neutron matterbordbig
and polarized symmetrical nuclear matterbordbig2 such as total energy,
magnetic susceptibility, pressure, etc using the microscopic calculations
employing LOCV method with the $AV_{18}$ potential. We have also concluded
that the spontaneous phase transition to a ferromagnetic state in the neutron
and asymmetrical nuclear matter does not occur. In this work, we intend to
calculate the properties of spin polarized asymmetrical nuclear matter and
neutron star matter using the LOCV method employing the $AV_{18}$ wiring ,
$Reid93$ R93 , $UV_{14}$ UV14 and $AV_{14}$ AV14 potentials.
## II LOCV Method
The LOCV method which was developed several years ago is a useful tool for the
determination of the properties of neutron, nuclear and asymmetric nuclear
matter at zero and finite temperature. The LOCV method is a fully self-
consistent formalism and it does not bring any free parameters into
calculation. It employs a normalization constraint to keep the higher order
term as small as possible. The functional minimization procedure represents an
enormous computational simplification over unconstrained methods that attempt
to go beyond lowest order bordb ; borda ; owen .
In this method we consider a trial many-body wave function of the form
$\displaystyle\psi=F\phi,$ (1)
where $\phi$ is the uncorrelated ground state wave function (simply the Slater
determinant of plane waves) of $A$ independent nucleon and $F=F(1\cdots A)$ is
an appropriate A-body correlation operator which can be replaced by a Jastrow
form i.e.,
$\displaystyle F=S\prod_{i>j}f(ij),$ (2)
in which S is a symmetrizing operator. We consider a cluster expansion of the
energy functional up to the two-body term,
$\displaystyle
E([f])=\frac{1}{A}\frac{\langle\psi|H\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(3)
The one-body term $E_{1}$ is total kinetic energy of the system. The two-body
energy $E_{2}$ is
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2A}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (4)
where
$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12)$,
$f(12)$ and $V(12)$ are the two-body correlation and potential. For the two-
body correlation function, $f(12)$, we consider the following form borda ;
bordb :
$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(12)O^{(k)}(12),$ (5)
where, the operators $O^{(k)}(12)$ are given by
$\displaystyle O^{(k=1-3)}(12)$ $\displaystyle=$ $\displaystyle 1,\
(\frac{2}{3}+\frac{1}{6}S_{12}),\ (\frac{1}{3}-\frac{1}{6}S_{12}),$ (6)
and $S_{12}$ is the tensor operator.
$\displaystyle S_{12}$ $\displaystyle=$ $\displaystyle
3(\bf{\sigma_{1}}.\hat{r})(\bf{\sigma_{2}}.\hat{r})-\bf{\sigma_{1}}.\bf{\sigma_{2}}$
(7)
Now, we can minimize the two-body energy Eq.(4), with respect to the
variations in the function ${f_{\alpha}}^{(i)}$ but subject to the
normalization constraint bordb ,
$\displaystyle\frac{1}{A}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0,$ (8)
where in the case of spin polarized neutron matter the function $h_{S_{z}}(r)$
is defined as
$\displaystyle h_{S_{z}}(r)$ $\displaystyle=$
$\displaystyle\left[1-\frac{9}{\nu}\left(\frac{J_{J}^{2}(k_{F}^{i})}{k_{F}^{i}}\right)^{2}\right]^{-1/2};\
S_{z}=\pm 1$ (9) $\displaystyle=$ $\displaystyle 1\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ;\ S_{z}=0$
here $\nu$ is the degeneracy of the system. From the minimization of the two-
body cluster energy, we get a set of coupled and uncoupled differential
equations the same as presented in Ref. bordb . We can get correlation
functions by solving the differential equations and in turn these functions
lead to the two-body energy.
## III Spin Polarized Asymmetrical Nuclear Matter
Spin polarized asymmetrical nuclear matter is an infinite system that is
composed of spin up and spin down neutrons with densities $\rho_{n}^{(1)}$ and
$\rho_{n}^{(2)}$ respectively, and spin up and spin down protons with
densities $\rho_{p}^{(1)}$ and $\rho_{p}^{(2)}$ . The total densities for
neutrons$(\rho_{n})$, protons $(\rho_{p})$, and nucleons $(\rho)$ are given
by:
$\displaystyle\rho_{p}=\rho_{p}^{(1)}+\rho_{p}^{(2)},\ \ \ \
\rho_{n}=\rho_{n}^{(1)}+\rho_{n}^{(2)},\ \ \ \ \rho$ $\displaystyle=$
$\displaystyle\rho_{p}+\rho_{n}$ (10)
Labels 1 and 2 are used instead of spin up and spin down nucleons,
respectively. One can use the following parameter to identify a given spin
polarized state of the system,
$\displaystyle\delta_{p}=\frac{\rho_{p}^{1}-\rho_{p}^{2}}{\rho},\ \ \
\delta_{n}=\frac{\rho_{n}^{1}-\rho_{n}^{2}}{\rho}$ (11)
$\delta_{p}$ and $\delta_{n}$ are proton and neutron spin asymmetry
parameters, respectively. Total polarization defined as,
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle\delta_{n}+\delta_{p}$
(12)
For the unpolarized case, we have $\delta_{n}=\delta_{p}=0$.
Asymmetry parameter describes isospin asymmetry of the system and is defined
as,
$\displaystyle\beta=\frac{\rho_{n}-\rho_{p}}{\rho}.$ (13)
Pure neutron matter is totally asymmetrical nuclear matter with $\beta=1$ and
symmetrical nuclear matter has $\beta=0$. In the case of spin polarized
asymmetrical nuclear matter, the energy per particle given by
$\displaystyle E(\rho,\delta,\beta)$ $\displaystyle=$ $\displaystyle
E_{1}(\rho,\delta,\beta)+E_{2}(\rho,\delta,\beta)\cdot$ (14)
where $E_{1}$, the kinetic energy contribution given by,
$\displaystyle E_{1}(\rho,\delta,\beta)$ $\displaystyle=$
$\displaystyle\frac{3}{20}\frac{\hbar^{2}k_{F}^{2}}{2m}\left\\{(1+\beta+2\delta_{n})^{\frac{5}{3}}+(1+\beta-2\delta_{n})^{\frac{5}{3}}\right.$
(15) $\displaystyle\left.\ \ \ \ \ \ \ \ \
+(1-\beta+2\delta_{p})^{\frac{5}{3}}+(1-\beta-2\delta_{p})^{\frac{5}{3}}\right\\},$
where $k_{F}=(3/2\pi^{2}\rho)^{1/3}$ is Fermi momentum of unpolarized
symmetrical nuclear matter. In expression (14), the two body energy can be
calculated by the semi-empirical mass formula bordb ,
$\displaystyle E_{2}(\rho,\delta,\beta)$ $\displaystyle=$
$\displaystyle\beta^{2}{E_{2}}_{neum}+(1-\beta^{2}){E_{2}}_{snucm},$ (16)
where the powers higher than quadratic are neglected. ${E_{2}}_{neum}$ is
polarized pure neutron matter potential energy, and ${E_{2}}_{snucm}$ is
polarized symmetrical nuclear matter potential energy that both of them have
been determined using the LOCV method employing a microscopic point of view
bordbig ; bordbig2 .
The energy per particle of the polarized asymmetrical nuclear matter versus
density at different values of the spin polarization and isospin asymmetry
parameter have been shown in Figs. 1-4 for the $AV_{18}$, $Reid93$, $UV_{14}$
and $AV_{14}$ potentials, respectively. As it can be seen from these figures,
the energy of polarized asymmetrical nuclear matter for various values of
isospin asymmetry parameters become more repulsive by increasing the
polarization for all relevant densities. There is no crossing of the energy
curves of different polarizations. Reversibly, by increasing density, the
difference between the energy of nuclear matter at different polarization
becomes more sizable. Indeed, the ground state of the system is that of
unpolarized matter in all ranges of density and isospin asymmetry considered.
This behavior is common for all potentials used in our calculations.
Another quantities that help us to understand ferromagnetic phase transition,
are magnetic susceptibility and Landau parameter. In what follows, we want to
consider these two quantities. The magnetic susceptibility, $\chi$, which
characterizes the response of a system to the magnetic field and gives a
measure of the energy required to produce a net spin alignment in the
direction of the magnetic field, is defined by
$\displaystyle\chi=\left(\frac{\partial M}{\partial H}\right)_{H=0},$ (17)
where $M$ is the magnetization of the system per unit volume and $H$ is the
magnetic field. We have calculated the magnetic susceptibility of the
polarized asymmetrical nuclear matter in the ratio ${{\chi}/{\chi_{F}}}$ form.
By using the Eq. (17) and some simplification, the ratio of $\chi$ to the
magnetic susceptibility for a degenerate free Fermi gas $\chi_{F}$ can be
written as
$\displaystyle\frac{\chi}{\chi_{F}}=\frac{2}{3}\frac{E_{F}}{\left(\frac{\partial^{2}E}{\partial\delta^{2}}\right)_{\delta=0}}\
,$ (18)
where $E_{F}={\hbar^{2}k_{F}^{2}}/{2m}$ is the Fermi energy. After doing some
algebra the ratio ${\chi}/{\chi_{F}}$ takes the form,
$\displaystyle\frac{\chi}{\chi_{F}}=\left[1+\frac{2\left(\beta^{2}\left\\{2^{\frac{2}{3}}({\chi_{F}}/{\chi})_{neum}-({\chi_{F}}/{\chi})_{snucm}\right\\}+({\chi_{F}}/{\chi})_{snucm}+\beta^{2}(1-2^{\frac{2}{3}})-1\right)}{(1+\beta)^{5/3}+(1-\beta)^{5/3}}\right]^{-1},$
(19)
where $({\chi}/{\chi_{F}})_{neum}$ and $({\chi}/{\chi_{F}})_{snucm}$ is
magnetic susceptibility of pure neutron matter bordbig , and asymmetry nuclear
matter bordbig2 , respectively. In Fig. 5, we have plotted the ratio
${\chi}/{\chi_{F}}$ as a function of density at several values of the isospin
asymmetry for the $AV_{18}$, $Reid93$, $UV_{14}$ and $AV_{14}$ potentials. For
all used potentials, we see that the value of the ratio ${\chi}/{\chi_{F}}$
decreases monotonically with the density, even at high densities. It shows,
there is no magnetic instability for any values of asymmetry parameter. If
such an instability existed, the value of ratio ${\chi}/{\chi_{F}}$ changes
suddenly unlike previous treatment.
As it is known, the Landau parameter, $G_{0}$, describes the spin density
fluctuation in the effective interaction. $G_{0}$ is simply related to the
magnetic susceptibility by the relation
$\displaystyle\frac{\chi}{\chi_{F}}=\frac{m^{*}}{1+G_{0}},$ (20)
where $m^{*}$ is the effective mass. A magnetic instability would require
$G_{0}<-1$. Our results for the Landau parameter have been presented in the
Fig. 6 for the $AV_{18}$, $Reid93$, $UV_{14}$ and $AV_{14}$ potentials. It is
seen that the value of $G_{0}$ is always positive and monotonically increases
by increasing the density. This shows that for all used potentials, the
spontaneous phase transition to a ferromagnetic state in the asymmetrical
nuclear matter does not occur.
The equation of state of polarized asymmetrical nuclear matter,
$P(\rho,\beta,\delta)$, can be obtained using
$\displaystyle P(\rho,\beta,\delta)=\rho^{2}\frac{\partial
E(\rho,\beta,\delta)}{\partial\rho}.$ (21)
In Fig. 7, we have shown the pressure of asymmetrical nuclear matter as a
function of density ($\rho$) at different polarizations for various choice of
asymmetry parameter ($\beta$) with the $AV_{18}$ potential. This figure shows
that the equation of state becomes stiffer by increasing the polarization for
all isospin asymmetry.
## IV Polarized Neutron Star Matter
Indeed the neutron star matter is charge neutral infinite system that is
mixture of asymmetrical nuclear matter and leptons, specially electrons and
muons. The energy per particle of polarized neutron star matter, $E_{nsm}$ can
be written as,
$\displaystyle E_{nsm}$ $\displaystyle=$ $\displaystyle E+E_{l}\ .$ (22)
where $E$ is the nucleonic energy contribution which given by Eq. (14) and
$E_{l}$ is leptonic energy contribution obtained as follows,
$\displaystyle E_{l}$ $\displaystyle=$ $\displaystyle\sum_{i=e,\ \mu}\
\sum_{k\leq k^{(F)}}[(m_{i}c^{2})^{2}+\hbar^{2}c^{2}k^{2}]^{1/2}\ .$ (23)
After some simplification the above equation leads to
$\displaystyle E_{l}=\frac{3}{8}\sum_{i=e,\
\mu}\frac{\rho_{i}}{\rho}\frac{m_{i}c^{2}}{{x_{i}^{(F)}}^{3}}\left[x_{i}^{(F)}(2{x_{i}^{(F)}}^{2}+1)\sqrt{{x_{i}^{(F)}}^{2}+1}\
-\sinh^{-1}x_{i}^{(F)}\right],$ (24)
where $x_{i}^{(F)}=\hbar k_{i}^{(F)}/m_{i}c$ . The conditions of charge
neutrality and beta equilibrium impose the following constraints on the
calculation of energy of neutron star matter shap ,
$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle\mu_{p}+\mu_{e}\ \ \ \ \
\ \ \mu_{e}=\mu_{\mu}\ .$ (25) $\displaystyle\rho_{p}$ $\displaystyle=$
$\displaystyle\rho_{e}+\rho_{\mu}\ .$ (26)
In Fig. 8, we have shown the energy per particle of the neutron star matter at
various values of spin polarization as a function of density for the
$AV_{18}$, $Reid93$, $UV_{14}$ and $AV_{14}$ potentials. The same as Figs.
1-4, this figure have also shown no phase transition to a ferromagnetic state.
The treatment of magnetic susceptibility and landau parameter of neutron star
matter versus density for different polarization, have been shown in Figs. 9
and 10, respectively. This treatments are the same as the treatments of the
polarized neutron matter, symmetrical and asymmetrical nuclear matter, and it
does not show any magnetic instability for the neutron star matter. This
behavior has been observed for all potentials used in our calculations.
For the $AV_{18}$ potential, the pressure of neutron star matter have been
presented as a function of density $\rho$ at different polarizations in Fig.
11. We see that the equation of state of neutron star matter becomes stiffer
by increasing the polarization.
## V Summary and Conclusions
The purpose of this paper was to calculate the properties of the spin
polarized asymmetrical nuclear matter and neutron star matter employing the
lowest order constrained variational technique with the $AV_{18}$, $Reid93$,
$UV_{14}$ and $AV_{14}$ nucleon-nucleon potentials. After introducing the LOCV
method briefly, we calculated the energy per particle of the polarized
asymmetrical nuclear matter and showed that the force becomes more repulsive
by increasing the polarization for all relevant densities. No crossing of the
energy curves was observed and the ground state of the system was found to be
that of the unpolarized matter in all ranges of densities and isospin
asymmetry. The magnetic susceptibility was computed and shown to remain almost
constant by increasing the asymmetry parameter. For the polarized neutron star
matter, we showed that there is no magnetic instability and the equation of
state becomes stiffer by increasing the polarization.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha. One os us (G.H. Bordbar) wish to thanks Shiraz
University Research Council.
## References
* (1) S. Shapiro and S. Teukolsky, Blak Holes, White Dwarfs and Neutron Stars, (Wiley-New york,1983).
* (2) F. Pacini, _Nature_ (London) 216 (1967) 567.
* (3) T. Gold, _Nature_ (London) 218 (1968) 731\.
* (4) D. H. Brownell and J. Callaway, _Nuovo Cimento_ B 60 (1969) 169.
* (5) M. J. Rice, _Phys. Lett_. A 29 (1969) 637.
* (6) J. W. Clark and N. C. Chao, _Lettere Nuovo Cimento_ 2 (1969) 185.
* (7) J. W. Clark, _Phys. Rev. Lett_. 23 (1969) 1463.
* (8) S. D. Silverstein, _Phys. Rev_. Lett. 23 (1969) 139.
* (9) E. Østgaard, _Nucl. Phys_. A 154 (1970) 202.
* (10) J. M. Pearson and G. Saunier, _Phys. Rev. Lett_. 24 (1970) 325.
* (11) V. R. Pandharipande, V. K. Garde and J. K. Srivastava, _Phys. Lett_. B 38 (1972) 485.
* (12) S. O. Backman and C. G. Kallman, _Phys. Lett_. B 43 (1973) 263.
* (13) P. Haensel, _Phys. Rev_. C 11 (1975) 1822.
* (14) A. D. Jackson, E. Krotscheck, D. E. Meltzer and R. A. Smith, _Nucl. Phys_. A 386 (1982)125.
* (15) M. Kutschera and W. W ojcik, _Phys. Lett_. B 223 (1989) 11.
* (16) S. Marcos, R. Niembro, M. L. Quelle and J. Navarro, _Phys. Lett_. B 271 (1991) 277.
* (17) P. Bernardos, S. Marcos, R. Niembro, M. L. Quelle, _Phys. Lett_. B 356 (1995) 175.
* (18) A. Vidaurre, J. Navarro and J. Bernabeu, _Astron. Astrophys_. 135 (1984) 361.
* (19) M. Kutschera and W. W ojcik, _Phys. Lett_. B 325 (1994) 271.
* (20) S. Fantoni, A. Sarsa and K. E. Schmidt, _Phys. Rev. Lett_. 87 (2001) 181101.
* (21) I. Vida na, A. Polls and A. Ramos, _Phys. Rev_. C 65 (2002) 035804.
* (22) I. Vida na and I. Bombaci, _Phys. Rev_. C 66 (2002) 045801.
* (23) W. Zuo, U. Lombardo and C.W. Shen, in Quark-Gluon Plasma and Heavy Ion Collisions, Ed. W.M. Alberico, M. Nardi and M.P. Lombardo, World Scientific, p. 192 (2002).
* (24) A. A. Isayev and J. Yang, _Phys. Rev_. C 69 (2004) 025801.
* (25) W. Zuo, U. Lombardo and C. W. Shen, nucl-th/0204056.
W. Zuo, C. W. Shen and U. Lombardo, Phys. Rev C 67 (2003) 037301\.
* (26) M. Baldo, G. Giansiracusa, U. Lombardo and H. Q. Song, Phys. Lett. B 473 (2000) 1.
* (27) A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58 (1998) 1804.
* (28) L. Engvik et al., Nucl. Phys. A 627 (1997) 85\.
* (29) A.Rios, A. Polls and I. Vidana, Phys. Rev. C 71(2005) 055802.
* (30) J. Navarro, E. S. Hern andez and D. Vautherin, _Phys. Rev_. C 60 (1999) 045801.
* (31) G. H. Bordbar and M. Modarres, _Phys. Rev._ C 57 (1998) 714.
* (32) R. B. Wiringa, V. Stoks and R. Schiavilla, _Phys. Rev._ C 75 (1995) 38.
* (33) G. H. Bordbar and M. Bigdeli, _Phys. Rev._ C 75 (2007) 045804.
* (34) G. H. Bordbar and M. Bigdeli, _Phys. Rev._ C 76 (2007) 035803.
* (35) V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J. de Swart, _Phys. Rev._ C 49 (1994) 2950.
* (36) I.E. Lagaris and V.R. Pandharipande, _Nucl. Phys._ A 359 (1981) 331.
* (37) R.B. Wiringa, R.A. Smith and T.L. Ainsworth,_Phys. Rev._ C 29 (1984) 1207.
* (38) G. H. Bordbar, M. Modarres, _J. Phys. G: Nucl. Part. Phys_. 23 (1997) 1631.
* (39) J. C. Owen, R. F. Bishop, and J. M. Irvine, _Nucl. Phys._ A 277 (1977) 45.
Figure 1: The energy per particle of the polarized asymmetrical nuclear matter
versus density($\rho$) for different values of the spin polarization
($\delta$) with the $AV_{18}$ potential at $\beta=0.0$ (a), $\beta=0.3$ (b),
$\beta=0.6$ (c), $\beta=1.0$ (d).
Figure 2: The energy per particle of the polarized asymmetrical nuclear matter
versus density($\rho$) for different values of the spin polarization
($\delta$) with the Reid93 potential at $\beta=0.0$ (a), $\beta=0.3$ (b),
$\beta=0.6$ (c), $\beta=1.0$ (d).
Figure 3: The energy per particle of the polarized asymmetrical nuclear matter
versus density($\rho$) for different values of the spin polarization
($\delta$) with the $UV_{14}$ potential at $\beta=0.0$ (a), $\beta=0.3$ (b),
$\beta=0.6$ (c), $\beta=1.0$ (d).
Figure 4: The energy per particle of the polarized asymmetrical nuclear matter
versus density($\rho$) for different values of the spin polarization
($\delta$) with the $AV_{14}$ potential at $\beta=0.0$ (a), $\beta=0.3$ (b),
$\beta=0.6$ (c), $\beta=1.0$ (d).
Figure 5: The magnetic susceptibility of the polarized asymmetrical nuclear
matter as the function of density ($\rho$) for different values of asymmetry
$(\beta)$ with the $AV_{18}$ (a), Reid93 (b), $UV_{14}$ (c) and $AV_{14}$ (d)
potentials.
Figure 6: The Landau parameter of the polarized asymmetrical nuclear matter as
the function of density ($\rho$) for different values of asymmetry $(\beta)$
with the $AV_{18}$ (a), Reid93 (b), $UV_{14}$ (c) and $AV_{14}$ (d)
potentials.
Figure 7: The equation of state of polarized asymmetrical nuclear matter for
different values of the spin polarization ($\delta$) with the $AV_{18}$
potential at $\beta=0.0$ (a), $\beta=0.3$ (b), $\beta=0.6$ (c), $\beta=1.0$
(d).
Figure 8: The energy per particle of the polarized neutron star matter versus
density ($\rho$) for different values of the spin polarization ($\delta$) with
the $AV_{18}$ (a), Reid93 (b), $UV_{14}$ (c) and $AV_{14}$ (d) potentials.
Figure 9: The magnetic susceptibility of the polarized neutron star matter
versus density ($\rho$) with the $AV_{18}$, Reid93, $UV_{14}$ and $AV_{14}$
potentials. Figure 10: The Landau parameter, $G_{0}$, of polarized neutron
star matter as function of density($\rho$) with the $AV_{18}$, Reid93,
$UV_{14}$ and $AV_{14}$ potentials. Figure 11: The equation of state of
polarized neutron star matter for different values of the spin polarization
($\delta$) with the $AV_{18}$ potential .
|
arxiv-papers
| 2008-09-20T11:30:43
|
2024-09-04T02:48:57.853548
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar and M. Bigdeli",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0809.3498"
}
|
0809.3538
|
# Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations
Michael Skeide
Università degli Studi del Molise
Dipartimento S.E.G.e S.
Via de Sanctis
86100 Campobasso, Italy
E-mail: skeide@unimol.it
Homepage: http://www.math.tu-cottbus.de/INSTITUT/lswas/_skeide.html
This work is supported by research funds of University of Molise and Italian
MIUR under PRIN 2007.
(September 2008)
###### Abstract
For many Markov semigroups dilations in the sense of Hudson and Parthasarathy,
that is a dilation which is a cocycle perturbation of a noise, have been
constructed with the help of quantum stochastic calculi. In these notes we
show that every Markov semigroup on the algebra of all bounded operators on a
separable Hilbert space that is spatial in the sense of Arveson, admits a
Hudson-Parthasarathy dilation. In a sense, the opposite is also true. The
proof is based on general results on the the relation between spatial
$E_{0}$–semigroups and their product systems.
## 1 Introduction
(Quantum) Markov semigroups are models for irreversible evolutions of
(quantum) physical systems. Dilating a Markov semigroup, means embedding the
irreversible system into a reversible one in such a way that the original
irreversible evolution can be recovered by projecting down (via a conditional
expectation) the reversible evolution to the subsystem.
Noises are models for reversible systems containing a subsystem. A noise is
actually a reversible evolution on a “big” system with a conditional
expectation onto a “small” subsystem that leaves the small system invariant.
One may think of a simultaneous description of a reversible system and the
small system, but with the interaction switched off. When the interaction is
switched on, the dynamics of the compound system changes and leaves the small
system no longer invariant. The projection back to the small subsystem
produces irreversible behavior.
Often, one tries to model the transition from the free dynamics (the noise) to
the real dynamics by perturbation of the noise with a unitary cocycle. This is
what we mean by a Hudson-Parthasarathy dilation. In practically all known
examples, such cocycles have been obtained by means of a quantum stochastic
calculus. The stochastic generator of the cocycle, is composed from the
generator of the Markov semigroup. Often, it may be interpreted in terms of an
interaction Hamiltonian. In these notes, we show in the case of
$\mathscr{B}(G)$ (the algebra of all bounded operators on a Hilbert space $G$)
without using any calculus, that a Markov semigroup admits a Hudson-
Parthasarathy dilation, if (and, in a sense, also only if) the Markov
semigroup is spatial. The case of a Markov semigroup on a general von Neumann
algebra (in particular, also on a commutative one, which corresponds to a
classical dynamical system) will be discussed in Skeide [Ske08a].
The necessary notions, spatial Markov semigroup, noise, Hudson-Parthasarathy
dilation, and so forth, are explained in Section 2 and then used to formulate
the result. In Section 3 we review the basic results about spatial
$E_{0}$–semigroups and spatial product systems, needed in the proof of the
result, and we proof the result. In Section 4, finally, we list several
natural questions.
Our proof makes heavy use of Arveson’s results [Arv89] on the classification
of $E_{0}$–semigroups (in particular, spatial ones) by tensor product systems
of Hilbert spaces. (In fact, the main scope of [Ske08a] is not to generalize
the present result to general von Neumann algebras, but to fill a long
standing gap, namely, to answer the question how Arveson’s classification of
$E_{0}$–semigroups by product systems generalizes to Hilbert modules.) We
shall assume that the reader is familiar with Arveson’s results at least in
the spatial case, and we shall also assume that the reader knows the works by
Bhat [Bha96] and Arveson [Arv97] on the relation with Markov semigroups via
the so-called minimal weak dilation (once more, in particular in the spatial
case).
Acknowledgements. I would like to thank B.V.R. Bhat, J.M. Lindsay, and O.M.
Shalit for several good comments and bibliographical hints. This work is
supported by research funds of University of Molise and Italian MIUR under
PRIN 2007.
## 2 Notations and statement of the result
$E_{0}$–Semigroups, in these notes, are strongly continuous one-parameter
semigroups of normal unital endomorphisms of $\mathscr{B}(H)$ where $H$ is
some separable infinite-dimensional Hilbert space.[1][1][1]Once for all other
cases, strongly continuous for a von Neumann algebra
$\mathcal{B}\subset\mathscr{B}(H)$ refers to the strong operator topology (or
the point strong topology). That is, a strongly continuous family $T_{t}$ of
operators on $\mathcal{B}$ satisfies that $t\mapsto T_{t}(b)h$ is norm
continuous for all $b\in\mathcal{B},h\in H$. Arveson [Arv89] associated with
every $E_{0}$–semigroup a tensor product system of Hilbert spaces (or, for
short, an Arveson system). He showed that two $E_{0}$–semigroups,
$\vartheta^{1}$ on $\mathscr{B}(H^{1})$ and $\vartheta^{2}$ on
$\mathscr{B}(H^{2})$ say, have isomorphic Arveson systems, if and only if they
are cocycle conjugate. That is, there exist a unitary $u\colon
H^{1}\rightarrow H^{2}$ and a strongly (and, therefore, $*$–strongly)
continuous family of unitaries $u_{t}\in\mathscr{B}(H^{2})$ fulfilling:
1. 1.
The $u_{t}$ form a left cocycle with respect to $\vartheta^{2}$, that is,
$u_{s+t}=u_{s}\vartheta^{2}_{s}(u_{t})$ for all $s,t\in\mathbb{R}_{+}$.
2. 2.
$u\vartheta^{1}_{t}(u^{*}au)u^{*}=u_{t}\vartheta^{2}_{t}(a)u_{t}^{*}$ for all
$t\in\mathbb{R}_{+},a\in\mathscr{B}(H^{2})$.
Markov semigroups, in these notes, are strongly continuous one-parameter
semigroups of normal unital completely positive maps (CP-maps) on
$\mathscr{B}(G)$ where $G$ is some separable Hilbert space. Bhat [Bha96,
Theorem 4.7] states that every Markov semigroup $T$ on $\mathscr{B}(G)$ admits
a weak dilation to an $E_{0}$–semigroup $\vartheta$ on $\mathscr{B}(H)$ in the
following sense: There is an isometry $\xi\colon G\rightarrow H$ such that
$T_{t}(b)~{}=~{}\xi^{*}\vartheta_{t}(\xi b\xi^{*})\xi$
for all $t\in\mathbb{R}_{+},b\in\mathscr{B}(G)$. This weak dilation can be
chosen minimal in the sense that the set
$\vartheta_{\mathbb{R}_{+}}(\xi\,\mathscr{B}(G)\xi^{*})\xi G$ is total in $H$.
The minimal weak dilation is unique up to suitable unitary equivalence,
namely, by a unitary that sends one $\xi$ to the other.[2][2][2]Roughly
speaking, we may identify $G$ with the subspace $\xi G$ of $H$, so that
$\mathscr{B}(G)$ becomes the corner
$\xi\,\mathscr{B}(G)\xi^{*}=p\mathscr{B}(H)p$ of $\mathscr{B}(H)$, where $p$
denotes the projection in $\mathscr{B}(H)$ onto $\xi G$. That leads to
slightly more readable formulae. But, in a minute, we will identify
$\mathscr{B}$ in another way as a unital subalgebra of $\mathscr{B}(H)$.
(Strictly speaking, for an $E_{0}$–semigroup as defined in the first
paragraph, $H$ should be infinite-dimensional. However, the missing case,
where the $H$ of the minimal weak dilation is finite-dimensional, happens if
and only if $G$ is finite-dimensional and $T$ consists of automorphisms. In
this case, for our theorem below there is nothing to prove, and we will
tacitly exclude it from the subsequent discusion.) Bhat [Bha96] defines the
Arveson system associated with the Markov semigroup to be the Arveson system
of its minimal weak dilation $\vartheta$.
By a noise over $\mathscr{B}(G)$ we understand an $E_{0}$–semigroup
$\mathscr{S}$ on some $\mathscr{B}(H)$ that contains $\mathscr{B}(G)$ as a
unital von Neumann subalgebra and an isometry $\omega\colon G\rightarrow H$,
fulfilling the following:
1. 1.
$\mathscr{S}$ leaves $\mathscr{B}(G)$ invariant, that is,
$\mathscr{S}_{t}(b)=b$ for all
$t\in\mathbb{R}_{+},b\in\mathscr{B}(G)\subset\mathscr{B}(H)$.
2. 2.
$\omega$ is $\mathscr{B}(G)$–$\mathbb{C}$–bilinear, that is, $b\omega=\omega
b$ for all $b\in\mathscr{B}(G)$.
3. 3.
$\mathscr{S}$ leaves $\mathbb{E}\colon a\mapsto\omega^{*}a\omega$ invariant,
that is, $\mathbb{E}\circ\mathscr{S}_{t}=\mathbb{E}$ for all
$t\in\mathbb{R}_{+}$.
By 2, $\mathbb{E}$ is a conditional expectation. By 3 (applied to $a=\omega
b\omega^{*}$), $\mathscr{S}$ with the isometry $\omega$ is a weak dilation of
the identity Markov semigroup on $\mathscr{B}(G)$. Therefore, the projection
$p:=\omega\omega^{*}\in\mathscr{B}(H)$ is increasing, that is,
$\mathscr{S}_{t}(p)\geq p$ for all $t\in\mathbb{R}_{+}$. A noise is
reversible, if all $\mathscr{S}_{t}$ are automorphisms. In this case,
$\mathscr{S}_{t}(p)=p$ for all $t\in\mathbb{R}_{+}$.[3][3][3]In fact, suppose
$\mathscr{S}$ is implemented as $\mathscr{S}_{t}=u_{t}\bullet u_{t}^{*}$ by a
unitary semigroup $u_{t}$. One easily checks that $p$ is increasing, if and
only if $\omega^{*}u_{t}^{*}\omega$ is an isometry. Since
$\mathscr{S}_{t}(b)=b$ we have $bu_{t}=u_{t}b$. Since also $b\omega=\omega b$,
it follows that $\omega^{*}u_{t}^{*}\omega$ is in the center of
$\mathscr{B}(G)$ and, therefore, a unitary. So, $p$ is also decreasing, thus,
constant.
This definition of noise is more or less from Skeide [Ske06]. In the scalar
case (that is, $G=\mathbb{C}$) it corresponds to noises in the sense of
Tsirelson [Tsi98, Tsi03]. A reversible noise is close to a Bernoulli shift in
the sense of Hellmich, Köstler and Kümmerer [HKK04].
Definition.A Hudson-Parthasarathy dilation of a Markov semigroup $T$ on
$\mathscr{B}(G)$ is a noise $(\mathscr{S},\omega)$ and a unitary left cocycle
$u_{t}$ with respect to $\mathscr{S}$, such that the cocycle conjugate
$E_{0}$–semigroup $\vartheta$ defined by setting
$\vartheta_{t}(a)=u_{t}\mathscr{S}_{t}(a)u_{t}^{*}$, fulfills
$\mathbb{E}\circ\vartheta_{t}\upharpoonright\mathscr{B}(G)~{}=~{}T_{t}$
for all $t\in\mathbb{R}_{+}$. The Hudson-Parthasarathy dilation is reversible,
if the underlying noise is reversible. In this case, also $\vartheta$ is an
automorphisms group.
Since the seminal work of Hudson and Parthasarathy [HP84a], the cocycles of
Hudson-Parthasarathy dilations have been obtained with the help of quantum
stochastic calculi as solutions of quantum stochastic differential equations.
[HP84a] dealt with a Lindblad generator with finite degree of freedom, while
[HP84b] considers a general (bounded) Lindblad generator. Chebotarev and
Fagnola [CF98] deal with a large class of unbounded generators. Versions for
general von Neumann algebras (Goswami and Sinha [GS99], Köstler [Kös00]) or
$C^{*}$–algebras (Skeide [Ske00]) require Hilbert modules. (Apart from the
fundamental monograph [Par92] by Parthasarathy, a still up-to-date reference
for everything that has to do with calculus based on Boson Fock spaces or
modules are Lindsay’s lecture notes [Lin05]. Results that use other types of
Fock constructions or abstract representation spaces are scattered over the
literature.)
In these notes we characterize the Markov semigroups on $\mathscr{B}(G)$ that
admit a Hudson-Parthasarathy dilation as those which are spatial in the sense
of Arveson [Arv97]. Unlike all other existing constructions of Hudson-
Parthasarathy dilations, the constructive part of our result will be by
general abstract methods without quantum stochastic calculus. It is based on
Arveson’s classification of $E_{0}$–semigroups by Arveson systems up to
cocycle conjugacy, and by the fact that it is easy to construct an
$E_{0}$–semigroup for an Arveson system provided the latter is spatial. (A
general reference for product systems of Hilbert spaces and $E_{0}$–semigroups
on $\mathscr{B}(H)$ is Arveson’s monograph [Arv03]. We will not in any way
enter the theory of product systems. Instead, when we use the well-known facts
about spatial $E_{0}$–semigroups and the relation between spatial Markov
semigroups and their dilating $E_{0}$–semigroups, which all have proofs based
on product systems, then we will refer the reader to [Arv03] or one of the
original contributions [Arv89, Bha96, Arv97, Bha01].) A version for arbitrary
von Neumann algebras will appear in Skeide [Ske08a]. The problem (and main
task in [Ske08a]) is to find the correct analogue of Arveson’s classification
result, in order to complete the theory of the classification of
$E_{0}$–semigroups acting on the algebra of adjointable operators on a Hilbert
$\mathcal{B}$–module by product systems of correspondences over $\mathcal{B}$.
The dilation result itself is just a corollary following the lines of the
present notes. We wish, however, to emphasize that the present notes have been
motivated by the ongoing work on [Ske08a], and not conversely. Several
ingredients have intriguing interpretations, when we use consequently the
language of Hilbert modules. We refer the interested reader to [Ske08a]. Here
we stay completely in the $\mathscr{B}(H)$–language, and use only results that
are known for this case since quite a while.
We now make the last definition and state the result.
Definition [Arv97, Definition 2.1]. A unit for a Markov semigroup $T$ on
$\mathscr{B}(G)$ is a strongly continuous semigroup $c$ in $\mathscr{B}(G)$
such that $T$ dominates the elementary CP-semigroup
$S_{t}(b):=c_{t}^{*}bc_{t}$, that is, the difference $T_{t}-S_{t}$ is
completely positive, too, for all $t\in\mathbb{R}_{+}$. A Markov semigroup on
$\mathscr{B}(G)$ is spatial, if it admits units.
A Markov semigroup on $\mathscr{B}(G)$ is spatial if and only if it admits a
Hudson-Parthasarathy dilation that is also a weak dilation with respect to the
isometry $\omega$. Such a Hudson-Parthasarathy dilation may be extended to a
reversible Hudson-Parthasarathy dilation.
We do not know, whether mere existence of a Hudson-Parthasarathy dilation
alone (without the requirement that it may be chosen to be also a weak
dilation), is already sufficient. For instance, a reversible noise has the
trivial (that is, the one-dimensional) Arveson system, and a cocycle does not
change the Arveson system. Hence, there is no chance that a cocycle
perturbation of a noise can be a weak dilation of a nonautomorphic Markov
semigroup.
## 3 Proof
We said, a Markov semigroup is spatial, if it admits a unit. Of course, also
an $E_{0}$–semigroup is a Markov semigroup. For $E_{0}$–semigroups there is
Powers’ definition [Pow87] of spatiality in terms of intertwining semigroups
of isometries, also refered to as (isometric) units. Bhat [Bha01, Section 6]
compared several notions of units. Also for Arveson systems there is the
concept of units and an Arveson system is spatial, if it admits a unit. We do
not repeat the definition of Arveson system nor that of a unit for an Arveson
system. (The discussion for general von Neumann algebras in [Ske08a] will be
much more self-contained, and many of the statements for $\mathscr{B}(G)$, we
simply quote here, will drop out very naturally without any effort.) What is
important to know is that all concepts of spatiality in the sense of existence
of units coincide: A semigroup, Markov or $E_{0}$, is spatial in whatsoever
sense if and only if its associated Arveson system is spatial. Moreover,
whether a weak dilation is spatial or not, does not depend on whether the
dilation is minimal, but only on whether the dilated Markov semigroup is
spatial or not; see [Bha01, Section 6] or [Arv03, Sections 8.9 and 8.10].
The fact that a Markov semigroup is spatial if and only if the Arveson system
of every of its weak dilations is spatial, has important consequences: For a
spatial Arveson system it is easy to construct an $E_{0}$–semigroup
$\mathfrak{S}$ on some $\mathscr{B}(\mathfrak{H})$ that has as associated
Arveson system the one we started with; see the appendix of [Arv89]. Moreover,
there exists a unit vector $\Omega\in\mathfrak{H}$ such that $\mathfrak{S}$
leaves the state $\varphi:=\langle\Omega,\bullet\Omega\rangle$ invariant (that
is, $\varphi\circ\mathfrak{S}_{t}=\varphi$ for all $t\in\mathbb{R}_{+}$). Two
$E_{0}$–semigroups are cocycle conjugate if and only they have isomorphic
Arveson systems; see the corollary of [Arv89, Definition 3.20]. So, a Markov
semigroup is spatial if and only if one (and, therefore, all) weak dilation(s)
is (are) cocycle conjugate to an $E_{0}$–semigroup with an invariant vector
state.
From $E_{0}$–semigroups with invariant vector states to noises and back, there
is only a small step. Suppose we have a noise $(\mathscr{S},\omega)$ over
$\mathscr{B}(G)$ on $\mathscr{B}(H)$. Clearly, a unital von Neumann subalgebra
$\mathscr{B}(G)$ decomposes $H$ into $G\otimes\mathfrak{H}$ for some
multiplicity space $\mathfrak{H}$, and $\mathscr{S}$ leaves
$\mathscr{B}(G)=\mathscr{B}(G)\otimes\operatorname{\text{\small$\textsf{id}$}}_{\mathfrak{H}}$
invariant, if and only if
$\mathscr{S}_{t}=\operatorname{\text{\small$\textsf{id}$}}_{\mathscr{B}(G)}\otimes\mathfrak{S}_{t}$
for a unique $E_{0}$–semigroup $\mathfrak{S}$ on
$\mathscr{B}(\mathfrak{H})$.[4][4][4]$\mathscr{S}_{t}(\operatorname{\scriptstyle\textsf{id}}_{G}\otimes
a)$ is in the commutant of $\mathscr{S}_{t}(\mathscr{B}(G))=\mathscr{B}(G)$.
So $\mathscr{S}_{t}$ leaves
$\operatorname{\scriptstyle\textsf{id}}_{G}\otimes\mathscr{B}(\mathfrak{H})$
invariant. For that the isometry $\omega$ intertwines the actions of
$\mathscr{B}(G)$, it necessarily has the form
$\omega=\operatorname{\text{\small$\textsf{id}$}}_{G}\otimes\Omega\colon
g\mapsto g\otimes\Omega$ for a unique unit vector $\Omega\in\mathfrak{H}$.
Clearly, $\mathscr{S}$ leaves the conditional expectation $\mathbb{E}$
invariant, if and only if $\mathfrak{S}$ leaves the vector state
$\varphi:=\langle\Omega,\bullet\Omega\rangle$ invariant. Conversely, if
$\mathfrak{S}$ is an $E_{0}$–semigroup with an invariant vector state
$\varphi$ induced by a unit vector $\Omega\in\mathfrak{H}$, then the
$E_{0}$–semigroup
$\mathscr{S}=\operatorname{\text{\small$\textsf{id}$}}_{\mathscr{B}(G)}\otimes\mathfrak{S}$
on $\mathscr{B}(G\otimes\mathfrak{H})$ with the isometry
$\omega:=\operatorname{\text{\small$\textsf{id}$}}_{G}\otimes\Omega$ is a
noise. On the other hand, $\mathscr{S}$ is just a multiple of $\mathfrak{S}$,
and multiplicity does not change the Arveson system; see [Arv89, Poposition
3.15]. Therefore, the Arveson system of a noise $\mathscr{S}$ is spatial.
Consequently, a Markov semigroup is spatial if and only if one (and,
therefore, all) weak dilation(s) is (are) cocycle conjugate to a noise.
This shows that a Markov semigroup that admits a Hudson-Parthasarathy
dilation, is necessarily spatial. For the opposite direction we need to choose
the noise (to which a weak dilation is cocycle conjugate) in such a way that
it admits a Hudson-Parthasarathy cocycle. Let us start with a spatial Markov
semigroup $T$. That is, let us assume that the Arveson system of the minimal
(or any other) weak dilation $\vartheta$ on $\mathscr{B}(H)$ of $T$, with the
isometry $\xi$, is spatial. To that Arveson system construct an
$E_{0}$–semigroup $\mathfrak{S}$ on a Hilbert space $\mathfrak{H}$ with an an
invariant vector state $\varphi=\langle\Omega,\bullet\Omega\rangle$. Tensor it
with the identity on $\mathscr{B}(G)$ as described before to obtain a noise
$(\mathscr{S},\omega)$ with the same Arveson system as $\vartheta$. We wish to
identify the two Hilbert spaces (infinite-dimensional and separable, unless
$T$ is an automorphism semigroup on $M_{n}$) by a unitary $u\colon
H\rightarrow G\otimes\mathfrak{H}$ in such a way that $u\xi=\omega$. But this
is easy. If $T$ is an $E_{0}$–semigroup, then, since $T$ is its own weak
dilation, there is nothing to show. If $T$ is not an $E_{0}$–semigroup, then
both $E_{0}$–semigroups, $\vartheta$ and $\mathfrak{S}$, are proper. We simply
fix a unitary $u\colon H\rightarrow G\otimes\mathfrak{H}$ that takes $\xi g$
to $g\otimes\Omega=\omega g$ and is arbitrary on the (infinite-dimensional!)
complements of $\xi G$ and $G\otimes\Omega$. There exists, then, a left
cocycle $u_{t}$ with respect to $\mathscr{S}$ that fulfills
$u\vartheta_{t}(u^{*}au)u^{*}=u_{t}\mathscr{S}_{t}(a)u_{t}^{*}.$
We find
$T_{t}(b)~{}=~{}\xi^{*}\vartheta_{t}(\xi
b\xi^{*})\xi~{}=~{}\xi^{*}u^{*}u\vartheta_{t}(u^{*}u\xi
b\xi^{*}u^{*}u)u^{*}u\xi\\\ ~{}=~{}\omega^{*}u\vartheta_{t}(u^{*}\omega
b\omega^{*}u)u^{*}\omega~{}=~{}\omega^{*}u_{t}\mathscr{S}_{t}(\omega
b\omega^{*})u_{t}^{*}\omega,$
so that $u_{t}\mathscr{S}_{t}(\bullet)u_{t}^{*}$ with the isometry $\omega$ is
a weak dilation of $T$. In particular, the projection $\omega\omega^{*}$ must
be increasing, that is,
$u_{t}\mathscr{S}_{t}(\omega\omega^{*})u_{t}^{*}\omega\omega^{*}=\omega\omega^{*}$
or $u_{t}\mathscr{S}_{t}(\omega\omega^{*})u_{t}^{*}\omega=\omega$. Now, by the
special property of $\omega$, we have $\omega
b\omega^{*}=(\omega\omega^{*})b(\omega\omega^{*})$. It follows
$T_{t}(b)~{}=~{}\omega^{*}u_{t}\mathscr{S}_{t}(\omega\omega^{*})u_{t}^{*}u_{t}\mathscr{S}_{t}(b)u_{t}^{*}u_{t}\mathscr{S}_{t}(\omega\omega^{*})u_{t}^{*}\omega~{}=~{}\omega^{*}u_{t}\mathscr{S}_{t}(b)u_{t}^{*}\omega,$
that is, the cocycle perturbation of the noise $(\mathscr{S},\omega)$ by the
cocycle $u_{t}$ is a Hudson-Parthasarathy dilation of $T$.
This concludes the proof of the first sentence of the theorem. To prove the
second sentence, we simply refer to the results in Skeide [Ske07] restricted
to the scalar case. In fact, the scalar-valued noise $\mathfrak{S}$ may be
obtained as a restriction of an inner automorphism group on $\mathscr{B}(K)$
for some “big” Hilbert space $K$ to a unital subalgebra
$\mathscr{B}(\mathfrak{H})\subset\mathscr{B}(K)$. Moreover, there is a vector
$w$ in $K$ such that the restriction of $\langle w,\bullet w\rangle$ to
$\mathscr{B}(\mathfrak{H})$ is $\varphi$, and the unitary semigroup $u_{t}$
implementing the automorphism semigroup can be chosen such that $u_{t}^{*}w=w$
for all $t\in\mathbb{R}_{+}$. From this, also the second sentence follows by
tensoring once more with $\mathscr{B}(G)$.
## 4 Remarks and outlook
Our construction of a Hudson-Parthasarathy dilation is by completely abstract
means. This leaves us with a bunch of natural questions.
Is our cocycle in any way adapted? Hudson-Parthasarathy cocycles obtained with
quantum stochastic calculus on the Boson Fock space are adapted in the sense
that $u_{t}$ is in the commutant of
$\mathscr{S}_{t}(\mathbf{1}\otimes\mathscr{B}(\mathfrak{H}))$ for each
$t\in\mathbb{R}_{+}$. Is our cocycle possibly adapted in the sense of [BS00,
Definition 7.4]? (Today, we would prefer to say weakly adapted. Roughly, this
means
$\mathscr{S}_{t}(\omega\omega^{*})u_{t}^{*}\omega\omega^{*}=u_{t}^{*}\omega\omega^{*}$,
so that the $u_{t}^{*}\omega\omega^{*}$ form a partially isometric cocycle.)
Instead of the minimal Arveson system of the minimal weak dilation, we could
have started the constructive part with the Arveson system associated with the
free flow generated by the spatial minimal Arveson system in a sense to be
worked out in [Ske08b]. (This has been outlined in Skeide [Ske06].) These free
flows come along with an own notion of adaptedness (see [KS92, Fow95, Ske00]),
and we may ask whether the cocycle is adapted in this sense.
In any case, quantum stochastic calculi, also the abstract one in [Kös00],
provide a relation between additive cocycles and multiplicative (unitary)
cocycles. Differentials of additive cocycles are, roughly, the differentials
of the quantum stochastic differential equation to be resolved. We may ask,
whether this relation holds for all spatial Markov semigroups, also if they
are not realized on the Fock spaces, that is, on type I or completely spatial
noises. The additive cocycles, usually, take their ingredients from the
generator of the Markov semigroup. If that generator is bounded, then one may
recognize the constituents of the Christensen-Evans generator, or, in the
$\mathscr{B}(G)$–case, of the Lindblad generator. This raises the problem to
characterize spatial Markov semigroups in terms of their generators. Do they
have generators that resemble in some sense the Lindblad form? Apparently the
most general form of unbounded Lindblad type generators of Markov semigroups
on $\mathscr{B}(G)$ has been discussed in [CF98]. But only a subclass of these
generators could be dilated by using Hudson-Parthasarathy calculus on the
Boson Fock space. It is natural to ask whether the others are also spatial.
Also, if they are spatial, is their product system type II (non-Fock) or is it
completely spatial (Fock)? In the latter case, can the solution be obtained
with a calculus? In any case, whenever for an example a solution of the
problem has been obtained with calculus, then we may ask, whether our abstract
cocycle (which, of course, can be written down explicitly; see [Ske08a]) can
be related to the concrete cocycle emerging from calculus. Generally, we may
ask, how two possible cocycles with respect to the same noise (adapted in some
sense or not) are related.
Last but not least, we ask, if there exist nontrivial examples of nonspatial
Markov semigroups. By this we mean Markov semigroups that are not type III
$E_{0}$–semigroups, or tensor products of such with a spatial Markov
semigroup.
## References
* [Arv89] W. Arveson, _Continuous analogues of Fock space_ , Mem. Amer. Math. Soc., no. 409, American Mathematical Society, 1989.
* [Arv97] , _The index of a quantum dynamical semigroup_ , J. Funct. Anal. 146 (1997), 557–588.
* [Arv03] , _Noncommutative dynamics and $E$–semigroups_, Monographs in Mathematics, Springer, 2003.
* [Bha96] B.V.R. Bhat, _An index theory for quantum dynamical semigroups_ , Trans. Amer. Math. Soc. 348 (1996), 561–583.
* [Bha01] , _Cocycles of CCR-flows_ , Mem. Amer. Math. Soc., no. 709, American Mathematical Society, 2001.
* [BS00] B.V.R. Bhat and M. Skeide, _Tensor product systems of Hilbert modules and dilations of completely positive semigroups_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519–575, (Rome, Volterra-Preprint 1999/0370).
* [CF98] A.M. Chebotarev and F. Fagnola, _Sufficient conditions for conservativity of minimal quantum dynamical semigroups_ , J. Funct. Anal. 153 (1998), 382–404.
* [Fow95] N.J. Fowler, _Free $E_{0}$–semigroups_, Can. J. Math. 47 (1995), 744–785.
* [GS99] D. Goswami and K.B. Sinha, _Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra_ , Commun. Math. Phys. 205 (1999), 377–403.
* [HKK04] J. Hellmich, C. Köstler, and B. Kümmerer, _Noncommutative continuous Bernoulli sifts_ , Preprint, arXiv:
math.OA/0411565, 2004.
* [HP84a] R.L. Hudson and K.R. Parthasarathy, _Quantum Ito’s formula and stochastic evolutions_ , Commun. Math. Phys. 93 (1984), 301–323.
* [HP84b] , _Stochastic dilations of uniformly continuous completely positive semigroups_ , Acta Appl. Math. 2 (1984), 353–378.
* [Kös00] C. Köstler, _Quanten-Markoff-Prozesse und Quanten-Brownsche Bewegungen_ , Ph.D. thesis, Stuttgart, 2000.
* [KS92] B. Kümmerer and R. Speicher, _Stochastic integration on the Cuntz algebra $O_{\infty}$_, J. Funct. Anal. 103 (1992), 372–408.
* [Lin05] J.M. Lindsay, _Quantum stochastic analysis — an introduction_ , Quantum independent increment processes I (M. Schürmann and U. Franz, eds.), Lect. Notes Math., no. 1865, Springer, 2005, pp. 181–271.
* [Par92] K.R. Parthasarathy, _An introduction to quantum stochastic calculus_ , Birkhäuser, 1992.
* [Pow87] R.T. Powers, _A non-spatial continuous semigroup of $*$–endomorphisms of $\mathscr{B}(\mathfrak{H})$_, Publ. Res. Inst. Math. Sci. 23 (1987), 1053–1069.
* [Ske00] M. Skeide, _Quantum stochastic calculus on full Fock modules_ , J. Funct. Anal. 173 (2000), 401–452, (Rome, Volterra-Preprint 1999/0374).
* [Ske06] , _The index of (white) noises and their product systems_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 617–655, (Rome, Volterra-Preprint 2001/0458, arXiv: math.OA/0601228).
* [Ske07] , _Spatial $E_{0}$–semigroups are restrictions of inner automorphismgroups_, Quantum Probability and Infinite Dimensional Analysis — Proceedings of the 26th Conference (L. Accardi, W. Freudenberg, and M. Schürmann, eds.), Quantum Probability and White Noise Analysis, no. XX, World Scientific, 2007, (arXiv: math.OA/0509323), pp. 348–355.
* [Ske08a] , _Classification of $E_{0}$–semigroups by product systems_, Preprint, in preparation, 2008.
* [Ske08b] , _Free product systems generated by spatial tensor product systems_ , Preprint, in preparation, 2008.
* [Tsi98] B. Tsirelson, _Unitary Brownian motions are linearizable_ , Preprint, arXiv: math.PR/9806112, 1998.
* [Tsi03] , _Scaling limit, noise, stability_ , Preprint, arXiv: math.PR/0301237, 2003.
|
arxiv-papers
| 2008-09-20T22:29:35
|
2024-09-04T02:48:57.857792
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Michael Skeide",
"submitter": "Michael Skeide",
"url": "https://arxiv.org/abs/0809.3538"
}
|
0809.3544
|
definition[equation]Definition remark[equation]Remark example[equation]Example
19D55 (primary), 55Q91 (secondary) The first and third authors were supported
in part by the National Science Foundation
# On the $K$-theory of truncated polynomial algebras
over the integers
Vigleik Angeltveit Teena Gerhardt Lars Hesselholt vigleik@math.uchicago.edu
tgerhard@indiana.edu larsh@math.nagoya-u.ac.jp
###### Abstract
We show that $K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is finite of order
$(mi)!(i!)^{m-2}$ and that $K_{2i+1}(\mathbb{Z}[x]/(x^{m}),(x))$ is free
abelian of rank $m-1$. This is accomplished by showing that the equivariant
homotopy groups $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ of the
topological Hochschild $\mathbb{T}$-spectrum $T(\mathbb{Z})$ are free abelian
for $q$ even, and finite for $q$ odd, and by determining their ranks and
orders, respectively.
## Introduction
It was proved by Soulé [13] and Staffeldt [14] that, for every non-negative
integer $q$, the abelian group $K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$ is finitely
generated and that its rank is either $0$ or $m-1$ according as $q$ is even or
odd. In this paper, we prove the following more precise result:
###### Theorem A
Let $m$ be a positive integer and $i$ a non-negative integer. Then:
1. (i)
The abelian group $K_{2i+1}(\mathbb{Z}[x]/(x^{m}),(x))$ is free of rank $m-1$.
2. (ii)
The abelian group $K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is finite of order
$(mi)!(i!)^{m-2}$.
In particular, the $p$-primary torsion subgroup of
$K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is zero, for every prime number $p>mi$. At
present, we do not know the group structure of the finite abelian group in
degree $2i$ except for small values of $i$ and $m$. We remark that the result
agrees with the calculation by Geller and Roberts [12] of the group in degree
$2$.
To prove Theorem A, we use the cyclotomic trace map of Bökstedt-Hsiang-Madsen
[4] from the $K$-groups in the statement to the corresponding topological
cyclic homology groups and a theorem of McCarthy [11] which shows that this
map becomes an isomorphism after pro-finite completion. The third author and
Madsen [7, Proposition 4.2.3], in turn, gave a formula for the topological
cyclic homology groups in question in terms of the equivariant homotopy groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(\mathbb{Z})]_{\mathbb{T}}$
of the topological Hochschild $\mathbb{T}$-spectrum $T(\mathbb{Z})$. Here
$\mathbb{T}$ is the multiplicative group of complex numbers of modulus $1$,
$C_{r}\subset\mathbb{T}$ is the finite subgroup of the indicated order,
$\lambda$ is a finite dimensional complex $\mathbb{T}$-representation, and
$S^{\lambda}$ is the one-point compactification of $\lambda$. Since the groups
$K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$ and
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ are finitely generated by [13,
14] and Lemma 1.2, respectively, these earlier results amount to a long exact
sequence
$\textstyle{{\cdots}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\lim_{R}^{\phantom{R}}\operatorname{TR}_{q-1-\lambda_{d}}^{r/m}(\mathbb{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{m}}$$\textstyle{{\lim_{R}^{\phantom{R}}\operatorname{TR}_{q-1-\lambda_{d}}^{r}(\mathbb{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{K_{q}(\mathbb{Z}[x]/(x^{m}),(x))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\cdots}}$
where $d=d(m,r)$ is the integer part of $(r-1)/m$, and where $\lambda_{d}$ is
the sum
$\lambda_{d}=\mathbb{C}(d)\oplus\mathbb{C}(d-1)\oplus\cdots\oplus\mathbb{C}(1)$
of the one-dimensional complex $\mathbb{T}$-representations defined by
$\mathbb{C}(i)=\mathbb{C}$ with $\mathbb{T}$ acting from the left by $z\cdot
w=z^{i}w$. The two limits range over the positive integers divisible by $m$
and the positive integers, respectively, ordered under division. The structure
maps $R$ and the map $V_{m}$ are explained in Section 1 below. In particular,
we show that for every integer $q$ there exists a positive integer $r=r(m,q)$
divisible by $m$ such that the canonical projections
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z}),\hskip
28.45274pt\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})$
are isomorphisms.
After localizing at a prime number $p$, the abelian groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ decompose as products of the
$p$-typical equivariant homotopy groups
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)=\operatorname{TR}_{q-\lambda}^{p^{n-1}}(\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},S^{\lambda}\wedge
T(\mathbb{Z})]_{\mathbb{T}}.$
In addition, the Verschiebung map $V_{m}$ which appears in the long exact
sequence above may be expressed in terms of the $p$-typical Verschiebung map
$V=V_{p}$. The corresponding $p$-typical equivariant homotopy groups with
$\mathbb{Z}/p\mathbb{Z}$-coefficients
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\lambda}\wedge T(\mathbb{Z})]_{\mathbb{T}}$
were evaluated by the first and second author [1] and by Tsalidis [16]. More
generally, the first and second author [1] evaluated the
$\operatorname{RO}(\mathbb{T})$-graded equivariant homotopy groups
$\operatorname{TR}_{\alpha}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{\beta}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\gamma}\wedge T(\mathbb{Z})]_{\mathbb{T}},$
where $\alpha\in\operatorname{RO}(\mathbb{T})$ is any virtual finite
dimensional orthogonal $\mathbb{T}$-representation, and where $\beta$ and
$\gamma$ are chosen actual representations with $\alpha=[\beta]-[\gamma]$.
Based on these results, we prove:
###### Theorem B
Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then:
1. (i)
For $q=2i$ even, $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ is a free
abelian group whose rank is equal to the number of integers $0\leqslant s<n$
such that $i=\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})$.
2. (ii)
For $q=2i-1$ odd, $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ is a
finite abelian group whose order is determined, recursively, by
$\lvert\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\rvert=\begin{cases}\lvert\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert\cdot
p^{n-1}(i-\dim_{\mathbb{C}}(\lambda))&\text{if\,
$i>\dim_{\mathbb{C}}(\lambda)$
}\cr\lvert\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert&\text{if\,
$i\leqslant\dim_{\mathbb{C}}(\lambda)$, }\cr\end{cases}$
where $\lambda^{\prime}=\rho_{p}^{*}\lambda^{C_{p}}$ is the
$\mathbb{T}/C_{p}$-representation $\lambda^{C_{p}}$ viewed as a
$\mathbb{T}$-representation via the isomorphism
$\rho_{p}\colon\mathbb{T}\to\mathbb{T}/C_{p}$ given by the $p$th root.
3. (iii)
For every integer $q$, the Verschiebung map
$V\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$
is injective, and for $q$ even the cokernel is a free abelian group.
We remark that for $\lambda=0$ the result is that
$\lvert\operatorname{TR}_{2i-1}^{n}(\mathbb{Z};p)\rvert=p^{n(n-1)/2}\,i^{n}$
while the even groups all are zero with the exception of
$\operatorname{TR}_{0}^{n}(\mathbb{Z};p)$ which is a free abelian group of
rank $n$. In the case $n=1$ which was proved by Bökstedt [3], the groups are
all cyclic. For $n>1$, this is not the case. It remains a very interesting
problem to determine the structure of these groups. We refer to [5, Theorem
18] for some partial results.
It follows from Theorem B that with $\mathbb{Z}/p\mathbb{Z}$-coefficients the
Verschiebung map
$V\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$
is injective for $q$ even. We do not know the value of this map for $q$ odd.
The calculation of this map, and hence, the groups
$K_{q}(\mathbb{Z}[x]/(x^{m}),(x);\mathbb{Z}/p\mathbb{Z})$ claimed in [16,
Proposition 7.7] is incorrect. Indeed, in loc. cit., it is only the map
induced by $V$ between the $E^{\infty}$-terms of two spectral sequences which
is evaluated.
The paper is organized as follows. In Section 1, we show that the groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ are finitely generated and
determine their ranks. In Section 2, we recall the results of [1] and [16] and
prove Theorem B. In the following Section 3, we evaluate the terms in the long
exact sequence above and prove Theorem A. Finally, in Section 4, we specialize
to the case of the dual numbers and determine the structure of the finite
group $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ of order $(2i)!$ in low degrees.
## 1 The groups $\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$
In this section, we recall the groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ and the Frobenius,
Verschiebung, and restriction operators that relate them. We refer to [8,
Section 1] and [5, Section 2] for further details.
Let $A$ be a unital associative ring. The topological Hochschild
$\mathbb{T}$-spectrum $T(A)$ is, in particular, an orthogonal
$\mathbb{T}$-spectrum in the sense of [10, Definition II.2.6]. Therefore, for
every finite dimensional orthogonal $\mathbb{T}$-representation $\lambda$ and
every finite subgroup $C_{r}\subset\mathbb{T}$, we have the equivariant
homotopy group given by the following abelian group of maps in the homotopy
category of orthogonal $\mathbb{T}$-spectra
$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}.$
For every divisor $s$ of $r$ with quotient $t=r/s$, there are maps
$\displaystyle F_{s}$
$\displaystyle\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda}^{t}(A)\hskip
19.91692pt\text{(Frobenius)}$ $\displaystyle V_{s}$
$\displaystyle\colon\operatorname{TR}_{q-\lambda}^{t}(A)\to\operatorname{TR}_{q-\lambda}^{r}(A)\hskip
19.91692pt\text{(Verschiebung)}$
induced by maps $f_{s}\colon(\mathbb{T}/C_{t})_{+}\to(\mathbb{T}/C_{r})_{+}$
and $v_{s}\colon(\mathbb{T}/C_{r})_{+}\to(\mathbb{T}/C_{t})_{+}$ in the
homotopy category of orthogonal $\mathbb{T}$-spectra. The map $f_{s}$ is the
map of suspension $\mathbb{T}$-spectra induced by the canonical projection
$\operatorname{pr}\colon\mathbb{T}/C_{t}\to\mathbb{T}/C_{r}$ and the map
$v_{s}$ is the corresponding transfer map defined as follows. Let
$\iota\colon\mathbb{T}/C_{t}\hookrightarrow\mu$ be an embedding into a finite
dimensional orthogonal $\mathbb{T}$-representation. Then the product embedding
$(\iota,\operatorname{pr})\colon\mathbb{T}/C_{t}\to\mu\times(\mathbb{T}/C_{r})$
has trivial normal bundle, and the linear structure of $\mu$ determines a
canonical trivialization. Therefore, the Pontryagin-Thom construction gives a
map of pointed $\mathbb{T}$-spaces
$S^{\mu}\wedge(\mathbb{T}/C_{r})_{+}\to S^{\mu}\wedge(\mathbb{T}/C_{t})_{+}.$
The induced map of suspension $\mathbb{T}$-spectra determines the homotopy
class of maps of orthogonal $\mathbb{T}$-spectra
$v_{s}\colon(\mathbb{T}/C_{r})_{+}\to(\mathbb{T}/C_{t})_{+}$ and this homotopy
class is independent of the choice of embedding $\iota$ as well as the choices
made in forming the Pontryagin-Thom construction.
The orthogonal $\mathbb{T}$-spectrum $T(A)$ has the additional structure of a
cyclotomic spectrum in the sense of [8, Definition 2.2.]. This implies that,
in the situation above, there is a map
$R_{s}\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)\hskip
19.91692pt\text{(restriction)},$
where $\lambda^{\prime}=\rho_{s}^{*}(\lambda^{C_{s}})$ is the
$\mathbb{T}/C_{s}$-representation $\lambda^{C_{s}}$ considered as a
$\mathbb{T}$-representation via the isomorphism
$\rho_{s}\colon\mathbb{T}\to\mathbb{T}/C_{s}$ defined by
$\rho_{s}(z)=z^{1/s}C_{s}$. Moreover, the map $R_{s}$ admits a canonical
factorization which we now explain. In general, let $G$ be a compact Lie group
and $\mathscr{F}$ a family of closed subgroups of $G$ stable under conjugation
and passage to subgroups. We recall that a universal $\mathscr{F}$-space is a
$G$-CW-complex $E\mathscr{F}$ with the property that, for every closed
subgroup $H\subset G$, the fixed point set $(E\mathscr{F})^{H}$ is
contractible if $H\in\mathscr{F}$ and empty if $H\notin\mathscr{F}$. It was
proved by tom Dieck [15, Satz 1] that a universal $\mathscr{F}$-space
$E\mathscr{F}$ exists and that, if both $E\mathscr{F}$ and
$E^{\prime}\\!\mathscr{F}$ are universal $\mathscr{F}$-spaces, then there
exists a unique $G$-homotopy class of $G$-homotopy equivalences $f\colon
E\mathscr{F}\to E^{\prime}\\!\mathscr{F}$. Given a universal
$\mathscr{F}$-space $E\mathscr{F}$, the pointed $G$-space
$\tilde{E}\mathscr{F}$ is defined to be the mapping cone of the map $\pi\colon
E\mathscr{F}_{+}\to S^{0}$ that collapses $E\mathscr{F}$ onto the non-base
point such that we have a cofibration sequence of pointed $G$-spaces
$E\mathscr{F}_{+}\xrightarrow{\pi}S^{0}\xrightarrow{\iota}\tilde{E}\mathscr{F}\xrightarrow{\delta}\Sigma
E\mathscr{F}_{+}.$
If $N\subset G$ is a closed normal subgroup, we denote by $\mathscr{F}[N]$ the
family of closed subgroups $H\subset G$ that do not contain $N$ as a subgroup.
Now, the map $R_{s}$ admits a factorization as the composition of the map
$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}\to[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{s}]\wedge
T(A)]_{\mathbb{T}}$
induced by the map $\iota\colon S^{0}\to\tilde{E}\mathscr{F}[C_{s}]$ and a
canonical isomorphism
$[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{s}]\wedge
T(A)]_{\mathbb{T}}\xrightarrow{\sim}[S^{q}\wedge(\mathbb{T}/C_{t})_{+},S^{\lambda^{\prime}}\wedge
T(A)]_{\mathbb{T}}=\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)$
induced from the cyclotomic structure of $T(A)$ and [10, Proposition V.4.17].
The group isomorphism $\rho_{s}\colon\mathbb{T}\to\mathbb{T}/C_{s}$ gives rise
to an equivalence of categories $\rho_{s}^{*}$ from the category of orthogonal
$\mathbb{T}/C_{s}$-spectra to the category of orthogonal $\mathbb{T}$-spectra
defined by
$(\rho_{s}^{*}T)({\lambda})=\rho_{s}^{*}(T({(\rho_{s}^{-1})^{*}\lambda})).$
The following result is a generalization of [8, Theorem 2.2].
###### Proposition 1.1
Let $A$ be a unital associative ring, $r$ a positive integer, and $\lambda$ a
finite dimensional orthogonal $\mathbb{T}$-representation. Let $p$ be a prime
number that divides $r$, and $u$ and $v$ positive integers with
$u+v=v_{p}(r)+1$. Then there is a natural long exact sequence
$\cdots\to\mathbb{H}_{q}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))\xrightarrow{N_{p^{u}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{R_{p^{v}}}\operatorname{TR}_{q-\lambda^{\prime}}^{r/p^{v}}(A)\xrightarrow{\partial}\mathbb{H}_{q-1}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))\to\cdots$
where the left-hand term is the $q$th Borel homology group of the group
$C_{p^{u}}$ with coefficients in the orthogonal $\mathbb{T}$-spectrum defined
by
$\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A)=\rho_{r/p^{u}}^{*}(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}.$
###### Proof 1.1.
The cofibration sequence of pointed $\mathbb{T}$-spaces
$E\mathscr{F}[C_{p^{v}}]_{+}\xrightarrow{\pi}S^{0}\xrightarrow{\iota}\tilde{E}\mathscr{F}[C_{p^{v}}]\xrightarrow{\delta}\Sigma
E\mathscr{F}[C_{p^{v}}]_{+}$
gives rise to a cofibration sequence of orthogonal $\mathbb{T}$-spectra
$E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)\xrightarrow{\pi}T(A)\xrightarrow{\iota}\tilde{E}\mathscr{F}[C_{p^{v}}]\wedge
T(A)\xrightarrow{\delta}\Sigma E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A).$
This, in turn, gives rise to a long exact sequence of equivariant homotopy
groups which we now identify with the long exact sequence of the statement. By
definition, we have
$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}},$
and, as recalled above, the restriction map $R_{p^{v}}$ factors through the
canonical isomorphism
$[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{p^{v}}]\wedge
T(A)]_{\mathbb{T}}\xrightarrow{\sim}\operatorname{TR}_{q-\lambda^{\prime}}^{r/p^{v}}(A).$
This identifies the middle and right-hand term of the long exact sequence. To
identify the left-hand term, we recall from [10, Proposition V.2.3] the
change-of-groups isomorphism
$[S^{q}\wedge(C_{r}/C_{r})_{+},S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)]_{C_{r}}\xrightarrow{\sim}[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A)]_{\mathbb{T}}.$
On the left-hand side, the family $\mathscr{F}[C_{p^{v}}]$ is equal to the
family of subgroups $C_{s}\subset C_{r}$ for which $v_{p}(s)<v$. Therefore, we
may choose the universal space $E\mathscr{F}[C_{p^{v}}]$ to be a $C_{r}$-CW-
complex that is non-equivariantly contractible and that only has cells of
orbit-type $C_{r}/C_{r/p^{u}}$. Indeed, in this case we have
$(E\mathscr{F}[C_{p^{v}}])^{C_{s}}=\begin{cases}E\mathscr{F}[C_{p^{v}}]&\text{if
$v_{p}(s)<v$}\cr\varnothing&\text{if $v_{p}(s)\geqslant v$}\cr\end{cases}$
as required. We then have canonical isomorphisms
$\displaystyle{[S^{q},S^{\lambda}\wedge E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)]_{C_{r}}}{}$ $\displaystyle\xleftarrow{\sim}[S^{q},(S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A))^{C_{r/p^{u}}}]_{C_{r}/C_{r/p^{u}}}$
$\displaystyle\xleftarrow{\sim}[S^{q},E\mathscr{F}[C_{p^{v}}]_{+}\wedge(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}]_{C_{r}/C_{r/p^{u}}}$
where for the second isomorphism we use that $E\mathscr{F}[C_{p^{v}}]$ is
chosen to be $C_{r/p^{u}}$-fixed. The group isomorphism $\rho_{r/p^{u}}\colon
C_{p^{u}}\to C_{r}/C_{r/p^{u}}$ induces an isomorphism of categories
$\rho_{r/p^{u}}^{*}$ from the category of orthogonal $C_{p^{u}}$-spectra to
the category of orthogonal $C_{r}/C_{r/p^{u}}$-spectra. In particular, this
gives an isomorphism of the lower group above to the group
$[S^{q},\rho_{r/p^{u}}^{*}E\mathscr{F}[C_{p^{v}}]_{+}\wedge\rho_{r/p^{u}}^{*}(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}]_{C_{p^{u}}}=\mathbb{H}_{q}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A)).$
This is indeed the desired Borel homology group, since
$\rho_{r/p^{u}}^{*}E\mathscr{F}[C_{p^{v}}]$ is a free $C_{p^{u}}$-CW-complex
which is non-equivariantly contractible.
We recall that the Borel homology groups that appear in the statement of
Proposition 1.1 are the abutment of the first quadrant skeleton spectral
sequence
$E_{s,t}^{2}=H_{s}(C_{p^{u}},\operatorname{TR}_{t-\lambda}^{r/p^{u}}(A))\Rightarrow\mathbb{H}_{s+t}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))$
(1.2)
from the group homology of $C_{p^{u}}$ with coefficients in the trivial
$C_{p^{u}}$-module $\operatorname{TR}_{t-\lambda}^{r/p^{u}}(A)$; see for
instance [5, Section 4].
We now specialize to the case $A=\mathbb{Z}$ and recall from Bökstedt [3] that
$\operatorname{TR}_{q}^{1}(\mathbb{Z})$ is zero if either $q$ is negative or
$q$ is positive and even, a free abelian group of rank one if $q=0$, and a
finite cyclic group of order $i$ if $q=2i-1$ is positive and odd; see also
[9].
###### Lemma 1.2.
Let $r$ be a positive integer, let $q$ be an integer, and let $\lambda$ be a
finite dimensional complex $\mathbb{T}$-representation. Then
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ is a finitely generated
abelian group whose rank is equal to the number of positive divisors $e$ of
$r$ for which $q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$. The group is zero for
$q<2\dim_{\mathbb{C}}(\lambda^{C_{r}})$.
###### Proof 1.3.
Let $\ell(r,q,\lambda)$ denote the number of positive divisors $e$ of $r$ with
$q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$ and note that $\ell(r,q,\lambda)$ is
zero, for $q$ odd. We prove by induction on the number $k$ of prime divisors
in $r$ that $\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ is a finitely
generated abelian group of rank $\ell(r,q,\lambda)$. If $k=0$, or
equivalently, if $r=1$ the statement follows from the result of Bökstedt which
we recalled above. Indeed, it follows from [10, Proposition V.2.3] that, up to
isomorphism,
$\operatorname{TR}_{q-\lambda}^{1}(\mathbb{Z})=\operatorname{TR}_{q-2\dim_{\mathbb{C}}(\lambda)}^{1}(\mathbb{Z}).$
So we let $k\geqslant 1$ and assume that the lemma has been proved, for all
$q$ and $\lambda$ as in the statement if $r$ has $k-1$ prime divisors. Let $p$
be a prime divisor of $r$ and write $r=p^{n}r^{\prime}$ with $r^{\prime}$ not
divisible by $p$. We consider the long exact sequence of Proposition 1.1 with
$u=n$ and $v=1$,
${\cdots}\to{\mathbb{H}_{q}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r^{\prime}}(\mathbb{Z}))}\xrightarrow{N_{p^{n}}}{\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})}\xrightarrow{R_{p}}{\operatorname{TR}_{q-\lambda^{\prime}}^{r/p}(\mathbb{Z})}\xrightarrow{\partial}{\mathbb{H}_{q-1}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r^{\prime}}(\mathbb{Z}))}\to{\cdots}$
Since $r^{\prime}$ has only $k-1$ prime divisors, the inductive hypothesis
implies that in the skeleton spectral sequence (1.2), $E_{0,q}^{2}$ is a
finitely generated abelian group of rank $\ell(q,r^{\prime},\lambda)$ and that
the groups $E_{s,t}^{2}$ with $s>0$ are finite. Hence, the left-hand group in
the long exact sequence is finitely generated of rank
$\ell(q,r^{\prime},\lambda)$. By further induction on $n\geqslant 0$, we may
assume that the group
$\operatorname{TR}_{q-\lambda^{\prime}}^{r/p}(\mathbb{Z})$ is finitely
generated of rank $\ell(q,r/p,\lambda^{\prime})$. The first part of the lemma
now follows from the formula
$\ell(q,r^{\prime},\lambda)+\ell(q,r/p,\lambda^{\prime})=\ell(q,r,\lambda)$
which holds since the two summands on the left-hand side count the number of
positive divisors $e$ of $r$ with $q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$ for
which $e$ is, respectively, prime to $p$ and divisible by $p$. The second part
of the lemma is proved in a similar manner.
###### Addendum 1.3
(i) Let $m,r\geqslant 1$, $0\leqslant\epsilon\leqslant 1$, and $i$ be
integers, and let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then the
canonical projection induces an isomorphism
$\lim_{R}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r}(\mathbb{Z})\xrightarrow{\sim}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r}(\mathbb{Z}),$
provided that $m(i+1)<p^{v_{p}(r)+1}$ for every prime number $p$.
(ii) Let $m\geqslant 1$, $0\leqslant\epsilon\leqslant 1$, and $i$ be integers,
let $r\geqslant 1$ be an integer divisible by $m$, and let $d=d(m,r)$. Then
the canonical projection induces an isomorphism
$\lim_{R}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r/m}(\mathbb{Z})\xrightarrow{\sim}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r/m}(\mathbb{Z})$
provided that $i+1<p^{v_{p}(r/m)+1}$ for every prime number $p$.
###### Proof 1.4.
We prove statement (i); the proof of statement (ii) is similar. It suffices to
show that for every prime number $p$ the restriction map
$R_{p}\colon\operatorname{TR}_{q-\lambda_{d(m,pr)}}^{pr}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d(m,r)}}^{r}(\mathbb{Z})$
is an isomorphism if $q=2i+\epsilon$ with $m(i+1)<p^{v_{p}(r)+1}$. We write
$r=p^{n-1}r^{\prime}$ with $r^{\prime}$ not divisible by $p$ and consider the
long exact sequence of Proposition 1.1 with $u=n$ and $v=1$,
$\cdots\to\mathbb{H}_{q}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{d(m,pr)}}^{r^{\prime}}(\mathbb{Z}))\xrightarrow{N_{p^{n}}}\operatorname{TR}_{q-\lambda_{d(m,pr)}}^{pr}(\mathbb{Z})\xrightarrow{R_{p}}\operatorname{TR}_{q-\lambda_{d(m,r)}}^{r}(\mathbb{Z})\to\cdots$
The skeleton spectral sequence (1.2) and Lemma 1.2 show that the left-hand
group vanishes, provided that
$q<2\dim_{\mathbb{C}}(\lambda_{d(m,pr)}^{C_{r^{\prime}}})$. Therefore, the map
$R_{p}$ is an isomorphism for
$i<\dim_{\mathbb{C}}(\lambda_{d(m,pr)}^{C_{r^{\prime}}})=\lfloor
d(m,pr)/r^{\prime}\rfloor.$
We claim that $d(m,p^{n})\leqslant\lfloor d(m,pr)/r^{\prime}\rfloor$. Indeed,
this inequality is equivalent to the inequality $d(m,p^{n})\leqslant
d(m,pr)/r^{\prime}$ which is equivalent to the inequality
$r^{\prime}d(m,p^{n})\leqslant d(m,pr)$ which, in turn, is equivalent to the
inequality $r^{\prime}d(m,p^{n})\leqslant(p^{n}r^{\prime}-1)/m$. We may
rewrite this inequality as $mr^{\prime}d(m,p^{n})\leqslant p^{n}r^{\prime}-1$
or $mr^{\prime}d(m,p^{n})<p^{n}r^{\prime}$ or $md(m,p^{n})<p^{n}$. But this
inequality is equivalent to the inequality $md(m,p^{n})\leqslant p^{n}-1$
which, in turn, is equivalent to the inequality
$d(m,p^{n})\leqslant(p^{n}-1)/m$ which holds. The claim follows. Finally, a
similar argument shows that the inequalities $i<d(m,p^{n})$ and $m(i+1)<p^{n}$
are equivalent.
## 2 The $p$-typical groups
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z}_{;}p)$
In this section, we prove Theorem B of the introduction. We first show that
after localization at the prime number $p$, the groups
$\operatorname{TR}_{q-\lambda}^{r}(A)$ decompose as products of the
$p$-typical groups
$\operatorname{TR}_{q-\lambda}^{n}(A;p)=\operatorname{TR}_{q-\lambda}^{p^{n-1}}(A)=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}.$
###### Proposition 1.
Let $A$ be a unital associative ring, $r\geqslant 1$ and $q$ integers, and
$\lambda$ a finite dimensional orthogonal $\mathbb{T}$-representation. Let $p$
be a prime number and write $r=p^{n-1}r^{\prime}$ with $r^{\prime}$ not
divisible by $p$. Then the map
$\gamma\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\textstyle{\prod_{j\mid
r^{\prime}}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)$
whose $j$th component is the composite map
$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{j}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{r/j}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{r^{\prime}/j}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{p^{n-1}}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{n^{\phantom{n}}}(A;p)}}$
becomes an isomorphism after localization at $p$.
###### Proof 2.1.
The proof is by induction on the number $k$ of positive divisors of
$r^{\prime}$. If $k=1$, or equivalently, if $r^{\prime}=1$ then $\gamma$ is
the identity map and the statement holds trivially. So we let $k\geqslant 2$
and assume that the statement holds whenever $r^{\prime}$ has at most $k-1$
divisors. Let $\ell$ be a prime divisor of $r^{\prime}$, and let
$v=v_{\ell}(r^{\prime})=v_{\ell}(r)$. We show that the map
$(R_{\ell},F_{\ell^{v}})\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda^{\prime}}^{r/\ell}(A)\times\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A)$
becomes an isomorphism after localization at $p$. This will prove the
induction step, since $r/\ell$ and $r/\ell^{v}$ have at most $k-1$ divisors
and $\gamma=(\gamma\times\gamma)\circ(R_{\ell},F_{\ell^{v}})$. Now, by
Proposition 1.1, we have the long exact sequence
$\cdots\to\mathbb{H}_{q}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\xrightarrow{N_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{R_{\ell}}\operatorname{TR}_{q-\lambda^{\prime}}^{r/\ell}(A)\xrightarrow{\partial}\mathbb{H}_{q-1}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\to\cdots$
Moreover, one readily shows that the composite map
$\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A)\xrightarrow{\epsilon}\mathbb{H}_{q}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\xrightarrow{N_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{F_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A),$
where $\epsilon$ is the edge homomorphism of the skeleton spectral sequence
(1.2), is equal to the composition $F_{\ell^{v}}V_{\ell^{v}}$ of the Frobenius
and Verschiebung maps which, in turn, is equal to the map given by
multiplication by $\ell^{v}$. Hence, after localization at $p$, $F_{\ell^{v}}$
is the projection onto a direct summand of the group
$\operatorname{TR}_{q-\lambda}^{r}(Z)$. The long exact sequence shows that the
map $(R_{\ell},F_{\ell^{v}})$ becomes an isomorphism after localization at $p$
as desired.
The maps $F_{s}$, $V_{s}$, and $R_{s}$ may also be expressed as products of
their $p$-typical analogs
$\displaystyle F=F_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n}(A;p)\to\operatorname{TR}_{q-\lambda}^{n-1}(A;p)\hskip
19.91692pt\text{(Frobenius)}\hfill$ $\displaystyle V=V_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n-1}(A;p)\to\operatorname{TR}_{q-\lambda}^{n}(A;p)\hskip
19.91692pt\text{(Verschiebung)}\hfill$ $\displaystyle R=R_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n}(A;p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(A;p)\hskip
17.92523pt\text{(restriction)}\hfill$
Suppose that $r=st$ and write $s=p^{v}s^{\prime}$ and $t=p^{n-v-1}t^{\prime}$
with $s^{\prime}$ and $t^{\prime}$ not divisible by $p$. Then there are three
commutative square diagrams
$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{F_{s}}$$\textstyle{{\prod_{j\mid
r^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{R_{s}}$$\textstyle{{\prod_{j\mid
r^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{t}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{V_{s}}$$\textstyle{{\prod_{j\mid
t^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n-v}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{\prod_{j\mid
t^{\prime}}\operatorname{TR}_{q-\lambda^{\prime\prime}}^{n-v}(A;p),}}$ (2.1)
where the maps $F_{s}^{\gamma}$, $V_{s}^{\gamma}$, and $R_{s}^{\gamma}$ are
defined as follows: The map $F_{s}^{\gamma}$ takes the factor indexed by a
divisor $j$ of $r^{\prime}$ that is divisible by $s^{\prime}$ to the factor
indexed by the divisor $j/s^{\prime}$ of $t^{\prime}$ by the map $F^{v}$ and
annihilates the remaining factors. The map $V_{s}^{\gamma}$ takes the factor
indexed by the divisor $j$ of $t^{\prime}$ to the factor indexed by the
divisor $s^{\prime}j$ of $r^{\prime}$ by the map $s^{\prime}V^{v}$. Finally,
the map $R_{s}^{\gamma}$ takes the factor indexed by a divisor $j$ of
$t^{\prime}$ to the factor indexed by the same divisor $j$ of $t^{\prime}$ by
the map $R^{v}$ and annihilates the factors indexed by divisors $j$ of
$r^{\prime}$ that do not divide $t^{\prime}$.
Let $M_{p}$ be the equivariant Moore spectrum defined by the mapping cone of
the multiplication by $p$ map on the sphere $\mathbb{T}$-spectrum. The
equivariant homotopy groups with $\mathbb{Z}/p\mathbb{Z}$-coefficients
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\lambda}\wedge T(\mathbb{Z})]_{\mathbb{T}}$
were evaluated for $p$ odd by Tsalidis [16], and for all $p$ by the first and
second authors [1]. We recall the result.
###### Theorem 2.1 ((Angeltveit-Gerhardt, Tsalidis)).
Let $p$ be a prime number, let $n$ be a positive integer, let $\lambda$ be a
finite dimensional complex $\mathbb{T}$-representation, and define
$\delta_{p}(\lambda)=(1-p)\sum_{s\geqslant
0}\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})p^{s}.$
Then the finite $\mathbb{Z}_{(p)}$-modules
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ have
the following structure:
1. (i)
For $q\geqslant 2\dim_{\mathbb{C}}(\lambda)$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
length $n$, if $q$ is congruent to $2\delta_{p}(\lambda)$ or
$2\delta_{p}(\lambda)-1$ modulo $2p^{n}$, and $n-1$, otherwise.
2. (ii)
For $2\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})\leqslant
q<2\dim_{\mathbb{C}}(\lambda^{C_{p^{s-1}}})$ with $1\leqslant s<n$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
length $n-s$, if $q$ is congruent to $2\delta_{p}(\lambda^{C_{p^{s}}})$ or
$2\delta_{p}(\lambda^{C_{p^{s}}})-1$ modulo $2p^{n-s}$, and $n-s-1$,
otherwise.
3. (iii)
For $q<2\dim_{\mathbb{C}}(\lambda^{C_{p^{n-1}}})$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ is
zero.
We show in Corollary 3 below that the groups
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ have
exponent $p$ for all prime numbers $p$.
###### Proof 2.2 (of Theorem B (i)).
By Lemma 1.2, $\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)$ is a finitely
generated abelian group, and hence, it suffices to show that it is torsion
free. We first show that the $p$-torsion subgroup is trivial. Comparing
Theorem 2.2 and Lemma 1.2, we find that for all integers $i$
$\operatorname{length}_{\mathbb{Z}_{(p)}}\\!\\!\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})-\operatorname{length}_{\mathbb{Z}_{(p)}}\\!\\!\operatorname{TR}_{2i-1-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=\operatorname{rk}_{\mathbb{Z}}\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p).$
Moreover, Lemma 1.2 shows that for every integer $i$,
$\operatorname{TR}_{2i-1-\lambda}^{n}(\mathbb{Z};p)_{(p)}$ is a finite
$p$-primary torsion group. By a Bockstein spectral sequence argument, we
conclude that $\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)_{(p)}$ is
torsion free; compare [5, Proposition 13]. This shows that the group
$\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)$ has no $p$-torsion. To see
that it has no prime to $p$ torsion, we use that, by Proposition 1, the map
$(R^{n-1-s}F^{s})\colon\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)\to\prod_{0\leqslant
s<n}\operatorname{TR}_{2i-\lambda^{(n-1-s)}}^{1}(\mathbb{Z};p)$
becomes an isomorphism after inverting $p$. Therefore, Bökstedt’s result
recalled earlier shows that also
$\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)[1/p]$ is torsion free. This
completes the proof.
###### Lemma 2.3.
Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then the restriction
map
$R\colon\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$
is surjective for every even integer $q$.
###### Proof 2.4.
We see as in the proof of Addendum 1.3 that the map of the statement is an
epimorphism, for $q\leqslant 2\dim_{\mathbb{C}}(\lambda)$. Moreover, Lemma 1.2
and Theorem B(i) show that
$\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$ is zero, for
$q>2\dim_{\mathbb{C}}(\lambda^{\prime})$ and even. The lemma follows, since
$\dim_{\mathbb{C}}(\lambda^{\prime})\leqslant\dim_{\mathbb{C}}(\lambda)$.
###### Proposition 2.
Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then in the skeleton
spectral sequence
$E_{s,t}^{2}=H_{s}(C_{p^{n-1}},\operatorname{TR}_{t-\lambda}^{1}(\mathbb{Z};p))\Rightarrow\mathbb{H}_{s+t}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p)),$
every non-zero differential $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is
supported in odd total degree.
###### Proof 2.5.
We must show that if $s+t$ is even then $d^{r}\colon E_{s,t}^{r}\to
E_{s-r,t+r-1}^{r}$ is zero. Suppose first that $s$ and $t$ are both even. Then
$E_{s,t}^{2}$ is zero unless $s=0$ and $t=2\dim_{\mathbb{C}}(\lambda)$.
Therefore, in this case, $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is
zero. Suppose next that $s$ and $t$ are both odd and that $r$ is even. Then
$E_{s,t}^{2}$ is zero unless $t>2\dim_{\mathbb{C}}(\lambda)$, and
$E_{s-r,t+r-1}^{2}$ is zero unless $t+r-1=2\dim_{\mathbb{C}}(\lambda)$. It
follows that, also in this case, $d^{r}\colon E_{s,t}^{r}\to
E_{s-r,t+r-1}^{r}$ is zero. It remains to prove that if $r$, $s$, and $t$ are
all odd then $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is zero.
To this end, we use that, for all integers $n^{\prime}\geqslant n\geqslant 1$,
the iterated Frobenius map
$F^{n^{\prime}-n}\colon\mathbb{H}_{q}(C_{p^{n^{\prime}-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))$
induces a map of skeleton spectral sequences that we write
$F^{n^{\prime}-n}\colon E_{s,t}^{r}(n^{\prime},\lambda)\to
E_{s,t}^{r}(n,\lambda).$
The map of $E^{2}$-terms is given by the transfer map in group homology and is
readily evaluated; see for instance [5, Lemma 6]. It is surjective if $s$ is
odd, and zero if is $s$ even, $t$ is odd, and $n^{\prime}-n$ is sufficiently
large. We prove by induction on $r\geqslant 2$ that, for all odd integers $s$
and $t$ and for all $n$ and $\lambda$ as in the statement, the differential
$d^{r}\colon E_{s,t}^{r}(n,\lambda)\to E_{s-r,t+r-1}^{r}(n,\lambda)$ is zero
and the Frobenius map $F\colon E_{s,t}^{r}(n+1,\lambda)\to
E_{s,t}^{r}(n,\lambda)$ surjective. The case $r=2$ was proved above. So we let
$r\geqslant 3$ and assume, inductively, that the statement has been proved for
$r-1$. Since the differential $d^{r-1}\colon E_{s,t}^{r-1}(n,\lambda)\to
E_{s-(r-1),t+r-2}^{r-1}(n,\lambda)$ is zero by the inductive hypothesis, the
canonical isomorphism
$H(E_{s+r-1,t-(r-2)}^{r-1}(n,\lambda)\xrightarrow{d^{r-1}}E_{s,t}^{r-1}(n,\lambda)\xrightarrow{d^{r-1}}E_{s-(r-1),t+r-2}^{r-1}(n,\lambda))\xrightarrow{\sim}E_{s,t}^{r}(n,\lambda)$
gives rise to a canonical surjection $\pi\colon
E_{s,t}^{r-1}(n,\lambda)\twoheadrightarrow E_{s,t}^{r}(n,\lambda)$. Moreover,
by naturality of the skeleton spectral sequence, the diagram
$\textstyle{{E_{s,t}^{r-1}(n+1,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{F}$$\textstyle{{E_{s,t}^{r}(n+1,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{{E_{s,t}^{r-1}(n,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{E_{s,t}^{r}(n,\lambda)}}$
commutes. Since the left-hand vertical map $F$ is surjective by the inductive
hypothesis, we conclude that the right-hand vertical map $F$ is surjective as
desired. It remains to prove that the differential $d^{r}\colon
E_{s,t}^{r}(n,\lambda)\to E_{s-r,t+r-1}^{r}(n,\lambda)$ is zero. The case
where $r$ is even was proved above, and in the case where $r$ is odd, we
consider the following commutative diagram:
$\textstyle{{E_{s,t}^{r}(n^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\scriptstyle{F^{n^{\prime}-n}}$$\textstyle{{E_{s-r,t+r-1}^{r}(n^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{n^{\prime}-n}}$$\textstyle{{E_{s,t}^{r}(n,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\textstyle{{E_{s-r,t+r-1}^{r}(n,\lambda)}}$
We have just proved that the left-hand vertical map is surjective. Moreover,
as recalled above, the right-hand vertical map is zero if $n^{\prime}-n$ is
sufficiently large. Indeed, $s-r$ is even and $t-r+1$ is odd. Hence, the lower
horizontal map $d^{r}$ is zero as desired. This proves the induction step and
the proposition.
###### Remark 2.6.
The proof of Proposition 2 also shows that in the Tate spectral sequence
$\hat{E}_{s,t}^{2}=\hat{H}^{-s}(C_{p^{n-1}},\operatorname{TR}_{t-\lambda}^{1}(\mathbb{Z};p))\Rightarrow\hat{\mathbb{H}}^{-s-t}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p)),$
every non-zero differential is supported in even total degree. It remains an
important problem to determine the differential structure of the two spectral
sequences.
###### Proof 2.7 (of Theorem B (ii)).
First, for $i\leqslant\dim_{\mathbb{C}}(\lambda)$, we see as in the proof of
Lemma 2.3 that the restriction map
$R\colon\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$
is an isomorphism, so the statement holds in this case. Next, for
$i=\dim_{\mathbb{C}}(\lambda)+1$, Proposition 1.1 and Lemmas 1.2 and 2.3 give
a short exact sequence
$0\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\to
0$
and the skeleton spectral sequence (1.2) shows that the left-hand group has
order $p^{n-1}$. So the statement also holds in this case. Finally, for
$i>\dim_{\mathbb{C}}(\lambda)+1$, Proposition 1.1 and Lemma 1.2 give a four-
term exact sequence
$\displaystyle 0$
$\displaystyle\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$
$\displaystyle\to\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to
0$
which shows that the orders of the four groups satisfy the equality
$\lvert\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\rvert\big{/}\lvert\operatorname{TR}_{q-1-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert=\lvert\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert\big{/}\lvert\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert.$
To evaluate the ratio on the right-hand side, we consider the skeleton
spectral sequence (1.2). We may write the ratio in question as
$\lvert\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert\big{/}\lvert\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert=\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{\infty}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{\infty}\rvert\big{)}.$
By Proposition 2, for all $r\geqslant 2$ and all $s$ and $t$ with $s+t$ is odd
we have an exact sequence
$0\to E_{s,t}^{r+1}\to E_{s,t}^{r}\xrightarrow{d^{r}}E_{s-r,t+r-1}^{r}\to
E_{s-r,t+r-1}^{r+1}\to 0.$
Hence, by induction on $r$, we find that
$\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{\infty}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{\infty}\rvert\big{)}=\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{2}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{2}\rvert\big{)}$
and the ratio on the right-hand side is readily seen to be equal to
$\lvert E_{0,q}^{2}\rvert\cdot\lvert
E_{q-2\dim_{\mathbb{C}}(\lambda),2\dim_{\mathbb{C}}(\lambda)}^{2}\rvert=(i-\dim_{\mathbb{C}}(\lambda))\cdot
p^{n-1}.$
This completes the proof.
###### Proof 2.8 (of Theorem B (iii)).
First, for $q$ odd, we use that the Verschiebung map in question is equal to
the composition of the edge homomorphism
$\epsilon\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)\to\mathbb{H}_{q}(C_{p},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))$
of the skeleton spectral sequence (1.2) and the norm map $N_{n-1}$ in the long
exact sequence
$\textstyle{{\cdots}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{q}(C_{p},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{n-1}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{n-1}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{(n-1)}}^{1}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\cdots}}$
from Proposition 1.1. Since $q$ is odd, Lemma 2.3 shows that the latter map is
injective. Hence, it will suffice to show that also the edge homomorphism is
injective, or equivalently, that in the skeleton spectral sequence, all
differentials of the form
$d^{r}\colon E_{r,q-r+1}^{r}\to E_{0,q}^{r}$
are zero. If $r$ is even, then $q-r+1$ is even and Theorem B (i) shows that
the group $E_{r,q-r+1}^{r}$ is zero. Hence, $d^{r}$ is zero in this case. If
$r$ is odd, we consider the iterated Frobenius map
$F^{v^{\prime}-v}\colon\mathbb{H}_{q}(C_{p^{v^{\prime}}},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))\to\mathbb{H}_{q}(C_{p^{v}},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p)).$
It induces a map of spectral sequences that we write
$F^{v^{\prime}-v}\colon E_{s,t}^{r}(v^{\prime},\lambda)\to
E_{s,t}^{r}(v,\lambda).$
As in the proof of Theorem B (ii), an induction on $r\geqslant 2$ shows that,
for all $v^{\prime}\geqslant v\geqslant 1$, the left-hand vertical map and the
horizontal maps in the diagram
$\textstyle{{E_{r,q-r+1}^{r}(v^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\scriptstyle{F^{v^{\prime}-v}}$$\textstyle{{E_{0,q}^{r}(v^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{v^{\prime}-v}}$$\textstyle{{E_{r,q-r+1}^{r}(v,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\textstyle{{E_{0,q}^{r}(v,\lambda)}}$
are surjective and zero, respectively. The proof of the induction step uses
that, for $v^{\prime}-v$ sufficiently large, the right-hand vertical map is
zero. This proves the statement for $q$ odd.
Finally, suppose that $q$ is even. We let $\mu$ be the direct sum of $\lambda$
and the one dimensional complex representation $\mathbb{C}(p^{n-2})$. Then [6,
Proposition 4.2] gives a long exact sequence
$\cdots\to\operatorname{TR}_{q+1-\mu}^{n}(\mathbb{Z};p)\xrightarrow{(F,-Fd)}\overset{\displaystyle{\operatorname{TR}_{q-1-\lambda}^{n-1}(\mathbb{Z};p)}}{\underset{\displaystyle{\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)}}{\oplus}}\xrightarrow{dV+V}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\xrightarrow{\iota_{*}}\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)\to\cdots$
Now, we proved in Thm B (i) that
$\operatorname{TR}_{q+1-\mu}^{n}(\mathbb{Z};p)$ and
$\operatorname{TR}_{q-1-\lambda}^{n-1}(\mathbb{Z};p)$ are finite abelian
groups while $\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$, and
$\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)$ are free abelian groups. Hence,
we obtain the exact sequence of free abelian groups
$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{{\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)}}$
which shows that for $q$ even, the Verschiebung map is the inclusion of a
direct summand as stated. This concludes the proof of Theorem B.
###### Corollary 3.
Let $n$ be a positive integer, $p$ a prime number, and $\lambda$ a finite
dimensional complex $\mathbb{T}$-representation. Then
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
exponent $p$ for every integer $q$.
###### Proof 2.9.
We first let $p=2$ and consider the coefficient long exact sequence
$\cdots\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{2}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\iota}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})\xrightarrow{\beta}\operatorname{TR}_{q-1-\lambda}^{n}(\mathbb{Z};2)\to\cdots.$
It is proved in [2, Theorem 1.1] that the composition
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})\xrightarrow{\beta}\operatorname{TR}_{q-1-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\eta}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\iota}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})$
is equal to multiplication by $2$. Now, Theorem B shows that for $q$ odd the
map $\beta$ is zero, and that for $q$ even the map $\eta$ is zero. Hence, the
group $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})$
is annihilated by multiplication by $2$ as stated. Finally, for $p$ odd, [2,
Theorem 1.1] shows that the multiplication by $p$ map on
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ is
equal to zero.
## 3 The groups $K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$
In this section, we prove Theorem A of the introduction.
###### Proposition 4.
Let $m$ and $r$ be positive integers, let $i$ be a non-negative integer, and
let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then:
1. (i)
The abelian group $\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})$
is free of rank $m$.
2. (ii)
The abelian group
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})$ is finite of
order $(mi)!(i!)^{m}$.
###### Proof 3.1.
It follows from Lemma 1.2 and Addendum 1.3 that the groups
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})$ are finitely
generated. Hence, it suffices to show that for every prime number $p$, the
$\mathbb{Z}_{(p)}$-module
$\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$ has finite
rank $m$ and the $\mathbb{Z}_{(p)}$-module
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$ has
finite length $v_{p}((mi)!(i!)^{m})$. We fix a prime number $p$ and let
$I_{p}$ be the set of positive integers not divisible by $p$. It follows from
Proposition 1 that there is a canonical isomorphism
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{j\in
I_{p}}\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{n}(\mathbb{Z};p)_{(p)},$
where, on the left-hand side, the limit ranges over the set of positive
integers ordered under division and $d=d(m,r)$, and where, on the right-hand
side, the limits range over the set of non-negative integers ordered
additively and $d=d(m,p^{n-1}j)$. Moreover, on the $j$th factor of the
product, the canonical projection
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{n}(\mathbb{Z};p)_{(p)}\to\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)}$
is an isomorphism for $q<2d(m,p^{s}j)$; see [6, Lemma 2.6]. The requirement
that $2i-2$ and $2i-1$ be strictly smaller than $2d(m,p^{s}j)$ is equivalent
to the requirement that $mi<p^{s}j$. Hence, for $q=2i-2$ or $q=2i-1$, we have
a canonical isomorphism
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$
where $s=s_{p}(m,i,j)$ is the unique integer such that
$p^{s-1}j\leqslant mi<p^{s}j.$
Now, from Theorem B (i) we find that
$\displaystyle\operatorname{rk}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\lvert\\{j\in
I_{p}\mid\text{$i=d(m,p^{n-1}j)$, for some $n\geqslant 1$}\\}\rvert$
$\displaystyle=\lvert\\{r\in\mathbb{N}\mid
i=d(m,r)\\}\rvert=\lvert\\{mi+1,mi+2,\dots,mi+m\\}\rvert=m$
which proves statement (i). Similarly, we have
$\operatorname{length}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\sum_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$
where $s=s_{p}(m,i,j)$, and Theorem B (ii) shows that the right-hand side is
equal to
$\displaystyle\sum_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\sum_{1\leqslant t\leqslant
s}\big{(}v_{p}(i-d(m,p^{t-1}j))+t-1\big{)}=\sum_{1\leqslant k\leqslant
mi}\big{(}v_{p}(i-d(m,k))+v_{p}(k)\big{)}$ $\displaystyle=m\sum_{0\leqslant
l<i}v_{p}(i-l)+\\!\\!\sum_{1\leqslant k\leqslant mi}v_{p}(k)=m\sum_{1\leqslant
k\leqslant i}v_{p}(k)+\\!\\!\sum_{1\leqslant k\leqslant
mi}v_{p}(k)=v_{p}((i!)^{m}(mi)!)$
which proves statement (ii).
###### Proposition 5.
Let $m$ and $r$ be positive integers, let $i$ be a non-negative integer, and
let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then:
1. (i)
The abelian group
$\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r/m}(\mathbb{Z})$ is free of rank
$1$.
2. (ii)
The abelian group
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r/m}(\mathbb{Z})$ is finite of
order $(i!)^{2}$.
###### Proof 3.2.
It follows from Lemma 1.2 and Addendum 1.3 that the groups
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})$ are finitely
generated. We fix a prime number $p$ and write $m=p^{v}m^{\prime}$ with
$m^{\prime}$ not divisible by $p$. Then for $q=2i-2$ and $q=2i-1$, there is a
canonical isomorphism
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)},$
where $s=s_{p}(m,i,j)$. From Theorem B (i) we find
$\displaystyle\operatorname{rk}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}=\lvert\\{j\in
m^{\prime}I_{p}\mid\text{$i=d(m,p^{n+v-1}j)$, for some $n\geqslant
1$}\\}\rvert$ $\displaystyle=\lvert\\{r\in m\mathbb{N}\mid
i=d(m,r)\\}\rvert=\lvert\\{mi+m\\}\rvert=1$
which proves statement (i). Similarly, from Theorem B (ii), we have
$\displaystyle\operatorname{length}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\sum_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\sum_{1\leqslant t\leqslant
s-v}\big{(}v_{p}(i-d(m,p^{t+v-1}j))+t-1\big{)}=\sum_{1\leqslant k\leqslant
i}\big{(}v_{p}(i-d(m,km))+v_{p}(k)\big{)}$ $\displaystyle=\sum_{0\leqslant
l<i}v_{p}(i-l)+\\!\\!\sum_{1\leqslant k\leqslant i}v_{p}(k)=2\sum_{1\leqslant
k\leqslant i}v_{p}(k)=v_{p}((i!)^{2})$
which proves statement (ii).
###### Proposition 6.
Let $m$ and $r$ be positive integers, and let $d=d(m,r)$ be the integer part
of $(r-1)/m$. Then the Verschiebung map
$V_{m}\colon\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})\to\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})$
is injective for all integers $q$, and has free abelian cokernel for all even
integers $q$.
###### Proof 3.3.
We fix a prime number $p$ and show that the Verschiebung map
$V_{m}\colon\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}\to\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$
is injective for all integers $q$, and has cokernel a free
$\mathbb{Z}_{(p)}$-module for all even integers $q$. We write
$m=p^{v}m^{\prime}$ with $m^{\prime}$ not divisible by $p$. Then for $q=2i-2$
and $q=2i-1$ the map $V_{m}$ is canonically isomorphic to the map
$m^{\prime}V^{v}\colon\prod_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\
j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)}\to\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$
where $s=s_{p}(m,i,j)$. The statement now follows from Theorem B (iii).
###### Proof 3.4 (of Theorem A).
The statement follows immediately from the long exact sequence recalled in the
introduction together with Propositions 4, 5, and 6.
## 4 The dual numbers
It follows from Theorem B that $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is a finite
abelian group of order $(2i)!$. In this section, we investigate the structure
of these groups in low degrees.
###### Theorem 4.0.
There are isomorphisms
$\displaystyle K_{2}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/2\mathbb{Z}$ $\displaystyle
K_{4}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$
$\displaystyle K_{6}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/9\mathbb{Z}\oplus\mathbb{Z}/5\mathbb{Z}$
###### Proof 4.1.
We know from Theorem B that the orders of these three groups are as stated.
Hence, it suffices to show that the $2$-primary torsion subgroup of the groups
in degree $4$ and $6$ and the $3$-primary torsion subgroup of the group in
degree $6$ are as stated.
We first consider the group in degree $4$ which is given by the short exact
sequence
$0\to\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{4}(\mathbb{Z}[x]/(x^{2}),(x))\to 0.$
The middle term in the short exact sequence decomposes $2$-locally as the
direct sum
$\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{3-\lambda_{1}}^{1}(\mathbb{Z};2)_{(2)},$
where the first and second summands on the right-hand side correspond to $j=1$
and $j=3$, respectively. Similarly, the left-hand term in the short exact
sequence decomposes $2$-locally as
$\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r/2}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{3-\lambda_{1}}^{0}(\mathbb{Z};2)_{(2)}.$
The summands corresponding to $j=3$ are both zero. Hence, the $2$-primary
torsion subgroup of $K_{4}(\mathbb{Z}[x]/(x^{2}),(x))$ is canonically
isomorphic to the cokernel of the Verschiebung map
$V\colon\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}.$
To evaluate this cokernel, we consider the following diagram
$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{3}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{2}}$$\textstyle{{\operatorname{TR}_{3}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{{\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$
where the rows are exact by [6, Proposition 4.2]. It follows from [5, Theorem
18] that this diagram is isomorphic to the diagram
$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}/2\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{(a,b)}$$\textstyle{{\mathbb{Z}/4\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}/2\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,4)}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\oplus
1}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$
where $a\in 2\mathbb{Z}/8\mathbb{Z}$ and $b\in(\mathbb{Z}/8\mathbb{Z})^{*}$.
Now, the cokernels of the middle and right-hand vertical maps are isomorphic
to $\mathbb{Z}/8\mathbb{Z}$. This shows that
$K_{4}(\mathbb{Z}[x]/(x^{2}),(x))_{(2)}$ is as stated.
We next consider the group in degree $6$ which is given by the short exact
sequence
$0\to\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{6}(\mathbb{Z}[x]/(x^{2}),(x))\to 0$
and begin by evaluating the $2$-primary torsion subgroup. The middle term in
the short exact sequence decomposes $2$-locally as the direct sum
$\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)},$
where the three summands on the right-hand side correspond to $j=1$, $j=3$,
and $j=5$, respectively. Similarly, the left-hand term in the short exact
sequence decomposes $2$-locally as
$\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r/2}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{0}(\mathbb{Z};2)_{(2)}.$
The summands corresponding to $j=5$ are both zero. Hence, the $2$-primary
torsion subgroup of $K_{6}(\mathbb{Z}[x]/(x^{2}),(x))$ is canonically
isomorphic to the direct sum of the cokernels of
$\displaystyle V$
$\displaystyle\colon\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$
$\displaystyle V$
$\displaystyle\colon\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{5-\lambda_{2}}^{2}(\mathbb{Z};2)_{(2)}.$
We show that these are isomorphic to
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and
$\mathbb{Z}/4\mathbb{Z}$, respectively. The statement for the latter cokernel
follows directly from Theorems B and 2.2. The two theorems also show that the
group $\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}$ is
isomorphic to $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and that
the group $\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$ is
isomorphic to either $\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ or
$\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. We will prove that the
latter group is isomorphic to
$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ by showing that it
contains $\mathbb{Z}/4\mathbb{Z}$ as a direct summand. To this end, we
consider the following diagram
$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{5}(C_{2},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{5}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{5}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{2}}$$\textstyle{{\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{2}}$$\textstyle{{\operatorname{TR}_{5}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}}$
where the rows, but not the columns, are exact. It follows from Theorem B that
the top middle and right-hand vertical maps $V$ are injective. Hence, also the
top left-hand vertical map $V$ is injective. Moreover, [6, Proposition 4.2]
and [5, Proposition 15] show that the bottom left-hand vertical map $F^{2}$ is
surjective. Hence, also the bottom right-hand vertical map $F^{2}$ is
surjective. The skeleton spectral sequence
$E_{s,t}^{2}=H_{s}(C_{4},\operatorname{TR}_{t-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}\Rightarrow\mathbb{H}_{s+t}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}$
shows that the middle left-hand group is an extension of
$E_{2,3}^{\infty}=\mathbb{Z}/4\mathbb{Z}$ by
$E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ and the diagram above shows that the
extension is split. It follows from [5, Lemma 6] that
$F\colon\mathbb{H}_{5}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}\to\mathbb{H}_{5}(C_{2},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}$
maps the generator of the summand $E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ to
zero. Hence, the lower left-hand vertical map $F^{2}$ in the diagram above
maps the generator of the summand $E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ to
zero. But the map $F^{2}$ is surjective, and therefore, maps a generator of
the summand $E_{3,2}^{\infty}=\mathbb{Z}/4\mathbb{Z}$ non-trivially. It
follows that $\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$
contains direct summand isomorphic to $\mathbb{Z}/4\mathbb{Z}$, and hence, is
isomorphic to $\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. This shows
that the cokernel of the upper middle vertical map $V$ is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, and hence, that
$K_{6}(\mathbb{Z}[x]/(x^{2}),(x))_{(2)}$ is as stated.
It remains to evaluate $K_{6}(\mathbb{Z}[x]/(x^{2}),(x))_{(3)}$. This group is
canonically isomorphic to the direct sum of
$\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};3)_{(3)}$ and
$\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};3)_{(3)}$. It follows from
Theorem B that the former group has order $9$ and that the latter group is
zero and from Theorem 2.2 that the former group is cyclic. This completes the
proof.
###### Theorem 4.1.
Let $p$ be an odd prime number. Then for $2i<p^{2}$ the $p$-primary torsion
subgroup of $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is isomorphic to
$(\mathbb{Z}/p\mathbb{Z})^{r_{1}}\oplus(\mathbb{Z}/p^{2}\mathbb{Z})^{r_{2}}$,
where
$(r_{1},r_{2})=\begin{cases}(0,\lfloor i/p\rfloor)&\text{if $2i+1\equiv 0$
modulo $p$}\cr(\lfloor 2i/p\rfloor-2,1)&\text{if $2i+1\equiv j$ modulo $p$
with $1\leqslant j\leqslant 2i/p$ odd}\cr(\lfloor
2i/p\rfloor,0)&\text{otherwise.}\cr\end{cases}$
Here $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
###### Proof 4.2.
After localizing at the odd prime number $p$, the short exact sequence
$0\to\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))\to 0$
induces a canonical isomorphism
$\bigoplus_{j}\operatorname{TR}_{2i-1-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)}\xrightarrow{\sim}K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))_{(p)},$
where the sum runs over integers $1\leqslant j\leqslant 2i$ coprime to both
$2$ and $p$ and $s=s_{p}(2,i,j)$ is the unique integer that satisfies
$p^{s-1}j\leqslant 2i<p^{s}j$. Since $2i<p^{2}$ we find
$s_{p}(2,i,j)=\begin{cases}2&\text{if $1\leqslant j\leqslant 2i/p$}\cr
1&\text{if $2i/p<j\leqslant 2i$.}\cr\end{cases}$
If $2i/p<j\leqslant 2i$ we have $d=(j-1)/2$, and hence
$\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{1}(\mathbb{Z};p)_{(p)}=v_{p}(i-d)=v_{p}(2i+1-j).$
The length is at most $1$ since $2i<p^{2}$. If $1\leqslant j\leqslant 2i/p$ we
have $d=(pj-1)/2$, and in this case Theorem B shows that
$\displaystyle\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{2}(\mathbb{Z};p)_{(p)}$
$\displaystyle=\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}^{\prime}}^{1}(\mathbb{Z};p)_{(p)}+v_{p}(i-d)+1$
$\displaystyle=v_{p}(2i+1-j)+v_{p}(2i+1)+1,$
where we have used that $\lambda_{d}^{\prime}=\lambda_{d^{\prime}}$ with
$d^{\prime}=(j-1)/2$. The length is at most $2$ since $2i<p^{2}$ and since $j$
is coprime to $p$. We claim that the group
$\operatorname{TR}_{2i-1-\lambda_{d}}^{2}(\mathbb{Z};p)_{(p)}$ is always
cyclic. By Theorem 2.2, the claim is equivalent to the congruence
$2i-1\not\equiv 2\delta_{p}(\lambda_{d})-1\mod 2p^{2}.$
We compute that modulo $p^{2}$
$\delta_{p}(\lambda_{d})\equiv(1-p)((pj-1)/2+(j-1)/2\cdot p)$
Hence, we have $2i-1\equiv 2\delta_{p}(\lambda_{d})-1$ modulo $2p^{2}$ if and
only if $2i+1\equiv 2pj+p^{2}$ modulo $2p^{2}$. This is possible only if
$2i+1$ is congruent to $0$ modulo $p$. If we write $2i+1=ap$ then $1\leqslant
a\leqslant p$ and $1\leqslant j<a$. Hence, $p\leqslant ap\leqslant p^{2}$ and
$2p+p^{2}\leqslant 2pj+p^{2}<2ap+p^{2}\leqslant 3p^{2}$ which implies that the
congruence $ap\equiv 2pj+p^{2}$ modulo $2p^{2}$ is equivalent to the equality
$ap+p^{2}=2pj$. But then $a+p=2j$ which contradicts that $j<a$. The claim
follows.
We now show that the integers $(r_{1},r_{2})$ are as stated. It follows from
Theorem A that
$r_{1}+2r_{2}=v_{p}((2i)!)=\lfloor 2i/p\rfloor.$
In the case where $2i+1$ is congruent to $0$ modulo $p$, we proved above that
$r_{1}=0$. If we write $2i+1=ap$ then $a$ is odd and
$r_{2}=\lfloor 2i/p\rfloor/2=(a-1)/2=\lfloor(a-1)/2+(p-1)/2p\rfloor=\lfloor
i/p\rfloor$
as stated. In the case where $2i+1\equiv j$ modulo $p$ with $1\leqslant
j\leqslant 2i/p$ odd, we proved above that $r_{2}=1$. Hence, $r_{1}=\lfloor
2i/p\rfloor-2$. Finally, in the remaining case, $r_{2}=0$, and hence,
$r_{1}=\lfloor 2i/p\rfloor$. This completes the proof.
###### Example 4.3.
Let $p$ be an odd prime number. We spell out the statement of Theorem 4.2, for
$i\leqslant p+1$. The $p$-primary torsion subgroup of
$K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is zero for $i\leqslant(p-1)/2$, is cyclic
of order $p$ for $(p+1)/2\leqslant i<p$, and is cyclic of order $p^{2}$ for
$i=p$. The structure of the $p$-primary torsion subgroup of
$K_{2p+2}(\mathbb{Z}[x]/(x^{2}),(x))$ depends on the odd prime $p$. It is
cyclic of order $9$ if $p=3$, and the direct sum of two cyclic groups of order
$p$ if $p\geqslant 5$.
## References
* [1] V. Angeltveit and T. Gerhardt, _$RO(S^{1})$ -graded $\operatorname{TR}$-groups of $\mathbb{F}_{p}$, $\mathbb{Z}$, and $\ell$_, arXiv:0811.1313.
* [2] S. Araki and H. Toda, _Multiplicative structures in $\operatorname{mod}q$ cohomology theories I_, Osaka J. Math. 2 (1965), 71–115.
* [3] M. Bökstedt, _Topological Hochschild homology of $\mathbb{F}_{p}$ and $\mathbb{Z}$_, Preprint, Bielefeld University, 1985.
* [4] M. Bökstedt, W.-C. Hsiang, and I. Madsen, _The cyclotomic trace and algebraic $K$-theory of spaces_, Invent. Math. 111 (1993), 465–540.
* [5] L. Hesselholt, _On the Whitehead spectrum of the circle_ , arXiv:0710.2823.
* [6] , _The tower of $K$-theory of truncated polynomial algebras_, J. Topol. 1 (2008), 87–114.
* [7] L. Hesselholt and I. Madsen, _Cyclic polytopes and the $K$-theory of truncated polynomial algebras_, Invent. Math. 130 (1997), 73–97.
* [8] , _On the $K$-theory of finite algebras over Witt vectors of perfect fields_, Topology 36 (1997), 29–102.
* [9] A. Lindenstrauss and I. Madsen, _Topological Hochschild homology of number rings_ , Trans. Amer. Math. Soc. 352 (2000), 2179–2204.
* [10] M. A. Mandell and J. P. May, _Equivariant orthogonal spectra and $S$-modules_, Mem. Amer. Math. Soc., vol. 159, Amer. Math. Soc., Providence, RI, 2002.
* [11] R. McCarthy, _Relative algebraic $K$-theory and topological cyclic homology_, Acta Math. 179 (1997), 197–222.
* [12] L. Roberts, _$K_{2}$ of some truncated polynomial rings. With a section written jointly with S. Geller_, Ring Theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978), Lecture Notes in Math., vol. 734, Springer-Verlag, New York, 1979, pp. 249–278.
* [13] C. Soulé, _Rational $K$-theory of the dual numbers of a ring of algebraic integers_, Algebraic $K$-theory (Evanston, Ill., 1980), Lecture Notes in Math., Springer-Verlag, New York, 1981, pp. 402–408.
* [14] R. E. Staffeldt, _Rational algebraic $K$-theory of certain truncated polynomial rings_, Prop. Amer. Math. Soc. 95 (1985), 191–198.
* [15] T. tom Dieck, _Orbittypen und äquivariante Homologie I_ , Arch. Math. (Basel) 26 (1975), 650–662.
* [16] S. Tsalidis, _On the algebraic $K$-theory of truncated polynomial rings_, Aberdeen Topology Center Preprint Series 9, 2002.
###### Acknowledgements.
This paper was written in part while the third author visited the Hausdorff
Center for Mathematics at the University of Bonn. He would like to thank the
center and in particular Stefan Schwede and Christian Ausoni for their
hospitality and support. Finally, we thank an anonymous referee for helpful
comments.
Vigleik Angeltveit
Department of Mathematics
The University of Chicago
Chicago, Illinois
USA
Teena Gerhardt
Department of Mathematics
Indiana University
Bloomington, Indiana
USA
Lars Hesselholt
Graduate School of Mathematics
Nagoya University
Chikusa-ku
Nagoya, Japan Japan
|
arxiv-papers
| 2008-09-21T01:13:02
|
2024-09-04T02:48:57.863196
|
{
"license": "Public Domain",
"authors": "Vigleik Angeltveit, Teena Gerhardt, Lars Hesselholt",
"submitter": "Lars Hesselholt",
"url": "https://arxiv.org/abs/0809.3544"
}
|
0809.3594
|
# Lepton flavor violating Higgs decays and unparticle physics
E. O. Iltan,
Physics Department, Middle East Technical University
Ankara, Turkey
E-mail address: eiltan@newton.physics.metu.edu.tr
###### Abstract
We predict the branching ratios of the lepton flavor violating Higgs decays
$H^{0}\rightarrow e^{\pm}\mu^{\pm}$, $H^{0}\rightarrow e^{\pm}\tau^{\pm}$ and
$H^{0}\rightarrow\mu^{\pm}\tau^{\pm}$ in the case that the lepton flavor
violation is carried by the scalar unparticle mediation. We observe that their
branching ratios are strongly sensitive to the unparticle scaling dimension
and they can reach to the values of the order of $10^{-4}$, for the heavy
lepton flavor case and for the small values of the scaling dimension.
The hunt of the Higgs boson $H^{0}$ in Large Hadron Collider (LHC) is one of
the main goals of physicists to test the standard model (SM), to get strong
information about the mechanism of the electroweak symmetry breaking, the
Higgs mass and to determine the scale of the new physics beyond. From
theoretical point of view, the couplings of Higgs boson with the fundamental
particles are well defined and the branching ratios (BRs) of its various
decays have been estimated as a function of the Higgs boson mass. There are
predictions on the Higgs mass limits from coupling to $Z/W_{\pm}$
$m_{H^{0}}>114.4\,CL\%95$ and, indirect one, from electroweak analysis
$m_{H^{0}}=129^{+74}_{-49}$ [1] ($m_{H^{0}}=114^{+69}_{-45}$ [2]).
The present work is devoted to the analysis of the lepton flavor violating
(LFV) Higgs boson decays in an appropriate range, $110-150\,(GeV)$, of the
Higgs boson mass. There are various analysis done on LFV Higgs boson decays in
the literature. In [3, 4] $H^{0}\rightarrow\tau\mu$ decay has been studied in
the framework of the 2HDM. In [3], large $BR$, of the order of magnitude of
$0.1-0.01$, has been obtained and in [4], the $BR$ was obtained in the
interval $0.001-0.01$ for the Higgs mass range $100-160\,(GeV)$, for the LFV
parameter $\lambda_{\mu\tau}=1$. [5] is devoted to the observable CP violating
asymmetries in the lepton flavor (LF) changing $H^{0}$ decays with $BR$s of
the order of $10^{-6}-10^{-5}$. The LFV $H^{0}\rightarrow l_{i}l_{j}$ decay
has been studied also in [6], in the framework of the two Higgs doublet model
type III. In these works the LF violation is carried by the lepton-lepton-new
Higgs boson couplings which are free parameters of the model used. In our
analysis we consider that the LF violation is carried by the scalar unparticle
(U)-lepton-lepton vertex and the scalar unparticle appears in the internal
line in the loop.
The unparticle idea, which is based on the interaction of the SM and the
ultraviolet sector, having non-trivial infrared fixed point at high energy
level, is introduced by Georgi [7, 8]. Georgi considers that the ultraviolet
sector comes out as new degrees of freedom, called unparticles, being massless
and having non integral scaling dimension $d_{u}$, around, $\Lambda_{U}\sim
1\,TeV$. The interactions of unparticles with the SM fields in the low energy
level is defined by the effective lagrangian
${\cal{L}}_{eff}\sim\frac{\eta}{\Lambda_{U}^{d_{u}+d_{SM}-n}}\,O_{SM}\,O_{U}\,,$
(1)
where $O_{U}$ is the unparticle operator, the parameter $\eta$ is related to
the energy scale of ultraviolet sector, the low energy one and the matching
coefficient [7, 8, 9] and $n$ is the space-time dimension.
In literature, the unparticle effect in the processes, which are induced at
least in one loop level, is studied in various works [11]-[23]. The process we
study exists at least in one loop level and the effective interaction
lagrangian, which drives the LFV decays in the low energy effective theory,
reads
$\displaystyle{\cal{L}}_{1}=\frac{1}{\Lambda_{U}^{du-1}}\Big{(}\lambda_{ij}^{S}\,\bar{l}_{i}\,l_{j}+\lambda_{ij}^{P}\,\bar{l}_{i}\,i\gamma_{5}\,l_{j}\Big{)}\,O_{U}\,,$
(2)
where $l$ is the lepton field and $\lambda_{ij}^{S}$ ($\lambda_{ij}^{P}$) is
the scalar (pseudoscalar) coupling. Notice that we consider the appropriate
operators with the lowest possible dimension in order to obtain the LFV
decays111The operators with the lowest possible dimension are chosen since
they have the most powerful effect in the low energy effective theory (see for
example [24])..
The $H^{0}\rightarrow l_{1}^{-}\,l_{2}^{+}$ decay (see Fig.1) exists at least
in one loop with the help of the scalar unparticle propagator, which is
obtained by using the scale invariance [8, 10]:
$\displaystyle\\!\\!\\!\int\,d^{4}x\,e^{ipx}\,<0|T\Big{(}O_{U}(x)\,O_{U}(0)\Big{)}0>=i\frac{A_{d_{u}}}{2\,\pi}\,\int_{0}^{\infty}\,ds\,\frac{s^{d_{u}-2}}{p^{2}-s+i\epsilon}=i\,\frac{A_{d_{u}}}{2\,sin\,(d_{u}\pi)}\,(-p^{2}-i\epsilon)^{d_{u}-2},$
(3)
with the factor $A_{d_{u}}$
$\displaystyle
A_{d_{u}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{u}}}\,\frac{\Gamma(d_{u}+\frac{1}{2})}{\Gamma(d_{u}-1)\,\Gamma(2\,d_{u})}\,.$
(4)
The function $\frac{1}{(-p^{2}-i\epsilon)^{2-d_{u}}}$ in eq. (3) becomes
$\displaystyle\frac{1}{(-p^{2}-i\epsilon)^{2-d_{u}}}\rightarrow\frac{e^{-i\,d_{u}\,\pi}}{(p^{2})^{2-d_{u}}}\,,$
(5)
for $p^{2}>0$ and a non-trivial phase appears as a result of non-integral
scaling dimension.
Now, we present the matrix element square of the LFV $H^{0}$ decay (see Fig. 1
for the possible self energy and vertex diagrams):
$\displaystyle|M|^{2}=2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}+m_{l_{2}^{+}})^{2}\Big{)}\,|A|^{2}+2\Big{(}m_{H^{0}}^{2}-(m_{l_{1}^{-}}-m_{l_{2}^{+}})^{2}\Big{)}\,|A^{\prime}|^{2}\,,$
(6)
where
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{S}\,,$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\,dx\,f_{self}^{\prime\,S}+\int^{1}_{0}\,dx\,\int^{1-x}_{0}\,dy\,f_{vert}^{\prime\,S}\,,$
and the explicit expressions of $f_{self}^{S}$, $f_{self}^{\prime\,S}$,
$f_{vert}^{S}$, $f_{vert}^{\prime\,S}$ read
$\displaystyle f_{self}^{S}$ $\displaystyle=$
$\displaystyle\frac{-i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}-m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{P}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}-m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{self}^{\prime\,S}$
$\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,(1-x)^{1-d_{u}}}{16\,\pi^{2}\,\Big{(}m_{l_{2}^{+}}+m_{l_{1}^{-}}\Big{)}\,(1-d_{u})}\,\sum_{i=1}^{3}\,\Big{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{l_{1}^{-}}\,m_{l_{2}^{+}}\,(1-x)$
$\displaystyle\times$ $\displaystyle\Big{(}L_{self}^{d_{u}-1}-L_{self}^{\prime
d_{u}-1}\Big{)}-(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S}-\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P})\,m_{i}\,\Big{(}m_{l_{2}^{+}}\,L_{self}^{d_{u}-1}+m_{l_{1}^{-}}\,L_{self}^{\prime
d_{u}-1}\Big{)}\Big{\\}}\,,$ $\displaystyle f_{vert}^{S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
$\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y-m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}-m_{i}^{2}\Big{\\}}$
$\displaystyle-$
$\displaystyle(\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(2\,y-1)\Big{)}\Bigg{\\}}\,,$
$\displaystyle f_{vert}^{\prime\,S}$ $\displaystyle=$
$\displaystyle\frac{i\,c_{1}\,m_{i}\,(1-x-y)^{1-d_{u}}}{16\,\pi^{2}}\,\sum_{i=1}^{3}\,\frac{1}{\,L_{vert}^{2-d_{u}}}\,\Bigg{\\{}(\lambda_{il_{1}}^{S}\,\lambda_{il_{2}}^{P}-\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,\Big{\\{}(1-x-y)$
(8) $\displaystyle\times$
$\displaystyle\Bigg{(}m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y+m_{l_{2}^{+}}\,m_{l_{1}^{-}}\Bigg{)}+x\,y\,m_{H^{0}}^{2}-\frac{2\,L_{vert}}{1-d_{u}}\Bigg{)}-m_{i}^{2}\Big{\\}}$
$\displaystyle+$
$\displaystyle(\lambda_{il_{1}}^{S}\lambda_{il_{2}}^{P}+\lambda_{il_{1}}^{P}\,\lambda_{il_{2}}^{S})\,m_{i}\,\Big{(}m_{l_{1}^{-}}\,(2\,x-1)+m_{l_{2}^{+}}\,(1-2\,y)\Big{)}\Bigg{\\}}\,,$
with
$\displaystyle L_{self}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{1}^{-}}^{2}\,(1-x)-m_{i}^{2}\Big{)}\,,$ $\displaystyle
L_{self}^{\prime}$ $\displaystyle=$ $\displaystyle
x\,\Big{(}m_{l_{2}^{+}}^{2}\,(1-x)-m_{i}^{2}\Big{)}\,,$ $\displaystyle
L_{vert}$ $\displaystyle=$
$\displaystyle(m_{l_{1}^{-}}^{2}\,x+m_{l_{2}^{+}}^{2}\,y)\,(1-x-y)-m_{i}^{2}\,(x+y)+m_{H^{0}}^{2}\,x\,y\,,$
(9)
and
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{g\,A_{d_{u}}}{4\,m_{W}\,sin\,(d_{u}\pi)\,\Lambda_{u}^{2\,(d_{u}-1)}}\,.$
(10)
In eq. (8), the scalar and pseudoscalar couplings $\lambda_{il_{1(2)}}^{S,P}$
represent the effective interaction between the internal lepton $i$,
($i=e,\mu,\tau$) and the outgoing $l_{1}^{-}\,(l_{2}^{+})$ lepton (anti
lepton). Finally, the BR for $H^{0}\rightarrow l_{1}^{-}\,l_{2}^{+}$ decay can
be obtained by using the matrix element square as
$\displaystyle BR(H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+})=\frac{1}{16\,\pi\,m_{H^{0}}}\,\frac{|M|^{2}}{\Gamma_{H^{0}}}\,,$
(11)
with the Higgs total decay width $\Gamma_{H^{0}}$. In the numerical analysis,
we consider the BR due to the production of sum of charged states, namely,
$\displaystyle BR(H^{0}\rightarrow
l_{1}^{\pm}\,l_{2}^{\pm})=\frac{\Gamma(H^{0}\rightarrow(\bar{l}_{1}\,l_{2}+\bar{l}_{2}\,l_{1}))}{\Gamma_{H^{0}}}\,.$
(12)
Discussion
In this section, we analyze the BRs of the LFV $H^{0}\rightarrow
l_{1}^{-}l_{2}^{+}$ decays with the assumption that the flavor violation is
induced by the scalar unparticle mediation. The U\- lepton-lepton vertex
drives the LF violation and the decays under consideration exist in the loop
level, in the effective theory. In the scenario studied there are number of
free parameters, namely, the scaling dimension of the scalar unparticle, the
couplings, the energy scale $\Lambda_{u}$, the Higgs mass and its total decay
width. These parameters should be restricted by using the current experimental
limits and the mathematical considerations. Here we choose the scaling
dimension in the range $1<d_{u}<2$, $d_{u}>1$ not to face with the non-
integrable singularity problem in the decay rate [8] and $d_{u}<2$ to obtain
convergent momentum integrals [13]. For the U\- lepton-lepton couplings
$\lambda^{S(P)}_{ij}$222We consider that the scalar $\lambda^{S}_{ij}$ and
pseudo scalar $\lambda^{P}_{ij}$ couplings have the same magnitude, namely
$\lambda^{S}_{ij}=\lambda^{P}_{ij}=\lambda_{ij}$. we consider two different
scenarios:
* •
the diagonal couplings $\lambda_{ii}$ respects the lepton family the
hierarchy, $\lambda_{\tau\tau}>\lambda_{\mu\mu}>\lambda_{ee}$, and the off-
diagonal couplings, $\lambda_{ij},i\neq j$ are family blind and universal.
Furthermore, we take the off diagonal couplings as,
$\lambda_{ij}=\kappa\lambda_{ee}$ with $\kappa<1$. In our numerical
calculations, we choose $\kappa=0.5$.
* •
the diagonal $\lambda_{ii}=\lambda_{0}$ and off diagonal
$\lambda_{ij}=\kappa\lambda_{0}$ couplings are family blind with $\kappa=0.5$.
The Higgs mass and its total decay width are other parameters existing in the
numerical calculations. The Higgs mass should lie in a certain range if the SM
is an acceptable theory. From the theoretical point of view, not to face with
the unitarity problem (the instability of the Higgs potential), one considers
the upper (lower) bound as $1.0\,TeV$ $(0.1\,TeV)$ [25]. On the other hand,
electroweak measurements results in the prediction of the Higgs mass as
$m_{H^{0}}=129^{+74}_{-49}$ [1], which is in the range of theoretical limits.
In our numerical calculation we choose the values $m_{H^{0}}=110\,(GeV)$,
$m_{H^{0}}=120\,(GeV)$ and $m_{H^{0}}=150\,(GeV)$ to observe the Higgs mass
dependence of the BRs of the LFV decays under consideration. The total Higgs
decay width is estimated by using the possible decays for the considered Higgs
mass. The light Higgs boson, $m_{H^{0}}\leq 130\,GeV$, mainly decays into
$b\bar{b}$ pair [26, 27]. However, its detection is difficult due to the QCD
background and the $t\bar{t}H^{0}$ channel, where the Higgs boson decays to
$b\bar{b}$, is the most promising one [28]. For a heavier Higgs boson
$m_{H^{0}}\sim 180\,GeV$, the suitable production exist via gluon fusion and
the leading decay mode is $H^{0}\rightarrow WW\rightarrow
l^{+}l^{\prime-}\nu_{l}\nu_{l^{\prime}}$ [29, 30, 31].
Notice that for the energy scale $\Lambda_{u}$ we take $\Lambda_{u}=10\,(TeV)$
and throughout our calculations we use the input values given in Table (1).
Parameter | Value
---|---
$m_{e}$ | $0.0005$ (GeV)
$m_{\mu}$ | $0.106$ (GeV)
$m_{\tau}$ | $1.780$ (GeV)
$\Gamma(H^{0})|_{m_{H^{0}}=110\,GeV}$ | $0.0026$ (GeV)
$\Gamma(H^{0})|_{m_{H^{0}}=120\,GeV}$ | $0.0029$ (GeV)
$\Gamma(H^{0})|_{m_{H^{0}}=150\,GeV}$ | $0.015$ (GeV)
$G_{F}$ | $1.1663710^{-5}(GeV^{-2})$
Table 1: The values of the input parameters used in the numerical
calculations.
In Fig.2, we present the BR $(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with
respect to the scale parameter $d_{u}$, for the couplings
$\lambda_{ee}=0.1\,\lambda_{\mu\mu}=0.01\,\lambda_{\tau\tau}$. Here, the lower
(upper) solid-dashed line represents the BR for $\lambda_{\tau\tau}=10\,(50)$
and $m_{H^{0}}=110-150\,(GeV)$ 333The BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ for $m_{H^{0}}=120\,(GeV)$ is slightly
smaller than the one for $m_{H^{0}}=110\,(GeV)$ and the difference enhances
with the increasing values of the parameter $d_{u}$.. The BR is strongly
sensitive to the scale $d_{u}$ and, it reaches to the values of the order of
$10^{-6}$ for strong couplings $\lambda_{ij}$, $d_{u}\sim 1.1$, especially,
for the light Higgs boson case. Fig.3 is devoted to the the BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ with respect to the flavor blind
coupling $\lambda_{0}$. Here, the lower (intermediate, upper) solid-dashed
line represents the BR for $d_{u}=1.3$ ($d_{u}=1.2$, $d_{u}=1.1$) and
$m_{H^{0}}=110-150\,(GeV)$. This figure shows that the BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ enhances considerably even for the
large values of the scale parameter $d_{u}$, in the case that the couplings
are flavor blind.
Fig.4, represents the BR $(H^{0}\rightarrow\tau^{\pm}\,e^{\pm})$ with respect
to the scale parameter $d_{u}$, for the couplings
$\lambda_{ee}=0.1\,\lambda_{\mu\mu}=0.01\,\lambda_{\tau\tau}$. Here, the lower
(intermediate, upper) solid-dashed line represents the BR for
$\lambda_{\tau\tau}=1.0\,(10,50)$ and $m_{H^{0}}=110-150\,(GeV)$. The BR
reaches to the values of the order of $10^{-6}$ even for weak couplings
$\lambda_{ij}$, $d_{u}\sim 1.1$. Fig.5 shows the BR
$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm})$ with respect to the coupling
$\lambda_{0}$. Here, the lower (intermediate, upper) solid-dashed line
represents the BR for $d_{u}=1.3$ ($d_{u}=1.2$, $d_{u}=1.1$) and
$m_{H^{0}}=110-150\,(GeV)$. We observe that the BR
$(H^{0}\rightarrow\tau^{\pm}\,e^{\pm})$ reaches to the values of the order of
$10^{-4}$ even for weak coupling $\lambda_{0}\sim 1.0$ in the case that the
couplings are flavor blind.
Finally, we analyze the the BR $(H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm})$ in
Figs.6 and 7. Fig.6 is devoted to the BR
$(H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm})$ with respect to the scale parameter
$d_{u}$, for the couplings
$\lambda_{ee}=0.1\,\lambda_{\mu\mu}=0.01\,\lambda_{\tau\tau}$. Here, the lower
(intermediate, upper) solid-dashed line represents the BR for
$\lambda_{\tau\tau}=1.0\,(10,50)$ and $m_{H^{0}}=110-150\,(GeV)$. The BR
reaches to the values of the order of $10^{-6}$ even for weak couplings
$\lambda_{ij}$ and $d_{u}\sim 1.1$, similar to the
$(H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm})$ decay. Fig.7 shows the BR
$(H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm})$ with respect to the coupling
$\lambda_{0}$. Here, the lower (intermediate, upper) solid-dashed line
represents the BR for $d_{u}=1.3$ ($d_{u}=1.2$, $d_{u}=1.1$) and
$m_{H^{0}}=110-150\,(GeV)$. It is observed that the BR
$(H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm})$ can get the values of the order of
$10^{-4}$ for the weak coupling $\lambda_{0}\sim 1.0$.
As a summary, the LFV decays of the Higgs boson $H^{0}$ are strongly sensitive
to the unparticle scaling dimension and, for its small values $d_{u}<1.1$, the
BRs enhance considerably, especially for heavy lepton output. In the case that
the U-lepton-lepton couplings are flavor blind, the BRs of the decays studied
reach to the values of the order of $10^{-4}$ even for weak couplings. The
possible production of the Higgs boson $H^{0}$ in LHC would stimulate one to
study its LFV decays and the near future experimental results would be
instructive to test the new physics which drives the flavor violation, here is
the unparticle physics.
## References
* [1] C. Amsler et al. (Particle Data Group),Phys. Lett. B667, 1 (2008).
* [2] LEP Electroweak Working Group 2004-08.
* [3] U. Cotti, L. Diaz-Cruz, C. Pagliarone, E. Vataga, hep-ph/0111236 (2001).
* [4] T. Han, D. Marfatia, Phys. Rev. Lett. D86, 1442 (2001).
* [5] J. G. Koerner,A. Pilaftsis, K. Schilcher, Phys. Rev. D47, 1080 (1993).
* [6] K. A. Assamagan, A. Deandrea, P.A. Delsart, Phys. Rev. D67 035001 (2003).
* [7] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007).
* [8] H. Georgi, Phys. Lett. B650, 275 (2007).
* [9] R. Zwicky, Phys. Rev. D77, 036004 (2008).
* [10] K. Cheung, W. Y. Keung, T. C. Yuan, Phys. Rev. Lett. 99, 051803 (2007).
* [11] A. Lenz, Phys. Rev. D76, 065006 (2007).
* [12] C. D. Lu, W. Wang, Y. M. Wang, Phys.Rev. D76, 077701 (2007).
* [13] Y. Liao, Phys. Rev. D76, 056006 (2007).
* [14] K. Cheung, W. Y. Keung, T. C. Yuan, Phys. Rev. D76, 055003 (2007).
* [15] D. Choudhury, D. K. Ghosh, hep-ph/0707.2074 (2007).
* [16] G. J. Ding, M. L. Yan, Phys. Rev. D77, 014005 (2008).
* [17] Y. Liao, hep-ph/0708.3327 (2007).
* [18] K. Cheung, T. W. Kephart, W. Y. Keung, T. C. Yuan, hep-ph/0801.1762 (2008).
* [19] E. O. Iltan, hep-ph/0710.2677 (2007).
* [20] E. O. Iltan, Eur. Phys. J C56 ,105 (2008).
* [21] E. O. Iltan, hep-ph/0804.2456 7 (2008).
* [22] X. G. He, L. Tsai, JHEP 0806, 074 (2008).
* [23] T.M. Aliev, O. Cakir, K.O. Ozansoy, hep-ph/0809.2327 (2008).
* [24] S. L. Chen, X. G. He, Phys. Rev. D76, 091702 (2007).
* [25] . K. G. Hagiawara, Particle Data Group Collaboration, Phys. Rev. D66, 010001 (2002).
* [26] A. Djouadi, J. Kalinowski, M. Spira, Comput. Phys. Commun. 108, 56 (1998).
* [27] M. Spira, P. Zerwas, hep-ph/ 9803257.
* [28] V. Drollinger, T. Muller, D. Denegri, hep-ph/0111312.
* [29] M. Carena, J. S. Conway, H. E. Haber, J. D. Hobbs, et. al., Physics at Run II: Supersymmery/Higgs workshop, hep-ph/0010338 (2000).
* [30] M. Dittmar, H. K. Dreiner, hep-ph/9703401 (1997).
* [31] M. Dittmar, H. K. Dreiner, Phys. Rev. D55, 167 (1997).
Figure 1: One loop diagrams contribute to $H^{0}\rightarrow
l_{1}^{-}\,l_{2}^{+}$ decay with scalar unparticle mediator. Solid line
represents the lepton field: $i$ represents the internal lepton, $l_{1}^{-}$
($l_{2}^{+}$) outgoing lepton (anti lepton), dashed line the Higgs field,
double dashed line the unparticle field.
Figure 2: The scale parameter $d_{u}$ dependence of the BR
$(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ for $\Lambda_{u}=10\,TeV$, the
couplings $\lambda_{ee}=0.1\,\lambda_{\mu\mu}=0.01\,\lambda_{\tau\tau}$. Here,
the lower (upper) solid-dashed line represents the BR for
$\lambda_{\tau\tau}=10\,(50)$ and $m_{H^{0}}=110-150\,(GeV)$. Figure 3:
$\lambda_{0}$ dependence of the BR $(H^{0}\rightarrow\mu^{\pm}\,e^{\pm})$ for
$\Lambda_{u}=10\,TeV$. Here, the lower (intermediate, upper) solid-dashed line
represents the BR for $d_{u}=1.3$ ($d_{u}=1.2$, $d_{u}=1.1$) and
$m_{H^{0}}=110-150\,(GeV)$. Figure 4: The scale parameter $d_{u}$ dependence
of the BR $(H^{0}\rightarrow\tau^{\pm}\,e^{\pm})$ for $\Lambda_{u}=10\,TeV$,
the couplings $\lambda_{ee}=0.1\,\lambda_{\mu\mu}=0.01\,\lambda_{\tau\tau}$.
Here, the lower (intermediate, upper) solid-dashed line represents the BR for
$\lambda_{\tau\tau}=1.0\,(10,50)$ and $m_{H^{0}}=110-150\,(GeV)$. Figure 5:
$\lambda_{0}$ dependence of the BR $(H^{0}\rightarrow\tau^{\pm}\,e^{\pm})$ for
$\Lambda_{u}=10\,TeV$. Here, the lower (intermediate, upper) solid-dashed line
represents the BR for $d_{u}=1.3$ ($d_{u}=1.2$, $d_{u}=1.1$) and
$m_{H^{0}}=110-150\,(GeV)$. Figure 6: The same as Fig.4 but for
$H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay. Figure 7: The same as Fig.5 but
for $H^{0}\rightarrow\tau^{\pm}\,\mu^{\pm}$ decay.
|
arxiv-papers
| 2008-09-21T17:15:22
|
2024-09-04T02:48:57.869508
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Iltan",
"submitter": "Erhan Iltan",
"url": "https://arxiv.org/abs/0809.3594"
}
|
0809.3620
|
# Spectral hardness evolution characteristics of tracking Gamma-ray Burst
pulses
Z. Y. Peng Department of Physics, Yunnan Normal University, Kunming 650092,
P. R. China pzy@ynao.ac.cn L. Ma∗ Department of Physics, Yunnan Normal
University, Kunming 650092, P. R. China Corresponding author,
astromali@126.com R. J. Lu Physical Science and Technology College, Guangxi
University, Nanning, Guangxi 530004, P. R. China L. M. Fang Department of
Physics, Guangdong Institute of Education, Guangzhou 510303, P. R. China Y.
Y. Bao Department of Physics, Yuxi Normal College, Yuxi 653100, P. R. China
Y. Yin Department of Physics, Yunnan Normal University, Kunming 650092, P. R.
China
###### Abstract
Employing a sample presented by Kaneko et al. (2006) and Kocevski et al.
(2003), we select 42 individual tracking pulses (here we defined tracking as
the cases in which the hardness follows the same pattern as the flux or count
rate time profile) within 36 Gamma-ray Bursts (GRBs) containing 527 time-
resolved spectra and investigate the spectral hardness, $E_{peak}$ (where
$E_{peak}$ is the maximum of the $\nu F_{\nu}$ spectrum), evolutionary
characteristics. The evolution of these pulses follow soft-to-hard-to-soft
(the phase of soft-to-hard and hard-to-soft are denoted by rise phase and
decay phase, respectively) with time. It is found that the overall
characteristics of $E_{peak}$ of our selected sample are: 1) the $E_{peak}$
evolution in the rise phase always start on the high state (the values of
$E_{peak}$ are always higher than 50 keV); 2) the spectra of rise phase
clearly start at higher energy (the median of $E_{peak}$ are about 300 keV),
whereas the spectra of decay phase end at much lower energy (the median of
$E_{peak}$ are about 200 keV); 3) the spectra of rise phase are harder than
that of the decay phase and the duration of rise phase are much shorter than
that of decay phase as well. In other words, for a complete pulse the initial
$E_{peak}$ is higher than the final $E_{peak}$ and the duration of initial
phase (rise phase) are much shorter than the final phase (decay phase). This
results are in good agreement with the predictions of Lu et al. (2007) and
current popular view on the production of GRBs. We argue that the spectral
evolution of tracking pulses may be relate to both kinematic and dynamic
process even if we currently can not provide further evidences to distinguish
which one is dominant. Moreover, our statistical results give some witnesses
to constrain the current GRB model.
###### keywords:
gamma-rays bursts; method: statistics; 98.70.Rz; 02.50.-r
## 1 Introduction
The origin of Gamma-ray bursts (GRBs) is still unclear even though much
progress has been made, especially the recent launch of Swift. The spectra of
GRB provide the most direct information about the emission process involved in
these enigmatic events. Early studies of burst spectra showed that they vary
within a given events as well as from event to event. Since the observed
spectra reflect the energy content and particle distributions within the
source’s emitting region, spectral variations are crucial diagnostics of
underlying physical processes within a burst, and may be a discriminant
between emission mechanisms as well.
Many authors have studied the spectral evolution since the discovery of GRBs.
These studies mainly focused on the “hardness” of bursts, measured either by
the ratio of counts in different energy channels or by more physical
variables, such as the peak energy $E_{peak}$, which is the maximum of the
$\nu F_{\nu}$ spectrum. The evolution has been studied over both the entire
burst, giving the overall behavior, and the individual pulse structures,
enabling us to better understand the physical mechanisms of the GRB prompt
emission process.
Golenetskii et al. (1983) compared two-channel data covering $\sim$ 40-700 keV
with 0.5 s time resolution from five bursts observed by the KONUS detector on
Venera 11 and Venera 12 and found that burst intensities and spectral hardness
were correlated, i.e. when burst intensity increased, the spectrum hardened.
Norris et al. (1986), however, found a hard-to-soft spectral evolution trend
across 10 bursts observed by the Gamma Ray Spectrometer (GRS) and the Hard
X-Ray Burst Spectrometer (HXRBS) on Solar Maximum Mission using hardness
ratio. Band et al. (1992) analyzed 9 bursts observed by BATSE SDs, confirming
the hard-to-soft spectral evolution. Similar result are found by many authors
(e.g. Bhat et al. 1994, Band 1997, Share & Matz, 1998, Preece et al. 1998).
Kargatis et al. (1994) studied the spectral evolution of 16 GRBs detected by
Franco-Soviet SIGNE. They found that there is no single characteristic of
spectral evolution: they saw hard-to-soft, soft-to-hard, luminosity-hardness
tracking, and chaotic evolution. In the Swift era, the spectral evolution is
also very ubiquitous. For example, focusing on GRBs 061121, 060614, and
060124, Butler & Kocevski (2007) found that the spectral evolution inferred
from fitting instead models used to fit GRBs demonstrates a common evolution-a
power-law hardness-intensity correlation and hard-to-soft evolution for GRBs
and the early X-ray afterglows and X-ray flares.
Qin et al. (2006) investigated the evolution of spectral hardness ratio of
counts in different energy channels and found the evolutionary curve of the
pure hardness ratio (when the background count is not included) would peak at
the very beginning of the curve, and then would undergo a drop-to-rise-to-
decay phase due to the curvature effect. Lu et al. (2007) also studied the
evolution of observed spectral hardness $E_{peak}$ based on the model of
highly symmetric expanding fireballs, where the Doppler effect of the
expanding fireball surface is the key factor concerned, and found that the
evolutionary curve of $E_{peak}$ also undergoes a drop to rise to decay
evolution.
In conclusion, such hardness parameters were typically found either to follow
a “hard-to-soft” trend, decreasing monotonically while the flux rises and
falls, or to “track” the flux during an individual pulse, with the spectral
hardness peaking on the leading edges of pulses (Wheaton et al. 1973; Norris
et al. 1986; Golenetskii et al. 1983; Laros et al. 1985; Kargatis et al. 1994;
Ford et al. 1995).
Kaneko et al. (2006, hereafter Paper I) made a systematic spectral analysis of
350 bright GRBs observed by BATSE with high temporal and spectral resolution.
Basing on their energy fluence or peak photon flux values to assure good
statistics, they selected 350 from 2704 BATSE GRBs. A thorough analysis was
performed on 350 time-integrated and 8459 time-resolved burst spectra using 5
different photon models. The Kaneko sample is the most comprehensive study of
spectral properties of GRB prompt emission to date. In the meantime, we also
analyse the spectra of weak bursts with the peak flux less than 10 photons
$s^{-1}cm^{-2}$ presented by Kocevski et al. (2003) using the same ways as
Paper I. Employing the two samples we investigate the evolution of $E_{peak}$
(the maximum of the $\nu F_{\nu}$ spectrum) confining individual pulses and
find that the evolution of $E_{peak}$ within a pulse follows hard-to-soft,
soft-to-hard-to-soft (flux and hardness tracking) and chaotic pattern. Similar
to Crider et al. (1997) we take the pulses as tracking if the rise and decay
of $E_{peak}$ coincides with those of count rate or flux to within 1 time bin
and if the rise lasts at least 3 time bins. Figure. 1 illustrates the
evolutions of $E_{peak}$ for the cases of hard-to-soft and tracking pulses,
where the photon flux (middle panels) interval is from 30 kev to 2 MeV.
The pattern of hard-to-soft is the most common and the tracking evolutionary
case is less common in Kaneko sample. As for the spectral evolution of hard-
to-soft pulses, many studies have been made on their origin (e.g. Liang et al.
1997). Whereas none has been made to investigate the characteristics of
spectral evolution of tracking pulses. In addition, since both Qin et al.
(2006) and Lu et al. (2007) found the spectral hardness indeed evolve in time
from drop to rise to decay due to curvature effect we first wonder what the
spectral evolutionary characteristics of the tracking pulse are. We would also
like to know whether the tracking pulses are interpreted by curvature effect.
Hence focusing on studying the $E_{peak}$ evolution of these tracking pulses
is our main purpose in this paper. We construct this paper as follows. In $\S$
2 the selection of Kaneko and our sample are introduced, respectively. In $\S$
3 we describe our analysis methods. The analysis results are presented in $\S$
4\. The conclusions and discussion are given in the last section.
Figure 1: Example plots of evolutions of the observed peak energy $E_{peak}$
of hard-to-soft (BATSE trigger 6397, left panel) and tracking (BATSE trigger
2083, right panel) pulse.
## 2 The Sample selection
### 2.1 The selection of Kaneko sample
Paper I first selected 350 GRBs according to some given criterions and then
made spectral analysis. In the following, we describe it simply. (For further
information about the sample, one can refer to Paper I.)
#### 2.1.1 The selection methodology of Kaneko sample
a) Burst sample selection: The Kaneko sample was selected from 2704 bursts
observed by BATSE. The burst selection criteria was a peak photon flux in 256
ms (50 $\sim$ 300 keV) greater than 10 photons $s^{-1}$ $cm^{-2}$ or a total
energy fluence in the summed energy range ($\sim$ 20-2000 keV) larger than 2.0
$\times$ $10^{-5}$ ergs $cm^{-2}$. b) Detector selection: In order to take
advantage of Large Area Detectors’ (LADs’) larger effective area Paper I only
selected LAD data. One of the purposes of selection one detector data was
keeps the analysis more uniform. c) Data type selection: Three LAD data types
were used in Paper I. In order of priority, they were High Energy Resolution
Burst data (HERB), Medium Energy Resolution data (MER), and Continuous data
(CONT). d) Time interval selection: In order to obtain the most statistical
analysis result, they used a minimum S/N of 45 for all data types. e) Energy
interval selection: The lowest seven channels of HERB and two channels of MER
and CONT were usually below the electronic lower energy cutoff and were
excluded. Likewise, the highest few channels of HERB and normally the very
highest channel of MER and CONT were unbounded energy overflow channels and
also not usable.
#### 2.1.2 energy spectra analysis of Kaneko sample
Kaneko et al. fitted 350 time-integrated spectra and 8459 time-resolved
spectra adopting a set of photon models that were usually used to fit GRB
spectra.
a) Spectral fitting software
They used specific spectral analysis software RMFIT developed by BATSE team
(Mallozzi, Preece & Briggs 2005), which incorporates a fitting algorithm MFIT
that employs the forwardfolding method (Briggs 1996), and the goodness of fit
is determined by $\chi^{2}$ minimization. One advantage of MFIT is that it
utilizes model variances instead of data variances, which enables more
accurate fitting even for low-count data (Ford et al. 1995).
b) Photon models
Kaneko et al. adopted 5 spectra models to fit BATSE GRB spectra. They are the
Power law model, the GRB model (BAND) (Band et al. 1993), the GRB model with
fixed $\beta$ (BETA), Comptonized model (COMP) and Smoothly Broken Power Law
(SBPL), respectively. Since there are many BATSE GRB spectra that lack high-
energy photons (Pendleton et al. 1997), and these no-high-energy spectra are
usually fitted well with COMP model, the only spectral model we actually use
in this work is this model. The COMP model is a low-energy power law with an
exponential high-energy cutoff. It is equivalent to the BAND model without a
high energy power law, the form of the COMP model is as follows:
$\displaystyle
f_{COMP}(E)=A(\frac{E}{E_{piv}})^{\alpha}\exp(-\frac{E(2+\alpha)}{E_{peak}}),$
(1)
Epiv was always fixed at 100 keV , therefore, the model consists of three
parameters: A, $\alpha$, and $E_{peak}$.
### 2.2 The selection of our sample
First, we download the Kaneko sample from BATSE public archive (http://ww
w.batse.msfc.nasa.gov/batse/grb/). As Paper I pointed out that the COMP model
tends to be preferable in fitting time-resolved spectra as the existence of
more spectra without high-energy component (Pendleton et al. 1997), as well as
the lower S/N in each spectrum compared with the time-integrated spectra.
Therefore, we can extract the time-resolved spectra fitted with the COMP model
and the following data points are excluded:
a) the resulting $\chi^{2}$ per degree of freedom is -1 when fitting each
time-resolved spectrum, because in this case the nonlinear fitting of
corresponding time-resolved spectra is failure.
b) $\alpha\leq-2$ for the COMP model. As Paper I pointed out that, the fitted
$E_{peak}$ represents the actual peak energy of the $\nu F_{\nu}$ spectrum
only in the case of $\alpha\leq-2$ for the COMP model.
c) the uncertainty of its corresponding $E_{peak}$ is larger 40% than itself.
In this way, we can obtain the best statistic.
With these criterions, we can obtain useful data points of $E_{peak}$ and flux
for all the bursts that Kaneko sample were selected. The count rates data of
these bursts are available in the BATSE public archive. These count rates data
were gathered by BATSE’s LADs, which provide discriminator rate with 64 ms
resolution from 2.048 s before the burst to several minutes after the trigger
(Fishman et al. 1994). For our analysis, we combine the data from the four
channels. Similar to Peng et al. (2006), we also subtract background from
initial count rates. Therefore, we can get the relationship between count
rates and time since trigger and that flux and time since trigger as well as
that between $E_{peak}$ and the time since trigger.
Then we present the three relationships in one figure for each burst, with the
upper panel showing the count rates versus time since trigger, middle panel
indicating the flux against time since trigger and the bottom one representing
the $E_{peak}$ versus time since trigger. In this manner, we can select
roughly the data points of $E_{peak}$ corresponding to pulses that we are
deemed “separable” by eyes and obtain 82 pulses. Since our work focus on the
$E_{peak}$ evolution of individual pulses, we must select those pulses
contaminated by other ones as few as possible. A single functional form is
used to fit these burst time profiles so that we can identified “separable”
pulses with pulse overlapping reduced. It is suspected that many pulses have a
shapes like FRED (fast rise and exponential decay). Similar to Ryde et al.
(2005) and Peng et al. (2006), we adopt the function presented in equation
(22) of Kocevski et al. (2003) (the KRL function) to fit those selected
background-subtracted pulses, combining the data from the BATSE four channels,
since the function can be well-presented the FRED pulses. In addition, a fifth
parameter $t_{0}$, which measures the offset between the start of the pulse
and the trigger time, is introduced. The KRL function is
$F(t)={F_{max}}(\frac{t+t_{0}}{t_{max}+t_{0}})^{r}[\frac{d}{d+r}+\frac{r}{d+r}(\frac{t+t_{0}}{t_{max}+t_{0}})^{(r+1)}]^{-\frac{r+d}{r+1}},$
(2)
where $t_{max}$ is the time of the pulse’s maximum flux, $F_{max}$; r and d
are the power-law rise and decay indexes, respectively. Note that equation (2)
holds for $t\geq-t_{0}$, when $t<-t_{0}$ we take $F(t)=0$.
Similar to Peng et al (2006) and Norris et al (1996), we developed and applied
an interactive graphical IDL routine for fitting pulses in bursts in order to
obtain an intuitive view of the result of the fit, which allows the user to
set and adjust the initial pulse parameter and the pulse position manually
before allowing the fitting routine to converge on the best-fitting model via
the reduced $\chi^{2}$ minimization. The MPFIT we used is a set of routines
for robust least-squares minimization (curve fitting), using arbitrary user
written IDL functions or procedures. It is based on the well-known and tested
MINPACK-1 FORTRAN package of routines available at www.netlib.org. Moreover,
MPFIT functions may permit you to fix any function parameters, as well as to
set simple upper and lower parameter bounds. There are five parameters in KRL
function in all. We first set the $F_{max}$ to the 90 percent of maximum pulse
intensity and the $t_{max}$ to the time of pulse’s maximum intensity and then
adjust the other 3 parameter according to the pulse’s shapes.
The fits are performed on the regions including a complete pulse and are
examined many times to ensure that they are indeed the best ones (the reduced
$\chi^{2}$, $\chi_{\nu}^{2}$, is the minimum). In addition, the data points of
$E_{peak}$ of each pulse must be larger than 6 to ensure both of the rise and
decay phase last at least 3 time bins.
In the course of fitting, we find the KRL function cannot well fit the pulses
with sharp peak though it can well fit that pulses with flat peak. Moreover,
it is not proven that all pulses have the shape of FRED. Therefore, we use
another function of equation (1) in Norris et al. (1996) (the Norris
function), which could be rewritten as follows:
$I(t)=A\left\\{\begin{array}[]{ll}(\exp(-(|t-t_{max}|/\sigma_{r})^{\nu})&t<t_{max},\\\
(\exp(-(|t-t_{max}|/\sigma_{d})^{\nu})&t>t_{max},\end{array}\right.$ (3)
where $t_{max}$ is the time of the pulse’s maximum intensity, A; $\sigma_{r}$
and $\sigma_{d}$ are the rise (t $<t_{max}$) and decay (t $>t_{max}$) time
constant, respectively; and $\nu$ is a measure of pulse sharpness.
Norris function also have 5 parameters and combined the rise, decay time
constant and pulse sharpness permit a wide variation of pulse shape. In
addition, when $\nu$ lower than unit we can yield spikier pulses.
We also first set the maximum intensity, A to the 90 percent of pulse
intensity and the $t_{max}$ to the time of pulse’s maximum intensity. Then we
adjust the other 3 parameter according to the pulse’s shapes.
The pulses we selected are fitted with the two functions, respectively. Then
we select the best fitted model parameters with smaller fitting
$\chi_{\nu}^{2}$ for each pulse, which can better present pulse profile. The
pulses with fitting $\chi_{\nu}^{2}$ larger than 2 are discarded. In this way,
we obtain 34 pulses in 29 GRBs.
Since the sample presented by Kaneko et al. are bright bursts with the peak
photon flux in 256 ms (50-300 keV) greater than 10 photons $s^{-1}cm^{-2}$, we
select weaker bursts with peak photon flux less than 10 photons
$s^{-1}cm^{-2}$ presented by Koceviski et al. (2003) to investigate their
${E_{peak}}$ evolutions in time because these bursts exhibit clean, single-
peaked or well-separated in multipeaked events. These burst spectral analysis
is also performed by RMFIT package. We always chose the data taken with
detector that are closest to line of sight to the GRB because it has the
strongest signal. We adopt the same means as Paper I to deal with these data.
Due to our study focus on the time-resolved spectra, we use, as Ryde &
Svensson (2002) did, a signal-to-noise ratio (S/N) of the observations of at
least $\geq$ 30 to get higher time resolution. We apply S/N $\sim$ 45 as much
as possible since Preece et al. (1998) has shown that S/N $\sim$ 45 is needed
to perform detailed time-resolved spectroscopy. For the weak bursts, we use
S/N $\sim$ 30, in which case we check that the results are consistent with
higher S/Ns. The spectra are modeled with the aforesaid COMP model. There are
34 bursts are strong enough to perform spectral analysis. Then we also remove
the data in the case of above a), b) and c). For the 34 weak bursts, only 8
pulses in 7 bursts whose ${E_{peak}}$ exhibit soft-to-hard-to-soft spectral
evolution, while the others are hard-to-soft. The trigger numbers of the 7
bursts are 1956, 3143, 4350, 5523, 5601, 6672, and 8111. We also fit them with
KRL and Norris function to get best fitting parameters.
Finally we obtain a sample consisting of 42 pulses in 36 GRBs, which contains
527 time-resolved spectra in all. Presented in Table 1 are our selected
bursts, in which include BATSE trigger number, $t_{max}$, fitted
$\chi_{\nu}^{2}$, FWHM (full width at half maximum), the ratio of rise width
to the decay width and the fitting function. Displayed in Figure 2 are the 4
typical examples for our selection results with two bright bursts and 2 weak
bursts, which are fitted by Norris function (the upper 2 panels) and KRL
function (the bottom 2 panels), respectively. We only afford the evolution of
count rate since it can better presents time profile than flux (Ryde &
Svensson 2002) because we defined tracking as the cases in which the hardness
follows the same pattern as the flux or count rate time profile. There are
four parts (for four pulses) to Figure 2: each part is composed of two panels,
with the upper panel and the bottom panel are the evolutionary curves of count
rate and $E_{peak}$, respectively. The distributions of the reduced $\chi^{2}$
for our selected sample is displayed in Figure 3.
Table 1: A list of burst sample with select parameters trigger | $\chi_{\nu}^{2}$ | Tmax | $FWHM$ | $ratio$ | fitting function
---|---|---|---|---|---
676 | 1.55 | 60.05 | 3.30 | 1.40 | N
1156 | 1.28 | 49.29 | 18.51 | 0.85 | K
1733 | 1.11 | 3.36 | 4.47 | 0.50 | K
1982 | 1.38 | 15.85 | 7.02 | 0.95 | N
2083:1 | 1.44 | 1.16 | 1.30 | 0.87 | N
2083:2 | 1.58 | 8.68 | 2.62 | 0.52 | K
2138:1 | 1.36 | 7.54 | 5.35 | 0.84 | K
2138:2 | 0.99 | 78.41 | 6.06 | 0.42 | K
2156 | 1.74 | 14.56 | 4.10 | 0.63 | K
2389 | 1.31 | 11.67 | 23.75 | 0.57 | N
2812 | 1.72 | 0.75 | 1.57 | 0.69 | K
2919 | 1.43 | 0.33 | 3.26 | 0.45 | K
3003 | 1.05 | 9.75 | 11.17 | 0.57 | K
3071 | 1.09 | 15.90 | 15.58 | 0.82 | N
3143 | 0.93 | 0.68 | 1.84 | 0.50 | K
3227 | 1.56 | 101.67 | 2.72 | 0.50 | N
3415:1 | 1.20 | 0.33 | 1.32 | 0.45 | K
Table 1: -Continued trigger | $\chi_{\nu}^{2}$ | Tmax | $FWHM$ | $ratio$ | fitting function
---|---|---|---|---|---
3415:2 | 1.66 | 11.53 | 1.46 | 0.39 | K
3415:3 | 1.60 | 44.72 | 1.09 | 1.13 | N
3491 | 1.74 | 7.74 | 1.86 | 0.42 | N
3765 | 1.40 | 66.15 | 1.65 | 0.48 | K
3891 | 1.89 | 33.26 | 0.62 | 0.31 | K
3954 | 1.11 | 0.77 | 2.87 | 0.54 | K
4350:1 | 1.71 | 13.98 | 3.40 | 0.28 | K
4350:2 | 1.18 | 34.11 | 6.51 | 0.52 | K
5523 | 0.98 | 0.85 | 2.79 | 0.38 | N
5601 | 1.31 | 7.70 | 3.74 | 0.59 | K
5621 | 1.73 | 3.93 | 0.83 | 0.73 | N
5773:1 | 1.33 | 8.23 | 5.923 | 0.59 | K
5773:2 | 1.65 | 17.16 | 5.64 | 0.84 | N
6100 | 1.03 | 8.27 | 2.03 | 0.42 | K
6414 | 0.99 | 6.13 | 13.31 | 0.58 | N
6581 | 1.49 | 47.71 | 0.49 | 0.40 | K
6672 | 1.19 | 0.81 | 2.08 | 0.29 | N
Table 1: -Continued
trigger | $\chi_{\nu}^{2}$ | Tmax | $FWHM$ | $ratio$ | fitting function
---|---|---|---|---|---
6763 | 1.15 | 7.87 | 7.795 | 0.54 | N
6891 | 1.18 | 12.68 | 8.62 | 0.85 | K
7113 | 1.57 | 19.71 | 0.50 | 1.37 | N
7360 | 1.64 | 40.29 | 12.30 | 1.30 | N
7491 | 1.02 | 18.68 | 0.54 | 1.21 | N
7515 | 1.08 | 9.10 | 8.08 | 0.61 | K
7549 | 1.45 | 127.23 | 1.29 | 1.14 | N
8111 | 1.12 | 4.96 | 2.50 | 0.32 | K
Note: N and K denotes the KRL and Norris function, respectively.
Figure 2: Example plots corresponding to BATSE trigger number 5523 (weak
burst), 6763 (bright burst), 3003 (bright burst), and 5601 (weak burst) of the
evolution of count rate (top panels) and $E_{peak}$ (bottom panels) fitted by
Norris function (the upper 2 panels) and KRL function (the bottom 2 panels),
where the dashed lines are the fitting curves and the dashed-dotted-dashed
lines are the boundary of fitted pulses.
Figure 3: Histograms for the distribution of $\chi_{\nu}^{2}$ (left panel) and
the ratios of the rise width to the decay width (right panel) in our selected
sample.
## 3 analysis method
In the previous section we have adopted KRL and Norris function to fit all
background-subtracted light curves of our selected sample and then obtained 5
fitting parameters. Therefore, the full width at half-maximum (FWHM) (see
Table 1), the rise width and decay width for each pulse are estimated. With
above preparation we make the following transformation.
Firstly, For the sake of comparison, let us re-scale the time since trigger of
each pulse by assigning $t_{max}$ for 0 so that the peak time of the
evolutionary curves of $E_{peak}$ almost locates at 0 and denote them as
shifttime. Secondly, let us sort all these data points of $E_{peak}$ in
shifttime order and then divide them into 10 groups evenly. For the every
group the histogram of $E_{peak}$ are plotted, respectively. Thirdly, we find
out the median of $E_{peak}$ of every group and indicate it by a line together
with the values of 50 keV, 100 keV, 200 keV in the corresponding histogram.
Fourthly, we calculate the ratios of above 200 keV, 100 keV and 50 keV,
respectively, for every group. Fifthly, we extract all the median and
corresponding shifttime (here the shifttime take the middle time of start and
end time for every group). Sixthly, in order to get a uniform time we
normalize shifttime in corresponding $FWHM$ of each pulse. This time are
denoted as normalizedshifttime. Then we repeat what the first to the fifth
step have done. Lastly, we examine the relationship between the rise width and
the decay width of each pulse to check if these pulse profiles are different.
## 4 analysis result
### 4.1 The evolutionary characteristic of $E_{peak}$ of all the pulses
We first would like to know what the evolutionary characteristics of all the
pulses are. So we study the evolution of $E_{peak}$ of 527 time-resolved
spectra in shifttime and normalizedshifttime, respectively.
Figure 4: The plots of $E_{peak}$ vs. shifttime (left panel) and
normalizedshifttime (right panel) for our selected sample, where the dot-line-
dot represents the evolution of trigger number 7515.
Figure 5: The plots of normalized $E_{peak}$ vs. shifttime (left panel) and
normalizedshifttime (right panel) for our selected sample, where the dot-line-
dot represents the evolution of trigger number 7515.
It is found in Figure 4. that the overall $E_{peak}$ evolution of our selected
pulses indeed follow soft-to-hard-to-soft pattern. We also give a example
event to show how the evolution proceeds. If the case when the $E_{peak}$ are
also normalized to maximum is different. We also study the evolution of
normalized $E_{peak}$ with shifttime and normalizedshifttime. The Figure 5
indicates the normalized $E_{peak}$ also follow the pattern of soft-to-hard-
to-soft. The example event clearly show the evolutionary trend.
In order to investigate the detailed characteristics of $E_{peak}$ evolution
of these pulses, we divide the 527 time-resolved spectra into 10 groups evenly
in the shifttime and normalizedshifttime order, respectively. Figure 6 and
Figure 7 show two example (the second and the sixth group) histograms of
$E_{peak}$ to the aforesaid two sorts of time, respectively. In every panel in
Figure 6 and 7 the median, 200 keV, 100 keV, and 50 keV are indicated. In the
meantime, we also give the histograms of all the $E_{peak}$ in our sample to
see if their distributions are different.
Figure 6: After our sample being divided into 10 groups in terms of shifttime,
the example histograms of $E_{peak}$ (the second and the sixth group). The
positions of median (dotted lines), 200 keV (long dashed-dotted lines), 100
keV (short dashed-dotted lines) and 50 keV (dashed lines) are also plotted.
where the histograms represented by dashed lines are all the $E_{peak}$, the
numbers on the top of the panels are the time interval of each group.
Figure 7: After our sample being divided into 10 groups in terms of
normalizedshifttime, the example histograms of $E_{peak}$ (the second and the
sixth group). The symbols are the same as those adopted in Fig. 6.
Both Figure 6 and Figure 7 indicate that the evolutionary trends of $E_{peak}$
and the changes of 200 keV, 100 keV and 50 keV. These histograms show that the
median of $E_{peak}$ first shift from low values to high ones then to even
lower than the first ones. The variation of position of 200 keV, 100 keV, and
50 keV are also seen.
In order to obtain a more intuitive view of these points, we make the
following scatter plots for all the groups that: median, the ratio of above
200 keV, the ratio of above 100 keV and the ratio 50 keV versus shifttime,
respectively. The scatter plots for the aforesaid four values against
normalizedshifttime are also made. These values are listed in Table 2 and
Table 3.
Figure 8: The plots of median of $E_{peak}$ vs. shifttime (left panel) and
normalizedshifttime (right panel) after our sample having been divided into 10
groups in terms of shifttime and normalizedshifttime.
Figure 9: The scatter plots of ratios of above 200 keV, 100 keV, and 50 keV vs. shifttime (the upper three panels) and normalizedshifttime (the lower three panels) after being divided into 10 groups in terms of shifttime and normalizedshifttime for our selected sample. Table 2: A list of the shifttime and normalizedshifttime versus median of $E_{peak}$, respectively. shifttime (s) | medianEp (keV) | normalizedshifttime | medianEp (keV)
---|---|---|---
-2.58 | 277.36 $\pm$ 30.17 | -0.43 | 278.24 $\pm$ 30.18
-0.96 | 319.85 $\pm$ 37.51 | -0.16 | 316.25 $\pm$ 39.02
-0.40 | 358.37 $\pm$ 38.63 | -0.05 | 353.46 $\pm$ 39.08
-0.10 | 363.03 $\pm$ 55.17 | -0.00 | 389.92 $\pm$ 39.02
0.28 | 322.78 $\pm$ 41.46 | 0.13 | 277.78 $\pm$ 37.14
0.55 | 307.78 $\pm$ 31.46 | 0.31 | 242.44 $\pm$ 34.81
0.76 | 199.57 $\pm$ 34.81 | 0.42 | 244.42 $\pm$ 38.47
1.28 | 201.15 $\pm$ 25.13 | 0.55 | 230.96 $\pm$ 26.63
2.12 | 183.98 $\pm$ 23.51 | 0.77 | 196.43 $\pm$ 24.83
4.39 | 165.67 $\pm$ 26.47 | 1.73 | 194.69 $\pm$ 30.52
Table 3: A list of the shifttime vs. the ratio of $E_{peak}$ larger than 200,
100, 50 keV and normalizedshifttime vs. the ratio of $E_{peak}$ larger than
200, 100, 50 keV, respectively.
shifttime (s) | rEl200 | rEl100 | rEl50 | normtime | rEl200 | rEl100 | rEl50
---|---|---|---|---|---|---|---
-11.63 | 0.68 | 0.90 | 1.00 | -1.22 | 0.65 | 0.96 | 1.00
-0.84 | 0.80 | 1.00 | 1.00 | -0.25 | 0.75 | 0.94 | 1.00
-0.27 | 0.84 | 1.00 | 1.00 | -0.01 | 0.79 | 1.00 | 1.00
-0.00 | 0.90 | 1.00 | 1.00 | -0.00 | 0.81 | 1.00 | 1.00
0.25 | 0.82 | 1.00 | 1.00 | 0.13 | 0.68 | 1.00 | 1.00
0.57 | 0.76 | 0.98 | 1.00 | 0.25 | 0.68 | 0.98 | 1.00
0.97 | 0.48 | 0.96 | 1.00 | 0.40 | 0.62 | 0.98 | 1.00
1.63 | 0.50 | 0.96 | 1.00 | 0.61 | 0.56 | 0.98 | 1.00
2.81 | 0.45 | 0.88 | 0.98 | 0.91 | 0.46 | 0.92 | 1.00
13.41 | 0.35 | 0.96 | 1.00 | 3.40 | 0.48 | 0.90 | 0.98
Note: normaltime, rEl200, rEl100 and rEl50 represent normalizedtime, ratio of
$E_{peak}$ larger than 200, ratio of $E_{peak}$ larger than 100, ratio of
$E_{peak}$ larger than 50, respectively.
It is clear that the evolution of median with shifttime and
normalizedshifttime are also soft-to-hard-to-soft from Figure 8, Table 2. In
addition, the phase of soft-to-hard (we denote it as rise phase) is shorter
than the phase of hard-to-soft (we denote it as decay phase), since the time
intervals of rise phase and decay phase are 2.58 s and 4.39 s, respectively,
for shifttime and 0.43, 1.73, respectively, for normalizedshifttime. The
softest spectra of rise phase (277.36 keV for shifttime and 278.24 keV for
normalizedshifttime) are harder than that of the decay phase (165.67 keV for
shifttime and 194.69 keV for normalizedshifttime).
From Figure 9 and Table 3, we find that: a) the ratios of above 50 keV almost
stay fixed in the whole phase; b) the ratios of above 100 keV change from
small to big at first and then to small in the end, besides there are three
bin time in the middle of the phase remain constant. The above results show
that the spectra of rise phase are harder than 50 keV, but some spectra are
softer than 100 keV. Whereas for the decay phase the softest spectra are lower
than 50 keV and there are many spectra are softer than 100 keV. The ratios of
above 200 keV are, however, similar to the variation of median, i.e. the value
of $E_{peak}$ larger than 200 keV have an asymmetrical distribution that they
vary from small-to-big-to-small, arriving at the smallest value in the end
phase. Clearly, our results are consistent with that of Preece et al. (2000),
who found that the $E_{peak}$ cluster on about 250 keV.
We do the Kolmogorov-Smirnov (K-S) test (Press et al. 1992) to check if this
$E_{peak}$ evolution is real because we would expect that the K-S tests would
also yield significant evidence that the divided samples are different. The
K-S test determines the parameter $D_{KS}$, which measures the maximum
difference in the cumulative probability distributions over parameter space,
and the significance probability $P_{KS}$ for the value of $D_{KS}$. A small
$P_{KS}$ indicates that the data sets are likely to be different (Press et al.
1992). We employ the K-S tests between group 4 (where $E_{peak}$ is maximal)
and all the groups in order to show if significant evolution is present.
Figure 10 indicates the variations of $P_{KS}$ for shifttime and
normalizedshittime. It is shown in the Figure 10 that the evolution of
$E_{peak}$ indeed exists.
Figure 10: The plots of $P_{KS}$ and groups for the cases of shifttime (left
panel) and normalizedshifttime (right panel), where the $P_{KS}$ are the
significance probability between group 4 and all the groups after our sample
being divided into 10 groups in terms of shifttime and normalizedshifttime.
Since many pulses have a shapes like FRED but one can not prove that all
pulses have such a shape. We check the ratios of rise width to decay width of
our selected sample (see Table 1). The ratios are obtained by using the best
model parameters due to it can reflect the profile of corresponding pulse. As
Koceviski et al. (2003) described that KRL function is an analytical function
based on physical first principles and well-established empirical descriptions
of GRB spectral evolution. These analytical profiles are independent of the
emission mechanism and can be fully model the FRED light curve. While Norris
et al. (1996) pointed out that the Norris function are more flexible model. It
can model that pulses with various shapes, especially for the sharp peak
pulse, and would not be constrained in the shapes of FRED. Therefore, we think
that the two models can well present the pulse shapes and can get satisfied
results. The histogram of ratios is displayed in Figure 3. It is found that
the ratios clustered at less than unit, which is consistent with the remarks
given by Norris et al. (1996) and Lee et al. (2000a, 2000b). With the ratios
less than unit we deem these pulses are similar to FRED pulses. Since the
pulse could be not presented by only one functional form, we consider one of
possible reasons of the ratios more than unit come from the functional form,
or could be cause by the pulses overlap. It is suspected that, to some extent,
the evolution of $E_{peak}$ of tracking pulses is related to the time profile.
## 5 conclusions and discussion
In this paper, we investigate a sample including 42 tracking pulses within 36
GRBs involved 527 time-resolved spectra and study the evolutionary
characteristics of $E_{peak}$. The sample consist of 29 bright and 7 weak
BATSE GRBs. In order to get good statistics, we use a S/N of the observations
of at least $\sim$ 30 and arrive at $\sim$ 45 as much as possible for the
time-resolved spectroscopy. Since the work focus on separate tracking pulse,
we adopt two pulse models to obtain better identification of the selected
pulses and discard that with large fitting $\chi_{\nu}^{2}$ ($>$2). Therefore,
we think that our sample is very representative of tracking pulses.
In order to make the time a relative uniform standard, we first make a
transformation of the time since trigger relative to the time of maximum
intensity of pulses (shifttime) and normalize the shifttime in the width of
pulse (normalizedshifttime). We find that the evolution of $E_{peak}$ indeed
follow soft-to-hard-to-soft with both of shifttime and normalizedshifttime
(see Figure 3). Then we divide evenly our sample into 10 groups according to
the two time order to study the evolution of median as well as the ratios of
above 200 keV, 100 keV and 50 keV. For this type of tracking pulse the
$E_{peak}$ of rise phase always larger than 50 keV, while some spectra in the
decay phase less than 50 keV. The spectra of rise phase are harder than that
of decay phase. In addition, we find that the rise phase of $E_{peak}$
evolution are shorter than that of the decay phase and this trends are
established in our selected pulses.
As the previous section pointed out the $E_{peak}$ of time resolved spectra
are fitted by COMP model. Is there a bias introduced in always using the COMP
model when the Band model is the appropriate model? Therefore we also
investigate the time resolved spectra of some pulses using Band and COMP model
and then compare the values of $E_{peak}$ when the fitting $\chi_{\nu}^{2}$ of
Band model are smaller than or comparable to that of COMP model. We find that
this lead to a little high $E_{peak}$ estimates and slightly high $E_{peak}$’s
during the rise phases.
The observed gamma-ray pulses are believed to be produced in a
relativistically expanding and collimated fireball because of the large
energies and the short time-scales involved. To account for the observed
spectra of bursts, the Doppler effect over the whole fireball surface (or the
curvature effect) would play an important role (e.g. Meszaros and Rees 1998;
Hailey et al. 1999; Qin 2002, 2003). The Doppler model is the model describing
the kinetic effect of the expanding fireball surface on the radiation
observed, where the variance of the Doppler factor and the time delay caused
by different emission areas on the fireball surface (or the spherical surface
of uniform jets) are the key factors to be considered (for a detailed
description, see Qin 2002 and Qin et al. 2004).
Qin et al. (2006) investigated the GRBs pulses and found that the curvature
effect influences the evolutionary curve of the corresponding hardness ratio.
They found the evolutionary curve of the pure hardness ratio would peak at the
very beginning of the curve, and then would undergo a drop-to-rise-to-decay
phase due to the curvature effect.
Based on the model of highly symmetric expanding fireballs, Lu et al. (2007)
investigated in detail the evolution of spectral hardness $E_{peak}$ of FRED
pulse caused by curvature effect. They first investigated the cases that the
local pulses are exponential rise and exponential decay and exponential rise,
respectively, and found that for both of the two local pulses the evolutionary
curves of $E_{peak}$ underwent drop-to-rise-to-decay evolution, which
corresponded to A, B, and C phases, respectively. Then they assumed that the
local pulses was exponential rise and exponential decay pulse as well as the
rest frame spectra varied with time, the same result were obtained. The B and
C phase correspond to the rise phase (soft-to-hard) and decay phase (hard-to-
soft). We can also find from Figure 1 and 3 in Lu et al. (2007) that the time
interval of B phase is shorter than that of C phase and the spectra of the B
phase are harder than that of the C phase. This situation are in good
agreement with the conclusions of the our selected pulses. Why the A phase in
our sample are not observed by BATSE? The main cause we consider that it
corresponds to the very onset of the light-curve pulse, where the real
emissions are always contaminated by the background.
Therefore, we think the evolution of the $E_{peak}$ in our selected pulses can
be mainly caused by Doppler effect and argue that kinematics effect may be
play important role in the course of spectral evolution.
In view of dynamics, the current popular views on the production of GRB is the
synchrotron shock model. Based on this model, a soft-to-hard co-moving
spectrum might be come into being in the case of a synchrotron radiation when
electrons radiate at the beginning of a shock gaining accelerations and then
arriving at the maximum speeds. The phase may be very short. After the hardest
spectrum appears, the electrons start to decelerate and the energy of
electrons become small. Moreover, the curvature effect must be at work because
the radiation come from different latitudes of fireball (or angles of line of
sight). Both of aforementioned two factors cause the observed spectra evolute
from hard to soft. The phase must be much longer than that of the soft-to-hard
(see Figure 8 and Table 2).
Kobayashi et al. (1997) discussed the possibility that GRBs result from
internal shocks in ultrarelativistic matter and provide the pulse profile of
internal shock in Figure 1. From Figure 1 given by Kobayshi et al. (1997) we
can find that the radiation power of internal shock indeed follow weak-to-
strong-to-weak, moreover, the time of weak-to-strong are shorter than that of
the strong-to-weak. This characteristic are consistent with that of $E_{peak}$
of the tracking pulses, which indicate that this type of tracking pulses also
are related to the process of internal shock. Therefore, our results for the
tracking pulses do clearly support the models of GRB shocks.
Consequently, we argue that the spectral evolution of tracking pulses may be
relate to both of kinematic and dynamic process. Maybe the two processes play
important roles together or only one is dominant, which are unclear and
deserve the further investigation. Our detailed statistical results of
$E_{peak}$ evolution of tracking pulses must be constrain the current
theoretic model for the fact that spectral properties of bursts can provide
powerful constrains on the detailed physical models.
In this work, we only concentrate our attention on the tracking pulses and
have not attempted to study non-tracking pulses or to show that all pulses
must be tracking pulses. We also consider and investigate whether all pulses
(or just most pulses) are hard-to-soft or tracking. For the 34 weak burst
pulses provided by Kocevski et al. (2003) we find they are either hard-to-soft
or tracking. In addition, the fraction of tracking pulses is about 24 percent.
However, we can not afford correctly the evolutionary forms and the fraction
of tracking pulses of most bright bursts because there are many short pulses
and the data points of $E_{p}eak$ are few. We only give the statistical
properties of tracking pulses, which will help us rule on the nature of GRB
pulses as tracking.
We thank the anonymous referee for constructive suggestions and Yi-Ping Qin
for his helpful discussions. Thanks are also given to Rorbet Preece and Yuki
Kaneko for their help with RMFIT. This work was supported by the Natural
Science Fund for Young Scholars of Yunnan Normal University (2008Z016 ), the
National Natural Science Foundation of China (No. 10778726, 10747001), and the
Natural Science Fund of Yunnan Province (2006A0027M).
## References
* Band et al., (1992) Band, D., et al. 1992, AIPC, 265, 169
* Band et al. (1993) Band, D., et al. 1993, ApJ, 413, 281
* Band et al. (1997) Band, D., 1997, ApJ, 486, 928
* 357Bhat, (1994) Bhat, P. N., et al., 1994, ApJ, 426, 604
* Briggs (1996) Briggs, M. S. 1996, in AIP Conf. Proc. 384, Gamma-Ray Bursts, 3rd Huntsville Symp., ed. C. Kouveliotou, M. Briggs, & G. Fishman (New York: AIP), 133
* Butler et al. (1997) Butler, N. R., Kocevski, D., 2007, ApJ, 663, 407
* Crider et al. (1997) Crider, A., et al. 1997, ApJ, 479, L39
* Fishman et al. (1994) Fishman, G., et al. 1994, ApJS, 92, 229
* 141Ford, (1995) Ford, L. A., et al. 1995, ApJ, 439, 307
* 353Golenetskii, (1983) Golenetskii, S. V., et al. 1983, Nature, 306, 451
* Hailey (1999) Hailey, C. J., et al. 1999, ApJ, 520, L25
* 358Kaneko, (2006) Kaneko, Y., et al. 2006, ApJS, 166, 298 (Paper I)
* 356Kargatis, (1994) Kargatis, V. E., et al. 1994, ApJ, 422, 260
* 310Kobayashi, (1997) Kobayashi, S., et al. 1997, ApJ, 490, 92
* Kocevski et al. (2003) Kocevski, D., et al. 2003, ApJ, 596, 389
* Laros et al. (2007) Laros, J. G., et al. 1985, ApJ, 290,728
* Lee et al. (2000) Lee, A., et al. 2000a, ApJS, 131, 1
* Lee et al. (2000) Lee, A., et al. 2000b, ApJS, 131, 21
* Liang et al. (1997) Liang, E. P., et al. 1997, ApJ, 476, L35
* Lu et al. (2007) Lu, R. J., et al. 2007, ApJ, 663, 1110
* Nemiroff (2000) Mallozzi, R. S., et al. 2005, RMFIT, A Lightcurve and Spectral Analysis Tool, ( Huntsville: Univ. Alabama)
* Maszaros, (1998) Maszaros, P., Rees, M. J. 1998, ApJ, 502, L105
* 355Norris, (1986) Norris, J. P., et al., 1986, ApJ, 301, 213
* Norris et al. (1996) Norris, J. P., et al. 1996, ApJ, 459, 393
* Pendleton (1997) Pendleton, G. N., et al. 1997, ApJ, 489, 175
* Peng et al. (2000) Peng, Z. Y., et al. 2006, MNRAS, 368, 1351
* Preece et al. (1998) Preece, R. D., et al. 1998, ApJ, 496, 849
* Preece et al. (2000) Preece, R. D., et al. 2000, ApJS, 126, 19
* Press et al. (1992) Press et al. 1992, Numerical Recipes in FORTRAN (2nd ed.; New York: Cambridge Univ. Press)
* Qin (2002) Qin, Y.-P. 2002, A&A, 396, 705
* Qin (2003) Qin, Y.-P. 2003, A&A, 407, 393
* Qin (2004) Qin, Y.-P., et al. 2004, ApJ, 617, 439
* Qin et al. (2006) Qin, Y.-P., et al. 2006, Phys. Rev. D, 74, 063005
* Ryde and Svensson (2002) Ryde, F., Svensson, R. 2002, ApJ, 566, 210
* Ryde and Petrosian (2002) Ryde, F., et al. 2005, A & A, 432, 105
* 130Share, (1998) Share, G. H., Matz, S. M., 1998, AIPC, 428, 354
* 130Share, (1998) Wheaton, W. A., et al. 1973, ApJ, 185, L57
|
arxiv-papers
| 2008-09-22T01:47:02
|
2024-09-04T02:48:57.874236
|
{
"license": "Public Domain",
"authors": "Z. Y. Peng, L. Ma, R. J. Lu, L. M. Fang, Y. Y. Bao, Y. Yin",
"submitter": "Peng Zhaoyang",
"url": "https://arxiv.org/abs/0809.3620"
}
|
0809.3671
|
# Performance of a fine-sampling electromagnetic calorimeter prototype in the
energy range from 1 to 19 GeV
Yu.V.Kharlov P.A.Semenov Yu.A.Matulenko O.P.Yushchenko Yu.I.Arestov
G.I.Britvich S.K.Chernichenko Yu.M.Goncharenko A.M.Davidenko
A.A.Derevschikov A.S.Konstantinov V.A.Kormilitsyn V.I.Kravtsov Yu.M.Melnik
A.P.Meschanin N.G.Minaev V.V.Mochalov D.A.Morozov A.V.Ryazantsev
I.V.Shein A.P.Soldatov L.F.Soloviev A.V.Soukhikh A.N.Vasiliev M.N.Ukhanov
V.G.Vasilchenko A.E.Yakutin corresponding author: Yuri.Kharlov@ihep.ru
Institute for High Energy Physics, Protvino, Russia
###### Abstract
The fine-sampling electromagnetic calorimeter prototype has been
experimentally tested using the $1-19$ GeV/$c$ tagged beams of negatively
charged particles at the U70 accelerator at IHEP, Protvino. The energy
resolution measured by electrons is $\Delta{E}/E=2.8\%/\sqrt{E}\oplus 1.3\%$.
The position resolution for electrons is $\Delta{x}=3.1\oplus 15.4/\sqrt{E}$
mm in the center of the cell. The lateral non-uniformity of the prototype
energy response to electrons and MIPs has turned out to be negligible.
Obtained experimental results are in a good agreement with Monte-Carlo
simulations.
###### keywords:
Electromagnetic calorimeter , fine sampling , plastic scintillators , energy
resolution
###### PACS:
00.11.–aa
, , , , , , , , , , , , , , , , , , , , , , , , , , ††thanks: deceased
## Introduction
Electromagnetic calorimeters are based on the total energy deposition of
photons and electrons in the active medium of detectors. Energy deposited by
secondary particles of an electromagnetic shower is detected either as a
Cherenkov radiation of electrons and positrons, like in the lead-glass
calorimeters [1], or as a scintillation light emitted by the active medium
[2]. Sampling calorimeters constructed from alternating layers of organic
scintillator and heavy absorber, have been used in high energy physics over
last tens of years [3]. The sampling of such calorimeters is determined by the
required lateral size of electromagnetic shower, expressed by a Molière radius
$R_{M}$, and the light yield provided by scintillator plates. Scintillation
light is absorbed, re-emitted and transported to a photodetector by wave-
length shifting (WLS) optical fibers penetrating through the calorimeter
modules longitudinally (along the beam direction). Typical stochastic term of
the energy resolution of all large electromagnetic calorimeters of the
sampling type was about 10% [4, 5, 6].
Recently the improved electromagnetic calorimeter modules with a very fine
sampling have been developed for KOPIO experiment at BNL [7]. The energy
resolution of these modules, measured with photons of energy $220-350$ MeV,
was about $3\%/\sqrt{E~{}\mbox{(GeV)}}$ [8]. Details of the improved modules
tested in the energy range of $50-1000$ MeV, including mechanical
construction, selection of WLS fibers and photodetectors as well as
development of a new scintillator with improved optical and mechanical
properties are described in [9].
Similar high-performance electromagnetic calorimeters are now being considered
for PANDA and CBM experiments [10, 11] at the future FAIR facility, which is
under construction at GSI, Darmstadt in Germany. The both fixed target
experimental setups require an ability to measure single photons, $\pi^{0}$’s
as well as $\eta$’s in the wide energy range with excellent energy and
position resolutions. Fine-sampling calorimeters, not very expensive and
meanwhile covering wide areas like 3 m2 in PANDA and 100 m2 in CBM, were
chosen to meet the requirements. The energy range in PANDA and CBM experiments
will be extended up to 15 and 35 GeV, respectively. It is essential for these
projects to study parameters of a fine-sampling calorimeter in the wide energy
region, significantly wider than the one with the existing data (only up to 1
GeV).
In this paper, we describe a fine-sampling electromagnetic calorimeter
prototype with lead absorber plates, which thickness is significantly smaller
than radiation length $X_{0}$ of lead. Such a small thickness of the absorber
layers results in a small interaction probability of the secondary shower
particles. The design of this prototype is close to the KOPIO one including
the same lateral size of cells. The results of miscellaneous studies of the
prototype in the energy range from 1 to 19 GeV are presented in this paper.
## 1 Design of the modules
The electromagnetic calorimeter modules with fine sampling were constructed at
IHEP, Protvino. A module design was based on the electromagnetic calorimeter
for the KOPIO experiment, with additional modification to high-energy range.
Details of the mechanical design of the modules can be found in [9], but a few
modifications were applied to the prototype under study. The KOPIO experiment
was aimed to low-energy photons, and the total radiation length $16X_{0}$ was
enough for their purposes. The current prototype is being proposed for CBM and
PANDA, where the photon energy extends up to 30 GeV, and the requirement to
provide the total radiation length of $20X_{0}$ was put to the design. The
modules were assembled from 380 alternating layers of lead and scintillator
plates. Lead plates were doped by 3% of antimony to improve their rigidity.
Scintillator plates were made of polystyrene doped by 1.5% of paraterphenile
and 0.04% of POPOP. Scintillator was manufactured at the scintillator workshop
of IHEP with the use of molding technology. The physical properties of the
modules are presented in Table 1.
lead plate thickness | 0.275 mm
---|---
scintillator plate thickness | 1.5 mm
number of layers | 380
effective radiation length, $X_{0}$ | 34 mm
total radiation length | $20X_{0}$
effective Moliere radius, $R_{M}$ | 59 mm
module size | $110\times 110\times 675~{}\mbox{mm}^{3}$
module weight | 18 kg
Table 1: Physical properties of the module.
The WLS optical fibers BCF-91A with a diameter of 1.2 mm were used in the
modules. Each fiber penetrated through the module along its longitudinal axis
twice, forming a loop at the face end of the module. Radius of the loop was 28
mm. In total 72 such looped fibers formed a grid of $12\times 12$ fibers per
module with spacing of 9.3 mm. All 144 fiber ends were assembled into a bundle
of a diameter about 10 mm, glued, cut, polished and attached to the
photodetector at the downstream end of the module. No optical grease was used
to provide an optical contact between the bundle cap and the photodetector,
thus there was a natural air gap between them. A photomutiplier Hamamatsu
R5800 was used as a photodetector for the prototype. The diameter of the
photocathode is 25.4 mm, the number of dynodes is 10, the applied high voltage
was about 1100 V. Each photomultiplier was monitored by LED light guided to
the photocathodes by a clear polystyrene fiber.
## 2 Experimental setup for the prototype studies
The prototype of electromagnetic calorimeter consisted of 9 modules assembled
into $3\times 3$ matrix installed on the remotely controlled $x,y$-moving
support positioned the prototype across the beam with a precision of 0.4 mm.
The beam line 2B of the U-70 accelerator was used to study performance of the
calorimeter prototype. The secondary beam of negatively charged particles of
momenta from 1 to 19 GeV/$c$ contained more than 70% of electrons mixed with
muons and hadrons (mainly $\pi^{-}$ and $K^{-}$). Particle identification was
not available at this beam line. A momentum spread of the beam was at the
level of 1 to 5% at energies from 45 to 1 GeV, respectively. However, the
momentum tagging system[12] gave a beam momentum resolution from 0.13% to 2%
in the same momentum range. The tagging system illustrated in Fig.1 consisted
of the dipole magnet M and 4 sets of 2-coordinate drift chambers DC1–DC4. A
bending angle of the magnet was 55 mrad.
Figure 1: Beam tagging system for the calorimeter prototype studies.
A trigger of the experimental setup used the coincidence of scintillator
counters S1, S2 and S3 installed upstream before the first drift chamber DC1,
and a scintillator counter S4 installed after the last drift chamber DC4 in
front of the calorimeter prototype ECAL.
An amplitude out of each prototype cell was measured by the 15-bit charge
sensitive ADC modules LRS2285 over 150-ns gate with a sensitivity of 30
fC/count. To read out time information from the drift chamber stations TDC,
the LRS3377 CAMAC modules were used. Data acquisition system included a couple
of crates with ADC and TDC modules as well as control modules to synchronize a
read-out process. VME crate with CAMAC parallel branch driver and PCI-VME
bridge linked all the electronics into the complete system. Detailed
description of the data acquisition system and front-end electronics can be
found in [13].
## 3 Monte Carlo simulations
The relevant simulation tools were developed. These tools, at the first stage,
are intended mainly for cross-check of experimental results as well as for
tuning of the reconstruction algorithms.
Having proved the consistency of Monte Carlo and the real data, we plan to use
these tools for further optimization of module design and reconstruction
algorithms to provide better performance of the photons and $\pi^{0}$’s
reconstruction. Simulation studies were performed with GEANT3 as a Monte Carlo
engine with detailed description of materials and module geometry.
The developing shower produces light which originates from two different
sources:
* •
scintillation in plastic plates due to continuous energy losses when charged
particles pass through the active calorimeter material,
* •
Cherenkov radiation when charged particles pass through the WLS fibers.
The simplified technique consists of counting energy deposition in the active
material (with some corrections to take into account light attenuation in the
fibers) and ignoring Cherenkov radiation inside the fibers. This method is
very fast while can not reproduce all details of the calorimeter response such
as non-uniformity due to fibers and cell borders.
For these studies, the detailed light propagation was applied taking into
account the optical properties of the materials, internal reflections at plate
borders, light capture by fibers with double cladding and the Cherenkov light
production and propagation inside the fibers. It was assumed that attenuation
length was 70 cm in the scintillator and 400 cm in the fiber, scintillator
refraction index is 1.59, total internal reflection efficiency at large
scintillator faces is 0.97 and reflection of diffusion type was assumed at
side scintillator faces with the same probability. The mean deposited energy
for one optical photon production in the scintillator was assumed to be 100 eV
and the Cherenkov photons were generated by GEANT.
## 4 Results
### 4.1 Calibration of the modules
The modules were calibrated by a 19-GeV/$c$ beam. Each module was exposed to
the beam using an $x,y$-moving support. The energy spectrum from one module
(Fig.2, left plot)
Figure 2: Energy deposited by the 19-GeV/c beam in one module.
shows a peak at 19 GeV corresponding to the energy deposited by electrons.
Another peak at low energies is due to minimum ionizing particles (MIP). A
broad distribution in the energy specrtum between the two peaks is due to
hadrons. Calibration of the modules was possible using both electron and MIP
signals, but the best relative calibration coefficients were found by
equalizing MIP signals, while the absolute calibration was obtained by setting
the total measured energy in the $3\times 3$ matrix to 19 GeV. Events when
only one module has an energy above the threshold of 100 MeV were selected for
the MIP calibration. The energy distribution around the MIP peak (Fig.2, right
plot) has two contributions. One is caused by the Landau distribution of
ionization energy loss, and another one is due to the finite energy resolution
of the calorimeter at low energy. The MIP peak was fitted by the Gaussian, and
the mean value of the fitting function served for the relative calibration.
### 4.2 Energy and position resolution
After some dedicated calibration runs, when each module was exposed to the
19-GeV/$c$ beam, the ECAL prototype was fixed so that the beam hit the central
module. It was exposed to beam at momenta 1, 2, 3.5, 5, 7, 10, 14 and 19
GeV/$c$. For each beam momentum, magnetic field in the spectrometric magnet M
was adjusted to provide the same bending angle of the beam. The momentum of
the beam particle $p$ was measured by the magnetic spectrometer, and the
energy $E$ measured in the calorimeter prototype is linearly correlated with
the momentum $p$, as illustrated by Fig.3.
Figure 3: Correlation between the energy measured in the calorimeter and the
beam momentum measured in the magnetic spectrometer.
Therefore, in order to obtain a true energy resolution, the measured energy
should be corrected by the beam momentum, or the energy resolution can be
represented by the width of the distribution of the $E/p$ ratio (Fig.4).
Figure 4: Ratio of the energy $E$ measured in the calorimeter to the momentum
$p$ measured by the magnetic spectrometer at 19 GeV/$c$.
The energy resolution is obtained from the Gaussian fit of the right peak
around $E/p=1$. The energy resolution $\Delta E/E$ measured by electrons at
energies from 1 to 19 GeV are shown in Fig.5.
Figure 5: Measured energy resolution.
The black bullets represent the experimentally measured points. The solid
curve is a result of a fit of these experimental points, and the dashed curve
is a result of a fit of the Monte Carlo points. The fitting function can be
represented by the equation (1):
$\frac{\Delta
E}{E}=\sqrt{\left(\frac{a}{E}\right)^{2}+\frac{b^{2}}{E}+c^{2}},$ (1)
where parameters $a$, $b$ and $c$ for the experimental and Monte Carlo fits
are shown in Table 2.
| $a$, $10^{-2}$ GeV | $b$, $10^{-2}$ GeV1/2 | $c$, $10^{-2}$
---|---|---|---
Experimental fit | $3.51\pm 0.28$ | $2.83\pm 0.22$ | $1.30\pm 0.04$
Monte Carlo fit | $3.33\pm 0.12$ | $3.07\pm 0.08$ | $1.24\pm 0.02$
Table 2: Fitting function parameters for the energy resolution.
A linear term $a$ of the energy resolution expansion is determined by a beam
spread rather than the calorimeter properties. As it was shown in previous
studies performed at this 2B beam channel [13], the main contribution to this
term comes from the electronics noise and the multiple scattering of the beam
particles on the beam pipe flanges and the drift chambers. The beam momentum
spread was introduced into Monte-Carlo simulations in order to fully reproduce
the experimental conditions. A simulated energy resolution is shown by the red
dashed line in Fig.5. The dotted line at this plot is a difference between the
experimental data fit and the Monte Carlo fit multiplied by 10. Thus a
deviation of the experimental result from the simulation one is less than
0.04%. The energy resolution obtained in Monte Carlo is in a good agreement
with the experimental data.
Position resolution has been determined by a comparison of the exact impact
coordinate of the beam particle, measured by the last drift chamber DC4, and
the center-of-gravity of electromagnetic shower developed in the calorimeter
prototype. Fig.6 shows a dependence of the measured coordinate $x_{\rm rec}$
on the true one $x_{0}$.
Figure 6: Center of gravity of the electromagnetic shower $X_{\rm rec}$ vs the
impact coordinate of the electron $x_{0}$.
A position resolution in the middle of the module is shown in Fig.7, where the
bullets represent the experimentally measured points, the solid curve is a
result of the experimental points fit, and and the red dashed curve is a
result of Monte Carlo points fit.
Figure 7: Measured position resolution.
The data were fitted by the function (2)
$\Delta x=\sqrt{a^{2}+\frac{b^{2}}{E}},$ (2)
where parameters $a$ and $b$ are given in Table 3.
| $a$, mm | $b$, mm GeV1/2
---|---|---
Experimental fit | $3.09\pm 0.16$ | $15.4\pm 0.3$
Monte Carlo fit | $3.40\pm 0.14$ | $14.5\pm 0.3$
Table 3: Fitting function parameters for the position resolution.
The dotted curve in Fig.7 stands for a deviation of the experimental data fit
results from the Monte Carlo fit results, which are consistent within 5% of
precision.
### 4.3 Lateral non-uniformity
Due to various mechanical inhomogeneities of the prototype one can expect to
observe the dependence of the energy $E$ deposited in the calorimeter on the
hit coordinates $(x,y)$. The “hot” zones, if any, should be seen at the WLS
fiber positions, at the steel strings, and at the boundaries between the
modules. A possible lateral non-uniformity of the energy response was studied
with the data collected in the 19-GeV/$c$-run. The last drift chamber DC4 was
used to measure the coordinate of the beam particle incidence onto the
calorimeter surface. As the beam contained several particle species which
interact differently with the calorimeter medium (see Fig.2), the mean
deposited energy was measured as a function of $(x,y$) for two energy
intervals, $E<0.5$ GeV and $16<E<22$ GeV corresponding to the MIP peak and
that of the electromagnetic shower, respectively. The relative energy response
profile for electrons vs $y$-coordinate at fixed $x$ is shown in Fig.8.
Figure 8: Relative energy response profile vs $y$-coordinate at fixed $x$.
As one can see, the fluctuations of the energy response do not exceed 1%, that
is no lateral non-uniformity of the energy response is observed within the
available statistics.
### 4.4 Light output measurement
Light output of the prototype modules, expressed as a number of photoelectrons
$N_{\rm p.e.}$, was evaluated with the highly stable LED pulses [14].
Fluctuations of the measured amplitude $A$ is determined by statistical
fluctuations of the number of detected photoelectrons and by fluctuations of
the photomultiplier gain $M$ [15]:
$\left(\frac{\sigma_{A}}{A}\right)^{2}=\frac{1}{N_{\rm
p.e.}}\left[1+\left(\frac{\sigma_{M}}{M}\right)^{2}\right].$ (3)
The gain fluctuation can be defined through the secondary emission factor of
the first dynode $\delta_{1}$ and the secondary emission factor of other
dynodes $\delta$:
$\left(\frac{\sigma_{M}}{M}\right)^{2}=\frac{\delta}{\delta_{1}}\cdot\frac{1}{\delta-1}.$
(4)
The total gain of the 10-dynode photomultiplier R5800 was equal to $10^{6}$
for the applied high voltage 1100 V, and the potential of the first dynode was
boosted to increase the secondary emission factor $\delta_{1}$ to
approximately 5. Thus, one can obtain the emission factor $\delta=3.9$ and the
equation (3) is derived to the number of the detected photoelectrons as a
function of the relative amplitude width:
$N_{\rm p.e.}\approx\frac{1.3}{(\sigma_{A}/A)^{2}}.$ (5)
A set of runs with six different LED amplitudes has been carries out. A
dependence of the number of photoelectrons on the LED amplitude for one cell
and a distribution of the light output for all 9 cells are shown in Fig.9.
Figure 9: Light output $N_{\rm p.e.}$ vs the LED amplitude for 1 modules
(left) and the light output distribution of all 9 modules (right).
These plots were fitted by the linear function, which slope represents the
number of photoelectrons per one ADC count. Being divided by the calibration
coefficient, one can obtain that the number of photoelectrons detected by the
the prototype modules is $4.8\pm 0.6$ p.e./MeV.
## Conclusion
The measurements of energy and position resolutions of the electromagnetic
calorimeter prototype of fine-sampling type for the PANDA and CBM experiments
at FAIR at Darmstadt have been carried out at the IHEP test beam facility at
the Protvino 70 GeV accelerator. The prototype consisted of a $3\times 3$
array with the cell sizes of $11\times 11$ cm2. Each cell had 380 layers with
1.5 mm scintillator and 0.3 mm lead. Scintillation light was collected by
optical fibers penetrating through the modules longitudinally along the beam
direction. The prototype was designed and assembled at the IHEP scintillator
workshop.
Studies were made in the electron beam energy range from 1 to 19 GeV. The
energy tagged has allowed us to measure the stochastic term in energy
resolution as $(2.8\pm 0.2)\times 10^{-2}~{}\mbox{GeV}^{1/2}$ which is
consistent with the one measured at BNL for the KOPIO project in the energy
range from 0.05 GeV to 1 GeV. Taking into account the effect of light
transmission in scintillator tiles and WLS fibers, photo statistics as well as
noise of the entire electronic chain resulted in good agreement between the
measured energy resolutions and the GEANT Monte Carlo simulations.
The stochastic term in the dependence of position resolution on energy in our
measurements is about $15.4\pm 0.3$ mm GeV1/2 which is in agreement with Monte
Carlo simulations. For 10 GeV electrons position resolution is 6 mm in the
center of the cell, and is 3 mm at a boundary between two cells.
The non-uniformity of the energy response of the prototype due to holes for
straight fibers studied with the use of electrons and MIPs has turned out to
be negligible. Monte-Carlo simulations are in a good agreement with the
obtained experimental results.
The characteristics experimentally determined for our calorimeter prototype
well meet the design goals of the PANDA and CBM experiments. However, the
final conclusion on lateral sizes of the cells as well as on Shashlyk
longitudinal sampling structure could be done only after studies of
reconstruction efficiency of $\pi^{0}$-mesons of different energies.
## Acknowledgment
This work was partially supported by the Rosatom grant with Ref. No.
N.4d.47.03.08.118, by the INTAS grants with Ref. No. 06-1000012-8845 and
06-1000012-8914, and by the RFBR grant 05-02-08009.
## References
* [1] D.Alde et al., Nucl.Instr.Meth., A240(1985),343.
* [2] O.V.Buyanov et al., Nucl.Instr.Meth., A349(1994),62.
* [3] M.Beddo et al., The STAR Barrel Electromagnetic Calorimeter. Nucl.Instr.Meth. A499 (2003) 725-739
* [4] “PHENIX calorimetrer”, L. Aphecetche et al., NIM A499 521-536 (2003)
* [5] HERA-B collaboration, HERA-B Design Report, DESY-PRC 95/01 January 1995.
* [6] LHCb Calorimeters Technical Design Report, CERN/LHCC 2000-0036, September 2000.
* [7] I.-H.Chiang et al., AGS Experimental Proposal 926, 1996\.
* [8] G.S.Atoian et al., Prepared for 11th International Conference on Calorimetry in High-Energy Physics (Calor 2004), Perugia, Italy, 28 Mar – 2 Apr 2004.
* [9] G.S.Atoian et al., NIM A 584 (2008) 291.
* [10] CBM technical status report, GSI, DOC-2005-Feb-447 (2005)
* [11] M.Kotulla et al., Technical Progress Report PANDA ”Strong Interaction Studies with Antiprotons”, February 2005.
* [12] V.A.Batarin et al., Development of a Momentum Determined Electron Beam in the 1-45 GeV Range, Nucl.Instr.Meth. A510 (2003) 211–218
* [13] V.A.Batarin et al., Precision Measurement of Energy and Position Resolutions of the BTeV Electromagnetic Calorimeter Prototype, Nucl. Instrum. and Meth. A510(2003), 248–261.
* [14] V.A.Batarin et al., LED monitoring system for the BTeV lead tungstate crystal calorimeter prototype. Nucl. Instrum. and Meth. A534(2004), 486–495.
* [15] E.Kowalski, Nuclear Electronics. Springer, Berlin (1970).
|
arxiv-papers
| 2008-09-22T12:22:56
|
2024-09-04T02:48:57.880532
|
{
"license": "Public Domain",
"authors": "Yu.V.Kharlov, P.A.Semenov, Yu.A.Matulenko, O.P.Yushchenko,\n Yu.I.Arestov, G.I.Britvich, S.K.Chernichenko, Yu.M.Goncharenko,\n A.M.Davidenko, A.A.Derevschikov, A.S.Konstantinov, V.A.Kormilitsyn,\n V.I.Kravtsov, Yu.M.Melnik, A.P.Meschanin, N.G.Minaev, V.V.Mochalov,\n D.A.Morozov, A.V.Ryazantsev, I.V.Shein, A.P.Soldatov, L.F.Soloviev,\n A.V.Soukhikh, A.N.Vasiliev, M.N.Ukhanov, V.G.Vasilchenko, A.E.Yakutin",
"submitter": "Yuri Kharlov",
"url": "https://arxiv.org/abs/0809.3671"
}
|
0809.3814
|
# Artinianness of local cohomology modules
Moharram Aghapournahr Moharram Aghapournahr
Arak University, Beheshti St, P.O. Box:879, Arak, Iran m-aghapour@araku.ac.ir
and Leif Melkersson Leif Melkersson
Department of Mathematics
Linköping University
S-581 83 Linköping
Sweden lemel@mai.liu.se
###### Abstract.
Let $A$ be a noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $M$ an
$A$–module. Some uniform theorems on the artinianness of certain local
cohomology modules are proven in a general situation. They generalize and
imply previous results about the artinianness of some special local cohomology
modules in the graded case.
###### Key words and phrases:
Local cohomology, artinian modules, cofinite modules.
###### 2000 Mathematics Subject Classification:
13D45, 13D07
## 1\. Introduction
Throughout $A$ is a commutative noetherian ring. As a general reference to
homological and commutative algebra we use [5] and [10]. One of the main
problems in the study of local cohomology modules is to determine when they
are artinian. Recently some results have been proved about the artinianness of
graded local cohomology modules in [2], [3], [13] and [14].
We prove some uniform results about artinianness of local cohomology modules
in the context of an arbitrary noetherian ring $A$. Their proofs are simple
and they have those special cases in the above references as immediate
consequences.
A Serre subcategory of the category of $A$–modules is a full subcategory
closed under taking submodules, quotient modules and extensions. An example is
given by the class of artinian $A$–modules. A useful method to prove that a
certain module belongs to such a Serre subcategory is to apply [12, Lemma
3.1].
An $A$–module $M$ is called $\mathfrak{a}$–cofinite if
$\operatorname{Supp}_{A}(M)\subset\operatorname{V}{(\mathfrak{a})}$ and
$\operatorname{Ext}^{i}_{A}(A/\mathfrak{a},M)$ is finite (finitely generated)
for all $i$. This notion was introduced by Hartshorne in [8]. For more
information about cofiniteness with respect to an ideal, see [9], [6] and
[12].
## 2\. Main results
###### Theorem 2.1.
Let $\mathcal{S}$ be a Serre subcategory of the category of $A$–modules. Let
$M$ be a finite $A$ module and let $\mathfrak{a}$ be an ideal of $A$ such that
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ belongs to $\mathcal{S}$ for all
$i>n$. If $\mathfrak{b}$ is an ideal of $A$ such that
$\operatorname{H}^{n}_{\mathfrak{a}}(M/{\mathfrak{b}}M)$ belongs to
$\mathcal{S}$, then the module
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
belongs to $\mathcal{S}$.
###### Proof.
Suppose
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
is not in $\mathcal{S}$. Let $N$ be a maximal submodule of $M$ such that
$\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\otimes_{A}{A/\mathfrak{b}}$ is not
in $\mathcal{S}$. Let $L\supset N$ be such that
$\operatorname{\Gamma}_{\mathfrak{b}}(M/N)=L/N$. Since
$\operatorname{Supp}_{R}(L/N)\subset{\operatorname{V}(\mathfrak{b})\cap\operatorname{Supp}_{R}(M)}$,
$\operatorname{H}^{i}_{\mathfrak{a}}(L/N)$ belongs to $\mathcal{S}$ for all
$i\geq n$ by [1, Theorem 3.1].
From the exact sequence $0\rightarrow L/N\rightarrow M/N\rightarrow
M/L\rightarrow 0$, we get the exact sequence
$\operatorname{H}^{n}_{\mathfrak{a}}(L/N)\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\overset{f}{\longrightarrow}\operatorname{H}^{n}_{\mathfrak{a}}(M/L)\longrightarrow\operatorname{H}^{n+1}_{\mathfrak{a}}(L/N)$.
$\operatorname{Tor}^{A}_{i}(A/\mathfrak{b},\operatorname{Ker}f)$ and
$\operatorname{Tor}^{A}_{i}(A/\mathfrak{b},\operatorname{Coker}f)$ are in
$\mathcal{S}$ for all $i$, because $\operatorname{Ker}f$ and
$\operatorname{Coker}f$ are in $\mathcal{S}$. It follows from [12, Lemma 3.1],
that $\operatorname{Ker}(f\otimes{A/\mathfrak{b}})$ and
$\operatorname{Coker}(f\otimes{A/\mathfrak{b}})$ are in $\mathcal{S}$. Since
$\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\otimes_{A}{A/\mathfrak{b}}$ is not
in $\mathcal{S}$, the module
$\operatorname{H}^{n}_{\mathfrak{a}}(M/L)\otimes_{A}{A/\mathfrak{b}}$ can not
be in $\mathcal{S}$. By the maximality of $N$, we get $L=N$. We have shown
that $\operatorname{\Gamma}_{\mathfrak{b}}(M/N)=0$ and therefore we can take
$x\in\mathfrak{b}$ such that the sequence $0\rightarrow
M/N\overset{x}{\rightarrow}M/N\rightarrow M/(N+{x}M)\rightarrow 0$ is exact.
Thus we get the exact sequence
$\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\overset{x}{\longrightarrow}\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/N+{x}M)\longrightarrow\operatorname{H}^{n+1}_{\mathfrak{a}}(M/N).$
This yields the exact sequence
$0\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/N)/{x}\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/N+{x}M)\longrightarrow
C\longrightarrow 0$,
where $C\subset\operatorname{H}^{n+1}_{\mathfrak{a}}(M/N)$ and thus $C$ is in
$\mathcal{S}$.
Note that $x\in\mathfrak{b}$. Hence we get the exact sequence
$\operatorname{Tor}^{A}_{1}(A/\mathfrak{b},C)\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\otimes_{A}{A/\mathfrak{b}}\longrightarrow\operatorname{H}^{n}_{\mathfrak{a}}(M/(N+{x}M))\otimes_{A}{A/\mathfrak{b}}$
However $N\subsetneqq{(N+{x}M)}$ and therefore
$\operatorname{H}^{n}_{\mathfrak{a}}(M/(N+{x}M))\otimes_{A}{A/\mathfrak{b}}$
belongs to $\mathcal{S}$ by the maximality of $N$. Consequently
$\operatorname{H}^{n}_{\mathfrak{a}}(M/N)\otimes_{A}{A/\mathfrak{b}}$ is in
$\mathcal{S}$ which is a contradiction. ∎
###### Corollary 2.2.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$. Let $M$ be a
finite $A$–module. If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is artinian for
$i>n$ and $\operatorname{H}^{n}_{\mathfrak{a}}(M/{\mathfrak{b}}M)$ is
artinian, then
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
is artinian.
###### Proof.
In 2.1 take $\mathcal{S}$ as the category of artinian $A$–modules. ∎
###### Corollary 2.3.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. If
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is artinian for $i>n$, then
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
is artinian.
###### Proof.
Note that
$\operatorname{H}^{n}_{\mathfrak{a}}(M/{\mathfrak{b}}M)\cong\operatorname{H}^{n}_{\mathfrak{a}+{\mathfrak{b}}}(M/{\mathfrak{b}}M)$
is artinian. ∎
###### Corollary 2.4.
If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is artinian for $i>n$, where
$n\geq 1$ then
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{a}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
is artinian.
###### Proof.
Note that $\operatorname{H}^{n}_{\mathfrak{a}}(M/{\mathfrak{a}}M)=0$ for
$n\geq 1$ ∎
###### Remark 2.5.
In 2.4 we must assume that $n\geq 1$. Take any ideal $\mathfrak{a}$ in a ring
$A$ such that $A/\mathfrak{a}$ is not artinian. Let $M=A/\mathfrak{a}$. Then
$\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for $i\geq 1$, and
$\operatorname{\Gamma}_{\mathfrak{a}}(M)=M$. On the other hand
$M/{\mathfrak{a}}M\cong M$. Thus
$\operatorname{H}^{0}_{\mathfrak{a}}(M)/{\mathfrak{a}}\operatorname{H}^{0}_{\mathfrak{a}}(M)$
is not artinian.
###### Corollary 2.6.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$. Let $M$ be a
finite $A$–module. If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is
$\mathfrak{a}$–cofinite artinian (resp. has finite support) for $i>n$, and
$\operatorname{H}^{n}_{\mathfrak{a}}(M/{\mathfrak{b}}M)$ is
$\mathfrak{a}$–cofinite artinian (resp. has finite support) then
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{b}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
is $\mathfrak{a}$–cofinite artinian (resp. has finite support). In particular,
if $n\geq 1$ then
$\operatorname{H}^{n}_{\mathfrak{a}}(M)/{\mathfrak{a}}\operatorname{H}^{n}_{\mathfrak{a}}(M)$
has finite length (resp. has finite support).
###### Corollary 2.7.
If $c=\operatorname{cd}(\mathfrak{a},M)>0$, then
$\operatorname{H}^{c}_{\mathfrak{a}}(M)={\mathfrak{a}}\operatorname{H}^{c}_{\mathfrak{a}}(M)$.
As a corollary we recover Yoshida’s theorem [15, Proposition 3.1].
###### Corollary 2.8.
$c=\operatorname{cd}(\mathfrak{a},M)>0$, then
$\operatorname{H}^{c}_{\mathfrak{a}}(M)$ is not finite.
###### Proof.
We may assume that $(A,\mathfrak{m})$ is a local ring. Use corollary 2.7 and
Nakayama’s lemma. ∎
###### Proposition 2.9.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of $A$ with
$A/{\mathfrak{a}}+\mathfrak{b}$ artinian. If the module $M$ is
$\mathfrak{a}$–cofinite, then $\operatorname{H}^{i}_{\mathfrak{b}}(M)$ is
artinian for all $i$.
###### Proof.
Use [12, Theorem 5.5 (i)$\Leftrightarrow$(iv)] and the fact that an
$\mathfrak{a}$–cofinite module has finite Bass numbers. ∎
###### Corollary 2.10.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. If $\mathfrak{a}$ is a principle
ideal and $M$ is a finite module, then
$\operatorname{H}^{i}_{\mathfrak{b}}(\operatorname{H}^{j}_{\mathfrak{a}}(M))$
is artinian for all $i,j$.
###### Proof.
Note that $\operatorname{H}^{j}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite
for all $j$ when $\mathfrak{a}$ is principle. ∎
###### Theorem 2.11.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. Let $M$ be a finite module such
that $\dim M/{\mathfrak{a}}M\leq 1$. Then
$\operatorname{H}^{i}_{\mathfrak{b}}(\operatorname{H}^{j}_{\mathfrak{a}}(M))$
is an artinian module for all $i,j$.
###### Proof.
Since $A/{\mathfrak{a}}+\mathfrak{b}$ is artinian,
$\operatorname{Supp}_{A}(\operatorname{H}^{j}_{\mathfrak{a}}(M))\subset\operatorname{V}(\mathfrak{a}+{\mathfrak{b}})$
and $\operatorname{V}(\mathfrak{a}+{\mathfrak{b}})$ consists of finitely many
maximal ideals. Thus we may assume that $(A,\mathfrak{m})$ is local.
Passing to the ring $A/\operatorname{Ann}(M)$ from [6, Theorem 1] and the
change of rings principle [12, Corollary 2.6], we get that
$\operatorname{H}^{j}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for all
$j$. We use theorem 2.9 to deduce that
$\operatorname{H}^{i}_{\mathfrak{b}}(\operatorname{H}^{j}_{\mathfrak{a}}(M))$
is artinian for all $i,j$ ∎
###### Theorem 2.12.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. Let $M$ be a finite module such
that $\dim M/{\mathfrak{a}}M\leq 2$. Then
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
is an artinian module for all $i$.
###### Proof.
We first reduce to the case $\operatorname{\Gamma}_{\mathfrak{b}}(M)=0$. From
the exact sequence
$0\longrightarrow\operatorname{\Gamma}_{\mathfrak{b}}(M)\longrightarrow
M\overset{f}{\longrightarrow}\overline{M}\longrightarrow 0$,
where $\overline{M}=M/\operatorname{\Gamma}_{\mathfrak{b}}(M)$, we get the
exact sequence
$\operatorname{H}^{i}_{\mathfrak{a}}(\operatorname{\Gamma}_{\mathfrak{b}}(M))\longrightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M)\overset{\operatorname{H}^{i}_{\mathfrak{a}}(f)}{\longrightarrow}\operatorname{H}^{i}_{\mathfrak{a}}(\overline{M})\longrightarrow\operatorname{H}^{i+1}_{\mathfrak{a}}(\operatorname{\Gamma}_{\mathfrak{b}}(M)).$
Note that
$\operatorname{H}^{i}_{\mathfrak{a}}(\operatorname{\Gamma}_{\mathfrak{b}}(M))\cong\operatorname{H}^{i}_{\mathfrak{a}+{\mathfrak{b}}}(\operatorname{\Gamma}_{\mathfrak{b}}(M))$,
and so is artinian for all $i$. Since
$\operatorname{Ker}\operatorname{H}^{i}_{\mathfrak{a}}(f)$ and
$\operatorname{Coker}\operatorname{H}^{i}_{\mathfrak{a}}(f)$ are artinian, it
follows from [12, Lemma 3.1] that
$\operatorname{Ker}\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(f))$
and
$\operatorname{Coker}\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(f))$
are artinian. Hence
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
is artinian if and only if
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(\overline{M}))$
is artinian.
Thus we may assume that $\operatorname{\Gamma}_{\mathfrak{b}}(M)=0$.
Take $x\in\mathfrak{b}$ outside all associated prime ideals of $M$ and outside
all prime ideals $\mathfrak{p}\supset\mathfrak{a}+\operatorname{Ann}(M)$, such
that $\dim A/\mathfrak{p}=2$. Then $\dim M/(\mathfrak{a}+{xA})M\leq 1$ and $x$
is a non-zerodivisor on $M$. Hence we get the long exact sequence
$\dots\rightarrow\operatorname{H}^{i-1}_{\mathfrak{a}}(M/{x}M)\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M)\overset{x}{\rightarrow}\operatorname{H}^{i}_{\mathfrak{a}}(M)\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M/{x}M)\rightarrow\dots$.
Consider the map $f$ which is multiplication by $x$ on
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$. We have the exact sequence
$0\longrightarrow\operatorname{H}^{i-1}_{\mathfrak{a}}(M)/{x}\operatorname{H}^{i-1}_{\mathfrak{a}}(M)\longrightarrow\operatorname{H}^{i-1}_{\mathfrak{a}}(M/{x}M)\longrightarrow\operatorname{Ker}f\longrightarrow
0$
and hence the exact sequence
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i-1}_{\mathfrak{a}}(M/{x}M))\longrightarrow\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{Ker}f)\longrightarrow\operatorname{H}^{2}_{\mathfrak{b}}(\operatorname{H}^{i-1}_{\mathfrak{a}}(M)/{x}\operatorname{H}^{i-1}_{\mathfrak{a}}(M))$.
However the last term is zero since
$\operatorname{Supp}_{A}(\operatorname{H}^{i-1}_{\mathfrak{a}}(M)/{x}\operatorname{H}^{i-1}_{\mathfrak{a}}(M))\subset\operatorname{Supp}_{A}(M/(\mathfrak{a}+{xA})M)$
and $\dim M/(\mathfrak{a}+{xA})M\leq 1$. Consequently
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{Ker}f)$ is artinian, since
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i-1}_{\mathfrak{a}}(M/{x}M))$
is artinian by theorem 2.11. From the exactness of
$0\rightarrow\operatorname{Coker}f\rightarrow\operatorname{H}^{i}_{\mathfrak{a}}(M/{x}M)$,
we get the exactness of
$0\rightarrow\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{Coker}f)\rightarrow\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M/{x}M))$.
Again we use 2.11 to deduce that
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M/{x}M)$
is artinan . Hence
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{Coker}f)$ is artinian.
Since we now have shown that
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{Ker}f)$ and
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{Coker}f)$ are artinian, we
are able to apply [12, Lemma 3.1] to deduce that
$\operatorname{Ker}\operatorname{H}^{1}_{\mathfrak{b}}(f)$ is artinian. But
$\operatorname{H}^{1}_{\mathfrak{b}}(f)$ is just multiplication by $x$ on
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$.
It follows from [4, Theorem 7.1.2] that
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
is artinian. ∎
The following lemma is a generalization of [11, Corollary 1.8].
###### Lemma 2.13.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. If $M$ is a module such that
$\operatorname{Supp}_{A}(M)\subset\operatorname{V}(\mathfrak{a})$ and
$0:_{M}{\mathfrak{a}}$ is finite, then
$\operatorname{\Gamma}_{\mathfrak{b}}(M)$ is $\mathfrak{a}$–cofinite artinian.
###### Proof.
The module $0:_{\operatorname{\Gamma}_{\mathfrak{b}}(M)}{\mathfrak{a}}\subset
0:_{M}{\mathfrak{a}}$ and is therefore finite. But the support of
$0:_{\operatorname{\Gamma}_{\mathfrak{b}}(M)}{\mathfrak{a}}$ is contained in
$\operatorname{V}({\mathfrak{a}+{\mathfrak{b}}})$, thus
$0:_{\operatorname{\Gamma}_{\mathfrak{b}}(M)}{\mathfrak{a}}$ has finite
length. Therefore $\operatorname{\Gamma}_{\mathfrak{b}}(M)$ is
$\mathfrak{a}$–cofinite artinian, by [12, Proposition 4.1]. ∎
###### Corollary 2.14.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. If $M$ is a module such that
$\operatorname{Ext}^{n}_{A}(A/\mathfrak{a},M)\text{ and
}\operatorname{Ext}^{n+1-j}_{A}(A/\mathfrak{a},\operatorname{H}^{j}_{\mathfrak{a}}(M))$
for all $j<n$, are finite, then
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{n}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$–cofinite artinian.
###### Proof.
Note that by [7, Theorem 6.3.9 (b)]
$\operatorname{Hom}_{A}(A/\mathfrak{a},\operatorname{H}^{n}_{\mathfrak{a}}(M))$
is finite, hence by lemma 2.13,
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{n}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$–cofinite artinian. ∎
###### Corollary 2.15.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. If $M$ is a finite module such
that $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for
$i<n$, then
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
is $\mathfrak{a}$–cofinite artinian for all $i\leq n$.
###### Theorem 2.16.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. For each finite module $M$, the
modules
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$
and
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$
are artinian.
###### Proof.
Corollary 2.15 with $n=1$ implies that
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$
is artinian. We may assume that $\operatorname{\Gamma}_{\mathfrak{a}}(M)=0$,
so there is an $M$–regular element $x$ in $\mathfrak{a}$. From the exact
sequence $0\rightarrow M\overset{x}{\rightarrow}M\rightarrow M/{x}M\rightarrow
0$, we get the exact sequence
(1)
$0\longrightarrow\operatorname{\Gamma}_{\mathfrak{a}}(M/{x}M)\longrightarrow\operatorname{H}^{1}_{\mathfrak{a}}(M)\overset{x}{\longrightarrow}\operatorname{H}^{1}_{\mathfrak{a}}(M)\longrightarrow\operatorname{H}^{1}_{\mathfrak{a}}(M/{x}M).$
Consider the map $f$ defined as multiplication with $x$ on
$\operatorname{H}^{1}_{\mathfrak{a}}(M)$ occuring in (1). We get
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{Ker}f)\cong\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{\Gamma}_{\mathfrak{a}}(M/{x}M))\cong\operatorname{H}^{1}_{\mathfrak{a}+{\mathfrak{b}}}(\operatorname{\Gamma}_{\mathfrak{a}}(M/{x}M))$,
which is artinian and the exact sequence
$0\longrightarrow\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{Coker}f)\longrightarrow\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M/{x}M))$
Since
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M/{x}M))$
artinian by 2.15,
$\operatorname{\Gamma}_{\mathfrak{b}}(\operatorname{Coker}f)$ is artinian. We
can use [12, Lemma 3.1] with $S=\operatorname{\Gamma}_{\mathfrak{b}}(-)$ and
$T=\operatorname{H}^{1}_{\mathfrak{b}}(-)$ to conclude that
$\operatorname{Ker}\operatorname{H}^{1}_{\mathfrak{b}}(f)$ is artinian. But
$\operatorname{H}^{1}_{\mathfrak{b}}(f)$ is multiplication by the element
$x\in\mathfrak{a}$ on
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$.
Again using [4, Theorem 7.1.2], we conclude that
$\operatorname{H}^{1}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$
is artinian. ∎
###### Lemma 2.17.
Let $f:L\longrightarrow M$ be $A$–linear and $\mathfrak{c}$ an ideal of $A$.
Let $\mathcal{S}$ be a Serre subcategory of the category of $A$–modules. If
$\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(f)$ and
$\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(f)$ are in
$\mathcal{S}$ for all $j$, then for each $i$:
$\operatorname{H}^{i+2}_{\mathfrak{c}}(\operatorname{Ker}f)$ is in
$\mathcal{S}$ if and only if
$\operatorname{H}^{i}_{\mathfrak{c}}(\operatorname{Coker}f)$ is in
$\mathcal{S}$.
###### Proof.
Let $K=\operatorname{Ker}f,\,I=\operatorname{Im}f$ and
$C=\operatorname{Coker}f$. Factorize $f$ as $f=h\circ{g}$, where
$0\longrightarrow K\longrightarrow
L\overset{g}{\longrightarrow}I\longrightarrow 0$
and
$0\longrightarrow I\overset{h}{\longrightarrow}M\longrightarrow
C\longrightarrow 0$
are exact. For each $j$, we have the exact sequence
$0\rightarrow\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(g)\rightarrow\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(f)\rightarrow\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(h)\rightarrow\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(g)\rightarrow\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(f)\rightarrow\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(h)\rightarrow
0.$
The hypothesis implies that
$\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(g)$ and
$\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(h)$ are in
$\mathcal{S}$ for each $j$. For each $j$,
$\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{c}}(h)$ is in $\mathcal{S}$
if and only if $\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{c}}(g)$ is
in $\mathcal{S}$. Moreover there are exact sequences
$\operatorname{H}^{i}_{\mathfrak{c}}(I)\overset{\operatorname{H}^{i}_{\mathfrak{c}}(h)}{\longrightarrow}\operatorname{H}^{i}_{\mathfrak{c}}(M)\longrightarrow\operatorname{H}^{i}_{\mathfrak{c}}(C)\longrightarrow\operatorname{H}^{i+1}_{\mathfrak{c}}(I)\overset{\operatorname{H}^{i+1}_{\mathfrak{c}}(h)}{\longrightarrow}\operatorname{H}^{i+1}_{\mathfrak{c}}(M)$
and
$\operatorname{H}^{i+1}_{\mathfrak{c}}(L)\overset{\operatorname{H}^{i+1}_{\mathfrak{c}}(g)}{\longrightarrow}\operatorname{H}^{i+1}_{\mathfrak{c}}(I)\longrightarrow\operatorname{H}^{i+2}_{\mathfrak{c}}(K)\longrightarrow\operatorname{H}^{i+2}_{\mathfrak{c}}(L)\overset{\operatorname{H}^{i+2}_{\mathfrak{c}}(g)}{\longrightarrow}\operatorname{H}^{i+2}_{\mathfrak{c}}(I).$
It follows that
$\begin{matrix}\operatorname{H}^{i}_{\mathfrak{c}}(C)\in\mathcal{S}&\Leftrightarrow&\operatorname{Ker}\operatorname{H}^{i+1}_{\mathfrak{c}}(h)\in\mathcal{S}\\\
&\Leftrightarrow&\operatorname{Coker}\operatorname{H}^{i+1}_{\mathfrak{c}}(g)\in\mathcal{S}\\\
&\Leftrightarrow&\operatorname{H}^{i+2}_{\mathfrak{c}}(K)\in\mathcal{S}&\end{matrix}$
. ∎
###### Theorem 2.18.
Let $\mathfrak{a}$ and $\mathfrak{b}$ be two ideals of $A$ such that
$A/{\mathfrak{a}}+\mathfrak{b}$ is artinian. Assume that
$\operatorname{ara}(\mathfrak{a})=2$. Then for each finite $A$–module $M$, the
module
$\operatorname{H}^{i}_{\mathfrak{b}}(\operatorname{H}^{2}_{\mathfrak{a}}(M))$
is artinian if and only if the module
$\operatorname{H}^{i+2}_{\mathfrak{b}}(\operatorname{H}^{1}_{\mathfrak{a}}(M))$
is artinian.
###### Proof.
We may assume that $\operatorname{\Gamma}_{\mathfrak{a}}(M)=0$. Then
$\mathfrak{a}$ can be generated by $M$–regular elements $x$ and $y$ on $M$.
(See [10, Exercise 16.8]). Then there is an exact sequence [4, Proposition
8.1.2]
$0\longrightarrow\operatorname{H}^{1}_{\mathfrak{a}}(M)\longrightarrow\operatorname{H}^{1}_{xA}(M)\overset{f}{\longrightarrow}\operatorname{H}^{1}_{xA}(M_{y})\longrightarrow\operatorname{H}^{2}_{\mathfrak{a}}(M)\longrightarrow
0.$
In order to apply lemma 2.17 we prove that
$\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{b}}(f)$ and
$\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{b}}(f)$ are artinian for
all $j$. By [4, Proposition 8.1.2], there is an exact sequence
$\operatorname{H}^{j}_{\mathfrak{b}+{y}A}(\operatorname{H}^{1}_{xA}(M))\longrightarrow\operatorname{H}^{j}_{\mathfrak{b}}(\operatorname{H}^{1}_{xA}(M))\overset{\operatorname{H}^{j}_{\mathfrak{b}}(f)}{\longrightarrow}\operatorname{H}^{j}_{\mathfrak{b}}(\operatorname{H}^{1}_{xA}(M)_{y})\longrightarrow\operatorname{H}^{j+1}_{\mathfrak{b}+{y}A}(\operatorname{H}^{1}_{xA}(M)).$
Since $\operatorname{H}^{1}_{xA}(M)$ is ${xA}$–cofinite, the outer modules are
artinian, by 2.9. Hence
$\operatorname{Ker}\operatorname{H}^{j}_{\mathfrak{b}}(f)$ and
$\operatorname{Coker}\operatorname{H}^{j}_{\mathfrak{b}}(f)$ are artinian for
all $j$. ∎
## References
* [1] M. Aghapournahr, L. Melkersson, Local cohomology and Serre subcategories, J. Algebra 320(2008), 1275-1287.
* [2] M. Brodmann, S. Fomasoli, R. Tajarod, Local cohomology over homogeneous rings with one-dimensional local base ring, Proc. Amer. Math. Soc. 131(2003), 2977-2985.
* [3] M. Brodmann, F. Rohrer, R. Sazeedeh, Multiplicities of graded components of local cohomology modules, J. Pure Appl. Alg. 197(2005), 249-278.
* [4] M.P. Brodmann, R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998.
* [5] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, revised ed., 1998.
* [6] D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Alg. 121(1997), 45-52.
* [7] M.T. Dibaei, S. Yassemi, Associated primes of the local cohomology modules. Abelian groups, rings, modules, and homological algebra, 49–56, Chapman and Hall/CRC, 2006.
* [8] R. Hartshorne, _Affine duality and cofiniteness_ , Invent. Math. 9 (1970), 145–164.
* [9] C. Huneke, J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc. 110(1991), 421-429.
* [10] H. Matsumura, _Commutative ring theory_ , Cambridge University Press, 1986.
* [11] L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Camb. Phil. Soc. 125(1999), 417-423.
* [12] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285(2005), 649-668.
* [13] R. Sazeedeh, Finiteness of graded local cohomology modules, J. Pure Appl. Alg. 212(2008), 275-280.
* [14] R. Sazeedeh, Artinianness of graded local cohomology modules, Proc. Amer. Math. Soc. 135(2007), 2339-2345.
* [15] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147(1997), 179-191.
|
arxiv-papers
| 2008-09-22T21:28:21
|
2024-09-04T02:48:57.886313
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moharram Aghapournahr and Leif Melkersson",
"submitter": "Moharram Aghapournahr",
"url": "https://arxiv.org/abs/0809.3814"
}
|
0809.3835
|
# Introduction to scattering for radial $3D$ NLKG below energy norm
Tristan Roy University of California, Los Angeles triroy@math.ucla.edu
###### Abstract.
We prove scattering for the radial nonlinear Klein-Gordon equation
$\left\\{\begin{array}[]{ccl}\partial_{tt}u-\Delta u+u&=&-|u|^{p-1}u\\\
u(0,x)&=&u_{0}(x)\\\ \partial_{t}u(0,x)&=&u_{1}(x)\end{array}\right.$
with $5>p>3$ and data $\left(u_{0},\,u_{1}\right)\in H^{s}\times H^{s-1}$,
$1>s>1-\frac{(5-p)(p-3)}{2(p-1)(p-2)}$ if $4\geq p>3$ and
$1>s>1-\frac{(5-p)^{2}}{2(p-1)(6-p)}$ if $5>p\geq 4$. First we prove
Strichartz-type estimates in $L_{t}^{q}L_{x}^{r}$ spaces. Then by using these
decays we establish some local bounds. By combining these results to a
Morawetz-type estimate and a radial Sobolev inequality we control the
variation of an almost conserved quantity on arbitrary large intervals. Once
we have showed that this quantity is controlled, we prove that some of these
local bounds can be upgraded to global bounds. This is enough to establish
scattering. All the estimates involved require a delicate analysis due to the
nature of the nonlinearity and the lack of scaling.
## 1\. Introduction
In this paper we consider the $p$\- defocusing Klein-Gordon equation on
$\mathbb{R}^{3}$
(1.1) $\begin{array}[]{ll}\partial_{tt}u-\Delta u+u&=-|u|^{p-1}u\\\
\end{array}$
with data $u(0)=u_{0}$, $\partial_{t}u(0)=u_{1}$ lying in $H^{s}$, $H^{s-1}$
respectively. Here $H^{s}$ is the standard inhomogeneous Sobolev space i.e
$H^{s}$ is the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^{3})$
with respect to the norm
(1.2)
$\begin{array}[]{ll}\|f\|_{H^{s}}&:=\|<D>^{s}f\|_{L^{2}(\mathbb{R}^{3})}\end{array}$
where $<D>$ is the operator defined by
(1.3)
$\begin{array}[]{ll}\widehat{<D>^{s}f}(\xi)&:=(1+|\xi|)^{s}\hat{f}(\xi)\end{array}$
and $\hat{f}$ denotes the Fourier transform
(1.4)
$\begin{array}[]{ll}\hat{f}(\xi)&:=\int_{\mathbb{R}^{3}}f(x)e^{-ix\cdot\xi}\,dx\end{array}$
We are interested in the strong solutions of the $p$\- defocusing Klein-Gordon
equation on some interval $[0,T]$ i.e maps $u$, $\partial_{t}u$ that lie in
$C\left([0,\,T],\,H^{s}(\mathbb{R}^{3})\right)$,
$C\left([0,\,T],\,H^{s-1}(\mathbb{R}^{3})\right)$ respectively and that
satisfy
(1.5)
$\begin{array}[]{ll}u(t)&=\cos{(t<D>)}u_{0}+\frac{\sin(t<D>)}{<D>}u_{1}-\int_{0}^{t}\frac{\sin\left((t-t^{{}^{\prime}})<D>\right)}{<D>}|u|^{p-1}(t^{{}^{\prime}})u(t^{{}^{\prime}})\,dt^{{}^{\prime}}\end{array}$
The $p$\- defocusing Klein-Gordon equation is closely related to the $p$\-
defocusing wave equation i.e
(1.6) $\begin{array}[]{ll}\partial_{tt}v-\triangle v=&-|v|^{p-1}v\end{array}$
with data $v(0)=v_{0}$, $\partial_{t}v(0)=v_{1}$. (1.6) enjoys the following
scaling property
(1.7)
$\begin{array}[]{ll}v(t,x)&\rightarrow\frac{1}{\lambda^{\frac{2}{p-1}}}u\left(\frac{t}{\lambda},\frac{x}{\lambda}\right)\\\
v_{0}(x)&\rightarrow\frac{1}{\lambda^{\frac{2}{p-1}}}u_{0}\left(\frac{x}{\lambda}\right)\\\
v_{1}(x)&\rightarrow\frac{1}{\lambda^{\frac{2}{p-1}+1}}u_{1}\left(\frac{x}{\lambda}\right)\end{array}$
We define the critical exponent $s_{c}:=\frac{3}{2}-\frac{2}{p-1}$. One can
check that the $\dot{H}^{s_{c}}\times\dot{H}^{s_{c}-1}$ norm of
$(u_{0},u_{1})$ is invariant under the transformation (1.7) 111 Here
$\dot{H}^{m}$ denotes the standard homogeneous Sobolev space endowed with the
norm $\|f\|_{\dot{H}^{m}}:=\|D^{m}f\|_{L^{2}(\mathbb{R}^{3})}$. (1.6) was
demonstrated to be locally well-posed by Lindblad and Sogge [7] in
$H^{s}\times H^{s-1}$, $s>\frac{3}{2}-\frac{2}{p-1}$, $p>3$ by using an
iterative argument. In fact their results extend immediately to (1.1) 222by
rewriting for example (1.1) in the ”wave” form $\partial_{tt}u-\triangle
u=-|u|^{p-1}u-u$.
If $p=5$ then $s_{c}=1$ and this is why we say that that the nonlinearity
$|u|^{p-1}u$ is $\dot{H}^{1}$ critical. If $3<p<5$ then $s_{c}<1$ and the
regime is $\dot{H}^{1}$ subcritical.
It is well-known that smooth solutions to (1.1) have a conserved energy
(1.8)
$\begin{array}[]{ll}E(u(t))&:=\frac{1}{2}\int_{\mathbb{R}^{3}}\left|\partial_{t}u(t,x)\right|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla
u(t,x)|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{3}}|u(t,x)|^{2}\,dx+\frac{1}{p+1}\int_{\mathbb{R}^{3}}|u(t,x)|^{p+1}\,dx\end{array}$
In fact by standard limit arguments the energy conservation law remains true
for solutions $(u,\partial_{t}u)\in H^{s}\times H^{s-1}$, $s\geq 1$.
Since the lifespan of the local solution depends only on the $H^{s}\times
H^{s-1}$ norm of the initial data $(u_{0},u_{1})$ (see [7]) then it suffices
to find an a priori pointwise in time bound in $H^{s}\times H^{s-1}$ of the
solution $(u,\partial_{t}u)$ to establish global well-posedness. The energy
captures the evolution in time of the $H^{1}\times L^{2}$ norm of the
solution. Since it is conserved we have global existence of (1.1).
The scattering theory (namely, the existence of the bijective wave operators)
in the energy space 333i.e with data $(u_{0},u_{1})\in H^{1}\times L^{2}$ for
(1.1) has been extensively studied for a large range of exponents $p$. In
particular Brenner [1, 2] was able to prove that if $\frac{7}{3}<p<5$, then
every solution scatters as $T$ goes to infinity. In fact he showed scattering
for all dimension $n$, $n\geq 3$ and for all exponent $p$ that is
$\dot{H}^{1}$ subcritical and $L^{2}$ supercritical 444since if
$p>1+\frac{4}{n}$ then $s_{c}>0$, i.e $1+\frac{4}{n}<p<1+\frac{4}{n-2}$. Later
Nakanishi ([10], [11]) was able to extend these results to $n=1$ and $2$.
In this paper we are interested in proving scattering results for data below
the energy norm i.e for $s<1$. We will assume that (1.1) has radial data. The
main result of this paper is the following one
###### Theorem 1.1.
The $p$-radial defocusing Klein-Gordon equation on $\mathbb{R}^{3}$ is
globally well-posed in $H^{s}\times H^{s-1}$, $1>s>s(p)$ and there exists a
scattering state $\left(u_{+,0},u_{+,1}\right)\in H^{s}\times H^{s-1}$ such
that
(1.9)
$\begin{array}[]{ll}\lim\limits_{T\rightarrow\infty}\|\left(u(T),\partial_{t}u(T)\right)-\left(\cos{(T<D>)}u_{+,0}+\frac{\sin{(T<D>)}}{<D>}u_{+,1}\right)\|_{H^{s}\times
H^{s-1}}&=0\end{array}$
Here $3<p<5$ and
(1.10)
$\begin{array}[]{ll}s_{p}&:=\left\\{\begin{array}[]{l}1-\frac{(5-p)(p-3)}{2(p-1)(p-2)},\,3<p\leq
4\\\ 1-\frac{(5-p)^{2}}{2(p-1)(6-p)},\,4\leq p<5\end{array}\right.\end{array}$
We set some notation that appear throughout the paper.
We write
$A=A\left(v,\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}},\,a_{1},...,a_{n},\,\right)$
if $A$ depends on a function $v$, the norm of the initial data and some
parameters $a_{1}$, … ,$a_{n}$. Given
$B=B\left(v,\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}},\,b_{1},\,b_{2},\,...,b_{l}\right)$
and
$C=\left(v,\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}},\,c_{1},\,c_{2},\,...,\,c_{m}\right)$,
$B\lesssim C$ means that there exists a constant
$K=K\left(b_{1},\,...,\,b_{l},\,c_{1},\,...,\,c_{m},\,\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}}\right)$
that does not depend on $v$ and such that $A\leq KB$. Sometimes we write
$A\lesssim_{q_{1},\,q_{2},\,...}B$ if we want to stress upon the fact that the
constant $K$ depends on $q_{1}$, $q_{2}$,…. We say that $K_{0}$ is the
constant determined by $\lesssim$ in the inequality $B\lesssim C$ if $K_{0}$
is the smallest $K$ such that $B\leq KC$ is true. We write $B\sim C$ when
$B\lesssim C$ and $C\lesssim B$. $A<<B$ denotes $A\leq KB$ for some universal
constant $K<\frac{1}{100}$. We say that a number $\alpha$ is small if there
exists a constant $\alpha_{0}=\alpha_{0}(p)$ 555with $p$ defined in (1.1) such
that $0<\alpha<\alpha_{0}$ and $\alpha_{0}<\frac{1}{100}$. Given $a$ and $M$
two real numbers we denote by $M^{a+}$, $M^{a-}$ the number $M^{a+f(\alpha)}$,
$M^{a-f(\alpha)}$ respectively with $\alpha$ small and $f$ some function such
that $f(\alpha)$ non negative and $\lim\limits_{\alpha\to 0^{+}}f(\alpha)=0$.
If an inequality involves $M^{a+}$ or $M^{a-}$ we do not try at first to
determine the function $f$ in order to avoid too many complicated
computations. However we also write the same inequality into square brackets
with an explicit formula for $f$ for the reader interested in the details. For
instance $f(\alpha)=\alpha(p+2)$ and $N^{+}$ means $N^{\alpha(p+2)}$ for
$\alpha$ small in the inequality (2.15): this is indicated in (2.16). If an
inequality involves several slight variations we might be interested in
comparing them in order to determine after simplification what the sign of the
total variation is. For instance assume that we want to simplify the fraction
$X:=\frac{N^{1+5\alpha}}{N^{2\alpha}}$. If we rewrite $X$ in the symbolic form
$\frac{N^{1+}}{N^{+}}$ then we cannot conclude. This is why we will write it
in the following form $\frac{N^{1++}}{N^{+}}$ so that we can conclude that
$X=N^{1+}=N^{1+3\alpha}$.
Let $\nabla$ denote the gradient operator. Let $s_{c}$,
$\theta_{1}$,…,$\theta_{3}$ denote the following numbers
(1.11) $\begin{array}[]{ll}s_{c}&:=\frac{3}{2}-\frac{2}{p-1}\end{array}$
(1.12)
$\begin{array}[]{ll}\theta_{1}&:=\left\\{\begin{array}[]{l}\frac{(2s-1)(4-p)}{s(p-1)(p-2)},\,3<p\leq
4\\\ \frac{(4s-1)(p-4)}{s(p-1)(6-p)},\,4\leq p<5\end{array}\right.\end{array}$
(1.13)
$\begin{array}[]{ll}\theta_{2}&:=\left\\{\begin{array}[]{l}\frac{(p+2)(p-3)}{(p-1)(p-2)},\,3<p\leq
4\\\ \frac{(p+2)(5-p)}{(6-p)(p-1)},\,4\leq p<5\end{array}\right.\end{array}$
and
(1.14)
$\begin{array}[]{ll}\theta_{3}&:=\left\\{\begin{array}[]{l}\frac{4-p}{s(p-1)(p-2)},\,3<p\leq
4\\\ \frac{p-4}{s(p-1)(6-p)},\,4\leq p<5\end{array}\right.\end{array}$
We write $F(v)$ for the following function
(1.15) $\begin{array}[]{ll}F(v)&:=|v|^{p-1}v\end{array}$
Let $I$ be the following multiplier
(1.16) $\begin{array}[]{ll}\widehat{If}(\xi)&:=m(\xi)\hat{f}(\xi)\end{array}$
where $m(\xi):=\eta\left(\frac{\xi}{N}\right)$, $\eta$ is a smooth, radial,
nonincreasing in $|\xi|$ such that
(1.17) $\begin{array}[]{ll}\eta(\xi)&:=\left\\{\begin{array}[]{l}1,\,|\xi|\leq
1\\\ \left(\frac{1}{|\xi|}\right)^{1-s},\,|\xi|\geq
2\end{array}\right.\end{array}$
and $N>>1$ is a dyadic number playing the role of a parameter to be chosen. We
shall abuse the notation and write $m(|\xi|)$ for $m(\xi)$, thus for instance
$m(N)=1$.
Some estimates that we establish throughout the paper require a Paley-
Littlewood decomposition. We set it up now. Let $\phi(\xi)$ be a real, radial,
nonincreasing function that is equal to $1$ on the unit ball
$\left\\{\xi\in\mathbb{R}^{3}:\,|\xi|\leq 1\right\\}$ and that that is
supported on $\left\\{\xi\in\mathbb{R}^{3}:\,|\xi|\leq 2\right\\}$. Let $\psi$
denote the function
(1.18) $\begin{array}[]{ll}\psi(\xi)&:=\phi(\xi)-\phi(2\xi)\end{array}$
If $(M,M_{1},M_{2})\in 2^{\mathbb{Z}}$ are dyadic numbers such that
$M_{2}>M_{1}$ we define the Paley-Littlewood operators in the Fourier domain
by
(1.19) $\begin{array}[]{ll}\widehat{P_{\leq
M}f}(\xi)&:=\phi\left(\frac{\xi}{M}\right)\hat{f}(\xi)\\\
\widehat{P_{M}f}(\xi)&:=\psi\left(\frac{\xi}{M}\right)\hat{f}(\xi)\\\
\widehat{P_{>M}f}(\xi)&:=\hat{f}(\xi)-\widehat{P_{\leq M}f}(\xi)\\\
\widehat{P_{<<M}f}(\xi)&:=\widehat{P_{\leq\frac{M}{128}}f}(\xi)\\\
\widehat{P_{\gtrsim M}f}(\xi)&:=\widehat{P_{>\frac{M}{128}}f}(\xi)\\\
P_{M_{1}<.\leq M_{2}}f&:=P_{\geq M_{2}}f-P_{<M_{1}}f\\\ \end{array}$
Since $\sum_{M\in 2^{\mathbb{Z}}}\psi\left(\frac{\xi}{M}\right)=1$ we have
(1.20) $\begin{array}[]{ll}f&=\sum_{M\in 2^{\mathbb{Z}}}P_{M}f\end{array}$
Notice also that
(1.21) $\begin{array}[]{ll}f&=P_{\lesssim M}f+P_{\gtrsim M}f\end{array}$
It $T$ is a multiplier with nonnegative symbol $m$ then $T^{\frac{1}{2}}$
denotes then multiplier with symbol $m^{\frac{1}{2}}$. For instance
$\widehat{P_{M}^{\frac{1}{2}}f}(\xi)=\psi^{\frac{1}{2}}\left(\frac{\xi}{M}\right)\widehat{f}(\xi)$.
Throughout this paper we constantly use Strichartz-type estimates . Notice
that some Strichartz estimates for the Klein-Gordon equation already exist in
Besov spaces [5]. Here we have chosen to work in the $L_{t}^{q}L_{x}^{r}$
spaces in order to avoid too many technicalities. The following proposition is
proved in Section 7
###### Proposition 1.2.
”Strichartz estimates for Klein-Gordon equations in $L_{t}^{q}L_{x}^{r}$
spaces” Assume that $u$ satisfies the following Klein-Gordon equation on
$\mathbb{R}^{d}$
(1.22) $\left\\{\begin{array}[]{ccl}\partial_{tt}u-\Delta u+u&=&Q\\\
u(0,x)&=&u_{0}(x)\\\ \partial_{t}u(0,x)&=&u_{1}(x)\end{array}\right.$
Let $T\geq 0$. Then
(1.23)
$\begin{array}[]{l}\|u\|_{L_{t}^{q}([0,T])L_{x}^{r}}+\|\partial_{t}<D>^{-1}u\|_{L_{t}^{q}([0,\,T])L_{x}^{q}}+\|u\|_{L_{t}^{\infty}([0,\,T])H^{m}}+\|\partial_{t}u\|_{L_{t}^{\infty}\left([0,\,T],\,H^{m-1}\right)}\\\
\\\
\lesssim\|u_{0}\|_{H^{m}}+\|u_{1}\|_{H^{m-1}}+\|Q\|_{L_{t}^{\tilde{q}}([0,T])L_{x}^{\tilde{r}}}\end{array}$
under the following assumptions
* •
$(q,r)$ is $m$\- wave admissible, i.e $(q,r)$ lies in the set $\mathcal{W}$ of
wave-admissible points
(1.24)
$\begin{array}[]{ll}\mathcal{W}&:=\left\\{(q,r):(q,r)\in(2,\infty]\times[2,\infty),\,\frac{1}{q}+\frac{d-1}{2r}\leq\frac{d-1}{4}\right\\}\end{array}$
it obeys the following constraint
(1.25) $\begin{array}[]{ll}\frac{1}{q}+\frac{d}{r}&=\frac{d}{2}-m\end{array}$
and, if $d\geq 3$
(1.26)
$\begin{array}[]{ll}(q,r)&\neq\left(2,\,\frac{2(d-1)}{d-3}\right)\end{array}$
* •
$(\tilde{q},\tilde{r})$ lies in the dual set $\widetilde{\mathcal{W}}$ of
$\mathcal{W}$ i.e
(1.27)
$\begin{array}[]{ll}\widetilde{\mathcal{W}}&:=\left\\{(\tilde{q},\tilde{r}):\frac{1}{\tilde{q}}+\frac{1}{q}=1,\,\frac{1}{\tilde{r}}+\frac{1}{r}=1\right\\}\end{array}$
and it satisfies the following inequality
(1.28)
$\begin{array}[]{ll}\frac{1}{\tilde{q}}+\frac{d}{\tilde{r}}-2&=\frac{1}{q}+\frac{d}{r}\end{array}$
###### Remark 1.3.
Notice that the constraints that $(q,r,\tilde{q},\tilde{r})$ must satisfy are
essentially the same to those in the Strichartz estimates for the wave
equation [7]. These similarities are not that surprising. Indeed the relevant
operator is $e^{it<D>}$, $e^{itD}$ for the Klein-Gordon, wave equations
respectively 666with $D$ multiplier defined by
$\widehat{Df}(\xi):=|\xi|\widehat{f}(\xi)$. They are similar to each other on
high frequencies.
Now we explain the main ideas of this paper.
Our first objective is to establish global well-posedness of (1.1) for data in
$H^{s}\times H^{s-1}$,$1>s>s(p)$. Unfortunately since the solution lies in
$H^{s}\times H^{s-1}$ pointwise in time the energy (1.8) is infinite.
Therefore we introduce the following mollified energy
(1.29)
$\begin{array}[]{ll}E\left(Iu(t)\right)&:=\frac{1}{2}\int_{\mathbb{R}^{3}}\left|\partial_{t}Iu(t,x)\right|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{3}}|DIu(t,x)|^{2}\,dx+\frac{1}{2}\int_{\mathbb{R}^{3}}|Iu(t,x)|^{2}\,dx+\frac{1}{p+1}\int_{\mathbb{R}^{3}}|Iu(t,x)|^{p+1}\,dx\end{array}$
This is the $I$ method originally designed by J. Colliander, M. Keel, G.
Staffilani, H. Takaoka and T. Tao [4] to study global existence for rough
solutions of semilinear Schrödinger equations. Since the multiplier gets
closer to the identity operator as the parameter $N$ goes to infinity
777formally speaking we expect the variation of the smoothed energy to
approach zero as $N$ grows. However it is not equal to zero and it needs to be
controlled on an arbitrary large interval. The semilinear Schrödinger and Wave
equations have a scaling property. In [4, 16] the authors were able after
scaling to make the mollified energy at time zero smaller than one. Then by
using the Strichartz estimates they locally bounded some numbers that allowed
them to find an upper bound of its local variation. Iterating the process they
managed to yield an upper bound 888depending on $N$, the time and the initial
data of its total variation. Choosing appropriately the parameter $N$ they
bounded it by a constant. Unfortunately the $p$-defocusing Klein-Gordon
equation does not have any scaling symmetry. We need to control the variation
of (1.29) by a fixed quantity. A natural choice is a constant $C>1$ multiplied
by the mollified energy $E(Iu_{0}):=E(Iu(0))$ at time zero. It occurs that
this is possible if $E(Iu_{0})$ is bounded by a constant depending on the
parameter $N$: see (2.20) and (2.22). But Proposition 2.1 shows that
$E(Iu_{0})$ is bounded by a power of $N$. Therefore we can choose $N$ to
control the mollified energy as long as $s>s(p)$. Since the pointwise in time
$H^{s}\times H^{s-1}$ norm of the solution is bounded by the mollified energy
(see (2.33)) we have global well-posedness.
Now we are interested in proving asymptotic completeness by using the
$I$-method. Notice that this method has already been used in [16] to prove
scattering below the energy norm for semilinear Schrödinger equations with a
power type nonlinearity. We would like to establish (1.9). Notice first that
if this result is true then it implies that the pointwise in time $H^{s}\times
H^{s-1}$ bound of the solution is bounded by a function that does not depend
on time. Therefore in view of the previous paragraph, the variation of the
smoothed energy should not depend on time $T$. To this end we use some tools.
Recall that this variation is estimated by using local bounds of some
quantities, namely some $Z_{m,s}$ $s$ (see Proposition 2.6). We divide the
whole interval $[0,T]$ into subintervals where the $L_{t}^{p+2}L_{x}^{p+2}$ of
$Iu$ is small and we control these numbers on them by the Strichartz estimates
and a continuity argument. Notice that in this process we are not allowed to
create powers of time $T$ 999by using Hölder locally in time since it will
eventually force us to choose $N$ as a function of $T$. We also need to
control the $L_{t}^{p+2}L_{x}^{p+2}$ norm of the solution on $[0,T]$. Morawetz
and Strauss [8, 9] proved a weighted long time estimate ( see (5.8)) depending
on the energy. Combining this result to a radial Sobolev inequality (see
(2.27)) 101010this is the only place where we rely crucially on the assumption
of spherical symmetry we can control the $L_{t}^{p+2}L_{x}^{p+2}$ norm of $u$
by some power of the energy. Of course since the solution lies in $H^{s}\times
H^{s-1}$, $s<1$ we cannot use this inequality as such. Instead we prove an
almost Morawetz-Strauss estimate (see Proposition 2.11 and Proposition 2.5 )
by substituting $u$ for $Iu$ in the establishment of (5.8). This approach was
already used in [15]. Notice here that the upper bound of (2.28) does not
depend on $T$ either. The almost conservation law (see Proposition 2.3) is
proved in Section 3 by performing a low-high frequency decomposition and using
the smoothness of $F$ 111111 namely $F$ is $C^{1}$ if $p>3$ when we estimate
the low frequency part of the variation. Combining all these tools we are able
to iterate and globally bound the mollified energy and the
$L_{t}^{p+2}L_{x}^{p+2}$ norm of $u$ by a function of $N$ and the data. These
global results allow us to update a local control of the $Z_{m,s}$ s to a
global one. It occurs that scattering holds if some integrals are finite. By
using the global control of the $Z_{m,s}$ s in the Cauchy criterion we prove
these facts. This is enough to establish scattering.
$\textbf{Acknowledgements}:$ The author would like to thank Terence Tao for
suggesting him this problem.
## 2\. Proof of Theorem 1.10
In this section we prove Theorem 1.10 assuming that the following propositions
are true.
###### Proposition 2.1.
”Mollified energy at time $0$ is bounded by $N^{2(1-s)}$” Assume that
$s_{c}<s<1$. Then
(2.1) $\begin{array}[]{ll}E(Iu_{0})&\lesssim
N^{2(1-s)}\left(\|u_{0}\|^{2}_{H^{s}}+\|u_{1}\|^{2}_{H^{s-1}}+\|u_{0}\|^{p+1}_{H^{s}}\right)\end{array}$
###### Proposition 2.2.
”Local Boundedness” Assume that $u$ satisfies (1.1). Let
$\mathcal{M}=[0,s]\cup\\{1-\\}$. There exists
$N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$ such that if $J$, time
interval satisfies
(2.2) $\begin{array}[]{ll}\sup_{t\in J}E(Iu(t))&\leq 3E(Iu_{0})\end{array}$
and
(2.3)
$\begin{array}[]{ll}\|Iu\|_{L_{t}^{p+2}(J)L_{x}^{p+2}}&\leq\frac{1}{N^{++}\left(E(Iu_{0})\right)^{\frac{1-\theta_{2}}{2\theta_{2}}}}\end{array}$
then
(2.4) $\begin{array}[]{ll}Z(J,u)&\lesssim E^{\frac{1}{2}}(Iu_{0})\end{array}$
Here $m=1-\alpha$, $N^{++}=N^{2\alpha}$ with $\alpha$ small and, given a
function $v$, $Z(J,v)$, $Z_{m,s}(J,v)$ denote the following quantities
(2.5)
$\begin{array}[]{ll}Z(J,v)&:=\sup_{m\in\mathcal{M}}Z_{m,s}(J,v)\end{array}$
and
(2.6) $\begin{array}[]{ll}Z_{m,s}(J,v)&:=\sup_{\begin{subarray}{c}(q,r)-m\\\
wave\,adm\end{subarray}}\|\partial_{t}<D>^{-m}Iv\|_{L_{t}^{q}(J)L_{x}^{r}}+\|<D>^{1-m}Iv\|_{L_{t}^{q}(J)L_{x}^{r}}\end{array}$
###### Proposition 2.3.
”Almost Conservation Law ” Assume that $u$ satisfies (1.1). Let $J=[a,b]$ be a
time interval. Let $3\leq p<5$ and $s\geq\frac{3p-5}{2p}$. Then
(2.7) $\begin{array}[]{ll}\left|\sup_{t\in
J}E(Iu(t))-E(Iu(a))\right|&\lesssim\frac{Z^{p+1}(J,u)}{N^{\frac{5-p}{2}-}}\end{array}$
Here $N^{\frac{5-p}{2}-}=N^{\frac{5-p}{2}-\alpha(p-1)}$ with $\alpha$ small.
###### Remark 2.4.
Notice that if $p=3$ then the upper bound is $O\left(\frac{1}{N^{1-}}\right)$
modulo $Z^{p+1}(J,u)$. This result has already been established in [15] for a
slighly different problem, i.e the defocusing cubic wave equation by using a
multilinear analysis.
###### Proposition 2.5.
”Estimate of integrals” Let $J$ be a time interval. Let $v$ be a function.
Then for $i=1,2$ we have
(2.8)
$\begin{array}[]{ll}|R_{i}(J,v)|&\lesssim\frac{Z^{p+1}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
with
(2.9)
$\begin{array}[]{ll}R_{1}(J,v):=\int_{J}\int_{\mathbb{R}^{3}}\frac{\nabla
Iv(t,x).x}{|x|}\left(F(Iv)-IF(v)\right)\,dxdt\end{array}$
and
(2.10)
$\begin{array}[]{ll}R_{2}(J,v):=\int_{J}\int_{\mathbb{R}^{3}}\frac{Iv(t,x)}{|x|}\left(F(Iv)-IF(v)\right)\,dxdt\end{array}$
Here $N^{\frac{5-p}{2}-}=N^{\frac{5-p}{2}-\alpha(p-1)}$ with $\alpha$ small.
###### Proposition 2.6.
”Almost Morawetz-Strauss Estimate” Let $u$ be a solution of (1.1) and let
$T\geq 0$. Then
(2.11)
$\begin{array}[]{ll}\int_{0}^{T}\int_{\mathbb{R}^{3}}\frac{|Iu(t,x)|^{p+1}}{|x|}\,dxdt&\lesssim\sup_{t\in[0,T]}E(Iu(t))+R_{1}([0,\,T],u)+R_{2}([0,\,T],u)\end{array}$
These propositions will be proved in the next sections. The proof of Theorem
1.10 is made of four steps
* •
Boundedness of the mollified energy and the quantity
$\|Iu\|_{L_{t}^{p+2}L_{x}^{p+2}}$. We will prove that we can control the
mollified energy $E(Iu)$ and the $L_{t}^{p+2}L_{x}^{p+2}$ norm of $Iu$ on
arbitrary large intervals $[0,T]$, $T\geq 0$. More precisely let
(2.12)
$\begin{array}[]{ll}F_{T}&:=\left\\{T^{{}^{\prime}}\in[0,\,T]:\sup_{t\in[0,\,T^{{}^{\prime}}]}E(Iu(t))\leq
2E(Iu_{0}),\,\|Iu\|^{p+2}_{L_{t}^{p+2}([0,\,T^{{}^{\prime}}])L_{x}^{p+2}}\leq
CE^{\frac{3}{2}}(Iu_{0})\right\\}\end{array}$
We claim that $F_{T}=[0,\,T]$ for some universal constant $C\geq 0$ and
$N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$ to be chosen later. Indeed
* –
$F_{T}\neq\emptyset$ since $0\in F_{T}$
* –
$F_{T}$ is closed by continuity
* –
$F_{T}$ is open. Let $\widetilde{T^{\prime}}\in F_{T}$. By continuity there
exists $\delta>0$ such that for all
$T^{{}^{\prime}}\in(\widetilde{T^{{}^{\prime}}}-\delta,\widetilde{T^{{}^{\prime}}}+\delta)\cap[0,T]$
we have
(2.13) $\begin{array}[]{ll}\sup_{t\in[0,T^{{}^{\prime}}]}E(Iu(t))&\leq
3E(Iu_{0})\end{array}$
and
(2.14)
$\begin{array}[]{ll}\|Iu\|^{p+2}_{L_{t}^{p+2}([0,\,T^{{}^{\prime}}])L_{x}^{p+2}}&\leq
2CE^{\frac{3}{2}}(Iu_{0})\end{array}$
Let $\mathcal{P}=(J_{j})_{1\leq j\leq l}$ be a partition of
$[0,\,T^{{}^{\prime}}]$ such that
$\|Iu\|_{L_{t}^{p+2}(J_{j})L_{x}^{p+2}}=\frac{1}{N^{++}E^{\frac{1-\theta_{2}}{2\theta_{2}}}(Iu_{0})}$
for all $j=1,..,l-1$ and
$\|Iu\|_{L_{t}^{p+2}(J_{l})L_{x}^{p+2}}\leq\frac{1}{N^{++}E^{\frac{1-\theta_{2}}{2\theta_{2}}}(Iu_{0})}$
with $N^{++}$ defined in Proposition 2.6. Then by (2.14)
(2.15) $\begin{array}[]{ll}l&\lesssim
E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}}(Iu_{0})N^{+}\end{array}$
(2.16) $\left[\begin{array}[]{l}l\lesssim
E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}}(Iu_{0})N^{\alpha(p+2)}\end{array}\right]$
By Proposition 2.6 and 2.3 we get after iteration
(2.17)
$\begin{array}[]{ll}\sup_{t\in[0,\,T]}E(Iu(t))-E(Iu_{0})&\lesssim\frac{E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}+\frac{p+1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}\end{array}$
(2.18)
$\left[\begin{array}[]{l}\sup_{t\in[0,\,T]}E(Iu(t))-E(Iu_{0})\lesssim\frac{E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}+\frac{p+1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-3\alpha(p-1)}}\end{array}\right]$
Let $C_{1}$ be the constant determined by (2.17). If we can choose $N>>1$ such
that
(2.19)
$\begin{array}[]{ll}C_{1}\frac{E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}+\frac{p+1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}\leq
E(Iu_{0})\end{array}$
then $\sup_{t\in[0,T^{{}^{\prime}}]}E(Iu(t))\leq 2E(Iu_{0})$. The constraint
(2.19) is equivalent to
(2.20)
$\begin{array}[]{ll}E(Iu_{0})&\leq\frac{N^{\frac{(5-p)(p-3)}{(p-1)(p-2)}-}}{C_{1}^{\frac{2(p-3)}{(p-1)(p-2)}}}\end{array}$
(2.21)
$\left[\begin{array}[]{l}E(Iu_{0})\leq\frac{E^{\frac{(5-p)(p-3)}{(p-1)(p-2)}-\frac{6\alpha(p-3)}{p-2}}}{C_{1}^{\frac{2(p-3)}{(p-1)(p-2)}}}\end{array}\right]$
if $3<p\leq 4$ and
(2.22)
$\begin{array}[]{ll}E(Iu_{0})&\leq\frac{N^{\frac{(5-p)^{2}}{(6-p)(p-1)}-}}{C_{1}^{\frac{2(5-p)}{(6-p)(p-1)}}}\end{array}$
(2.23)
$\left[\begin{array}[]{l}E(Iu_{0})\leq\frac{E^{\frac{(5-p)^{2}}{(6-p)(p-1)}-\frac{6\alpha(5-p)}{6-p}}}{C_{1}^{\frac{2(5-p)}{(6-p)(p-1)}}}\end{array}\right]$
if $4\leq p<5$ after plugging (1.13) into (2.19). By Proposition 2.1 it
suffices to prove that there exists
$N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$ such that
(2.24)
$\begin{array}[]{ll}N^{2(1-s)}\max(\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}},\,\|u_{0}\|_{H^{s}}^{p+1})&\lesssim
N^{\frac{(5-p)(p-3)}{(p-1)(p-2)}-}\end{array}$
in order to satisfy (2.20) and
(2.25)
$\begin{array}[]{ll}N^{2(1-s)}\max(\|u_{0}\|_{H^{s}},\,\|u_{1}\|_{H^{s-1}},\,\|u_{0}\|_{H^{s}}^{p+1})&\lesssim
N^{\frac{(5-p)^{2}}{(6-p)(p-1)}-}\end{array}$
in order to satisfy (2.22) 121212with $N^{\frac{(5-p)(p-3)}{(p-1)(p-2)}-}$,
$N^{\frac{(5-p){2}}{(6-p)(p-1)}-}$ defined in (2.21), (2.23) respectively.
Such a choice is possible if and only if $s>s(p)$. By Proposition 2.11,
Proposition 2.5 and (2.13) we get
(2.26)
$\begin{array}[]{ll}\int_{0}^{T^{{}^{\prime}}}\int_{\mathbb{R}^{3}}\frac{|Iu(t,x)|^{p+1}}{|x|}\,dxdt&\lesssim
E(Iu_{0})+\frac{E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}+\frac{3}{2}+\frac{p+1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}\\\
&\lesssim E(Iu_{0})\end{array}$
Combining (2.26) to the following pointwise radial Sobolev inequality
(2.27)
$\begin{array}[]{ll}|Iu(t,x)|&\lesssim\frac{\|Iu(t,.)\|_{H^{1}}}{|x|}\end{array}$
we have
(2.28)
$\begin{array}[]{ll}\|Iu\|^{p+2}_{L_{t}^{p+2}([0,T^{{}^{\prime}}])L_{x}^{p+2}}&\lesssim
E^{\frac{3}{2}}(Iu_{0})\end{array}$
and we assign to $C$ the constant determined by $\lesssim$ in (2.28).
* •
_Global existence_ We have just proved that
(2.29) $\begin{array}[]{ll}\sup_{t\in[0,T]}E(Iu(t))&\leq
2E(Iu_{0})\end{array}$
and
(2.30) $\begin{array}[]{ll}\|Iu\|^{p+2}_{L_{t}^{p+2}([0,T])_{x}^{p+2}}\leq
CE^{\frac{3}{2}}(Iu_{0})\end{array}$
for some well-chosen $N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$ and
$1>s>s(p)$. Therefore by Proposition 2.1
(2.31)
$\begin{array}[]{ll}\sup_{t\in[0,T]}E(Iu(t))&\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
and
(2.32)
$\begin{array}[]{ll}\|Iu\|^{p+2}_{L_{t}^{p+2}([0,T])L_{x}^{p+2}}\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
Now by Plancherel and (2.31)
(2.33) $\begin{array}[]{ll}\|(u(T),\partial_{t}u(T))\|_{H^{s}\times
H^{s-1}}&\lesssim E(Iu(T))\\\ &\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
This proves global well-posedness of (1.1) with data $(u_{0},u_{1})\in
H^{s}\times H^{s-1}$, $1>s>s(p)$. More over by continuity we have
(2.34)
$\begin{array}[]{ll}\sup_{t\in\mathbb{R}}E(Iu(t))&\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
and
(2.35)
$\begin{array}[]{ll}\|Iu\|^{p+2}_{L_{t}^{p+2}(\mathbb{R})L_{x}^{p+2}}\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
* •
Global estimates
Let $\mathcal{P}:=(\widetilde{J}_{j}=[a_{j},\,b_{j}])_{1\leq
j\leq\widetilde{l}}$ be a partition of $[0,\infty)$ such that
(2.36)
$\begin{array}[]{ll}\|Iu\|_{L_{t}^{p+2}(J_{j})L_{x}^{p+2}}&\leq\frac{1}{N^{++}\left(E(Iu_{0})\right)^{\frac{1-\theta_{2}}{2\theta_{2}}}}\end{array}$
with $N^{++}$ defined in Proposition 2.6. Notice that from Proposition 2.1 and
(2.35) the number of intervals $\widetilde{l}$ satisfies
(2.37) $\begin{array}[]{ll}\tilde{l}&\lesssim
E^{\frac{(p+2)(1-\theta_{2})}{2\theta_{2}}}(Iu_{0})\\\ \\\
&\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
Moreover by slightly modifying the steps between (4.5) and (4.13) and by
(2.34) we have
(2.38) $\begin{array}[]{ll}Z_{s,s}(J_{j},u)&\lesssim
E^{\frac{1}{2}}(Iu(a_{j}))+C_{1}Z_{s,s}^{\theta_{3}(p-1)+1}(J_{j},u)+C_{2}Z_{s,s}^{\theta(p-1)+1}(J_{j},u)\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+C_{1}Z_{s,s}^{\theta_{3}(p-1)+1}(J_{j},u)+C_{2}Z_{s,s}^{\theta(p-1)+1}(J_{j},u)\end{array}$
with $C_{1}$, $C_{2}$, and $\theta$ defined in (4.14), (4.15) and (4.10)
respectively. Even if it means increasing the value of
$N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$ in (2.24) and (2.25) we can
assume that (4.16) and (4.17) hold. Therefore by Lemma 4.3 and Proposition 2.1
we have
(2.39) $\begin{array}[]{ll}Z_{s,s}(J_{j},u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})\\\ &\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
By (2.39) and (2.37) we have
(2.40) $\begin{array}[]{ll}Z_{s,s}(\mathbb{R},u)&\lesssim_{{\underset{\hskip
8.53581pt\|u_{1}\|_{H^{s-1}}}{\|u_{0}\|_{H^{s}}}}}1\end{array}$
* •
Scattering
Let
(2.41) $\begin{array}[]{ll}v(t)&:=\left(\begin{array}[]{l}u(t)\\\
\partial_{t}u(t)\end{array}\right)\end{array}$
(2.42) $\begin{array}[]{ll}v_{0}:=\left(\begin{array}[]{l}u_{0}\\\
u_{1}\end{array}\right)\end{array}$
(2.43)
$\begin{array}[]{ll}\mathbf{K}(t)&:=\left(\begin{array}[]{cc}\cos{(t<D>)}&\frac{\sin{(t<D>)}}{<D>}\\\
&\\\ -<D>\sin{(t<D>)}&\cos{(t<D>)}\end{array}\right)\end{array}$
and
(2.44)
$\begin{array}[]{ll}\mathbf{u_{nl}}(t)&=\left(\begin{array}[]{l}\int_{0}^{t}\frac{\sin{\left((t-t^{{}^{\prime}})<D>\right)}}{<D>}\left(|u|^{p-1}(t^{{}^{\prime}})u(t^{{}^{\prime}})\right)\,dt^{{}^{\prime}}\\\
\\\
\int_{0}^{t}\cos{\left((t-t^{{}^{\prime}})<D>\right)}\left(|u|^{p-1}(t^{{}^{\prime}})u(t^{{}^{\prime}})\right)\,dt^{{}^{\prime}}\end{array}\right)\end{array}$
Then we get from (1.5)
(2.45)
$\begin{array}[]{ll}\mathbf{v}(t)&=\mathbf{K}(t)\mathbf{v_{0}}-\mathbf{u_{nl}}(t)\end{array}$
Recall that the solution $u$ scatters in $H^{s}\times H^{s-1}$ if there exists
(2.46)
$\begin{array}[]{ll}\mathbf{v_{+,0}}&:=\left(\begin{array}[]{l}u_{0,+}\\\
u_{1,+}\end{array}\right)\end{array}$
such that
(2.47)
$\left\|\mathbf{v}(t)-\mathbf{K}(t)\mathbf{v_{+,0}}\right\|_{(H^{s},H^{s-1})}$
has a limit as $t\rightarrow\infty$ and the limit is equal to $0$. In other
words since $K$ is bounded on $H^{s}\times H^{s-1}$ it suffices to prove that
the quantity
(2.48)
$\left\|\mathbf{K}^{-1}(t)\mathbf{v}(t)-\mathbf{v_{+,0}}\right\|_{H^{s}\times
H^{s-1}}$
has a limit as $t\rightarrow\infty$ and the limit is equal to $0$. A
computation shows that
(2.49)
$\begin{array}[]{ll}\mathbf{K}^{-1}(t)&=\left(\begin{array}[]{cc}\cos{(t<D>)}&-\frac{\sin{(t<D>)}}{<D>}\\\
&\\\ <D>\sin{(t<D>)}&\cos{(t<D>)}\end{array}\right)\end{array}$
But
(2.50)
$\begin{array}[]{ll}\mathbf{K}^{-1}(t)\mathbf{v}(t)&=\mathbf{v_{0}}-\mathbf{K}^{-1}(t)\mathbf{u_{nl}}(t)\end{array}$
By Proposition 1.2
(2.51)
$\begin{array}[]{ll}\|\mathbf{K}^{-1}(t_{1})\mathbf{u_{nl}}(t_{1})-\mathbf{K}^{-1}(t_{2})\mathbf{u_{nl}}(t_{2})\|_{H^{s}\times
H^{s-1}}&\lesssim\|\mathbf{u_{nl}}(t_{1})-\mathbf{u_{nl}}(t_{2})\|_{H^{s}\times
H^{s-1}}\\\
&\lesssim\||u|^{p-1}u\|_{L_{t}^{\frac{2}{1+s}}([t_{1},\,t_{2}])L_{x}^{\frac{2}{2-s}}}\\\
&\lesssim\|<D>^{1-s}I\left(|u|^{p-1}u\right)\|_{L_{t}^{\frac{2}{1+s}}([t_{1},\,t_{2}])L_{x}^{\frac{2}{2-s}}}\end{array}$
If we let $J:=[t_{1},t_{2}]$ in (4.5) and follow the same steps up to (4.13)
we get from (2.29)
(2.52)
$\begin{array}[]{ll}\|<D>^{1-s}I\left(|u|^{p-1}u\right)\|_{L_{t}^{\frac{2}{1+s}}([t_{1},\,t_{2}])L_{x}^{\frac{2}{2-s}}}&\lesssim
C_{1}Z_{s,s}^{\theta_{3}(p-1)+1}([t_{1},\,t_{2}],u)+C_{2}Z_{s,s}^{\theta(p-1)+1}([t_{1},\,t_{2}],u)\end{array}$
By (2.40), (2.51) and (2.52)
(2.53)
$\begin{array}[]{ll}\lim\limits_{t_{1}\rightarrow\infty}\|\mathbf{K}^{-1}(t_{1})\mathbf{u_{nl}}(t_{1})-\mathbf{K}^{-1}(t_{2})\mathbf{u_{nl}}(t_{2})\|_{H^{s}\times
H^{s-1}}&=0\end{array}$
uniformly in $t_{2}$. This proves that $\mathbf{K}^{-1}(t)v(t)$ has a limit in
$H^{s}\times H^{s-1}$ as $t$ goes to infinity. Moreover
(2.54)
$\begin{array}[]{ll}\lim\limits_{t\rightarrow\infty}\left\|\mathbf{v}(t)-\mathbf{K}(t)\mathbf{v_{+,0}}\right\|_{(H^{s},H^{s-1})}&=0\end{array}$
with $\mathbf{v_{+,0}}$ defined in (2.46),
(2.55)
$\begin{array}[]{ll}u_{+,0}&:=u_{0}+\int_{0}^{\infty}\frac{\sin{(t^{{}^{\prime}}<D>)}}{<D>}\left(|u|^{p-1}(t^{{}^{\prime}})u(t^{{}^{\prime}})\right)\,dt^{{}^{\prime}}\end{array}$
and
(2.56)
$\begin{array}[]{ll}u_{+,1}&:=u_{1}-\int_{0}^{\infty}\cos{(t^{{}^{\prime}}<D>)}\left(|u|^{p-1}(t^{{}^{\prime}})u(t^{{}^{\prime}})\right)\,dt^{{}^{\prime}}\end{array}$
## 3\. Proof of ”Mollified energy at time 0 is bounded by $N^{2(1-s)}$”
In this section we aim at proving Proposition 2.1. By Plancherel we have
(3.1) $\begin{array}[]{ll}\|Iu_{1}\|^{2}_{L^{2}}&\lesssim\int_{|\xi|\leq
2N}|\widehat{u_{1}}(\xi)|^{2}\,d\xi+\int_{|\xi|\geq
2N}\frac{N^{2(1-s)}}{|\xi|^{2(1-s)}}|\widehat{u_{1}}(\xi)|^{2}\,d\xi\\\
&\lesssim N^{2(1-s)}\|u_{1}\|_{H^{s-1}}^{2}\end{array}$
Similarly
(3.2) $\begin{array}[]{ll}\|\nabla
Iu_{0}\|^{2}_{L^{2}}&\lesssim\int_{|\xi|\leq
2N}|\xi|^{2}|\widehat{u_{0}}(t,\xi)|^{2}\,d\xi+\int_{|\xi|\geq
2N}|\xi|^{2}\frac{N^{2(1-s)}}{|\xi|^{2(1-s)}}|\widehat{u_{0}}(\xi)|^{2}\,d\xi\\\
&\lesssim N^{2(1-s)}\|u_{0}\|_{H^{s}}^{2}\end{array}$
Moreover by the assumption $s>s_{c}$
(3.3)
$\begin{array}[]{ll}\|u_{0}\|^{p+1}_{L^{p+1}}&\lesssim\|P_{<<N}u_{0}\|^{p+1}_{L^{p+1}}+\|P_{\gtrsim
N}u_{0}\|^{p+1}_{L^{p+1}}\\\ &\lesssim
N^{(p+1)\left(\frac{3(p-1)}{2(p+1)}-s\right)}\|u_{0}\|^{p+1}_{H^{s}}\\\
&\lesssim N^{2(1-s)}\|u_{0}\|_{H^{s}}^{p+1}\end{array}$
## 4\. Proof of ”Local Boundedness”
Before attacking the proof of Proposition 2.6 let us prove a short lemma
###### Lemma 4.1.
Let $x(t)$ be a nonnegative continuous function of time $t$ such that
$x(0)=0$. Let $X$ be a positive constant and let $\alpha_{i},\,C_{i}$,
$i\in\\{1,..,m\\}$ be nonnegative constants such that
(4.1) $\begin{array}[]{ll}C_{i}X^{\alpha_{i}-1}&<<1\end{array}$
and
(4.2) $\begin{array}[]{ll}x(t)&\lesssim
X+\displaystyle{\sum_{i=1}^{m}}C_{i}x^{\alpha_{i}}(t)\end{array}$
Then
(4.3) $\begin{array}[]{ll}x(t)&\lesssim X\end{array}$
###### Proof.
If we let $\overline{x}(t):=\frac{x(t)}{X}$ then we have
(4.4) $\begin{array}[]{ll}\overline{x}(t)&\lesssim
1+\displaystyle{\sum_{i=1}^{m}}C_{i}X^{\alpha_{i}-1}\overline{x}^{\alpha_{i}}(t)\end{array}$
and $\overline{x}(0)=0$. Applying a continuity argument to $\overline{x}$ we
have $\overline{x}(t)\lesssim 1$. This implies (4.3).
∎
Plugging $<D>^{1-m}I$ into (1.23) we have
(4.5) $\begin{array}[]{ll}Z_{m,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-m}I(|u|^{p-1}u)\|_{L_{t}^{\frac{2}{1+m}}(J)L_{x}^{\frac{2}{2-m}}}\end{array}$
There are three cases
* •
$m=s$. By (4.5), the fractional Leibnitz rule and Hölder inequality
(4.6) $\begin{array}[]{ll}Z_{s,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-s}Iu\|_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}\||u|^{p-1}\|_{L_{t}^{2}(J)L_{x}^{2}}\\\
&\\\ &\lesssim
E^{\frac{1}{2}}(Iu_{0})+Z_{s,s}(J,u)\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\
&\\\ &\lesssim
E^{\frac{1}{2}}(Iu_{0})+Z_{s,s}(J,u)\left(\|P_{<<N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}+\|P_{\gtrsim
N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\right)\\\ \end{array}$
We are interested in estimating
$\|P_{<<N}u\|^{p-1}_{L_{t}^{2(p-1)}L_{x}^{2(p-1)}}$. There are two cases
* –
$3\leq p\leq 4$. By interpolation and (2.3) we have
(4.7)
$\begin{array}[]{ll}\|P_{<<N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\|P_{<<N}u\|^{\theta_{1}(p-1)}_{L_{t}^{\infty}(J)L_{x}^{2}}\|P_{<<N}u\|^{\theta_{2}(p-1)}_{L_{t}^{p+2}(J)L_{x}^{p+2}}\|P_{<<N}u\|^{\theta_{3}(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}\\\
&\lesssim\|Iu\|^{\theta_{1}(p-1)}_{L_{t}^{\infty}(J)L_{x}^{2}}\|Iu\|^{\theta_{2}(p-1)}_{L_{t}^{p+2}(J)L_{x}^{p+2}}\|<D>^{1-s}Iu\|^{\theta_{3}(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}\\\
&\\\
&\lesssim\frac{E^{\frac{(\theta_{1}+\theta_{2}-1)(p-1)}{2}}(Iu_{0})}{N^{++}}\,Z_{s,s}^{\theta_{3}(p-1)}(J,u)\\\
&\\\
&\lesssim\frac{E^{\frac{(-\theta_{3})(p-1)}{2}}(Iu_{0})}{N^{++}}\,Z_{s,s}^{\theta_{3}(p-1)}(J,u)\end{array}$
(4.8)
$\left[\begin{array}[]{ll}\|P_{<<N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\frac{E^{\frac{(-\theta_{3})(p-1)}{2}}(Iu_{0})}{N^{2\alpha(p-1)}}\,Z_{s,s}^{\theta_{3}(p-1)}(J,u)\end{array}\right]$
* –
$p>4$. By interpolation, Sobolev inequality and (2.3) we have
(4.9)
$\begin{array}[]{ll}\|P_{<<N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\left(\|P_{<<N}u\|^{\theta_{1}(p-1)}_{L_{t}^{\infty}(J)L_{x}^{6}}\|P_{<<N}u\|^{\theta_{2}(p-1)}_{L_{t}^{p+2}(J)L_{x}^{p+2}}\|P_{<<N}u\|^{\theta_{3}(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{6}{1-s}}}\right)\\\
&\\\ &\lesssim\left(\|\nabla
Iu\|^{\theta_{1}(p-1)}_{L_{t}^{\infty}(J)L_{x}^{2}}\|Iu\|^{\theta_{2}(p-1)}_{L_{t}^{p+2}(J)L_{x}^{p+2}}\|<D>^{1-s}Iu\|^{\theta_{3}(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}\right)\\\
&\\\
&\lesssim\frac{E^{\frac{(\theta_{1}+\theta_{2}-1)(p-1)}{2}}(Iu_{0})}{N^{++}}\,Z_{s,s}^{\theta_{3}(p-1)}(J,u)\\\
&\\\
&\lesssim\frac{E^{\frac{(-\theta_{3})(p-1)}{2}}(Iu_{0})}{N^{++}}\,Z_{s,s}^{\theta_{3}(p-1)}(J,u)\end{array}$
See (4.8) for an explicit formula of $N^{++}$ in (4.9).
Now we estimate $\|P_{\gtrsim N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}$.
Let
(4.10) $\begin{array}[]{ll}\theta:=&\frac{1}{s(p-1)}\end{array}$
By interpolation we have
(4.11) $\begin{array}[]{ll}\|P_{\gtrsim
N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\|P_{\gtrsim
N}u\|^{\theta(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}\|P_{\gtrsim
N}u\|^{(1-\theta)(p-1)}_{L_{t}^{\infty}(J)L_{x}^{\frac{2\left(s(p-1)-1\right)}{2s-1}}}\\\
&\\\
&\lesssim\frac{\|<D>^{1-s}Iu\|^{\theta(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}}{N^{(1-s)\theta(p-1)}}\frac{\|<D>Iu\|^{(1-\theta)(p-1)}_{L_{t}^{\infty}(J)L_{x}^{2}}}{N^{(p-1)(1-\theta)(1-s)}N^{++}}\\\
&\\\ &\lesssim
E^{\frac{(1-\theta)(p-1)}{2}}(Iu_{0})\frac{Z_{s,s}^{\theta(p-1)}(J,u)}{N^{(1-s)(p-1)}N^{++}}\end{array}$
since $s>s_{c}\geq\frac{1}{p-1}$.
(4.12) $\left[\begin{array}[]{ll}\|P_{\gtrsim
N}u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\frac{\|<D>^{1-s}Iu\|^{\theta(p-1)}_{L_{t}^{\frac{2}{s}}(J)L_{x}^{\frac{2}{1-s}}}}{N^{(1-s)\theta(p-1)}}\frac{\|<D>Iu\|^{(1-\theta)(p-1)}_{L_{t}^{\infty}(J)L_{x}^{2}}}{N^{(p-1)(1-\theta)(1-s)}N^{2\alpha(p-1)}}\\\
&\lesssim
E^{\frac{(1-\theta)(p-1)}{2}}(Iu_{0})\frac{Z_{s,s}^{\theta(p-1)}(J,u)}{N^{(1-s)(p-1)}N^{2\alpha(p-1)}}\end{array}\right]$
Therefore we get from (4.5), (4.7), (4.9) and (4.11)
(4.13) $\begin{array}[]{ll}Z_{m,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+C_{1}Z_{s,s}^{\theta_{3}(p-1)+1}(J,u)+C_{2}Z_{s,s}^{\theta(p-1)+1}(J,u)\end{array}$
with
(4.14)
$\begin{array}[]{ll}C_{1}&:=\frac{E^{\frac{-\theta_{3}(p-1)}{2}}(Iu_{0})}{N^{++}}\end{array}$
and
(4.15)
$\begin{array}[]{ll}C_{2}&:=\frac{E^{\frac{(1-\theta)(p-1)}{2}}(Iu_{0})}{N^{(1-s)(p-1)}N^{++}}\end{array}$
Notice that by Proposition 2.1
(4.16)
$\begin{array}[]{ll}C_{1}E^{\frac{\theta_{3}(p-1)}{2}}(Iu_{0})&\lesssim\frac{1}{N^{++}}\\\
&<<1\end{array}$
and
(4.17)
$\begin{array}[]{ll}C_{2}E^{\frac{\theta(p-1)}{2}}(Iu_{0})&\lesssim\frac{1}{N^{++}}\\\
&<<1\end{array}$
if we choose $N=N(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}})>>1$. Applying Lemma
4.3, we get
(4.18) $\begin{array}[]{ll}Z_{s,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})\end{array}$
* •
$m<s$ Notice that by (4.7), (4.9), (4.11), (4.16), (4.17) and (4.18)
(4.19)
$\begin{array}[]{ll}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}&\lesssim\frac{1}{N^{++}}\\\
&<<1\end{array}$
Moreover
(4.20) $\begin{array}[]{ll}Z_{m,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-m}I(|u|^{p-1}u)\|_{L_{t}^{\frac{2}{1+m}(J)}L_{x}^{\frac{2}{2-m}}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-m}Iu\|_{L_{t}^{\frac{2}{m}}(J)L_{x}^{\frac{2}{1-m}}}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+Z_{m,s}(J,u)\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\end{array}$
By (4.19) and (4.20) and Lemma 4.3, we get (2.4).
* •
$m=1-=1-\alpha$ with $\alpha$ small. We have
(4.21) $\begin{array}[]{ll}Z_{m,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-(1-)}I\left(|u|^{p-1}u\right)\|_{L_{t}^{1+}(J)L_{x}^{2-}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+N^{+}\||u|^{p-1}u\|_{L_{t}^{1^{+}}(J)L_{x}^{2-}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\,N^{+}\|\,|P_{<<N}u|^{p-1}P_{<<N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}+\,N^{+}\|\,|P_{<<N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}\\\ &+\,N^{+}\|\,|P_{\gtrsim
N}u|^{p-1}P_{<<N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}+\,N^{+}\|\,|P_{\gtrsim
N}u|^{p-1}P_{\gtrsim N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}\end{array}$
(4.22) $\left[\begin{array}[]{ll}Z_{m,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\|<D>^{1-(1-\alpha)}I\left(|u|^{p-1}u\right)\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+N^{\alpha}\||u|^{p-1}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\\\
&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\,N^{\alpha}\|\,|P_{<<N}u|^{p-1}P_{<<N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\\\
&+\,N^{\alpha}\|\,|P_{<<N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\\\
&+\,N^{\alpha}\|\,|P_{\gtrsim
N}u|^{p-1}P_{<<N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\\\
&+\,N^{\alpha}\|\,|P_{\gtrsim N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}\end{array}\right]$
But by (4.19) we have
(4.23)
$\begin{array}[]{ll}N^{+}\||P_{<<N}u|^{p-1}P_{<<N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}&\lesssim
N^{+}\|Iu\|_{L_{t}^{2+}(J)L_{x}^{\infty-}}\|Iu\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\
&\lesssim
N^{+}\|<D>^{1-(1-)}Iu\|_{{L_{t}^{2+}(J)L_{x}^{\infty-}}}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\
&\lesssim N^{+}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}Z_{1-,s}(J,u)\\\
&\lesssim\frac{Z_{1-,s}(J,u)}{N^{+}}\end{array}$
(4.24)
$\left[\begin{array}[]{ll}N^{\alpha}\|\,|P_{<<N}u|^{p-1}P_{<<N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}&\lesssim
N^{\alpha}\|Iu\|_{L_{t}^{\frac{2}{1-\alpha}}(J)L_{x}^{\frac{2}{1-(1-\alpha)}}}\\\
&\|Iu\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\ &\lesssim
N^{\alpha}\|<D>^{1-(1-\alpha)}Iu\|_{L_{t}^{\frac{2}{1-\alpha}}(J)L_{x}^{\frac{2}{1-(1-\alpha)}}}\\\
&\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}\\\ &\lesssim
N^{\alpha}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}Z_{1-\alpha,s}(J,u)\\\
&\lesssim\frac{Z_{1-\alpha,s}(J,u)}{N^{\alpha(2p-3)}}\end{array}\right]$
Similarly
(4.25) $\begin{array}[]{ll}\|N^{+}|P_{\gtrsim
N}u|^{p-1}P_{<<N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}&\lesssim\frac{Z_{1-,s}(J,u)}{N^{+}}\end{array}$
(4.26) $\left[\begin{array}[]{ll}\|N^{\alpha}|P_{\gtrsim
N}u|^{p-1}P_{<<N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}&\lesssim
N^{\alpha}\|u\|^{p-1}_{L_{t}^{2(p-1)}(J)L_{x}^{2(p-1)}}Z_{1-\alpha,s}(J,u)\\\
&\lesssim\frac{Z_{1-\alpha,s}(J,u)}{N^{\alpha(2p-3)}}\end{array}\right]$
Moreover since $s>\frac{p-3}{2}$
(4.27) $\begin{array}[]{ll}N^{+}\|\,|P_{<<N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{1+}L_{x}^{2-}}&\lesssim
N^{+}\|P_{<<N}u\|^{p-1}_{L_{t}^{(p-1)+}(J)L_{x}^{\frac{6(p-1)}{p-3}-}}\|P_{\gtrsim
N}u\|_{L_{t}^{\infty}(J)L_{x}^{\frac{6}{6-p}-}}\\\ \\\ &\lesssim
N^{+}Z^{p-1}_{1-,s}(J,u)\frac{\|<D>Iu\|_{L_{t}^{\infty}(J)L_{x}^{2}}}{N^{\frac{5-p}{2}-}}\\\
\\\
&\lesssim\frac{E^{\frac{1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}Z^{p-1}_{1-,s}(J,u)\\\
\end{array}$
(4.28) $\left[\begin{array}[]{ll}N^{\alpha}\|\,|P_{<<N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}&\lesssim
N^{\alpha}\|P_{<<N}u\|^{p-1}_{L_{t}^{\frac{2(p-1)}{2-\alpha}}(J)L_{x}^{\frac{6(p-1)}{p-3+\alpha}}}\\\
\\\ &\|P_{\gtrsim N}u\|_{L_{t}^{\infty}(J)L_{x}^{\frac{6}{6-p+2\alpha}}}\\\
\\\
&\lesssim\frac{\|<D>Iu\|_{L_{t}^{\infty}(J)L_{x}^{2}}}{N^{\frac{5-p}{2}-\alpha}}Z^{p-1}_{1-\alpha,s}(J,u)\end{array}\right]$
By Proposition 2.1 we have for
$N=N\left(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}}\right)>>1$
(4.29) $\begin{array}[]{ll}N^{+}\|\,|P_{\gtrsim N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{1+}(J)L_{x}^{2-}}&\lesssim
N^{+}\frac{\|<D>^{1-\left(\frac{2}{p}-\right)}Iu\|^{p}_{L_{t}^{p+}(J)L_{x}^{\frac{2p}{p-2}-}}}{N^{\frac{5-p}{2}}}\\\
\\\ &\lesssim\frac{Z^{p}_{\frac{2}{p}-,s}(J,u)}{N^{\frac{5-p}{2}-}}\\\ \\\
&\lesssim\frac{E^{\frac{p}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}\\\ \\\ &\lesssim
E^{\frac{1}{2}}(Iu_{0})\end{array}$
since $s>\frac{2}{p}>s_{c}$.
(4.30) $\left[\begin{array}[]{ll}N^{\alpha}\|\,|P_{\gtrsim
N}u|^{p-1}P_{\gtrsim
N}u\|_{L_{t}^{\frac{2}{1+(1-\alpha)}}(J)L_{x}^{\frac{2}{2-(1-\alpha)}}}&\lesssim
N^{\alpha}\frac{\|<D>^{1-\left(\frac{2(2-\alpha)}{2p}\right)}Iu\|^{p}_{L_{t}^{\frac{2p}{2-\alpha}}(J)L_{x}^{\frac{2p}{p-2+\alpha}}}}{N^{\frac{5-p}{2}}}\\\
&\lesssim
N^{\alpha}\frac{Z^{p}_{\frac{2}{p}-\frac{\alpha}{p},s}(J,u)}{N^{\frac{5-p}{2}}}\\\
&\\\ &\lesssim\frac{E^{\frac{p}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-\alpha}}\\\ \\\
&\lesssim E^{\frac{1}{2}}(Iu_{0})\end{array}\right]$
Now by (4.21), (4.25), (4.27) and (4.30)
(4.31) $\begin{array}[]{ll}Z_{1-,s}(J,u)&\lesssim
E^{\frac{1}{2}}(Iu_{0})+\frac{Z_{1-,s}(J,u)}{N^{+}}+\frac{E^{\frac{1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}Z^{p-1}_{1-,s}(J,u)\end{array}$
Let $C_{3}:=\frac{E^{\frac{1}{2}}(Iu_{0})}{N^{\frac{5-p}{2}-}}$. Then by
Proposition 2.1
(4.32) $\begin{array}[]{ll}C_{3}E^{\frac{p-2}{2}}(Iu_{0})&<<1\end{array}$
and
(4.33) $\begin{array}[]{ll}\frac{1}{N^{+}}&<<1\end{array}$
if $N=N\left(\|u_{0}\|_{H^{s}},\|u_{1}\|_{H^{s-1}}\right)>>1$. From Lemma 4.3,
(4.19), (4.31) (4.32) and (4.33) we get (2.4).
## 5\. Proof of ”Almost Morawetz-Strauss estimate”
In this section we prove Proposition 2.11.
First we recall the proof of the Morawetz-Strauss estimate based upon the
important equality [8, 9, 14]
(5.1) $\begin{array}[]{l}\Re\left(\left(\frac{\nabla\bar{u}\cdot
x}{|x|}+\frac{\bar{u}}{|x|}\right)\left(\partial_{tt}u-\triangle
u+u+|u|^{p-1}u\right)\right)=\partial_{t}\left(\Re\left(\frac{\nabla\bar{u}\cdot
x}{|x|}+\frac{\bar{u}}{|x|}\right)\partial_{t}u\right)\\\ \\\
+\operatorname{div}\left(-\frac{|\partial_{t}u|^{2}\cdot
x}{2|x|}-\frac{|u|^{2}\cdot
x}{|x|^{3}}-\Re\left(\left(\frac{\nabla\bar{u}\cdot
x}{|x|}+\frac{\overline{u}}{|x|}\right)\nabla u\right)+\frac{|\nabla
u|^{2}}{2|x|}+\frac{|u|^{p+1}x}{(p+1)|x|}+\frac{|u|^{2}x}{2}\right)\\\
+\frac{p-1}{p+1}\frac{|u|^{p+1}}{|x|}+\frac{1}{|x|}\left(|\nabla
u|^{2}-\frac{|\nabla u\cdot x|^{2}}{|x|}\right)\end{array}$
Integrating (5.1) with respect to space and time we have
(5.2)
$\begin{array}[]{l}\frac{p-1}{p+1}\int_{0}^{T}\int_{\mathbb{R}^{3}}\frac{|u|^{p+1}(t,x)}{|x|}\,dxdt+2\pi\int_{0}^{T}|u|^{2}(0,t)\,dt\\\
=-\int_{\mathbb{R}^{3}}\Re\left(\frac{\nabla\overline{u}(T,x)\cdot
x}{|x|}+\frac{\overline{u}(T,x)}{|x|}\right)\partial_{t}u(T,x)\,dx+\int_{\mathbb{R}^{3}}\Re\left(\frac{\nabla\overline{u}(0,x)\cdot
x}{|x|}+\frac{\overline{u}(0,x)}{|x|}\right)\partial_{t}u(0,x)\,dx\end{array}$
if $u$ satisfies (1.1). By Cauchy-Schwartz
(5.3)
$\begin{array}[]{ll}\left|\int_{\mathbb{R}^{3}}\Re\left(\left(\frac{\nabla\overline{u}\cdot
x}{|x|}+\frac{\overline{u}}{|x|}\right)\partial_{t}u(T,x)\right)\,dx\right|&\lesssim
E^{\frac{1}{2}}(u)\left(\int_{\mathbb{R}^{3}}\left|\frac{\nabla\overline{u}(T,x).\cdot
x}{|x|}+\frac{\overline{u}}{|x|}\right|^{2}\,dx\right)^{\frac{1}{2}}\end{array}$
After expansion we have
(5.4)
$\begin{array}[]{ll}\int_{\mathbb{R}^{3}}\left|\frac{\nabla\overline{u}(T,x).\cdot
x}{|x|}+\frac{\overline{u}(T,x)}{|x|}\right|^{2}\,dx&=\int_{\mathbb{R}^{3}}\left|\frac{\nabla
u(T,x)\cdot
x}{|x|}\right|^{2}\,dx+2\int_{\mathbb{R}^{3}}\frac{\nabla\left(\frac{|u|^{2}(T,x)}{2}\right)\cdot
x}{|x|^{2}}\,dx+\int_{\mathbb{R}^{3}}\frac{|u|^{2}(T,x)}{|x|^{2}}\,dx\\\
&=\int_{\mathbb{R}^{3}}\left|\frac{\nabla u(T,x)\cdot
x}{|x|}\right|^{2}\,dx\\\ &\lesssim E(u)\end{array}$
Here we used the identity
(5.5)
$\begin{array}[]{ll}\operatorname{div}\left(\frac{|u|^{2}(T,x)x}{2|x|^{2}}\right)&=\frac{\nabla\left(\frac{|u|^{2}(T,x)}{2}\right)\cdot
x}{|x|^{2}}+\frac{|u|^{2}(T,x)}{2|x|^{2}}\end{array}$
Combining (5.2) and (5.6) we get
(5.6)
$\begin{array}[]{ll}\int_{\mathbb{R}^{3}}\left|\frac{\nabla\overline{u}(T,x)\cdot
x}{|x|}+\frac{\overline{u}(T,x)}{|x|}\right|^{2}\,dx&\lesssim E(u)\end{array}$
Similarly
(5.7)
$\begin{array}[]{ll}\int_{\mathbb{R}^{3}}\left|\frac{\nabla\overline{u}(0,x).\cdot
x}{|x|}+\frac{\overline{u}(0,x)}{|x|}\right|^{2}\,dx&\lesssim E(u)\end{array}$
We get from (5.2), (5.6) and (5.7) the Morawetz-Strauss estimate
(5.8)
$\begin{array}[]{ll}\int_{0}^{T}\int_{\mathbb{R}^{3}}\frac{|u|^{p+1}(t,x)}{|x|}\,dxdt&\lesssim
E(u)\end{array}$
Now we plug the multiplier $I$ into (5.1) and we redo the computations. We get
(2.11).
## 6\. Proof of ”Almost conservation law” and ”Estimate of integrals”
The proof of Proposition 2.3, 2.5 relies on the following lemma
###### Lemma 6.1.
Let $G$ such that $\|G\|_{L_{t}^{\infty}(J)L_{x}^{2}}\lesssim Z(J,v)$. If
$s\geq\frac{3p-5}{2p}>s_{c}$ and $3\leq p<5$ then
(6.1)
$\begin{array}[]{ll}\int_{J}\int_{\mathbb{R}^{3}}\left|G\,(F(Iv)-IF(v))\right|\,dxdt&\lesssim\frac{Z^{p+1}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
###### Proof.
We have
(6.2)
$\begin{array}[]{l}\int_{J}\int_{\mathbb{R}^{3}}\left|G\,(F(Iv)-IF(v))\right|\,dxdt\\\
\\\
\lesssim\|G\|_{L_{t}^{\infty}(J)L_{x}^{2}}\|F(Iv)-F(v)\|_{L_{t}^{1}(J)L_{x}^{2}}+\|G\|_{L_{t}^{\infty}(J)L_{x}^{2}}\|F(v)-IF(v)\|_{L_{t}^{1}(J)L_{x}^{2}}\\\
\\\ \lesssim
Z(J,v)\left(\|F(Iv)-F(v)\|_{L_{t}^{1}(J)L_{x}^{2}}+\|F(v)-IF(v)\|_{L_{t}^{1}(J)L_{x}^{2}}\right)\end{array}$
Let
(6.3)
$\begin{array}[]{ll}X_{1}:=&\|F(Iv)-F(v)\|_{L_{t}^{1}(J)L_{x}^{2}}\end{array}$
and
(6.4) $\begin{array}[]{ll}X_{2}:=&\|F(v)-IF(v)\|_{L_{t}^{1}(J)L_{x}^{2}}\\\
\end{array}$
We are interested in estimating $X_{1}$. By the fundamental theorem of
calculus we have the pointwise bound
(6.5)
$\begin{array}[]{ll}|F(Iv)-F(v)|&\lesssim\max{\left(|Iv|,|v|\right)}^{p-1}|Iv-v|\\\
\end{array}$
Plugging this bound into $X_{1}$ we get
(6.6)
$\begin{array}[]{ll}X_{1}&\lesssim\|P_{<<N}v\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p}+}(J)L_{x}^{\frac{4(p-1)}{p-3}-}}\|P_{\gtrsim
N}v\|_{L_{t}^{\frac{4}{p-3}-}(J)L_{x}^{\frac{4}{5-p}+}}+\|P_{\gtrsim
N}v\|^{p-1}_{L_{t}^{p}(J)L_{x}^{2p}}\|P_{\gtrsim
N}v\|_{L_{t}^{p}(J)L_{x}^{2p}}\\\ &\\\
&\lesssim\frac{1}{N^{\frac{5-p}{2}-}}\left(\begin{array}[]{l}\|<D>^{1-(1-)}Iv\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p}+}(J)L_{x}^{\frac{4(p-1)}{p-3}-}}\|<D>^{1-(\frac{p-3}{2}+)}Iv\|_{L_{t}^{\frac{4}{p-3}-}(J)L_{x}^{\frac{4}{5-p}+}}+\\\
\|<D>^{1-\frac{3p-5}{2p}}Iv\|^{p}_{L_{t}^{p}(J)L_{x}^{2p}}\end{array}\right)\\\
&\\\
&\lesssim\frac{Z^{p-1}_{1-,s}(J,v)Z_{\frac{p-3}{2}+,s}(J,v)\,+\,Z^{p}_{\frac{3p-5}{2p},s}(J,v)}{N^{\frac{5-p}{2}-}}\\\
&\\\ &\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
(6.7)
$\left[\begin{array}[]{ll}X_{1}&\lesssim\|P_{<<N}v\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p-2\alpha(p-1)}}(J)L_{x}^{\frac{4(p-1)}{p-3+2\alpha(p-1)}}}\|P_{\gtrsim
N}v\|_{L_{t}^{\frac{4}{p-3+2\alpha(p-1)}}(J)L_{x}^{\frac{4}{5-p-2\alpha(p-1)}}}\\\
&+\|P_{\gtrsim N}v\|_{L_{t}^{p}(J)L_{x}^{2p}}^{p-1}\|P_{\gtrsim
N}v\|_{L_{t}^{p}(J)L_{x}^{2p}}\\\ &\\\
&\lesssim\frac{1}{N^{\frac{5-p}{2}-\alpha(p-1)}}\left(\begin{array}[]{l}\|<D>^{1-(1-\alpha)}Iv\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p-2\alpha(p-1)}}(J)L_{x}^{\frac{4(p-1)}{p-3+2\alpha(p-1)}}}\\\
\|<D>^{1-\left(\frac{p-3}{2}+\alpha(p-1)\right)}Iv\|_{L_{t}^{\frac{4}{p-3+2\alpha(p-1)}}(J)L_{x}^{\frac{4}{5-p-2\alpha(p-1)}}}\\\
+\|<D>^{1-\frac{3p-5}{2p}}Iv\|^{p}_{L_{t}^{p}(J)L_{x}^{2p}}\end{array}\right)\\\
&\\\
&\lesssim\frac{Z^{p-1}_{1-\alpha,s}(J,v)Z_{\frac{p-3}{2}+\alpha(p-1),s}(J,v)\,+\,Z^{p}_{\frac{3p-5}{2p},s}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\\\
&\\\
&\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\end{array}\right]$
Now we turn to $X_{2}$. On low frequencies we use the smoothness of $F$
whereas on high frequencies we take advantage of the regularity of $u$, lying
in $H^{s}$. More precisely by the fundamental theorem of calculus we have
(6.8) $\begin{array}[]{ll}F(v)&=F(P_{<<N}v+P_{\gtrsim N}v)\\\ &\\\
&=F(P_{<<N}v)+\left(\int_{0}^{1}|P_{<<N}v+sP_{\gtrsim
N}v|^{p-1}\,ds\right)P_{\gtrsim N}v\\\
&+\left(\int_{0}^{1}\frac{P_{<<N}v+sP_{\gtrsim
N}v}{\vspace{0.2mm}\overline{P_{<<N}v+sP_{\gtrsim N}v}}|P_{<<N}v+sP_{\gtrsim
N}v|^{p-1}\,ds\right)\overline{P_{\gtrsim N}v}\end{array}$
Therefore
(6.9) $\begin{array}[]{ll}X_{2}&\lesssim\|P_{\gtrsim
N}F(v)\|_{L_{t}^{1}(J)L_{x}^{2}}\\\ &\\\ &\lesssim\|P_{\gtrsim
N}F(P_{<<N}v)\|_{L_{t}^{1}(J)L_{x}^{2}}+\|\,|P_{<<N}v|^{p-1}\,P_{\gtrsim
N}v\|_{L_{t}^{1}(J)L_{x}^{2}}+\|\,|P_{\gtrsim N}v|^{p-1}\,P_{\gtrsim
N}v\|_{L_{t}^{1}(J)L_{x}^{2}}\\\ &\\\ &\lesssim
X_{2,1}+X_{2,2}+X_{2,3}\end{array}$
with $X_{2,1}:=\|P_{\gtrsim N}F(P_{<<N}v)\|_{L_{t}^{1}(J)L_{x}^{2}}$,
$X_{2,2}:=\|\,|P_{<<N}v|^{p-1}\,P_{\gtrsim N}v\|_{L_{t}^{1}(J)L_{x}^{2}}$ and
$X_{2,3}:=\|\,|P_{\gtrsim N}v|^{p-1}\,P_{\gtrsim
N}v\|_{L_{t}^{1}(J)L_{x}^{2}}$. But again by the fundamental theorem of
calculus
(6.10) $\begin{array}[]{ll}X_{2,1}&\lesssim\frac{1}{N}\|\nabla
F(P_{<<N}v)\|_{L_{t}^{1}(J)L_{x}^{2}}\\\ &\\\
&\lesssim\frac{1}{N}\|\,|P_{<<N}v|^{p-1}\nabla
P_{<<N}v+\frac{|P_{<<N}v|^{p-1}P_{<<N}v}{\overline{P_{<<N}v}}\overline{\nabla
P_{<<N}v}\|_{L_{t}^{1}(J)L_{x}^{2}}\end{array}$
Therefore
(6.11)
$\begin{array}[]{ll}X_{2,1}&\lesssim\frac{1}{N}\|P_{<<N}v\|_{L_{t}^{\frac{4(p-1)}{7-p}+}(J)L_{x}^{\frac{4(p-1)}{p-3}-}}^{p-1}\|\nabla
P_{<<N}v\|_{L_{t}^{\frac{4}{p-3}-}(J)L_{x}^{\frac{4}{5-p}+}}\\\ &\\\
&\lesssim\frac{1}{N^{\frac{5-p}{2}-}}\|<D>^{1-(1-)}Iv\|_{L_{t}^{\frac{4(p-1)}{7-p}+}(J)L_{x}^{\frac{4(p-1)}{p-3}-}}^{p-1}\|<D>^{1-\left(\frac{p-3}{2}+\right)}Iv\|_{L_{t}^{\frac{4}{p-3}-}(J)L_{x}^{\frac{4}{5-p}+}}\\\
&\\\
&\lesssim\frac{Z^{p-1}_{1-,s}(J,v)Z_{\frac{p-3}{2}+,s}(J,v)}{N^{\frac{5-p}{2}-}}\\\
&\\\ &\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
(6.12)
$\left[\begin{array}[]{ll}X_{2,1}&\lesssim\frac{1}{N}\|P_{<<N}v\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p-2\alpha(p-1)}}(J)L_{x}^{\frac{4(p-1)}{p-3+2\alpha(p-1)}}}\|\nabla
P_{<<N}v\|_{L_{t}^{\frac{4}{p-3+2\alpha(p-1)}}(J)L_{x}^{\frac{4}{5-p-2\alpha(p-1)}}}\\\
&\lesssim\frac{1}{N^{\frac{5-p}{2}-\alpha(p-1)}}\|<D>^{1-(1-\alpha)}Iv\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p-2\alpha(p-1)}}(J)L_{x}^{\frac{4(p-1)}{p-3+2\alpha(p-1)}}}\\\
&\|<D>^{1-\left(\frac{p-3}{2}+\alpha(p-1)\right)}Iv\|_{L_{t}^{\frac{4}{p-3+2\alpha(p-1)}}(J)L_{x}^{\frac{4}{5-p-2\alpha(p-1)}}}\\\
&\lesssim\frac{Z^{p-1}_{1-\alpha,s}(J,v)Z_{\frac{p-3}{2}+\alpha(p-1),s}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\\\
&\\\
&\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\end{array}\right]$
Moreover
(6.13)
$\begin{array}[]{ll}X_{2,2}&\lesssim\frac{1}{N^{\frac{5-p}{2}-}}\|<D>^{1-(1-)}Iv\|_{L_{t}^{\frac{4(p-1)}{7-p}+}(J)L_{x}^{\frac{4(p-1)}{p-3}-}}^{p-1}\|<D>^{1-\left(\frac{p-3}{2}+\right)}Iv\|_{L_{t}^{\frac{4}{p-3}-}(J)L_{x}^{\frac{4}{5-p}+}}\\\
&\\\
&\lesssim\frac{Z^{p-1}_{1-,s}(J,v)Z_{\frac{p-3}{2}+,s}(J,v)}{N^{\frac{5-p}{2}-}}\\\
&\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
(6.14)
$\left[\begin{array}[]{ll}X_{2,2}&\lesssim\frac{1}{N^{\frac{5-p}{2}-\alpha(p-1)}}\|<D>^{1-(1-\alpha)}Iv\|^{p-1}_{L_{t}^{\frac{4(p-1)}{7-p-2\alpha(p-1)}}(J)L_{x}^{\frac{4(p-1)}{p-3+2\alpha(p-1)}}}\\\
&\|<D>^{1-\left(\frac{p-3}{2}+\alpha(p-1)\right)}Iv\|_{L_{t}^{\frac{4}{p-3+2\alpha(p-1)}}(J)L_{x}^{\frac{4}{5-p-2\alpha(p-1)}}}\\\
&\lesssim\frac{Z^{p-1}_{1-\alpha,s}(J,v)Z_{\frac{p-3}{2}+\alpha(p-1),s}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\\\
&\\\
&\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-\alpha(p-1)}}\end{array}\right]$
As for $X_{2,3}$ we have
(6.15) $\begin{array}[]{ll}X_{2,3}&\lesssim\|P_{\gtrsim
N}v\|^{p}_{L_{t}^{p}(J)L_{x}^{2p}}\\\ &\\\
&\lesssim\frac{\|<D>^{1-\frac{3p-5}{2p}}Iv\|^{p}_{L_{t}^{p}(J)L_{x}^{2p}}}{N^{\frac{5-p}{2}}}\\\
&\\\ &\lesssim\frac{Z_{\frac{3p-5}{2},s}^{p}(J,v)}{N^{\frac{5-p}{2}}}\\\ &\\\
&\lesssim\frac{Z^{p}(J,v)}{N^{\frac{5-p}{2}-}}\end{array}$
∎
Let $t^{{}^{\prime}}\in J=[a,b]$. Then if $u$ is a solution to (1.1) then
(6.16)
$\begin{array}[]{ll}\left|E(Iu(t^{{}^{\prime}}))-E(Iu(a))\right|&=\left|\int_{[a,t^{{}^{\prime}}]}\int_{\mathbb{R}^{3}}\Re(\overline{\partial_{t}Iu}(F(Iu)-IF(u)))\right|\\\
&\lesssim\int_{[a,t^{{}^{\prime}}]}\int_{\mathbb{R}^{3}}\left|\overline{\partial_{t}Iu}(F(Iu)-IF(u))\right|\end{array}$
Notice that
(6.17)
$\begin{array}[]{ll}\|\partial_{t}Iu\|_{L_{t}^{\infty}(J)L_{x}^{2}}&\lesssim
Z_{0,s}(J,u)\end{array}$
Applying Lemma 6.1 with $G:=\partial_{t}Iu$ to (6.16) we get (2.7). Notice
also that
(6.18) $\begin{array}[]{ll}\|\frac{\nabla Iv\cdot
x}{|x|}\|_{L_{t}^{\infty}(J)L_{x}^{2}}&\lesssim\|\nabla
Iv\|_{L_{t}^{\infty}(J)L_{x}^{2}}\\\ &\lesssim Z_{0,s}(J,v)\end{array}$
and that
(6.19)
$\begin{array}[]{ll}\|\frac{Iv}{|x|}\|_{L_{t}^{\infty}(J)L_{x}^{2}}&\lesssim\|\nabla
Iv\|_{L_{t}^{\infty}(J)L_{x}^{2}}\\\ &\lesssim Z_{0,s}(J,v)\end{array}$
by Hardy inequality. Letting $G(t,x):=\frac{\nabla Iv(t,x).x}{|x|}$ we get
(2.8) from (6.18) and Lemma 6.1 for $i=1$. Similarly (2.8) holds for $i=2$ if
we let $G(t,x):=\frac{Iv(t,x)}{|x|}$.
## 7\. Strichartz estimates for $NLKG$ in $L_{t}^{q}L_{x}^{r}$ spaces
The techniques used in the proof of these estimates are, broadly speaking,
standard [7, 6]. However some subtleties appear because unlike the homogeneous
Schrodinger and wave equations the homogeneous defocusing Klein-Gordon
equation does not enjoy any scaling property. Now we mention them. Regarding
the estimates involving the homogeneous part of the solution we apply, broadly
speaking, a $"TT^{*}"$ argument to the truncated cone operators localized at
all the frequencies 131313i.e to $e^{it<D>}P_{M}$, $M\in 2^{\mathbb{Z}}$ : see
(7.15) instead of applying it at frequency equal to one and then use a scaling
argument for the other frequencies. The inhomogeneous estimates are slightly
more complicated to establish. In the first place we try to reduce the
estimates (see (7.38)) localized at all frequencies to the estimate at
frequency one (see (7.46)). This strategy does not totally work because of the
lack of scaling. However the remaining estimate (see 7.50), after duality is
equivalent to an homogeneous estimate on high frequencies (see (7.52)) that
has already been established.
Let $u$ be the solution of (1.22) with data $(u_{0},u_{1})$. Since
$u\phi(\frac{t}{2T})$ satisfies (1.22) on $[0,T]$ it suffices to prove (1.23)
with $[0,\,T]$ substituted for $\mathbb{R}$.
Let $u_{l}:=\cos{(t<D>)}u_{0}+\frac{\sin{(t<D>)}}{<D>}u_{1}$ and
$u_{nl}:=-\int_{0}^{t}\frac{\sin{(t-t^{{}^{\prime}})<D>}}{<D>}Q(t^{{}^{\prime}})$.
We need to show
(7.1)
$\begin{array}[]{ll}\|u_{l}\|_{L_{t}^{\infty}H^{m}}+\|\partial_{t}u_{l}(t)\|_{L_{t}^{\infty}H^{m-1}}&\lesssim\|u_{0}\|_{H^{m}}+\|u_{1}\|_{H^{m-1}}\end{array}$
(7.2)
$\begin{array}[]{ll}\|u_{l}\|_{L_{t}^{q}L_{x}^{r}}+\|\partial_{t}<D>^{-1}u_{l}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|u_{0}\|_{H^{m}}+\|u_{1}\|_{H^{m-1}}\end{array}$
(7.3)
$\begin{array}[]{ll}\|u_{nl}\|_{L_{t}^{q}L_{x}^{r}}+\|\partial_{t}<D>^{-1}u_{nl}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
and
(7.4)
$\begin{array}[]{ll}\|u_{nl}\|_{L_{t}^{\infty}H^{m}}+\|\partial_{t}u_{nl}\|_{L_{t}^{\infty}H^{m-1}}&\lesssim\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
By Plancherel theorem we have (7.1). We prove (7.2), (7.3) and (7.4) in the
next subsections.
### 7.1. Proof of (7.2)
By decomposition and substitution it suffices to prove
(7.5)
$\begin{array}[]{ll}\|e^{it<D>}u_{0}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|u_{0}\|_{H^{m}}\end{array}$
If we could prove for every Schwartz function $f$
(7.6) $\begin{array}[]{ll}\|e^{it<D>}P_{\leq
1}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{L^{2}}\end{array}$
and
(7.7) $\begin{array}[]{ll}\|e^{it<D>}P_{M}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M^{m}\|f\|_{L^{2}}\\\ \end{array}$
for $M\in 2^{\mathbb{Z}}$, $M>1$, then (7.2) would follow. Indeed let
$\widetilde{P_{M}}:=P_{\frac{M}{2}\leq\,\leq 2M}$ and $\widetilde{P_{\leq
1}}:=P_{\leq 2}$. Applying (7.7) to $f:=\widetilde{P_{M}}f$ we have
(7.8) $\begin{array}[]{ll}\|e^{it<D>}P_{M}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M^{m}\|\widetilde{P_{M}}f\|_{L^{2}}\\\
&\lesssim\|\widetilde{P_{M}}f\|_{\dot{H^{m}}}\end{array}$
Similarly plugging $f:=\widetilde{P_{\leq 1}}f$ into (7.6) we have
(7.9) $\begin{array}[]{ll}\|e^{it<D>}P_{\leq
1}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|\widetilde{P_{\leq 1}}f\|_{L^{2}}\\\
&\lesssim\|f\|_{H^{m}}\end{array}$
Before moving forward, we recall the fundamental Paley-Littlewood equality
[13]: if $1<p<\infty$ and $h$ is Schwartz then
(7.10) $\begin{array}[]{ll}\|h\|_{L^{p}}&\sim\|\left(\sum_{M\in
2^{\mathbb{Z}}}|P_{M}h|^{2}\right)^{\frac{1}{2}}\|_{L^{p}}\end{array}$
We plug $h:=P_{>1}f$ into (7.10). Hence by Minkowski inequality and Plancherel
theorem
(7.11)
$\begin{array}[]{ll}\|e^{it<D>}P_{>1}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|\left(\sum_{M\geq
1}|e^{it<D>}P_{M}f|^{2}\right)^{\frac{1}{2}}\|_{L_{t}^{q}L_{x}^{r}}\\\
&\lesssim\left(\sum_{M\geq
1}\|e^{it<D>}P_{M}f\|^{2}_{L_{t}^{q}L_{x}^{r}}\right)^{\frac{1}{2}}\\\
&\lesssim\left(\sum_{M\geq
1}\|\widetilde{P_{M}}f\|^{2}_{\dot{H}^{m}}\right)^{\frac{1}{2}}\\\
&\lesssim\|f\|_{H^{m}}\end{array}$
since $q\geq 2$ and $r\geq 2$. Combining (7.9) to (7.11) we get (7.5). It
remains to prove (7.6) and (7.7). Let $T_{1}(f):=e^{it<D>}P_{\leq 1}f$ and
$T_{M}(f):=e^{it<D>}P_{M}f$, $M\in 2^{\mathbb{Z}}$, $M>1$. We have
(7.12)
$\begin{array}[]{ll}T_{1}(f)(t,x)&:=\int_{\mathbb{R}^{3}}\phi(\xi)e^{it<\xi>}\hat{f}(\xi)e^{i\xi.x}\,d\xi\end{array}$
and if $M\in 2^{\mathbb{Z}}$, $M>1$ let
(7.13)
$\begin{array}[]{ll}T_{M}(f)(t,x)&:=\int_{\mathbb{R}^{3}}\psi\left(\frac{\xi}{M}\right)e^{it<\xi>}\hat{f}(\xi)e^{i\xi.x}\,d\xi\end{array}$
We would like to prove
(7.14)
$\begin{array}[]{ll}\|T_{1}(f)\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{L^{2}}\end{array}$
and
(7.15)
$\begin{array}[]{ll}\|T_{M}(f)\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{H^{m}}\end{array}$
By a $"TT^{*}"$ argument we are reduced showing for every continuous in time
Schwartz in space function $g$
(7.16)
$\begin{array}[]{ll}\|T_{1}T_{1}^{*}(g)\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
and similarly
(7.17) $\begin{array}[]{ll}\|T_{M}T^{*}_{M}(g)\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M^{2m}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
with $\frac{1}{q}+\frac{1}{q^{{}^{\prime}}}=1$ and
$\frac{1}{r}+\frac{1}{r^{{}^{\prime}}}=1$. But a computation shows that
(7.18) $\begin{array}[]{ll}T_{1}T_{1}^{*}(g)&=K_{1}\ast g\\\ &=\int
K_{1}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\,dt^{{}^{\prime}}\end{array}$
and
(7.19) $\begin{array}[]{ll}T_{M}T_{M}^{*}(g)&=K_{M}\ast g\\\ &=\int
K_{M}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\,dt^{{}^{\prime}}\end{array}$
with
(7.20)
$\begin{array}[]{ll}K_{1}(t-t^{{}^{\prime}},x)&:=\int_{\mathbb{R}^{3}}|\phi(\xi)|^{2}e^{i<\xi>(t-t^{{}^{\prime}})}e^{i\xi\cdot
x}\,d\xi\end{array}$
and
(7.21)
$\begin{array}[]{ll}K_{M}(t-t^{{}^{\prime}},x)&:=\int_{\mathbb{R}^{3}}\left|\psi\left(\frac{\xi}{M}\right)\right|^{2}e^{i<\xi>(t-t^{{}^{\prime}})}e^{i\xi\cdot
x}\,d\xi\end{array}$
One one hand by Plancherel equality we have
(7.22) $\begin{array}[]{ll}\|K_{M}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\|_{L^{2}}&\lesssim\|g(t^{{}^{\prime}},.)\|_{L^{2}}\end{array}$
On the other hand
(7.23) $\begin{array}[]{ll}\|K_{M}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\|_{L^{\infty}}&\lesssim\|K_{M}(t-t^{{}^{\prime}},.)\|_{L^{\infty}}\|g(t^{{}^{\prime}},.)\|_{L^{1}}\end{array}$
where $\|K_{M}(t-t^{{}^{\prime}},.)\|_{L^{\infty}}$ is estimated by the
stationary phase method [5], p 441
(7.24)
$\begin{array}[]{ll}\|K_{M}(t-t^{{}^{\prime}},.)\|_{L^{\infty}}&\lesssim
M^{d}\min{\left(1,\frac{1}{\left(M|t-t^{{}^{\prime}}|\right)^{\frac{d-1}{2}}}\right)}\min{\left(1,\left(\frac{M}{|t-t^{{}^{\prime}}|}\right)^{\frac{1}{2}}\right)}\end{array}$
and
(7.25)
$\begin{array}[]{ll}\|K_{1}(t-t^{{}^{\prime}},.)\|_{L^{\infty}}&\lesssim\min{\left(1,\frac{1}{|t-t^{{}^{\prime}}|^{\frac{d}{2}}}\right)}\end{array}$
By complex interpolation we have
(7.26) $\begin{array}[]{ll}\|K_{1}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\|_{L^{r}}&\lesssim\left(\min{\left(1,\frac{1}{|t-t^{{}^{\prime}}|^{\frac{d}{2}}}\right)}\right)^{1-\frac{2}{r}}\|g(t^{{}^{\prime}},.)\|_{L^{r{{}^{\prime}}}}\end{array}$
and
(7.27) $\begin{array}[]{ll}\|K_{M}(t-t^{{}^{\prime}},.)\ast
g(t^{{}^{\prime}},.)\|_{L^{r}}&\lesssim\widetilde{K_{M}}(t-t^{{}^{\prime}})\|g(t^{{}^{\prime}},.)\|_{L^{r^{{}^{\prime}}}}\end{array}$
with
(7.28)
$\begin{array}[]{ll}\widetilde{K_{M}}(t)&:=\left(M^{d}\min{\left(1,\frac{1}{\left(M|t|\right)^{\frac{d-1}{2}}}\right)}\min{\left(1,\left(\frac{M}{|t|}\right)^{\frac{1}{2}}\right)}\right)^{1-\frac{2}{r}}\end{array}$
and $r^{{}^{\prime}}$ such that $\frac{1}{r}+\frac{1}{r^{{}^{\prime}}}=1$.
Observe that if $(q,r)$ is wave admissible and $(q,r)\neq(\infty,2)$ then
$\frac{1}{q}+\frac{d}{2r}<\frac{d}{4}$. Therefore there are two cases
First we estimate $\|T_{1}T_{1}^{*}\|_{L_{t}^{q}L_{x}^{r}}$. There are two
cases
* •
Case 1: $r>2$. Then since $(q,r)$ is wave admissible and $(q,r)\neq(\infty,2)$
we also have $\frac{1}{q}+\frac{d}{2r}<\frac{d}{4}$ and by (7.18), Young’s
inequality and (7.26)
(7.29)
$\begin{array}[]{ll}\|T_{1}T_{1}^{\ast}g\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|\min{\left(1,\frac{1}{|t|^{\frac{d}{2}}}\right)}^{1-\frac{2}{r}}\|_{L_{t}^{\frac{q}{2}}}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\\\
&\lesssim\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
* •
Case 2: $r=2$. Then $q=\infty$. Then by (7.18) and (7.22) we get (7.16).
We turn to (7.17). We write
$\widetilde{K_{M}}=\widetilde{K_{M,a}}+\widetilde{K_{M,b}}+\widetilde{K_{M,c}}$
in (7.27) with
$\widetilde{K_{M,a}}:=\widetilde{K_{M}}\chi_{{}_{|t|\leq\frac{1}{M}}}$,
$\widetilde{K_{M,b}}:=\widetilde{K_{M}}\chi_{{}_{\frac{1}{M}\leq|t|\leq M}}$
and $\widetilde{K_{M,c}}:=\widetilde{K_{M}}\chi_{{}_{|t|\geq M}}$. We have by
Young’s inequality and (1.25)
(7.30)
$\begin{array}[]{ll}\left\|\widetilde{K_{M,a}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L_{x}^{r^{{}^{\prime}}}}\right\|_{L_{t}^{q}}&\lesssim
M^{d\left(1-\frac{2}{r}\right)}\|\chi_{|t|\leq\frac{1}{M}}\|_{L_{t}^{\frac{q}{2}}}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\\\
&\lesssim
M^{2m}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
To estimate
$\|\widetilde{K_{M,b}}(t-t^{{}^{\prime}})\ast\|g(t^{{}^{\prime}},.)\|_{L^{r^{{}^{\prime}}}}\|_{L_{t}^{q}}$
there are two cases
* •
Case 1: $\frac{1}{q}+\frac{d-1}{2r}<\frac{d-1}{4}$ By Young’s inequality,
(7.28) and (1.25) we have
(7.31)
$\begin{array}[]{ll}\left\|\widetilde{K_{M,b}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L_{x}^{r^{{}^{\prime}}}}\right\|_{L_{t}^{q}}&\lesssim\|\chi_{\frac{1}{M}\leq
t\leq
M}\frac{M^{2\left(1-\frac{2}{r}\right)}}{(Mt)^{\frac{d-1}{2}}}\|_{L_{t}^{\frac{q}{2}}}\|g\|_{L^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\\\
&\lesssim M^{2m}\|g\|_{L^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
* •
Case 2: $\frac{1}{q}+\frac{d-1}{2r}=\frac{d-1}{4}$. By (7.28) we have
(7.32)
$\begin{array}[]{ll}\widetilde{K_{M,b}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L_{x}^{r^{{}^{\prime}}}}&\lesssim
M^{\frac{d+1}{2}\left(1-\frac{2}{r}\right)}\int_{\mathbb{R}}\frac{\|g(t^{{}^{\prime}},.)\|_{L^{r^{{}^{\prime}}}}}{|t-t^{{}^{\prime}}|^{\frac{d-1}{2}\left(1-\frac{2}{r}\right)}}\,dt^{{}^{\prime}}\\\
&\lesssim
M^{d\left(1-\frac{2}{r}\right)-\frac{2}{q}}\int_{\mathbb{R}}\frac{\|g(t^{{}^{\prime}},.)\|_{L^{r^{{}^{\prime}}}}}{|t-t^{{}^{\prime}}|^{\frac{2}{q}}}\,dt^{{}^{\prime}}\\\
\end{array}$
By (1.26), (1.25) and Hardy-Littlewood-Sobolev inequality [13]
(7.33)
$\begin{array}[]{ll}\left\|\widetilde{K_{M,b}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L_{x}^{r^{{}^{\prime}}}}\right\|_{L_{t}^{q}}&\lesssim
M^{2m}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
We estimate
$\left\|\widetilde{K_{M,c}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L^{r^{{}^{\prime}}}}\right\|_{L_{t}^{q}}$
by applying Young inequality, (1.25) and (1.24) i.e
(7.34)
$\begin{array}[]{ll}\left\|\widetilde{K_{M,c}}(t-t^{{}^{\prime}})\,\|g(t^{{}^{\prime}},.)\|_{L_{x}^{r^{{}^{\prime}}}}\right\|_{L_{t}^{q}}&\lesssim
M^{\left(\frac{d}{2}+1\right)\left(1-\frac{2}{r}\right)}\|\chi_{{}_{t\geq
M}}\frac{1}{t^{\frac{d}{2}\left(1-\frac{2}{r}\right)}}\|_{L_{t}^{\frac{q}{2}}}\\\
&\lesssim
M^{\frac{2}{q}+1-\frac{2}{r}}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\\\
&\lesssim
M^{2m}\|g\|_{L_{t}^{q^{{}^{\prime}}}L_{x}^{r^{{}^{\prime}}}}\end{array}$
By (7.27), (7.30), (7.31), (7.33) and (7.34) we get (7.17).
### 7.2. Proof of (7.3)
By decomposition and substitution it suffices to prove
(7.35)
$\begin{array}[]{ll}\|\int_{t^{{}^{\prime}}<t}e^{i(t-t^{{}^{\prime}})<D>}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|<D>Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
By Christ-Kisilev lemma [12] 141414an original proof of this lemma can be
found in [3] it suffices in fact to prove
(7.36) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|<D>Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
If we could prove
(7.37) $\begin{array}[]{ll}\|\int e^{i(t-t^{{}^{\prime}})<D>}P_{\leq
1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
and
(7.38) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
then (7.1) would follow. Indeed introducing $\widetilde{P_{\leq 1}}$ and
$\widetilde{P_{M}}$ as in the previous subsection we have
(7.39) $\begin{array}[]{ll}\|\int e^{i(t-t^{{}^{\prime}})<D>}P_{\leq
1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|\widetilde{P_{\leq
1}}Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\\\
&\lesssim\|<D>Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
and
(7.40) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M\|\widetilde{P_{M}}Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\\\
&\lesssim\|\widetilde{P_{M}}<D>Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
Therefore we have
(7.41) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{>1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\left\|\left(\sum\limits_{\begin{subarray}{c}M\in
2^{\mathbb{Z}}\\\ M>1\end{subarray}}|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t{{}^{\prime}})\,dt^{{}^{\prime}}|^{2}\right)^{\frac{1}{2}}\right\|_{L_{t}^{q}L_{x}^{r}}\\\
&\lesssim\left(\sum\limits_{\begin{subarray}{c}M\in 2^{\mathbb{Z}}\\\
M>1\end{subarray}}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|^{2}_{L_{t}^{q}L_{x}^{r}}\right)^{\frac{1}{2}}\\\
&\lesssim\left(\sum\limits_{\begin{subarray}{c}M\in 2^{\mathbb{Z}}\\\
M>1\end{subarray}}\|P_{M}<D>Q\|^{2}_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\right)^{\frac{1}{2}}\\\
&\lesssim\left\|\left(\sum\limits_{\begin{subarray}{c}M\in 2^{\mathbb{Z}}\\\
M>1\end{subarray}}|P_{M}<D>Q|^{2}\right)^{\frac{1}{2}}\right\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\\\
&\lesssim\|<D>Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
Now we establish (7.37). It is not difficult to see from the proof of (7.6)
and (7.7) that we also have
(7.42) $\begin{array}[]{ll}\|e^{it<D>}P^{\frac{1}{2}}_{\leq
4}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{L^{2}}\end{array}$
and
(7.43) $\begin{array}[]{ll}\|e^{it<D>}P^{\frac{1}{2}}_{\leq
1}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{L^{2}}\end{array}$
for every Schwartz function $f$. A dual statement of (7.43) is
(7.44) $\begin{array}[]{ll}\|\int
e^{-it^{{}^{\prime}}<D>}P^{\frac{1}{2}}_{\leq
1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L^{2}}&\lesssim\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
Composing (7.42) with (7.44) we get (7.37).
We turn to (7.38). We need to prove
(7.45) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}\psi\left(\frac{\xi}{M}\right)\widehat{Q}(t^{{}^{\prime}},\xi)\,dt^{{}^{\prime}}\,e^{i\xi\cdot
x}\,d\xi\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
By the change of variable
$\left(\xi,\,t^{{}^{\prime}}\right)\rightarrow\left(\frac{\xi}{M},\,Mt^{{}^{\prime}}\right)$
we are reduced showing
(7.46) $\begin{array}[]{ll}\|\int
e^{i(Mt-t^{{}^{\prime}})\left(|\xi|^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}\psi(\xi)\widehat{Q\left(\frac{t^{{}^{\prime}}}{M},\,\frac{.}{M}\right)}(\xi)\,dt^{{}^{\prime}}\,e^{iMx\cdot\xi}\,d\xi\|_{L_{t}^{q}L_{x}^{r}}&\lesssim
M^{2}\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
If we could prove that for every Schwartz function $G$
(7.47)
$\begin{array}[]{ll}\|S_{M}G\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|G\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
with
(7.48) $\begin{array}[]{ll}S_{M}G&:=\int
e^{i(t-t^{{}^{\prime}})\left(|\xi|^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}\psi(\xi)\widehat{G}(t^{{}^{\prime}},\xi)\,dt^{{}^{\prime}}\,e^{i\xi\cdot
x}d\xi\end{array}$
then (7.46) would hold. Indeed by (1.28) we have
(7.49) $\begin{array}[]{ll}\|\int
e^{i(Mt-t^{{}^{\prime}})\left(|\xi|^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}\psi(\xi)\widehat{Q\left(\frac{t^{{}^{\prime}}}{M},\,\frac{.}{M}\right)}(\xi)\,dt^{{}^{\prime}}\,e^{iMx\cdot\xi}\,d\xi\|_{L_{t}^{q}L_{x}^{r}}&=\|S_{M}\left(Q\left(\frac{.}{M},\frac{.}{M}\right)\right)(Mt,\,Mx)\|_{L_{t}^{q}L_{x}^{r}}\\\
&\lesssim
M^{\frac{1}{\widetilde{q}}+\frac{d}{\widetilde{r}}-\frac{1}{q}-\frac{d}{r}}\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\\\
&\lesssim M^{2}\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
By duality and composition with (7.42) it suffices to show
(7.50)
$\begin{array}[]{ll}\|e^{it\left(D^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}P^{\frac{1}{2}}_{1}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{L^{2}}\end{array}$
Again it is not difficult to see from the proof of (7.15) that
(7.51)
$\begin{array}[]{ll}\|e^{it<D>}P^{\frac{1}{2}}_{M}f\|_{L_{t}^{q}L_{x}^{r}}&\lesssim\|f\|_{H^{m}}\end{array}$
But after performing the change of variable $\xi\rightarrow M\xi$ we have by
(1.25) and (7.51)
(7.52)
$\begin{array}[]{ll}\|e^{it\left(D^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}P^{\frac{1}{2}}_{1}f\|_{L_{t}^{q}L_{x}^{r}}&=\|\int
e^{i\frac{t}{M}\left(|\xi|^{2}+1\right)^{\frac{1}{2}}}\psi^{\frac{1}{2}}\left(\frac{\xi}{M}\right)\widehat{f(M\,.)}(\xi)e^{i\frac{x}{M}\cdot\xi}\,d\xi\|_{L_{t}^{q}L_{x}^{r}}\\\
&=\|\left(e^{it<D>}P^{\frac{1}{2}}_{M}\right)\left(\widetilde{P_{M}}^{\frac{1}{2}}f(M.)\right)\left(\frac{t}{M},\frac{x}{M}\right)\|_{L_{t}^{q}L_{x}^{r}}\\\
&\lesssim
M^{\frac{1}{q}+\frac{d}{r}}\|\widetilde{P_{M}}^{\frac{1}{2}}f(M\,.)\|_{H^{m}}\\\
&\lesssim M^{\frac{1}{q}+\frac{d}{r}-\frac{d}{2}+m}\|f\|_{L^{2}}\\\
&\lesssim\|f\|_{L^{2}}\end{array}$
### 7.3. Proof of (7.4)
By decomposition, substitution and Christ-Kisilev lemma [12] it suffices to
prove
(7.53) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}Q\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim\|<D>^{1-m}Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
If we could prove
(7.54) $\begin{array}[]{ll}\|\int e^{i(t-t^{{}^{\prime}})<D>}P_{\leq
1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
and
(7.55) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim
M^{1-m}\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
then (7.53) would follow. Indeed
(7.56) $\begin{array}[]{ll}\|\int e^{i(t-t^{{}^{\prime}})<D>}P_{\leq
1}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim\|\widetilde{P_{\leq
1}}Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\\\
&\lesssim\|<D>^{1-m}Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
and
(7.57) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}P_{M}Q(t^{{}^{\prime}})\,dt^{{}^{\prime}}\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim
M^{1-m}\|\widetilde{P_{M}}Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\\\
&\lesssim\|\widetilde{P_{M}}<D>^{1-m}Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
Therefore following the same steps to those in (7.41) we get (7.53).
(7.54) follows from the composition of the trivial inequality
$\|e^{it<D>}P^{\frac{1}{2}}_{\leq
1}f\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim\|f\|_{L^{2}}$ and (7.44).
We turn to (7.55). We need to prove
(7.58) $\begin{array}[]{ll}\|\int
e^{i(t-t^{{}^{\prime}})<D>}\psi\left(\frac{\xi}{M}\right)\widehat{Q}(t^{{}^{\prime}},\xi)\,dt^{{}^{\prime}}\,e^{i\xi\cdot
x}\,d\xi\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim
M^{1-m}\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
Again by the change of variable
$\left(\xi,\,t^{{}^{\prime}}\right)\rightarrow\left(\frac{\xi}{M},\,Mt^{{}^{\prime}}\right)$
it suffices to show
(7.59) $\begin{array}[]{ll}\|\int
e^{i(Mt-t^{{}^{\prime}})\left(|\xi|^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}\psi(\xi)\widehat{Q\left(\frac{t^{{}^{\prime}}}{M},\,\frac{.}{M}\right)}(\xi)\,dt^{{}^{\prime}}\,e^{iMx\cdot\xi}\,d\xi\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim
M^{2-m}\|Q\|_{L_{t}^{\widetilde{q}}L_{x}^{\widetilde{r}}}\end{array}$
If we could prove for any Schwartz function
(7.60)
$\begin{array}[]{ll}\|S_{M}G\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim\|G\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
with $S_{M}$ defined in (7.48) then substituting $q$, $r$ for $\infty$, $2$
respectively in (7.49) we have
(7.61) $\begin{array}[]{ll}\|\int
e^{i(Mt-t^{{}^{\prime}})\left(|\xi|^{2}+\frac{1}{M^{2}}\right)^{\frac{1}{2}}}\psi(\xi)\widehat{Q\left(\frac{t^{{}^{\prime}}}{M},\,\frac{.}{M}\right)}(\xi)\,dt^{{}^{\prime}}\,e^{iMx\cdot\xi}\,d\xi\|_{L_{t}^{\infty}L_{x}^{2}}&\lesssim
M^{\frac{1}{\tilde{q}}+\frac{d}{\tilde{r}}-\frac{d}{2}}\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{q}}}\\\
&\lesssim M^{2-m}\|Q\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}}\end{array}$
where in the last inequality we used (1.25) and (1.28). It remains to prove
(7.60). By duality and composition with the trivial inequality
$\|e^{it<D>}P^{\frac{1}{2}}_{\leq
4}f\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim\|f\|_{L^{2}}$ it suffices to show
(7.50), which has already been established.
## References
* [1] P.Brenner, _On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations_ , Math. Z. 186 (1984), 383-391
* [2] P. Brenner, _On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon Equations_ , J. Differential Equations 56 (1985), 310-344
* [3] M. Christ, A. Kisilev, _Maximal operators associated to filtrations_ , J. Func. Anal. 179 (2001)
* [4] J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao, _Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation_ , Math. Res. Letters 9 (2002), pp. 659-682
* [5] J. Ginebre and G. Velo, _Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equation_ , Ann. Inst. H. Poincare Phys Theor 43 (1985), 399-442
* [6] M. Keel and T. Tao, _Endpoint Strichartz estimates_ , Amer. J. Math, 120 (1998), 955-980
* [7] H. Lindblad, C. D Sogge, _On existence and scattering with minimal regularity for semilinear wave equations_ , J. Func.Anal 219 (1995), 227-252
* [8] C. Morawetz, _Time decay for the nonlinear Klein-Gordon equation_ , Proc. Roy. Soc. A 306 (1968), 291-296
* [9] C. Morawetz and W. Strauss, _Decay and scattering of solutions of a nonlinear relativistic wave equation_ , Comm. Pure Appl. Math. 25 (1972), pp 1-31
* [10] K. Nakanishi, _Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$_, Journal of Functional Analysis 169 (1999), 201-225
* [11] K. Nakanishi, _Remarks on the energy scattering for nonlinear Klein-Gordon and Schrodinger equations_ , Tohoku Math J. 53 (2001), 285-303
* [12] H. Smith, C.D. Sogge, _Global Strichartz estimates for nontrapping perturbations of the Laplacian_ , Commun. PDE, 25 (2000), 2171-2183
* [13] E. M. Stein, _Harmonic Analysis_ , Princeton University Press, 1993
* [14] W. A. Strauss, _Nonlinear wave equations_ , CBMS Regional Conference Series in Mathematics, no 73, Amer. Math. Soc. Providence, RI, 1989
* [15] T. Roy, _Global well-posedness for the radial defocusing cubic wave equation on $\mathbb{R}^{3}$ and for rough data_, EJDE, 166, 2007, 1-22
* [16] M. Visan and X. Zhang, _Global well-posedness and scattering for a class of nonlinear Schrödinger equation below the energy space_ , preprint, available at http://www.arxiv.org/abs/math.AP/0606611
|
arxiv-papers
| 2008-09-23T01:28:40
|
2024-09-04T02:48:57.892704
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Tristan Roy",
"submitter": "Tristan Roy",
"url": "https://arxiv.org/abs/0809.3835"
}
|
0809.3886
|
# Soft processes at high energy without soft Pomeron:
a QCD motivated model.
A. Kormilitzin and E. Levin
Department of Particle Physics, School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Science
Tel Aviv University, Tel Aviv, 69978, Israel Email:
andreyk1@post.tau.ac.ilEmail: leving@post.tau.ac.il, levin@mail.desy.de;
###### Abstract:
In this paper we develop a QCD motivated model for both hard and soft
interactions at high energies. In this model the long distance behavior of the
scattering amplitude is determined by the approximate solution to the non-
linear evolution equation for parton system in the saturation domain. All
phenomenological parameters for dipole-proton interaction were fitted from the
deep inelastic scattering data and the soft processes are described with only
one new parameter, related to the wave function of hadron. It turns out that
we do not need to introduce the so called soft Pomeron that has been used in
high energy phenomenology for four decades. The model described all data on
soft interactions: the values of total, elastic and diffractive cross sections
as well as their $s$ and $t$ behavior. The value for the survival probability
of the diffractive Higgs production is calculated being less 1% for the LHC
energy range.
High density QCD , saturation, single diffraction, Pomeron structure
††preprint: TAUP -2884-08
## 1 Introduction
For three decades our microscopic theory-QCD, has been unable to describe soft
(long distance) interactions. It happens, partly, because of the embryonic
stage of our understanding of the most challenging problem in QCD: confinement
of quarks and gluons. We believe, however, that the main features of high
energy scattering at long distances can be described in the framework of QCD,
and such a description is based on the phenomenon of saturation of the parton
densities in QCD [1, 2, 3]. Indeed, at short distances the parton densities
increase with the growth of energy due to gluon emission, which is described
by the QCD linear evolution equations [4, 5]. Such an emission is proportional
to the density of emitters (partons). The process of annihilation is
suppressed since it’s strength is proportional to the parton density squared,
reflecting the fact that two partons have to meet each other in the small
volume of interaction. However, at high energy the density of partons becomes
so large that the annihilation term tends to be of the same order as the
emission term. It leads to a slow down in the increase of the parton density,
and the system of partons lives in the dynamical equilibrium with the critical
value of density (it is saturated). We have several theoretical approaches for
a QCD description of the parton system in the saturation domain (see Ref. [1,
2, 3, 6, 7, 8, 9, 10, 11, 12], but the equations that have been proposed turn
out to be so complicated, that our knowledge about this domain mostly stems
from numerical solutions of these equations [13].
The alternative approach that has been developed during the past decade is to
build models [14, 15] that incorporate the main qualitative properties of
these solutions, such as the geometrical scaling behavior of the scattering
amplitude [19, 20, 21] and the existence of the new dimensional scale (the so
called saturation scale) [1, 2, 3], which increases in the region of low $x$
(high energy). Such models successfully described all the data on deep
inelastic scattering, including the behavior of inclusive and diffractive
cross sections, at a low value of photon virtuality [14].
The main goal of this paper is to describe the processes that occur at long
distances (soft processes), where we cannot apply perturbative QCD, using the
saturation model. In this model all the needed phenomenological parameters, in
particular the energy behavior and the value of the saturation scale, were
fitted using the DIS data. The main ingredient of the high energy
phenomenology during the past four decades, namely the soft Pomeron, is not
introduced in this model. In simple words, we replace the soft Pomeron by the
scattering amplitude that describes the behavior of the saturated parton
system. The idea, that the matching between soft and hard interaction occurs
in the saturation region, and can be described by the amplitude that
originates from the high density QCD approach, is not new. This idea has been
in the air for a number of years and have been advocated in several lectures
on the subject [22]. The first attempt of a practical application of this idea
was done in Ref. [24] (see also Ref. [25]). The result of this paper was
encouraging since the estimates gave a reasonably good agreement with the
experimental data on the total cross section of the pion, kaon, and proton
interaction with the proton target. In this paper we continue to develop these
ideas and we give the comparison with the experiment data for the full set of
the experimental data of soft interactions, that include the energy behavior
of the total, elastic and diffraction cross sections as well as a $t$\-
behavior of the elastic cross sections, and the mass behavior for the cross
sections of diffraction production.
## 2 The main idea
As has been mentioned, our main idea is to describe the so called ‘soft’
interaction in the high parton density QCD (hdQCD). In hdQCD we are dealing
with the dense system of partons which cannot be treated in the perturbative
QCD approach. Being non-perturbative in its nature, hdQCD leads to typical
distances $r=1/Q_{s}(x)\ll R\sim 1/\Lambda_{QCD}$ where $R$ is the size of a
hadron. The saturation momentum $Q_{s}(x)$ is a new dimensionful scale which
increases with the growth of the energy ( or at $x\to 0$)[1, 2, 3]. Therefore,
the QCD coupling $\alpha_{S}\left(Q_{s}(x)\right)$ is small and, because of
this, we are able to develop a theoretical approach for such a dense system
having only a limited input from the unknown confinement region. To illustrate
the limitation of our approach, it is enough to look at the expression for the
saturation scale in hdQCD[1, 26]
$\displaystyle Q^{2}_{s}(Y)\,\,$ $\displaystyle=$
$\displaystyle\,\,Q^{2}_{s}(Y_{0})\,\exp\left(\bar{\alpha}_{S}\,\frac{\chi(\gamma_{cr})}{1-\gamma_{cr}}\,\,(Y-Y_{0})\,\,-\,\,\frac{3}{2(1-\gamma_{cr})}\,\ln(Y/Y_{0})-\right.$
$\displaystyle-$
$\displaystyle\,\left.\frac{3}{(1-\gamma_{cr})^{2}}\,\sqrt{\frac{2\,\pi}{\bar{\alpha}_{S}\,\chi^{\prime\prime}(\gamma_{cr})}}\,(\frac{1}{\sqrt{Y}}\,-\,\frac{1}{\sqrt{Y_{0}}}\,)\,,+\,O(\frac{1}{Y})\right)$
(2.1)
In Eq. (2) the energy dependence of $Q_{s}(Y=\ln(1/x))$ is predicted from
hdQCD, ($\chi(\gamma)$ is the Mellin transform of the BFKL kernel) but the
scale $,Q^{2}_{s}(Y_{0})$ stems from the confinement region, demonstrating a
need for some input from this theoretically unknown region. Our hope that such
input will be a limited number of constants.
Let us consider the deep inelastic electron-proton scattering (DIS), to
illustrate our point. If the virtuality of the photon $Q^{2}$ is very large,
$Q^{2}\gg Q^{2}_{s}(x)$, the typical distances $r\approx 1/Q$, and we can
safely apply the perturbative QCD approach based on the operator product
expansion, and the DGLAP evolution equation. However, if $\Lambda^{2}_{QCD}\ll
Q^{2}\ll Q^{2}_{s}(x)$, the situation changes crucially. All terms in the
operator product expansion become of the same order, and we have to use the
hdQCD approach. In this approach, the cross section $\sigma^{\gamma^{*}p}$
shows a geometrical scaling behaviour[27], namely,
$\sigma^{\gamma^{*}p}=\left(1/Q^{2}_{s}(x)\right)\,F\left(Q^{2}/Q^{2}_{s}(x)\right)$
which has been confirmed experimentally. This means that the typical distances
$r\approx 1/Q_{s}(x)\ll 1/\Lambda_{QCD}$, which are short.
Our idea is that we can describe $Q^{2}\approx\Lambda^{2}_{QCD}$, in the
framework of the same hdQCD approach. Having the geometrical scaling behavior
in mind, such an idea does not look crazy. However, in DIS we are dealing with
the total cross sections which are related to the amplitude integrated over
the impact parameters ($b$). For a treatment of the soft interaction
observables, we need to know the $b$ dependence. As was shown in Ref. [28],
hdQCD predicts the power-like decrease at large value of $b$, which
contradicts both the theoretical estimates and experimental observations.
Therefore, we need an input from the confinement region to specify the $b$
dependence of the scattering amplitude. Let us first discuss large $b\geq
1/\Lambda_{QCD}\gg 1/Q_{s}(x)$. The scattering amplitude for DIS
$A\left(Q^{2},W;t\right)$ depends on the photon virtuality $Q^{2}$, energy
$W=\sqrt{s}=Q/\sqrt{x}$ and on the momentum transferred $t=-q^{2}<0$. Using
the dispersion relation in the $t$-channel we have
$A\left(Q^{2},W;t\right)\,\,=\,\,\frac{1}{\pi}\,\int^{\infty}_{4m^{2}_{\pi}}\,\frac{dt^{\prime}\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)}{t^{\prime}+q^{2}}$
(2.2)
Calculating the amplitude in the impact parameter representation we can reduce
Eq. (2.2) to the form
$\displaystyle A\left(Q^{2},W;b\right)\,\,$ $\displaystyle=$
$\displaystyle\,\,\frac{1}{(2\pi)^{2}}\,\int\,d^{2}\vec{q}\,A\left(Q^{2},W;t\right)\,e^{i\vec{q}\cdot\vec{b}}\,\,=\,\,\frac{1}{2\pi^{2}}\,\int^{\infty}_{4m^{2}_{\pi}}\,dt^{\prime}\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)\,\int\,\frac{q\,dq\,\,J_{0}\left(qb\right)}{t^{\prime}+q^{2}}\,$
$\displaystyle=$
$\displaystyle\,\,\frac{1}{2\pi^{2}}\,\int^{\infty}_{4m^{2}_{\pi}}\,dt^{\prime}\,K_{0}\left(\sqrt{t^{\prime}}\,b\right)\,\,Im_{t}A\left(Q^{2},W;t^{\prime}\right)\,\,\
\stackrel{{\scriptstyle b\geq
1/(2m_{\pi})}}{{\longrightarrow}}\,C\,Im_{t}A\left(Q^{2},W;t^{\prime}=4m^{2}_{\pi}\right)\,K_{0}\left(2m_{\pi}b\right)$
where in the constant $C$, all the numerical factors have been absorbed.
In Eq. (2), the $b$-dependence is determined by the mass of the lightest
hadron (pion), and cannot not be reproduced in perturbative or/and high
density QCD. However, one can see from Eq. (2) that we have a factorization of
the $b$ dependence, and the dependence on the photon virtuality and energy.
As it is well known (see Ref. [12] and references therein), in the kinematic
region $\alpha_{S}\,\leq\alpha_{S}\,\ln(1/x)\leq 1/\alpha_{S}$ in the
saturation region the hdQCD approach reduces to the BFKL Pomeron calculus. We
generalize Eq. (2) for the BFKL Pomeron in the following way
$A_{BFKL}\left(Q^{2},x;b\right)\,\,\,=\,\,\,\stackrel{{\scriptstyle\mbox{
\small short
distances}}}{{A_{BFKL}\left(Q^{2},x;t=0\right)}}\,\times\,\stackrel{{\scriptstyle\mbox{\small
long distances}}}{{S(b)}}$ (2.4)
One can see that Eq. (2.4) claims a kind of factorization between short and
long distances. The short distance part we can describe in hdQCD, while the
long distance part has to be taken from the confinement domain. In a more
general way, we can re-write Eq. (2.4) in the form
$A_{BFKL}\left(Q^{2},x;\vec{b}\right)\,\,\,=\,\,\,\int
d^{2}\vec{b^{\prime}}\stackrel{{\scriptstyle\mbox{ \small short
distances}}}{{A_{BFKL}\left(Q^{2},x;\vec{b^{\prime}}\right)}}\,\times\,\stackrel{{\scriptstyle\mbox{\small
long distances}}}{{S(\vec{b}-\vec{b}^{\prime})}}$ (2.5)
where $b^{\prime}$ in the short distance part is of the order of
$b^{\prime}\approx 1/Q_{s}(x)$.
The approach of Eq. (2.4) and Eq. (2.5), at first sight contradicts the high
energy phenomenology based on the soft Pomeron and Reggeons, as well as well
as the lattice calculations [30]. In both approaches, the Pomeron trajectory
has $\alpha^{\prime}_{{I\\!\\!P}}>0$, and at positive $t$ the glueballs lie on
this trajectory. In Eq. (2.4) and Eq. (2.5), the Pomeron slope
$\alpha^{\prime}_{{I\\!\\!P}}=0$. We will show below that we will be able
reproduce the experimental data on energy dependence of the elastic slope
which used to consider as the argument for
$\alpha^{\prime}_{{I\\!\\!P}}=0.25\,GeV^{-2}$. On the theoretical side, the
only theory which can treat at the moment on the same footing both short and
long distances: N=4 SYM with AdS/CFT correspondence, leads to a picture which
is in striking agreement with our approach[29]. Namely, in this approach at
$t>0$ ( resonance region) we have a normal soft Pomeron with
$\alpha^{\prime}_{{I\\!\\!P}}>0$ while at $t<0$ (scattering region)
$\alpha^{\prime}_{{I\\!\\!P}}=0$.
## 3 The model
### 3.1 Motivation
Our building of the model is based on the dipole approach to high energy QCD
[31]. In this approach, the evolution of the parton (colorless dipole) system
can be written in the form of the equation for the generating functional, with
a transparent probabilistic interpretation for them. The generating functional
is defined as [31]
$Z\left(Y\,-\,Y_{0};\,[u]\right)\,\,\equiv\,\,\sum_{n=1}\,\int\,\,P_{n}\left(Y\,-\,Y_{0};\,x_{1},y_{1};\dots;x_{i},y_{i};\dots;x_{n},y_{n}\right)\,\,\prod^{n}_{i=1}\,u(x_{i},y_{i})\,d^{2}\,x_{i}\,d^{2}\,y_{i}$
(3.6)
where $u(x_{i},y_{i})\equiv u_{i}$ is an arbitrary function of $x_{i}$ and
$y_{i}$. The coordinates $(x_{i},y_{i})$ describe the colorless pair of gluons
or a dipole. $P_{n}$ is a probability density to find $n$ dipoles with the
size $x_{i}-y_{i}$, and with impact parameter $(x_{i}+y_{i})/2$. Assuming that
we have only a decay of one dipole to two dipoles, directly from the physical
meaning of $P_{n}$ it follows [31, 32, 33]
$\frac{\partial\,P_{n}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})}{\partial
Y}\,\,\,=$ (3.7) $=\,\,\sum_{i}\ V_{1\to
2}\bigotimes\left(P_{n-1}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})\,-\,P_{n}(Y;\dots;x_{i},y_{i};\dots;x_{n},y_{n})\right)$
where $\bigotimes$ denotes all necessary integrations and $V_{1\to 2}$ is the
vertex for the decay of one dipole to two dipoles. This vertex is equal to
$V_{1\to
2}\left((x,y)\to(x,z)+(z,y)\right)\,\,=\,\,\frac{\bar{\alpha}_{S}}{2\pi}\,\frac{(\vec{x}-\vec{y})^{2}}{(\vec{x}-\vec{z})^{2}\,(\vec{z}-\vec{y})^{2}}$
(3.8)
Eq. (3.7) is a typical Markov’s chain which takes into account the $s$ \-
channel unitarity on each step of evolution since it has two terms: the birth
of the new dipole due to the decay of the parent dipole (positive term in Eq.
(3.7)) and the death of the dipole due to the same process.
Eq. (3.7) can be rewritten as the equation for the generating functional [32,
33], namely,
$\frac{\partial\,Z\,\left(Y-Y_{0};[\,u\,]\right)}{\partial\,Y}\,\,=\,\,\int\,d^{2}x\,d^{2}y\,d^{2}z\,\,V_{1\to
2}\left((x,y)\to(x,z)+(z,y)\right)\,\frac{\partial}{\partial\,u(x,y)}\,\,Z\,\left(Y-Y_{0};[\,u\,]\right)$
(3.9)
$Z\,\left(Y-Y_{0};[\,u\,]\right)$ satisfies the initial and boundary
conditions:
Initial conditions: $\displaystyle Z\,\left(Y-Y_{0}=0;[\,u\,]\right)$
$\displaystyle=\,\,\,u(x,y)\,;$ (3.10) Boundary conditions: $\displaystyle
Z\,\left(Y-Y_{0};[\,u=1\,]\right)$ $\displaystyle=\,\,\,1\,;$ (3.11)
The advantage of the generating functional is that we can write it in terms of
this simple functional formulae for the scattering amplitude, which has an
obvious partonic interpretation. Indeed, the scattering amplitude in the lab.
frame can be written in the form [6, 32, 33]
$\displaystyle N\left(Y-Y_{0};r,b\right)\,\,=$ (3.12)
$\displaystyle\,\,\sum^{\infty}_{n=1}\,\frac{(-1)^{n}}{n!}\,\,\int\,\prod^{n}_{i=1}\,\,d^{2}\,x_{i}\,d^{2}\,y_{i}\,\,\rho_{n}\left(r,b;\\{x_{i},y_{i}\\},Y-Y_{0}\right)\gamma_{n}\left(r,b;\\{x_{i},y_{i}\\},Y_{0}\right)$
where $Y-Y_{0}\,\,\gg\,\,1$ but $Y_{0}\,\,\approx\,\,1$ and
$\rho_{n}\left(Y-Y_{0},\\{x_{i},y_{i}\\}\right)\,\,\,=\,\,\,\prod^{n}_{i=1}\,\frac{\delta}{\delta
u(x_{i},y_{i})}Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)|_{u(x_{i},y_{i})=1}$
(3.13)
and $\gamma_{n}$ is the amplitude of interaction of $n$ dipoles at low energy
( small values of rapidity $Y_{0}$) with the coordinates $x_{i},y_{i}$ with
the target with size $r$ and impact parameter $b$. In the parton language the
generating functional determines the parton wave function for which we have
the evolution equation Eq. (3.9) in QCD. $\gamma_{n}$ can have a non-
perturbative origin. We assume that
$\gamma_{n}\left(r,b;\\{x_{i},y_{i}\\},Y_{0}\right)\,\,=\,\,\prod^{n}_{i=1}\gamma(x_{i},y_{i};Y_{0})$.
This assumption means that the dipoles inside the target have no correlation,
which is correct, as far as we know, only for a nucleus target.
In spite of the transparent physics behind our equations, the equation for the
generating functional is difficult to solve analytically, especially if we
generalize Eq. (3.9) to the case of the so called Pomeron loops [35, 36, 37].
As has been mentioned, we wish to find a model solution to Eq. (3.9). We will
do this using two key observations of Ref.[12]. The first one is the fact that
in the huge kinematic region of $\bar{\alpha}_{S}Y\,\leq 1/\bar{\alpha}_{S}$,
the system of partons that we are dealing with turns out to be a system of
non-interactive partons. In other words, it means that the generating
functional of Eq. (3.6) can be rewritten in a simpler form, namely.
$Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)\,\,\,=\,\,\,\sum^{\infty}_{n=1}\,\frac{C_{n}}{n!}\,\,\left(\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,P(Y-Y_{0};x,y;x^{\prime},y^{\prime})\,\,\left(u(x^{\prime},y^{\prime})\,-\,1\right)\right)^{n}$
(3.14)
where $P(Y-Y_{0};x,y;x^{\prime},y^{\prime})$ is the amplitude for one BFKL
Pomeron exchange, between dipoles $(x,y)$ and $(x^{\prime},y^{\prime})$.
In Ref.[12] it was shown , using the analytical solution of Ref. [34] , that
actually $C_{n}\,=\,1$ for the dipoles with the size $r\sim 1/Q^{2}_{s}$ where
$Q_{s}$ is the saturation momentum.
Therefore, Eq. (3.14) can be simplified and rewritten in the form
$Z\,\left(Y-Y_{0};\\{\,u(x_{i},y_{i})\,\\}\right)\,\,\,=\,\,\,\exp\left(\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,P(Y-Y_{0};x,y;x^{\prime},y^{\prime})\,\left(u(x^{\prime},y^{\prime})\,-\,1\right)\right)$
(3.15)
and the amplitude has the form
$N\left(Y-Y_{0};x,y\right)\,\,=\,\,1-\exp\left(-\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,\,P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\right)$
(3.16)
where $\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)$ is the scattering
amplitude of the dipole $(x^{\prime},y^{\prime})$, with the target at low
energy.
### 3.2 Main formulae and assumptions
Eq. (3.16) is the main formula for constructing our model. The first
ingredient of the model is the choice of the amplitude $\gamma$ and the value
of $Y_{0}$. We choose this amplitude in the form [38]
$\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\,\,\frac{\pi^{2}\,\alpha_{S}(\mu^{2})}{3}\,r^{2}\,x_{0}G^{DGLAP}(x_{0},\mu^{2})\,S(b)$
(3.17)
where $\vec{r}\,=\,\vec{x}-\vec{y}$ and
$\vec{b}=\frac{1}{2}\,(\vec{x}+\vec{y})$ with $x_{0}=10^{-2}$. $xG$ in Eq.
(3.17) is the solution of the DGLAP evolution equation which describes the DIS
experimental data. $\mu$ is equal to
$\mu^{2}\,\,=\,\,\mu^{2}_{0}\,\,+\,\,\frac{C}{r^{2}}$ (3.18)
where $\mu_{0}$ and $C$ are phenomenological parameters which has to be found
fitting the experimental data on DIS (see Ref.[15] for details). Eq. (3.18)
means that $\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)$ at large
dipole size $r\mu_{0}\gg 1$ is determined by the following expression
$\gamma_{mod}\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,=\,\,\,\,\frac{\pi^{2}\,\alpha_{S}(\mu^{2}_{0})}{3}\,r^{2}\,x_{0}G^{DGLAP}(x_{0},\mu^{2}_{0})\,S(b)$
(3.19)
Of course Eq. (3.19) has no theoretical basis, and our hope is that the
amplitude at low $x$ will be not very sensitive to this kinematic region, in
the initial condition.
$S(b)$ is the impact parameter profile function, which is chosen as the
Fourier image of the proton form factor, namely,
$S(b)\,\,=\,\,\frac{2}{\pi\,R^{2}}\left(\frac{\sqrt{8}\,b}{R}\right)\,K_{1}\left(\frac{\sqrt{8}\,b}{R}\right)$
(3.20)
where $R$ is the proton radius ($R=0.89\,fm$).
In Eq. (3.16) $P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)$ stands for the
BFKL Pomeron Green’s function. This Green’s function describes the energy
($x$) evolution of the gluon system in the region of high energy (low $x$).
This evolution allows us to find the dipole scattering amplitude or parton
density at low $x$ from the initial condition: the amplitude at lower
$x=x_{0}$, but for any value of the dipole size ($r$).
However, we know that the BFKL Pomeron alone cannot describe the experimental
data, since it determines the anomalous dimension only at low $x$. The
experimental data shows a good agreement with the DGLAP evolution equation,
which is written as the evolution in $\ln(\mu^{2})$ (see Eq. (3.17)), which
gives us the parton density from the boundary condition: the parton density at
$\mu^{2}=\mu^{2}_{0}$ but at any value of $x$ including the region of low $x$.
Strictly speaking, it cannot be used in Eq. (3.16), but assuming Eq. (3.18)
for the scale $\mu^{2}$, we impose the condition that at long distances, the
typical momentum scale in the linear evolution equation is equal to
$\mu^{2}_{0}$. In the framework of this assumption, it looks reasonable to
assume that
$\displaystyle\Omega(Y,r,b)\,\,\,$ $\displaystyle=$
$\displaystyle\,\,2\,\int\,d^{2}x^{\prime}\,d^{2}y^{\prime}\,\,P\left(Y-Y_{0};x,y;x^{\prime},y^{\prime}\right)\gamma\left(Y_{0};x^{\prime},y^{\prime}\right)\,\,$
(3.21) $\displaystyle=$
$\displaystyle\frac{\pi^{2}\,\alpha_{S}(\mu^{2})}{3}\,r^{2}\,xG^{DGLAP}(x,\mu)\,S(b)$
Finally, we can write the following model expression for the scattering
amplitude;
$N\left(x;r,b\right)\,\,\,=\,\,1\,\,\,-\,\,\exp\left(-\frac{1}{2}\Omega\left(x,r,b\right)\right)$
(3.22)
where $\Omega\left(x,r,b\right)$ is given by Eq. (3.21).
### 3.3 Fixing the values of the phenomenological parameters
In Eq. (3.18) we have introduced two phenomenological parameters:
$\mu^{2}_{0}$ and $C$. The first determines the value of the virtuality of the
probe in DIS processes from which we can start using the perturbative QCD
evolution equations. The second ($C$) gives the relation between scales in the
momentum representation and in the coordinate representation. In the case of
the DGLAP evolution, $C=4$ ( see Ref. [38]), but since the amplitude has a
more complicated form than Eq. (3.22), the value of $C$ can be different.
Two more parameters stem from the initial condition for the gluon density in
Eq. (3.21), namely,
$xG^{DGLAP}\left(x,Q_{0}^{2}\right)\,\,\,=\,\,\frac{A}{x^{\omega_{0}}}(1-x)^{6}\,$
(3.23)
which corresponds to the exchange of the soft Pomeron with the intercept
$\omega_{0}$ at the initial hard scale $\mu^{2}=Q^{2}_{0}$ in traditional high
energy phenomenology, and the factor $(1-x)^{6}$ reflects the valence quark
dependance. In our approach we do not assume that the soft Pomeron exists, and
we view Eq. (3.23) just as the simplest function that reflects the behavior of
the experimental data in DIS.
We use the observation of Ref. [39], that the anomalous dimension in the
leading order can be written in a very simple form, namely,
$\gamma(\omega)\,\,=\,\,\bar{\alpha}_{S}\left(\frac{1}{\omega}\,\,-\,\,1\right)$
(3.24)
The explicit solution to the DGLAP evolution equation with $\gamma(\omega)$ of
Eq. (3.24), and with the initial condition of Eq. (3.23), has been found in
Ref. [15] and looks as follows
$xG^{DGLAP}(x,t)\,=\,\sum_{k=0}^{6}\binom{6}{k}\,(-1)^{k}\,xG^{(k)}(x,t,\omega_{k})$
(3.25)
where
$xG^{(k)}\left(y\equiv\ln(1/x),t,\omega_{k}\right)\,\,=\,\,A\,e^{-t\,+\,\omega_{k}\,y}\,\,\left\\{\int^{y}_{0}\,\,dy^{\prime}\,e^{-\omega_{k}\,y^{\prime}}\,\sqrt{\frac{t}{y^{\prime}}}\,\,I_{1}\left(2\sqrt{t\,y^{\prime}}\right)\,\,\,+\,\,\,1\right\\}$
(3.26)
with
$\omega_{k}=\omega_{0}-k\,;\;\;\;\;t\,\,\,\equiv\,\,\,\frac{4N_{c}}{b_{0}}\,\ln\frac{\ln\left(\mu^{2}/\Lambda^{2}\right)}{\ln\left(\mu^{2}_{0}/\Lambda^{2}\right)}\,;\,\,\,\,\,\,\,\,b_{0}\,\,=\,\,11-\frac{2n_{f}}{3}\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,y\,=\,\ln(1/x)$
(3.27)
In Eq. (3.21) the typical scale of hardness in the gluon structure function is
determined by the process of gluon emission while the factor
$\alpha_{S}\,r^{2}$ takes into account the integration over the wave function
of the dipole with the size $r$. Having this in mind, we introduce a different
scale for $\alpha_{S}$ in Eq. (3.21). Finally, we use
$\Omega(x,r,b)\,\,\,=\,\,\,\frac{\pi^{2}\,\alpha_{S}(\tilde{\mu}^{2})}{3}\,r^{2}\,xG^{DGLAP}(x,\mu)\,S(b)\,\,\,\mbox{with}\,\,\,\tilde{\mu}^{2}\,\,=\,\,\tilde{\mu}^{2}_{0}\,\,+\,\,\frac{C}{r^{2}}$
(3.28)
where $xG^{DGLAP}(x,\mu)$ is given by Eq. (3.26) and the initial hard scale
$\widetilde{\mu}^{2}_{0}$ is different from that of $\mu_{0}^{2}$. We perform
a fit of the saturation model, to the DIS experimental data and fixing the
parameters, we wish to describe the soft data without using the concept of the
”soft” pomeron. The fit procedure was performed by using the minuit routine,
the statistical and systematical errors have been added in quadrature. We have
used all of the recent data on deep inelastic scattering processes, from
different collaborations. The most suitable were H1 [16] and ZEUS [17, 18]. It
was observed, that H1 data for medium and large values of $Q^{2}$ coincides
with that from the ZEUS col., but up to a scaling factor of 1.05. Hence, for
the fitting procedure, we took only the ZEUS data for vary small, medium and
large values of $Q^{2}$. Since we are interested in the high energy
description, we took data below $x<0.01$ and $0.045<Q^{2}<150GeV^{2}$. The
upper limit on virtuality is originated from the upper limit on the $x$
variable. Fitting with H1 scaled data, yields almost the same result. The
total number of experimental points was 170. During the fitting procedure, we
observed a strong correlation between different parameters, so we decided to
fix a number of them. We choose to fix a parameter $C$ and the initial hard
scale $Q^{2}_{0}$. The masses of the quarks were taken as follows:
$m_{light}\;=\;0.25\;GeV$ and $m_{charm}\;=\;1.3\;GeV$. $\chi^{2}$ for this
parametrization is $\thicksim 1.05$. The summary of the fitting procedure is
given in the table 1
Model | $\widetilde{\mu}_{0}^{2}$ | $\mu_{0}^{2}$ | $Q_{0}^{2}$ | $C$ | $\omega_{0}$ | $A$ | $\chi^{2}/d.o.f.$
---|---|---|---|---|---|---|---
A | 0.23 | 1.23 | 1.0 | 1.0 | 0.028 | 1.81 | 1.04
B | 0.069 | 1.23 | 0.58 | 1.0 | 0.087 | 0.6 | 3.08
Table 1: _Resulting parameters from the fit to DIS data Model A gives a very
good description of the DIS data, while model B reproduces quite well the data
on soft processes_.
The bolded font corresponds to the fixed parameters $C$ and $Q^{2}_{0}$, which
were chooses to be equal to $1.0$. The value of the hard scale
$\mu^{2}_{0},\;\widetilde{\mu}_{0}^{2}$. is in the units of energy $GeV$.
As one can see, $\chi^{2}/d.o.f.$ is close to 1 and fit gives very good
description of all experimental data on DIS (see Fig. 1). However, we present
in Table 1 a second fit which leads to worse $\chi^{2}/d.o.f.$ but reproduces
the data on soft interaction quite well as we will see below. It should be
stressed that both fits describe the DIS data at large values of photon
virtualities with small $\chi^{2}/d.o.f.$ . Therefore, we can consider model
the A as an attempt to describe the experimental data on DIS and soft
processes, assuming that Eq. (3.22) is correct in the saturation region. In
model B we explore a different idea: Eq. (3.22) is only approximate formula
that incorporates the main qualitative properties of scattering amplitude for
both DIS and soft scattering in the saturation region. The question which we
try to answer using model B is the following: is it possible to give the
unique description of long distance physics both in DIS and soft interaction
based on the QCD amplitudes at short distances.
| |
---|---|---
| |
| |
Figure 1: Description of the DIS experimental data in our models. Dotted line
describes the model B while solid line corresponds to Model A.
## 4 Cross sections of hadron-hadron interaction at high energy
### 4.1 General approach
Eq. (3.22) gives a smooth continuation to the long distance physics describing
the DIS data at very low values of the photon virtualities. We wish to extend
this description to the hadron-hadron interaction without assuming something
in addition, for example, the existence of the soft Pomeron. The simplest
formula we can write for the total cross section, is a straightforward
generalization of the formula for the DIS cross section with the replacement
of $\Psi_{\gamma^{*}}(r)\to\Psi_{hadron}(\\{r_{i}\\})$ , namely
$\sigma_{tot}\,\,=\,\,2\,\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)$
(4.29)
where $n$ is the number of dipoles that we need to introduce to describe a
hadron. For example, for a meson we need only one dipole, while for the proton
we have to introduce at least two colorless dipoles.
The total elastic cross section can be written in the form:
$\sigma_{el}\,\,=\,\,\int\,d^{2}b\left(\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\right)^{2}$
(4.30)
For the differential elastic cross section we have the following expression
$\frac{d\sigma_{el}}{dt}\,\,=\,\,\left(\frac{1}{(2\pi)^{2}}\,\int
J_{0}(qb)d^{2}b\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i},b)\right)^{2}$
(4.31)
where $J_{0}$ is the Bessel function. It is even easier to calculate the slope
in $t$ at $t=0$. It is equal to
$B_{el}\,\,=\,\,\frac{d\ln
d\sigma_{el}/dt}{dt}|_{t=0}\,\,=\,\,\frac{1}{2}\,\frac{\int\,b^{2}\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)}{\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)}$
(4.32)
The process of diffractive dissociation is a more complicated phenomenon.
Indeed, the eikonal type formula of Eq. (3.22) leads to diffractive
dissociation in the state of $n$ free dipoles which, being a system with a
limited number of dipoles, cannot create a system of hadrons with large mass.
In other words, this diffractive production falls down as a function of
produced mass. Therefore, we can calculate diffraction in the region of the
small mass using the Good-Walker formula [41] , namely
$\displaystyle\sigma^{low\;M}_{diff}\,\,\,=\,\,\,$ (4.33)
$\displaystyle\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N^{2}(x,r_{i};b)\,\,-\,\,\int\,d^{2}b\left(\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\right)^{2}$
the first term is the cross section for the production of the system of $n$
free dipoles, while the second is the elastic cross section. We subtracted
this term to find the cross section of the hadron state, which is different
from the initial one.
Figure 2: Simple Pomeron diagrams for the total cross section (Fig. 2-a) and
for the diffractive production in the region of large mass
($Y-Y_{1}=\ln(M^{2}/s_{0})$, Fig. 2-b). The dashed line shows the cut Pomeron.
. For large mass diffraction, we have to develop a new approach. The simple
form of the generating functional of Eq. (3.15) stems from the fact that the
system of interacting Pomerons can be reduced to the exchange of non-
interacting Pomerons after integration over rapidity $Y_{1}$, as it is shown
in Fig. 2-a. In the case of diffractive production we can also calculate the
Pomeron diagrams with one cut Pomeron (shown by dashed line in Fig. 2-b) but
the value of $Y-Y_{1}$ is fixed, namely, $Y-Y_{1}=\ln(M^{2}/s_{0})$ where $M$
is the mass of the diffractively produced system and $s_{0}$ is the energy
scale ($s_{0}\approx 1GeV^{2}$). In more complicated diagrams, we have to
integrate over rapidities (for example, over $Y^{\prime}_{1}$ and
$Y^{\prime}_{2}$ in Fig. 2-b). For the Pomeron with the intercept larger than
1, as in our case, these integrations result in three diagrams in Fig. 2-b. It
should be stressed that the integral over the produced mass gives the dominant
contribution at $Y-Y_{1}=\ln(M^{2}/s_{0})\,\approx 1/\bar{\alpha}_{S}$.
However, it has not been taken into account in our eikonal - type model, which
only describes the G-W contribution to the region of small mass.
Therefore, for $d\sigma/dM^{2}$ at large $M^{2}\gg s_{0}$, for one dipole
$(x,y)$ we can write a generic formula (see Refs. [44])
$\displaystyle
M^{2}\,\frac{d\sigma^{high\;M}_{diff}}{dM^{2}}\,\,\,=\,\,\,V_{1\to
2}\left((x^{\prime},y^{\prime})\to(x^{\prime},z)+(z,y^{\prime})\right)\,\,\bigotimes\,\,\,\,\exp\left(-\Omega(s;x,y)\right)$
(4.34)
$\displaystyle\left(1-\exp\left(-\tilde{\Omega}(M^{2};x,y;x^{\prime},y^{\prime})\right)\right)\,\left[\exp\left(-\frac{1}{2}\left\\{\Omega(s/M^{2};x^{\prime},z)+\Omega(s/M^{2};z,y^{\prime})\right\\}\right)\,-\,\exp\left(-\frac{1}{2}\Omega(s;x^{\prime},y^{\prime})\right)\right]^{2}$
where $\bigotimes$ denotes all needed integrations, and $\tilde{\Omega}$ is a
new opacity for the interaction of two dipoles: $(x,y)$ and
$(x^{\prime},y^{\prime})$, which we can build using the same approach as in
Eq. (3.21). We start to analyze Eq. (4.34) considering the emission of an
extra gluon (see Fig. 4).
Figure 3: The process of diffractive production in the region of large mass in
perturbative QCD.
Figure 4: Feedback to the process of elastic scattering of dipole$(x,y)$ due
to the emission of an extra gluon.
One can see that our process has three well separated stages. The first one is
a penetration of the dipole $(x,y)$ through the target without inelastic
interaction. We introduce the factor ”$\exp\left(-\Omega\right)$” to describe
this stage. This factor sums all Pomeron diagrams with Pomerons that carry the
total energy of the process( rapidity $Y$). See the second diagram in Fig.
2-b. The second stage is the emission of one extra gluon. The dipole decay is
responsible for this stage with the probability given by Eq. (3.8). The last,
third stage is the interaction of two produced dipoles $(x^{\prime},z)$ and
$z,y^{\prime})$ with the target. This stage is taken into account in Eq.
(4.34) by the factor in the brackets. This factor has been discussed in Ref.
[43].
In the first diagrams (see Fig. 4) the second stage is very simple: it is just
the perturbative emission of one extra gluon. The simplified formula for this
diagram has the same structure as Eq. (4.34), but with a simple expression for
$\exp\left(-\Omega(s;x,y)\right)\left(1-\exp\left(-\tilde{\Omega}(M^{2};x,y;x^{\prime},y^{\prime})\right)\right)$,
namely
$\displaystyle\sigma^{high\;M}_{diff}({Fig.~{}\ref{gem1}})\,\,=\,\,\frac{\bar{\alpha}_{S}\,n}{2}\,\,\int^{\infty}_{M^{2}_{0}}\,\frac{dM^{2}}{M^{2}}\,\,\,\int
d^{2}b\int\prod^{n}_{i=1}\,dr^{2}_{i}\,\,\exp\left(-\sum^{n}_{i=2}\Omega(s/s_{0};r_{i},b)\right)\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,$
$\displaystyle\left\\{\exp\left(-\Omega(s/s_{0};r_{1},b)\right)\,\,r^{2}_{1}\int^{\infty}_{r^{2}_{1}}\frac{dr^{2}}{r^{4}}\,\left(1-\exp\left(-\left\\{\Omega(s/M^{2};r;b)\,-\,\frac{1}{2}\Omega(s/M^{2},r_{1};b)\right\\}\right)\right)^{2}\,\,\right.$
$\displaystyle\left.-\,\,r^{2}_{1}\,\int\frac{d^{2}r}{2\pi\,r^{2}\,(\vec{r}_{1}-\vec{r})^{2}}\,\left(1\,\,-\,\,\exp\left(-\Omega(s/s_{0};r_{1},b)\right)\right)^{2}\right\\}$
(4.35)
where $n$ is the number of dipoles in a proton and $\vec{r}=\vec{x}-\vec{z}$ .
In Eq. (4.1) we consider the region of integration where $r_{i}\ll r$. The
second term in curly brackets takes into account a change for elastic
scattering of the dipole $r_{1}$ due to the emission of one extra gluon (see
Fig. 4.) Eq. (4.1) is written in the leading log approximation of perturbative
QCD in which we consider $\bar{\alpha}_{S}\ln(M^{2}/s_{0})\,\approx\,1$ while
$\alpha_{S}\ll 1$. Factor
$\left(1-\exp\left(-\Omega(s/M^{2};r)+\frac{1}{2}\Omega(s/M^{2},r_{l};b)\right)\right)$
is the amplitude of gluon-dipole scattering that has been discussed in Ref.
[43]. The formulae for diffraction production is well known (see Refs. [44]
for details). In the same approximation Eq. (4.34) has the form
$\displaystyle
M^{2}\,\frac{d\sigma^{high\;M}_{diff}}{dM^{2}}\,\,\,=\,\,\,\frac{\bar{\alpha}_{S}\,n}{2}\,\,\,\int\,d^{2}b\,\int\prod^{n}_{i=1}\,d^{2}r_{i}\,\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\left(\exp\left(-\Omega(s/s_{0};r_{i},b)\right)\right)\,$
(4.36) $\displaystyle\int\,d^{2}r\,\,r^{2}\,\,\int
d^{2}b^{\prime}\,\,\,\left(1-\exp\left(-\Omega^{BFKL}\left(M^{2}/s_{0};r_{i},r\right)\right)\right)\,\,\int^{\infty}_{r^{2}_{i}}\frac{dr^{\prime
2}}{r^{\prime
4}}\,\,\,\left(1-\exp\left(-\Omega(s/M^{2};r^{\prime};b)\right)\right)^{2},$
where $\Omega^{BFKL}\left(M^{2}/s_{0};r_{i},r\right)$ is the solution of the
BFKL equation with the initial condition:
$\frac{d\,\,\Omega^{BFKL}\left(M^{2}/s_{0}=1;r_{i},r\right)}{d\ln(M^{2}/s_{0})}\,\,\,=\,\,\,\frac{\bar{\alpha}_{S}}{2}\,\delta(\vec{r}-\vec{r}^{\prime})$
(4.37)
This $\Omega$ is equal to [5]
$\displaystyle\Omega^{BFKL}\left(M^{2}/s_{0}=1;r_{i},r\right)\,\,$
$\displaystyle=$
$\displaystyle\,\,\int\frac{d\gamma}{2\pi\,i\,\omega(\gamma)}\,e^{\omega(\gamma)\ln(M^{2}/s_{0})\,\,+\gamma\ln(r^{2}_{i}/r^{2})}\,\,$
$\displaystyle=$
$\displaystyle\,\,\sqrt{\frac{r_{i}^{2}}{r^{2}}}\,\frac{1}{\omega(0)}\,\sqrt{\frac{2\pi}{\omega"(0)\ln(M^{2}/s_{0})}}\,\,e^{\omega(0)\ln(M^{2}/s_{0})\,\,-\,\,\frac{\ln^{2}(r^{2}/r^{2}_{i})}{2\omega"(0)\,\ln(M^{2}/s_{0})}}$
where the eigenvalues of the BFKL equation $\omega(\nu)$ can be found in Ref.
[5].
The factor $1-\exp\left(-\Omega^{BFKL}\right)$ describes the inelastic cross
section which in terms of the Pomeron diagrams of Fig. 2, reflects the
possibility that in the fourth of Fig. 2-b, we can have two cut upper
Pomerons.
We would like to stress again that Eq. (4.1) describes the dependence of the
diffractive production cross section, in the region of large mass, while Eq.
(4.33) is written for the low mass diffractive cross section. Therefore, to
calculate the cross section for diffractive production, we need to calculate
$\sigma_{diff}\,\,\,=\,\,\sigma^{low\;M}_{diff}\left({Eq.~{}(\ref{SDXSSM})}\right)\,\,+\,\,\sigma^{high\;M}_{diff}({Eq.~{}(\ref{SDLM})})$
(4.39)
### 4.2 Hadronic wave functions
As seen from the formulae in the previous subsection, we need to know the
hadronic wave functions. We have made an assumption in writing these formulae,
that the correct degrees of freedom at long distances are the colorless
dipoles at least at high energy[23, 24]. This is one of the strongest
assumptions in our approach. It is enough to recall that for a long time the
constituent quarks have been considered as a good candidate for the correct
degrees of freedom in the entire range of energy. The only support for such an
approach, can be seeen in the success of the Heildelberg group (see Refs. [23]
and references therein), in the description of soft interactions using this
ansatz.
In the case of mesons, we have only one colorless dipole and we take the
transverse wave function in the form of a simple Gaussian, namely
$|\Psi_{meson}\left(r\right)|^{2}\,\,\,=\,\,\frac{1}{\pi
S^{2}_{M}}\,e^{-\frac{r^{2}}{S^{2}_{M}}}$ (4.40)
where $S_{M}$ is a parameter that can be found from the electromagnetic radii,
namely, $R_{\pi}=0.66\pm 0.01fm$ and $R_{K}=0.58\pm 0.04fm$ [40]. Using
$S_{M}=\sqrt{\frac{8}{3}}R_{M}$ we obtain $S_{\pi}=1.08fm$ and $S_{K}=0.95fm$.
Figure 5: Electromagnetic (Fig. 5-a) and two gluon (Fig. 5-b) form factors.
for a meson. Fig. 5-c shows the interaction of a virtual photon with a parton.
However, electromagnetic form factors gives us the space distribution of the
electric charge inside of the hadron, while in our case the distribution of
the density of partons (dipoles) is probed by the two gluon interaction (see
Fig. 5-b). This distribution could be different from the charged one. We
prefer to find the value of $S_{M}$ from the experimental data and, therefore,
$S_{M}$ as well as $S_{p}$ (see below), will be the only fitting parameters in
our description of the soft experimental data.
For the baryon, the situation is more complicated: we can have two or even
three dipoles. Follow Ref. [23], we choose the proton wave function in the
simple form for the two dipole model
$|\Psi_{proton}(r_{1},r_{2})|^{2}\,\,=\,\,\frac{1}{(\pi\,S_{p})^{2}}\,\,e^{-\frac{r^{2}_{1}+r^{2}_{2}}{S^{2}_{p}}}$
(4.41)
where two dipoles are defined as $\vec{r}_{1}=\vec{R}_{1}-\vec{R}_{2}$ and
$\vec{r}_{2}=\vec{R}_{3}-\frac{1}{2}\left(\vec{R}_{1}+\vec{R}_{2}\right)$
where $\vec{R}_{i}$ is the position of the constituent quark $i$. In Eq.
(4.41) $S_{P}=\sqrt{\frac{3}{2}}\,R_{p}\,=\,1.05fm$ for $R_{p}=0.862\pm
0.012fm$[40].
However, for large $N_{c}$ it is proven that the baryon consists of $N_{c}$
dipoles [42]. Therefore, we consider the alternative assumption for
$\Psi_{proton}(r_{1},r_{2},r_{3})$, namely
$|\Psi_{proton}(r_{1},r_{2},r_{3})|^{2}\,\,=\,\,\frac{1}{(\pi\,S_{p})^{3}}\,\,e^{-\frac{r^{2}_{1}+r^{2}_{2}\,+\,r^{2}_{3}}{S^{2}_{p}}}$
(4.42)
where dipoles have the size
$\vec{R}_{i}-\frac{1}{3}(\vec{R}_{1}+\vec{R}_{2}+\vec{R}_{3})$. In this case,
from the electromagnetic radius of the proton, follows the value of
$S_{p}=R_{p}=0.862\pm 0.012fm$.
In Eq. (4.41), we take the size of two dipoles to be equal. We did this for
simplicity, since even in the constituent quark model (CQM),
$<|r^{2}_{2}|>=4/3<|r^{2}_{1}|>$. Our hope is that the hadron interaction will
be determined by the behavior of the dipole amplitude in the saturation
domain, where the sensitivity to the size of dipoles is expected to be weak.
On the other hand, the experimental ratio
$\sigma_{tot}(\pi-p)/\sigma_{tot}(p-p)\,\approx\,2/3$ in the CQM stems from
quark counting. In our approach, the dipole counting leads to the value of
this ration $1/2$ (for the two dipole model for a proton) or even 1/3 (in the
three dipole model) which contradicts the experimental data. The only way to
obtain a reasonable description, is to hope that the perturbative QCD
dependence of the dipole cross section on the size of the dipoles
$\sigma\propto r^{2}$ will remain in the entire accessible range of energies.
Fortunately, Ref. [24] demonstrates that this is the case, and this result
encourages us to search for the description of the soft processes using the
dipole hypothesis for the hadronic wave functions.
### 4.3 Energy variable in the model
In section 2.2 and 2.3 we have discussed our approach introducing the typical
energy variable for deep inelastic scattering: $x_{Bjorken}\equiv x=Q^{2}/s$
where $Q^{2}$ is the photon virtuality and $s$ is the energy. However, with
this variable we cannot discuss the soft processes which have $Q^{2}=0$. We
reconsider the derivation of the Bjorken variable to introduce a new energy
variable which will have a limit $x_{soft}\to x$ at $Q^{2}\gg\mu^{2}$ where
$\mu$ is the scale for the soft processes. From Fig. 5-c we have a relation
$\displaystyle(q+k)^{2}$
$\displaystyle=-Q^{2}+x_{soft}s\,\,+\,\,k^{2}\,\,=\,\,0;$ $\displaystyle
x_{soft}$ $\displaystyle=\left(Q^{2}+k^{2}_{\perp}\right)/s$ (4.43)
where $q$ is the momentum of virtual photon ($q^{2}-Q^{2}$).
In Eq. (4.43) we used the fact that at high energy, $k^{2}=-k^{2}_{\perp}$.
Since in the saturation domain $k^{2}_{\perp}\,\,=\,\,Q^{2}_{s}(x)$, we obtain
the final expression for our energy variable
$\displaystyle x_{soft}$
$\displaystyle\,\,=\,\,\frac{Q^{2}+Q^{2}_{s}\left(x_{soft}\right)}{s}\,\,\xrightarrow{Q^{2}\rightarrow
0}\,\,\frac{Q^{2}_{s}\left(x_{soft}\right)}{s};$ $\displaystyle x_{soft}$
$\displaystyle\,\,=\,\,\frac{Q^{2}+Q^{2}_{s}\left(x_{soft}\right)}{s}\,\,\xrightarrow{Q^{2}\gg
Q^{2}_{s}}\,\,x_{Bjorken};$ (4.44)
We believe that one of the main advantages of the saturation approach is the
natural choice of the energy variable given by Eq. (4.44), which depends on
the new scale: the saturation momentum.
The value of the saturation momentum in our model, we find by resolving the
following equation:
$\Omega\left(x_{soft},r^{2}_{sat}\,b=0\right)\,\,=\,\,1\,\,\,\mbox{with}\,\,r^{2}_{sat}=4/Q^{2}_{S}\left(x_{soft}\right)$
(4.45)
Partons (dipoles) with size $r_{sat}$ are populated densely in the hadron
disc, and $Q_{s}$ gives the new scale which shows that at $Q^{2}<Q^{2}_{s}$
the hard process reaches the saturation domain.
### 4.4 Final formulae for proton (antiproton) - proton scattering
Using the form of the proton wave function given by Eq. (4.41), we can rewrite
Eq. (4.29) - Eq. (4.32) in a more accurate form, taking into account the
simultaneous interaction of two dipoles (see Eq. (6)).
Figure 6: Total cross section for proton( antiproton) -proton collision
taking into account the interaction of two dipoles.
For the total cross section, the general formula has the form which follows
directly from Fig. 6, namely
$\displaystyle\sigma_{tot}\,\,=\,\,2\,\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2}\right)|^{2}$
(4.46)
$\displaystyle\,\,\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$
where $\vec{b}$ is the impact parameter or the distance between
$\frac{1}{3}(\vec{R}_{1}+\vec{R}_{2}+\vec{R}_{3})$ and the position of the
target nucleon. One can see that we need to replace
$\displaystyle\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\,\,\longrightarrow\,\,\,|\Psi_{hadron}(r_{1},r_{2})|^{2}$
(4.47)
$\displaystyle\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$
in all of the equations of section 3.1.
Using the explicit form of the proton wave function of Eq. (4.41), we can
easily rewrite the integrals in Eq. (4.29) - Eq. (4.32), in the form
$\displaystyle\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2}\right)|^{2}$
$\displaystyle\left(N(x,r_{1};\vec{b}-\frac{1}{3}\vec{r}_{2})\,\,+\,\,N(x,r_{2};\vec{b}-\frac{1}{6}\vec{r}_{2})\,\,-\,\,N(x,r_{1};\vec{b}+\frac{1}{3}\vec{r}_{2})\,N(x,r_{2};\vec{b}+\frac{1}{6}\vec{r}_{2})\right)$
$\displaystyle=$ $\displaystyle\int
dr^{2}_{1}\,db^{2}_{1}\,db^{2}_{2}\,\frac{1}{S_{p}^{2}}\,\exp\left(-\frac{r^{2}_{1}+4\,b^{2}_{1}+4\,b^{2}_{2}}{S^{2}_{p}}\right)\,I_{0}\left(\frac{4}{S^{2}_{p}}b_{1}\,b_{2}\right)\,\,$
$\displaystyle\times$
$\displaystyle\left(N(x,r_{1};b_{1})\,\,\,+\,\,\,N(x,r_{1};b_{2})\,\,\,-\,\,N(x,r_{1};b_{1})\,\,N(x,r_{1};b_{2})\right)$
In Eq. (4.4), we used the formula 8.431(3) of Ref. [61].
Since the typical impact parameter increases with energy, (see the next
section) and the $r^{2}/3$ is small (about $1/3\,fm$ ), we can safely neglect
the shift in the definition of the impact parameters in different amplitudes,
replacing
$\displaystyle\sum^{n}_{i=1}\,\,|\Psi_{hadron}(\\{r_{i}\\})|^{2}\,\,N(x,r_{i};b)\,\,\longrightarrow\,\,\,$
(4.49)
$\displaystyle|\Psi_{hadron}(r_{1},r_{2})|^{2}\,\,\left(N(x,r_{1};\vec{b})\,\,+\,\,N(x,r_{2};\vec{b})\,\,-\,\,N(x,r_{1};\vec{b})\,N(x,r_{2};\vec{b})\right)$
$\displaystyle|\Psi_{hadron}(r_{1},r_{2})|^{2}\,\,\left(1\,\,\,-\,\,\,\exp\left[-\frac{1}{2}\left(\Omega\left(x,r_{1};\vec{b}\right)+\Omega\left(x,r_{2};\vec{b}\right)\right)\right]\right)$
(4.50)
This simplified formula works quite well in all the observables of section
3.1, except for $d\sigma_{el}/dt$ at large values of $t$. The formula of Eq.
(4.50) has a simple generalization to the case of the three dipole proton
model . In the three dipole model, for a proton the cross section has the
form:
$\displaystyle\sigma_{tot}\,\,=$ (4.51) $\displaystyle
2\,\int\,d^{2}b\,d^{2}r_{1}\,d^{2}r_{2}\,\,|\Psi_{proton}\left(r_{1},r_{2},r_{3}\right)|^{2}\,\,\left(1\,\,\,-\,\,\,\exp\left[-\frac{1}{2}\left(\Omega\left(x,r_{1};\vec{b}\right)+\Omega\left(x,r_{2};\vec{b}\right)+\Omega\left(x,r_{3};\vec{b}\right)\right)\right]\right)$
It should be stressed that the second term in Eq. (4.46) plays a very
important role at high energies, since it provides the observation that the
cross section approaches a black disc limit. Indeed, only with this non-linear
term in the region where the dipole amplitude tends to unity, ($N\to 1$) the
factor in parenthesis ($2N-N^{2}$) also approaches 1 leading in the black disc
regime.
### 4.5 Description of the soft cross sections
In the previous sections we have built our model and fitted all needed
parameters using the DIS experimental data. In this section, we compare the
model with the experimental data without any additional fitting parameters.
#### 4.5.1 Total cross sections
Using Eq. (4.29) , Eq. (4.40) and Eq. (4.41) we calculate the cross sections
of pion-proton, kaon -proton, proton-proton and antiproton - proton
scattering. The results are presented in Fig. 7 and Fig. 8. In addition to Eq.
(4.29), we add a contribution of the secondary Regge trajectories using the
Donnachie and Landshoff parametrization [45]. The agreement with the data is
good, and shows that the saturation can replace the soft Pomeron, which has
been used for fitting the soft experimental data. However, as has been
mentioned, we still have one fitting parameter, namely, $S_{M}$ and $S_{p}$ in
the hadron wave function. The values of $S_{\pi}$,$S_{K}$ and $S_{p}$ is given
in Table 2. It is interesting to notice that the relation between the fitted
values of $S$, is the same since it stems from the electromagnetic radii.
Model | $S_{\pi}$ (fm) | $S_{K}$ (fm) | $S_{p}$(fm)
---|---|---|---
A(2D) | 0.515 | 0.552 | 0.561
A(3D) | 0.515 | 0.552 | 0.458
B(2D) | 0.556 | 0.597 | 0.615
B(3D) | 0.556 | 0.597 | 0.519
Table 2: _Values of parameters in hadronic wave function that give description
of the soft data_.
One can see that we are able to describe both the value and the energy
behavior of the total cross section for meson-proton and baryon - proton
interaction in our two models. However,the model B leads to a better
description demonstrating, in our opinion, that the main idea to replace the
phenomenological soft Pomeron by the behavior of the ’hard’ amplitude in the
saturation region, is fruitful . As has been discussed in the previous
subsection, the most surprising fact is that the model describes both the
proton-proton and meson-proton cross sections.
|
---|---
Fig. 7 \- a | Fig. 7 -b
Figure 7: Total cross section for $\pi-p$ (Fig. 7-a) and $K-p$ (Fig. 7-b) as
a function of energy. The solid line (DIS) corresponds to the model A which
describes the DIS data with $\chi^{2}/d.o.f.\,<\,1$, while the dotted line
presents the model B (soft) with a worse $\chi^{2}/d.o.f.\,\approx 3$, but
which leads to a better description for DIS, and soft processes data together.
The number of dipoles in the proton wave function is shown in the legend.
Figure 8: Total cross section for $p\bar{p}$ scattering versus energy in our
approach. Solid lines correspond to the model A, which describes the DIS data
with $\chi^{2}/d.o.f.\,<\,1$, while the dotted lines present the model B
(soft) with worse $\chi^{2}/d.o.f.\,\approx 3$ but which leads to a better
description for DIS, and soft processes data together.
This fact can be explained only by the dependence of the dipole cross section,
on the size of the dipole which the model reproduces. Since such a sensitivity
can be only outside the saturation region, or close to the boundary between
the saturation domain, and the perturbative QCD region. Therefore, we conclude
that the soft data allow us to obtain new information on this very important
transition region between the saturation and perturbative QCD region. It
should be noticed that we also reproduce the difference between the
interaction of pions and kaons. Since our QCD dipole cross section does not
depend on the mass of the quarks, this difference also reflects the difference
in the size of the dipoles, inside pion and kaon.
Figure 9: Elastic cross section for proton-proton scattering versus energy in
our approach. Solid line shows the comparison with the model A with 3 dipoles
in the proton wave function which described DIS with $\chi^{2}/d.o.f.\,<\,1$
while in the dotted line we plot the comparison with the experimental data the
model B with $\chi^{2}/d.o.f.\,\approx\,3$ for DIS data for the two dipole
model of proton.
Figure 10: Slope for the cross section for proton-proton scattering versus
energy in our approach. Notations are the same as in Fig. 9.
#### 4.5.2 Elastic cross section: energy dependence
We concentrate our efforts on the proton(antiproton) interactions since meson-
proton collisions have been studied experimentally only in the limited energy
range (see Fig. 8). The energy dependence of the elastic cross section is
presented in Fig. 9. We use Eq. (4.30) for performing our calculations. In
Fig. 9 \- a we compare with the experimental data our model B, while in Fig. 9
\- b, is plotted our prediction for model A (three quarks). One can see that
we obtain a good agreement with the experimental data, without any fitting
parameter.
#### 4.5.3 Elastic cross section: t-dependence
Using Eq. (4.32) we calculate the slope for the elastic cross section (see
Fig. 10). One can see that in the model B, we reproduce the value and energy
behavior of the slope at high energy, but overshoot the experimental value of
this slope at sufficiently low energy. We did not include the secondary
Reggeons, since we need a piece of information on the slope of the
contribution of the secondary Reggeons. We also need to take into account the
fact that our profile function $S(b)$ corresponds to a power-like form factor
in the $t$ representation, and the value of the slope for such a function is
sensitive on the range of $t$, where the slope was found experimentally. We
calculate the slope at $t=0$, while the slope at an average value of
$|t|=|t_{0}|>0$ is less than at $t=0$. Therefore, it is better to compare with
the experimental data the $t$ dependence of the differential elastic cross
section, given by Eq. (4.31). Such a comparison, one can see in Fig. 11, where
the data at small $t$ is described by our model even at low energies
($\sqrt{s}=23.5GeV\,\,\mbox{and}\,\,62.5GeV$). Notice that at such low
energies, the slope, that we predict, exceeds the experimental one (see Fig.
10). However, in model A, the slope is too small and, actually, this is the
key difference between the two approaches. The origin for such a difference in
the framework of our approach is clear: a different energy dependence of the
dipole-proton amplitude in the transition region between the saturation
domain, and the perturbative QCD region. Indeed, Fig. 1 shows that model B has
a steeper behavior in this region in comparison with model A.
In Fig. 10 we compare our prediction with the standard phenomenological
parametrization for the slope in the soft Pomeron model:
$B_{el}=B_{0}+2\alpha^{\prime}\ln(s/s_{0})$ with
$\alpha^{\prime}=0.25\,GeV^{-2}$ [45] and $B_{0}=9\,GeV^{-2}$. One can see
that our prediction in model B is in perfect agreement with this model at high
energies, while we predict a large slope at low energies. However, model A
leads to a smaller slope than is seen in experiment. It should be noticed, the
comparison with the experimental data (see Fig. 11) shows that model A fits
the experimental data.
Figure 11: $t$-dependence of the elastic cross section for different energies
at $t\,\leq\,0.6GeV^{2}$. The solid line corresponds to Model A predictions,
while the dotted line describes Model B behavior of the differential cross
section. Both models reproduce small $t$ behavior of the experimental data in
spite of the poor description of the elastic slope in model A.
In Fig. 11 we plot the $t$-dependence for the different energy. We would like
to draw your attention to two interesting features of our model B: it
describes the experimental data quite well at small values of ${t}\leq
0.6\,GeV^{2}$; and it predicts the minimum at the LHC energy at rather small
values of $t$ ($|t|\approx 0.45\,GeV^{2}$). The appearance of this minimum
shows that our model reproduces the effective shrinkage of the diffractive
peak which results in moving the typical diffractive minima in the region of
smaller $t$, in comparison with lower energies. Since we did not include the
real part of our scattering amplitude, we see a deep minimum. However, the
real part of the amplitude will make our minima more shallow. Model _A_ leads
to a smaller value of $B_{el}$, which actually leads to a dip at large values
of $t$, as compared to model _B_. For our rather crude models, such a
description looks successful and , we hope, will encourage others to look
seriously into attempts involving the description of the experimental, data
based on the saturation regime of QCD.
#### 4.5.4 Diffraction production: energy behavior of the cross section
Figure 12: Total cross section for meson-proton and proton(antiproton)-proton
scattering scattering at high energies (extrapolation). Notations are the same
as in Fig. 9.
In Fig. 13 the total cross section (integrated over the masses of the produced
particles) of diffractive production is plotted. We use Eq. (4.34) to
calculate this cross section. However, we multiply this formula by 2 since the
two protons: projectile and target, can dissociate diffractively. One can see
from this picture that we reproduce values and energy dependence in agreement
with the experimental data. The qualitative behavior of the experimental data
on $\sigma_{diff}$ versus energy is characterized by the slow dependence on
energy in the region of high energies. This fact is perfectly reproduced by
our model.
Figure 13: Cross section for single diffraction production for
proton(antiproton)-proton scattering versus energy. The upper curves are the
total cross section for diffractive production while the low curves show the
energy behavior of the low mass contribution to the diffractive cross section
from Eq. (4.33). Notations are the same as in Fig. 9.
However, one can see from Fig. 13 that our approach cannot reproduce
$\sigma_{SD}$ in the region of low energies (small values of produced mass).
There are at least two reasons for such a failure: first, we do not take into
account the exchange of the secondary Reggeons whose contribution is
essential, as we see on the example of the energy behavior of the total cross
section, and, second, the contribution of the one extra gluon emission has
been calculated in leading log(1/x) order, which cannot describe the low mass
diffraction. The experimental data on single diffraction (see Refs. [53, 52])
can be described assuming, in addition to the triple Pomeron vertex, that we
modeled by extra gluon emission, a significant contribution from the Reggeon-
Pomeron-Reggeon vertex in Regge phenomenology, which we cannot include in our
approach.
### 4.6 Predictions for the LHC range of energies
As we have seen, the model is able to describe the available experimental data
quite satisfactorily (and Model B even quite well) both for soft and hard
interactions. Therefore, we can rely on the model B for the predictions in the
LHC energy range. From Fig. 8-Fig. 13 we see that at the LHC energy we have
$\sigma_{tot}=101.3\,mb$, $\sigma_{el}=28.84\,mb$ , $B_{el}=18.4\,GeV^{-2}$
and $\sigma_{diff}=10.5\,mb$ .
Model | $\sigma_{tot}$ | $\sigma_{el}$ | $\sigma_{diff}$ | $B_{el}$ | $A_{el}(b=0)$ | $A_{el}(b=0)$
---|---|---|---|---|---|---
| mb | mb | mb | $GeV^{-2}$ | (Tevatron) | (LHC)
Our A(3D) | 83.0 | 23.54 | 10 | 16.67 | 0.94 | 0.98
Our B(2D) | 101.3 | 28.84 | 10.5 | 18.4 | 0.94 | 0.98
GLM1[25] | 110.5 | 25.3 | 11.6 | 20.5 | 0.6 | 0.7
GLM2[56] | 91.7 | 20.9 | 11.8 | 17.3 | 0.94 | 0.95
RMK[55] | 88.0 (86.3) | 20.1 (18.1) | 13.3 (16.1) | 19 | 0.89 | 0.92
Table 3: Cross sections and elastic slope at the LHC energy in different
models. For the RMK model, we put two parameterizations and the value for
$B_{el}$ directly from the curve in Fig.18 of Ref.[55]
Table 3 shows that our prediction for LHC energy is close to ones that are
given by the models that fitted the experimental data. The RMK and GLM2 models
are based on soft Pomeron phenomenology while the GLM1 model is close to our
approach: in this model the existence of soft Pomeron is not assumed and the
parametrization was chosen for from the same ideas as our approach here.
However, in the GLM1 model all parameters are found from fitting soft data. In
spite the fact that the values of cross section are close the different models
are different in more detailed characteristics. For example, one can see from
the Table 1 they predict different values for $A_{el}(b=0)$ . From unitarity
$A_{el}(b=0)\,\leq\,1$. Our model predicts that at the LHC energy our proton-
proton collision is close to the black disc regime. The same we can see in
GLM2 model, but in the RMK model we are not so close to this regime at the
LHC, while in the GLM1 models the proton-proton interaction is far away from
the black disc limit.
As you can see from Eq. (8)-c we have some information on the total cross
section from the cosmic ray experiment. However, to extract the value of the
total cross section from such an experiment, we need to know the meson-proton
cross section as well. In Fig. 12, we plotted our predictions which we hope
will be useful for discussing the cosmic ray experiment*** We thank S.
Nussinov for drawing our attention to the necessary knowledge of the meson-
proton total cross sections at high energy..
## 5 Survival probability for diffractive Higgs production
Figure 14: Diffractive Higgs production.
The diffractive Higgs production, is the reaction which has the best
experimental signature for the discovery of the Higgs boson at the LHC. At
fist sight this process occurs at short distances of the order of $1/M_{H}$,
where $M_{H}$ is the mass of Higgs boson ($M_{h}\geq 100GeV$) and can be
calculated in perturbative QCD (see Fig. 14-a and Ref.[46] for calculations).
However, as was noticed a long ago [47, 48] that it is not enough to calculate
the diagram of Fig. 14-a which describes the Higgs boson production from one
parton shower. We have to multiply this cross section by a probability that
two parton showers (or more) will not interact with the target since they
could produced hadrons that will fill up the large rapidity gaps between Higgs
and protons in the final state (see Fig. 14-b). This probability we call the
survival probability. In the case of our model with the eikonal type formula
for the scattering amplitude, the expression for the survival probability is
very simple (see Fig. 14-c), namely,[47, 48]
$\langle|S^{2}|\rangle\,\,\,=\,\,\frac{\int d^{2}b\,\,\left(\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}\,e^{-\frac{1}{2}\Omega(s,r_{1},b)}\right)^{4}\,\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)}{\int
d^{2}b\,\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)}$ (5.52)
The appearance of the factor $\exp\left(-\frac{1}{2}\Omega(s,r_{1},b)\right)$
in Eq. (5.52) is clear from Fig. 14-c) but the power of 4 requires discussion.
Actually this power reflects the fact that we have to find the probability
that neither dipole 1 nor dipole 2 could scatter inelastically. The
probability that one dipole does not scatter inelastically is equal to
$\left(\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}\,e^{-\frac{1}{2}\Omega(s,r_{1},b)}\right)^{2}\,\,\equiv\,\,\left(\langle|e^{-\frac{1}{2}\Omega(s,r_{1},b)}|\rangle\right)^{2}$
Therefore for the probability that two dipoles cannot scatter inelastically we
obtain $\left(\langle|e^{-\frac{1}{2}\Omega(s,r_{1},b)}|\rangle\right)^{4}$.
Figure 15: The value of the survival probability for diffractive Higgs
production versus energy. Solid lines show the survival probability in Model A
that describes the DIS data with a very good $\chi^{2}/d.o.f<1$, while dashed
lines correspond to $<S^{2}>$ in Model B which gives a good description of all
soft data.
We can see from Eq. (5.52) that for the calculation of the survival
probability we need to know the $b$ \- dependence of the hard cross section.
This dependence has been discussed in detail in Ref.[49], where it was
extracted from the process of diffraction production of J$/\Psi$ with the
proton (elastic) and the state with mass larger than the proton (inelastic).
These two processes have two different slopes in the $t$ behavior:
$B_{el}=4\,GeV^{-2}$ and $B_{in}=1.86\,GeV^{-2}$[50] . The dependence of the
vertex for photon to J/$\Psi$ transition was extracted from these reactions in
Ref. [51] leading to the values of slopes for the vertices of the transition
of proton to proton (elastic), and proton to non-proton final state
(inelastic): $B_{el}=3.6\,GeV^{-2}$ and $B_{in}=1.46\,GeV^{-2}$. In our model,
the projectile proton first goes to the state of two free dipoles (see Fig.
14-c) , these two dipoles can produce two and more parton showers; and finally
two free dipoles in hard processes create a proton. Therefore, at least the
upper vertex is the same as in the inelastic diffractive production of
J$/\Psi$ in DIS. Considering the lower vertex being elastic, we can write
$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)$ in the form
$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)\,\,\,=\,\,\,\frac{\sigma_{0}}{\pi
R_{H}}\,e^{-\frac{b^{2}}{R^{2}_{H}}}$ (5.53)
with $R^{2}_{H}=B_{in}+B_{el}=5.06\,GeV^{-2}$. However, it is our model
disadvantage that we describe differently projectile and target protons. In
any case the eikonal type model that we use, does not mean that only the
elastic rescattering contributes to the shadowing corrections. Treating both
protons on the same ground we take $R^{2}_{H}=2B_{in}=2.92\,GeV^{-2}$.
Since our model naturally includes both soft and hard interactions we can hope
that the hard process of diffractive Higgs production can be described in the
same way as we did in the model using the same amplitude $\Omega$. However, we
need to take into account that in this case
$\sigma_{H}(b;{Fig.~{}\ref{spH}}-a)\,\,\,=\,\,\sigma_{0}\,S^{2}(b)$ (5.54)
where $S(b)$ is given by Eq. (3.20).
Our results on the survival probability is shown in Fig. 15, and in Table 2.
Table 2 demonstrates that our value for the survival probability turns out to
be much smaller than the values in the phenomenological models, that fitted
the experimental data using the soft Pomeron approach (see Table 2, the review
of [54] and Ref. [55]). We think that the difference stems not from the
details of the models, but from the key ingredient of our model: the fact that
we took into account the interaction at short distances. Therefore, we confirm
that the short distances give a substantial contribution to the value of
survival probability, as has been noticed in Refs. [57, 58]. The Table 2 also
shows that the value of the survival probability crucially depends on the
model for the $b$-dependence of the hard cross section. Our model with Eq.
(5.54) for the hard cross section is similar to the one that was used in the
GLM model [25]. It is interesting that our model confirms the general tendency
to obtain a smaller value for the survival probability advocated in Ref. [25].
Model | $\sigma_{H}(b)$ | $\langle|S^{2}|\rangle$
---|---|---
Our Model A(3D) | Eq. (5.53) $R^{2}_{H}=5.06\,GeV^{-2}$ | 0.24%
| Eq. (5.53) $R^{2}_{H}=2.92\,GeV^{-2}$ | 0.02%
| Eq. (5.54) | 0.89%
Our Model B(2D) | Eq. (5.53) $R^{2}_{H}=5.06\,GeV^{-2}$ | 0.24%
| Eq. (5.53) $R^{2}_{H}=2.92\,GeV^{-2}$ | 0.096%
| Eq. (5.54) | 0.57%
GLM1[25] | Two channel model for $\sigma_{H}$ | 2% (0.7%)
GLM2[56] | Two channel model for $\sigma_{H}$ | 0.21%
RMK[55] | Eq. (5.53) $R^{2}_{H}=11\,GeV^{-2}$ | 3.2% (2.3%)
| Eq. (5.53) $R^{2}_{H}\,=\,8\,GeV^{-2}$ | 1.7% (1.2%)
Table 4: The survival probability for diffractive Higgs production at the LHC
energy, in different models. In the GLM model, for $\sigma_{H}(b)$ is used in
the two channel model, with the same $B_{in}$ and $B_{el}$ for the diffractive
production of J/$\Psi$ in DIS.
## 6 Lessons from the model
Long distance physics is very complicated, non-perturbative phenomenon, and we
certainly do not pretend that we are able to describe it in its full richness.
However, we demonstrate in this paper that the gap between this physics and
the short distance physics which is under full control of perturbative QCD, is
not so huge that it would be a hopeless task to build a bridge. Comparing this
with the experimental data, we showed that the soft data depends on the
transition region between saturation domain and perturbative QCD region and,
therefore, can give valuable information on this transition, checking our
theoretical approaches to it. The widely used, phenomenological soft Pomeron
do not appear in our approach, and we hope that the reader will ask the
question: do we need a soft Pomeron, having in mind our negative answer.
Our description of the experimental data, is not worse (in the case of Model
B) than the one in the models which fitted the data on the basis of the soft
Pomeron phenomenology (see Refs. [60, 59, 55, 25]). Recalling that we fitted
all parameters of our model from DIS processes, we interpret this success in
the way that the high energy soft scattering processes are determined by QCD,
at short distances of the order of $1/Q_{s}$, where $Q_{s}$ is the saturation
momentum. Model A describes the data worse than Model B, but we consider this
description quite satisfactory remembering the crude character of this
approach. It should be stressed, once more that the difference between our
model A and model B, is in our attitude to the simple formula of Eq. (3.22):
in model A we trust Eq. (3.22) in the entire kinematic region of accessible
distances, while in model B we view this formula as a kind of qualitative
description, that includes the main features of the saturation regime.
Therefore, we were searching for the parameters of model B, in the way that
describes all the data both on DIS, and on soft interactions, in the best
possible way.
In our model, we used several assumptions which have a different theoretical
status. The first assumption is the exponential form of the dipole scattering
amplitude (see Eq. (3.16) and Eq. (3.22). We have discussed the theoretical
arguments for such an assumption, namely, this form has been proven for the
transition region between perturbative QCD, and the saturation domain (see
also Ref. [12]). It should be stressed that the soft data, are sensitive to
the transition region as we have discussed.
The second assumption, is the expression of Eq. (3.21) for
$\Omega\left(x,r,b\right)$. This assumption is a compromise between what we
should do, and what we can do. We can check it by describing the DIS
experimental data, at large values of the photon virtualities, where the
difference between our formula for $\Omega$, and the DGLAP expression for it
should be large.
The third assumption is the impact parameter dependence of
$\Omega\left(x,r,b\right)$ (see Eq. (3.19) and Eq. (3.20)). This is a pure
phenomenological ansatz, since the current stage of our theory does not allow
us to find $b$ dependence [28]. Actually, only soft data allows us to check
this dependence. Indeed, our $\Omega$ describes the $t$-dependence of the
elastic cross section while , for example, the Gaussian $b$ dependence results
in the appearance of the structure of maxima and minima, at small values of
$t$, which contradicts the experimental behavior of $d\sigma_{el}/dt$.
In Fig. 16-a we plot the dipole amplitude averaged with the parton wave
function
$\langle|N|\rangle|\,\,\,\equiv\,\,\,\int
d^{2}r_{1}d^{2}r_{2}|\Psi_{proton}(r_{1},r_{2})|^{2}N(x,r_{1},\vec{b}=0)$
(6.55)
One can see, that this amplitude increases and it approaches 1. However, it
happens at ultra high energies and at the Tevatron energy, for example, this
amplitude is only $0.8$ at $b=0$. Such an average scattering amplitude, leads
to the elastic amplitude of proton-proton scattering $2N-N^{2}=0.96$. The
averaged slowly increases with energy, and reaches 1 at energy. At the LHC we
expect $\langle|N|\rangle=0.915$ and the average elastic amplitude is close to
unity $\langle|2N-N^{2}|\rangle=0.99$. Such a behavior shows, that the values
as well as the dependence on energy crucially depends on the behavior of our
dipole amplitude, in the vicinity of the saturation scale. To illustrate the
strength of the saturation effect, we plot the average $\Omega$ at $b=$ as a
function of energy (see Fig. 16-b), which is a considerably overshoots the
average scattering amplitude. It should be stressed, that in spite of the fact
that the elastic amplitude is very close to unity at the LHC energy, one can
see from Fig. 16-a that $\langle|N|\rangle$ is only 0.9. and the asymptotic
behavior with $\langle|N|\rangle$ close to 1, starts from
$s=10^{12}\,GeV^{2}$.
|
---|---
Fig. 16-a | Fig. 16-b
Figure 16: Average dipole amplitude ($\langle{N}\rangle$ see Eq. (6.55)),
average proton- proton amplitude (both in Fig. 16-a) and average opacity
$\Omega$(see Fig. 16-b) at $b=0$ as a function of energy in our model.
The principle difference between our models and the models of soft
interactions, is the fact that we predict all observables fitting all
parameters from the DIS data. The GLM1 model[25] is close to our approach
ideologically, because it does not assume the existence of the soft Pomeron.
However, at first sight this model has two major shortcomings: minima at small
$t$ which have not been seen experimentally; and the slow fall down of the
cross section of diffractive production, as a function of the mass of produced
system of hadrons. Our model shows, that the $t$ dependence can be easily
heeled by assuming the exponential form for the profile function, instead of
the Gaussian one that has been used in the GLM1 model. As far as large mass
diffraction that has been neglected in the GLM1 model[25], we show that this
diffraction is essential. In this respect, we are close to the RMK model[55]
and to the GLM2 model [56] which are based on the soft Pomeron exchange. Our
model can be considered as an argument that the multi Pomeron exchanges, and
the Pomeron interactions could be essential for high mass diffraction.
In general we demonstrated in this paper that the distances, essential in so
called soft interactions, is not so long, but rather about $1/Q_{s}\ll R_{h}$,
where $R_{h}$ is the hadron radius. This conclusion stems both from the
success of our model in the description of the data, using the parameters
fitted in DIS, and from the use of the energy variable $x_{soft}$, which is
determined by the saturation scale.
We hope that our model will generate deeper theoretical ideas on the matching
between soft and hard interactions, based on high parton density QCD.
## Acknowledgements
We are grateful to Jochen Bartels, Errol Gotsman, Lev Lipatov, Uri Maor and
Misha Ryskin for fruitful discussions on the subject. This research was
supported in part by the Israel Science Foundation, founded by the Israeli
Academy of Science and Humanities, by BSF grant $\\#$ 20004019 and by a grant
from Israel Ministry of Science, Culture and Sport and the Foundation for
Basic Research of the Russian Federation.
## References
* [1] L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100, 1 (1983).
* [2] A. H. Mueller and J. Qiu, Nucl. Phys.,427 B 268 (1986) .
* [3] L. McLerran and R. Venugopalan, Phys. Rev. D 49,2233, 3352 (1994); D 50,2225 (1994); D 53,458 (1996); D 59,09400 (1999).
* [4] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys 15 (1972) 438;
G. Altarelli and G. Parisi,Nucl. Phys. B 126 (1977) 298;
Yu. l. Dokshitser, Sov. Phys. JETP 46 (1977) 641.
* [5] E. A. Kuraev, L. N. Lipatov, and F. S. Fadin, Sov. Phys. JETP 45, 199 (1977); Ya. Ya. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 22 (1978).
* [6] I. Balitsky, [arXiv:hep-ph/9509348]; Phys. Rev. D60, 014020 (1999) [arXiv:hep-ph/9812311] Y. V. Kovchegov, Phys. Rev. D60, 034008 (1999), [arXiv:hep-ph/9901281].
* [7] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Phys. Rev. D59, 014014 (1999), [arXiv:hep-ph/9706377]; Nucl. Phys. B504, 415 (1997), [arXiv:hep-ph/9701284]; J. Jalilian-Marian, A. Kovner and H. Weigert, Phys. Rev. D59, 014015 (1999), [arXiv:hep-ph/9709432]; A. Kovner, J. G. Milhano and H. Weigert, Phys. Rev. D62, 114005 (2000), [arXiv:hep-ph/0004014] ; E. Iancu, A. Leonidov and L. D. McLerran, Phys. Lett. B510, 133 (2001); [arXiv:hep-ph/0102009]; Nucl. Phys. A692, 583 (2001), [arXiv:hep-ph/0011241]; E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nucl. Phys. A703, 489 (2002), [arXiv:hep-ph/0109115]; H. Weigert, Nucl. Phys. A703, 823 (2002), [arXiv:hep-ph/0004044].
* [8] A. Kovner and M. Lublinsky, Phys. Rev. D 71, 085004 (2005) [arXiv:hep-ph/0501198].
* [9] Y. Hatta, E. Iancu, L. McLerran, A. Stasto and D. N. Triantafyllopoulos, Nucl. Phys. A764, 423 (2006) [arXiv:hep-ph/0504182].
* [10] E. Iancu, A. H. Mueller and S. Munier, Phys. Lett. B606 (2005) 342 [arXiv:hep-ph/0410018]; E. Brunet, B. Derrida, A. H. Mueller and S. Munier, arXiv:cond-mat/0603160; Phys. Rev. E73 (2006) 056126 [arXiv:cond-mat/0512021].
* [11] R. Enberg, K. Golec-Biernat and S. Munier, Phys. Rev. D72 (2005) 074021 [arXiv:hep-ph/0505101]. S. Munier, Phys. Rev. , D 75 (2007) 034009 [arXiv:hep-ph/0608036].
* [12] E. Levin, J. Miller and A. Prygarin, “Summing Pomeron loops in the dipole approach,” Nucl.Phys. A (in press); arXiv:0706.2944 [hep-ph].
* [13] N. Armesto and M. A. Braun, Eur. Phys. J. C20, 517 (2001) [arXiv:hep-ph/0104038]; M. Lublinsky, Eur. Phys. J. C21, 513 (2001) [arXiv:hep-ph/0106112]; E. Levin and M. Lublinsky, Nucl. Phys. A712, 95 (2002) [arXiv:hep-ph/0207374]; Nucl. Phys. A712, 95 (2002) [arXiv:hep-ph/0207374]; Eur. Phys. J. C22, 647 (2002) [arXiv:hep-ph/0108239]; M. Lublinsky, E. Gotsman, E. Levin and U. Maor, Nucl. Phys. A696, 851 (2001) [arXiv:hep-ph/0102321]; Eur. Phys. J. C27, 411 (2003) [arXiv:hep-ph/0209074]; K. Golec-Biernat, L. Motyka and A.Stasto, Phys. Rev. D65, 074037 (2002) [arXiv:hep-ph/0110325]; E. Iancu, K. Itakura and S. Munier, Phys. Lett. B590 (2004) 199 [arXiv:hep-ph/0310338]. K. Rummukainen and H. Weigert, Nucl. Phys. A739, 183 (2004) [arXiv:hep-ph/0309306]; K. Golec-Biernat and A. M. Stasto, Nucl. Phys. B668, 345 (2003) [arXiv:hep-ph/0306279]; E. Gotsman, M. Kozlov, E. Levin, U. Maor and E. Naftali, Nucl. Phys. A742, 55 (2004) [arXiv:hep-ph/0401021]; K. Kutak and A. M. Stasto, Eur. Phys. J. C41, 343 (2005) [arXiv:hep-ph/0408117]; G. Chachamis, M. Lublinsky and A. Sabio Vera, Nucl. Phys. A748, 649 (2005) [arXiv:hep-ph/0408333]; J. L. Albacete, N. Armesto, J. G. Milhano, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D71, 014003 (2005) [arXiv:hep-ph/0408216]; E. Gotsman, E. Levin, U. Maor and E. Naftali, Nucl. Phys. A750 (2005) 391 [arXiv:hep-ph/0411242].
* [14] K. J. Golec-Biernat and M. Wusthoff, Phys. Rev. D 59 (1999) 014017 [arXiv:hep-ph/9807513] ,̇ Phys. Rev. D 60 (1999) 114023 [arXiv:hep-ph/9903358] ; E. Gotsman, E. Levin, M. Lublinsky, U. Maor, E. Naftali and K. Tuchin, J. Phys. G 27 (2001) 2297 [arXiv:hep-ph/0010198] ; H. Kowalski and D. Teaney, Phys. Rev. D 68 (2003) 114005 [arXiv:hep-ph/0304189] ; J. Bartels, K. J. Golec-Biernat and H. Kowalski, Phys. Rev. D 66 (2002) 014001 [arXiv:hep-ph/0203258].
* [15] A. Kormilitzin, “Saturation model in the non-Glauber approach,” arXiv:0707.2202 [hep-ph].
* [16] C. Adloff et al. [H1 Collaboration], Eur. Phys. J. C 21, 33 (2001) [arXiv:hep-ex/0012053].
* [17] S. Chekanov et al. [ZEUS Collaboration], Eur. Phys. J. C 21, 443 (2001) [arXiv:hep-ex/0105090].
* [18] J. Breitweg et al. [ZEUS Collaboration], Phys. Lett. B 487, 53 (2000) [arXiv:hep-ex/0005018].
* [19] J. Bartels and E. Levin, Nucl. Phys. B387 (1992) 617.
* [20] J. Kwiecinski and A. M. Stasto, Acta Phys. Polon. B33 (2002) 3439; Phys. Rev. D66 (2002) 014013 [arXiv:hep-ph/0203030]; A. M. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596 arXiv:hep-ph/0007192].
* [21] E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A708 (2002) 327 [arXiv:hep-ph/0203137].
* [22] L. McLerran, “Some comments about the high energy limit of QCD,” Acta Phys. Polon. B 37 (2006) 3237 [arXiv:hep-ph/0702] and references therein; E. Ferreiro, E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A 710 (2002) 373 [arXiv:hep-ph/0206241]; E. Iancu, A. Leonidov and L. McLerran, “The colour glass condensate: An introduction,” arXiv:hep-ph/0202270; E. Levin, “Saturation 2005 (mini-review),” AIP Conf. Proc. 792 (2005) 536 [arXiv:hep-ph/0506161]; “An introduction to pomerons,” arXiv:hep-ph/9808486; E. Laenen and E. Levin, “Parton Densities At High-Energy,” Ann. Rev. Nucl. Part. Sci. 44 (1994) 199; E. M. Levin and M. G. Ryskin, “High-Energy Hadron Collisions In QCD,” Phys. Rept. 189 (1990) 267.
* [23] H. G. Dosch, E. Ferreira and A. Kramer, Phys. Rev. D 50 (1994) 1992 [arXiv:hep-ph/9405237] and references therein.
* [24] J. Bartels, E. Gotsman, E. Levin, M. Lublinsky and U. Maor, Phys. Rev. D 68 (2003) 054008 [arXiv:hep-ph/0304166]; Phys. Lett. B 556 (2003) 114 [arXiv:hep-ph/0212284].
* [25] E. Gotsman, E. Levin and U. Maor, “A Soft Interaction Model at Ultra High Energies: Amplitudes, Cross Sections and Survival Probabilities,” arXiv:0708.1506 [hep-ph].
* [26] S. Munier and R. Peschanski, “Universality and tree structure of high energy QCD,” arXiv:hep-ph/0401215; Phys. Rev. D69 (2004) 034008 [arXiv:hep-ph/0310357]; Phys. Rev. Lett. 91 (2003) 232001 [arXiv:hep-ph/0309177]; A. H. Mueller and V. N. Triantafyllopoulos, Nucl.Phys. B640, 331 (2002); D. N. Triantafyllopoulos, Nucl. Phys. B 648, 293 (2003).
* [27] J. Bartels and E. Levin, Nucl. Phys. B387 (1992) 617; A. M. Stasto, K. Golec-Biernat and J. Kwiecinski, Phys. Rev. Lett., 86, 596 (2001); E. Levin and K. Tuchin, Nucl. Phys. A693 (2001) 787, [arXiv:hep-ph/0101275] ; A691 (2001) 779,[arXiv:hep-ph/0012167]; B573 (2000) 833, [arXiv:hep-ph/9908317]; E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A708, 327 (2002).
* [28] A. Kovner and U. A. Wiedemann, Phys. Lett. B 551 (2003) 311 [arXiv:hep-ph/0207335]; Phys. Rev. D 66 (2002) 034031 [arXiv:hep-ph/0204277]; Phys. Rev. D 66 (2002) 051502 [arXiv:hep-ph/0112140].
* [29] R. C. Brower, J. Polchinski, M. J. Strassler and C. I. Tan, JHEP 0712 (2007) 005 [arXiv:hep-th/0603115].
* [30] C.J. Morninstar and M. J. Peardon, Phys. Rev. D60 (2005) 344, [arXiv:hep-lat/9901004].
* [31] A. H. Mueller, Nucl. Phys. B415, 373 (1994); ibid B437, 107 (1995).
* [32] E. Levin and M. Lublinsky, Nucl. Phys. A730, 191 (2004) [arXiv:hep-ph/0308279].
* [33] E. Levin and M. Lublinsky, Phys. Lett. B607, 131 (2005) [arXiv:hep-ph/0411121].
* [34] E. Levin and K. Tuchin, Nucl. Phys. A693 (2001) 787 [arXiv:hep-ph/0101275]; A691 (2001) 779 [arXiv:hep-ph/0012167]; B573 (2000) 833 [arXiv:hep-ph/9908317].
* [35] A. H. Mueller, A. I. Shoshi and S. M. H. Wong, Nucl. Phys. B 715, 440 (2005) [arXiv:hep-ph/0501088].
* [36] E. Levin and M. Lublinsky, Nucl. Phys. A 763, 172 (2005) [arXiv:hep-ph/0501173].
* [37] E. Iancu and D. N. Triantafyllopoulos, Nucl. Phys. A756, 419 (2005) [arXiv:hep-ph/0411405]; Phys. Lett. B610, 253 (2005) [arXiv:hep-ph/0501193].
* [38] E. Gotsman, E. Levin and U. Maor, Nucl. Phys. B 464 (1996) 251 [arXiv:hep-ph/9509286] and references therein.
* [39] K. Ellis, Z. Kunst and E. Levin: Phys. Rev. D 50 (1994) 1992.
* [40] G.G. Simon et al.,Z. Naturforschung 35 A (1980) 1; S.R. Amendola et al.,Nucl. Phys. B 277 (1985) 168; Phys.Lett. B 178 (1986) 435.
* [41] M. L. Good and W. D. Walker, Phys. Rev. 120 (1960) 1857.
* [42] E. E. Jenkins, Ann. Rev. Nucl. Part. Sci. 48 (1998) 81 [arXiv:hep-ph/9803349] and references therein.
* [43] Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 65 (2002) 074026 [arXiv:hep-ph/0111362].
* [44] C. Marquet, Phys. Rev. D 76 (2007) 094017 [arXiv:0706.2682 [hep-ph]]; H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D 74, 074016 (2006) [arXiv:hep-ph/0606272]; Y. V. Kovchegov, Phys. Rev. D 64, 114016 (2001) [Erratum-ibid. D 68, 039901 (2003)] [arXiv:hep-ph/0107256]; Y. V. Kovchegov and L. D. McLerran, Phys. Rev. D 60, 054025 (1999) [Erratum-ibid. D 62, 019901 (2000)] [arXiv:hep-ph/9903246]; Y. V. Kovchegov, Phys. Rev. D 64, 114016 (2001) [Erratum-ibid. D 68, 039901 (2003)] [arXiv:hep-ph/0107256]; J. R. Forshaw, R. Sandapen and G. Shaw, Phys. Lett. B 594, 283 (2004) [arXiv:hep-ph/0404192]; Y. V. Kovchegov and L. D. McLerran, Phys. Rev. D 60, 054025 (1999) [Erratum-ibid. D 62, 019901 (2000)] [arXiv:hep-ph/9903246]; M. Wusthoff, Phys. Rev. D 56, 4311 (1997) [arXiv:hep-ph/9702201]; K. J. Golec-Biernat and M. Wusthoff, Phys. Rev. D 59, 014017 (1999) [arXiv:hep-ph/9807513]. E. Levin and M. Wusthoff, Phys. Rev. D 50, 4306 (1994).
* [45] A. Donnachie and P.V. Landshoff, Nucl. Phys. B231, (1984) 189; Phys. Lett. B296, (1992) 227; Zeit. Phys. C61, (1994) 139.
* [46] V. A. Khoze, A. D. Martin and M. G. Ryskin, Phys. Lett. B 650 (2007) 41 [arXiv:hep-ph/0702213]; Eur. Phys. J. C 24 (2002) 459 [arXiv:hep-ph/0201301]; Eur. Phys. J. C 26 (2002) 229 [arXiv:hep-ph/0207313] Eur. Phys. J. C 14 (2000) 525 [arXiv:hep-ph/0002072];
* [47] J. D. Bjorken, Int. J. Mod. Phys. A7, (1992) 4189; Phys. Rev. D47, (1993) 101.
* [48] E. Gotsman, E.M. Levin and U. Maor, Phys. Lett. B309, (1993) 199.
* [49] E. Gotsman, H. Kowalski, E. Levin, U. Maor and A. Prygarin, Eur. Phys. J. C47, (2006) 655.
* [50] ZEUS Collaboration, Nucl. Phys. B695 (2004) 3; Eur. Phys. J. C24 (2002) 345.
* [51] H. Kowalski and D. Teaney, Phys. Rev. D 68 (2003) 114005 [arXiv:hep-ph/0304189].
* [52] K. Goulianos and J. Montanha, Phys. Rev. D59 (1999) 114017.
* [53] F. Abe et al.(CDF Collaboration), Phys. Rev. D50 (1994) 5535.
* [54] E. Gotsman, E. Levin, U. Maor, E. Naftali and A. Prygarin, ”HERA and the LHC - A workshop on the implications of HERA for LHC physics: Proceedings Part A” (2005) 221. (arXiv:hep-ph/0511060[hep-ph]).
* [55] M. G. Ryskin, A. D. Martin and V. A. Khoze, “Soft diffraction at the LHC: a partonic interpretation,” arXiv:0710.2494 [hep-ph].
* [56] E. Gotsman, E. Levin, U. Maor and J. S. Miller, “A QCD motivated model for soft interactions at high energies,” arXiv:0805.2799 [hep-ph].
* [57] J. Bartels, S. Bondarenko, K. Kutak and L. Motyka, Phys. Rev. D73 (2006) 093004.
* [58] J. S. Miller, “Survival probability in diffractive Higgs production in high density QCD,” Eur. Phys. J. (in press) arXiv:hep-ph/0610427.
* [59] E. Gotsman, E. Levin and U. Maor, Phys. Lett. B452, (1999) 387.
* [60] E. Gotsman, E. Levin and U. Maor, Phys. Rev. D49, (1994) R4321
* [61] I. Gradstein and I. Ryzhik, “ Tables of Series, Products, and Integrals”, Verlag MIR, Moskau,1981.
|
arxiv-papers
| 2008-09-23T10:46:32
|
2024-09-04T02:48:57.902692
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Kormilitzin and E. Levin (Tel Aviv Un.)",
"submitter": "Eugene Levin",
"url": "https://arxiv.org/abs/0809.3886"
}
|
0809.3973
|
# Symmetric homogeneous diophantine equations of odd degree
M. A. Reynya Ahva St. 15/14, Haifa, Israel misha_371@mail.ru
###### Abstract.
We find a parametric solution of an arbitrary symmetric homogeneous
diophantine equation of 5th degree in 6 variables using two primitive
solutions. We then generalize this approach to symmetric forms of any odd
degree by proving the following results.
(1) Every symmetric form of odd degree $n\geq 5$ in $6\cdot 2^{n-5}$ variables
has a rational parametric solution depending on $2n-8$ parameters.
(2) Let $F(x_{1},\dots,x_{N})$ be a symmetric form of odd degree $n\geq 5$ in
$N=6\cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the
equation $F(x_{i})=q$ has a rational parametric solution depending on $2n-6$
parameters.
The latter result can be viewed as a solution of a problem of Waring type for
this class of forms.
## 1\. Introduction
Given any form $F(x_{1},\dots,x_{N})$ with rational coefficients, one can look
for conditions under which the equation $F(x_{1},\dots,x_{N})=0$ has a nonzero
rational solution. It is known that if $N$ is big enough with respect to the
degree of $F$ (which is assumed odd), such a solution always exists (see
[Bi1], [Bi2]; more recent papers [W1], [W2] provide explicit estimates for
$N$). In the present paper, we consider symmetric forms
$F(x_{1},\dots,x_{N})$, i.e. forms invariant under the natural action of the
symmetric group $S_{N}$ on variables. For such a form the existence of a
nonzero rational solution is guaranteed, and we are interested in getting
parametric solutions and estimating the number of parameters. More precisely,
for any given integers $n$ and $k$ we want to find an integer $N$ such that
every symmetric form in $N$ variables of degree $2n+1$ with rational
coefficients has a parametric solution depending on $k$ parameters.
Our method, being essentially elementary, is inspired by a geometric approach
proposed by Manin [M] for studying rational points on cubic hypersurfaces:
given a few rational points on such a variety, one can use a composition law
in order to produce new rational points. We implement a similar idea in the
case of symmetric forms of odd degree: starting from two “primitive” rational
points, we propose an algorithm producing lots of rational points.
To explain our method, we begin with the case of quintics in 6 variables (see
Section 2). We prove that any such quintic has a parametric solution
$(x_{1},\dots,x_{6})$ such that $x_{1}+\dots+x_{6}=0$ (Theorem 2.4). Note that
to the best of our knowledge, such a result did not appear in the literature,
even in the simplest case of diagonal quintics. Our method is significantly
different both from the elementary approach in [R], [C] and computer
investigation of [LP], [LPS], [E]. It can be compared with geometric
constructions in [Br] which led to some new insight on parametric solutions
obtained in [SwD].
Next, in Section 3, we extend our method to symmetric forms of arbitrary odd
degree. Our main result (Theorem 3.1) states that every symmetric form of odd
degree $n\geq 5$ in $6\cdot 2^{n-5}$ variables has a rational parametric
solution depending on $2n-8$ parameters. As an application, we obtain a
solution of the following version of Waring’s problem (Theorem 3.2):
Let $F(x_{1},\dots,x_{N})$ be a symmetric form of odd degree $n\geq 5$ in
$N=6\cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the
equation $F(x_{i})=q$ has a rational parametric solution depending on $2n-6$
parameters.
## 2\. Symmetric quintics in 6 variables
We begin with a simple general observation:
###### Lemma 2.1.
Every symmetric form of degree $2n+1$ in $2N\geq 6$ variables has a zero of
the form
$(a_{1},-a_{1},\dots,a_{N-1},-a_{N-1},1,-1)$ (1)
with $a_{i}\neq a_{j}\neq 1$.
###### Proof.
By a well-known algebraic theorem, every symmetric polynomial $P\in
k[x_{1},\dots,x_{N}]^{S_{N}}$ can be represented as a polynomial in elementary
symmetric polynomials $e_{1}(x_{1},\dots,x_{N})$, …,
$e_{N}(x_{1},\dots,x_{N})$. In particular, every symmetric form can be
represented as
$F=\sum_{i_{1}+\dots+i_{k}=2n+1}\alpha_{i_{1},\dots,i_{k}}e_{i_{1}}\dots
e_{i_{k}}$
where $e_{i_{m}}(x_{1},\dots,x_{N})$ stands for the elementary symmetric
polynomial of degree $i_{m}$. Each summand contains a factor of odd degree,
therefore they all vanish at any point of the form (1). ∎
###### Definition 2.2.
We call any point of the form (1), as well as any its image with respect to
the action of the symmetric group, a primitive solution of $F=0$.
First consider the case of diagonal quintics.
###### Proposition 2.3.
The equation $x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}+x_{5}^{5}+x_{6}^{5}=0$
has a parametric solution, where $x_{i}$ are polynomials in two parameters $a$
and $b$, and $x_{1}+\dots+x_{6}=0$.
###### Proof.
This equation has primitive solutions $(a,-a,b,-b,1,-1)$ and
$(-1,c,-c,d,-d,1)$. Let us look for a new solution in the form
$(a\cdot t-1)^{5}+(-a\cdot t+c)^{5}+(b\cdot t-c)^{5}+(-b\cdot t+d)^{5}+(1\cdot
t-d)^{5}+(-1\cdot t+1)^{5}=0.$ (2)
This gives an equation in $t$. For every $t$ we have ${x_{1}+\dots+x_{6}=0}$.
When $t$ runs over the projective line, this equation has 5 roots: $t_{1}=0$,
$t_{2}=\infty$, and 3 additional roots: $t_{3}$, $t_{4}$, $t_{5}$.
The main idea is as follows: to get these 3 remaining roots $t_{3}$, $t_{4}$,
$t_{5}$, we have an equation in $t$ of degree $3$ obtained from (2):
$A\cdot t^{3}+B\cdot t^{2}+C\cdot t+D=0$ (3)
where $A$, $B$, $C$, $D$ are functions in $a$, $b$, $c$, $d$; let us require
$A=0$, $B=0$. Solving this system of two equations, we get $c$, $d$ as
functions in $a$ and $b$. Then $t=-D/C$, $D$ and $C$ are also functions in $a$
and $b$. We have
$\displaystyle A$ $\displaystyle=$ $\displaystyle-a^{4}+a^{4}\cdot
c-b^{4}\cdot c+b^{4}\cdot d-d+1,$ $\displaystyle B$ $\displaystyle=$
$\displaystyle a^{3}-a^{3}\cdot c^{2}+b^{3}\cdot c^{2}-b^{3}\cdot
d^{2}+d^{2}-1.$
If $A=0$ then $c=(d\cdot(1-b^{4})+a^{4}-1)/(a^{4}-b^{4})$.
If $B=0$ then $c^{2}=(d^{2}\cdot(b^{3}-1)+1-a^{3})/(b^{3}-a^{3})$.
We have a system of two equations, where $c$ and $d$ are unknowns. It is easy
to transform this system into one equation of degree 2 in $d$; then this
equation evidently has a solution $d=1$, because $c=d=1$ is a solution of this
system, so the second root is $d=f(a,b)$. ∎
This method can be extended to more complicated cases.
###### Theorem 2.4.
Every symmetric quintic equation $F(x_{1},\dots,x_{6})=0$ has a parametric
solution where $x_{i}$ are polynomials in two parameters $a$ and $b$, and
$x_{1}+\dots+x_{6}=0$.
###### Proof.
Under the assumption $x_{1}+\dots+x_{6}=0$, every quintic is representable as
$A_{1}\cdot(x_{1}^{5}+\dots+x_{6}^{5})+A_{2}\cdot(x_{1}^{3}+\dots+x_{6}^{3})\cdot(x_{1}^{2}+\dots+x_{6}^{2}).$
Indeed, any symmetric quintic can be represented as a polynomial in diagonal
forms:
$\displaystyle A_{1}(x_{1}^{5}+\dots+x_{6}^{5})$
$\displaystyle+A_{2}(x_{1}^{3}+\dots+x_{6}^{3})\cdot(x_{1}^{2}+\dots+x_{6}^{2})+A_{3}(x_{1}^{4}+\dots+x_{6}^{4})\cdot(x_{1}+\dots+x_{6})$
$\displaystyle+A_{4}(x_{1}^{3}+\dots+x_{6}^{3})\cdot(x_{1}+\dots+x_{6})^{2}+A_{5}(x_{1}+\dots+x_{6})\cdot(x_{1}^{2}+\dots+x_{6}^{2})^{2}$
$\displaystyle+A_{6}(x_{1}+\dots+x_{6})^{3}\cdot(x_{1}^{2}+\dots+x_{6}^{2})+A_{7}(x_{1}+\dots+x_{6})^{5},$
and all terms, except for the first two ones, vanish. To find a parametric
solution of the resulting quintic, we use exactly the same method as for the
diagonal quintic in Proposition 2.3.
This quintic has primitive solutions
$(a,-a,b,-b,1,-1),(-1,c,-c,d,-d,1).$
Let us look for a new solution in the form:
$A_{1}\cdot((a\cdot t-1)^{5}+(-a\cdot t+c)^{5}+(b\cdot t-c)^{5}+(-b\cdot
t+d)^{5}+(1\cdot t-d)^{5}+(-1\cdot t+1)^{5})+A_{2}((a\cdot t-1)^{3}+(-a\cdot
t+c)^{3}+(b\cdot t-c)^{3}+(-b\cdot t+d)^{3}+(1\cdot t-d)^{3}+(-1\cdot
t+1)^{3})((a\cdot t-1)^{2}+(-a\cdot t+c)^{2}+(b\cdot t-c)^{2}+(-b\cdot
t+d)^{2}+(1\cdot t-d)^{2}+(-1\cdot t+1)^{2})=0.$
This gives an equation in $t$. We use the same idea: this equation in $t$ has
5 roots: $0$, $\infty$, $t_{1}$, $t_{2}$, $t_{3}$. For $t_{1}$, $t_{2}$,
$t_{3}$ we have an equation:
$R_{1}\cdot t^{3}+R_{2}\cdot t^{2}+R_{3}\cdot t+R_{4}=0$
where $R_{1}$, $R_{2}$, $R_{3}$, $R_{4}$ are functions in $a$, $b$, $c$, $d$,
$A_{1}$, $A_{2}$. Now we require: $R_{1}=0$, $R_{2}=0$. This is a system of
two equations with unknowns $c$ and $d$. The equation $R_{1}=0$ is linear in
$c$ and $d$, and the equation $R_{2}=0$ is quadratic in $c$ and $d$. It is
easy to see that if we transform this system to one equation of degree $2$ in
$d$, this equation evidently has the root $d_{1}=1$ corresponding to the
primitive solution
$(a\cdot t-1,-a\cdot t+1,b\cdot t-1,-b\cdot t+1,t-1,-t+1).$
The second root is $d_{2}=F(a,b)$. So the equation in $t$ reduces to the
equation:
$K_{1}(a,b)\cdot t+K_{2}(a,b)=0$
from which we can express $t$ as a function in $a$ and $b$. We thus find a
parametric solution for any symmetric diophantine equation of degree 5 in 6
variables: $x_{i}=f_{i}(a,b)$ for which we have $x_{1}+\dots+x_{6}=0.$ ∎
## 3\. General results
In this section we generalize Theorem 2.4 to the case of a form of an
arbitrary odd degree.
###### Theorem 3.1.
Let $F$ be a symmetric form in $N$ variables of odd degree $n\geq 5$ with
rational coefficients. If $N\geq 6\cdot 2^{n-5}$ then the equation
$F(x_{1},\dots,x_{N})=0$ has a parametric solution where $x_{i}$ are
polynomials in $s=2n-8$ parameters, and $x_{1}+\dots+x_{N}=0$.
###### Proof.
First consider the equation
$\displaystyle(x_{1}^{5}+\dots+x_{6}^{5})$
$\displaystyle+(x_{1}^{4}-x_{2}^{4}+x_{3}^{4}-x_{4}^{4}+x_{5}^{4}-x_{6}^{4})\cdot
D_{1}(x_{i})$ (4)
$\displaystyle+(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3})\cdot
D_{2}(x_{i})$
$\displaystyle+(x_{1}^{2}-x_{2}^{2}+x_{3}^{2}-x_{4}^{2}+x_{5}^{2}-x_{6}^{2})\cdot
D_{3}(x_{i})=0,$
where $D_{1}(x_{i})$ is a polynomial in $x_{i}$ of degree at most 1,
$D_{2}(x_{i})$ is a polynomial in $x_{i}$ of degree at most 2, $D_{3}(x_{i})$
is a polynomial in $x_{i}$ of degree at most 3. It is easy to see that this
equation has solutions:
$(a,-a,b,-b,1,-1),(-1,c,-c,d,-d,1).$
Let us try to find a new solution by the same method:
$x_{1}=a\cdot t-1,x_{2}=-a\cdot t+c,x_{3}=b\cdot t-c,x_{4}=-b\cdot
t+d,x_{5}=t-d,x_{6}=-t+1,$
so equation (4) in $t$ has roots $t=0$, $t=\infty$, and hence can be
transformed to an equation of degree $3$
$S_{1}\cdot t^{3}+S_{2}\cdot t^{2}+S_{3}\cdot t+S_{4}=0$
(here $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$ are functions in $a$, $b$, $c$, $d$).
Now, as above, we have two equations for $c$ and $d$: $S_{1}=0$, $S_{2}=0$.
The equation $S_{1}=0$ is linear in $c$ and $d$, and $S_{2}=0$ is quadratic in
$c$ and $d$. From the equation $S_{1}=0$ it follows that if $d=1$ then $c=1$,
because for $c=1$, $d=1$, equation (4) becomes an identity. Therefore the
equation $S_{2}=0$ must have the root $d_{2}=1$, $d_{1}=F(a,b)$, and equation
(4) is transformed to $S_{3}\cdot t+S_{4}=0$. So we can set $t=-S_{4}/S_{3}$
and find a parametric solution of equation (4).
Now suppose that we have an arbitrary symmetric form $F$ of degree $n=2k+1$,
and the number of variables is $N=4s$. For every quadruple of variables we use
a transformation of the form
$z_{1}=x_{1}+c_{1},z_{2}=-x_{2}+d_{1},z_{3}=-x_{1}-d_{1},z_{4}=x_{2}-c_{1}.$
(5)
We represent this form as follows:
$z_{1}^{2n+1}+\dots+z_{N}^{2n+1}+A_{1}(z_{1}^{2n-1}+\dots+z_{N}^{2n-1})R_{1}(z_{i})+\dots+A_{k}(z_{1}+\dots+z_{N})R_{k}(z_{i}),$
where $R_{j}(z_{i})$ are symmetric polynomials. Let us now look how this
transformation works.
It is easy to see that we obtain a form of degree $2n$ whose coefficients are
functions in $c_{N}$, $d_{N}$, but we obtain a new symmetric construction of
the form
$x_{1}^{2n}-x_{2}^{2n}+x_{3}^{2n}-x_{4}^{2n}+\dots$ (6)
Under transformation (5), polynomials of this type go over to linear
combinations of symmetric diagonal polynomials of odd degree and polynomials
of the form (6) of even degree. The degree of these polynomials is smaller
than $2n+1$. Repeating this transformation several times, we obtain an
equation of the form
$\displaystyle(x_{1}^{5}+\dots+x_{6}^{5})$
$\displaystyle+(x_{1}^{4}-x_{2}^{4}+x_{3}^{4}-x_{4}^{4}+x_{5}^{4}-x_{6}^{4})\cdot
D_{1}(x_{i})$
$\displaystyle+(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3})\cdot
D_{2}(x_{i})$
$\displaystyle+(x_{1}^{2}-x_{2}^{2}+x_{3}^{2}-x_{4}^{2}+x_{5}^{2}-x_{6}^{2})\cdot
D_{3}(x_{i})=0,$
but for this equation the existence of a solution depending on two parameters
$a$ and $b$ is proved in Theorem 2.4. So if a symmetric form has $6\cdot
2^{n-5}$ ($n\geq 5$ is an odd number) variables, it has a parametric solution
in $s=2n-8$ parameters. ∎
One can use the same approach as in Theorem 3.1 for finding parametric
solutions of the following problem of Waring’s type.
###### Theorem 3.2.
Let $n\geq 5$ be an odd integer, $N=6\cdot 2^{n-4}$, let $F$ be a symmetric
form of degree $n$ with rational coefficients, and let $q$ be a fixed rational
number. Then the equation $F(x_{1},\dots,x_{N})=q$ has a parametric solution
depending on $s=2n-6$ parameters.
###### Proof.
The proof follows, almost word for word, the arguments of Theorem 3.1, and we
leave the details to a scrupulous reader. ∎
###### Remark 3.3.
One can pose the following problem: for given integers $n$ and $k$ to find an
integer $N$ such that every symmetric form in $N$ variables of degree $2n+1$
with rational coefficients has a parametric solution depending on $k$
parameters. Such a problem can be treated using the approach proposed in this
paper. In particular, it is very easy to generalize the results of this paper
from symmetric quintics in 6 variables to symmetric quintics in any even
number of variables. One can use primitive solutions to construct, for every
symmetric quintic in $2P$ variables, a parametric solution depending on $P-1$
parameters. Using transformation (5), one can then generalize this result for
every symmetric form of odd degree. Finally, for every given integers $n$ and
$k$ one can find an integer $N$ such that every symmetric form in $N$
variables of degree $2n+1$ with rational coefficients has a parametric
solution depending on $s$ parameters with $s>k$. If in this solution we
substitute arbitrary rational numbers instead of $s-k$ parameters, we obtain a
desired parametric solution depending on $k$ parameters.
## References
* [Bi1] B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957) 102–105.
* [Bi2] B. J. Birch, Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/1962) 245–263.
* [Br] A. A. Bremner, Geometric approach to equal sums of fifth powers, J. Number Theory 13 (1981) 337–354.
* [C] A. Choudhry, On equal sums of fifth powers, Indian J. Pure Appl. Math. 28 (1997) 1443–1450.
* [E] R. L. Ekl, New results in equal sums of like powers, Math. Comput. 67 (1998) 1309–1315.
* [LP] L. J. Lander and T. R. Parkin, A counterexample to Euler’s sum of powers conjecture, Math. Comput. 21 (1967) 101–103.
* [LPS] L. J. Lander, T. R. Parkin, and J. L. Selfridge, A survey of equal sums of like powers, Math. Comput. 21 (1967) 446–459.
* [M] Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, Nauka, Moscow, 1972; 2nd ed. of the English transl. North-Holland, Amsterdam, 1986.
* [R] K. S. Rao, On sums of fifth powers, J. London Math. Soc. 9 (1934) 170–171.
* [SwD] H. P. F. Swinnerton-Dyer, A solution of $A^{5}+B^{5}+C^{5}=D^{5}+E^{5}+F^{5}$, Proc. Cambridge. Phil. Soc. 48 (1952) 516–518.
* [W1] T. D. Wooley, Forms in many variables, Analytic Number Theory (Kyoto, 1996), London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 361–376.
* [W2] T. D. Wooley, An explicit version of Birch’s theorem, Acta Arith. 85 (1998) 79–96.
|
arxiv-papers
| 2008-09-23T17:35:47
|
2024-09-04T02:48:57.911416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. A. Reynya",
"submitter": "Misha Arcadii Reynya",
"url": "https://arxiv.org/abs/0809.3973"
}
|
0809.4045
|
Nonlinear properties of split-ring resonators
Bingnan Wang1, Jiangfeng Zhou1, Thomas Koschny1,2 and
Costas M. Soukoulis1,2,∗
1 Ames Laboratory and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA
2 Institute of Electronic Structure and Laser, FORTH, and Department of
Materials Science and Technology, University of Crete, 71110 Heraklion, Crete,
Greece
∗soukoulis@ameslab.gov
###### Abstract
In this letter, the properties of split-ring resonators (SRRs) loaded with
high-Q capacitors and nonlinear varactors are theoretically analyzed and
experimentally measured. We demonstrate that the resonance frequency $f_{m}$
of the nonlinear SRRs can be tuned by increasing the incident power. $f_{m}$
moves to lower and higher frequencies for the SRR loaded with one varactor and
two back-to-back varactors, respectively. For high incident powers, we observe
bistable tunable metamaterials and hysteresis effects. Moreover, the coupling
between two nonlinear SRRs is also discussed.
OCIS codes: (350.4010) Microwaves; (999.9999) Metamaterials.
## References and links
* [1] D. R. Smith, J. B. Pendry and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004)
* [2] C. M. Soukoulis, S. Linden and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science 315, 47-49 (2007)
* [3] J. B. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave. Theory Technol. 47, 2075-2084 (1999)
* [4] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$,” Sov. Phys. Usp. 10, 509-514 (1968)
* [5] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000)
* [6] P. Markos and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622-8 (2002)
* [7] R. A. Shelby, D. R. Smith and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77-79 (2002)
* [8] A. A. Zharov, I. V. Shadrivov and Y. S. Kivshar, “Nonlinear Properties of Left-handed Materials,” Phys. Rev. Lett. 91, 037401-4 (2003)
* [9] I. V. Shadrivov, S. K. Morrison and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14, 9344-9349 (2006)
* [10] D. A. Powell, I. V. Shadrivov, Y. S. Kivshar and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett. 91, 144107 (2007)
* [11] I. Gil, J. Bonache, J. Garcia-Garcia and F. Martin, “Tunable metamaterial transmission lines based on varactor loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech 54, 2665-2674 (2007)
* [12] D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. 92, 264104 (2008)
* [13] E. Ozbay, K. Aydin, S. Butun, K. Kolodziejak and D. Pawlak, “Ferroelectric based tuneable SRR based metamaterial for microwave applications,” in Proceedings of the 37th European Microwave Conference, 497-9 (2007)
* [14] A. Degiron, J. J. Mock and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express 15, 1115-1127 (2007)
* [15] L. Kang, Q. Zhao, H. Zhao and J. Zhou, “Magnetically tunable negative permeability metamaterial composed by split ring resonators and ferrite rods,” Opt. Express 16, 8825-8834 (2008)
* [16] D. Wang, L. Ran, H. Chen, M. Mu, J. A. Kong and B. Wu, “Active left-handed material collaborated with microwave varactors,” Appl. Phys. Lett. 91, 164101 (2007)
* [17] J. Carbonell, V. E. Boria and D. Lippens, “Nonlinear effects in split-ring resonators loaded with heterostructure barrier varactors,” Microwave Opt. Technol. Lett. 50, 474-479 (2008)
* [18] K. Aydin, and E. Ozbay, “Capacitor-loaded split ring resonators as tunable metamaterial components,” Journal of Applied Physics, 101, 024911 (2007)
* [19] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd Edition (Course of Theoretical Physics Vol. 1)
## 1 Introduction
Electromagnetic Metamaterials are periodically arranged artificial structures
that show peculiar behaviors such as negative refraction not seen in natural
materials[1, 2]. The photonic atoms, or the element structures of the
metamaterials, are typically much smaller in size than the working wavelengths
such that the metamaterials can be considered to be homogeneous and
macroscopic parameters such as electrical permittivity and magnetic
permeability can be used to describe the electromagnetic properties of the
metamaterials [3]. By carefully engineering the photonic atoms, both the
permittivity and permeability can be made negative such that negative
refraction can be achieved from the metamaterial [4, 5]. The most widely used
structures for this purpose is the composite of short metallic wires and
split-ring resonators (SRRs) [6, 7]. While the array of short wires gives a
negative permittivity in the wide frequency range below the effective plasma
frequency, the array of SRRs gives a negative permeability in the narrow
frequency range just above the resonance frequency so that the effective index
of refraction can be negative in a narrow frequency band. While most of the
research in this area is in a linear regime, where the electromagnetic
responses are independent of the external fields, some effort has been made to
study the nonlinear effects of the metamaterials, especially the nonlinear
tunability of the SRRs [8-12]. The SRRs are essentially LC resonators and the
resonance frequency is determined by the geometry of the rings. To tune the
magnetic responses of the SRRs, extra components or materials need to be
introduced into the SRRs.
In Ref. [13], ferroelectric films are added to the substrate of the SRRs and
the magnetic tunability is achieved by controlling the electric permittivity
of the ferroelectric films with the change of temperature. In Ref. [14], low-
doped semiconductors are photodoped within the slits of the SRRs and the
magnetic response is tuned by varying the conductivity of the semiconductors
with an external light source. In Ref. [15], ferrite rods are introduced to
ambient the SRR unit cells and the magnetic resonance is modulated by
magnetically tuning the inductance of the ferrite rods by an external magnetic
field. Compared to these methods, the use of varactors is more feasible in
microwave applications in that the tunability can either be realized by a
small DC bias voltage [9, 11, 16] or self-tuned by the intensity of the
applied electromagnetic fields without biasing [10, 17]. Tunable
metamaterials, based on the nonlinear SRRs with varactors, have been tested
experimentally in both transmission line form [11, 12] and bulk form [16].
In this letter, experiments are completed to analyze the properties of the
SRRs loaded with linear and nonlinear elements. The multivalue effect [10, 12]
of nonlinear SRRs with one varactor is analyzed. The nonlinear properties of
SRRs with one varactor and two varactors are compared. The coupling effect of
two nonlinear SRRs are also discussed.
## 2 Nonlinear properties of a single nonlinear SRR
The geometry of the SRRs is a single ring with the outer diameter of $7$ mm,
the inner diameter of $6$ mm, a slit width of $0.7$ mm, and fabricated on a PC
board substrate of $0.8$ mm thick. The resonance frequency of the SRR is
around 4 GHz. When a high Q capacitor with capacitance 2 pF is mounted into
the slit of the SRR, the resonance frequency is brought down to 0.9 GHz. This
makes the ratio between the wavelength and the SRR size as large as 50, which
is useful to miniaturize the size of potential microwave devices [18].
To study the nonlinear magnetic response of the SRR, a varactor is mounted
onto the slit of the SRR. The reflection is measured with a loop antenna and a
vector network analyzer. The small loop antenna is made by a semi-rigid
coaxial cable and the diameter of the loop is 8 mm. The antenna is kept 2 mm
away and on top of the SRR (see the inset of Fig. 1(a)). The measured
reflection at different input power levels is shown in Fig. 1(a). As the
incident power from the vector network analyzer increases, the resonance
frequency of the nonlinear SRRs moves to lower frequencies. We will able to
tune the resonance frequency of the SRR by $10\%$ by increasing the incident
power by $6$ dB. If the incident power increases more, one can see clearly
from Fig. 1(b) that the reflection coefficient starts showing jumps. In
addition, we have observed strong hysteresis and bistable behavior in the
nonlinear SRRs. (As one can see from Fig. 1(a), as the incident power
increases the resonance frequency shifts to lower frequencies and, in
addition, we observe a decrease in the quality factor of the nonlinear SRR.)
Fig. 1: (a) The reflection of the SRR loaded with one varactor, measured by a
loop antenna. The curves show the measurement results at input power from -15
dBm to 9 dBm in 3 dBm steps. The inset shows the picture of the loop antenna
and the sample. (b) The hysteresis effect at high input power levels. The blue
curves are measured for forward sweep (the source frequency of the network
analyzer is scanned from low to high) and the red dashed curves are measured
for reverse sweep (frequency is scanned from high to low).
Below, we present some analytical arguments to explain the tunability and the
hysteresis effects of the nonlinear SRRs observed experimentally.
The varactor, mounted onto the slit of the SRR, is a Skyworks SMV1231-079
hyperabrupt tuning varactor. The nonlinear voltage-dependent depletion layer
capacitance $C(V_{D})$ ($V_{D}$ is the voltage across the diode) is described
as the following, provided by the manufacturer SPICE model.
$C(V_{D})=C_{0}\left(1-V_{D}/V_{p}\right)^{-M}$, where $C_{0}=2.2$ pF is the
DC rest capacitance, $V_{p}=1.5$ V is the intrinsic potential and $M=0.8$. The
dissipative current is given by
$I_{D}(V_{D})=I_{0}\left(e^{(V_{D}/V_{T})}-1\right)$. From
$C(V_{D})=\frac{\,dQ_{D}}{\,dV_{D}}$, we can determine the time-dependent
charge,
$Q_{D}=\frac{C_{0}V_{p}}{1-M}\left[1-\left(1-V_{D}/V_{p}\right)^{1-M}\right]$.
Assume $V_{D}<V_{p}$, then the voltage across the diode can be expressed by
the charge,
$V_{D}(q)=V_{p}\left[1-\left(1-q\frac{1-M}{V_{p}}\right)^{\frac{1}{1-M}}\right]$,
where the renormalized voltage is defined as $q=Q_{D}/C_{0}$.
The SRR can be modeled as an RLC circuit with external excitation and the
voltage equation can be expressed by
$-L\frac{\,dI}{\,dt}-R_{S}-V_{D}=\varepsilon(t)$, where $I$ is the current in
the resonator, $L$ is the inductance of the resonator determined by the ring
geometry, $R_{S}$ is the resistance, and $\varepsilon$ is the driven term
provided by the loop antenna in the experiment. For small excitations, $I_{D}$
can be neglected so the current can be estimated by $I\approx\,dQ_{D}/\,dt$.
The equation of motion is now
$\frac{\,d^{2}q}{\,dt^{2}}+\gamma\frac{\,dq}{\,dt}+\omega_{0}^{2}V_{D}=-\omega_{0}^{2}\varepsilon(t)$,
where $\omega_{0}^{2}=1/(LC_{0})$ and $\gamma=\omega_{0}^{2}R_{S}C_{0}$.
Expand the restoring term $V_{D}$ by the Taylor series for small oscillations
(the oscillation amplitude satisfies $(1-M)|q|<V_{p}$) and omit the higher
order terms, $V_{D}(q)\approx
q-\frac{M}{2V_{p}}q^{2}+\frac{M(2M-1)}{6V_{p}^{2}}q^{3}$. Assume harmonic
excitation so that $\varepsilon(t)=f\cos(\omega t)$, where $f$ is the
excitation amplitude and $\omega$ is the excitation frequency, the equation of
motion is further estimated by
$\frac{\,d^{2}q}{\,dt^{2}}+\gamma\frac{\,dq}{\,dt}+\omega_{0}^{2}q+\alpha
q^{2}+\beta q^{3}=-\omega_{0}^{2}f\cos(\omega t)$, where
$\alpha=-\frac{\omega_{0}^{2}M}{2V_{p}}$,
$\beta=\frac{\omega_{0}^{2}M(2M-1)}{6V_{p}^{2}}$. This is now a nonlinear
driven oscillator problem [19].
The driven frequency can be written as $\omega=\omega_{0}+\delta$. When
$\delta$ is small, the driven frequency is close to the resonance frequency.
Without the $q^{2}$ and $q^{3}$ term, the oscillator is linear and the
amplitude of oscillation, $b$, is given by
$b^{2}\left(\delta^{2}+\gamma^{2}/4\right)=\omega_{0}^{2}f^{2}/4$. The
nonlinear $q^{2}$ and $q^{3}$ terms make the eigen-frequency amplitude
dependent, $\omega_{0}\to\omega_{0}+\kappa b^{2}$, where
$\kappa=\frac{3\beta}{8\omega_{0}}-\frac{5\alpha^{2}}{12\omega_{0}^{3}}$. Then
$\delta\to\delta-\kappa b^{2}$, and the oscillation amplitude satisfies the
equation $b^{2}\left[\left(\delta-\kappa
b^{2}\right)^{2}+\gamma^{2}/4\right]=\omega_{0}^{2}f^{2}/4$. This is a cubic
equation about $q^{2}$ and the real roots give the amplitude of oscillations.
When the external excitation is small, the oscillation amplitude is also small
and the higher orders of $b$ may be neglected and the oscillation can be
considered to be linear. When the excitation power is larger, the curve is
distorted and the resonance shifts to a lower frequency, since, in our case,
$\kappa$ is negative. When the excitation power is large enough, there are
three real roots of $b^{2}$ and the curve is folded over, see Fig. 1(b). The
branch in the middle is unstable and the oscillation tends to go to the other
two branches. In experiment, the oscillation follows the lower branch until it
jumps to the higher branch for forward sweep, and follows the higher branch
until it jumps to the lower branch. So, the hysteresis effect is observed in
our experimental measurements. Note, when the voltage on the varactor is
larger than 0.5 V, the nonlinear DC dissipative current (
$I_{D}(V_{D})=I_{0}\left(e^{V/V_{T}}-1\right)$, where $V_{T}=k_{B}T/e$ is the
thermal voltage, $k_{B}$ is the Boltzmann constant, $T$ is the temperature,
and $e$ is the electron charge) sets in and this increases the loss on the SRR
and the reflection dip measured is not as strong as the small oscillation
case. See Fig. 1(a), the reflection minimum increases from around -40 dB to -3
dB when the input power is increased from -15 dBm to 9 dBm. This is not
covered in the simplified model of the nonlinear oscillator model.
Fig. 2: The oscillation amplitude vs. source frequency calculated from the
simplified nonlinear oscillator model. The curves from bottom to top
correspond to excitation power from low to high. The blue arrow shows the jump
for forward sweep of the top curve and the red arrow shows the jump for
reverse sweep of the top curve.
To remove the nonlinear DC current and obtain a better self-tuning effect, a
new SRR is fabricated. The SRR ring is of the same geometry, but with another
identical cut on the other side of the ring and a varactor is mounted onto
each of the cuts, see inset in Fig. 3. The varactors are arranged back-to-back
such that no DC current can circulate in the SRR and the effective capacitance
characteristics $C(V)$ of the two varactors is now symmetric. This
configuration has the same effect of one heterostructure barrier varator (HBV)
diode [17]. The two varactors can now be regarded as two tunable capacitors
connected in a series. The total capacitance is smaller than a single varactor
at the same power level, so that the resonance frequency shifts to a higher
region (see Fig. 3). When the input power increases, the effective capacitance
decreases and the resonance frequency increases. Also see from Fig. 3 that the
resonance strength and the quality factor are almost the same for different
input power levels. The only problem of this configuration is that the
varactors discharge very slowly, due to the lack of a circulating current.
Fig. 3: The reflection of the SRR loaded with two back-to-back varactors,
measured by a loop antenna. The curves show the measurement results at power
levels from -14 dBm to 2 dBm, in 2 dBm steps. The inset shows the sample.
## 3 Mutual coupling between two nonlinear SRRs
To make nonlinear metamaterials out of the nonlinear SRRs, the mutual coupling
of the varactor loaded SRRs must to be studied. Since the mutual coupling of
two coplane SRRs are very weak, we studied the case of two parallel SRRs with
the same axis (solenoid case). The loop antenna is also parallel to the two
SRRs and has the same axis. The distance of the antenna and the first SRR is
fixed, and the second SRR is moved away from the first SRR. Reflection is
measured by the loop antenna for different distances between the two SRRs. In
Fig. 4(a), we present the frequency-dependence of the reflection coefficient
for two linear SRRs, as the distance between the two SRRs increases. As one
can see from Fig. 4(a), if the distance between the SRRs is large, only one
reflection resonance is observed and as the two SRRs move closer, the mutual
coupling becomes stronger and the reflection splits. The closer the two SRRs,
the wider the split. For two linear SRRs, the mutual coupling can be
calculated analytically with a simple LC model.
Fig. 4: The reflection of (a) two linear SRRs and (b) two nonlinear SRRs at
input power of $-15$ dBm. The legend shows the distance between the two SRRs.
The distance between the first SRR and the antenna is fixed.
In Fig. 4(b), we present the frequency-dependence of the reflection
coefficient for two nonlinear SRRs, as the distance between the two SRRs
increases. The incident power is $-15$ dBm, which is relatively low and the
results presented in Fig. 4(b) are almost equivalent to the linear SRRs
presented in Fig. 4(a), except that the reflection dip is not as deep as the
linear case, due to a higher loss in the varactor than the high-Q capacitor.
When the input power is low, the nonlinear SRRs behaves like the linear ones
and the splitting is nicely seen in the -15 dBm case of the 2d plot in Fig. 5.
When the input power gets higher, the resonance of the SRR gets broadened, due
to higher loss and the splitting becomes worse, as seen in Fig. 5. One can
clearly see from Fig. 5 that the hybridization gets very weak as the incident
power increases.
Fig. 5: The 2d plot of the reflection of two coupled nonlinear SRRs. The
x-axis shows the distance between the two SRRs and the y-axis shows the
frequency. The four figures from left to right display the result at input
power from -15 dBm to 0 dBm. The SRRs and the antenna are arranged such that
they have the same axis and the planes of the rings are parallel to each
other. The distance between the loop antenna and the first SRR is fixed and
the second SRR is movable along its axis, as seen from the inset.
## 4 Conclusion
In conclusion, we have demonstrated experimentally dynamic tunability,
hysteresis, and bistable behavior in nonlinear SRRs. The nonlinear SRR has a
typical SRR design and we have soldered in the gap of the SRR a commercial
varactor diode. Tunability of the resonance frequency was completed
dynamically by increasing the incident power of the vector network analyzer.
We have introduced different nonlinear designs and observed experimentally
that the resonance frequency can decrease or increase by increasing the
incident power. This way, we are able to change the sign of the nonlinearity.
A theoretical model was given that explained all the observed nonlinear
effects. Finally, we study the hybridization effects of the linear and
nonlinear SRR.
## Acknowledgments
Work at Ames Laboratory was supported by the Department of Energy (Basic
Energy Sciences) under contract No. DE-AC02-07CH11358. This work was partially
supported by the office of Naval Research (Award No. N00014-07-1-D359), and
European Community FET project PHOME (Contract No. 213390).
|
arxiv-papers
| 2008-09-24T15:10:13
|
2024-09-04T02:48:57.916186
|
{
"license": "Public Domain",
"authors": "Bingnan Wang, Jiangfeng Zhou, Thomas Koschny and Costas M. Soukoulis",
"submitter": "Bingnan Wang",
"url": "https://arxiv.org/abs/0809.4045"
}
|
0809.4089
|
# On Base Station Localization for State Estimation over Lossy Networks
Ufuk Topcu, Kenneth Hsu, and Kameshwar Poolla The authors contributed equally
to this work.This work was supported in part by the NSF under Grant ECS
03-02554.Authors are with the Department of Mechanical Engineering at the
University of California at Berkeley.
###### Abstract
We consider a state estimation problem where observations are made by multiple
sensors. These observations are communicated over a lossy wireless network to
a central base station that computes estimates via a Kalman filter. The goal
is to determine the optimal location of the base station under a certain class
of packet loss probability models. It is shown in the two sensor case that the
base station is optimally located at one of the sensor locations. Empirical
evidence suggests that the result holds in some generality.
## 1 Introduction
The recent confluence of low-cost sensing, communication, and computation
technologies has led to much interest in the development of wireless sensor
networks for estimation and control. These wireless sensor networks provide
for an economical means of extracting greater performance and efficiency in a
variety of applications, such as manufacturing and chemical plant processes
[Moyne], indoor climate control (HVAC) [Pakzad], environment monitoring
[Szewczyk], electrical power distribution [Chong], and automatic traffic flow
[Giridhar].
There are many advantages that wireless sensor networks hold over their wired
counterparts. For example, the fact that communication is performed wirelessly
eliminates any physical connection between the various nodes in the network.
This enables wireless sensor networks to be implemented without any major
infrastructure overhauls. This also allows for the design of networks that can
gracefully accept dynamic changes to its structure, such as the addition or
subtraction of nodes, or in the case of mobile sensor networks, changing
communication topologies.
While the physical disconnection within the network allows for many practical
and performance related advantages, it also introduces many fundamental issues
(e.g. loss/delay of information, limits on communication bandwidth, power
constraints [Culler]) that are of lesser concern in wired configurations.
Issues such as packet loss/delay are crucial in estimation and control
problems where information flow is assumed to be uninterrupted, and much work
has been done to examine performance in the presence of these complications
[Sinopoli, Schenato, Hespanha].
In this paper, we consider the problem where (noisy) observations of a dynamic
process are made by multiple sensors at fixed locations. These sensors
communicate over a lossy wireless network to a base station that computes the
optimal state estimate by processing the data via a Kalman filter. Since the
physical distance between the sensors and the base station affects the packet
loss probability (which in turn affects the quality of the estimates), our
goal is to determine the optimal position of the base station. A similar
problem of locating control logic has been studied in [Kumar].
There has been much recent work on decentralized estimation problems where no
single node is burdened with the majority of the computational tasks [Grime,
Spanos]. However, implementation of these ideas require “smart” sensors that
have local computational abilities. While these approaches have been shown to
possess various performance advantages, they are not always realizable. In
order to examine more fundamental issues regarding the implementation of
wireless sensor networks, we restrict our attention to the case where
stationary sensors merely _sense_ and _communicate_ , and possess no
computational power. We leave the more general problem for future work.
The remainder of the paper is organized as follows. In Section 2, we give a
brief overview of Kalman filtering. Section 3 contains our problem formulation
and main results. Some discussion regarding more general cases can be found in
Section 4.
## 2 Preliminaries
### 2.1 Kalman Filtering
We begin by giving a brief overview of the canonical state estimation problem.
Consider a process whose dynamics are modeled by the (possibly time-varying)
state space equation
$x_{k+1}=Ax_{k}+w_{k},$ (1)
where $x_{k}\in\mathds{R}^{n}$ is the state vector at time $k$, and $w$ is a
sequence of independent Gaussian random vectors with zero mean and covariance
matrix $Q$. The initial condition $x_{0}$ is unknown and modeled as a Gaussian
random variable with zero mean and covariance matrix $P_{0}$.
We monitor this process via measurements of the form
$y_{k}=Cx_{k}+v_{k},$
where $C$ is possibly time-varying, and $v$ is a sequence of independent
Gaussian random vectors with zero mean and covariance matrix $R$. The
sequences $w$ and $v$ are assumed to be uncorrelated. The minimum variance
estimate of the sequence $x$ is then given recursively by the Kalman filter
[Kalman],
$\displaystyle K_{k}$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}(CP_{k}C^{*}+R)^{-1}$ (2a) $\displaystyle\hat{x}_{k+1}$
$\displaystyle=$ $\displaystyle A\hat{x}_{k}+K_{k}(y_{k}-C\hat{x}_{k})$ (2b)
$\displaystyle P_{k+1}$ $\displaystyle=$ $\displaystyle
AP_{k}A^{*}+Q-K_{k}CP_{k}A^{*}.$ (2c)
Here, $M^{*}$ denotes the complex conjugate transpose of a matrix $M$.
Due to the increasing availability of high quality, low-cost sensors, we are
often confronted with the situation where we have access to measurements from
multiple sensors. That is, suppose that we monitor the process (1) via $N$
sensors, whose measurements are represented as
$\displaystyle y_{k}^{1}$ $\displaystyle=$ $\displaystyle
C_{1}x_{k}+v_{k}^{1}$ $\displaystyle\vdots$ $\displaystyle y_{k}^{N}$
$\displaystyle=$ $\displaystyle C_{N}x_{k}+v_{k}^{N}.$
Here, $v^{1},\dots,v^{2}$ are sequences of independent Gaussian random vectors
with zero mean and covariance matrices $R_{1},\dots,R_{N}$, respectively. The
sequences $v^{1},\dots,v^{N}$ and $w$ are assumed to be uncorrelated. The
resulting state estimates are again given by (2), where we now define
$\displaystyle C$ $\displaystyle=$ $\displaystyle\mbox{blkcol}(C_{j}),\ \
j=1,\dots,N$ $\displaystyle R$ $\displaystyle=$
$\displaystyle\mbox{blkdiag}(R_{j}),\ \ j=1,\dots,N.$
Here, blkcol denotes the vertical stacking of its arguments, and blkdiag
denotes a block diagonal matrix of its arguments.
### 2.2 Kalman Filtering with Packet Losses
We now discuss the minimum variance state estimation problem in the presence
of packet losses. Let $\Omega$ be the power set of $\\{1,\dots,N\\}$ and let
each subset $\omega\in\Omega$ be given the standard simple order. Since the
Kalman filtering equations (2) yield the minimum variance state estimate for
linear time-varying systems, we obtain the following optimal filter for state
estimation with packet losses
$\left.\begin{array}[]{l}K_{k}=AP_{k}F_{\omega_{k}}^{*}(F_{\omega_{k}}P_{k}F_{\omega_{k}}^{*}+G_{\omega_{k}})^{-1}\\\
\hat{x}_{k+1}=A\hat{x}_{k}+K_{k}(y_{k}-F_{\omega_{k}}\hat{x}_{k})\\\
P_{k+1}=AP_{k}A^{*}+Q-K_{k}F_{\omega_{k}}P_{k}A^{*}\end{array}\right\\}\mbox{
for }\omega_{k}\in\Omega\backslash\phi,$
and
$\left.\begin{array}[]{l}\hat{x}_{k+1}=A\hat{x}_{k}\\\
P_{k+1}=AP_{k}A^{*}+Q\end{array}\right\\}\mbox{ for }\omega_{k}=\phi,$
where
$\displaystyle F_{\omega_{k}}$ $\displaystyle=$
$\displaystyle\mbox{blkcol}(C_{j}),\ \ j\in\omega_{k}$ $\displaystyle
G_{\omega_{k}}$ $\displaystyle=$ $\displaystyle\mbox{blkdiag}(R_{j}),\ \
j\in\omega_{k}.$
For a similar treatment, see [Liu, Hounkpevi].
## 3 Base Station Placement
### 3.1 Problem Formulation
We will model the packet arrival process for each sensor as a sequence of
independent binomial random variables. More specifically, for $i=1,\dots,N$,
let $\lambda_{i}$ be the probability that the observation from sensor $i$ is
received. Since the sensor locations are fixed, a given base station location
gives rise to distances $d_{1},\dots,d_{N}$. We will assume that the
probability of packet arrival decreases as the distance $d_{i}$ between a
sensor and the base station increases.
Note that the sequences $(F_{\omega})_{k}$ and $(G_{\omega})_{k}$ both depend
on whether packets have been received or lost, and hence the sequence
$(P_{k})_{k}$ is itself random. We will then aim to minimize (with respect to
the Loewner ordering [Horn], i.e., $X\succeq Y$ if $X-Y$ is positive semi-
definite). the expected estimation error covariance. That is, we consider the
problem
$\min{\bf E}\left[P_{k+1}|P_{k}\right],$
where the minimization is performed over the set of possible base station
locations. If we define for each nonempty $\neq\omega\in\Omega$
$\displaystyle\alpha_{\omega}$ $\displaystyle=$
$\displaystyle\prod_{j\in\omega}\lambda_{j}\prod_{j\in\\{1,\dots,N\\}\backslash\omega}(1-\lambda_{j})$
$\displaystyle K_{k}$ $\displaystyle=$ $\displaystyle
AP_{k}F_{\omega}^{*}(F_{\omega}P_{k}F_{\omega}^{*}+G_{\omega})^{-1},$
and $\alpha_{\omega}=0$ for $\omega=\phi$, we can write
${\bf
E}\left[P_{k+1}|P_{k}\right]=AP_{k}A^{*}+Q-\sum_{\omega\in\Omega}\alpha_{\omega}K_{k}F_{\omega}P_{k}A^{*}.$
As stated, the above optimization problem is a difficult nonlinear programming
problem that can be attacked by the appropriate solvers. Since it is not
feasible for the location of the base station to change over time, we will
insist that the solution to the above problem minimizes ${\bf
E}\left[P_{k+1}|P_{k}\right]$ for any $P_{k}$. Consequently, we shall
hereafter restrict our attention to an illuminating instance of the above
problem.
### 3.2 The 2 Sensor Case
In this section, we consider the case where measurements are obtained from $2$
homogeneous sensors, $s_{1}$ and $s_{2}$. Without loss of generality, these
sensors are located at $\xi_{1}=0$ and $\xi_{2}=1$ on the real line. The goal
is to find the best location $d\in[0,1]$ for the base station. Note that
optimal solution must lie in the interval $[0,1]$. The situation $d=0$ (or
$d=1)$ corresponds to the base station being physically wired to one of the
sensors, and wireless communication is performed only by the other sensor.
Although there may be many factors that influence the packet loss probability,
we shall consider only the effects due to the physical distance between the
communicating nodes. Since the sensors have identical broadcasting
capabilities, it is then natural to model the probability of packet loss in
the following manner. Let $f:[0,1]\rightarrow[0,1]$ be convex and decreasing,
with $f(0)=1$. When the base station is located at some $d\in[0,1]$, the
probability that a packet is received from sensor $s_{1}$ is $f(d)$, and the
probability that a packet is received from sensor $s_{2}$ is $f(1-d)$. The
constraint that $f(0)=1$ captures the implicit assumption that no packet is
lost when a sensor is affixed to the base station and communication is not
performed wirelessly.
#### 3.2.1 Vector-valued Measurements, Same Covariances
We will first restrict our attention to the case where
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
C_{2}=C\in\mathds{R}^{m\times n}$ $\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle R_{2}=R\in\mathds{R}^{m\times m}.$
If we model the probability of packet loss as described above, we can examine
the estimation error covariance as a function of $d$. For ease of notation,
let us define
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}(CP_{k}C^{*}+R)^{-1}CP_{k}A^{*}$ $\displaystyle M_{2}$
$\displaystyle=$ $\displaystyle AP_{k}C^{*}(CP_{k}C^{*}+R/2)^{-1}CP_{k}A^{*}$
$\displaystyle S$ $\displaystyle=$ $\displaystyle AP_{k}A^{*}+Q.$
The estimation error covariance $P_{k+1}$ then assumes values as follows:
$\begin{array}[]{|c|c|}\hline\cr P_{k+1}&\mbox{Probability}\\\ \hline\cr
S-M_{2}&\ \ f(d)f(1-d)\\\ \hline\cr\ \ S-M_{1}&\ \ f(d)-2f(d)f(1-d))+f(1-d)\\\
\hline\cr S&(1-f(d))(1-f(1-d))\\\ \hline\cr\end{array}$
Note that in the case where packets from both sensors are lost, the resulting
covariance corresponds to that of propagating the previous state estimate. The
expected error covariance can then be computed as
$\displaystyle{\bf E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$
$\displaystyle S-f(d)f(1-d)M_{1}$ $\displaystyle\hskip
3.61371pt-(f(d)-2f(d)f(1-d)+f(1-d))M_{2}.$
We have the following result.
###### Proposition 1
Let $C_{1}=C_{2}=C\in\mathds{R}^{m\times n}$ and
$R_{1}=R_{2}=R\in\mathds{R}^{m\times m}$. Let $f:[0,1]\rightarrow[0,1]$ be
twice differentiable, convex, and decreasing, with $f(0)=1$. Then, ${\bf
E}\left[P_{k+1}|P_{k}\right]$ is matrix concave in $d$ on $[0,1]$.
Proof Define
$\displaystyle J_{k+1}(d)$ $\displaystyle=$ $\displaystyle{\bf
E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$ $\displaystyle
AP_{k}A^{*}+Q-f(d)f(1-d)M_{2}$ $\displaystyle\hskip
10.84006pt-(f(d)-2f(d)f(1-d)+f(1-d))M_{1}.$
Taking the second derivative yields
$\displaystyle J_{k+1}^{\prime\prime}(d)$ $\displaystyle=$ $\displaystyle
f^{\prime\prime}(d)(f(1-d)-1)M_{1}$
$\displaystyle+f^{\prime\prime}(1-d)(f(d)-1)M_{1}$
$\displaystyle-2f^{\prime}(d)f^{\prime}(1-d)(2M_{1}-M_{2})$
$\displaystyle+(f^{\prime\prime}(d)f(1-d)+f(d)f^{\prime\prime}(1-d))(M_{1}-M_{2}).$
Since $f$ is convex and decreasing with $f(0)=1$, we have $f^{\prime}(x)\leq
0$ and $f^{\prime\prime}(x)\geq 0$ for $x\in[0,1]$. Consequently,
$J^{\prime\prime}(d)\preceq 0$, and hence ${\bf E}\left[P_{k+1}|P_{k}\right]$
is matrix concave on $[0,1]$ (see [Boyd], p.110). $\blacksquare$
###### Corollary 1
Let $C_{1}=C_{2}=C\in\mathds{R}^{m\times n}$ and
$R_{1}=R_{2}=R\in\mathds{R}^{m\times m}$. Let $f:[0,1]\rightarrow[0,1]$ be
twice differentiable, convex, and decreasing, with $f(0)=1$. Then, the base
station is optimally located at one of the sensor positions.
#### 3.2.2 Scalar-valued Measurements, Same Covariances
A similar result may be obtained in the case where
$C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and the noise covariances $R_{1}$ and
$R_{2}$ are not necessarily equal. We will need the following preliminary
result.
###### Lemma 1
Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and define
$\displaystyle T$ $\displaystyle=$ $\displaystyle CP_{k}C^{*}$ $\displaystyle
M_{0}$ $\displaystyle=$ $\displaystyle AP\begin{bmatrix}C\\\
C\end{bmatrix}^{*}\begin{bmatrix}T+R_{1}&T\\\
T&T+R_{2}\end{bmatrix}^{-1}\begin{bmatrix}C\\\ C\end{bmatrix}PA^{*}$
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
APC^{*}(T+R_{1})^{-1}CPA^{*}$ $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle APC^{*}(T+R_{2})^{-1}CPA^{*}.$
Then, $M_{0}-M_{1}-M_{2}\succeq 0$.
Proof Let us define
$\begin{bmatrix}\alpha&\beta\\\
\beta&\gamma\end{bmatrix}=\begin{bmatrix}T+R_{1}&T\\\ T&T+R_{2}\end{bmatrix},\
\mbox{ and }\ U=\begin{bmatrix}C\\\ C\end{bmatrix}P_{k}A^{*}.$
Then, $M_{0}-M_{1}-M_{2}$ can be written as
$\displaystyle M_{0}-$ $\displaystyle M_{1}$ $\displaystyle-M_{2}$
$\displaystyle=$ $\displaystyle U^{*}\left(\begin{bmatrix}\alpha&\beta\\\
\beta&\gamma\end{bmatrix}^{-1}-\begin{bmatrix}\alpha&0\\\
0&\gamma\end{bmatrix}^{-1}\right)U$ $\displaystyle=$ $\displaystyle
U^{*}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}&-\frac{\beta\gamma^{-1}}{\alpha-\beta^{2}\gamma^{-1}}\\\
-\frac{\gamma^{-1}\beta}{\alpha-\beta^{2}\gamma^{-1}}&\frac{\gamma^{-2}\beta^{2}}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}U$
$\displaystyle=$ $\displaystyle
U^{*}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}\\\
-\frac{\gamma^{-1}\beta}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}\begin{bmatrix}\frac{1}{\alpha-\beta^{2}\gamma^{-1}}&-\frac{\beta\gamma^{-1}}{\alpha-\beta^{2}\gamma^{-1}}\end{bmatrix}U$
$\displaystyle\hskip 10.84006pt-U^{*}\begin{bmatrix}-\alpha^{-1}&0\\\
0&0\end{bmatrix}U$ $\displaystyle=$ $\displaystyle
AP_{k}C^{*}\left(\frac{(1-\gamma^{-1}\beta)^{2}}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}\right)CP_{k}A^{*},$
where the second equality follows from the matrix inversion lemma [Horn]. Now,
let
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{(1-\gamma^{-1}\beta)^{2}}{\alpha-\beta^{2}\gamma^{-1}}-\alpha^{-1}$
$\displaystyle=$
$\displaystyle\frac{(1-\gamma^{-1}\beta)^{2}-\alpha^{-1}(\alpha-\beta^{2}\gamma^{-1})}{\alpha-\beta^{2}\gamma^{-1}}.$
The desired result follows from noticing that the denominator is positive and
the numerator satisfies
$\displaystyle(1-\gamma^{-1}\beta)^{2}-\alpha^{-1}(\alpha-\beta^{2}\gamma^{-1})$
$\displaystyle\hskip
25.29494pt=-\beta\gamma^{-1}\left(2-\frac{CP_{k}C^{*}}{CP_{k}C^{*}+R_{2}}-\frac{CP_{k}C^{*}}{CP_{k}C^{*}+R_{1}}\right)$
$\displaystyle\hskip 25.29494pt\leq 0.$
$\blacksquare$
###### Proposition 2
Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and suppose that $R_{1}$ and
$R_{2}$ are not necessarily equal. Let $f:[0,1]\rightarrow[0,1]$ be twice
differentiable, convex, and decreasing, with $f(0)=1$. Then, ${\bf
E}\left[P_{k+1}|P_{k}\right]$ is matrix concave in $d$ on $[0,1]$.
Proof Define
$\displaystyle T$ $\displaystyle=$ $\displaystyle CP_{k}C^{*}$ $\displaystyle
M_{0}$ $\displaystyle=$ $\displaystyle AP\begin{bmatrix}C\\\
C\end{bmatrix}^{*}\begin{bmatrix}T+R_{1}&T\\\
T&T+R_{2}\end{bmatrix}^{-1}\begin{bmatrix}C\\\ C\end{bmatrix}PA^{*}$
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle
APC^{*}(T+R_{1})^{-1}CPA^{*}$ $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle APC^{*}(T+R_{2})^{-1}CPA^{*}.$
We can then write
$\displaystyle J_{k+1}(d)$ $\displaystyle=$ $\displaystyle{\bf
E}\left[P_{k+1}|P_{k}\right]$ $\displaystyle=$ $\displaystyle AP_{k}A^{*}+Q$
$\displaystyle\hskip 10.84006pt-f(d)f(1-d)M_{0}-f(d)(1-f(1-d))M_{1}$
$\displaystyle\hskip 10.84006pt-f(1-d)(1-f(d))M_{2}$
Taking the second derivative yields
$\displaystyle J_{k+1}^{\prime\prime}(d)$ $\displaystyle=$ $\displaystyle
f^{\prime\prime}(d)(f(1-d)-1)M_{1}$
$\displaystyle+f^{\prime\prime}(1-d)(f(d)-1)M_{2}$
$\displaystyle-2f^{\prime}(d)f^{\prime}(1-d)(M_{1}+M_{2}-M_{0})$
$\displaystyle+f^{\prime\prime}(d)f(1-d)(M_{2}-M_{0})$
$\displaystyle+f(d)f^{\prime\prime}(1-d)(M_{1}-M_{0}).$
Since $f$ is convex and decreasing with $f(0)=1$, we have $f^{\prime}(x)\leq
0$ and $f^{\prime\prime}(x)\geq 0$ for $x\in[0,1]$. From Lemma 1 we have
$M_{1}+M_{2}-M_{0}\preceq 0$. As a result, $J^{\prime\prime}(d)\preceq 0$, so
${\bf E}\left[P_{k+1}|P_{k}\right]$ is matrix concave on $[0,1]$ (see [Boyd],
p.110). $\blacksquare$
###### Corollary 2
Let $C_{1}=C_{2}=C\in\mathds{R}^{1\times n}$ and suppose that $R_{1}$ and
$R_{2}$ are not necessarily equal. Let $f:[0,1]\rightarrow[0,1]$ be twice
differentiable, convex, and decreasing, with $f(0)=1$. Then, the base station
is optimally located at one of the sensor positions. If $R_{1}\succeq R_{2}$,
then the base station is optimally located at the second sensor.
#### 3.2.3 An Example
We now offer an illustrative example to complement our main results.
Suppose that there are two sensors, located respectively at the points $0$ and
$1$ on the real line. Let us model our packet arrival probability function as
$f(d)=e^{-d}.$
Note that $f$ is convex and decreasing on $[0,1]$ with $f(0)=1$. The process
being monitored is described by the state space equations in (1), where all
the eigenvalues of $A\in\mathds{R}^{2\times 2}$ lie in the unit circle and
$C\in\mathds{R}^{1\times 2}$. The trace of the estimation error covariance
matrices is plotted in Figure 1 for the case where the base station is
situated at the point $0$ and the case where it is situated at the point
$0.5$.
Figure 1: Covariances for $d=0$ (dashed) and $d=0.5$ (solid).
## 4 General Cases
We now offer some commentary on extending our main results.
Although the previous results were confined to the case $C_{1}=C_{2}$,
simulations suggest that they also hold in more generality. Consider the case
where $N$ sensors located colinearly are available to make observations. In
general, we may have different values for $C_{1},\dots,C_{N}$ and different
values for $R_{1},\dots,R_{N}$. In this case, empirical evidence supports the
extension of our main result in the sense that the expected error covariance
is piecewise concave over the intervals defined by the location of the
sensors. Figure 2 illustrates this situation for the case $N=3$.
Figure 2: Piecewise concavity for multiple colinear sensors.
In the case where the convex hull of the sensor locations has a nonempty
interior in $\mathds{R}^{2}$, it can be shown that the optimal base station
location may lie at a point other than the sensor locations. Consider 3
sensors, located at the vertices of an equilateral triangle with side length
$0.5$. For $P_{k}=I$, $f(d)=e^{-d}$, $C_{1}=C_{2}=C_{3}=0.6$, and unit noise
covariances, the expected error covariance (as a function of base station
location) has a local minimum at the centroid, as illustrated in Figure 3.
Figure 3: Local minimum in the interior of the convex hull.
## 5 Conclusion
We considered the problem of positioning a central computational node within a
wireless sensor network for the purpose of state estimation. In the presence
of potential packet losses, the optimal estimate is given by a time-varying
Kalman filter. We then strived to determine the optimal location of the base
station so as to minimize the expected conditional estimation error
covariance. It was shown for the case of a 2-sensor configuration that under a
certain class of packet loss probability models, the base station is optimally
located at one of the sensor positions.
|
arxiv-papers
| 2008-09-24T05:36:06
|
2024-09-04T02:48:57.921456
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ufuk Topcu, Kenneth Hsu, and Kameshwar Poolla",
"submitter": "Ufuk Topcu",
"url": "https://arxiv.org/abs/0809.4089"
}
|
0809.4104
|
# Topological defects in two-dimensional crystals
Yong Chen Email: ychen@gmail.com Institute of Theoretical Physics, Lanzhou
University, Lanzhou $730000$, China Wei-Kai Qi Email: weikaiqi@gmail.com
Institute of Theoretical Physics, Lanzhou University, Lanzhou $730000$, China
###### Abstract
By using topological current theory, we study the inner topological structure
of the topological defects in two-dimensional (2D) crystal. We find that there
are two elementary point defects topological current in two-dimensional
crystal, one for dislocations and the other for disclinations. The topological
quantization and evolution of topological defects in two-dimensional crystals
are discussed. Finally, We compare our theory with Brownian-dynamics
simulations in 2D Yukawa systems.
###### pacs:
61.72.Cc, 61.72.Lk, 64.70.D-
In 1970’s, Kosterlitz and Thouless construct a detailed and complete theory of
superfluidity on two-dimensions (2D) kt . They indicate vortices pair
unbinding will lead to a second-order transition in superfluid films. Later, a
microscopic scenario of 2D melting has been posited in the form of the
Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory hny . The KTHNY
theory predicts the unbinding of topological defects to break the symmetry in
the two-stage transitions. Topological defects, which are a necessary
consequence of broken continuous symmetry, play a important role in two-
dimensional phase transition Dn ; Cl .
In 2D crystal, the evolution of topological defects have been studied in
experiments and computer simulations. A serial experiments were performed to
calculate dislocations and disclinations dynamic of two-dimensional colloidal
systems, and dissociation of dislocations and disclinations were observed
td001 . During the years, a large number of computer simulations indicated
that exist a two-stage melting as prescribed by KTHNY theory, however, results
are still controversial KnNs . Our previous work found that exist a hexatic-
isotropic liquid phase coexistence during the melting of soft Yukawa systems
Qi . By Voronoi polygons analysis, the behavior of piont defects in the
coexistence is very complicated. The evolution of topological defects during
the melting of two-dimensional system still a open question.
It is interesting to consider the appropriate form for the point defect
densities when expressed in terms of the vector order parameter field
$\phi(\vec{r},t)$. This has been carried out by Halperin Ha , and exploited by
Liu and Mazenko LM . However, their analysis is incomplete Duan00 . In two-
dimensional system, a gauge field-theoretic formalism developed by Kleinert HK
. The gauge theory of topological quantum melting in 2+1 dimensions Bose
system was developed by Nussinov et al, and the Superfluidity and
superconductivity can arise in a strict quantum field-theoretic setting Z1 .
Recently, a topological field theory for topological defect developed by Duan
et al Duan . By using $\phi$-mapping method and topological current theory,
the evolution of topological defect which relate to sigularities of the order-
paramter field, such as vortex in BEC Duan04 and superconductivity Duan05 ,
was studied. In this Letter, we will develop a topological current theory of
discloations and disclination in 2D crystals. By using the topological field
approach, the topological quantization and evolution of topological defects in
2D crystals was discussed. We compared our theory with Brownian-dynamics
simulations in Yukawa systems.
In continuum elasticity theory, the homogeneous equation in 2D triangular
solid is given by Landau ; Nelson
$\nabla^{4}\chi/Y=\epsilon_{kl}\partial_{k}\partial_{l}\theta+\epsilon_{ik}\partial_{k}(\epsilon_{jl}\partial_{l}\partial_{j}u_{i})$
(1)
where $Y=4\mu(\mu+\lambda)/(2\mu+\lambda)$ is the two-dimensional Young’s
modulus and $\sigma_{0}=\lambda/(2\mu+\lambda)$ the two-dimensional Poisson
ratio. The defects associated with the continuum elastic theory of a solid are
dislocations and disclinations. In the following we consider only the
triangular lattice since it is the most densely packed one in two-dimensional
and favored by Nature.
Figure 1: Voronoi polygons analysis in 2D Yukawa systems.
Topological defects associated with the continuum elastic theory of a two-
dimensional solid are dislocations and disclinations (see Fig. 1).
Disclinations can be introduced into the theory in a way similar to the
discussion of vortices in two-dimensional XY model QY . Since the bond angle
is defined only up to $2\pi/6$, it implies
$\oint d\theta=\oint\nabla\theta d\vec{s}=\frac{2\pi}{6}W^{6},\quad
W^{6}=0,\pm 1,\ldots,n.$ (2)
Where $W^{6}$ is the winding number of disclinations. Disclinations, which can
influence on the decay of orientational order, have a much higher energy than
dislocations. They are defined in terms of the bond angle field
$\theta(r)\equiv 1/2(\partial u_{y}/\partial x-\partial u_{x}/\partial y),$
which measures the bonds orientational order. It is convenient to define an
order parameter for bond orientations, which for the triangular lattices is
$\psi(r)=\psi_{0}e^{i6\theta(r)}$. However, the bond angle field is undefined
at the disclination cores, i.e., the zero points of the order parameter. We
rewrite the orientation order parameter $\psi(r)=\phi_{6}^{1}+i\phi_{6}^{2}$
instead of $\psi(r)=\psi_{0}e^{i6\theta(r)}$. We can define the unit vector
field $\vec{n}_{6}(\textbf{x})$ as $n_{6}^{a}=\phi_{6}^{a}/||\phi_{6}||$ and
$||\phi_{6}||=\sqrt{\phi_{6}^{a}\phi_{6}^{a}}$, $a=1,2$. We can construct a
topological current of the orientation order parameter field in two-
dimensional crystal, which carries the topological information of
$\vec{\phi}(x)$,
$j_{disc}^{k}=\frac{1}{6}\epsilon^{ijk}\epsilon_{ab}\partial_{i}n_{6}^{a}\partial_{j}n_{6}^{b}.$
(3)
It is the $\phi$-mapping topological current for disclinations. We will see
that this current associate with the singularities of the unit field
$\vec{n}_{6}(x)$. It is easy to see that this topological current is
identically conserved, i.e., $\partial_{k}j_{disc}^{k}=0$.
We consider only the triangular lattice since it is the most densely packed
one in two-dimensional and favored by Nature. The dislocation is characterized
by a Burgers vector $\vec{b}$, which can be determined by considering a loop
enclosing the dislocation. In mathematical language,
$\oint du_{l}=\oint\partial_{i}u_{l}dx_{i}=b_{l}.$ (4)
By introduce two-dimensional local transition order parameters as
$\rho_{x}=\rho_{0x}e^{iG_{x}u_{x}}=\phi_{x}^{1}+i\phi_{x}^{2}$ and
$\rho_{y}=\rho_{0y}e^{iG_{y}u_{y}}=\phi_{y}^{1}+i\phi_{y}^{2}$, where G is any
reciprocal lattice vector and $\rho_{0}=e^{iG_{0}r_{0}}$. We can define a unit
vector field $\vec{n}_{l}(x)$, where
$\vec{n}_{l}(\textbf{x})=e^{iG_{l}u_{l}}=n_{l}^{1}+in_{l}^{2}$ ($l=x,y$). It
can be further expressed as
$n_{l}^{a}(\textbf{x})=\frac{\phi_{l}^{a}(\textbf{x})}{||\phi_{l}(\textbf{x})||},\quad||\phi_{l}(\textbf{x})||=\sqrt{\phi_{l}^{a}(\textbf{x})\phi_{l}^{a}(\textbf{x})},$
obviously, $n_{l}^{a}n_{l}^{a}=1$. We can construct topological currents of
the order parameter field in two-dimensional crystal, which carries the
topological information of $\vec{\phi}_{l}(x)$,
$j_{diso}^{kl}=\frac{1}{G_{l}}\epsilon^{ijk}\epsilon_{ab}\partial_{i}n_{l}^{a}\partial_{j}n_{l}^{b}.$
(5)
where the Burgers vectors satisfy $b_{l}G_{l}=2\pi$. It is the $\phi$-mapping
current for dislocations. Dislocations can be quite effective at breaking up
translational order. However, they are less disruptive of orientational
correlations. The disclination can be seen as a bound of dislocations with
parallel Burger vectors. On the hand, disclinations, which play a important
role in the disruptive of the orientational order in two-dimensional solid,
can be seen as bound disclination and anti-disclination pairs.
From the relationship between the bond angle field and displacement vector
field that $\partial_{i}\theta=-\epsilon_{ab}\partial_{a}\partial_{i}(u^{b})$.
Using the topological current theory, we can obtain
$\partial_{k}j_{diso}^{kl}=-\epsilon_{li}j_{disc}^{i},$ (6)
it indicate that disclinations can be seen as sources for dislocations. Since
there aren’t free disclinations in two-dimensional crystal at the low
temperature phase, the dislocation currents are conserved as
$\partial_{k}J_{k}^{l}=0$.
By using Green function relation and substitute $\phi_{6}^{a}$ to the
homogeneous equation Duan , we can rewrite the homogeneous equation into a
$\delta$-function form as
$\frac{1}{Y}\nabla^{4}\chi=\frac{2\pi}{6}\delta^{2}(\vec{\phi_{6}})J\big{(}\frac{\phi_{6}}{x}\big{)}+\epsilon_{kl}\partial_{k}\frac{2\pi}{G_{l}}\delta^{2}(\vec{\phi_{l}})J\big{(}\frac{\phi_{l}}{x}\big{)}.$
(7)
With the $\delta$-function theory, the homogeneous equation can deduce into
$\frac{\nabla^{4}\chi}{Y}=\sum_{i}\beta_{i}\eta_{i}\delta^{2}(\vec{r}-\vec{z_{i}}(t))+\sum_{j}\beta_{j}^{l}\eta_{j}^{l}\epsilon_{kl}\partial_{k}\delta^{2}(\vec{r}-\vec{z_{j}}(t)),$
(8)
where $\eta_{i}=\pm 1$ is Brouwer degree, the positive integer $\beta_{i}$ is
called the Hopf index of map $x\rightarrow\vec{\phi}$. The topological charge
of disclination and dislication is $\beta_{i}\eta_{i}=2\pi/6$ and
$\beta_{j}^{l}\eta_{j}^{l}=b_{j}^{l}$. The first term is related to
dislocation, and the second term is related to disclination. Also, it can
simply seen as disclination current, if dislocations are thought of as
disclination dipole pairs.
We explore what will happen to the disclination at the zero point
$(t^{*},\vec{z_{l}})$. According to the values of the vector Jacobian at zero
points of the order parameter, there are limit points ($J^{0}(\phi_{6}/x)\neq
0$) and bifurcation points ($J^{0}(\phi_{6}/x)=0$). Each kind corresponds to
different cases of branch processes. The limit points are determined by
$\left.J^{0}\left(\frac{\phi_{6}}{x}\right)\right|_{t^{*},\vec{z_{l}}}=0,\quad\left.J^{1}\left(\frac{\phi_{6}}{x}\right)\right|_{t^{*},\vec{z_{l}}}\neq
0.$ (9)
Considering the condition (9) and making use of the implicit function theorem,
the solution in the neighborhood of the point ($t^{*},\vec{z_{l}}$) is
$t=t(x),~{}y=y(x)$, where $t^{*}=t(z_{l}^{1})$. In this case, one can see that
$\left.dx/dt\right|_{(t^{*},\vec{z_{l}})}=\infty$, the Taylor expansion of
$t=t(x)$ at the limit points $(t^{*},\vec{z_{l}})$ is
$t-t^{*}=\left.\frac{1}{2}\frac{d^{2}t}{dx^{2}}\right|_{t^{*},\vec{z_{l}}}(x-z_{l}^{1})^{2},$
(10)
which is a parabola in the x-t plane. From this equation, we can obtain two
solutions $x_{1}(t)$ and $x_{2}(t)$, which give two branch solutions (World
lines of disclinations). If $d^{2}t/dx^{2}|_{(t^{*},\vec{z_{l}})}>0,$ we have
the branch solutions for $t>t^{*}$. It is related to the origin of a
disclination pair. Otherwise, we have the branch solutions for $t<t^{*}$,
which related to the annihilation of a disclination pair. Since the
topological current is identically conserved, the topological charges of these
two generated or annihilated disclinations must be opposite at the limit
points, say $\beta_{1}\eta_{1}+\beta_{2}\eta_{2}=0$. For a limit point it is
required that $J^{1}(\phi/x)|_{t^{*},\vec{z_{l}}}\neq 0$.
At the neighborhood of the limit point,one can obtain the scaling law as Xu
$v\varpropto(t-t^{*})^{-1/2}$. The growth or annihilation is parameterized in
terms of a relevant characteristic length $\xi(t)$, which is also
characteristic the mean separation distance of disclinations pair. The
relevant length $\xi(t)$ obeys that $\xi(t)\sim(t-t^{*})^{1/2}$. It is the
dynamic scaling law of the topological defects pairs. In the low temperature
equilibrium phase has essentially no disclination pairs, and $\xi(t)$ is
infinite. The relationship between the relevant length and the metastable
disclination density which below the critical temperature
is$\xi(t)=\sqrt{1/\rho_{v}}$. then the number of disclinations satisfy the
power law $N\propto t^{-1}$.
Consider in which the restrictions on zero point $(t^{*},\vec{z_{l}})$ are
$J^{k}(\phi_{6}/x)\big{|}_{(t^{*},\vec{z_{l}})}=0,(k=0,1,2)$, which imply an
important fact that the function relationship between t and x or y is not
unique in the neighborhood of the bifurcation point $(t^{*},\vec{z_{l}})$.
With the aim of finding the different directions of all branch curves at the
bifurcation point, we suppose $\partial\phi_{6}^{1}/\partial
y\big{|}_{t^{*},\vec{z_{l}}}\neq 0$. According to the $\phi$-mapping theory,
the Taylor expansion of the solution of the zeros of the order parameter field
in the neighborhood of $(t^{*},\vec{z_{l}})$ can be expressed as
$A(x-z_{l}^{1})^{2}+2B(x-z_{l}^{1})(t-t^{*})+C(t-t^{*})^{2}+\cdots=0,$ (11)
where A, B, and C are constants determined by the order parameter. The
solutions of Eq. (11) give different directions of the branch curves (world
line of disclinations) at the bifurcation point. There are four possible
cases, which will show the physical meanings of the bifurcation points.
First we consider the case that $A\neq 0$. If $\Delta=4(B^{2}-AC)>0$, from
Eq.(11) we get two different motion directions of the core of disclination
$dx/dt\big{|}_{1,2}=(-B\pm\sqrt{B^{2}-AC})/A$, where two topological defects
encounter and the depart at the bifurcation point. However, when
$\Delta=4(B^{2}-AC)=0$, form Eq. (11), we obtain only one motion direction of
the core of disclination$dx/dt\big{|}_{1,2}=-B/A$, which includes three
important cases. (i) Two disclinations tangentially encounter at the
bifurcation point. (ii) Two disclinations merge into one disclination at the
bifurcation point. (iii) One disclinations splits into two disclinations at
the bifurcation point.
Then we consider the case that $A=0$, If $C\neq 0$ and $\Delta=4(B^{2}-AC)=0$,
from Eq. (11) we have $dt/dx\big{|}_{1,2}=(-B\pm\sqrt{B^{2}-AC})/C=0,-(2B)/C$.
There are two important cases. (i) One disclination split into three
disclinations at the bifurcation point. (ii)Three disclinations merge into one
disclination at the bifurcation point. From the above discuss, we can then
obtain the growth or annihilation velocity of the vortices as $v\propto
const$. The approximation asymptotic relation of is $\xi(t)\propto(t-t^{*})$.
If $C=0$, Equations (11) give $dx/dt=0$ or $dt/dx=0$. This case shows that two
worldlines intersect normally at the bifurcation point, which is similar to
the case that $A=0$ and $C\neq 0$. We can obtain $\xi(t)=const,~{}v=0$.
It is obvious that vortices are relatively at rest when $\xi(t)=const$.
Through our topological current theory, we can get the dynamical scaling law.
Moreover, it worth to emphasizing that the dynamical scaling law of vortices,
only depend on topological properties of the order parameter field.
Figure 2: (a)The fractions of $6$-coordinated, $5$-coordinated, and
$7$-coordinated particles. At low temperature the number of defects is very
small. When the temperature above $0.500$, $N_{6}/N$ decreases rapidly.
We performed Brownian dynamics simulations on melting of two-dimensional
colloidal crystal in which particles interact with Yukawa potential Qi . We
have observed the mechanisms of defects in the two-dimensional Yukawa system
mv . Fig. 2 plots the fractions of $6$-coordinated, $5$-coordinated, and
$7$-coordinated particles. At low temperature, all particles are nearly six-
coordinated, and the number of defects is very small. When the temperature
reaches $0.500$, $N_{6}/N$ behaves in a rapidly decreasing. At $T^{*}=6.05$,
almost $20\%$ of particles are attached to the defects, which consisted with
the results in Ref. Ta. As the defects fraction rises above $30\%$, the system
melt into a liquid phase. At the low temperature, the paired dislocation is
formed or annihilated. With the temperature rises to the hexatic phase, the
dislocation pair comes into dissociated. In the solid phase, the unbinding of
dislocation is unstable, and these unbind dislocation will bind soon, whereas
the stable free dislocation is found in the hexatic phase. The above solutions
form topological current theory reveal the evolution of disclinations.
Besides the encountering of disclinations, splitting and merging of
disclinations are also included. When a multicharged disclinations, such as
$8$-coordinated particles, pass the bifurcation point, it may split into
several disclinations with lower value of Burgers vector along different
branch curves. On the other hand, several disclinations can merge into one
disclination at the bifurcation point. As before, since the topological
current of disclinations is identically conserved, the sum of the topological
charge of final disclinations must be equal to that the initial vortices at
the bifurcation point, i.e.
$\sum_{f}\beta_{lf}\eta_{lf}=\sum_{i}\beta_{li}\eta_{li}$ for fixed l. It
indicates that, vortices with a higher value of Burgers vector can evolve to
the lower value of Burgers vector, or vortices with a lower value of Burgers
vector can evolve to the higher value of Burgers vector through the
bifurcation process. But due to Such case was observed in our BD simulation in
Yukawa systems. That is why at the liquid phase the number of $5$-coordinated
particles is much more than $7$-coordinated particles due to the emergence of
the $8$-coordinated particles. The similar experimental result was observed in
melting of two-dimensional tunable-diameter colloidal crystals Han .
###### Acknowledgements.
We would like to thank Y. X. Liu for his helpful discussions. This work was
supported by the National Natural Science Foundation of China and by the SRF
for ROCS, SEM.
## References
* (1) J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).
* (2) D. R. Nelson and B. I. Halperin, Phys. Rev. B. 19, 2457 (1979); A. P. Young, Phys. Rev. B 19, 1855 (1979); K. J. Strandburg, Rev. Mod. Phys. 60, 161 (1988).
* (3) D. R. Nelson, Defects and Geometry in Condensed Matter Physics, (Cambridge University Press, Cambridge, 2002).
* (4) P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics, (Cambridge University Press, Cambridge, 1995).
* (5) C. A. Murry and D. H. Van Winkle, Phys. Rev. Lett. 58, 1200 (1987); Y. Tang, A. J. Armstrong, R. C. Mockler, and W. J. O’Sullivan, Phys. Rev. Lett. 62, 2401 (1989); A. H. Marcus and S. A. Rice, Phys. Rev. E 55, 637 (1997); H. H. von Grünberg, P. Keim, and G. Maret, Soft Matter ( Vol.3): Colloidal Order from Entropic and Surface Forces, Edited by G. Gompper and M. Schick, Wiley-VCH (2007).
* (6) K. Chen, T. Kaplan, and N. A. Clark, Phys. Rev. Lett. 74, 4019 (1995); K. J. Naidoo and J. Schnitker, J. Chem. Phys. 100, 3114 (1994).
* (7) W. K. Qi, S. M. Qin, X. Y. Zhao, and Y. Chen, J. Phys.: condensed Matter 20, 245102 (2008).
* (8) B. I. Halperin, Physics of Defects, edited by R. Balian et al. (North-Holland, Amsterdam, 1981).
* (9) F. Liu and G. F. Mazenko, Phys. Rev. B 46, 5963 (1992).
* (10) Y. S. Duan, H. Zhang, and L. B. Fu, Phys. Rev. E 59, 528 (1999).
* (11) H. Kleinert, Gauge Theory in Condensed Matter, (World Scientific, Singapore, 1989).
* (12) J. Zaanen, Z. Nussinov and S. I. Mukhin, Ann. Phys (New York) 310, 181 (2004); V. Cvetkovic, Z. Nussinov, S. Mukhin, and J. Zaanen, Europhys. Lett. 81, 27001 (2007).
* (13) Y. S. Duan and S. L. Zhang, Int. J. Eng. Sci. 29, 689 (1991); G. H. Yang and Y. S. Duan, Int. J. Theor. Phys. 37, 2371 (1998); Y. S. Duan and H. Zhang, Phys. Rev. E 60, 2568 (1999).
* (14) Y. S. Duan and H. Zhang, Eur. Phys. J. D 5, 47 (1999).
* (15) Y. S. Duan, H. Zhang and S. Li, Phys. Rev. B. 58, 125 (1998).
* (16) L. D. Landau and E. M. Lifshitz, Theory of Elasticity, (Pergamon, New York, 1970)
* (17) H. S. Seung and D. R. Nelson, Phys. Rev. A 38 1005 (1984).
* (18) W. K. Qi and Y. Chen, e-prints arXiv: 0809.0348.
* (19) T. Xu, Phys. Rev. E. 72, 036303 (2005).
* (20) A movie is archived at the website: $http://210.26.51.253/tiki/tiki-download_{f}ile.php?fileId=17$.
* (21) Y. Han, N. Y. Ha, A. M. Alsayed, and A. G. Yodh, Phys. Rev. E 77, 041406 (2008).
|
arxiv-papers
| 2008-09-24T06:56:10
|
2024-09-04T02:48:57.926323
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong Chen and Wei-Kai Qi",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/0809.4104"
}
|
0809.4248
|
# Searching for modifications to the exponential radioactive decay law with
the Cassini spacecraft
Peter. S. Cooper
Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A
###### Abstract
Data from the power output of the radioisotope thermoelectric generators
aboard the Cassini spacecraft are used to test the conjecture that small
deviations observed in terrestrial measurements of the exponential radioactive
decay law are correlated with the Earth-Sun distance. No significant
deviations from exponential decay are observed over a range of $0.7-1.6A.U.$ A
$90\%$ Cl upper limit of $0.84\times 10^{-4}$ is set on a term in the decay
rate of ${}^{238}Pu$ proportional to $1/R^{2}$ and $0.99\times 10^{-4}$ for a
term proportional to $1/R$.
PACS numbers: 23.60.+e, 23.40.-s, 95.55.Pe, 96.60.Vg, 0620.Jr.
A recent archive preprint reports evidence for a correlation between nuclear
decay rates and the Earth-Sun distanceFischbach . This correlation is
extracted from an annual modulation in the observed decay rates of
${}^{32}Si/^{36}Cl$, both $\beta$ emitters, and ${}^{226}Ra$, an $\alpha$
emitter. Reference 1 analyzes this as a correlation with $1/R^{2}(t)$, the
variation in the Earth-Sun distance due to the eccentricity of the Earth’s
orbit. To set a scale for this correlation the amplitude of the decay rate
variation is $\sim 0.1\%$ roughly in phase with the $3\%$ annual modulation in
$1/R^{2}(t)$ suggesting a $(3\times 10^{-2})/R^{2}$ term in the decay rate.
In the conclusion of this paper the authors observe that: _These conclusions
can be tested…_ [by] _measurements on radioactive samples carried aboard
spacecraft to other planets_ [which] _would be very useful since the sample-
Sun distance would vary over a much wider range_. I report here the results of
exactly such a measurement based on the power output of the Radioisotope
Thermoelectric Generators (RTG) aboard the Cassini spacecraft which launched
in 1997 and reached Saturn in 2004.
Cassini is powered by three RTGs each of which is a very large ($7.7Kg$,
$130KCu$) ${}^{238}Pu$ radioactive source, an $\alpha$ emitter with an $87.7y$
half lifeNNDC . The heat from these sources are converted to electric power
with thermoelectric piles. Together these sources produced $878w$ of
electrical power from $\sim 13Kw$ of radioactive decay heat at launch. The
power output of these RTGs were literally the lifeblood of the Cassini
mission. Their power output was monitored carefully and often.
Figure 1: Left, suppressed zero, scale: solid red curve, Heliocentric distance
[$R(t)(A.U.)$], Black points are the 5 times when R(t)=1 (green line), Right,
suppressed zero, scale: blue diamonds, RTG electrical power.
The trajectory of the Cassini spacecraft is available on the webTRAJ . I’ve
used these data, converted to astronomical units ($A.U.$), to compute
$1/R(t)^{2}=R_{e}^{2}/[x(t)^{2}+y(t)^{2}+z(t)^{2}]$. Over the first 2 years
Cassini went from R(0)=1 at launch, made 2 visits to Venus at R=0.7 and
crossed the orbit of the earth a total of 4 times before finally gaining
enough speed to reach Saturn at R=9.
JPL kindly providedJPL P(t), the total electrical power from the three RTGs
aboard, measured daily since launch and the expected power output from their
RTG modeling. The distance of Cassini from the Sun and the electrical power
output are plotted in Figure 1 for the first $2$ years of the mission. R
ranged from $0.7-1.6A.U.$ $(0.35<1/R^{2}<2.2)$. The power dropped from $878w$
at launch to $815w$ over this period.
The thermal power output of an RTG is directly proportional to the decay rate
of the radioisotope generating the heat: $P_{th}(t)=N_{0}\lambda exp(-\lambda
t)E_{d}$, where $E_{d}$ is the energy released per decay. The electrical power
output, $P(t)=P_{th}(t)\epsilon_{0}\epsilon(t)$ is modified by an initial
efficiency, $\epsilon_{0}$, and a time (or power) dependent thermoelectric
conversion efficiency; $\epsilon(t),\epsilon(0)=1$.
I cannot safely use the RTG efficiency modelDEGRA here since it assumed only
exponential behavior for radioactive decay. Any new physics effect might be
inadvertently subsumed into that model. The Cassini trajectory provides a
natural calibration for $\epsilon(t)$ using the measured power at the 5 points
(shown in Figure 1) where the spacecraft was $1A.U.$ from the sun. Fitting
these measurements to $P_{R=1}(t)=P(0)2^{(-t/87.7y)}2^{(-t/T_{eff})}$ yields
$T_{eff}=21.2\pm 1.9y$. This simple model agrees in shape with Reference 4 and
obeys the requirements of Carnot thermal efficiency: as the power decreases,
and the temperature difference across the thermoelectric piles decrease, the
efficiency can only decrease. Something as complicated as a space-born
thermoelectric pile requires more than one parameter to accurately describe
its behavior. The single exponential reduces a $5\%$ power drop in the first 2
years to a $\pm 1\%$ variation from unity.
Following Reference 1 I’ve plotted $\epsilon(t)U(t)=P(t)/P(0)2^{t/87.7y}$, the
normalized electrical power corrected for ${}^{238}Pu$ decay, and U(t), the
normalized thermal power, as a functions of time since launch in Figure 2 for
the first 2 years of Cassini’s voyage to Saturn.
Figure 2: RTG power; green open circles, decay corrected [$\epsilon(t)U(t)$],
black solid circles, efficiency and decay corrected [$U(t)$], cyan curve, 3rd
order polynomial correction function described the in text. red solid line,
expected effect extrapolated from Reference 1: $1-(1/R^{2}(t)-1)/30$.
To compare to the rough magnitude of the effect reported by reference 1 I’ve
plotted $U_{ref1}(t)=1+(0.1\%/3\%)(1/R(t)^{2}-1)$ on Figure 2 to extrapolate
the small R variation of reference 1 for comparison with the larger R range
available in this study. Extrapolating a $0.1\%$ decay rate change for a $3\%$
change in $1/R^{2}$ to a $50\%$ change in $1/R^{2}$ ($R=0.7A.U.$, $t=0.43y$)
should cause a $+4\%$ change in the power output of the RTG 5 months after
launch. In fact $U(t)$ decreases by $\sim 1\%$. Changes this large are
excluded by the Cassini efficiency corrected data both in magnitude and by the
absence of any reflection of the shape of $1/R^{2}(t)$ in either normalized
power curve.
In order to set quantitative limits I’ve fit the efficiency and half-life
corrected $U(t)$ normalized power data (suppressing the two obvious outlying
points in Figure 1) to $P_{3}(t)$, a 3th order polynomial in time, to
phenomenologically describe the last $1\%$ variation in the $U(t)$. As shown
in Figure 2, this polynomial smoothly interpolates the $U(t)$ measurements. A
3th order polynomial make a very poor fit to $1/R^{2}(t)$; these two shapes
are approximately orthogonal. The error assigned to each U(t) measurement by
requiring this fit to have $\chi^{2}/\nu=1$ is $0.0015$. The individual
relative power measurements have a resolution of $0.15\%$ and the polynomial
is a $\sim 6\sigma$ systematic correction beyond the simple exponential
efficiency model.
Figure 3: Black points; $\Delta(t)$, RTG normalized thermal power less a 3rd
order polynomial correction, Red curve; $10$ times the $90\%$ CL limit,
$10\alpha/R^{2}$, Blue curve; $10$ times the $90\%$ CL limit, $10\beta/R$.
The difference of the data from the polynomial fit; $\Delta(t)=U(t)-P_{3}(t)$
are plotted in Figure 3. Some structure at the $1\sigma$ level and some
outlying measurements remain. These difference data are fit to $\alpha/R^{2}$
and $\beta/R$ to give limits on the contribution of a term in the ${}^{238}Pu$
decay rate dependent on the Earth-Sun distance. The $90\%CL$ limit on from
these fits are $|\alpha|<0.84\times 10^{-4}$ and $|\beta|<0.99\times 10^{-4}$
respectively. The limiting functions, scale up by a factor of $10$ for
visibility, are also shown in Figure 3. $\alpha$ is to be compared with the
correlation seen in Reference 1 for ${}^{226}Rn$ decay of $\sim+3\times
10^{-2}$.
The Cassini RTG power data exclude any variation of the ${}^{238}Pu$ nuclear
decay rate correlated with the distance of the source from the Sun to a level
$350\times$ smaller than the effect reported by Reference 1. ${}^{238}Pu$ and
${}^{226}Ra$ are similar $\alpha$ emitters. Another physical or experimental
cause of the reported annual variations in nuclear decay rates appears to be
necessary. More generally Rutherford, Chadwick, and Ellis’s 1930 conclusion
that _The rate of transformation of an element has been found to be constant
under all conditions._Rutherford now has solid experimental support at least
from Venus (R=0.7) to Mars (R=1.5).
I am indebted to several of my colleagues for calling this paper and physics
issue to my attention; Chris Quigg and Martin Hu of Fermilab and Jurgen
Engelfried of the Universidad Autǿnoma de San Luis Potosí, Mexico. I am also
thankful for very useful lunchtime conversations on this subject with several
of my Fermilab colleagues. I an indebted to Richard Ewell and Torrence Johnson
of JPL for making the RTG data available and Stephen Parke of Fermilab for
critical comments on this manuscript.
## References
* (1) Jere H. Jenkins et al., _Evidence for Correlations Between Nuclear Decay Rates and the Earth-Sun Distance_ , arXiv:astro-ph:0808.3283v1.
* (2) National Nuclear Data Center, http://www.nndc.bnl.gov/.
* (3) http://www.lepp.cornell.edu/ seb/celestia/cassini-all.zip
* (4) private communication, R. Ewell, JPL.
* (5) R. Ewell, D. Hanks, J. Lozano, V. Shields, $\&$ E. Wood, _DEGRA - A computer Model for Predicting Long Term Thermoelectric Generator Performance_ , Space Technologies and Applications International Forum (STAIF), Albuquerque, New Mexico, February 12-16, 2005., [http://hdl.handle.net/2014/38760]
* (6) S. E. Rutherford, J. Chadwick, and C. Ellis, _Radiations from Radioactive Substances_ (Cambridge University Press, 1930).
|
arxiv-papers
| 2008-09-24T18:58:46
|
2024-09-04T02:48:57.931572
|
{
"license": "Public Domain",
"authors": "Peter S. Cooper",
"submitter": "Peter Cooper",
"url": "https://arxiv.org/abs/0809.4248"
}
|
0809.4258
|
# Borcherds-Kac-Moody Symmetry of $\CN=4$ Dyons
Miranda C. N. Cheng${}^{~{}1}$ and Atish Dabholkar${}^{~{}2,~{}3}$
1Jefferson Physical Laboratory,
Harvard University, Cambridge, MA 02138, USA
2Department of Theoretical Physics
Tata Institute of Fundamental Research
Homi Bhabha Rd, Mumbai 400 005, India
3Laboratoire de Physique Théorique et Hautes Energies (LPTHE)
Université Pierre et Marie Curie-Paris 6; CNRS UMR 7589
Tour 24-25, 5${}^{\grave{e}me}$ étage, Boite 126, 4 Place Jussieu
75252 Paris Cedex 05, France
###### Abstract:
We consider compactifications of heterotic string theory to four dimensions on
CHL orbifolds of the type $T^{6}/\mathbb{Z}_{N}$ with $\CN=4$ supersymmetry.
The exact partition functions of the quarter-BPS dyons in these models are
given in terms of genus-two Siegel modular forms. Only the $N=1,2,3$ models
satisfy a certain finiteness condition, and in these cases one can identify a
Borcherds-Kac-Moody superalgebra underlying the symmetry structure of the dyon
spectrum. We identify the real roots, and find that the corresponding Cartan
matrices exhaust a known classification. We show that the Siegel modular form
satisfies the Weyl denominator identity of the algebra, which enables the
determination of all root multiplicities. Furthermore, the Weyl group
determines the structure of wall-crossings and the attractor flows of the
theory. For $N>4$, no such interpretation appears to be possible.
black holes, superstrings, dyons
††preprint: TIFR/TH/08-36
## 1 Introduction
The spectrum of quarter-BPS dyons in various string theory compactifications
with ${\cal N}=4$ supersymmetry in four dimensions has revealed a surprisingly
rich structure [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. The first
surprise is of course that the exact degeneracy of these non-perturbative
states can be computed at all, especially given the fact that these dyons are
in general complicated bound states of not only D-branes but also NS5-branes,
KK-monopoles, strings and momenta. The second surprise is that the spectrum is
now known for all duality orbits of charges [14, 16, 17, 18, 19] at all points
in the moduli space, and not just in weakly coupled corners of the moduli
space. This is all the more surprising because the degeneracy of the quarter-
BPS states jumps across infinitely many “walls of marginal stability” in the
moduli space. Therefore, unlike in the case of half-BPS states, one cannot
naively analytically continue the weak coupling results to other regions in
the moduli space[13, 14, 15].
Such an exact, non-perturbative knowledge of the full quarter-BPS spectrum at
all points in the moduli space is sure to illuminate the structure of the
theory in unexpected ways. One may hope to learn something about the theory in
the middle of the moduli space where there is no weakly coupled description.
Already, it has allowed us to address a number of physical questions in far
greater detail than is possible for the ${\cal N}=2$ dyons. For example, from
the microscopic analysis of the exact partition function, one can analyze very
precisely the structure of the walls of marginal stability [13, 14, 15, 19]
and the sub-leading corrections to the entropy of the corresponding dyonic
black holes [20, 9]. These are found to be in beautiful agreement with the
independent macroscopic supergravity analysis [21, 22, 23, 20, 9, 13, 14, 15,
16, 17, 18].
One important motivation for studying the spectrum of dyons in such detail is
to explore whether there are deeper underlying structures. Indeed, duality
symmetries in field theory were first noticed in the dyon spectrum. In string
theory, the analysis of half-BPS states was already powerful enough to suggest
the intricate web of dualities. It is quite likely that the symmetries of
string theory are a further generalization of known symmetries such as gauge
invariance and general coordinate invariance, since duality relates different
string and brane charges to each other. Exact information about the spectrum
of non-perturbative dyons may suggest more concrete structures which might
enable us to develop the necessary tools to come to grips with such a
symmetry.
For the toroidally compactified heterotic string theory, the partition
function of BPS dyons [1] is given in terms of the so-called ‘Igusa cusp form’
which is the unique weight ten Siegel modular form of the group
$Sp(2,\mathbb{Z})$. It has been long known that the square root of the Igusa
form equals the denominator of a generalized (or Borcherds-) Kac-Moody
superalgebra [24]. Recently, a more physical interpretation of this symmetry
was found in the observation that the Weyl group of this superalgebra controls
the structure of wall-crossing and discrete attractor flows [25]. Furthermore,
a physical role of the full algebra was proposed in an equivalence between the
dyon degeneracy and a second-quantized multiplicity of a moduli-dependent
highest weight of the algebra [25].
Before drawing broader conclusions, it is important to know which of the
features mentioned above are of general validity and which are specific to
this very special model. To this end we study a larger class of models with
${\cal N}=4$ supersymmetry which are obtained as CHL orbifolds of the
toroidally compactified theory. Also in these models, the partition function
of BPS dyons is known exactly and is given similarly in terms of Siegel
modular forms of congruence subgroups of $Sp(2,\mathbb{Z})$ [6, 7, 8, 9, 10,
12]. The question we would like to address is whether these Siegel forms can
be related to the denominator of some generalized Borcherds-Kac-Moody
superalgebra and the whether the Weyl group of this algebra can be given a
physical interpretation in terms of attractor flows as in the case of the un-
orbifolded theory111This question has been addressed independently in a recent
paper [26]. However, the results reported here are different in essential
ways, particularly regarding the role of the Weyl group..
Generalized hyperbolic Kac-Moody algebras of similar type have made their
appearance in string theory in other related contexts. Harvey and Moore have
proposed that a generalized Kac-Moody superalgebra can be associated with BPS
states in general, and especially in the context of perturbative half-BPS
states [27, 28]. Another hyperbolic algebra $E_{10}$ has made its appearance
as the duality group of one-dimensional theory obtained from the M-theory
compactification on 10-torus. More generally, the role played by the Weyl
groups of hyperbolic Kac-Moody algebras in various gravitational theories in
the background close to a spacelike singularity has led to conjectures about
the presence of some hyperbolic Kac-Moody symmetries underlying these
(super-)gravity theories. See [29, 30, 31, 32, 33, 34] and references
therein222The Weyl groups which appear in these works are very similar to the
Weyl groups of the algebras discussed in the present paper. However, an
important difference is that the hyperbolic Kac-Moody algebras discussed there
are not Borcherds-Kac-Moody algebras, in that they do not have imaginary
simple roots., and also [35, 36] for earlier discussions of an infinite
dimensional gauge algebra underlying the toroidally compactified heterotic
string theory.
One difficulty in dealing with the hyperbolic algebras in general is that
while their formal structure can be defined in parallel with finite or affine
Lie algebras, very often it is not easy to determine all root multiplicities.
Without such a knowledge, it is hard to even begin to find any use of these
algebras. As we will see, the situation in the context of ${\cal N}=4$ dyons
is better. We will show explicitly that the Weyl denominator identity is
satisfied and provide a complete list of all roots with an algorithm for
computing the root multiplicities. We hope that this explicit construction of
the algebra, together with the analysis of its role in the BPS partition
function and in the physics of crossing the walls of marginal stability, will
be a step towards untangling further symmetries in the ${\cal N}=4$
superstring theories.
The paper is organized as follows. In section $\S$2, we summarize the
background and our results. In $\S$3 we perform the macroscopic supergravity
analysis of the walls of marginal stability and show that there is a
qualitative difference between ${\mathbb{Z}}_{\SS N}$ models with $N<4$ and
$N\geq 4$. For $N=1,2,3$, we determine the Weyl group, the simple real roots,
the real root part of the generalized Cartan matrix, Weyl group, and the Weyl
chamber of a Borcherds-Kac-Moody superalgebra associated with the dyon
spectrum. In $\S$4 we turn to the microscopic analysis and review the relevant
properties of the exact partition function of BPS dyons for the CHL as Siegel
modular forms and write down their expressions as infinite products and
infinite sums. $\S$5 contains some of the main results of the paper, where we
show that the automorphic form $\Phi_{t}$ can be interpreted as the
denominator in the Weyl-Kac character formula for the same algebra. In
particular, we show that the Weyl denominator identity is satisfied using the
sum and the product representations of this automorphic form. This enables us
to determine the full root system in principle. After constructing the
algebras we briefly summarize their physical significance. Finally we conclude
with a discussion in $\S$6.
## 2 Summary of Results
We consider compactifications of the heterotic string to four dimensions on
CHL orbifolds [37, 38, 39] of the type
$\frac{T^{5}\times S^{1}}{\mathbb{Z}_{\SS N}}\,,$ (1)
or equivalently their Type-II duals [40, 41], where the generator of the group
$\mathbb{Z}_{\SS N}$ acts on the heterotic theory by a $1/N$-shift along the
circle $S^{1}$ and an order-$N$ left-moving twist symmetry of the
$T^{5}$-compactified string. Now, the un-orbifolded toroidally compactified
theory has ${\cal N}=4$ supersymmetry and its massless spectrum consists of
one graviton multiplet together with $22$ vector multiplets. Since the
orbifold symmetry acts trivially on the right-moving fermions, all sixteen
supercharges are preserved by the orbifold. On the other hand, since the twist
symmetry acts nontrivially on the left-moving gauge degrees of freedom, some
of the $22$ vector multiplets will be projected out. Furthermore, because of
the $1/N$ shift, the twisted states have a $1/N$ fractional winding along the
circle and hence all twisted states are massive. As a result, the orbifolded
theory has fewer
$n_{v}=\frac{48}{N+1}-2$ (2)
massless vector multiplets. The CHL models of interest here are given by
$N=1,2,3,5,7$ and have $n_{v}=22,16,12,8,4$ vector multiplets, respectively.
The S-duality group of the $\mathbb{Z}_{\SS N}$ model is the following
congruence subgroup $\Gamma_{1}(N)$ of $PSL(2,\mathbb{Z})$
$\Gamma_{1}(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\Big{\lvert}\quad
ad-bc=1,\;\quad c=0\,\textrm{mod}\,N,\quad
a=1\,\textrm{mod}\,N\right\\}/\\{\pm{\mathds{1}}\\}\,.$ (3)
The T-duality group of this theory is a subgroup of
$O(n_{v},6;\mathbb{Z}).$ (4)
A dyonic charge of this theory is specified by a charge vector
$\begin{pmatrix}Q\\\ P\end{pmatrix}\,,$ (5)
where the components $Q$ and $P$ can be regarded as the electric and magnetic
charge vectors of the dyon respectively, each of which transforms in the
$(n_{v}+6)$-component vector representation of the $O(n_{v},6;{\mathbb{Z}})$
T-duality group.
The T-moduli live in the Narain moduli space and are parametrized by a
vierbein matrix $\mu$ which specifies the right-moving and left-moving
projections of lattice vectors like $Q$ and $P$. For example,
$Q_{R}^{a}=\mu^{a}_{I}Q^{I},\quad
Q_{L}^{\tilde{a}}=\mu^{\tilde{a}}_{I}Q^{I},\quad I=1,\ldots,6+n_{v},\quad
a=1,\ldots,6,\quad{\tilde{a}}=1,\ldots,n_{v},$ (6)
so that $Q^{2}=Q_{R}^{2}-Q_{L}^{2}$ as usual. Similarly, the S-moduli live on
the upper half-plane and are parametrized by the axion-dilaton field
$\lambda$.
The S-duality group (3) acts on the charges and the S-modulus $\lambda$ as
$\left(\begin{array}[]{c}Q\\\ P\\\
\end{array}\right)\rightarrow\left(\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}\right)\left(\begin{array}[]{c}Q\\\ P\\\
\end{array}\right)\;,\;\;\lambda\to\frac{a\lambda+b}{c\lambda+b}\;\;,\;\;\;\;\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{1}(N).$ (7)
One can check that restriction on the integers $a$ and $c$ in (3) arises from
requirement of preserving the lattice (9) [41, 42, 43, 6]. We will see later
how this S-duality group can be nicely combined with the parity-change
transformation into a larger “extended S-duality group”.
The degeneracy of BPS dyons in this class of theories is summarized by a
partition function which is the inverse of a Siegel modular form
$(\Phi_{t})^{2}$ of a congruence subgroup of $Sp(2,\mathbb{Z})$ of weight
$2t=\frac{24}{N+1}-2$. For the cases $N=1,2,3$ of most interest for our
purposes, the Siegel form has a square-root $\Phi_{t}$ which is a Siegel
modular form with integral weights $t=5,3,2$.
From the charge vector $\big{(}\begin{smallmatrix}Q\\\
P\end{smallmatrix}\big{)}$, the three quadratic T-duality invariants can be
organized as a $2\times 2$ symmetric matrix333Apart from the three quadratic
invariants listed here, the duality orbits of general ${\cal N}=4$ dyons can
depend on additional discrete invariants such as $I=\gcd{Q\wedge P}$ [14, 16,
17, 18]. We restrict our attention here to the case of $I=1$.
$\Lambda_{Q,P}=\begin{pmatrix}Q\cdot Q&Q\cdot P\\\ Q\cdot P&P\cdot
P\end{pmatrix}\;.$ (8)
For the twisted states, the T-duality invariants are quantized so that
$\Lambda_{Q,P}$ is of the form
$\left(\begin{array}[]{cc}2n/N&\ell\\\ \ell&2m\\\ \end{array}\right),$ (9)
with $n,m,\ell$ all integers.
Recall that the S-duality groups (3) are subgroups of $PSL(2,{\mathbb{Z}})$,
which is the same as the time-orientation preserving component
$SO^{+}(2,1;{\mathbb{Z}})$ of the Lorentz group acting on a Lorentzian space
${\mathbb{R}}^{2,1}$. Since the charge vector $\big{(}\begin{smallmatrix}Q\\\
P\end{smallmatrix}\big{)}$ transforms as a doublet in the spinor
representation of $SO(2,1)$, the vector $\Lambda_{Q,P}$ of T-duality
invariants transforms as a triplet in the vector representation of this
Lorentz group. Therefore, the matrices (8) with the quantization (9) form a
lattice in a Lorentzian space ${\mathbb{R}}^{2,1}$. One of our main results is
to show that, in some cases, this Lorentzian lattice spanned by all possible
T-duality invariants $\Lambda_{Q,P}$ can be interpreted as the root lattice of
a Borcherds-Kac-Moody superalgebra. In this respect, the models with $N<4$ and
$N\geq 4$ are qualitatively different both physically and mathematically, as
will be explained in detail in $\S$3. More specifically, only in the models
with $N<4$, the walls of marginal stability for the dyons partition the moduli
space into compartments bounded by finitely many walls.
### 2.1 Models with $N<4$
From the macroscopic supergravity analysis it is known that for the $N<4$
models, the walls divide the moduli space into connected domains each bounded
by a finite number of walls for a given set of total charges[13]. We will
combine these results with a microscopic analysis to identify a Borcherds-Kac-
Moody superalgebra with the following features.
* •
In $\S$3, we identify a special set of lattice vectors
$\\{\alpha_{i}^{\scriptscriptstyle{(N)}}\\}$ with $i=1,\ldots
r^{\scriptscriptstyle{(N)}}$ where $r^{\scriptscriptstyle{(N)}}=3,4,6$ for
$N=1,2,3$ respectively. These lattice vectors will eventually be identified as
the real simple roots of the relevant Borcherds-Kac-Moody algebra.
* •
For each model, the matrix of inner products of the simple real roots defines
a symmetric Cartan matrix
$A^{\scriptscriptstyle{(N)}}_{ij}=(\alpha_{i}^{\scriptscriptstyle{(N)}},\alpha_{j}^{\scriptscriptstyle{(N)}});\quad
i,j=1,\ldots r^{\scriptscriptstyle{(N)}}\,.$ (10)
The Cartan matrices in the three cases are given by
$A^{(1)}=\left(\begin{array}[]{rrr}2&-2&-2\\\ -2&2&-2\\\
-2&-2&2\end{array}\right),$ (11)
$A^{(2)}=\left(\begin{array}[]{rrrr}2&-2&-6&-2\\\ -2&2&-2&-6\\\ -6&-2&2&-2\\\
-2&-6&-2&2\end{array}\right),$ (12)
$A^{(3)}=\left(\begin{array}[]{rrrrrr}2&-2&-10&-14&-10&-2\\\
-2&2&-2&-10&-14&-10\\\ -10&-2&2&-2&-10&-14\\\ -14&-10&-2&2&-2&-10\\\
-10&-14&-10&-2&2&-2\\\ -2&-10&-14&-10&-2&2\end{array}\right),$ (13)
respectively in the three cases $N=1,2,3$.
* •
All three Cartan matrices are ‘hyperbolic’ in that they have one negative
eigenvalue. All have rank three. Now, rank three, hyperbolic Cartan matrices
have been classified by Gritsenko and Nikulin [44, 45] with two additional
‘niceness’ assumptions:
(a) that they are ‘elliptic’ in that the fundamental Weyl chamber has finite
volume,
(b) that they admit a lattice Weyl vector defined by (85).
The set of such hyperbolic Cartan matrices is a finite list and the three
Cartan matrices $A^{(1)},A^{(2)},A^{(3)}$ above correspond to
$A_{II,1},A_{II,2},A_{II,3}$ in this classification. Moreover, they are unique
in that these are the only Cartan matrices in the classification with all the
vertices of the Weyl chamber at infinity. In our context, the condition of
having all vertices at infinity follows from the physical consideration that
the walls of marginal stability of the theories under consideration have all
intersections at the boundary of the moduli space and they do not terminate in
the middle of the moduli space.
* •
The group of reflections with respect to the simple real roots defines the
Weyl group $W^{\scriptscriptstyle{(N)}}$ of the model for each $N$. As
mentioned earlier, for a given set of charges, the moduli space of these
models is divided into connected domains each bounded by a finite number of
walls. All domains can be mapped to the ‘fundamental Weyl chamber’ by the
action of this Weyl group. Physically, the ‘fundamental Weyl chamber’ is the
attractor region in the moduli space where no multi-centered solution which
can decay exists. From the number of simple real roots $r^{\SS(N)}=3,4,6$, we
see that the fundamental Weyl chamber mapped onto the Poincaré disk is
triangular, square, and hexagonal for $N=1,2,3$ respectively (Fig. 1).
* •
The Weyl group, generated by the reflections
$s_{i}^{\scriptscriptstyle{(N)}}:X\to
X-2\alpha_{i}^{\scriptscriptstyle{(N)}}\frac{(X,\alpha_{i}^{\scriptscriptstyle{(N)}})}{(\alpha_{i}^{\scriptscriptstyle{(N)}},\alpha_{i}^{\scriptscriptstyle{(N)}})}\;\;,\;\;i=1,\dotsi,r^{\SS(N)}\;,$
(14)
can be directly related to the physical symmetry group of the theory as
follows. The extended physical symmetry group of each $\mathbb{Z}_{N}$ model
is the following congruence subgroup of $PGL(2,{\mathbb{Z}})$
$\tilde{\Gamma}_{1}(N)=\bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{\lvert}\,ad-bc=\pm 1,c=0\text{ mod }N\;,a=\pm 1\text{ mod
}N\bigg{\\}}/\big{\\{}\pm{\mathds{1}}\big{\\}}\;$ (15)
which is generated by the ${\mathbb{Z}}_{2}$-symmetry of parity change and the
S-duality group. This physical symmetry group turns out to be also the
symmetry group of the root system of the relevant Borcherds-Kac-Moody algebra.
More precisely, it is related to the Weyl group as
$\tilde{\Gamma}_{1}(N)=W^{\scriptscriptstyle{(N)}}\rtimes\text{\small
Sym}({\cal W}^{\scriptscriptstyle{(N)}})\;\;,\;\;N=1,2,3\;,$ (16)
where $\text{\small Sym}({\cal W}^{\scriptscriptstyle{(N)}})$ is a finite
group of symmetries of the corresponding fundamental Weyl chamber preserving
the relevant lattice structure (9).
* •
We show that for each theory there exists a Weyl vector satisfying (85). On
the other hand, the dyon degeneracies of $N=1,2,3$ models are given in terms
of a genus two Siegel modular form $\Phi_{t}(\Omega)$ as in (69). With this
Weyl vector, the Siegel modular form $\Phi_{t}$ can be interpreted as the
denominator of a character of a representation of a Borcherds-Kac-Moody
superalgebra, and the Weyl vector is related to the level-matching condition
of the heterotic string. Moreover, the Weyl denominator identity is manifestly
satisfied by the Siegel modular form $\Phi_{t}$, as can be seen from its
product and sum representation. Some roots are bosonic and others are
fermionic and they lead to a large cancelation in the index. This identity
provides an algorithm for determining the multiplicities of both imaginary and
real roots of this BKM superalgebra. In $\S$5 we explicitly determine the root
multiplicities of some low-lying roots.
* •
Based on this analysis, we propose a microscopic model for the quarter-BPS
dyons for all three cases along the lines of [25]. The Weyl group of this
algebra is identified with the group of attractor flows. The dyon degeneracy
is given by the “second-quantized multiplicity” of a charge- and moduli-
dependent highest weight vector. Specifically, apart from being consistent
with the expected asymptotic growth of degeneracies in the large charge
regime, such a proposal predicts a jump in the (graded) dyon degeneracy when
the moduli cross a wall of marginal stability that precisely reproduces the
result from macroscopic analysis.
### 2.2 Models with $N\geq 4$
For the ${\mathbb{Z}}_{\scriptscriptstyle N}$ models with $N\geq 4$, the
situation is much less clear. From the macroscopic supergravity analysis, we
conclude that the walls are so severely removed due to the orbifold action
such that the partition of the moduli space by the walls of marginal stability
yields compartments which are not bounded by finitely many walls. On the other
hand, on the microscopic side, partition functions for quarter-BPS dyons have
been proposed for prime numbers $N=5,7$ in terms of Siegel modular forms [6].
From our analysis, it appears impossible to relate these partition functions
to the denominators of any generalized Kac-Moody algebra.
It is not clear to us how to interpret these results for $N\geq 4$. One
possibility is that in these cases, the symmetry is of an even more general
type. If this is the case, then the symmetry might be too general to be
particularly illuminating. A more radical possibility is that these models are
non-perturbatively inconsistent even though they are perfectly sensible as
perturbative orbifolds.
In any case, the $N<4$ models provide a class of models for which the BPS dyon
spectrum exhibits an immense symmetry. To construct and interpret the
Borcherds-Kac-Moody algebra for this class of models will be the focus of the
present paper.
## 3 Discrete Attractor Flows and Weyl Groups
In this section we first quickly review the analysis of the relationship
between the real roots of the algebra and the walls of marginal stability in
the un-orbifolded theory [25]. After introducing the setup, we redo the
supergravity analysis [13] of the walls of marginal stability for the CHL
models from a more group-theoretic and algebraic point of view. In particular
we study how the walls divide the future light-cone, and recover the result
that the moduli space is divided into adjoining domains each bounded by a
finite number of walls if and only if $N<4$. From the structure of the walls
we can then define the group of wall-crossings, generated by the reflections
with respect to the set of walls bounding one domain. As was argued in [25],
the attractor flow of the moduli fields in a black hole solution with the
given charges has to follow the ordering of such a group, and by construction
the full moduli dependence of the BPS spectrum is encoded in this “group of
discrete attractor flow”.
### 3.1 The Walls of Marginal Stability
To make the Lorentz group action of the duality group
$PSL(2,{\mathbb{Z}})\cong SO^{+}(2,1,{\mathbb{Z}})$ of the toroidally
compactified heterotic theory more manifest, we identify the Lorentzian space
${\mathbb{R}}^{2,1}$ with the space of $2\times 2$ symmetric matrices $X$ with
real entries equipped with the metric
$(X,X)=-2\,\text{\small det}X\;,$ (17)
and we choose the time-orientation such that the future light-cone is given by
$V^{+}=\left\\{X\Big{\lvert}\;X=\bigg{(}\begin{array}[]{ll}x^{+}&x\\\
x&x^{-}\end{array}\bigg{)},\;X=X^{T}\;,{\rm Tr}X>0\;,\text{\small
det}X>0\right\\}\;.$ (18)
Note that with $x^{\pm}=t\pm y$, the metric
$(X,X)=-2\det X=2(-t^{2}+x^{2}+y^{2})$ (19)
is indeed the usual Lorentzian metric in $\mathbb{R}^{2,1}$. The normalization
factor of $2$ is chosen consistent with the group theory convention that
simple roots have length-squared two. For later notational convenience we also
define $|X|^{2}=-(X,X)/2=\text{\small det}X$.
In this space, a discrete Lorentz transformation is given by
$X\to\gamma(X):=\gamma X\gamma^{T}$ (20)
for some $PSL(2,{\mathbb{Z}})$ matrix $\gamma$. The vector of T-duality
invariants is given by the symmetric $2\times 2$ matrix $\Lambda_{Q,P}$ (8).
When a charge vector transforms as
$\begin{pmatrix}Q\\\ P\end{pmatrix}\rightarrow\gamma\begin{pmatrix}Q\\\
P\end{pmatrix}\;,$ (21)
the matrix of T-duality invariants transforms as
$\Lambda_{Q,P}\rightarrow\gamma(\Lambda_{Q,P})\,.$
The length of this vector is related to the Bekenstein-Hawking entropy
$S(P,Q)=\pi|\Lambda_{Q,P}|=\pi\sqrt{Q^{2}P^{2}-(Q\cdot P)^{2}}$ (22)
of the black hole with charge $(Q,P)$. Note that half-BPS states correspond to
vectors on the boundary the future light-cone whereas quarter-BPS states
correspond to vectors in its interior.
Besides the charge vector $\Lambda_{Q,P}$, there are other vectors that live
naturally in the future light-cone. First, one can define the right-moving
charge vector
$\Lambda_{Q_{R},P_{R}}=\begin{pmatrix}Q_{R}\cdot Q_{R}&Q_{R}\cdot P_{R}\\\
Q_{R}\cdot P_{R}&P_{R}\cdot P_{R}\end{pmatrix}\;.$ (23)
which implicitly depends on the T-moduli $\mu$ through (6). One can then
define two natural vectors associated to the axion-dilaton S-moduli $\lambda$
and the T-moduli $\mu$ respectively by
${\cal
S}=\frac{1}{\mbox{Im}\lambda}\begin{pmatrix}|\lambda|^{2}&\mbox{Re}\lambda\\\
\mbox{Re}\lambda&1\end{pmatrix}\quad,\quad{\cal
T}=\frac{1}{{|\Lambda_{Q_{R},P_{R}}|}}\Lambda_{Q_{R},P_{R}}\;.$ (24)
Both matrices are normalized to unity $|X|^{2}=1$ and transform as
$X\to\gamma(X)$ under S-duality transformation (39). In terms of them, we can
construct the moduli-dependent ‘central charge vector’
${\mathcal{Z}}={\scriptsize\sqrt{|\Lambda_{Q_{R},P_{R}}|}}\,\big{(}{\cal
S}+{\cal T}\big{)},$ (25)
whose norm equals the BPS mass
$M_{Q,P}=|{\mathcal{Z}}|$ (26)
and whose orientation satisfies
${\mathcal{Z}}\lvert_{\text{attr.}}\sim\Lambda_{Q,P}$ at the attractor moduli
of the black hole of charge $(Q,P)$.
Because the central charge vector enters the BPS mass formula, it is probably
not surprising that it encodes all stability information of all potential
multi-centered configurations with the total charge $(Q,P)$ which are relevant
for the counting. One can show that these are the two-centered solutions with
each center carrying $1/2$-BPS charges [46] in the following form [13, 25]
$\displaystyle\begin{pmatrix}Q\\\ P\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}Q_{1}\\\
P_{1}\end{pmatrix}+\begin{pmatrix}Q_{2}\\\ P_{2}\end{pmatrix}$ (27)
$\displaystyle=$ $\displaystyle\pm\left\\{(aQ-bP)\begin{pmatrix}d\\\
c\end{pmatrix}+(-cQ+dP)\begin{pmatrix}b\\\
a\end{pmatrix}\right\\}\;\;,\;a,b,c,d\in{\mathbb{Z}},ad-bc=\pm 1\;.$
As promised, the stability condition on the moduli for the corresponding two-
centered solution to exist can be expressed solely in terms of the moduli
vector ${\mathcal{Z}}$:
$(\alpha,{\mathcal{Z}})(\alpha,\Lambda_{Q,P})<0$ (28)
with
$\alpha=\begin{pmatrix}2bd&ad+bc\\\ ad+bc&2ac\end{pmatrix}\;.$ (29)
The boundary of the above stability region is given by the marginal stability
condition $M_{Q,P}=M_{Q_{1},P_{1}}+M_{Q_{2},P_{2}}$, which can be rewritten as
$({\cal Z},\alpha)=0$ (30)
and which defines a surface of codimension one in the moduli space called the
“wall of marginal stability” for the two-centered solution with charges
$(Q_{1},P_{1})$ and $(Q_{2},P_{2})$444See also [47, 48, 49, 50] for studies of
the wall of marginal stability in a ${\cal N}=2$ setting.. In this form it is
manifest that none of these two-centered solutions exist when the moduli are
at their attractor values, since ${\mathcal{Z}}\sim\Lambda_{Q,P}$ and equation
(28) cannot be satisfied.
It was observed in [25] that the above vectors $\alpha$ can be identified with
the positive real roots of the underlying Borcherds-Kac-Moody algebra. Recall
that there are different choices of positive roots, or equivalently different
choices of simple roots in the root system, which one can make in a Lie
algebra and which are related to one another by a Weyl reflection. For
counting dyons, the choice is fixed in terms of the total charges by the
condition
$(\Lambda_{Q,P},\alpha)<0\;\;\text{for all positive roots
}\alpha\in\Delta_{+}\;,$ (31)
or equivalently that the charge vector $\Lambda_{Q,P}$ lies inside the
“fundamental Weyl chamber” 555Note that this is actually a condition on the
choice of real simple roots only, since the condition is automatically
satisfied as long as both vectors $\Lambda_{Q,P}$ and $\alpha$ are in the
future light-cone $V^{+}$. Unless otherwise noted, we will restrict our
attention to charges with a classical horizon ($\Lambda_{Q,P}\in V^{+}$) and
also exclude those charges which admit two-centered scaling solutions
($(\Lambda_{Q,P},\alpha)=0$ for some real root $\alpha$) in this paper.
${\cal W}=\left\\{X\Big{\lvert}\;(X,\alpha)\leq 0\text{ for all positive roots
}\alpha\right\\}\;.$ (32)
In a more abstract notation, the positive real roots $\alpha$ of the algebra
are in one-to-one correspondence with the two-centered solutions with the
charges given by
$\displaystyle\Lambda_{Q_{1},P_{1}}$ $\displaystyle=$ $\displaystyle
P_{\alpha}^{2}\,\alpha^{+}\quad,\quad\Lambda_{Q_{2},P_{2}}=Q_{\alpha}^{2}\,\alpha^{-}$
$\displaystyle\Lambda_{Q,P}$ $\displaystyle=$ $\displaystyle
P_{\alpha}^{2}\,\alpha^{+}+Q_{\alpha}^{2}\,\alpha^{-}-|(P\cdot
Q)_{\alpha}|\,\alpha\;,$ (33)
where the lightlike vectors
$\\{\alpha^{+},\alpha^{-}\\}=\Bigg{\\{}\begin{pmatrix}b^{2}&ab\\\
ab&a^{2}\end{pmatrix},\begin{pmatrix}d^{2}&cd\\\
cd&c^{2}\end{pmatrix}\Bigg{\\}}$ (34)
are determined as the vectors on the intersection of the light-cone with the
plane of marginal stability (30) and normalized to have inner product
$(\alpha^{+},\alpha^{-})=-1$.
To relate the walls of marginal stability in the moduli space to the objects
of the BKM algebra, recall that the Weyl group, which we will call $W$ (or
$W^{\SS(N)}$ for general $N>1$), is defined to be the group of reflections
with respect to the real roots and divides the Lorentzian root space into
different Weyl chambers. In particular, the fundamental Weyl chamber ${\cal
W}$ is defined as in (32). In our case, when we identify the Minkowski space
${\mathbb{R}}^{2,1}$ with the projected moduli space of the moduli vector
${\cal Z}$, the above statement means that the walls of the Weyl chambers are
exactly the physical walls of marginal stability and crossing a wall can be
described as a Weyl reflection with respect to the positive root corresponding
to the two-centered configuration in question. Specifically, the fundamental
Weyl chamber ${\cal W}$ is the so-called “attractor region” where no relevant
two-centered configuration exists, and an attractor flow can be identified
with a series of Weyl reflections from elsewhere in the future light-cone to
${\cal W}$.
Another physical way to understand the appearance of the Weyl group lies in
its relation to the physical duality group. The walls of marginal stability
cuts the future light-cone into different domains bounded by sets of, in the
present un-orbifolded case, three walls. Put differently, when projecting the
future light-cone onto a constant-length hyperboloid, the Weyl group gives a
triangular tessellation of the Poincaré disk in which the Weyl chamber ${\cal
W}$ is the triangle bounded by three planes of orthogonality to the three
simple real roots of the algebra, whose matrix of inner products (11) is the
real root part of Cartan matrix of the Borcherds-Kac-Moody algebra. See Figure
1.
Then we find that the full symmetry group of the root system is
$PGL(2,{\mathbb{Z}})=W\rtimes\text{\small Sym}({\cal W})\;,$ where
$\text{\small Sym}({\cal W})$, the symmetry group of the fundamental domain
which leaves the appropriate lattice structure intact, is in this case the
symmetry group $D_{3}$ of a regular triangle. On the other hand, the group
$PGL(2,{\mathbb{Z}})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{\lvert}\;\;ad-bc=\pm
1\right\\}/\big{\\{}\pm{\mathds{1}}\big{\\}}$ (35)
is nothing but the duality group $PSL(2,{\mathbb{Z}})$ discussed before, now
extended with the spacetime parity change operation generated by
$R:\,\begin{pmatrix}Q\\\ P\end{pmatrix}\to\begin{pmatrix}Q\\\
-P\end{pmatrix}\;\;,\;\;\lambda\to-\overline{\lambda}$ (36)
and acts on the charges and the heterotic axion-dilaton as
$\displaystyle\begin{pmatrix}Q\\\ P\end{pmatrix}\to\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}Q\\\ P\end{pmatrix}$
$\displaystyle\lambda\to\left\\{\begin{array}[]{ll}\displaystyle\frac{a\lambda+b}{c\lambda+d}&\;\;\text{when}\;\;ad-
bc=1\\\\[11.38109pt]
\displaystyle\frac{a\overline{\lambda}+b}{c\overline{\lambda}+d}&\;\;\text{when}\;\;ad-
bc=-1\end{array}\right.\ .$ (39)
This concludes our review of the role of the positive real roots, the Weyl
group, and the Weyl chamber of a Borcherds-Kac-Moody algebra in the un-
orbifolded theory.
### 3.2 Three Arguments for Two Classes of Theories
For the $N=1$ case discussed in [25], the set of three simple real roots which
gives the Cartan matrix (11) can be chosen to be
$\alpha_{1}=\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix},\;\;\alpha_{2}=\begin{pmatrix}2&1\\\
1&0\end{pmatrix},\;\;\alpha_{3}=\begin{pmatrix}0&1\\\ 1&2\end{pmatrix}\;.$
(40)
The corresponding positive real roots take the form
$\alpha=\begin{pmatrix}2n&\ell\\\
\ell&2m\end{pmatrix},\;(\alpha,\alpha)=2\;\;,\;\;(n,m,\ell)>0$ (41)
where the condition $(n,m,\ell)>0$ on the integers means $n,m\geq 0$,
$\ell\in{\mathbb{Z}}$ and $\ell<0$ when $n=m=0$. From the condition that the
charge vector $\Lambda_{Q,P}$ has to lie in the fundamental Weyl chamber (32),
this choice of the simple roots is actually only suitable for dyon charges
whose T-duality invariants satisfy $Q^{2},P^{2}>Q\cdot P>0$. But from the
relation between the Weyl group and the physical duality group (16), we see
that it is always possible to map the charges such that the above set of
simple roots is the right choice for the charges.
When we take the orbifold, because the T-duality invariants are now quantized
as (9) due to the presence of the twisted states, not all the splits of
charges in (27) are allowed for forming a two-centered solution. Instead we
have to restrict ourselves to elements of the following congruence subgroup of
$PGL(2,{\mathbb{Z}})$
$\tilde{\Gamma}_{0}(N)=\bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{\lvert}\,ad-bc=\pm 1,c=0\text{ mod
}N\;\bigg{\\}}/\big{\\{}\pm{\mathds{1}}\big{\\}}\;.$ (42)
Notice that there is an extra constraint on the group element for the extended
S-duality group $\tilde{\Gamma}_{1}(N)$ since it has to leave invariant the
spacing of the lattice (9) of all possible vectors $\Lambda_{Q,P}$ of
T-duality invariants. Only for $N<4$ these two groups $\tilde{\Gamma}_{0}(N)$
and $\tilde{\Gamma}_{1}(N)$ happen to be the same.
From the relation between the split of charges and the root (29), the charge
quantization imposes that the roots relevant for the wall-crossing must be of
the form
$\alpha^{\scriptscriptstyle{(N)}}=\begin{pmatrix}2*&*\\\
*&2N*\end{pmatrix}\;\;,\;\;(\alpha,\alpha)=2\;.$ (43)
Therefore, we expect that the “positive real roots” of the ${\mathbb{Z}}_{\SS
N}$-orbifolded theory will be of the form666To avoid proliferation of
notations and names, we are already using in our macroscopic analysis the
terminology for the microscopic algebra which will be defined in later
sections.
$\alpha^{\scriptscriptstyle{(N)}}=\begin{pmatrix}2n&\ell\\\
\ell&2m\end{pmatrix},\;(\alpha,\alpha)=2\;,\;\;(n,m,\ell)>0\;,m=0\text{ mod
}N\;\;\;.$ (44)
From this we immediately see that, of the three simple roots of the un-
orbifolded theory (40), two of them $\alpha_{1,2}$ always survive the
orbifolding while the third one gets removed whenever $N>1$. In other words,
if we denote $\\{\alpha_{i}^{\scriptscriptstyle{(N)}}\\}$ the set of “simple
real roots” whose walls of orthogonality bound a domain in the moduli space,
then we have
$\alpha_{1}^{\scriptscriptstyle{(N)}}=\alpha_{1}\;\;,\;\;\alpha_{2}^{\scriptscriptstyle{(N)}}=\alpha_{2}\;\;\text{
for all }N\;.$ (45)
A particular symmetry of the positive real roots will help us to find the rest
of the simple roots corresponding to the other walls which bound a domain in
the moduli space together with the two walls $(\alpha_{1,2},{\cal Z})=0$ found
above. Notice that there is a special element of the $\tilde{\Gamma}_{1}(N)$
group
$\gamma^{\scriptscriptstyle{(N)}}=\begin{pmatrix}1&-1\\\
N&1-N\end{pmatrix}\;,$ (46)
which has the property that when it acts on a positive root of the form (44),
the image is again a positive root satisfying the conditions in (44). This
means that this transformation permutes simple roots among themselves and must
therefore be a symmetry of the fundamental Weyl chamber ${\cal
W}^{{\scriptscriptstyle(N)}}$. For example, for $N=1$ one can easily check
that $\gamma^{(1)}$ acts as a permutation of the three simple roots
$\alpha_{1},\alpha_{2},\alpha_{3}$ and is indeed a symmetry of the simple root
system.
Now we are ready to show how the existence of such a symmetry predicts that
the walls of marginal stability fail to partition the moduli space into finite
compartments when $N>3$. Concretely, the existence of this symmetry implies
that
$\gamma^{{\scriptscriptstyle(N)}}(\alpha_{1}),\gamma^{{\scriptscriptstyle(N)}}(\alpha_{2})\in\\{\alpha^{{\scriptscriptstyle(N)}}_{i}\\}\;,$
(47)
so a necessary condition for the set of walls bounding the fundamental Weyl
chamber to be finite is that
$(\gamma^{{\scriptscriptstyle(N)}})^{k}=\mathds{1}$ for some finite integer
$k$. There are many ways to see that it is not the case for $N>3$. We will
sketch three ways to understand this which give us different insights into the
topology of the walls of marginal stability of the CHL theories.
#### 3.2.1 A Group Theoretic Argument
Recall that the group $PSL(2,{\mathbb{Z}})$ is generated by the two generators
$S$ and $ST$, and the only relations among them are
$S^{2}=(ST)^{3}={\mathds{1}}\;.$ (48)
For convenience let us write $B=ST$. Now observe that
$\gamma^{{\scriptscriptstyle(N)}}=(SB^{2})^{N-1}SBS=(SB^{2})^{N-1}S(SB^{2})^{-1}$
(49)
and thus
$\displaystyle(\gamma^{{\scriptscriptstyle(N)}})^{k}$ $\displaystyle=$
$\displaystyle(SB^{2})^{N-1}[S(SB^{2})^{N-2}]^{k-1}S(SB^{2})^{-1}$ (50)
$\displaystyle=$
$\displaystyle(SB^{2})^{N-1}[B^{2}(SB^{2})^{N-3}]^{k-1}S(SB^{2})^{-1}\;,$ (51)
from this we immediately see that
$(\gamma^{{\scriptscriptstyle(2)}})^{2}=(\gamma^{{\scriptscriptstyle(3)}})^{3}={\mathds{1}}\;,$
(52)
but there is no $k<\infty$ such that
$(\gamma^{{\scriptscriptstyle(N)}})^{k}={\mathds{1}}$ for any $N\geq 4$ .
#### 3.2.2 A Geometric Argument
Recall that the matrix $\gamma=\big{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\big{)}$ defines a Möbius transformation on the Riemann
sphere
$z\to\frac{az+b}{cz+d}\;.$ (53)
By inspecting the eigenvalues of $\gamma^{{\scriptscriptstyle(N)}}$
$\gamma^{{\scriptscriptstyle(N)}}\xrightarrow[\begin{subarray}{c}similarity\\\
transf.\end{subarray}]{}\begin{pmatrix}\zeta&0\\\
0&\zeta^{-1}\end{pmatrix}\;,$ (54)
it is easy to see that the corresponding Möbius transformation is elliptic,
parabolic, or hyperbolic when
$({\rm
Tr}\gamma^{{\scriptscriptstyle(N)}})^{2}=(N-2)^{2}\;\;\begin{cases}<4\\\ =4\\\
>4\end{cases}\;.$ (55)
In other words, when viewed as a Möbius transformation of the Riemann sphere,
$\gamma^{\SS(N)}$ is conjugate to
$z\to\begin{cases}e^{i\theta}z\\\ z+\theta\\\
e^{\theta}z\end{cases}\;\;\;\text{when}\;\;\;N\;\;\begin{cases}<4\\\ =4\\\
>4\end{cases}\;\text{ for some }\theta\in{\mathbb{R}}\;,$ (56)
and clearly does not return to itself whenever $N\geq 4$.
#### 3.2.3 An Arithmetic Argument
In the un-orbifolded theory, the relevant two-centered solutions (27) are
given by positive roots of the form (29), or equivalently the lightlike
vectors $\alpha^{\pm}$ given in (34). Therefore, with each wall of marginal
stability one can uniquely associate a pair of rational numbers of the
following form $\\{\textstyle{\frac{b}{a},\frac{d}{c}}\\}$, with the
normalization fixed by the condition that $a,b,c,d$ being integers from which
$a$ and $c$ are nonnegative and satisfying $ad-bc=1$[25].
This corresponds to compactifying the real line by identifying $+\infty$ with
$-\infty$ into a circle, which is then identified with the boundary of the
Poincaré disk, or equivalently the boundary of the future light-cone $V^{+}$
before projecting it onto a constant-length slice. Then the walls of marginal
stability of the $N=1$ theory are in one-to-one correspondence with the
geodesic lines connecting two neighboring rational numbers in the so-called
Stern-Brocot tree
$\begin{array}[]{ccccccccccccc}\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\frac{-1}{0}&\frac{-2}{1}&\frac{-1}{1}&\frac{-1}{2}&\frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\\
&&&&&&&&&&&&\\\
\frac{-1}{0}&&\frac{-1}{1}&&\frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\\
&&&&&&&&&&&&\\\
\frac{-1}{0}&&&&\frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\end{array}$ (57)
which is formed by successively taking the “mediant” $\frac{b+d}{a+c}$ of the
previous pair of rationals $\\{\textstyle{\frac{b}{a},\frac{d}{c}}\\}$
starting from $\frac{\pm 1}{0}$, $\frac{0}{1}$, and which contains all the
rational numbers. For example, the fundamental Weyl chamber in the un-
orbifolded theory is bounded by walls connecting the three rational numbers
$\frac{\pm 1}{0}$, $\frac{0}{1},\frac{1}{1}$. See Figure 1.
From this point of view of the rational numbers, the effect of orbifold is
that some lines connecting certain pairs of the rationals will be removed by
the orbifold and the fundamental Weyl chamber will correspond to a finer
spacing of the real line by the rational numbers, such that the walls
correspond to the lines connecting two pairs of rational numbers
$\\{\textstyle{\frac{b}{a},\frac{d}{c}}\\}$ with the property that the product
of the two denominators are divisible by $N$. Namely, the walls of the
${\mathbb{Z}}_{\SS N}$ theory will correspond to lines connecting pairs of
rational numbers of the form
$\Big{\\{}\frac{b}{a},\frac{d}{c}\Big{\\}}\quad,\quad ad-bc=1\quad,\quad
a,c\geq 0,\;ac=0\text{ mod }N\;.$ (58)
In particular, the fact that $\alpha_{1}$, $\alpha_{2}$ are left untouched
while $\alpha_{3}$, corresponding to the pair
$\\{\textstyle{\frac{0}{1},\frac{1}{1}}\\}$, is removed for all $N>1$, is
translated to the statement that the fundamental Weyl chamber will correspond
to a finer spacing of the segment $[0,1]\in{\mathbb{R}}$ of the real line.
Therefore we will now focus on the following middle part of the Stern-Brocot
tree starting from 0 and ending at ${1}$
$\begin{array}[]{ccccc}\vdots&\vdots&\vdots&\vdots&\vdots\\\
\frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}\\\ &&&&\\\
\frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}\\\ &&&&\\\
\frac{0}{1}&&&&\frac{1}{1}\end{array}\;.$ (59)
From the symmetry consideration, there is another thing we can say about the
division of the segment $[0,1]\in{\mathbb{R}}$ by the walls of a fundamental
Weyl chamber ${\cal W}^{{\scriptscriptstyle(N)}}$. That is, we expect the
fundamental Weyl chamber to be a regular polygon, or to say that the Gram
matrix $A^{\SS(N)}$ of inner products of the simple roots should be invariant
under a cyclic permutation of them. This will correspond to a row of in the
above tree (59). For example, as we will see later, the second row
$\\{\frac{0}{1},\frac{1}{2},\frac{1}{1}\\}$ and the third row
$\\{\frac{0}{1},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{1}{1}\\}$ will give
us the real simple roots of the $N=2$ and $N=3$ model respectively. But there
is no such a row (58) with finitely many elements such that the neighboring
rational numbers all have the product of their denominators divisible by $N$,
when $N\geq 4$. This excludes the existence of a domain in the moduli space
bounded by a finite number of walls in these theories777According to our
macroscopic analysis, for the $N=4$ model the walls of marginal stability
divide the moduli space into domains bounded by an infinite number of walls
which are related to one another by some S-duality transformation. Partition
function for the dyons in such a model has been proposed in [11] but the
analysis is complicated by the fact that $N$ is not a prime in this case and
we will not discuss it in the present paper..
### 3.3 The Finite Cases
We now focus on the $N<4$ models in which the walls of marginal stability
partition the moduli space into compartments each bounded by a finite number
of walls. We would like to identify the real simple roots, the Weyl group and
the fundamental Weyl chamber of the relevant algebra which will be shown to
generate the BPS dyon spectrum of the theory.
Figure 1: The structure of walls when projected onto the Poincaré disk. The
fundamental Weyl Chamber is bounded by three walls labeled by
$\alpha_{1},\alpha_{2},\alpha_{3}$ for $N=1$; by four walls labeled by
$\alpha_{1},\alpha_{2},\alpha^{(2)}_{3},\alpha^{(2)}_{4}$ for $N=2$; and by
six walls labeled by
$\alpha_{1},\alpha_{2},\alpha^{(3)}_{3},\alpha^{(3)}_{4},\alpha^{(3)}_{5},\alpha^{(3)}_{6}$
for $N=3$. Recall that each triangle is really equivalent and any triangle can
be mapped to the central one by some conformal transformation.
#### 3.3.1 $N=1$
This case was analyzed in [25], where it was found that three simple roots
given by (40) and the physical extended S-duality group is also the symmetry
group of the root system, which is related to the Weyl group as (16).
#### 3.3.2 $N=2$
As mentioned earlier, the simplest way to read off the real simple roots for a
given $N$ is to read them off from the relevant part of the Stern-Brocot tree
of rational numbers. From (29) and the second row in (59) we see that, apart
from $\alpha^{\scriptscriptstyle(2)}_{1}=\alpha_{1}$ and
$\alpha^{\scriptscriptstyle(2)}_{2}=\alpha_{2}$ as in the un-orbifolded case
in (40), there are two other real simple roots
$\alpha_{3}^{(2)}=\begin{pmatrix}2&3\\\
3&4\end{pmatrix}\;\;,\;\;\alpha_{4}^{(2)}=\begin{pmatrix}0&1\\\
1&4\end{pmatrix}\;.$ (60)
One can then verify that their inner product matrix, which will turn out to be
the real root part of the generalized Cartan matrix of the algebra we are
going to define in the next section, is given by (12). The fundamental Weyl
chamber, defined by (32), is therefore a regular square when projected on the
Poincaré disk. See Figure 1.
By our construction, the Weyl group $W^{(2)}$ again plays a physical role as
the group of wall-crossings or the group of the discrete attractor flow of the
theory. One can verify that the four generators of the corresponding Weyl
group $W^{(2)}$ (14) have no relations among themselves other than that
$(s_{i}^{\scriptscriptstyle{(2)}})^{2}={\mathds{1}}$. This is an important
consistency condition since this property ensures that there is a unique
“shortest-path ordering”, the so-called weak Bruhat ordering, among the group
elements, a property that an attractor flow always exhibits. See [25] for
details about the ordering and its precise relationship to the attractor flow.
The S-duality group of the theory is related to the Weyl group in a similar
way as in the un-orbifolded case (16), namely
$\tilde{\Gamma}_{1}(2)=W^{\scriptscriptstyle{(2)}}\rtimes\text{\small
Sym}({\cal W}^{\scriptscriptstyle{(2)}})\;,$ (61)
where the group $\text{\small Sym}({\cal W}^{\scriptscriptstyle{(2)}})$ is the
group of symmetry of the fundamental square ${\cal W}^{\SS(2)}$ which is
compatible with the lattice structure (9). In this case it is the four-element
group generated by the order-two reflection element
$TR:\alpha^{\scriptscriptstyle{(2)}}_{i}\leftrightarrow\alpha^{\scriptscriptstyle{(2)}}_{3-i\text{
mod }4}$ (62)
and the order-two next-next-neighbor rotation
$\gamma^{\scriptscriptstyle{(2)}}:\alpha^{\scriptscriptstyle{(2)}}_{i}\to\alpha^{\scriptscriptstyle{(2)}}_{i-2\text{
mod }4}\;.$ (63)
#### 3.3.3 $N=3$
A very similar story holds for the $N=3$ case. For the ${\mathbb{Z}}_{3}$
orbifold theory, the third row of the Stern-Brocot tree gives us six simple
real roots (59). Apart from $\alpha^{\scriptscriptstyle(3)}_{1}=\alpha_{1}$
and $\alpha^{\scriptscriptstyle(3)}_{2}=\alpha_{2}$, they are
$\displaystyle\alpha_{3}^{(3)}$ $\displaystyle=\begin{pmatrix}4&5\\\
5&6\end{pmatrix}\;\;,\;\;\;\alpha_{4}^{(3)}$
$\displaystyle=\begin{pmatrix}4&7\\\ 7&12\end{pmatrix}$ (64)
$\displaystyle\alpha_{5}^{(3)}$ $\displaystyle=\begin{pmatrix}2&5\\\
5&12\end{pmatrix}\;\;,\;\;\;\alpha_{6}^{(3)}$
$\displaystyle=\begin{pmatrix}0&1\\\ 1&6\end{pmatrix}\;.$
Their matrix of the inner products is given as (13). From this we can see that
they define a fundamental Weyl chamber ${\cal W}^{\scriptscriptstyle(3)}$
which is a regular hexagon, and a Weyl group $W^{\scriptscriptstyle(3)}$ which
is generated by six generators $s_{i}^{\scriptscriptstyle(3)}$ with no
relations other than $(s_{i}^{\scriptscriptstyle(3)})^{2}={\mathds{1}}$. See
Figure 1. The S-duality group of the theory is
$\tilde{\Gamma}_{1}(3)=W^{\scriptscriptstyle{(3)}}\rtimes\text{\small
Sym}({\cal W}^{\scriptscriptstyle{(3)}})\;,$ (65)
where the group $\text{\small Sym}({\cal W}^{\scriptscriptstyle{(3)}})$ is the
group of symmetry of the fundamental hexagon, generated by again the order-two
reflection element
$TR:\alpha^{\scriptscriptstyle{(3)}}_{i}\leftrightarrow\alpha^{\scriptscriptstyle{(3)}}_{3-i\text{
mod }6}$ (66)
and the order-three next-next-neighbor rotation
$\gamma^{\scriptscriptstyle{(3)}}:\alpha^{\scriptscriptstyle{(3)}}_{i}\to\alpha^{\scriptscriptstyle{(3)}}_{i-2\text{
mod }6}\;.$ (67)
## 4 Exact Partition Function for CHL Dyons
In this section, we will start by introducing the partition function for
supersymmetric dyons in the ${\mathbb{Z}}_{\SS N}$ CHL models with $N=1,2,3$
as proposed in [1, 6]. We then see how they are related to an automorphic form
of weight
$t=\frac{12}{N+1}-1$ (68)
of a congruence subgroup $G_{0}(N)$ of the modular group $Sp(2,{\mathbb{Z}})$.
In the second subsection we give explicit expressions for this automorphic
form $\Phi_{t}(\Omega)$ both as an infinite product and as an infinite Fourier
sum. These properties will enable us to construct a Borcherds-Kac-Moody
algebra for each $N$ using these automorphic forms.
### 4.1 The Dyon Partition Function
In a $\mathbb{Z}_{\SS N}$ CHL model, the degeneracy of dyons with given
charges $(Q,P)$ and at a given point of the moduli space, whose coordinates
are given by $\mu$ and $\lambda$, is given by the following contour integral
[1, 6]
$D^{\SS{(N)}}(Q,P)\lvert_{\mu,\lambda}\,=(-1)^{Q\cdot
P+1}\frac{1}{N}\,\oint_{\cal C}d\Omega\,\frac{e^{\pi
i(\Lambda_{Q,P}.\Omega)}}{(\Phi_{t}(\Omega))^{2}}\quad,\quad\Omega=\begin{pmatrix}\rho&\nu\\\
\nu&\sigma\end{pmatrix}$ (69)
The integration contour is given by [15]
${\cal C}={\cal
C}(Q,P)\lvert_{\mu,\lambda}=\\{\mathrm{Im}\Omega=\varepsilon^{-1}{\cal
Z},\,0\leq\mathrm{Re}\rho,\tfrac{1}{N}\mathrm{Re}\sigma,\mathrm{Re}\nu<1\\},$
(70)
where $\varepsilon\ll 1$ is any small positive number playing the role of a
regulator. For a given set of charges, the contour depends on the moduli
$\mu,\lambda$ through the definition of the central charge vector (25). A
remarkable property of the spectrum of ${\cal N}=4$ dyons is that the entire
moduli dependence of the degeneracy is captured completely by the moduli
dependence of the choice of the contour.
To understand the properties of the above partition function, let us recall a
few facts about the congruence subgroups of $Sp(2,\mathbb{Z})$ and their
Siegel modular forms. By definition, an $Sp(2,\mathbb{Z})$ element can be
represented by a $(4\times 4)$ matrix which leaves invariant the symplectic
form
$J=\begin{pmatrix}0&-{\mathds{1}}\\\ {\mathds{1}}&0\end{pmatrix}\;.$ (71)
When expressed in terms of ($2\times 2$) blocks, they are
$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\;\;\;,\text{
with}\;\;\;\;AB^{T}=BA^{T},\;CD^{T}=DC^{T},\;AD^{T}-BC^{T}=\mathds{1}$ (72)
and they act on the matrix of chemical potentials for the charge vector
$\Lambda_{Q,P}$, or the “period matrix”, as
$\Omega\to(A\Omega+B)(C\Omega+D)^{-1}\;.$ (73)
The following subgroup of $Sp(2,\mathbb{Z})$, which we denote by $G_{0}(N)$,
will be of special interest to us. In terms of matrices, these are elements
satisfying the extra condition that they have the form
$U_{0}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}U_{0}^{-1}\;\;,\;\;\;C=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}\text{ mod }N\;,$ (74)
where
$U_{0}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ 0&-1&0&0\end{pmatrix}$
(75)
can be thought of an $Sp(2,{\mathbb{Z}})$ counterpart of
$S=\big{(}\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\big{)}$ in
$SL(2,{\mathbb{Z}})\sim Sp(1,{\mathbb{Z}})$. A special family of elements in
$G_{0}(N)$ is given when one takes
$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}a&0&0&b\\\ 0&a&b&0\\\
0&c&d&0\\\ c&0&0&d\end{pmatrix}\;,\;\;\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{0}(N)\;,$ (76)
then one can easily check that the corresponding $G_{0}(N)$ element acts on
the period matrix as
$\Omega\to\gamma\,\Omega\,\gamma^{T}$ (77)
and this gives a natural embedding of $\Gamma_{0}(N)\subset
SL(2,{\mathbb{Z}})$ into $G_{0}(N)\subset Sp(2,{\mathbb{Z}})$.
For the partition function to converge, the matrix $\Omega$ has to lie in the
so-called Siegel upper-half plane, given by the condition $\mbox{Im}\Omega\in
V^{+}$. Furthermore, the object $\Phi_{t}(\Omega)$ turns out to be an
automorphic form of weight $t$ under the subgroup $G_{0}(N)$ of
$Sp(2,{\mathbb{Z}})$, namely they transform as
$\Phi_{t}[(A\Omega+B)(C\Omega+D)^{-1}]=\pm\\{\det{(C\Omega+D)}\\}^{t}\,\Phi_{t}(\Omega)$
(78)
when $\big{(}\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\big{)}\in
G_{0}(N)$. In particular, it is independently invariant under the following
three transformations
$\rho\to\rho+1\;,\;\;\nu\to\nu+1\;,\;\;\sigma\to\sigma+N\;.$ (79)
This invariance explains the real part of the contour of integration in (70).
This mathematical property of the automorphism of the partition function turns
out to be closely related to the physics of crossing the walls of marginal
stability in the moduli space. The degeneracy formula as we wrote them in
(69), is moduli-dependent through the moduli dependence of the contour of
integration. With such a contour prescription (70), changing the moduli causes
a deformation of the contour. As long as the contour does not encounter a pole
of the partition function, the degeneracy does not change and hence is a
smooth function in a region of moduli space. However, the partition function
$1/(\Phi_{t})^{2}$ is not entirely holomorphic and has poles in the Siegel
upper-half plane. It turns out that the class of poles that one can hit when
deforming the contour are in one-to-one correspondence with the two-centered
solutions that can decay and are related to each other by
$\tilde{\Gamma}_{0}(N)$ transformations as we have seen in the last section.
Indeed, crossing a wall in the moduli space corresponds to crossing a pole in
deforming a contour in accordance with the change of moduli so the degeneracy
jumps in a way consistent with the macroscopic analysis [13, 14, 15]. What is
remarkable with these ${\cal N}=4$ theories is that even though the degeneracy
jumps from one region to another, it is possible to define a partition
function globally over the moduli space such that its pole structure
completely captures the intricate structure of walls of marginal stability.
As is suggested by our choice of notation, it turns out that in all cases of
our interest, $N=1,2,3$, the inverse partition function
$(\Phi_{t}(\Omega))^{2}$ is a complete square of the automorphic form
$\Phi_{t}(\Omega)$ of integral weight $t$ of the congruence subgroup
$G_{0}(N)$ of $Sp(2,{\mathbb{Z}})$888It is not always true that the inverse
partition function is a square of some automorphic form with integral weight.
In particular, for $N=7$ the partition function cannot be written as the
square of some infinite product with integral exponents.. In particular,
taking $A=(D^{-1})^{T}=\gamma\in\Gamma_{0}(N)$ in (78), we see that it is
invariant or anti-invariant (invariant up to a minus sign) under the
transformation $\Omega\to\gamma(\Omega),\,\gamma\in\Gamma_{0}(N)$. It is this
form $\Phi_{t}$, with $t=5,3,2$ for $N=1,2,3$, which will enable us to
construct the Borcherds-Kac-Moody algebra underlying the dyon spectrum in
$\S$5.
### 4.2 The Siegel Modular Forms
In order for us to construct the generalized Kac-Moody algebra relevant for
the dyon spectrum of the CHL model, and to be have an explicit knowledge about
the set of imaginary simple roots and the multiplicities of all positive
roots, it is crucial that we have the knowledge of the denominator of its
characters both as an infinite product and as an infinite sum. The automorphic
form $\Phi_{t}(\Omega)$ discussed above is known to have an infinite product
presentation, which can be derived using the type IIB realization of the dyons
as D1-D5-momenta system in the KK-monopole background [7, 9]. The exponents in
the product formula (82) are the Fourier coefficients of certain weak Jacobi
forms $\chi^{(n,m)}(\tau,z)$ of zero weight and index $1$ of the congruence
subgroup $\Gamma_{0}(N)$ of $SL(2,{\mathbb{Z}})$
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)=\sum_{\begin{subarray}{c}k,\ell\in{\mathbb{Z}}\\\
k>0\end{subarray}}\,c^{\scriptscriptstyle{(n,m)}}(\tfrac{4k}{N}-\ell^{2})\,q^{k/{N}}y^{\ell}\;\;,\;\;q=e^{2\pi
i\tau},\,y=e^{2\pi iz}\;.$ (80)
These weak Jacobi forms are periodic with period $N$ in both $n$ and $m$
(106). Physically, these weak Jacobi forms $\chi^{(n,m)}(\tau,z)$ are related
to the orbifold-invariant part of the elliptic genus of the two-dimensional
CFT with target space $K3/{\mathbb{Z}}_{\scriptscriptstyle N}$, restricted to
the $n^{\prime}$-th twisted sector, where $0\leq n^{\prime}<N$ and
$n^{\prime}=n$ mod $N$. The explicit expression of them and in particular the
proof of the integrality of their Fourier coefficients can be found in
Appendix A. This relation between the product representation of an automorphic
form and a modular or weak Jacobi form is sometimes known as the
“multiplicative lift” or the “Borcherds lift”[51, 24, 45].
Moreover, the automorphic form $\Phi_{t}(\Omega)$ has also a simple Fourier
sum, with the Fourier coefficients given by the Fourier coefficients of the
Jacobi form
$\displaystyle\phi_{t,1/2}(\tau,z)=i\,\eta^{\frac{12}{N+1}-3}(\tau)\,\eta^{\frac{12}{N+1}}({\tau}/{N})\,\theta_{1,1}(\tau,z)=\sum_{\begin{subarray}{c}k>0\\\
k,\ell\in{\mathbb{Z}}\end{subarray}}C(k,\ell)\,q^{\frac{k}{2N}}y^{\frac{\ell}{2}}$
(81)
of the same weight $t$ and index $1/2$ of a subgroup of $SL(2,{\mathbb{Z}})$
which is related to an S-duality group $\Gamma_{1}(N)$ by a conjugation by the
S-transformation (114). This relation between the Fourier coefficients of the
automorphic forms and those of a modular form is sometimes called the
“additive lift” or the “arithmetic lift” [52, 53].
More explicitly, we have the following two expressions for the automorphic
form $\Phi_{t}(\Omega)$ as an infinite sum and as an infinite product
$\displaystyle\Phi_{t}(\Omega)$ $\displaystyle=$ $\displaystyle e^{2\pi
i(\frac{1}{2N}\sigma+\frac{1}{2}\rho+\frac{1}{2}\nu)}\,\prod_{\begin{subarray}{c}n,m,\ell\in{\mathbb{Z}}\\\
(n,m,\ell)>0\end{subarray}}\big{(}1-e^{2\pi
i(\frac{n}{N}\sigma+m\rho+\ell\nu)}\big{)}^{c^{\scriptscriptstyle{(n,m)}}(\frac{4nm}{N}-\ell^{2})}$
(82) $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\tilde{n},\tilde{m},\tilde{\ell}\in
2{\mathbb{Z}}+1\\\ \tilde{n},\tilde{m}>0\end{subarray}}e^{2\pi
i(\frac{\tilde{n}}{2N}\sigma+\frac{\tilde{m}}{2}\rho+\frac{\tilde{\ell}}{2}\nu)}\sum_{\begin{subarray}{c}\delta\lvert(\tilde{n},\tilde{m},\tilde{\ell})\\\
\delta>0\end{subarray}}\delta^{t-1}\,d(\delta)\;C\big{(}\frac{\tilde{n}\tilde{m}}{\delta^{2}},\frac{\tilde{\ell}}{\delta}\big{)}\;,$
where the Dirichlet characters in the sum formula are $d(\delta)=1$ for $N=1$,
$d(\delta)=\begin{cases}0&,\,\delta=0{\text{ mod }}2\\\ 1&,\,\delta=1{\text{
mod }}2\end{cases}$ (83)
for $N=2$ and
$d(\delta)=\begin{cases}0&,\,\delta=0{\text{ mod }}3\\\ 1&,\,\delta=1{\text{
mod }}3\\\ -1&,\,\delta=-1{\text{ mod }}3\end{cases}$ (84)
for $N=3$. In section 5.2 we will see how the above formula encodes all the
information about the simple roots and the resulting root multiplicities of
the algebra for CHL dyons.
## 5 The Algebra for CHL Dyons
The objective of this section is to first construct the generalized Kac-Moody
algebras from the partition functions of the CHL dyons and subsequently
elucidate the role of this algebra in the supersymmetric dyon spectrum of the
CHL models. To achieve this we first analyze the Weyl vector of the algebra
and explain how the physical “niceness” condition on the walls of marginal
stability is translated into the mathematical “niceness” condition of the
existence of a time-like Weyl vector of [44, 45]. We then see, for models
satisfying these “niceness” conditions, the sum and product representations of
the automorphic form $\Phi_{t}(\Omega)$ in the last section gives us the
denominator identity of the algebra and therefore provides us with complete
knowledge of both the simple roots as well as all the root multiplicities.
Finally we see how the dyon spectrum furnishes a representation of the algebra
and in what sense the algebra can been seen as an extra symmetry in the
supersymmetric sector of the CHL theories.
### 5.1 The Weyl Vector
One of the important objects in the theory of finite and affine Lie algebra is
the Weyl vector. The Weyl vector of a Borcherds-Kac-Moody algebra is defined
analogously as the vector in the root space which has inner product $-1$ with
all simple real roots of the algebra
$(\varrho,\alpha_{i})=-\frac{1}{2}(\alpha_{i},\alpha_{i})\quad\text{for all
simple real roots}\quad\alpha_{i}\,.$ (85)
In the present physical context, the Weyl vector is directly related to the
level-matching condition of the orbifolds. Mathematically it is crucial for
the automorphic properties of the partition function which will play an
important role in the definition of the algebra. In this subsection we will
see how this Weyl vector carries information about the topology of the system
of walls of marginal stability of the theory. In particular, only if this
vector is time-like, the theory will belong the finite class of models
discussed in $\S$3.2.
To begin with, from the above definition of the Weyl vector, we expect this
vector to be invariant under the action of elements of the symmetry group
$\text{\small Sym}({\cal W}^{(N)})$ of the fundamental Weyl chamber, and in
particular under the action of the $\gamma^{{\scriptscriptstyle(N)}}$ given in
(46). This consistency condition actually fixes for us the Weyl vector up to a
normalization factor and we are left with the following unique choice
$\varrho^{{\scriptscriptstyle(N)}}=\begin{pmatrix}1/N&1/2\\\
1/2&1\end{pmatrix}\;.$ (86)
Notice that this vector is space-like, light-like, or time-like exactly when
the symmetry generator $\gamma^{{\scriptscriptstyle(N)}}$ is hyperbolic,
parabolic or elliptic respectively. To understand this, recall that there is a
map between a complex number $z$ in the interior of the upper-half plane to a
ray in the future light-cone Minkowski space ${\mathbb{R}}^{2,1}$ by
$v(z)\sim\frac{1}{\mathrm{Im}z}\begin{pmatrix}|z|^{2}&\mbox{Re}z\\\
\mbox{Re}z&1\end{pmatrix}\;,\;\;\mbox{Im}z>0$ (87)
such that the vector transforms as
$v(\frac{az+b}{cz+d})=\gamma\,v(z)\,\gamma^{T}\;\;\;,\;\;\;\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\;.$ (88)
This is the familiar map (24) which is often used to write down a manifestly
S-duality action for supergravity, see for example [54]. Furthermore, we have
seen in $\S$3.2 how this map can be extended to a map between the rational
numbers $z=B/A,g.c.d.(A,B)=1$ on the boundary of the upper-half plane ${\cal
H}$ and the boundary of the future light-cone, given by
$v({\frac{B}{A}})\sim\begin{pmatrix}B^{2}&AB\\\ AB&A^{2}\end{pmatrix}\;$ (89)
such that the same transformation rule (88) applies. Now the question is, what
about the rest of the real line, which is the boundary of the upper-half
plane? Consider complex numbers of the form
$z=q_{1}+q_{2}\sqrt{D}\;,\;\;q_{1},q_{2}\in{\mathbb{Q}}$ (90)
where $D$ is some square-free integer, notice that the transformed number
$\frac{az+b}{cz+d}$ is again of this above form. This is why these numbers are
sometimes said to be in the “quadratic number field” ${\mathbb{Q}}(\sqrt{D})$.
Now we can write down the following map of these numbers to the rays in
${\mathbb{R}}^{2,1}$
$v(z)\sim\frac{1}{|q_{2}\sqrt{D}|}\begin{pmatrix}q_{1}^{2}-q_{2}^{2}D&q_{1}\\\
q_{1}&1\end{pmatrix}$ (91)
such that the same transformation rule (88) again applies and the vector is
time- or space-like depending on whether the integer $D$ is negative or
positive.
Finally, not surprisingly the Weyl vectors are simply the image of the fixed
point of the Möbius transformation (53) given by the symmetry transformation
$\gamma^{\scriptscriptstyle{(N)}}$ under the above map. Recall that we have
shown that the number of walls are infinite for $N\geq 4$ because the symmetry
generator $\gamma^{\scriptscriptstyle{(N)}}$ has real fixed points, we see
that the infinitely many walls and the absence of a time-like Weyl vector is
really one and the same thing. Therefore, the physical requirement that each
domain in the moduli space should be bounded by a finite number of walls leads
us to consider the root lattice admitting a time-like Weyl vector as discussed
in [44, 45].
### 5.2 Partition Function as a Denominator Formula
In this section we would like to answer the following question: can the
automorphic form $\Phi_{t}(\Omega)$ appearing in the dyon degeneracy formula
for the CHL models (69) be used to define a “automorphic form corrected”
Borcherds-Kac-Moody algebra? As announced in $\S$2, the answer is positive for
${\mathbb{Z}}_{\SS N}$ models with $N<4$ and negative for the $N>4$ model. The
obstruction for the $N>4$ case, whose partition function has also been
proposed in [6], basically lies in the fact that the would-be Weyl vector is
space-like which brings in other inconsistencies with it. This is in turn
related to the property of the physical theory that there is no domain in the
moduli space bounded by finitely many walls of marginal stability. We will
focus in this section on the $N<4$ cases and see how the automorphic forms
$\Phi_{t}(\Omega)$ specify all the data of the algebra for us. Our analysis
will be analogous to the work of Gritsenko and Nikulin[24] in which
$\Phi_{5}(\Omega)$ was used to construct a Borcherds-Kac-Moody superalgebra.
A Borcherds-Kac-Moody superalgebra is most conveniently defined using its
Chevalley basis, subject to a set of (anti-)commutation relations specified by
the Cartan matrix. Therefore, we can conveniently decompose it into
$\mathfrak{g}=\sum_{\alpha\in\Delta^{+}}\mathfrak{g}_{-\alpha}\oplus\mathfrak{h}\oplus\sum_{\alpha\in\Delta^{+}}\mathfrak{g}_{\alpha}$
where $\Delta^{+}$ denotes the set of positive roots. We will not write down
their definitions nor discuss their properties here. The reader might find
them in, for example, [55, 24, 25].
Here we are in particular interested in the (super-) denominator identity of
such a superalgebra, which reads
$e(-\varrho)\prod_{\alpha\in\Delta_{+}}(1-e(-\alpha)\,)^{\text{mult}\alpha}=\sum_{w\in
W}\,\text{\small det}(w)\,w\bigl{(}e(-\varrho)\,\Sigma\bigr{)}\;,$ (92)
where $\text{\small det}(w)=1$ (-1) if the group element $w$ can be written as
an even (odd) number of reflections, and mult$\alpha$ denotes the graded
(fermionic root weighted with $-1$) multiplicity of the positive root
$\alpha$. When the algebra has imaginary simple roots (and therefore is
“generalized” or “Borcherds-”), the sum involves the following correction term
given by
$\Sigma=\sum_{s}\epsilon(s)\,e(-s)\,,$ (93)
where $s$ denotes a sum of imaginary simple roots, $\
\epsilon(s)=(-1)^{n_{\overline{0}}}$ if $s$ is a sum of $n_{\overline{0}}$
number of pairwise perpendicular even (bosonic) imaginary simple roots and
$n_{\overline{1}}$ number of odd (fermionic) imaginary simple roots, which are
all distinct unless it is lightlike, and $\epsilon(s)=0$ otherwise. In the
above formula, we have used the abstract exponentials $e(\alpha)$ satisfying
the relation $e(\alpha)e(\alpha^{\prime})=e(\alpha+\alpha^{\prime})$999These
abstract exponentials can be thought of as linear functions on the
complexified root space. In our case it maps the space of symmetric $2\times
2$ complex matrices to ${\mathbb{C}}$ by $e(\alpha):v\mapsto e^{(\alpha,v)}$,
where the exponential on the right-hand side is the ordinary one. Choosing
$v=\pi i\Omega$ gives a ‘specialization’ of the denominator formula (92) of
the form (96) in our context..
In the present case, the vector space where the roots live is of a Lorentzian
signature with only one time-like direction. Using the fact that no two
vectors in the future light-cone are perpendicular to each other, we can
rewrite the above equation as
$e(-\varrho)\prod_{\alpha\in\Delta_{+}}(1-e(-\alpha)\,)^{\text{mult}\alpha}=\sum_{w\in
W}\,\text{\small
det}(w)\,\bigg{(}e\bigl{(}w(\varrho)\bigr{)}-\sum_{\alpha\in{\cal
W}}M(\alpha)\,e\bigl{(}w(\varrho+\alpha)\bigr{)}\bigg{)}\;,$ (94)
where ${\cal W}$ again denotes the fundamental Weyl chamber. The Fourier
coefficients $M(\alpha)$ contain all the information of the imaginary simple
roots of the algebra as summarized in Table LABEL:imaginary_r_table. The
graded degeneracy of a simple root $\alpha$ is the number of times it is
repeated in the set of simple roots counted with plus/minus sign for $\alpha$
in the set of bosonic/fermionic simple roots. It is just given by the Fourier
coefficients $M(\alpha)$ for time-like simple roots. For light-like ones it is
given by the integers ${\tilde{M}}(\alpha)$, related to $M(\alpha)$ by
$1-\sum_{n\in{\mathbb{N}}}M(n\alpha)q^{n}=\prod_{k\in{\mathbb{N}}}(1-q^{k})^{\tilde{M}(k\alpha)}\;.$
(95)
Table 1: The List of Imaginary Simple Roots category | condition | graded degeneracy
---|---|---
bosonic time-like | $(\alpha,\alpha)<0,\;M(\alpha)>0$ | $M(\alpha)$
bosonic light-like | $(\alpha,\alpha)=0,\;M(\alpha)>0$ | ${\tilde{M}}(\alpha)$
fermionic time-like | $(\alpha,\alpha)<0,\;M(\alpha)<0$ | $M(\alpha)$
fermionic light-like | $(\alpha,\alpha)=0,\;M(\alpha)<0$ | ${\tilde{M}}(\alpha)$
We would like to argue that the sum and product representations of the
automorphic form $\Phi_{t}(\Omega)$ (82) give the right-hand side and the
left-hand side of the above denominator identity (94) respectively, and
therefore defines for us the Borcherds-Kac-Moody algebras we are looking for.
If true, this means that the Fourier coefficients of the Jacobi forms
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ and $\phi_{t,1/2}(\tau,z)$ encode
the information about the root multiplicities and the list of imaginary simple
roots respectively. For this interpretation to be correct, the following
consistency conditions must be satisfied (82) which are easy to verify.
* •
On the product side, we show that
1. 1.
all exponents $c^{\SS(n,m)}(\frac{4nm}{N}-\ell^{2})$ are integers and
therefore might be interpreted as the graded root multiplicities,
2. 2.
the product has the form of the product side of the denominator (94) with Weyl
vector $\varrho=\varrho^{\SS(N)}$ as derived in (86) from symmetry
consideration,
3. 3.
all positive real roots are of the form (44) and therefore are indeed in one-
to-one correspondence with the walls of marginal stability of the theory.
Therefore the corresponding real simple roots are indeed those derived in
$\S$3.3 by analyzing the physical walls of marginal stability of the theory.
* •
On the sum side, we show that
1. 1.
all Fourier coefficients are integers and therefore might be interpreted as
the degeneracies of the imaginary simple roots,
2. 2.
the automorphic form has the correct transformation property
$\Phi_{t}(w(\Omega))=\text{\small det}(w)\,\Phi_{t}(\Omega)$ for all elements
$w$ of the Weyl group,
3. 3.
the sum involves only vectors in the future light-cone $w(\varrho+\alpha)\in
V^{+}$, with $\alpha$ being a vector in the fundamental Weyl chamber and the
Weyl vector $\varrho$ again given by same vector $\varrho^{\SS(N)}$ (86) as in
the product expression.
The details of the proof of these statements can be found in Appendix A.
In other words, we see that the square root of the inverse partition function
for the $N=1,2,3$ CHL model (82) can be expressed in terms of the data of a
Borcherds-Kac-Moody algebra as follows
$\displaystyle\Phi_{t}(\Omega)$ $\displaystyle=$ $\displaystyle\sum_{w\in
W}\text{\small det}(w)\,\left(e^{-\pi
i(w(\varrho),\Omega)}-\sum_{\alpha\in\Delta_{s}^{im}}M(\alpha)e^{-\pi
i(w(\varrho+\alpha),\Omega)}\right)$ (96) $\displaystyle=$ $\displaystyle
e^{-\pi i(\varrho,\Omega)}\prod_{\alpha\in\Delta_{+}}\bigg{(}1-e^{-\pi
i(\alpha,\Omega)}\bigg{)}^{\text{mult}\,\alpha}\;.$
For the readability we have suppressed the label “$N$” denoting the different
models. For example, $\varrho$ stands for the Weyl vector
$\varrho^{\scriptscriptstyle(N)}$ (86) of the algebra for a given value of $N$
and $W$ stands for its Weyl group $W^{\scriptscriptstyle{(N)}}$. Similarly
$\Delta_{s}^{im}$ and $\Delta_{+}$ stands for the set of imaginary simple
roots and the positive roots of the $N$-th algebra respectively.
Explicitly, writing the vector $\alpha$ in the root lattice in its components
as follows
$\alpha=\begin{pmatrix}2n/N&\ell\\\ \ell&2m\end{pmatrix}\;,$ (97)
then the multiplicities of the positive roots are given by the Fourier
coefficients of the weak Jacobi form $\chi^{\scriptscriptstyle(n,m)}(\tau,z)$
$\text{\small mult
}\alpha=c^{\scriptscriptstyle(n,m)}(|\alpha|^{2})=c^{\scriptscriptstyle(n,m)}(\tfrac{4nm}{N}-\ell^{2})\\\
$ (98)
and the degeneracies of imaginary simple roots are given by the Fourier
coefficients of the Jacobi form $\phi_{t,1/2}(\tau,z)$:
$\displaystyle
M(\alpha)=-\sum_{\begin{subarray}{c}\delta\lvert(2n+1,2m+1,2\ell+1)\\\
\delta>0\end{subarray}}\delta^{t-1}\,d(\delta)\,C(\tfrac{(2n+1)(2m+1)}{\delta^{2}},\tfrac{2\ell+1}{\delta})$
(99)
The first few values of $\text{\small mult }\alpha$ and $M(\alpha)$ can be
found in Appendix A.
### 5.3 Dyon Spectrum as a Representation of the Algebra
After having constructed a Borcherds-Kac-Moody algebra from the partition
function of the CHL models and verified that the Weyl group structure is the
group structure of crossing the walls of marginal stability in the theory, we
would like to ask what is the role of the rest of the algebra in the physical
theory. It turns out that the algebra we constructed for $N=2,3$ orbifold
theory plays a basically identical role in the CHL models as the algebra
constructed by Gritsenko and Nikulin [24] in the un-orbifolded theory, as
recently discussed in [25]. We will thus be rather brief in this part of the
discussion and refer the readers to [25] for proofs and details of various
statements in this subsection.
First, recall that a Verma module of a Borcherds-Kac-Moody superalgebra is an
infinite-dimensional representation with a highest weight $\Lambda$. Its
super-character is given by
$\text{sch}\,{\mathfrak{M}}(\Lambda)=\frac{e(-\varrho+\Lambda)}{e(-\varrho)\prod_{\alpha\in\Delta^{+}}\big{(}1-e(-\alpha)\big{)}^{\text{mult}\,\alpha}}\;.$
(100)
The integrand in the degeneracy formula (69) can thus be identified with the
square of the character of the Verma module with a charge- and moduli-
dependent highest weight
$\Lambda_{w}=\varrho+\frac{1}{2}\,w^{-1}(\Lambda_{Q,P})\;,$ (101)
where $w\in W$ is the Weyl group element given by the moduli such that ${\cal
Z}\in w({\cal W})$. In particular, the highest weight is simply
$\varrho+\frac{1}{2}\,\Lambda_{Q,P}$ when the moduli are fixed at their
attractor values. Only in this case is the Verma module a “dominant weight
module”, meaning that the highest weight of the module is inside the
fundamental Weyl chamber. Then the effect of the contour integration in
computing the dyon degeneracy (69) is simply to compute the (graded) dimension
of the zero-weight subspace of the Verma module. In other words, the
degeneracy of dyons at a given moduli is simply the number of different ways
the vector $2\Lambda_{w}$ can be written as a sum of two copies of positive
roots, or a “second-quantized” degeneracy of the weight $2\Lambda_{w}$.
Second, an attractor flow in a black hole background towards the attractor
point of the moduli is described by a sequence of Weyl group elements
following the so-called “weak Bruhat ordering” of the group. This means,
starting from a given point in the moduli space ${\cal Z}\in w({\cal W})$,
there is a natural RG-flow-like sequence of Weyl group elements $w=w_{n}\to
w_{n-1}\to\dotsi\to w_{0}={\mathds{1}}$ and the moduli vector ${\cal Z}$
follows the path $w_{n}({\cal W})\to w_{n-1}({\cal W})\to\dotsi\to w_{1}({\cal
W})\to{\cal W}$ along the attractor flow. The corresponding Verma modules form
a sequence of modules which decreases in size and terminates only when the
highest weight is dominant
${\mathfrak{M}}(\Lambda_{w_{n}})\supset{\mathfrak{M}}(\Lambda_{w_{n-1}})\supset\dotsi\supset{\mathfrak{M}}(\Lambda_{w_{0}=\mathds{1}})\;.$
(102)
Combined with the relationship between the Verma modules and the dyon
degeneracies discussed in the previous paragraph, this sequence gives a
prescription of how to compute the difference in dyon degeneracies when a wall
of marginal stability is crossed. This prescription gives a microscopic
derivation of the so-called wall-crossing formula for the present ${\cal N}=4$
theories.
Finally, we would like to comment on the role of the Borcherds-Kac-Moody
algebra we constructed as a symmetry of the supersymmetric dyon spectrum. As
implied in our discussion about the Verma modules, the dyon spectrum is
generated by a set of freely-acting bosonic and fermionic oscillators, with
each positive root of the algebra corresponding to two copies of such
oscillators and the Weyl vector $\varrho$ stipulating the vacuum of the
system. More explicitly, with a given choice of simple roots we can rewrite
(69) as
$\frac{1}{e(-2\varrho)\prod_{\alpha\in\Delta^{+}}\big{(}1-e(-\alpha)\big{)}^{2\,\text{mult}\alpha}}=\sum_{w\in
W,\tilde{\Lambda}\in{\cal W}}D(w(\tilde{\Lambda}))\,e(w(\tilde{\Lambda}))\;,$
(103)
and the coefficient $D(w(\tilde{\Lambda}))$ denotes the degeneracy
$D(w(\tilde{\Lambda}))=D(Q,P)\lvert_{\mu,\lambda}\,\,,\quad\tilde{\Lambda}=\Lambda_{Q,P},\quad{\cal
Z}\in w^{-1}({\cal W})\;.$ (104)
Notice that the degeneracies of any other charges with
$\Lambda_{Q,P}\not\in{\cal W}$ can be obtained by a simultaneous S-duality
transformation (39) of the charges and the moduli of the formula above. What
the above formula implies is, by acting on a dyon microstate by an element of
${\mathfrak{g}}_{\alpha}$ one obtains another dyon state as long as $\alpha$
is a bosonic positive root. The same goes for fermionic positive roots except
for the fact that one has to be more careful with the exclusion principle as
usual. In this sense, the Borcherds-Kac-Moody algebra we constructed plays the
role as a spectrum-generating symmetry of the BPS spectrum. While the physical
relevance of the Weyl group symmetry has been elucidated in $\S$3, a physical
understanding of this larger symmetry is yet to be developed.
## 6 Discussion
We have identified a Borcherds-Kac-Moody superalgebra underlying the spectrum
of dyon in $T^{6}/\mathbb{Z}_{N}$ CHL orbifolds for $N=1,2,3$. Analogous to
the $N=1$ case of toroidal compactification discussed in [25], we conclude
that the algebra we constructed in the present paper plays the following role
in the physical theory. First, the Weyl group gives the underlying group
structure of crossing the walls of marginal stability. Second, the spectrum is
generated by a set of bosonic and fermionic freely-acting oscillators with
charge- and moduli-dependent oscillation levels. In particular, this algebraic
microscopic model provides an alternative derivation of the wall-crossing
formula.
Two important questions remain unanswered. The first question is: what is the
microscopic model for the ${\mathbb{Z}}_{\SS N}$ CHL models with $N\geq 4$? As
we have seen, on the macroscopic side the walls of marginal stability of these
theories do not satisfy the natural “niceness” condition, namely they do not
render a partition of the moduli space into domains bounded by finitely many
walls. Related to this, we expect the relevant lattice to have a light- or
space-like lattice Weyl vector [44, 45]. On the microscopic side, we conclude
that the partition functions proposed in [6] for these models cannot be
related to the denominator formula of some Borcherds-Kac-Moody algebra in a
similar way as in the $N<4$ models. This fact seems to be related to the
appearance of some very light states (more precisely, the so-called “polar
states” which are not heavy enough to form black holes) in the spectrum. By
way of an answer, one can entertain several possibilities. Either the
microscopic models exist but are much more involved than in the case of the
finite models, or these theories are non-perturbatively inconsistent in some
yet unknown way, or the BKM symmetry is only a spectrum-generating symmetry
which may or may not exist in a given model.
The second unanswered question is: how should we understand physically the
appearance of these algebras underlying the dyon spectrum? What we have
achieved, apart from constructing these new algebras, is that we have now
understood the significance of all real roots of the algebra as corresponding
to the multi-centered solutions which cause the jump in the spectrum (see also
[25]). Furthermore, we have given a prescription of how the “highest weight”
of the relevant Verma module (101), or the “oscillation level” in the
partition function (103), is given by the total charges and the moduli. This
gives us a microscopic model of the supersymmetric dyons in the ${\cal N}=4$
CHL models. We have also shown that the models are very likely to be correct,
since they reproduce the asymptotic growth as expected from the semi-classical
Bekenstein-Hawking entropy and all its known corrections, and, on a much finer
level, they reproduce the wall-crossing formula as predicted from the
supergravity analysis (see also [20, 1, 25]). But we do not yet understood the
physical origin of this Borcherds-Kac-Moody symmetry. Understanding this
symmetry will lead to a far more complete understanding of the non-
perturbative supersymmetric states in these theories than has been possible in
other four-dimensional gravitational theories less supersymmetry.
Finally, having identified an algebra, it is natural to consider the symmetry
obtained by the ‘exponentiation’ of this algebra. One would like to know
whether this symmetry is a global symmetry or a gauge symmetry, whether it is
unbroken or broken, and whether it is a genuine symmetry or a spectrum-
generating symmetry. Clearly, the symmetry includes the Weyl group which is
essentially the physical duality group. Now, at a generic point in the moduli
space, duality symmetry is a spontaneously broken discrete gauge symmetry to
the extent that one identifies theories at different points related by
duality. Therefore, at least a part of this symmetry is a genuine,
spontaneously broken, gauge symmetry. It would be interesting to better
understand the nature of the full symmetry and in particular whether it
corresponds to a large spontaneously broken gauge symmetry.
## Acknowledgments
It is a pleasure to thank Frederik Denef, Valery Gritsenko, Axel Kleinschmidt,
Andy Neitzke, Daniel Persson, Ashoke Sen, Erik Verlinde, Xi Yin for useful
discussions. We would like to thank the ICTS and the organizers of the
‘Monsoon Workshop in String Theory’ at the TIFR, and IPMU of the Tokyo
University for hospitality where part of this research was completed. The work
of M.C. is supported by the Netherlands Organization for Scientific Research
(NWO); and of A. D. is supported in part by the Excellence Chair of the Agence
Nationale de la Recherche (ANR).
## Appendix A Root Multiplicities from Weak Jacobi Forms
### A.1 The Product Representation and the Positive Roots
In the type IIB duality frame, the quarter-BPS dyons can be realized as bound
states of D1-D5-P and the KK monopole together with momenta along the KK
monopole circle. By quantizing the relevant degrees of freedom, the product
expression (82) of the partition function $(\Phi_{t}(\Omega))^{2}$ has been
derived in [9]. As mentioned in (80), the exponents of the products, or the
graded multiplicity of the positive roots (96), are given by the Fourier
coefficients of the following weak Jacobi form
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)=\frac{1}{2N}\sum_{s\in{\mathbb{Z}}\text{
mod }\scriptscriptstyle{N}}e^{2\pi
ism}F^{\scriptscriptstyle{(n,s)}}(\tau,z)=\sum_{\begin{subarray}{c}k,\ell\in{\mathbb{Z}}\\\
k>0\end{subarray}}\,c^{\scriptscriptstyle{(n,m)}}(\tfrac{4k}{N}-\ell^{2})\,q^{k/{N}}y^{\ell}\;,$
(105)
where $F^{\scriptscriptstyle{(n,s)}}(\tau,z)$ have the following
interpretation in the D1-D5 CFT: it is the $n$-th twisted elliptic genus of
the two-dimensional $(4,4)$ supersymmetric conformal field theory with target
space $K3/{\mathbb{Z}}_{\scriptscriptstyle{N}}$, with the insertion of the
orbifold generator to the $s$-th power. The sum
$\frac{1}{N}\sum_{s\in{\mathbb{Z}}\text{ mod }\scriptscriptstyle{N}}e^{2\pi
ism}F^{\scriptscriptstyle{(n,s)}}(\tau,z)$ therefore projects out the states
that are not invariant under the orbifold action. From this we immediately see
that the pair of integers $(n,m)$ in the above formula can be seen as only
defined up to mod $N$:
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)=\chi^{\scriptscriptstyle{(n+N,m)}}(\tau,z)=\chi^{\scriptscriptstyle{(n,m+N)}}(\tau,z)\;.$
(106)
The explicit expressions for $F^{\scriptscriptstyle{(n,s)}}(\tau,z)$ can be
found in [9], using which we derive the following expression for the weak
Jacobi forms $\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$
* •
$n,m=0$ mod $N$
$\displaystyle\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ $\displaystyle=$
$\displaystyle\frac{1}{N+1}\bigg{(}2\,\phi_{0,1}(\tau,z)+(N-1)\phi_{-2,1}(\tau,z)\,\phi_{2}^{\scriptscriptstyle(N)}(\tau)\bigg{)}$
(107) $\displaystyle=$
$\displaystyle\phi_{-2,1}(\tau,z)+\frac{24}{N+1}\bigg{(}\varphi(\tau,z)+\phi_{-2,1}(\tau,z)\,\sum_{k=1}^{\infty}\sigma_{1}(k)\,(q^{k}-Nq^{kN})\bigg{)}$
* •
$n$ or $m=0$ mod $N$
$\displaystyle\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ $\displaystyle=$
$\displaystyle\frac{1}{N+1}\bigg{(}\phi_{0,1}(\tau,z)-\phi_{-2,1}(\tau,z)\,\phi_{2}^{\scriptscriptstyle(N)}(\tau)\bigg{)}$
(108) $\displaystyle=$
$\displaystyle\frac{12}{N+1}\bigg{(}\varphi(\tau,z)-\frac{2}{N-1}\phi_{-2,1}(\tau,z)\,\sum_{k=1}^{\infty}\sigma_{1}(k)\,(q^{k}-Nq^{kN})\bigg{)}$
* •
$n,m\neq 0$ mod $N$
$\displaystyle\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ $\displaystyle=$
$\displaystyle-\frac{1}{N(N+1)}\phi_{-2,1}(\tau,z)\,\sum_{k\in{\mathbb{Z}}\text{
mod }\scriptscriptstyle N}\,e^{-2\pi i\frac{nmk}{\scriptscriptstyle
N}}\phi_{2}^{\scriptscriptstyle(N)}(\tfrac{\tau+k}{N})$ (109) $\displaystyle=$
$\displaystyle-\frac{24}{N^{2}-1}\;\phi_{-2,1}(\tau,z)\sum_{k=nm\text{ mod
}N}\sigma_{1}(k)\,q^{k/N}\;.$
In the above formulas, $\phi_{2}^{\scriptscriptstyle(N)}(\tau)$ is a weight 2
modular form under the subgroup $\Gamma_{1}(N)$ and
$\sigma_{x}(n)=\sum_{d\lvert n}d^{x}$ (110)
denotes the divisor function. The precise expressions for the weak Jacobi
forms and modular forms appearing in the above formula can be found in
Appendix A.3.
The first lines of the above expressions for
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ makes it manifest that they are
weak Jacobi forms of zero weight and index 1 with respect to the congruence
subgroup $\Gamma_{1}(N)$. In other words, from the modular invariance and the
spectral flow invariance of the CFT partition functions
$F^{\SS(n,m)}(\tau,z)$, one can show that they satisfy the two defining
conditions for weak Jacobi forms
$\displaystyle\chi^{\scriptscriptstyle{(n,m)}}(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d})$
$\displaystyle=$ $\displaystyle\epsilon\,e^{2\pi
i\frac{cz^{2}}{c\tau+d}}\,\chi^{(n,m)}(\tau,z)\;\;\;,\;\epsilon^{N}=1\;\;,\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{1}(N)$
$\displaystyle\chi^{\scriptscriptstyle{(n,m)}}(\tau,z+\ell\tau+m)$
$\displaystyle=$ $\displaystyle e^{-2\pi i(\ell^{2}\tau+2\ell
z)}\chi^{(n,m)}(\tau,z)\;,\quad\ell,m\in{\mathbb{Z}}.$ (111)
On the other hand, the second lines of the above expressions make it manifest
that all the Fourier coefficients
$c^{\scriptscriptstyle{(n,m)}}(\frac{4nm}{N}-\ell^{2})$ are indeed all
integers and are amenable to the interpretation as the graded multiplicity of
the positive roots. See Appendix A.3 for a detailed proof of the integrality
property.
Now we can straightforwardly read out the (graded) root multiplicity
$c^{\scriptscriptstyle{(n,m)}}(|\alpha|^{2})$ of positive roots. The few
lowest lying ones are listed in table 2.
Table 2: Root multiplicity of root $\alpha$ in the form (5.13) | $N=1$ | $N=2$ | $N=3$
---|---|---|---
| $|\alpha|^{2}$ | mult $\alpha$ | $|\alpha|^{2}$ | mult $\alpha$ | $|\alpha|^{2}$ | mult$\,\alpha$
$n,m=0$ mod $N$ | -1 | 1 | -1 | 1 | -1 | 1
0 | 10 | 0 | 6 | 0 | 4
3 | -64 | 3 | -32 | 3 | -22
4 | 108 | 4 | 52 | 4 | 36
7 | -513 | 7 | -257 | 7 | -171
8 | 808 | 8 | 408 | 8 | 268
$n$ or $m=0$ mod $N$ | | | 0 | 4 | 0 | 3
| | 3 | -32 | 3 | -21
| | 4 | 56 | 4 | 36
| | 7 | -256 | 7 | -171
| | 8 | 400 | 8 | 270
| | 11 | -1376 | 11 | -918
$n$, $m\neq 0$ mod $N$ | | | 1 | -8 | 1/3 | -3
| | 2 | 16 | 4/3 | 6
| | 3 | -24 | 5/3 | -9
| | 4 | 48 | 8/3 | 18
| | 5 | -96 | 3 | -12
| | | 6 | 160 | 4 | 24
In particular, we find that the positive real roots are exactly those vectors
labeling the walls of marginal stability of the theory (44) with multiplicity
one, and the light-like positive roots have the degeneracies
$\text{\small mult}\,\alpha=\begin{cases}\frac{24}{N+1}-2&\text{when
}n,m=0\text{ mod }N\\\ \frac{12}{N+1}&\text{when }n\text{ or }m=0\text{ mod
}N\\\ 0&\text{otherwise }\;.\end{cases}$ (112)
Furthermore, when $N>1$, the lowest-lying fermionic roots have length
$|\alpha|^{2}=|\varrho^{\SS(N)}|^{2}=\frac{4}{N}-1$ (time-like) and graded
multiplicity $-\frac{24}{N^{2}-1}$.
### A.2 The Sum Representation and the Imaginary Simple Roots
In this appendix we will analyze the Fourier sum expression of the automorphic
form (82). First we will show that it can be written as the sum expression of
the denominator formula of a Borcherds-Kac-Moody formula (94), and then
discuss some properties of the imaginary simple roots following from such an
expression.
As mentioned in (81), the Fourier coefficients of the automorphic form
$\Phi_{t}(\Omega)$ are given by the Fourier coefficients of the following
weight $t$, index $1/2$ Jacobi form
$\displaystyle\phi_{t,1/2}(\tau,z)$ $\displaystyle=$ $\displaystyle
i\eta^{\frac{12}{N+1}-3}(\tau)\eta^{\frac{12}{N+1}}({\tau}/{\textstyle
N})\,\theta_{1,1}(\tau,z)$ (113) $\displaystyle=$ $\displaystyle
q^{\frac{1}{2N}}y^{\frac{1}{2}}\prod_{n\geq
1}(1-q^{n})^{\frac{12}{N+1}-2}(1-q^{n/N})^{\frac{12}{N+1}}(1-q^{n}y)(1-q^{n-1}y^{-1})$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}k>0\\\
k,\ell\in{\mathbb{Z}}\end{subarray}}C(k,\ell)\,q^{\frac{k}{2N}}y^{\frac{\ell}{2}}\;.$
It is a Jacobi form with respect to the congruence subgroup $\Gamma_{0}(N)$
conjugated by the $S$-transformation. Namely, we have
$\phi_{t,1/2}\big{(}\,\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\,\big{)}=e^{\pi
i\frac{z^{2}}{c\tau+d}}\,\,[-i(c\tau+d)]^{t}\,\phi_{t,1/2}(\tau,z)\;\;,\;$
(114)
if
$\;\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
PSL(2,{\mathbb{Z}})\;,\;\;b=0\text{ mod }N\;.$ (115)
As mentioned in the text before formula (96), there are three conditions we
should check for the sum formula of the automorphic form to have an
interpretation as the sum formula in the denominator identity of certain
generalized Kac-Moody algebra.
First of all, the integrality of the Fourier coefficients $C(k,\ell)$ is
manifest in the above definition (113) of the Jacobi form
$\phi_{t,1/2}(\tau,z)$. Second, the property of the theta-function
$\theta_{1,1}(\tau,-z)=-\theta_{1,1}(\tau,z)$ (116)
translates into the transformation property
$\Phi_{t}(s_{1}(\Omega))=-\Phi_{t}(\Omega)$ (117)
of the automorphic form $\Phi_{t}$, where $s_{1}$ denotes the reflection with
respect to the simple real root $\alpha_{1}$ (40). Conjugated with the
symmetry generator of the fundamental Weyl chamber, for example
$(\gamma^{\SS(N)})^{-1}s_{1}\gamma^{\SS(N)}=s_{3}^{\SS(N)}$, we get
$\Phi_{t}(s_{i}(\Omega))=-\Phi_{t}(\Omega)\;.$ (118)
That is to say, the automorphic form is anti-invariant under any Weyl
reflection (14). The composition of more than one reflections then gives the
transformation property of the automorphic form $\Phi_{t}(\Omega)$ under the
action of a general Weyl group element
$\Phi_{t}(w(\Omega))=\text{\small det}(w)\Phi_{t}(\Omega)\;.$ (119)
This is the property that enables us to write the automorphic form as a graded
sum of the images of the same thing under the Weyl group action as in the
denominator formula (92). Third, from (113) it is easy to check that the
Fourier coefficients have the property
$C(k,\ell)=0\;\;\text{ if
}\;\;\frac{4k}{N}-\ell^{2}<\rvert\varrho^{\scriptscriptstyle(N)}\lvert^{2}=\frac{1}{N}-\frac{1}{4}\;.$
(120)
In particular, the coefficients of the term $e^{-\pi i(\beta,\Omega)}$ in the
Fourier expansion of $\Phi_{t}(\Omega)$ is non-zero only when $\beta$ is
inside the future light-cone. Using (119) we can then concentrate on the part
of the sum with $\beta\in{\cal W}$. Notice that $\beta$ cannot be on the
boundary of any Weyl chamber, basically because the Weyl vector is not in the
root lattice, and therefore $(\varrho+\alpha,\alpha^{\prime})=1$ mod $2$ for
any root $\alpha$, $\alpha^{\prime}$. Now write $\beta=\varrho+\alpha$,
following the definition of the Weyl vector (85) we conclude that $\alpha$
must also lie within the fundamental Weyl chamber. This is the third condition
we mentioned in $\S$5.2 for the sum expression for the automorphic form
$\Phi_{t}(\Omega)$ to be written as the sum expression for the denominator of
a certain Borcherds-Kac-Moody algebra.
It is now a straightforward task to read out the degeneracies of the imaginary
simple roots we are interested in. For example, from the factors of the
Dedekind eta functions in (81) we can see that the multiplicities (95) of the
light-like simple roots are
$\tilde{M}(k\alpha)=\begin{cases}\frac{24}{N+1}-3&\text{ when }\;n,m=0\text{
mod }N\\\ \frac{12}{N+1}&\text{ when }\;n\text{ or }m=0\text{ mod }N\\\
0&\text{ otherwise }\end{cases}\;\;\quad,\quad\text{ for all
}k\in{\mathbb{N}}\;.$ (121)
Comparing it with the multiplicity of the light-like positive roots
$\alpha=\big{(}\begin{smallmatrix}2n/N&\ell\\\
\ell&2m\end{smallmatrix}\big{)}$ we found in the previous subsection (112), we
see that the two are indeed consistent, taking into account the fact that the
same vector $\alpha$ can also be written as a combination involving two real
roots when $n,m=0$ mod $N$. For example, the light-like simple root
$\big{(}\begin{smallmatrix}2&2\\\ 2&2\end{smallmatrix}\big{)}$ has degeneracy
$9$, together with the possibility of writing it as the sum
$\alpha_{2}+\alpha_{3}$ of two real simple roots, we conclude that the
multiplicity of the positive root $\big{(}\begin{smallmatrix}2&2\\\
2&2\end{smallmatrix}\big{)}$ is $10=c^{\SS(0,0)}(0)$.
As the next example, let us consider the multiplicity of the lowest lying
fermionic roots $2\varrho^{\scriptscriptstyle(N)}$ in the $N>1$ cases. For the
$N=2$ case, from $C({1},1)=1$, $C({9},3)=-1$ and using (99) we conclude that
the multiplicity of this simple root should be $-(-1+3^{2})=-8$. For $N=3$,
from $C(9,3)=3$ we conclude that the multiplicity should be $-3$. This is
indeed consistent with the result of a root multiplicity $-\frac{24}{N^{2}-1}$
from the product representation.
It is worth noting that from the Fourier expansion of (113) that the numerical
values of the degeneracies of the imaginary simple roots grow very slowly.
They are still of order $\lesssim 10^{2}$ when $|\alpha|^{2}\sim 10^{2}$, in
contrast with the root multiplicities encoded in the weak Jacobi form (105),
which are of order $\sim 10^{10}$ when $|\alpha|^{2}\sim 10^{2}$. The
denominator identity still holds thanks to the impressive cancelation between
the contribution from the fermionic and the bosonic roots, which translates in
the partition function into the large cancelation between the contribution
from the fermionic and bosonic states to the index.
### A.3 More on Weak Jacobi Forms
In this subsection we collect various definitions and formulas of the modular
forms we have used in this paper. In particular we will explain the objects
involved in the construction of the weak Jacobi forms
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ in Appendix A.1, and prove the
integrality of their Fourier coefficients. For details of modular forms and
weak Jacobi forms with respect to congruence subgroups, one can consult, for
example, [56, 57].
Let us begin by recalling the two Eisenstein series and their relationship to
the modular discriminant $\Delta(\tau)$
$\displaystyle E_{4}(\tau)$ $\displaystyle=$ $\displaystyle
1-\frac{8}{B_{4}}\sum_{n\geq 1}\sigma_{3}(n)q^{n}=1+240\sum_{n\geq
1}\sigma_{3}(n)q^{n}$ $\displaystyle E_{6}(\tau)$ $\displaystyle=$
$\displaystyle 1-\frac{12}{B_{6}}\sum_{n\geq
1}\sigma_{5}(n)q^{n}=1-504\sum_{n\geq 1}\sigma_{5}(n)q^{n}$ $\displaystyle
E_{4}^{3}(\tau)-E_{6}^{2}(\tau)$ $\displaystyle=$ $\displaystyle
1728\,\Delta(\tau)=1728\,\eta^{24}(\tau)\;,$ (122)
where $\sigma_{x}(n)$ are again the divisor functions defined as in (110).
Using these and the their Jacobi form counterparts $E_{4,1}(\tau,z)$ and
$E_{6,1}(\tau,z)$ (see [57]), we can write down the following weight $0$,
weight $-2$ and index 1 weak Jacobi forms with which we have constructed our
weak Jacobi forms $\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$
$\displaystyle\phi_{-2,1}(\tau,z)$ $\displaystyle=$
$\displaystyle-\eta^{-6}(\tau)\,\theta^{2}_{1}(\tau,z)=\frac{E_{6}(\tau)E_{4,1}(\tau,z)-E_{4}(\tau)E_{6,1}(\tau,z)}{144\Delta(\tau)}$
$\displaystyle=$ $\displaystyle(y^{-1}-2+y)+(-2y^{-2}+8y^{-1}-12+8y-2y^{2})q$
$\displaystyle+(y^{-3}-12y^{-2}+39y^{-1}-56+39y-12y^{2}+y^{3})q^{2}+{\cal
O}(q^{3})$ $\displaystyle\phi_{0,1}(\tau,z)$ $\displaystyle=$
$\displaystyle\frac{\theta^{2}_{2}(\tau,z)}{\theta^{2}_{2}(\tau,0)}+\frac{\theta^{2}_{3}(\tau,z)}{\theta^{2}_{3}(\tau,0)}+\frac{\theta^{2}_{4}(\tau,z)}{\theta^{2}_{4}(\tau,0)}=\frac{E_{4}^{2}(\tau)E_{4,1}(\tau,z)-E_{6}(\tau)E_{6,1}(\tau,z)}{144\Delta(\tau)}$
$\displaystyle=$
$\displaystyle(y^{-1}+10+y)+(10y^{-2}-64y^{-1}+108-64y+10y^{2})q$
$\displaystyle+(y^{-3}+108y^{-2}-513y^{-1}+808-513y+108y^{2}+y^{3})q^{2}+{\cal
O}(q^{3})\;,$
These two weak Jacobi forms $\phi_{-2,1}(\tau,z)$ and $\phi_{0,1}(\tau,z)$,
together with the two Eisenstein series $E_{4}(\tau)$, $E_{6}(\tau)$
introduced above, generate the ring of weak Jacobi forms of even weight and
arbitrary integral indices.
Another element we need is the following weight two modular form under the
congruence subgroup $\Gamma_{0}(N)$ when $N$ is prime
$\displaystyle\phi_{2}^{\scriptscriptstyle(N)}(\tau)$ $\displaystyle=$
$\displaystyle
q\partial_{q}\log\bigg{(}\frac{\Delta(N\tau)}{\Delta(\tau)}\bigg{)}^{\frac{1}{N-1}}$
$\displaystyle=$ $\displaystyle
1+\frac{24}{N-1}\sum_{k=1}^{\infty}\sigma_{1}(k)\,(q^{k}-Nq^{kN})\;.$
Using the modular properties of these modular and Jacobi forms we see that
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ given in Appendix A.1 are indeed
weak Jacobi forms of weight zero and index one. But it is not clear whether
they will have integral Fourier coefficients since they involve non-integral
combinations of the above forms.
To show that they nevertheless have integral coefficients, let us rewrite them
in a slightly different form as in (107)-(109), where we have defined the
following object which does not have nice modular properties
$\displaystyle\varphi(\tau,z)$ $\displaystyle=$
$\displaystyle\frac{1}{12}(\phi_{0,1}(\tau,z)-\phi_{-2,1}(\tau,z))$
$\displaystyle=$ $\displaystyle 1+(y^{-2}-6y^{-1}+10-6y+y^{2})q$
$\displaystyle+(10y^{-2}-46y^{-1}+72-46y+10y^{2})q^{2}+{\cal O}(q^{3})\;.$
To show that it has integral coefficients, notice that (122)
$\frac{1}{12}\big{(}E_{4}(\tau)\phi_{0,1}-E_{6}(\tau)\phi_{-2,1}(\tau,z)\big{)}=E_{4,1}(\tau,z)\;.$
(123)
Since all the coefficients of $E_{4}(\tau)$ and $E_{6}(\tau)$ are divisible by
12 except for the constant terms (122), namely $E_{4}(\tau),E_{6}(\tau)=1$ mod
$12$, we conclude that $\varphi(\tau,z)$ and therefore
$\chi^{\scriptscriptstyle{(n,m)}}(\tau,z)$ always have integral Fourier
coefficients. This is of course a necessary condition for the product
representation to have an interpretation as the denominator formula of a
certain BKM algebra, as mentioned in the main text in $\S$5.2.
## Appendix B Notations and Definitions
For the convenience of the readers, we collect the definitions of various
objects frequently used in the main text.
* •
Vectors in ${\mathbb{R}}^{2,1}$
$\Lambda_{Q,P}\;\;$
The vector of T-duality invariants. See (8).
${\cal Z}\;\;$
The central charge like vector encoding all the moduli dependence. See (25).
$\Omega\;\;$
The vector of chemical potentials/integration variables. See (69).
* •
Groups
$PSL(2,{\mathbb{Z}})$
The S-duality group of the un-orbifolded theory.
$PGL(2,{\mathbb{Z}})$
The extended S-duality group of the un-orbifolded theory. See (35).
$\Gamma_{1}(N)$
The S-duality group of the ${\mathbb{Z}}_{\SS N}$ orbifold theory. A subgroup
of $PSL(2,{\mathbb{Z}})$. See (3).
$\tilde{\Gamma}_{1}(N)$
The extended S-duality group of the ${\mathbb{Z}}_{\SS N}$ orbifold theory. A
subgroup of $PGL(2,{\mathbb{Z}})$. See (3).
$\tilde{\Gamma}_{0}(N)$
The group relating different walls of marginal stability of the
${\mathbb{Z}}_{\SS N}$ orbifold theory. A subgroup of $PGL(2,{\mathbb{Z}})$.
See (42). For $N<4$ we have $\tilde{\Gamma}_{0}(N)=\tilde{\Gamma}_{1}(N)$.
$G_{0}(N)$
The $Sp(2,{\mathbb{Z}})$ subgroup of which the Siegel modular form
$\Phi_{t}(\Omega)$ is an automorphic form. See (74).
* •
Objects of the Algebra
$\varrho\;\;$
The Weyl vector. See (85).
$\varrho^{\SS(N)}\;\;$
The Weyl vector of the algebra for the ${\mathbb{Z}}_{\SS N}$ theory. See
(86).
$\Delta_{+}\;\;$
The set of all positive roots.
$\Delta_{s}^{im}\;\;$
The set of all imaginary simple roots.
${\mathfrak{M}}(\Lambda)$
Verma module with highest weight $\Lambda$. See (100).
$\alpha_{i}^{\SS(N)}\;\;$
The $i$-th simple real root of the algebra for the ${\mathbb{Z}}_{\SS N}$
theory. See $\S$3.3.
$s_{i}^{\SS(N)}\;\;$
The reflection with respect to the $i$-th simple real root of the algebra for
the ${\mathbb{Z}}_{\SS N}$ theory. See (14).
${W}^{\SS(N)}\;\;$
The Weyl group of the algebra for the ${\mathbb{Z}}_{\SS N}$ theory, which is
the reflection group generated by $s_{i}^{\SS(N)}$.
${\cal W}^{\SS(N)}\;\;$
The fundamental Weyl chamber of the algebra for the ${\mathbb{Z}}_{\SS N}$
theory. See (32).
$\gamma^{\SS(N)}$
The generator of a particular symmetry of the fundamental Weyl chamber of the
algebra for the ${\mathbb{Z}}_{\SS N}$ theory. See (46).
## References
* [1] R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B484 (1997) 543–561, [hep-th/9607026].
* [2] D. Gaiotto, A. Strominger, and X. Yin, New connections between 4D and 5D black holes, JHEP 02 (2006) 024, [hep-th/0503217].
* [3] D. Shih, A. Strominger, and X. Yin, Recounting dyons in N = 4 string theory, JHEP 10 (2006) 087, [hep-th/0505094].
* [4] D. Gaiotto, Re-recounting dyons in N = 4 string theory, hep-th/0506249.
* [5] D. Shih and X. Yin, Exact black hole degeneracies and the topological string, JHEP 04 (2006) 034, [hep-th/0508174].
* [6] D. P. Jatkar and A. Sen, Dyon spectrum in CHL models, hep-th/0510147.
* [7] J. R. David, D. P. Jatkar, and A. Sen, Product representation of dyon partition function in CHL models, JHEP 06 (2006) 064, [hep-th/0602254].
* [8] A. Dabholkar and S. Nampuri, Spectrum of Dyons and Black Holes in CHL orbifolds using Borcherds Lift, JHEP 11 (2007) 077, [hep-th/0603066].
* [9] J. R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, JHEP 11 (2006) 072, [hep-th/0605210].
* [10] J. R. David, D. P. Jatkar, and A. Sen, Dyon spectrum in N = 4 supersymmetric type II string theories, JHEP 11 (2006) 073, [hep-th/0607155].
* [11] J. R. David, D. P. Jatkar, and A. Sen, Dyon spectrum in generic N = 4 supersymmetric Z(N) orbifolds, JHEP 01 (2007) 016, [hep-th/0609109].
* [12] A. Dabholkar and D. Gaiotto, Spectrum of CHL dyons from genus-two partition function, JHEP 12 (2007) 087, [hep-th/0612011].
* [13] A. Sen, Walls of marginal stability and dyon spectrum in N=4 supersymmetric string theories, JHEP 05 (2007) 039, [hep-th/0702141].
* [14] A. Dabholkar, D. Gaiotto, and S. Nampuri, Comments on the spectrum of CHL dyons, JHEP 01 (2008) 023, [hep-th/0702150].
* [15] M. C. N. Cheng and E. Verlinde, Dying dyons don’t count, 0706.2363.
* [16] S. Banerjee, A. Sen, and Y. K. Srivastava, Generalities of quarter BPS dyon partition function and dyons of torsion two, 0802.0544.
* [17] S. Banerjee, A. Sen, and Y. K. Srivastava, Partition functions of torsion $>1$ dyons in heterotic string theory on $T^{6}$, 0802.1556.
* [18] S. Banerjee and A. Sen, S-duality action on discrete T-duality invariants, 0801.0149.
* [19] A. Dabholkar, J. Gomes, and S. Murthy, Counting all dyons in N =4 string theory, arXiv:0803.2692.
* [20] G. Lopes Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy, JHEP 12 (2004) 075, [hep-th/0412287].
* [21] G. Lopes Cardoso, B. de Wit, and T. Mohaupt, Corrections to macroscopic supersymmetric black-hole entropy, Phys. Lett. B451 (1999) 309–316, [hep-th/9812082].
* [22] G. Lopes Cardoso, B. de Wit, and T. Mohaupt, Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes, Nucl. Phys. B567 (2000) 87–110, [hep-th/9906094].
* [23] G. Lopes Cardoso, B. de Wit, and T. Mohaupt, Area law corrections from state counting and supergravity, Class. Quant. Grav. 17 (2000) 1007–1015, [hep-th/9910179].
* [24] V. Gritsenko and V. Nikulin, Siegel automorphic form corrections of some lorentz kac-moody algebras, Amer.J.Math. 119 (1991) 181.
* [25] M. C. N. Cheng and E. P. Verlinde, Wall Crossing, Discrete Attractor Flow, and Borcherds Algebra, arXiv:0806.2337.
* [26] S. Govindarajan and K. Gopala Krishna, Generalized Kac-Moody Algebras from CHL dyons, arXiv:0807.4451.
* [27] J. A. Harvey and G. W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489–519, [hep-th/9609017].
* [28] J. A. Harvey and G. W. Moore, Algebras, BPS states, and strings, Nucl. Phys. B463 (1996) 315–368, [hep-th/9510182].
* [29] R. W. Gebert and H. Nicolai, E(10) for beginners, hep-th/9411188.
* [30] T. Damour, M. Henneaux, B. Julia, and H. Nicolai, Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B509 (2001) 323–330, [hep-th/0103094].
* [31] T. Damour, M. Henneaux, and H. Nicolai, E(10) and a ’small tension expansion’ of M theory, Phys. Rev. Lett. 89 (2002) 221601, [hep-th/0207267].
* [32] T. Damour, M. Henneaux, and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145–R200, [hep-th/0212256].
* [33] M. Henneaux, D. Persson, and P. Spindel, Spacelike Singularities and Hidden Symmetries of Gravity, Living Rev. Rel. 11 (2008) 1, [arXiv:0710.1818].
* [34] M. Henneaux, D. Persson, and D. H. Wesley, Coxeter group structure of cosmological billiards on compact spatial manifolds, arXiv:0805.3793.
* [35] A. Giveon and M. Porrati, Duality invariant string algebra and D = 4 effective actions, Nucl. Phys. B355 (1991) 422–454.
* [36] A. Giveon, M. Porrati, and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77–202, [hep-th/9401139].
* [37] S. Chaudhuri, G. Hockney, and J. D. Lykken, Maximally supersymmetric string theories in d ¡ 10, Phys. Rev. Lett. 75 (1995) 2264–2267, [hep-th/9505054].
* [38] S. Chaudhuri and J. Polchinski, Moduli space of chl strings, Phys. Rev. D52 (1995) 7168–7173, [hep-th/9506048].
* [39] S. Chaudhuri and D. A. Lowe, Type iia heterotic duals with maximal supersymmetry, Nucl. Phys. B459 (1996) 113–124, [hep-th/9508144].
* [40] J. H. Schwarz and A. Sen, Type IIA dual of the six-dimensional CHL compactification, Phys. Lett. B357 (1995) 323–328, [hep-th/9507027].
* [41] C. Vafa and E. Witten, Dual string pairs with n = 1 and n = 2 supersymmetry in four dimensions, Nucl. Phys. Proc. Suppl. 46 (1996) 225–247, [hep-th/9507050].
* [42] P. S. Aspinwall, Some relationships between dualities in string theory, Nucl. Phys. Proc. Suppl. 46 (1996) 30–38, [hep-th/9508154].
* [43] A. Sen, Entropy function for heterotic black holes, hep-th/0508042.
* [44] V. A. Gritsenko and V. V. Nikulin, Automorphic forms and lorentzian kac–moody algebras (i), .
* [45] V. A. Gritsenko and V. V. Nikulin, Automorphic forms and lorentzian kac-moody algebras (ii), .
* [46] A. Sen, Rare decay modes of quarter BPS dyons, 0707.1563.
* [47] F. Denef, Supergravity flows and d-brane stability, JHEP 08 (2000) 050, [hep-th/0005049].
* [48] F. Denef, Quantum quivers and hall/hole halos, JHEP 10 (2002) 023, [hep-th/0206072].
* [49] F. Denef and G. W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146.
* [50] D. Gaiotto, G. W. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723.
* [51] R. Borcherds, Automorphic forms on $o_{s+2,2}(r)$ and infinite products, Invent.Math 120 (1995) 161–213.
* [52] H. Maaß, Über einer spezialschar von modulformen zweiten grades i,ii,iii, Invent. math. 52,53 (1979) 95–104,249–253,255–265.
* [53] V. Gritsenko, Modular forms and moduli spaces of abelian and k3 surfaces, St. Petersburg Math. Jour. 6:6 (1995) 1179–1208.
* [54] E. Bergshoeff, H. J. Boonstra, and T. Ortin, S duality and dyonic p-brane solutions in type II string theory, Phys. Rev. D53 (1996) 7206–7212, [hep-th/9508091].
* [55] U. Ray, Automorphic Forms and Lie Superalgebra. Springer, 2006.
* [56] T. Apostol, Modular Functions and Dirichlet Series in Number Theory. Springer-Verlag, 1990.
* [57] M. Eichler and D. Zagier, The Theory of Jacobi Forms. Birkhäuser, 1985.
|
arxiv-papers
| 2008-09-25T18:39:34
|
2024-09-04T02:48:57.936470
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miranda C. N. Cheng and Atish Dabholkar",
"submitter": "Miranda C.N. Cheng",
"url": "https://arxiv.org/abs/0809.4258"
}
|
0809.4322
|
# Non-Standard Analysis, Multiplication of Schwartz Distributions, and Delta-
Like Solution of Hopf’s Equation
###### Abstract
Non-Standard Analysis, Multiplication of Schwartz Distributions, and Delta-
Like Solution of Hopf’s Equation
Guy Berger
We construct an algebra of generalized functions
${}^{*}\mathcal{E}(\mathbb{R}^{d})$. We also construct an embedding of the
space of Schwartz distributions $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ into
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ and thus present a solution of the problem
of multiplication of Schwartz distributions which improves J.F. Colombeau’s
solution. As an application we prove the existence of a weak delta-like
solution in ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ of the Hopf equation. This
solution does not have a counterpart in the classical theory of partial
differential equations. Our result improves a similar result by M. Radyna
obtained in the framework of perturbation theory.
A Thesis Presented to the Faculty of
California Polytechnic State University
San Luis Obispo
In Partial Fulfillment of the Requirements for the Degree
Master of Science in Mathematics
by
Guy Berger
September 2005
AUTHORIZATION FOR REPRODUCTION OF MASTER’S THESIS
I grant permission for the reproduction of this thesis in its entirety or any
of its parts, without further authorization from me.
$\overline{\text{Signature \quad\quad\quad\quad\quad}}$
$\overline{\text{Date \quad\quad\quad\quad\quad\quad\quad}}$
APPROVAL PAGE
TITLE: Non-Standard Analysis, Multiplication of Schwartz Distributions, and
Delta-Like Solution of Hopf’s Equation
AUTHOR: Guy Berger
DATE SUBMITTED: September 16, 2005
$\overline{\text{Adviser\quad\quad\quad\quad\quad\quad\quad\quad}}$ | $\overline{\text{Signature\quad\quad\quad\quad\quad}}$
---|---
$\overline{\text{Committee Member\quad\quad\,\;\;}}$ | $\overline{\text{Signature\quad\quad\quad\quad\quad}}$
$\overline{\text{Committee Member\quad\quad\,\;\;}}$ | $\overline{\text{Signature\quad\quad\quad\quad\quad}}$
Acknowledgements
I would like to thank the faculty and staff of the Cal Poly Mathematics
Department, who have made my experience over the past two years a positive
one.
I am especially grateful to my supervisor, Dr. Todor D. Todorov, for his help,
guidance, and support throughout the past year.
Key words and phrases: Schwartz distributions, multiplication of Schwartz
distributions, Colombeau’s algebra of generalized functions, non-standard
analysis, saturation principle, conservation law, Hopf equation, weak
solution, shock wave.
AMS Subject Classification: 26E35, 30G06, 46F10, 46F30, 46S10, 46S20, 35D05,
35L67, 35L65.
###### Contents
1. 1 Introduction
2. 2 Ordered Fields
3. 3 Filters and Ultrafilters
4. 4 Ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$
5. 5 Non-Standard Numbers
6. 6 Internal Sets
7. 7 Saturation Principle in ${}^{*}\mathbb{C}$
8. 8 Non-Standard Smooth Functions
9. 9 Internal Sets and Saturation Principle in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$
10. 10 Weak Equality
11. 11 Schwartz Distributions
12. 12 Embedding of Schwartz Distributions in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$
13. 13 Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$ and the Hopf Equation
14. 14 Generalized Delta-like Solution of the Hopf Equation
## 1 Introduction
In what follows
$\mathcal{E}(\mathbb{R}^{d})=\mathcal{C}^{\infty}(\mathbb{R}^{d})$ denotes the
class of $\mathcal{C}^{\infty}$-functions on $\mathbb{R}^{d}$. Also
$\mathcal{D}(\mathbb{R}^{d})=\mathcal{C}_{0}^{\infty}(\mathbb{R}^{d})$ denotes
the class of test functions on $\mathbb{R}^{d}$ and
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ stands for the space of Schwartz
distributions (Schwartz generalized functions) on $\mathbb{R}^{d}$ (H.
Bremermann [1]).
The algebra of generalized functions ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is
a particular non-standard extension of the class
$\mathcal{E}(\mathbb{R}^{d})$. The field of the scalars ${}^{*}\mathbb{C}$ of
the algebra ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is a particular non-standard
extension of the field of complex numbers $\mathbb{C}$ and the field of the
real scalars ${}^{*}\mathbb{R}$ is a non-standard extension of $\mathbb{R}$.
That means that both ${}^{*}\mathbb{C}$ and ${}^{*}\mathbb{R}$ are non-
Archimedean fields containing non-zero infinitesimals, i.e. generalized
numbers $h$ such that $0<|h|<1/n$ for all $n\in\mathbb{N}$. Since the
involvement of non-Archimedean fields in applied mathematics is somewhat
unusual, we start with a summary of the relevant definitions and results in
the theory of ordered fields and non-Archimedean fields (Section 2).
In Sections 3-4 we present the basic facts of the theory of free filters and
ultrafilters (C. C. Chang and H. J. Keisler [2]). We construct a particular
ultrafilter on the space of test functions $\mathcal{D}(\mathbb{R}^{d})$ which
is important for the embedding of Schwartz distributions in the algebra
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$.
In Sections 5-6 we present the construction of the fields of the complex and
real non-standard numbers ${}^{*}\mathbb{C}$ and ${}^{*}\mathbb{R}$. In
Section 7 we prove the Saturation Principle in ${}^{*}\mathbb{C}$ which plays
a role in non-standard analysis similar to the role of the completeness of
$\mathbb{R}$ and $\mathbb{C}$ in usual (standard) analysis. These sections
might be viewed as an introduction to non-standard analysis (A. Robinson
[12]). We should note that our exposition of non-standard analysis does not
require any background in mathematical logic or model theory.
The construction of the algebra ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is
presented in Section 8; in short, ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is a
differential associative commutative algebra of generalized functions similar
to (but much larger than) the class
$\mathcal{E}(\mathbb{R}^{d})=\mathcal{C}^{\infty}(\mathbb{R}^{d})$. In Section
9 we state the Saturation Principle for ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$,
playing the role of the completeness property.
In Section 12 we construct the chain of embeddings
$\mathcal{E}(\mathbb{R}^{d})\subset\mathcal{D}^{\prime}(\mathbb{R}^{d})\subset{{}^{*}\mathcal{E}(\mathbb{R}^{d})}$.
These embeddings presents a solution of the problem of multiplication of
Schwartz distributions similar to but different from Colombeau’s solution of
the same problem (J.F. Colombeau [3]). The problem of multiplication of
Schwartz distributions has an interesting and dramatic history. Soon after the
distribution theory was invented by L. Schwartz, he proved that the space of
distributions $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ cannot be supplied with
an associative and commutative product that reproduces the usual product in
the spaces $\mathcal{C}^{k}(\mathbb{R}^{d}),k=0,1,2,\dots$. This negative
result, known as Schwartz Impossibilities Result (L. Schwartz [14]), was the
reason this problem was considered for a long time as unsolvable. In the late
1980’s Jean F. Colombeau offered a solution of the problem of multiplication
of distributions by constructing an algebra of generalized functions
$\mathcal{G}(\mathbb{R}^{d})$ with the chain of algebraic embeddings
$\mathcal{E}(\mathbb{R}^{d})\subset\mathcal{D}^{\prime}(\mathbb{R}^{d})\subset\mathcal{G}(\mathbb{R}^{d})$
thus avoiding Schwartz Impossibilities Result (since $k=\infty$). One
(slightly disturbing) feature of Colombeau’s solution is that the set of
scalars $\overline{\mathbb{C}}$ of the algebra $\mathcal{G}(\mathbb{R}^{d})$
is a ring with zero divisors, not a field as any set of scalars should be. In
this respect our solution of the problem of multiplication of Schwartz
distributions presents an important improvement of Colombeau’s theory: the set
of scalars ${}^{*}\mathbb{C}$ of the algebra
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ is an algebraically complete
$c^{+}$-saturated field (Section 7). As a consequence, the set of the real
scalars ${}^{*}\mathbb{R}$ is a real closed Cantor complete field. We should
notice that the fact that ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ is a
differential algebra (not merely a linear space) is important for our goals in
applied mathematics, in particular, for studying generalized solutions of non-
linear partial differential equations such as shock-wave and delta-like
solutions. Notice that these are the solutions after the formation of the
shock in many conservation law type equations.
In Section 14 we prove the existence of a weak delta-like solution of the Hopf
equation $u_{t}(x,t)+u(x,t)u_{x}(x,t)=0$ in the framework of
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$. This solution has counterparts neither
in the spaces of classical functions such as
$\mathcal{C}^{k}(\mathbb{R}^{d}),\;k=1,2,\dots,\infty$, nor in the spaces of
Schwartz distributions such as $\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Our
result improves a similar result by M. Radyna [11] obtained in the spirit of
perturbation theory.
## 2 Ordered Fields
We will begin by defining ordered fields and giving examples of some well
known and some lesser known orderings on the (non-archimedean) fields of
rational functions and Laurent series.
###### Definition 2.1
Let $\mathbb{K}$ be a field (ring). $\mathbb{K}$ is called orderable if there
exists a nonempty subset $\mathbb{K}_{+}\subset\mathbb{K}$ such that
* (1)
$0\notin\mathbb{K}_{+}$
* (2)
$x,y\in\mathbb{K}_{+}\Longrightarrow x+y,xy\in\mathbb{K}_{+}$
* (3)
For every non-zero $x\in\mathbb{K}$, either $x\in\mathbb{K}_{+}$ or
$-x\in\mathbb{K}_{+}$
$\mathbb{K}_{+}$ generates an order relation $<_{\mathbb{K}}$ on $\mathbb{K}$
as follows: $x<_{\mathbb{K}}y$ i f f $y-x\in\mathbb{K}_{+}.$
$(\mathbb{K},<_{\mathbb{K}})$ is called a _totally ordered field_ or simply an
_ordered field_.
We will write $<$ instead of $<_{\mathbb{K}}$ when it is clear from context
which field’s order relation we are referring to.
In addition to $(\mathbb{R},<)$ (where $<$ is the usual order on $\mathbb{R}$)
there are many (more interesting) examples of ordered fields. But first let us
make a short detour:
###### Example 2.1
$\mathbb{C}$, the set of complex numbers, is not orderable.
_Proof_ Suppose there exists a subset $\mathbb{C}_{+}$ satisfying Definition
2.1. Then consider:
Case 1 Suppose $i\in\mathbb{C}_{+}$. Then $i\cdot i=-1\in\mathbb{C}_{+}$,
implying $(-1)\cdot(-1)=1\in\mathbb{C}_{+}$. This is impossible since
$-1+1=0\notin\mathbb{C}_{+}$.
Case 2 Suppose $i\notin\mathbb{C}_{+}$. Then $-i\in\mathbb{C}_{+}$, implying
$(-i)\cdot(-i)=-1\in\mathbb{C}_{+}$, leading to the same contradiction as in
Case 1. $\blacktriangle$
The previous example can be generalized as follows:
###### Theorem 2.1
A field $\mathbb{K}$ is orderable i f f it is formally real. This means that
for every $n\in\mathbb{N}$ and every $x_{k}\in\mathbb{K}$
$\sum_{k=1}^{n}x_{k}^{2}=0\quad implies\quad x_{k}=0\quad for\quad all\quad
k.$
For details on the subject of formally real fields and the proof of this
theorem, see (Van Der Waerden [18], Chapter 11).
###### Definition 2.2
Let $\mathbb{K}$ and $\mathbb{L}$ be ordered fields. If
$\varphi:\;\mathbb{K}\longrightarrow\mathbb{L}$ is a field homomorphism that
preserves order, i.e. $x<_{\mathbb{K}}y$ implies
$\varphi(x)<_{\mathbb{L}}\varphi(y)$, then $\varphi$ is said to be an ordered
field homomorphism.
Ordered field isomorphisms and ordered field embeddings are defined similarly.
###### Remark 2.1
There exists an ordered field embedding from $\mathbb{Q}$ into any ordered
field $\mathbb{K}$. We call it the canonical embedding of $\mathbb{Q}$ into
$\mathbb{K}$ and it is defined by: $\sigma(0)=0$, $\sigma(n)=n\cdot 1$ and
$\sigma(-n)=-\sigma(n)$ for $n\in\mathbb{N}$, and
$\sigma(p/q)=\sigma(p)/\sigma(q)$ for $p,\,q\in\mathbb{Z},\,q\neq 0$.
From now on, if $x\in\mathbb{N},\mathbb{Z},$ or $\mathbb{Q}$, we will refer to
$x$ and $\sigma(x)\in\mathbb{K}$ interchangeably.
###### Example 2.2
Let $(\mathbb{R},<)$ be the field of real numbers with the usual order, and
let $\mathbb{R}(x)$ be the set of rational functions in the variable $x$ with
coefficients in $\mathbb{R}$. Note that we may think of $\mathbb{R}$ as a
subfield of $\mathbb{R}(x)$, as represented by the constant functions. Then
define $\mathbb{R}(x)_{+}=\\{R(x):R(x)\in\mathbb{R}(x)$ and there exists
$x_{0}\in\mathbb{R}$ such that $R(x)>0$ whenever $x>x_{0}\\}.$
The ordered field generated by $\mathbb{R}(x)_{+}$, which we will refer to
simply as $\mathbb{R}(x)$, has some surprising properties. Namely:
* (i)
$\mathbb{R}(x)$ contains infinitely large elements like $f(x)=x$. This means
that $f(x)>n$ for all $n\in\mathbb{N}$ (let $x_{0}=n$).
* (ii)
$\mathbb{R}(x)$ contains positive infinitesimals like $g(x)=\frac{1}{x}$. This
means that $0<g(x)<\frac{1}{n}$ for all $n\in\mathbb{N}$ (let $x_{0}=n$).
###### Remark 2.2
Let $\mathbb{L}$ be an ordered integral domain (an ordered ring without zero
divisors) and $\mathbb{K}$ be the field of fractions of $\mathbb{L}$. Define
$\mathbb{K}_{+}=\\{\frac{x}{y}:x,y\in\mathbb{L}_{+}\;or\;-x,-y\in\mathbb{L}_{+}\\}$
The order generated by $\mathbb{K}_{+}$ is the only one which extends the
order in $\mathbb{L}$. It is said to be the order inherited from $\mathbb{L}$.
###### Example 2.3
With Remark (2.2) in mind, we may revisit Example (2.2).
If $\mathbb{R}[x]$ is the ring of polynomials over $\mathbb{R}$, we may define
$\mathbb{R}[x]_{+}=\\{P(x)\in\mathbb{R}[x]$: $lead(P)>0\\}$, where $lead(P)$
is the leading coefficient of $P(x)$.
The order generated on $\mathbb{R}[x]$ by $\mathbb{R}[x]_{+}$ can be extended
to $\mathbb{R}(x)$ since $\mathbb{R}(x)$ is the field of fractions of
$\mathbb{R}[x]$. That is, we may redefine
$\mathbb{R}(x)_{+}=\\{\frac{P(x)}{Q(x)}:P(x),Q(x)\in\mathbb{R}[x]\;and\;lead(P),lead(Q)>0\;or\;lead(P),lead(Q)<0\\}$
This definition is equivalent to that given previously and the orders
generated by the two are in fact one and the same. Now we may plainly see that
$f(x)=x$ is indeed an infinitely large element since $lead(x-n)=1>0$ for all
$n\in\mathbb{N}$.
Similarly, $g(x)=\frac{1}{x}$ is a positive infinitesimal because
$\frac{1}{n}-\frac{1}{x}=\frac{x-n}{nx}$ and $lead(x-n),lead(nx)>0$ for all
$n\in\mathbb{N}$.
###### Example 2.4
The set
$\mathbb{R}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}\frac{a_{n}}{x^{n}}:\quad
a_{n}\in\mathbb{R}\quad,\quad m\in\mathbb{Z},\quad and\quad a_{m}\neq 0\\}$
of Laurent series with coefficients in $\mathbb{R}$ is a field under normal
polynomial addition and multiplication.
We may define an order on $\mathbb{R}(x^{\mathbb{Z}})$ by
$\mathbb{R}_{+}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}\frac{a_{n}}{x^{n}}\in\mathbb{R}(x^{\mathbb{Z}}):\quad
a_{m}>0\\}$
Here, an element such as
$\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+\ldots$
is infinitesimal.
###### Example 2.5
The field of (formal) Laurent Series may also be defined as follows:
$\mathbb{R}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}a_{n}{x^{n}}:\quad
a_{n}\in\mathbb{R},\quad m\in\mathbb{Z},\quad and\quad a_{m}\neq 0\\}$
If we now let
$\mathbb{R}_{+}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}a_{n}{x^{n}}\in\mathbb{R}(x^{\mathbb{Z}}):\quad
a_{m}>0\\}$
then even a series that is divergent for all x, such as
$x+2x^{2}+6x^{3}+\ldots+n!x^{n}+\ldots$
is an infinitesimal in this field.
## 3 Filters and Ultrafilters
In this section we define and give examples of filters on an arbitrary
infinite set. Having done so, we will prove the existence of an ultrafilter
using the axiom of choice.
###### Definition 3.1
Let $I$ be an infinite set and let
$\mathcal{F}\subset\mathcal{P}(I),\;\mathcal{F}\neq\varnothing$. If
$\mathcal{F}$ satisfies:
* (F1)
If $A\in\mathcal{F}$ and $A\subset B\subset I$, then $B\in\mathcal{F}$.
* (F2)
$A,\,B\in\mathcal{F}$ implies $A\cap B\in\mathcal{F}$
* (F3)
$\varnothing\notin\mathcal{F}$
then $\mathcal{F}$ is a filter on $I$. If it is also true that
* (F4)
$\bigcap_{A\in\mathcal{F}}A=\varnothing$
then $\mathcal{F}$ is called a free filter on $I$. A filter $\mathcal{F}$ is
called countably incomplete if:
* (F5)
There exists a sequence of decreasing sets $I=I_{0}\supset I_{1}\supset
I_{2}\supset\ldots$ in $\mathcal{F}$ such that
$\bigcap_{n=0}^{\infty}I_{n}=\varnothing$.
If $\mathcal{F}$ is a filter on $I$, it follows immediately from the
definition that:
* (i)
$I\in\mathcal{F}$
* (ii)
$\mathcal{F}$ is closed under finite intersections.
* (iii)
If $A\in\mathcal{F}$ then $I\setminus A\notin\mathcal{F}$
* (iv)
If $\mathcal{F}$ is countably incomplete, then $\mathcal{F}$ is free.
###### Definition 3.2
A filter $\mathcal{U}$ on a set $I$ is called an ultrafilter if for every
filter $\mathcal{F}$ on $I$, $\mathcal{U}$ is not a proper subset of
$\mathcal{F}$. That is, there is no filter $\mathcal{F}$ on $I$ that properly
contains $\mathcal{U}$.
###### Theorem 3.1
Let $\mathcal{U}$ be a filter on $I$. Then $\mathcal{U}$ is an ultrafilter on
$I$ i f f for every $A\subset I$ either $A\in\mathcal{U}$ or $I\setminus
A\in\mathcal{U}$.
_Proof_ Suppose $\mathcal{U}$ is an ultrafilter on $I$ and $A,I\setminus
A\notin\mathcal{U}$. Let $\widehat{\mathcal{U}}=\\{X\;:\;A\cup
X\in\mathcal{U}\\}$. It is not hard to check that $\widehat{\mathcal{U}}$ is a
filter that properly contains $\mathcal{U}$, since $I\setminus
A\in\widehat{\mathcal{U}}$. Thus $\mathcal{U}$ cannot be an ultrafilter. To
prove the other direction, suppose, to the contrary, that $\mathcal{U}$ is not
an ultrafilter. Then there exists a filter $\mathcal{V}$ which is a proper
extension of $\mathcal{U}$. Let $A\in\mathcal{V}\setminus\mathcal{U}$. Then,
since $\mathcal{U}\subset\mathcal{V}$, we have that $A\cap X\neq\varnothing$
for all $X\in\mathcal{U}$. But $I\setminus A\in\mathcal{U}$ by assumption, so
$A\cap\left(I\setminus A\right)\neq\varnothing$, a contradiction.
$\blacktriangle$.
###### Example 3.1
Let $I=\mathbb{N}$, and fix $n\in\mathbb{N}$. Then
$\mathcal{U}=\\{X\;:\;X\subset\mathbb{N},\;n\in X\\}$ is an ultrafilter on
$\mathbb{N}$. However, $\mathcal{U}$ is clearly not free.
###### Example 3.2
The filter $\mathcal{F}_{r}(\mathbb{N})$ consisting of all cofinite sets of
natural numbers is called the Fréchet filter on $\mathbb{N}$. It is free and
countably incomplete since
$\bigcap_{n=1}^{\infty}\left(\mathbb{N}\setminus\\{n\\}\right)=\varnothing$
However, since neither the set of even numbers nor the set of odd numbers is
in $\mathcal{F}_{r}(\mathbb{N})$, by Theorem (3.1), it is not an ultrafilter.
###### Theorem 3.2
Let $I$ be an infinite set and $\mathcal{F}$ a free filter on $I$. Then there
exists a free ultrafilter $\mathcal{U}$ on $I$ such that
$\mathcal{F}\subseteq\mathcal{U}$.
_Proof_ Let $\widehat{\mathcal{F}}$ be the set of all filters on $I$ that
contain $\mathcal{F}$. $\widehat{\mathcal{F}}\neq\varnothing$ since
$\mathcal{F}\in\widehat{\mathcal{F}}$. Let $\widehat{\mathcal{F}}$ be ordered
by set inclusion, and consider a linearly ordered subset
$\mathcal{M}\subseteq\widehat{\mathcal{F}}$. Define
$\widehat{\mathcal{M}}=\bigcup_{M\in\mathcal{M}}M$. Note that if
$A\in\widehat{\mathcal{M}}$ then $A\in M$ for some $M\in\mathcal{M}$. Thus, if
$A\subset B\subset I$, it follows that $B\in M$, implying
$B\in\widehat{\mathcal{M}}$. Also, if $A,B\in\widehat{\mathcal{M}}$, then we
must have that $A\in M_{1}$ and $B\in M_{2}$ for some
$M_{1},M_{2}\in\mathcal{M}$. Since $\mathcal{M}$ is linearly ordered, we may
assume without loss of generality that $M_{1}\subset M_{2}$. Thus $A,B\in
M_{2}$. Hence $A\cap B\in M_{2}$, implying $A\cap B\in\widehat{\mathcal{M}}$.
Finally, we have that $\varnothing\notin\widehat{\mathcal{M}}$ because
otherwise $\varnothing$ would be an element of some filter $M\in\mathcal{M}$,
which is impossible. We have just shown that $\widehat{\mathcal{M}}$ is itself
a filter. But since the choice of $\mathcal{M}$ was arbitrary, we can conclude
that every linearly ordered subset of $\widehat{\mathcal{F}}$ has an upper
bound in $\widehat{\mathcal{F}}$. Thus, by Zorn’s Lemma,
$\widehat{\mathcal{F}}$ has a maximal element, $\mathcal{U}$, which is an
ultrafilter on $I$ containing $\mathcal{F}$. Also, since
$\mathcal{F}\subset\mathcal{U}$ and $\mathcal{F}$ is free, we have that
$\bigcap_{A\in\mathcal{U}}A\subseteq\bigcap_{A\in\mathcal{F}}A=\varnothing$.
Therefore $\mathcal{U}$ is free. $\blacktriangle$.
## 4 Ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$
Here we define an ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$, the set of
test functions, in order to construct ordered, non-archimedean fields of non-
standard real and complex numbers, ${}^{*}\mathbb{R}$ and ${}^{*}\mathbb{C}$,
respectively.
###### Definition 4.1
Let $\mathcal{D}(\mathbb{R}^{d})$ be the set of test functions on
$\mathbb{R}^{d}$. That is,
$\mathcal{D}(\mathbb{R}^{d})=\mathcal{C}^{\infty}_{0}(\mathbb{R}^{d})$. For
every $n\in\mathbb{N}$, define the basic set $\mathcal{B}_{n}$ by
$\displaystyle\mathcal{B}_{n}=\\{$
$\displaystyle\varphi\in\mathcal{D}(\mathbb{R}^{d})\,:\ $
$\displaystyle\varphi\text{ is real-valued and symmetric},$
$\displaystyle\varphi(x)=0\text{ for all }x\in\mathbb{R}^{d},\|x\|\geq 1/n,$
$\displaystyle\int\varphi=1$ $\displaystyle\int x^{\alpha}\varphi=0\text{ for
all }\alpha\in\mathbb{N}^{d}_{0},1\leq|\alpha|\leq n,$ $\displaystyle
1\leq\int|\varphi|<1+\frac{1}{n}\\}$
and $\mathcal{B}_{0}=\mathcal{D}(\mathbb{R}^{d})$.
###### Theorem 4.1
* (i)
$\mathcal{B}_{n}\neq\varnothing$ for all n.
* (ii)
$\mathcal{B}_{0}\supset\mathcal{B}_{1}\supset\mathcal{B}_{2}\supset\ldots$
* (iii)
$\bigcap_{n}\mathcal{B}_{n}=\varnothing$
_Proof_ For the proof of (i), see (Oberguggenberger and Todorov [10]). (ii)
follows from Definition (4.1). For (iii), suppose there were a function
$\varphi$ such that $\varphi\in\bigcap_{n}\mathcal{B}_{n}$ for all $n$. Then
consider $\widehat{\varphi}(\xi)=\int\varphi(x)e^{i\xi x}dx$, the Fourier
transform of $\varphi$. Since $\varphi\in\mathcal{D}(\mathbb{R}^{d})$,
$\widehat{\varphi}$ is entire (_Bremermann_ [1] Lemma 8.11, p.85). Therefore,
we can write
$\widehat{\varphi}(\xi)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}\frac{(\partial^{\alpha}\widehat{\varphi})(0)}{\alpha!}\xi^{\alpha}$.
But $0=i^{|\alpha|}\int x^{\alpha}\varphi(x)dx=\left.i^{|\alpha|}\int
x^{\alpha}\varphi(x)e^{i\xi
x}dx\right|_{\xi=0}=(\partial^{\alpha}\widehat{\varphi})(0)$ for all
$\alpha\neq 0$. It follows that $\widehat{\varphi}$ is constant. However, by
the same lemma as before, we also have that
$\lim_{|\xi|\to\infty}\widehat{\varphi}(\xi)=0$. Thus $\widehat{\varphi}=0$,
implying $\varphi(x)=0$. This contradicts the property that $\int\varphi=1$.
Hence $\bigcap_{n}\mathcal{B}_{n}=\varnothing$. $\blacktriangle$.
###### Definition 4.2
Define the basic filter $\mathcal{F}_{\mathcal{B}}$ on
$\mathcal{D}(\mathbb{R}^{d})$ by
$\mathcal{F}_{\mathcal{B}}=\\{\Phi\subseteq\mathcal{D}(\mathbb{R}^{d})\;:\;\mathcal{B}_{n}\subseteq\Phi\text{
for some }n\in\mathbb{N}\\}.$
Since each $\mathcal{B}_{n}$ is itself an element of
$\mathcal{F}_{\mathcal{B}}$, it follows from Theorem (4.1) that
$\mathcal{F}_{\mathcal{B}}$ is countably incomplete, and therefore free. Thus,
by Theorem (3.2), there exists an ultrafilter $\mathcal{U}$ on
$\mathcal{D}(\mathbb{R}^{d})$ containing $\mathcal{F}_{\mathcal{B}}$. We shall
keep $\mathcal{U}$ fixed in what follows.
## 5 Non-Standard Numbers
We will now use the ultrafilter defined in the previous section to construct
fields of non-standard real and complex numbers.
###### Definition 5.1
Let $\mathcal{U}$ be as before, and let
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ be the ring of functionals from
$\mathcal{D}(\mathbb{R}^{d})$ to $\mathbb{C}$ supplied with pointwise addition
and multiplication. We shall denote these functionals as “families”
$(A_{\varphi})$ and treat the domain $\mathcal{D}(\mathbb{R}^{d})$ as an
“index set”.
We may define the operations of absolute value, real part mapping, imaginary
part mapping, and complex conjugation on the elements of
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by:
$\displaystyle|(A_{\varphi})|=(|A_{\varphi}|)$
$\displaystyle\Re(A_{\varphi})=(\Re A_{\varphi})$
$\displaystyle\Im(A_{\varphi})=(\Im A_{\varphi})$
$\displaystyle\overline{(A_{\varphi})}=(\overline{A_{\varphi}})$
Also, we may define an embedding of $\mathbb{C}$ into
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by $c\rightarrow(C_{\varphi})$
where $C_{\varphi}=c$ for all $\varphi\in\mathcal{D}(\mathbb{R}^{d})$.
Define an equivalence relation $\sim_{\mathcal{U}}$ on
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by
$(A_{\varphi})\sim_{\mathcal{U}}(B_{\varphi})\;\;if\;\;\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}=B_{\varphi}\\}\in\mathcal{U}$
Finally, let
${}^{*}\mathbb{C}=\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}/\sim_{\mathcal{U}}$.
That is, ${}^{*}\mathbb{C}$ consists of equivalence classes of functionals in
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. We may write
$\langle(A_{\varphi})\rangle$ to represent these classes, but to simplify
notation we will denote by $\langle A_{\varphi}\rangle\in$ ${}^{*}\mathbb{C}$
the non-standard number (equivalence class of functionals) with representative
$(A_{\varphi})$. ${}^{*}\mathbb{C}$ is called a field of complex non-standard
numbers.
${}^{*}\mathbb{C}$ inherits the operations and embedding mentioned above from
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. With the embedding in mind, we
shall treat elements of $\mathbb{C}$ as their images in ${}^{*}\mathbb{C}$.
A non-standard number $\langle A_{\varphi}\rangle$ is called real if
$\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}\in\mathbb{R}\\}\in\mathcal{U}$
We denote the set of all real non-standard numbers by ${}^{*}\mathbb{R}$ and
supply it with an order relation as follows:
$\langle A_{\varphi}\rangle>_{{}^{*}\mathbb{R}}0\;\text{ if
}\;\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}>0\\}\in\mathcal{U}$
###### Theorem 5.1
* (i)
Every number $\gamma\in\ ^{*}\mathbb{C}$ can be uniquely represented in the
form $\gamma=\alpha+\beta i$ where $\alpha,\beta\in$ ${}^{*}\mathbb{R}$ and
$\alpha=\Re\gamma$, $\beta=\Im\gamma$, and
$|\gamma|=\sqrt{\alpha^{2}+\beta^{2}}$.
* (ii)
${}^{*}\mathbb{C}$ is an algebraically closed non-Archimedean field of
characteristic zero. $\mathbb{C}$ is a subfield of ${}^{*}\mathbb{C}$.
* (iii)
${}^{*}\mathbb{R}$ is a totally ordered non-Archimedean real closed field.
Moreover, $\alpha>0$ in ${}^{*}\mathbb{R}$ i f f $\alpha=\beta^{2}$ for some
$\beta\in$ ${}^{*}\mathbb{R}$, $\beta\neq 0$. $\mathbb{R}$ is an ordered
subfield of ${}^{*}\mathbb{R}$.
_Proof_ (i) Let $\gamma\in\,^{*}\mathbb{C}$. Then $\gamma=\langle
C_{\varphi}\rangle$ for some
$(C_{\varphi})\in\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. But for each
$\varphi$, $C_{\varphi}=A_{\varphi}+B_{\varphi}i$, where $A_{\varphi}=\Re
C_{\varphi}$ and $B_{\varphi}=\Im C_{\varphi}$. Thus $\langle
C_{\varphi}\rangle=\langle A_{\varphi}\rangle+\langle B_{\varphi}\rangle i$.
To prove uniqueness, suppose that $\langle C_{\varphi}\rangle=\langle
D_{\varphi}\rangle+\langle E_{\varphi}\rangle i$ also. Then
$\\{\varphi:\Re C_{\varphi}=D_{\varphi}\\}\cap\\{\varphi:\Re
C_{\varphi}=A_{\varphi}\\}=\\{\varphi:A_{\varphi}=D_{\varphi}\\}\in\mathcal{U}$
because $\mathcal{U}$ is closed under intersections. Therefore $\langle
A_{\varphi}\rangle=\langle D_{\varphi}\rangle$.
The same argument can be applied to show that $\langle
B_{\varphi}\rangle=\langle E_{\varphi}\rangle$.
The proof for $|\gamma|$ is similar.
(ii) It is not hard to check that $\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$
really is a ring, and that $\sim_{\mathcal{U}}$ really is an equivalence
relation. It follows that ${}^{*}\mathbb{C}$ is a (commutative) ring. To prove
that ${}^{*}\mathbb{C}$ is a field, we must show that each non-zero element
has a multiplicative inverse. For any non-zero $\gamma\in\,^{*}\mathbb{C}$, we
may choose a representative $(C_{\varphi})$ such that $C_{\varphi}\neq 0$ for
all $\varphi$. Let $D_{\varphi}=1/C_{\varphi}$ and $\delta=\langle
D_{\varphi}\rangle$. Then $\delta\gamma=\langle 1\rangle$.
Let
$P(x)=\sum_{k=0}^{n}\alpha_{k}x^{k},\;\alpha_{k}\in\,^{*}\mathbb{C}\text{ for
all k }$
be a polynomial in ${}^{*}\mathbb{C}[x]$. Define
$P_{\varphi}(x)=\sum_{k=0}^{n}A_{k,\varphi}x^{k}$
where $\alpha_{k}=\langle A_{k,\varphi}\rangle$ for each k. Since each
$P_{\varphi}(x)$ is a polynomial over $\mathbb{C}$, there exists a number
$C_{\varphi}\in\mathbb{C}$ such that $P_{\varphi}(C_{\varphi})=0$. If we let
$\gamma=\langle C_{\varphi}\rangle$, it follows that $P(\gamma)=0$ in
${}^{*}\mathbb{C}$.
That $\mathbb{C}$ is a subfield of ${}^{*}\mathbb{C}$ is clear from the
embedding.
(iii) The trichotomy of the order relation on ${}^{*}\mathbb{R}$ follows from
the trichotomy of the order relation on $\mathbb{R}$. For suppose
$\mathcal{A}=\\{\varphi:A_{\varphi}<B_{\varphi}\\}$,
$\mathcal{B}=\\{\varphi:A_{\varphi}=B_{\varphi}\\}$, and
$\mathcal{C}=\\{\varphi:A_{\varphi}>B_{\varphi}\\}$, for some non-standard
real numbers $\langle A_{\varphi}\rangle$, $\langle B_{\varphi}\rangle$. Note
that $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are mutually disjoint.
Therefore, at most one of $\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ can
be in $\mathcal{U}$. Also,
$\mathcal{A}\cup\mathcal{B}\cup\mathcal{C}=\mathcal{D}(\mathbb{R}^{d})\in\mathcal{U}$.
We can use this to prove that one of $\mathcal{A}$, $\mathcal{B}$, or
$\mathcal{C}$ must be in $\mathcal{U}$. For suppose that none of
$\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ is in $\mathcal{U}$. Then by
Theorem (3.1), $\mathcal{B}\cup\mathcal{C}\in\mathcal{U}$ and
$\mathcal{A}\cup\mathcal{C}\in\mathcal{U}$. Taking the intersection of these
two sets, we would have $\mathcal{C}\in\mathcal{U}$, a contradiction.
$\blacktriangle$
###### Definition 5.2
Define the sets of infinitesimal, finite, and infinitely large numbers as
follows:
$\mathcal{I}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|<1/n\text{ for all
}n\in\mathbb{N}\\}$
$\mathcal{F}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|<n\text{ for some
}n\in\mathbb{N}\\}$
$\mathcal{L}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|>n\text{ for all
}n\in\mathbb{N}\\}$
It is not hard to prove that $\mathcal{F}(^{*}\mathbb{C})$ is a subring of
${}^{*}\mathbb{C}$ and $\mathcal{I}(^{*}\mathbb{C})$ is a maximal ideal in
$\mathcal{F}(^{*}\mathbb{C})$.
###### Example 5.1
Define $(R_{\varphi})\in\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by
$R_{\varphi}=\sup\\{\|x\|:x\in{\rm{supp}}\varphi\\}$
where ${\rm{supp}}\varphi=\overline{\\{x\in\mathbb{R}^{d}:\varphi(x)\neq
0\\}}$ is the support of $\varphi$. The non-standard number $\rho=\langle
R_{\varphi}\rangle$ is a (positive) infinitesimal. For let
$\mathcal{A}=\\{\varphi:0<R_{\varphi}<1/n\\}$. Then for any
$\varphi\in\mathcal{B}_{n+1}$, we have $\varphi\in\mathcal{A}$, by the
definition of $\mathcal{B}_{n+1}$. Thus $\mathcal{B}_{n+1}\subset\mathcal{A}$,
implying $\mathcal{A}\in\mathcal{U}$. $\rho$ is called the canonical
infinitesimal in ${}^{*}\mathbb{C}$.
###### Definition 5.3
Define the standard part mapping $st:$
${}^{*}\mathbb{R}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ by
$st(x)=\begin{cases}\sup\\{r\in\mathbb{R}:r<x\\}&\text{if
$x\in\mathcal{F}(^{*}\mathbb{R})$}\\\ \infty&\text{if
$x\in\mathcal{L}(^{*}\mathbb{R}_{+})$}\\\ -\infty&\text{if
$x\in\mathcal{L}(^{*}\mathbb{R}_{-})$}\end{cases}$
We may extend this definition to ${}^{*}\mathbb{C}$ by
$st(x+yi)=st(x)+st(y)i$.
###### Theorem 5.2
If $x\in\mathcal{F}(^{*}\mathbb{C})$ then $x$ has a unique asymptotic
expansion: $x=r+dx$ where $r\in\mathbb{C}$ and
$dx\in\mathcal{I}(^{*}\mathbb{C})$. In fact, $r=st(x)$.
_Proof_ We will prove the case for $x\in\mathcal{F}(^{*}\mathbb{R})$. The
general result will follow. Let $x\in\mathcal{F}(^{*}\mathbb{R})$. First note
that $x-st(x)\in\mathcal{I}(^{*}\mathbb{R})$, for otherwise we would have
$|x-st(x)|>1/n$ for some n, implying either that $st(x)>x$ or that
$st(x)+1/2n<x$. In either case, this is a contradiction to Definition (5.3).
To prove uniqueness, suppose that $x=r+dx$ and $x=s+dy$ are two expansions of
$x$. Then we would have $r-s=dx-dy$, implying that
$r-s\in\mathcal{I}(^{*}\mathbb{R})$. But since $r-s\in\mathbb{R}$, $r-s=0$.
Hence $r=s$. Therefore $r+dx=r+dy$, implying $dx=dy$. $\blacktriangle$
## 6 Internal Sets
In non-standard analysis, internal sets play the role of the “good” sets, in a
similar way to the measurable sets in Lebesgue theory.
In what follows we will use the abbreviation a.e. to mean that the set of
functions for which some statement is true is in $\mathcal{U}$.
###### Definition 6.1
Let $\mathbb{A}\subseteq\mathbb{C}$. The non-standard extension of
$\mathbb{A}$ is
${}^{*}\mathbb{A}=\\{\langle
A_{\varphi}\rangle\in\,^{*}\mathbb{C}:A_{\varphi}\in\mathbb{A}\;a.e.\\}$
A set $\mathcal{A}$ of non-standard numbers is called internal standard if it
is the non-standard extension of some subset of $\mathbb{C}$. The set of all
internal standard sets is denoted by ${}^{\sigma}\mathcal{P}(\mathbb{C})$.
###### Example 6.1
The non-standard extensions of the intervals $(a,\,b)$, $[a,b]$, $(a,\infty)$,
etc. are
${}^{*}(a,b)=\\{x\in\ ^{*}\mathbb{R}:\,a<x<b\\}$ ${}^{*}[a,b]=\\{x\in\
^{*}\mathbb{R}:\,a\leq x\leq b\\}$ ${}^{*}(a,\infty)=\\{x\in\
^{*}\mathbb{R}:\,a<x\\},\;etc.$
###### Definition 6.2
Let
$(\mathbb{A}_{\varphi})\in\mathcal{P}(\mathbb{C})^{\mathcal{D}(\mathbb{R}^{d})}$
be a family of subsets of $\mathbb{C}$. We define the internal set generated
by $(\mathbb{A}_{\varphi})$ by
$\langle\mathbb{A}_{\varphi}\rangle=\\{\langle
A_{\varphi}\rangle\in\,^{*}\mathbb{C}:A_{\varphi}\in\mathbb{A}_{\varphi}\;a.e.\\}$
A set is called external if it is not internal.
###### Example 6.2
Let $\mathbb{A}_{\varphi}=(0,R_{\varphi})$, where $R_{\varphi}$ is as in
Example (5.1). Then the internal set $\langle\mathbb{A}_{\varphi}\rangle$
generated by $(\mathbb{A}_{\varphi})$ is the internal interval $(0,\rho)$. It
is important to note (and easy to check) that this coincides with the more
natural definition for $(0,\rho)$ given by
$(0,\rho)=\\{x\in\,^{*}\mathbb{R}:0<x<\rho\\}$
A set $S\subset\mathbb{R}^{d}$ is called relatively compact if its closure
$\overline{S}$ is compact in $\mathbb{R}^{d}$. Unless it is specified
otherwise, we shall call Lebesgue measurable sets of $\mathbb{R}^{d}$ simply
measurable sets.
###### Definition 6.3
An internal set $\large\langle\mathbb{A}_{\varphi}\large\rangle$ of
${}^{*}\mathbb{R}^{d}$ is called $*$-measurable ($*$-compact, $*$-relatively-
compact, $*$-closed, $*$-open, etc.) if
$\mathbb{A}_{\varphi}\text{\;is measurable (compact, relatively compact,
closed, etc.) in \;}\mathbb{R}^{d}\text{ for a.e. }\varphi$
Let $\rho\in{{}^{*}\mathbb{R}}$ denote a positive infinitesimal in
${}^{*}\mathbb{R}$ (for example, $\rho$ might be the positive infinitesimal
defined in (Example 5.1)). We shall keep $\rho$ fixed in what follows.
###### Definition 6.4
Let $\rho$ be a positive infinitesimal in ${}^{*}\mathbb{R}$. We define the
following (external) sets of non-standard numbers:
$\displaystyle\mathcal{M}_{\rho}({{}^{*}\mathbb{C}})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|\leq\rho^{-n}\text{\>
for some\;}n\in\mathbb{N}\\}$
$\displaystyle\mathcal{N}_{\rho}({{}^{*}\mathbb{C}})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|<\rho^{n}\text{\>
for all\;}n\in\mathbb{N}\\}$
$\displaystyle\mathcal{F}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|<1/\sqrt[n]{\rho}\text{\;
for all\; }n\in\mathbb{N}\\},$
$\displaystyle\mathcal{I}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|\leq\sqrt[n]{\rho}\text{\;
for some\; }n\in\mathbb{N}\\},$
$\displaystyle\mathcal{C}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;\sqrt[n]{\rho}<|x|<1/\sqrt[n]{\rho}\text{\;
for all\; }n\in\mathbb{N}\\}.$
The numbers in $\mathcal{M}_{\rho}(^{*}\mathbb{C})$ and
$\mathcal{N}_{\rho}(^{*}\mathbb{C})$ are called $\rho$-moderate and
$\rho$-null non-standard numbers, respectively. Similarly, the numbers in
$\mathcal{F}_{\rho}(^{*}\mathbb{C})$, $\mathcal{I}_{\rho}(^{*}\mathbb{C})$ and
$\mathcal{C}_{\rho}(^{*}\mathbb{C})$ are called $\rho$-finite,
$\rho$-infinitesimal and $\rho$-constant, respectively.
## 7 Saturation Principle in ${}^{*}\mathbb{C}$
###### Theorem 7.1
Let $\\{\mathcal{A}_{n}\\}$ be a sequence of internal sets in
${}^{*}\mathbb{C}$ such that
$\bigcap_{n=0}^{m}\mathcal{A}_{n}\neq\varnothing$
for all $m\in\mathbb{N}$. (The sequence $\\{\mathcal{A}_{n}\\}$ satisfies the
finite intersection property.) Then
$\bigcap_{n=0}^{\infty}\mathcal{A}_{n}\neq\varnothing.$
_Proof_ Since each $\mathcal{A}_{n}$ is internal,
$\mathcal{A}_{n}=\langle\mathbb{A}_{n,\varphi}\rangle\text{ ,
$\mathbb{A}_{n,\varphi}\subseteq\mathbb{C}$}$
Also, since for each $m$ it is given that
$\bigcap_{n=0}^{m}\mathcal{A}_{n}\neq\varnothing$, this implies that for each
$m$ there exists a non-standard number $\langle C_{m,\varphi}\rangle\in$
${}^{*}\mathbb{C}$ such that
$\langle
C_{m,\varphi}\rangle\in\bigcap_{n=0}^{m}\langle\mathbb{A}_{n,\varphi}\rangle$
or, in other words,
$\langle C_{m,\varphi}\rangle\in\langle\mathbb{A}_{n,\varphi}\rangle\text{ for
}0\leq n\leq m$
This means that for a.e. $\varphi$ and $0\leq n\leq m$,
$C_{m,\varphi}\in\mathbb{A}_{n,\varphi}$
Remembering that $\mathcal{U}$ is closed under finite intersections, we see
that for a.e. $\varphi$,
$C_{m,\varphi}\in\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}$
Hence for each $m$,
$\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\;a.e.$
We may assume without loss of generality that $\mathbb{A}_{0,\varphi}$ is non-
empty for all $\varphi$. (Else define
$\mathbb{A^{\prime}}_{0,\varphi}=\mathbb{A}_{0,\varphi}$ if
$\mathbb{A}_{0,\varphi}\neq\varnothing$ and
$\mathbb{A^{\prime}}_{0,\varphi}=\mathbb{C}$ otherwise. Then it will still be
true that $\mathcal{A}_{0}=\langle\mathbb{A^{\prime}}_{0,\varphi}\rangle$ .)
Next define a function
$\mu:\mathcal{D}(\mathbb{R}^{D})\longrightarrow\mathbb{N}\cup\\{\infty\\}$ by
$\mu(\varphi)=\max\\{m\in\mathbb{N}_{0}\cup\\{\infty\\}\,|\,\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\\}$
Notice that $\mu$ is defined for all $\varphi$ due to our assumption for
$\mathbb{A}_{0,\varphi}$.
Thus we have
$\bigcap_{n=0}^{\mu(\varphi)}\mathbb{A}_{n,\varphi}\neq\varnothing\text{ for
all }\varphi\in\mathcal{D}(\mathbb{R}^{d})$
Hence for every $\varphi\in\mathcal{D}(\mathbb{R}^{d})$ there exists (by Axiom
of Choice) $A_{\varphi}$ such that
$A_{\varphi}\in\bigcap_{n=0}^{\mu(\varphi)}\mathbb{A}_{n,\varphi}$.
We intend to show that
$\langle A_{\varphi}\rangle\in\bigcap_{n=0}^{\infty}\mathcal{A}_{n}$
or, equivalently, that for every $m$, $A_{\varphi}\in\mathbb{A}_{m,\varphi}$
for a.e. $\varphi$.
If $\varphi$ is such that
$\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing$, this implies that
$0\leq m\leq\mu(\varphi)$. Thus $A_{\varphi}\in\mathbb{A}_{m,\varphi}$, by the
choice of $A_{\varphi}$. Therefore,
$\\{\varphi\,|\,\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\\}\subseteq\\{\varphi\,|\,A_{\varphi}\in\mathbb{A}_{m,\varphi}\\}$
But the set on the left is in $\mathcal{U}$, and so the set on the right is
also, as required. $\blacktriangle$
## 8 Non-Standard Smooth Functions
Having constructed the fields ${}^{*}\mathbb{R}$ and ${}^{*}\mathbb{C}$, the
natural next step is to look at functions on these fields. However, for our
purposes we will focus on a certain class of function contained in
${}^{*}\mathbb{C}^{{}^{*}\mathbb{R}}$. In what follows
$\mathcal{E}(\mathbb{R})$ is the set of $\mathcal{C}^{\infty}$-functions from
$\mathbb{R}$ into $\mathbb{C}$.
###### Definition 8.1
A function $f\in\,^{*}\mathbb{C}^{{}^{*}\mathbb{R}}$ is called internal smooth
if there exists a family
$(f_{\varphi})\in\mathcal{E}(\mathbb{R})^{\mathcal{D}(\mathbb{R}^{d})}$ such
that for every $x=\langle X_{\varphi}\rangle\in\,^{*}\mathbb{R}$
$f(x)=\langle f_{\varphi}(X_{\varphi})\rangle$
The set of all internal smooth functions will be denoted by
${}^{*}\mathcal{E}(\mathbb{R})$.
###### Remark 8.1
${}^{*}\mathcal{E}(\mathbb{R})$ may equivalently be defined as the set of
equivalence classes $\langle f_{\varphi}\rangle$ of families of functions in
$\mathcal{E}(\mathbb{R})^{\mathcal{D}(\mathbb{R}^{d})}$, where the equivalence
relation is as usual:
$(f_{\varphi})\sim_{\mathcal{U}}(g_{\varphi})\text{ if
}f_{\varphi}=g_{\varphi}\text{ for a.e. }\varphi$
It is not hard to prove that the value of an internal function does not depend
on the choice of representatives. If $\langle X_{\varphi}\rangle=\langle
Y_{\varphi}\rangle\in\,^{*}\mathbb{R}$ and $\langle f_{\varphi}\rangle=\langle
g_{\varphi}\rangle\in\,^{*}\mathcal{E}(\mathbb{R})$, then
$\\{\varphi\,|\,f_{\varphi}(X_{\varphi})=g_{\varphi}(X_{\varphi})\\}\cap\\{\varphi\,|\,g_{\varphi}(X_{\varphi})=g_{\varphi}(Y_{\varphi})\\}\subseteq\\{\varphi\,|\,f_{\varphi}(X_{\varphi})=g_{\varphi}(Y_{\varphi})\\}$
Since $\mathcal{U}$ is closed under intersections, $\langle
f_{\varphi}(X_{\varphi})\rangle=\langle g_{\varphi}(Y_{\varphi})\rangle$.
The operations of addition, multiplication, and partial differentiation in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ are inherited from
$\mathcal{E}(\mathbb{R}^{d})$. Also, $\mathcal{E}(\mathbb{R}^{d})$ is embedded
in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ by $f\longrightarrow\,^{*}f$ where
${}^{*}f=\langle f_{\varphi}\rangle$, $f_{\varphi}=f$ for all
$\varphi\in\mathcal{D}(\mathbb{R}^{d})$.
In what follows integrable means Lebesgue integrable.
###### Definition 8.2
Let
$\large\langle\mathbb{X}_{\varphi}\large\rangle\subseteq\,^{*}\mathbb{R}^{d}$
be a $*$-measurable internal set and let $\large\langle
f_{\varphi}\large\rangle\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ be an internal
function. We say that $\large\langle f_{\varphi}\large\rangle$ is
$*$-integrable over $\large\langle\mathbb{X}_{\varphi}\large\rangle$ if
$f_{\varphi}\text{\; is integrable over\; }\mathbb{X}_{\varphi}\text{ for a.e.
}\varphi$
If $\large\langle f_{\varphi}\large\rangle$ is $*$-integrable over
$\large\langle\mathbb{X}_{\varphi}\large\rangle$, we define the integral:
(1) $\int_{\large\langle\mathbb{X}_{\varphi}\large\rangle}\large\langle
f_{\varphi}\large\rangle(x)\,dx=\left<\int_{\mathbb{X}_{\varphi}}f_{\varphi}(x)\,dx\right>.$
We also say that the integral converges in ${}^{*}\mathbb{C}$ (since it is a
number in ${}^{*}\mathbb{C}$). Notice that as long as the integral converges
for a.e. $\varphi$, we include this object in the equivalence class, even if
the integral diverges for other $\varphi$.
## 9 Internal Sets and Saturation Principle in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$
We define internal sets in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ similarly to
those of ${}^{*}\mathbb{C}$.
###### Definition 9.1
* (i)
Let
$(\mathbb{F}_{\varphi})\in\mathcal{P}(\mathcal{E}(\mathbb{R}^{d}))^{\mathcal{D}(\mathbb{R}^{d})}$
be a family of subsets of $\mathcal{E}(\mathbb{R}^{d})$. We define the
internal set generated by $(\mathbb{F}_{\varphi})$ by
$\langle\mathbb{F}_{\varphi}\rangle=\\{\langle
f_{\varphi}\rangle\in\,^{*}\mathcal{E}(\mathbb{R}^{d}):f_{\varphi}\in\mathbb{F}_{\varphi}\;a.e.\\}$
A set is called external if it is not internal.
* (ii)
An internal set $\mathcal{F}$ is called standard if there exists
$\mathbb{F}\subset\mathcal{E}(\mathbb{R}^{d})$ such that
$\mathcal{F}=\langle\mathbb{F}\rangle$. In this case we may also write
$\mathcal{F}=\,^{*}\mathbb{F}$.
###### Theorem 9.1
Let $\\{\mathcal{F}_{n}\\}$ be a sequence of internal sets in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ such that
$\bigcap_{n=0}^{m}\mathcal{F}_{n}\neq\varnothing$
for all $m\in\mathbb{N}$. (The sequence $\\{\mathcal{F}_{n}\\}$ satisfies the
finite intersection property.) Then
$\bigcap_{n=0}^{\infty}\mathcal{F}_{n}\neq\varnothing.$
_Proof_ The proof is almost identical to that of (Theorem 7.1).
###### Definition 9.2
We define the following (external) subsets of
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$:
$\displaystyle\mathcal{F}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{F}(^{*}\mathbb{C})\right]\\},$
$\displaystyle\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{I}(^{*}\mathbb{C})\right]\\},$
$\displaystyle\mathcal{M}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{M}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{N}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{F}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{F}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{I}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{I}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{C}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{C}_{\rho}(^{*}\mathbb{C})\right]\right\\}.$
The functions in
$\mathcal{F}(^{*}\mathcal{E}(\mathbb{R}^{d})),\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))$,
$\mathcal{M}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))$,
$\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$,
$\mathcal{F}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$,
$\mathcal{I}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$, and
$\mathcal{C}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$ are called finite,
infinitesimal, $\rho$-moderate, $\rho$-null, $\rho$\- finite,
$\rho$-infinitesimal and $\rho$-constant functions, respectively. For more
details we refer to (Lightstone and Robinson [7]) and (Wolf and Todorov [17].
## 10 Weak Equality
###### Definition 10.1
Let $x,y\in\,^{*}\mathbb{C}$, $f,\,g\in\,^{*}\mathcal{E}(\mathbb{R}^{d})$
* (i)
$x\approx y$ if $x-y\in\mathcal{I}(^{*}\mathbb{C})$
* (ii)
$x\overset{\rho}{=}y$ if $x-y\in\mathcal{N}_{\rho}(^{*}\mathbb{C})$
* (iii)
$f\approx g$ if $f-g\in\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))$
* (iv)
$f\overset{\rho}{=}g$ if
$f-g\in\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))$
* (v)
$f\cong g$ if $\int f(x)\tau(x)\,dx=\int g(x)\tau(x)\,dx$ for every
$\tau\in\mathcal{D}(\mathbb{R}^{d})$
* (vi)
$f\overset{\rho}{\simeq}g$ if $\int f(x)\tau(x)\,dx\overset{\rho}{=}\int
g(x)\tau(x)\,dx$ for every
$\tau\in\mathcal{D}(\mathbb{R}^{d})$
* (vii)
$f\approxeq g$ if $\int f(x)\tau(x)\,dx\approx\int g(x)\tau(x)\,dx$ for every
$\tau\in\mathcal{D}(\mathbb{R}^{d})$
It is not hard to prove that each of these weak equalities forms an
equivalence relation in its respective space. Many results in non-standard
analysis hold weakly in the sense of one of these weak equalities.
## 11 Schwartz Distributions
At this point, we must take a short detour to present some basic definitions
and results from the Schwartz theory.
###### Definition 11.1
A distribution is a mapping
$F:\mathcal{D}(\mathbb{R}^{d})\longrightarrow\mathbb{C}$ that satisfies the
following conditions:
* (i)
Linearity:
$F[c_{1}\tau_{1}+c_{2}\tau_{2}]=c_{1}F[\tau_{1}]+c_{2}F[\tau_{2}]$ for all
$c_{1},c_{2}\in\mathbb{C}$ and
$\tau_{1},\tau_{2}\in\mathcal{D}(\mathbb{R}^{d})$.
* (ii)
Continuity: Let $\\{\tau_{k}\\}$ be a sequence in
$\mathcal{D}(\mathbb{R}^{d})$. Suppose there exists $R$ such that
${\rm{supp}}\tau_{k}\subseteq\\{x:\;|x|<R\\}$ for all $k$. Also, suppose there
exists $\tau\in\mathcal{D}(\mathbb{R}^{d})$ such that
$\partial^{\alpha}\tau_{k}\longrightarrow\partial^{\alpha}\tau$ for all
$\alpha\in\mathbb{N}_{0}^{d}$ uniformly as $k\longrightarrow\infty$. Then
$F[\tau_{k}]\longrightarrow F[\tau]$.
We will denote by $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ the set of all such
distributions.
We supply $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ with the usual pointwise
addition and scalar multiplication. In addition, we define partial
differentiation by
$(\partial^{\alpha}F)[\tau]=(-1)^{|\alpha|}F[\partial^{\alpha}\tau]$
and multiplication by a smooth function $g\in\mathcal{E}(\mathbb{R}^{d})$ by
$(gF)[\tau]=F[g\tau]$
Both of these operations are well-defined since
$\partial^{\alpha}\tau,\,g\tau\in\mathcal{D}(\mathbb{R}^{d})$.
$\mathcal{L}_{loc}(\mathbb{R}^{d})$, the set of locally integrable functions,
is embedded in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ by the mapping
$S(f)=\int f(t)\tau(t)dt$
It is not hard to show that this embedding preserves the operations mentioned
above.
Finally, we define the convolution of a distribution with a test function by
$(F\ast\tau)(x)=F[\tau(x-t)]$
###### Theorem 11.1
If $F\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$ and
$\tau\in\mathcal{D}(\mathbb{R}^{d})$ then
$(F\ast\tau)(x)\in\mathcal{E}(\mathbb{R}^{d})$ and
$\partial^{\alpha}(F\ast\tau)=F\ast\partial^{\alpha}\tau$.
Before proving this theorem, we will state (without proof) a result from
analysis. See (Rudin [13] p.148):
###### Lemma 11.1
Suppose
$\lim_{n\to\infty}f_{n}(x)=f(x)\quad(x\in E).$
Put
$M_{n}=\sup_{x\in E}|f_{n}(x)-f(x)|.$
Then $f_{n}\longrightarrow f$ uniformly on $E$ if and only if
$M_{n}\longrightarrow 0$ as $n\longrightarrow\infty$ .
_Proof of the theorem_ We will prove the theorem for the case d=1. The general
result will follow.
Let $f(x)=(F\ast\tau)(x)$. Fixing $x$, we wish to show that
$f(x+h)-f(x)\longrightarrow 0\text{ as }h\longrightarrow 0$
Note that
$\displaystyle f(x+h)-f(x)$ $\displaystyle=(F\ast\tau)(x+h)-(F\ast\tau)(x)$
$\displaystyle=F[\tau(x+h-t)]-F[\tau(x-t)]$
$\displaystyle=F[\tau(x+h-t)-\tau(x-t)]$
by the linearity of $F$. Let
$\psi(t)=\tau(x+h-t)-\tau(x-t)$
$\psi$ is itself a test function, and if we restrict $|h|<1$, then the support
of $\psi$ and all its derivatives is contained in $E=\\{y\,|\,|y|\leq
r+|x|+1\\}$, where $r$ is the radius of the support of $\tau$. It is clear
that any sequence $\\{\psi_{h_{n}}\\}$ where $h_{n}\longrightarrow 0$ as
$n\longrightarrow\infty$ converges pointwise to 0 for all $x$ and $t$ (by the
uniform continuity of $\tau$). Also, since one compact set, $E$, contains the
support of $\psi_{h_{n}}$ for all n, and since each $\psi_{h_{n}}$ is
continuous, $M_{n}=\sup_{t\in E}|\psi_{h_{n}}(t)|$ is achieved by
$\psi_{h_{n}}$ for each $n$. Thus $M_{n}\to 0$, implying that
$\\{\psi_{h_{n}}\\}\to 0$ uniformly, by the Lemma. Therefore, since $F$ is
continuous in the sense of (Definition 11.1),
$F[\psi_{h_{n}}]\longrightarrow F[0]=0$
Since $x$ was chosen arbitrarily, this proves that $f=F\ast\tau$ is
continuous.
To prove that $f^{\prime}$ exists and that
$f^{\prime}(x)=(F\ast\tau^{\prime})(x)$, we must show that
$\frac{f(x+h)-f(x)}{h}-(F\ast\tau^{\prime})(x)\longrightarrow 0\text{ as
}h\longrightarrow 0$
Note that
$\displaystyle\frac{f(x+h)-f(x)}{h}-(F\ast\tau^{\prime})(x)$
$\displaystyle=\frac{(F\ast\tau)(x+h)-(F\ast\tau)(x)}{h}-(F\ast\tau^{\prime})(x)$
$\displaystyle=F\left[\frac{\tau(x+h-t)-\tau(x-t)}{h}-\tau^{\prime}(x-t)\right]$
Now, if we let
$\chi(t)=\frac{\tau(x+h-t)-\tau(x-t)}{h}-\tau^{\prime}(x-t)$
we can use the same argument as before to show that
$F[\chi_{h_{n}}(t)]\longrightarrow F[0]=0\text{ as }h\longrightarrow 0$
This proves that $(F\ast\tau)^{\prime}=F\ast\tau^{\prime}$. Since
$\tau^{\prime}$ is itself a test function, the same proof works to show that
$(F\ast\tau)^{\prime\prime}=F\ast\tau^{\prime\prime}$ and so on. For functions
of several variables, the same argument can be applied in each variable to
show the general result. $\blacktriangle$
Before we can prove the embedding of the distributions in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$, we need a result showing that
distributions can be “approximated” in a way by a certain sequence of test
functions.
###### Theorem 11.2
Let $\\{\delta_{n}\\}$ be a sequence in $\mathcal{D}(\mathbb{R}^{d})$ such
that $\delta_{n}\in\mathcal{B}_{n}$ for every $n$. Then for any distribution
$T\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, $T\ast\delta_{n}\to T$ weakly. (A
sequence of distributions $\\{F_{k}\\}$ converges weakly to a distribution $F$
if $F_{k}[\tau]\to F[\tau]$ for all test functions $\tau$.)
Before proving this theorem, we need two lemmas:
###### Lemma 11.2
Let $\\{\delta_{n}\\}$ be as above and let $\tau$ be any test function. Then
there exists $R$ such that ${\rm{supp}}(\delta_{n}\ast\tau)\subset\\{x:|x|\leq
R\\}$. Also, $\partial^{\alpha}(\delta_{n}\ast\tau)\to\tau$ uniformly for
every $\alpha\in\mathbb{N}_{0}^{d}$.
_Proof_ For each $n$, ${\rm{supp}}\delta_{n}\subset\\{x:|x|\leq 1/n\\}$. In
particular, ${\rm{supp}}\delta_{n}\subset\\{x:|x|\leq 1\\}$. If we let
$R_{\tau}$ be the radius of the support of $\tau$ and set $R=R_{\tau}+1$, then
it is not hard to see that ${\rm{supp}}(\delta_{n}\ast\tau)\subset\\{x:|x|\leq
R\\}$.
As before, to show the uniform convergence it is enough to prove that
$\sup_{|x|\leq R}|(\delta_{n}\ast\tau)(x)-\tau(x)|\to 0$
Recalling that $\int\delta_{n}=1$, we see that
$\displaystyle\sup_{|x|\leq R}|(\delta_{n}\ast\tau)(x)-\tau(x)|$
$\displaystyle=\sup_{|x|\leq
R}\left|\int\delta_{n}(t)\tau(x-t)dt-\tau(x)\int\delta_{n}(t)dt\right|$
$\displaystyle=\sup_{|x|\leq R}\left|\int_{|t|\leq
1/n}\delta_{n}(t)[\tau(x-t)-\tau(x)]dt\right|$
By the mean value theorem for integrals, there exists $|t_{n}|\leq 1/n$ such
that
$=\sup_{|x|\leq R}\left|[\tau(x-t_{n})-\tau(x)]\int\delta_{n}(t)dt\right|$
and by the extreme value theorem there exists $|x_{n}|\leq R$ such that
$=\left|\tau(x_{n}-t_{n})-\tau(x_{n})\right|$
This last expression vanishes as $n\to\infty$ since $\tau$ is uniformly
continuous. The case $\alpha\neq 0$ is similar. $\blacktriangle$
For the proof of the next lemma see (Folland [5] p.318):
###### Lemma 11.3
Suppose $F$ is a distribution and $\phi$ and $\psi$ are test functions. Then
$(F\ast\phi)[\psi]=F[\tilde{\phi}\ast\psi]$, where $\tilde{\phi}(x)=\phi(-x)$.
_Proof of the theorem_ We must show that for any distribution $T$ and any test
function $\tau$,
$(T\ast\delta_{n})[\tau(x)]\to T[\tau(x)]$
Using (Lemma 11.3) and remembering that $\delta_{n}$ is symmetric for all $n$,
$(T\ast\delta_{n})[\tau(x)]=T[(\delta_{n}\ast\tau)(x)]\to T[\tau(x)]$
by (Lemma 11.2) and the continuity of $T$. $\blacktriangle$
## 12 Embedding of Schwartz Distributions in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$
Finally, we are ready to define the embedding $\Sigma$ of
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ into
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ as follows:
$\Sigma(T)=\langle T\ast\varphi\rangle$
By (theorem 11.1), $\Sigma(T)\in\,^{*}\mathcal{E}(\mathbb{R}^{d})$. From the
definition of the convolution, it is clear that $\Sigma$ is linear. It remains
to prove that $\Sigma$ is injective.
###### Lemma 12.1
$\Sigma$ is injective.
_Proof_ Since $\Sigma$ is linear, it is enough to show that $\Sigma(T)=0$
implies $T=0$.
If $\Sigma(T)=0$, we have that $T\ast\varphi=0$ a.e. That is,
$\Phi=\\{\varphi\,|\,T\ast\varphi=0\\}\in\mathcal{U}$. Thus
$\varnothing\neq\Phi\cap\mathcal{B}_{n}\in\mathcal{U}$ for each $n$, where
$\mathcal{B}_{n}$ are the basic sets. Therefore we can construct a sequence
$\\{\varphi_{n}\\}$ such that $\varphi_{n}\in\Phi\cap\mathcal{B}_{n}$ for each
$n$. Then by (Theorem 11.2), we have that $T=0$ since $T\ast\varphi_{n}=0$ for
every $n$. $\blacktriangle$
###### Theorem 12.1
* (i)
$\langle P\ast\varphi\rangle=\,^{*}P$ for every polynomial
$P\in\mathbb{C}[x_{1},\ldots,x_{d}]$
* (ii)
$\langle f\ast\varphi\rangle\overset{\rho}{=}\,^{*}f$ for all
$f\in\mathcal{E}(\mathbb{R}^{d})$
_Proof_ (i) Let $P\in\mathbb{C}[x_{1},\ldots,x_{d}]$ be a polynomial of degree
$p$. By the Taylor formula,
$P(x-t)=P(x)+\sum_{|\alpha|=1}^{p}\frac{(-1^{|\alpha|})\partial^{\alpha}P(x)}{\alpha!}t^{\alpha}$
It follows that for every test function $\varphi$ and $x\in\mathbb{R}^{d}$
$(P\ast\varphi)(x)=\int
P(x-t)\varphi(t)dt=P(x)\int\varphi(t)dt+\sum_{|\alpha|=1}^{p}\frac{(-1^{|\alpha|})\partial^{\alpha}P(x)}{\alpha!}\int
t^{\alpha}\varphi(t)dt$
Notice that if $\varphi\in\mathcal{B}_{n}$ for some $n\geq p$, then
$\int\varphi(t)dt=1$ and $\int t^{\alpha}\varphi(t)dt=0$,
$|\alpha|=1,2,\ldots,p$. Thus we have
$\mathcal{B}_{n}\subseteq\\{\varphi\,|\,P\ast\varphi=P\\}$
implying that $P\ast\varphi=P$ a.e. as required.
(ii) Let $\xi\in\mathcal{F}(^{*}\mathbb{R}^{d})$, $n\in\mathbb{N}$, and
$\alpha$ be a multi-index. We have to show that
$|\partial^{\alpha}(f\ast\varphi)(\xi)-\partial^{\alpha}f(\xi)|<\rho^{n}$. We
will show this for the case $\alpha=0$, the general result will follow.
Since $st(\xi)\in\mathbb{R}^{d}$ we can find an open relatively compact set
$\mathcal{O}\subset\mathbb{R}^{d}$ such that $st(\xi)\in\mathcal{O}$ and by
(Robinson [12], p.90 Theorem 4.1.4) $\xi\in\,^{*}\mathcal{O}$ and hence
$\xi\in\,^{*}\overline{\mathcal{O}}$.
As before, the Taylor formula gives
$f(x-t)=f(x)+\sum_{|\alpha|=1}^{n}\frac{(-1^{|\alpha|})\partial^{\alpha}f(x)}{\alpha!}t^{\alpha}+\sum_{|\alpha|=n+1}\frac{(-1)^{|\alpha|}\partial^{\alpha}f(\eta(x,t))}{\alpha!}t^{\alpha}$
where $\eta(x,t)$ is a point in $\mathbb{R}^{d}$ ”between $x$ and $t$”. It
follows that for every $\varphi\in\mathcal{D}(\mathbb{R}^{d})$
$(f\ast\varphi)(x)=f(x)\int\varphi(t)dt+\sum_{|\alpha|=1}^{n}\frac{(-1)^{|\alpha|}\partial^{\alpha}f(x)}{\alpha!}\int
t^{\alpha}\varphi(t)dt+$
$\sum_{|\alpha|=n+1}\int\frac{(-1)^{|\alpha|}\partial^{\alpha}f(\eta(x,t))}{\alpha!}t^{\alpha}\varphi(t)dt$
Letting
$M\overset{def}{=}2\sum_{|\alpha|=n+1}\sup_{x\in
K}\sup_{t\in{\rm{supp}}(\varphi)}\left|\frac{\partial^{\alpha}f(\eta(x,t))}{\alpha!}\right|$
we have that $M\rho^{n+1}<\rho^{n}$ since $M\in\mathbb{R}$ and $\rho$ is a
positive infinitesimal. In other words, if $\rho=\langle R_{\varphi}\rangle$,
then $MR_{\varphi}^{n+1}<R_{\varphi}^{n}$ a.e.
By the properties of $\mathcal{B}_{n}$, it follows that for a.e. $\varphi$,
$x\in K$,
$\displaystyle|(f\ast\varphi)(x)-f(x)|<\sup_{x\in K}|(f\ast\varphi)(x)-f(x)|$
$\displaystyle<\sum_{|\alpha|=n+1}\sup_{x\in
K}\sup_{t\in{\rm{supp}}(\varphi)}\left|\frac{\partial^{\alpha}f(\eta(x,t))}{\alpha!}\right|\left(\sup_{t\in{\rm{supp}}(\varphi)}\|t\|^{n-1}\right)\int|\varphi(t)|dt$
$\displaystyle\leq MR_{\varphi}^{n+1}<R_{\varphi}^{n}$
Finally, since $\xi\in\,^{*}K$, we have that
$|(f\ast\varphi)(\xi)-f(\xi)|<\rho^{n}$, as required. The general result
follows from the case $\alpha=0$ and the fact that
$\partial^{\alpha}(f\ast\varphi)=(\partial^{\alpha}f)\ast\varphi$.
$\blacktriangle$
## 13 Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$ and the Hopf Equation
The embedding in the previous section is done deliberately, with the intent of
showing that ${}^{*}\mathcal{E}$ is a natural extension of
$\mathcal{D}^{\prime}$ and an appropriate setting for the study of weak
solutions to non-linear partial differential equations, an abundance of which
arise from the conservation law of physics.
###### Theorem 13.1 (Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$)
Let $L\in\mathbb{R}_{+}\cup\\{\infty\\}$,
$F\in\mathcal{C}^{\infty}(\mathbb{C})$ and let ${}^{*}F$ be the non-standard
extension of $F$. Let $u\in{{}^{*}\mathcal{E}(\Omega)}$, where
$\Omega=(0,L)\times(0,\infty)$. Then the following are equivalent:
(i) $u_{t}(x,t)+\left[{}^{*}F(u(x,t))\right]_{x}=0$ for all
$x,t\in{{}^{*}\mathbb{R}},\;0<x<L,\;t>0$.
(ii) $u_{t}(x,t)+{{}^{*}F^{\prime}\left(u(x,t)\right)}\,u_{x}=0$ for all
$x,t\in{{}^{*}\mathbb{R}},\;0<x<L,\;t>0$.
(iii)
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx={{}^{*}F(u(a,t))}-{{}^{*}F(u(b,t))}$ for
every $a,b,t\in{{}^{*}\mathbb{R}},\;0<a<b<L,\;t>0$.
###### Remark 13.1
The term “conservation law” is due to (iii) which in a classical setting is
given by
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx={F(u(a,t))}-{F(u(b,t))}.$
Here $u(x,t)$ stands for the density of a physical quantity (the density of
the mass of a fluid, the density of the heat energy, etc.) in a rod of length
$L$ and $F(u(x,t))$ stands for the flux of the quantity from left to right
through the $x$-cross section. Then the above equality expresses the
conservation of this quantity in any $(a,b)$-segment of the rod. Recall that,
according to the classical theory, (i)-(iii) are equivalent for solutions $u$
in the class $\mathcal{C}^{2}(\Omega)$ and for all $x,t,a,b\in\mathbb{R}$ in
the corresponding intervals. The proof which follows can be generalized
(without new complications) in the case of more complicated flux $F(u,u_{x})$
or even $F(u,u_{x},u_{xx})$.
_Proof_ (i) $\Leftrightarrow$ (ii): The equivalency between (i) and (ii)
follows immediately from the fact that the partial differentiation and
extension mapping $*$ commute in ${}^{*}\mathcal{E}(\Omega)$ and the fact that
${}^{*}\mathcal{E}(\Omega)$ is a differential algebra (with Leibniz rule for
differentiation of products and chain rule). So, we have
$\left[{}^{*}F(u(x,t))\right]_{x}={(^{*}F)^{\prime}(u(x,t))\,u_{x}(x,t)}={{}^{*}F^{\prime}(u(x,t))\,u_{x}(x,t)},$
as required.
(i) $\Rightarrow$ (iii): We have
$a=\left<a_{\varphi}\right>,b=\left<b_{\varphi}\right>$ and
$u=\left<u_{\varphi}\right>$ for some families of real numbers
$(a_{\varphi}),(b_{\varphi})\in\mathbb{R}^{\mathcal{D}(\mathbb{R}^{2})}$ and
some family of smooth functions
$(u_{\varphi})\in\mathcal{E}(\Omega)^{\mathcal{D}(\mathbb{R}^{2})}$. We have
$\Phi=\\{\varphi\mid a_{\varphi}<b_{\varphi}\\}\in\mathcal{U}$ since $a<b$ in
${}^{*}\mathbb{R}$, by assumption. Thus (involving the classical arguments in
the framework of $\mathcal{E}(\Omega)$) we have $\Phi\subseteq\Phi_{1}$, where
$\Phi_{1}=\\{\varphi\mid\frac{d}{dt}\int_{a_{\varphi}}^{b_{\varphi}}\,u_{\varphi}(x,t)\,dx={F(u_{\varphi}(a_{\varphi},t))}-{F(u_{\varphi}(b_{\varphi},t))}\text{\;
for all\;}t\in\mathbb{R}_{+}\\}.$
The latter implies $\Phi_{1}\in\mathcal{U}$ which implies (iii), as required,
after transferring the result from representatives to the corresponding
equivalence classes.
(i) $\Leftarrow$ (iii): Suppose (on the contrary) that there exist
$\xi,\tau\in{{}^{*}\mathbb{R}},\,0<\xi<L,\,\tau>0$, such that
$u_{t}(\xi,\tau)+\left[{}^{*}F(u(\xi,\tau))\right]_{x}\not=0$ in
${}^{*}\mathbb{C}$. We have $\xi=\left<\xi_{\varphi}\right>$ and
$\tau=\left<\tau_{\varphi}\right>$ for some
$(\xi_{\varphi}),(\tau_{\varphi})\in\mathbb{R}^{\mathcal{D}(\mathbb{R}^{2})}$.
We denote
$\Phi=\\{\;\varphi\mid(u_{\varphi})_{t}(\xi_{\varphi},\tau_{\varphi})+\left[F(u_{\varphi}(\xi_{\varphi},\tau_{\varphi}))\right]_{x}\not=0\;\\}$
and observe that $\Phi\in\mathcal{U}$ (by our assumption). Also, we let
$\displaystyle\Phi_{\varphi}=\\{\;(\alpha,\beta,\gamma)\in\mathbb{R}^{3}\mid$
$\displaystyle\;\frac{d}{dt}\int_{\alpha}^{\beta}u_{\varphi}(x,\gamma)\,dx\not=F(u_{\varphi}(\alpha,\gamma))-F(u_{\varphi}(\beta,\gamma)),$
$\displaystyle\;0<\alpha<\beta<L,\;\gamma>0\;\\},$
and observe that $\Phi_{\varphi}\not=\varnothing$ for all $\varphi\in\Phi$ (by
the classical theory in the framework of $\mathcal{E}(\Omega)$). By axiom of
choice, there exist families
$(a_{\varphi}),(b_{\varphi}),(\gamma_{\varphi})\in\mathbb{R}^{\Phi}$ such that
$(a_{\varphi},b_{\varphi},\gamma_{\varphi})\in\Phi_{\varphi}$ for all
$\varphi\in\Phi$. If $\varphi\in\mathcal{D}(\mathbb{R}^{2})\setminus\Phi$, we
define $(a_{\varphi},b_{\varphi},\gamma_{\varphi})$ anyhow (say, by
$a_{\varphi}=b_{\varphi}=\gamma_{\varphi}=1$). These families (of real
numbers) determine the non-standard real numbers
$a=\left<a_{\varphi}\right>,b=\left<b_{\varphi}\right>$ and
$t=\left<\gamma_{\varphi}\right>$. Next, we observe that $\Phi\subseteq\Psi$
(by the definition of $\Phi_{\varphi}$), where
$\displaystyle\Psi=\\{\;\varphi\mid\;$
$\displaystyle\frac{d}{dt}\int_{a_{\varphi}}^{b_{\varphi}}u_{\varphi}(x,\gamma_{\varphi})\,dx\not=F(u_{\varphi}(a_{\varphi},\gamma_{\varphi}))-F(u_{\varphi}(b_{\varphi},\gamma_{\varphi})),$
$\displaystyle 0<a_{\varphi}<b_{\varphi}<L,\;\gamma_{\varphi}>0\\}.$
Next, $\Phi\in\mathcal{U}$ implies $\Psi\in\mathcal{U}$ which implies
$\frac{d}{dt}\int_{a}^{b}u(x,t)\,dx\not=F(u(a,t))-F(u(b,t)),\;0<a<b<L,\;t>0,$
in the framework of ${}^{*}\mathbb{C}$, contradicting (iii). The proof is
complete. $\blacktriangle$
###### Example 13.1 (Hopf Equation)
To appreciate the result of Theorem 13.1 we recall that (i)-(iii) might or
might not be equivalent in classes of classical functions and Schwartz
distributions larger than $\mathcal{C}^{2}(\Omega)$, where the (important for
the theory and applications) fundamental solutions and shock wave solutions
belong. For a discussion we refer to (J. David Logan [9], p. 309-310). Here is
an example: Let $F(u)=\frac{1}{2}u^{2}$ and $L=\infty$, so we have
$\Omega=\mathbb{R}^{2}_{+}$. In this case (i)-(iii) become:
(i) $u_{t}(x,t)+[\frac{1}{2}u(x,t)^{2}]_{x}=0$ for all
$x,t\in{\mathbb{R}},\;x>0,\;t>0$.
(ii) $u_{t}(x,t)+u(x,t)u_{x}(x,t)=0$ (Hopf equation) for all
$x,t\in{\mathbb{R}},\;x>0,\;t>0$.
(iii)
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx=\frac{1}{2}\left[u^{2}(a,t)-u^{2}(b,t)\right]$
for every $a,b,t\in{\mathbb{R}},\;0<a<b,\;t>0$,
respectively. Let $v\in\mathbb{R}_{+}$, $H$ be the Heaviside step function and
let $u(x,t)=2vH(x-vt)$ be a shock wave. The next analysis shows that (i), (ii)
and (iii) are not equivalent in the spaces of classical functions and Schwartz
distributions:
(i) Since $u=2vH(x-vt)\notin\mathcal{C}^{2}(\mathbb{R}^{2}_{+})$ this function
can not be a classical solution of (i). However, $u=2vH(x-vt)$ is a
(generalized) solution of (i) in the framework of the class of Schwartz
distributions $\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$. Indeed, for the
first term of (i) we have $u_{t}(x,t)=-2v^{2}\delta(x-vt)$, where $\delta(x)$
is the Dirac delta function. For the second term we have
$[\frac{1}{2}u(x,t)^{2}]_{x}=[\frac{4v^{2}}{2}H(x-vt)^{2}]_{x}=[2v^{2}H(x-vt)]_{x}=2v^{2}\delta(x-vt)$.
Thus $u(x,t)=2vH(x-vt)$ is a (generalized) solution of (i). We should notice
that $u(x,t)=2vH(x-vt)$ is also a weak solution of (i) in the framework of
$\mathcal{L}_{loc}(\mathbb{R}^{2}_{+})$ (see the remark below).
(ii) $u(x,t)=2vH(x-vt)$ is clearly not a solution of (ii) in classical sense.
Neither is it a (generalized) solution of (ii) in the class
$\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$ because the term
$uu_{x}=4v^{2}H(x-vt)\delta(x-vt)$ does not make sense within
$\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$ (recall that there is no
multiplication in $\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$).
(iii) For the LHS of (iii) we have
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx=\frac{d}{dt}\int_{a}^{b}\,2vH(x-vt)\,dx=2v\frac{d}{dt}\int_{a-vt}^{b-vt}\,H(x)\,dx=-2v^{2}H(vt-a)$.
For the RHS of (iii) we have
$\frac{1}{2}\left[u^{2}(a,t)-u^{2}(b,t)\right]=2v^{2}\left[H(a-vt)-H(b-vt)\right]=-2v^{2}H(vt-a)$.
Thus $u(x,t)=2vH(x-vt)$ is a solution of (iii).
###### Remark 13.2 (Weak Solution)
Suppose the $u\in\mathcal{L}_{loc}(\Omega)$ is a solution of
$u_{t}(x,t)+\left[F(u(x,t))\right]_{x}=0$ in the framework of
$\mathcal{D}^{\prime}(\Omega)$ (that means that both $u_{t}(x,t)$ and
$\left[F(u(x,t))\right]_{x}$ are in $\mathcal{D}^{\prime}(\Omega)$). These
solutions are often called weak solutions because they satisfy the weak
equality:
$\iint_{\Omega}\left[u(x,t)\tau_{t}(x,t)+F(u(x,t))\tau_{x}(x,t)\right]\,dx\,dt=0,$
for all test functions $\tau\in\mathcal{D}(\Omega)$. In
$\mathcal{D}^{\prime}(\Omega)$ we have:
$\iint_{\Omega}\left[u(x,t)\tau_{t}(x,t)+F(u(x,t))\tau_{x}(x,t)\right]\,dx\,dt=\left<u_{t}(x,t)+\left[F(u(x,t))\right]_{x},\tau(x,t)\right>,$
where $\left<\;,\;\right>$ stands for the pairing between
$\mathcal{D}^{\prime}(\Omega)$ and $\mathcal{D}(\Omega)$. Thus every
(generalized) solution in $\mathcal{D}^{\prime}(\Omega)$ is also weak solution
in $\mathcal{L}_{loc}(\Omega)$. In particular the function $u(x,t)=2vH(x-vt)$
in the example above is both a generalized and a weak solution of (i).
## 14 Generalized Delta-like Solution of the Hopf Equation
In this section we will prove the existence of a delta-like weak solution to
the Hopf equation of the type $\overset{\rho}{\simeq}$. That is, we are
looking for a function of the form
$u(x,t)=u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)$
where $\Theta\in\,^{*}\mathcal{S}(\mathbb{R})$ ($\mathcal{S}(\mathbb{R})$ is
the class of rapidly decreasing functions, such as $e^{-x^{2}}$,
${}^{*}\mathcal{S}$ is its non-standard extension, defined similarly to
${}^{*}\mathcal{E}$), $\int\Theta(x)dx=1$,
$u_{0},A,v\in\mathcal{M}_{\rho}(^{*}\mathbb{R})$ (we consider A to be the
amplitude of the soliton and $v$ its velocity), and for all $t>0$,
$u_{t}+uu_{x}\overset{\rho}{\simeq}0$
That is, for all $t>0$, $\tau\in\mathcal{D}^{\prime}(\mathbb{R})$,
$\int[u_{t}+uu_{x}]\tau(x)dx\overset{\rho}{=}0$
In addition, we would like this function to satisfy the conservation law, so
that for all $a,b\in\mathbb{R}$, $t>0$,
$\frac{d}{dt}\int_{a}^{b}u(x,t)dx\overset{\rho}{=}\frac{1}{2}[u^{2}(a,t)-u^{2}(b,t)]$
Calculating $u_{t}+uu_{x}$ we get
$u_{t}+uu_{x}=-\frac{Av}{\rho^{2}}\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)+\frac{u_{0}A}{\rho^{2}}\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)+\frac{A^{2}}{\rho^{3}}\Theta\left(\frac{x-vt}{\rho}\right)\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)$
Simplifying and letting
$\Theta\left(\frac{x-vt}{\rho}\right)\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)=\frac{\rho}{2}\left(\Theta^{2}\left(\frac{x-vt}{\rho}\right)\right)_{x}$
gives us
$\displaystyle\int[u_{t}+uu_{x}]\tau(x)dx$
$\displaystyle=\frac{(u_{0}-v)A}{\rho^{2}}\int\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)\tau(x)dx+\frac{A^{2}}{2\rho^{2}}\int\left(\Theta^{2}\left(\frac{x-vt}{\rho}\right)\right)_{x}\tau(x)dx$
Integrating by parts and making the substitution $y=\frac{x-vt}{\rho}$ gives
$=\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]\tau^{\prime}(vt+\rho
y)dy$
Finally, using the Taylor formula for $\tau^{\prime}(vt+\rho y)$, we have that
for each $m\in\mathbb{N}$,
$=\sum_{n=0}^{m}\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}\frac{\tau^{(n+1)}(vt)}{n!}\rho^{n}\,dy+R_{m}(\tau)$
where the remainder term is
$R_{m}(\tau)=\rho^{m+1}\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]\frac{\tau^{(m+2)}(\eta(y,t))}{(m+1)!}y^{m+1}dy$
We would like to find a function $\Theta$ such that for every $m$,
$\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}dy=0,\,0\leq
n\leq m$
and $|R_{m}(\tau)|<\rho^{m+k}$ (for some fixed k).
When $m=0$, we have that
$A=\frac{2\rho(v-u_{0})}{\int\Theta^{2}(y)dy}$
(remembering that $\int\Theta(y)dy=1$) Replacing this value of A, we have that
for every $m$,
$\int\Theta^{2}(y)dy\int\Theta(y)y^{n}dy=\int\Theta^{2}(y)y^{n}dy,\,0\leq
n\leq m$
Define
$S_{m}=\\{f\in\mathcal{S}\,:\,\int f(x)x^{n}dx=\frac{\int
f^{2}(x)x^{n}dx}{\int f^{2}(x)dx},\,0\leq n\leq m\\}$
For each $m$, $S_{m}$ is non-empty by (M. Radyna [11] p. 275).
Now let
$\displaystyle\overline{S}_{m}=\\{f\in\,^{*}S_{m}\,:$ $\displaystyle f(0)=0$
$\displaystyle|\ln\rho|^{-1}\int|f(x)x^{n}|<1$
$\displaystyle|\ln\rho|^{-1}\int|f^{2}(x)x^{n}|dx<1,\,0\leq n\leq m\\}$
The standard functions in ${}^{*}S_{m}$ certainly satisfy the second and third
conditions, since their integrals will be standard (and therefore finite) and
$|\ln\rho|^{-1}$ is infinitely small. As for the first condition, we can say
with certainty that if a function $f\in\,^{*}S_{m}$ has at least one zero, say
$f(-k)=0$, then $g(x)\overset{def}{=}f(x-k)\in\,^{*}S_{m}$ and $g(0)=0$ by the
following lemma:
###### Lemma 14.1
Suppose $f(x)$ satisfies
$\int f^{2}(x)dx\int f(x)x^{n}dx=\int f^{2}(x)x^{n}dx$
Then $g(x)=f(x-k)$ also satisfies
$\int g^{2}(x)dx\int g(x)x^{n}dx=\int g^{2}(x)x^{n}dx$
_Proof_ Substituting $y=x-k$, we get
$\displaystyle\int g^{2}(x)dx\int g(x)x^{n}dx$ $\displaystyle=\int
f^{2}(y)dy\int f(y)(y+k)^{n}dy$
$\displaystyle=\sum_{j=0}^{n}\begin{pmatrix}n\\\ j\end{pmatrix}k^{n-j}\int
f^{2}(y)dy\int f(y)y^{j}dy$ $\displaystyle=\sum_{j=0}^{n}\begin{pmatrix}n\\\
j\end{pmatrix}k^{n-j}\int f^{2}(y)y^{j}dy$ $\displaystyle=\int
f^{2}(y)\sum_{j=0}^{n}\begin{pmatrix}n\\\ j\end{pmatrix}y^{j}k^{n-j}dy$
$\displaystyle=\int f^{2}(y)(y+k)^{n}dy$ $\displaystyle=\int
f^{2}(x-k)x^{n}dx$ $\displaystyle=\int
g^{2}(x)x^{n}dx\quad\quad\blacktriangle$
Thus, if at least one function in ${}^{*}S_{m}$ has at least one zero, then
$\overline{S}_{m}$ will be non-empty. In addition, the sets $\overline{S}_{m}$
are internal and $\overline{S}_{0}\supset\,\overline{S}_{1}\supset\ldots$.
Therefore, by the saturation principle, there exists a function
$\Theta(x)\in\bigcap_{n=0}^{\infty}\overline{S}_{n}$.
Noting that $\int|\Theta(x)x^{n}|dx$ and $\int|\Theta^{2}(x)x^{n}|dx$ are at
most $\mathcal{C}_{\rho}(\mathbb{{}^{*}C})$, we have that for this $\Theta$,
$|R_{m}(\tau)|<\frac{\rho^{m+1}}{(m+1)!}\sup_{x\in\mathbb{R}}|\tau^{(m+2)}(x)|\int\left|\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}\right|dy<\rho^{m+k}$
where $k$ is some real constant that depends on $v,u_{0},$ and $A$. Therefore
$u(x,t)=u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)$ satisfies
the Hopf equation weakly, in the sense of $\overset{\rho}{\simeq}$.
If, in addition, it is true that $\Theta(0)=0$, then $u(x,t)$ will also
satisfy the conservation law:
for all $a,b\in\mathbb{R}$, $t>0$,
$\frac{d}{dt}\int_{a}^{b}u(x,t)dx\overset{\rho}{=}\frac{1}{2}[u^{2}(a,t)-u^{2}(b,t)]$
Let us prove this by first calculating the left side:
$\displaystyle\frac{d}{dt}\int_{a}^{b}\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)\right]dx$
$\displaystyle=\frac{d}{dt}\left[u_{0}(b-a)+\frac{A}{\rho}\int_{a}^{b}\Theta\left(\frac{x-vt}{\rho}\right)dx\right]$
$\displaystyle=\frac{d}{dt}\left[A\int_{\frac{a-vt}{\rho}}^{\frac{b-vt}{\rho}}\Theta(y)dy\right]$
$\displaystyle=A\left[\Theta\left(\frac{b-vt}{\rho}\right)\left(-\frac{v}{\rho}\right)-\Theta\left(\frac{a-vt}{\rho}\right)\left(-\frac{v}{\rho}\right)\right]$
$\displaystyle=A\frac{v}{\rho}\left[\Theta\left(\frac{a-vt}{\rho}\right)-\Theta\left(\frac{b-vt}{\rho}\right)\right]$
Since $\Theta\in\,^{*}\mathcal{S}$, if $a\neq vt$ and $b\neq vt$ then this
quantity vanishes. However, if $a=vt$ or $b=vt$ (but not both) then we have
that the left side equals $\pm\frac{Av}{\rho}\Theta(0)$, respectively. If
$\Theta(0)=0$, this also vanishes.
Calculating the right side, we have:
$\displaystyle\frac{1}{2}\left[\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{a-vt}{\rho}\right)\right]^{2}-\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{b-vt}{\rho}\right)\right]^{2}\right]$
$\displaystyle=\frac{1}{2}u_{0}^{2}+\frac{u_{0}A}{\rho}\Theta\left(\frac{a-vt}{\rho}\right)+\frac{A^{2}}{2\rho^{2}}\Theta^{2}\left(\frac{a-vt}{\rho}\right)$
$\displaystyle-\frac{1}{2}u_{0}^{2}-\frac{u_{0}A}{\rho}\Theta\left(\frac{b-vt}{\rho}\right)-\frac{A^{2}}{2\rho^{2}}\Theta^{2}\left(\frac{b-vt}{\rho}\right)$
$\displaystyle=\frac{u_{0}A}{\rho}\left[\Theta\left(\frac{a-vt}{\rho}\right)-\Theta\left(\frac{b-vt}{\rho}\right)\right]+\frac{A^{2}}{2\rho^{2}}\left[\Theta^{2}\left(\frac{a-vt}{\rho}\right)-\Theta^{2}\left(\frac{b-vt}{\rho}\right)\right]$
Again, we have that if $a\neq vt$ and $b\neq vt$ then this quantity vanishes.
If $a=vt$ or $b=vt$ (but not both) then the right side equals
$\pm\left(\frac{u_{0}A}{\rho}\Theta(0)+\frac{A^{2}}{2\rho^{2}}\Theta^{2}(0)\right)$,
respectively. Here also, if $\Theta(0)=0$ the right side vanishes, and so the
conservation law holds.
In conclusion, we may make some conjectures based on the relation
$A=\frac{2\rho(v-u_{0})}{\int\Theta^{2}(y)dy}$
There are many possibilities here, but if we assume (for simplicity) that
$\int\Theta^{2}(y)dy$ is finite (and not infinitesimal) and $u_{0}=0$, then
there are at least the following two particular cases:
* (i)
u has infinitesimal amplitude with finite or infinitely large velocity,
resembling a small signal, or
* (ii)
u has non-infinitesimal, finitely large amplitude, and infinitely large
velocity, resembling an explosion.
###### Remark 14.1 (Connection with Perturbation Theory)
The closest to our result is the work by M. Radyna [11] in the framework of
perturbation theory. M. Radyna proves the following result: For every
$n\in\mathbb{N}$ there exists a function
$\Theta_{n}\in\mathcal{S}(\mathbb{R})$ such that the function
$u(x,t)=\frac{A}{\epsilon}\Theta_{n}(\frac{x-vt}{\epsilon})$ satisfies:
$\left|\int_{\mathbb{R}}[u_{t}(x,t)+u(x,t)u_{x}(x,t)]\tau(x)\,dx\right|<\epsilon^{n},$
for every test function $\tau\in\mathcal{D}(\mathbb{R})$, every
$t\in\mathbb{R}$ and all sufficiently small $\epsilon\in\mathbb{R}$. For
comparison we mention the following:
(a) Instead of a small real parameter $\epsilon$ we use a proper positive
infinitesimal $\rho$. Our framework is, of course, quite different from M.
Radyna’s theory.
(b) In contrast to M. Radyna’s result, we have proved the existence of a
function $\Theta\in{{}^{*}\mathcal{S}(\mathbb{R})}$ (not depending on $n$)
such that the function $u(x,t)=\frac{A}{\rho}\Theta(\frac{x-vt}{\rho})$
satisfies:
$\left|\int_{\mathbb{R}}[u_{t}(x,t)+u(x,t)u_{x}(x,t))\tau(x)\,dx\right|<\rho^{n},$
for every test function $\tau\in\mathcal{D}(\mathbb{R})$, every
$t\in\mathbb{R}$ and for all $n\in\mathbb{N}$.
## References
* [1] H. Bremermann, Distributions, Complex Variables, and Fourier Transforms, Addison-Wesley Publ. Co., Inc., Palo Alto, 1965\.
* [2] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, 1973.
* [3] Colombeau, J. F.: New Generalized Functions and Multiplication of Distributions, North-Holland Math. Studies 84, North - Holland, Amsterdam, 1984.
* [4] R. Estrada and R. P. Kanwal, Asymptotic Analysis: A Distributional Approach, Birkhäuser, Boston, Basel, Berlin, 1994.
* [5] G. B. Folland, Fourier Analysis and its Applications, Brooks/Cole Publ. Co., New York, 1992.
* [6] T. Levi-Civita, Sugli Infiniti ed Infinitesimi Attuali Quali Elementi Analitici (1892-1893), Opere Mathematiche, vol. 1 (1954), p. 1-39.
* [7] A. H. Lightstone and A. Robinson, Nonarchimedean Fields and Asymptotic Expansions, North-Holland, Amsterdam, 1975.
* [8] T. Lindstrøm, An invitation to nonstandard analysis, in: Nonstandard Analysis and its Applications, N. Cutland (Ed), Cambridge U. Press, 1988, p. 1-105.
* [9] J. David Logan, Applied Mathematics, John Wiley & Sons, Inc, New York, 2nd Ed., 1996.
* [10] M. Oberguggenberger and T. Todorov, An embedding of Schwartz distributions in the algebra of asymptotic functions, Int’l. J. Math. and Math. Sci. 21 (1998), p.. 417-428.
* [11] Mikalai Radyna, New Model of Generalized Functions and Its Applcations to Hopf Equation, in A. Delcroix, M. Hasler, J.-A. Marti, V. Valmorin (Eds.), Nonlinear Algebraic Analysis and Applications, Proceedings of the International Conference on Generalized Functions (ICGF 2000), Cambridge Scientific Publ., Cottenham 2004, p. 269-288.
* [12] A. Robinson, Nonstandard Analysis, North Holland, Amsterdam, 1966.
* [13] W. Rudin, Principles of Mathematical Analysis (3rd ed.), McGraw-Hill, New York, 1976.
* [14] L. Schwartz, Sur l’impossibilit de la multiplication des distributions, C.R.Acad.Sci., Paris 239 (1954), p. 847-848.
* [15] Todor Todorov, An Existence of Solutions for Linear PDE with $C^{\infty}$-Coefficients in an Algebra of Generalized Functions, in Transactions of the American Mathematical Society, Vol. 348, 2, Feb. 1996, p. 673–689.
* [16] Todor Todorov, An Existence Result for a Class of Partial Differential Equations with Smooth Coefficients, In S. Albeverio, W.A.J. Luxemburg, M.P.H. Wolff (Eds.), “Advances in Analysis, Probability and Mathematical Physics; Contributions to Nonstandard Analysis”, Kluwer Acad. Publ., Dordrecht, Vol. 314, 1995, p. 107–121.
* [17] Todor Todorov and Robert Wolf, Hahn Field Representation of A. Robinson’s Asymptotic Numbers, in A. Delcroix, M. Hasler, J.-A. Marti, V. Valmorin (Eds.), Nonlinear Algebraic Analysis and Applications, Proceedings of the International Conference on Generalized Functions (ICGF 2000), Cambridge Scientific Publ., Cottenham 2004, p.357-374.
* [18] B. L. Van Der Waerden, Modern Algebra, Ungar Publishing, New York, third printing, 1964.
|
arxiv-papers
| 2008-09-25T05:52:34
|
2024-09-04T02:48:57.946849
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guy Berger",
"submitter": "Todor Todorov D.",
"url": "https://arxiv.org/abs/0809.4322"
}
|
0809.4553
|
# Numerical revision of the universal amplitude ratios for the two-dimensional
4-state Potts model
Lev N. Shchur∗,∗∗, Bertrand Berche∗∗ and Paolo Butera∗∗∗
∗ Landau Institute for Theoretical Physics,
Russian Academy of Sciences,
Chernogolovka 142432, Russia
∗∗ Laboratoire de Physique des Matériaux,
Université Henri Poincaré, Nancy 1
BP 239, F-54506 Vandœuvre les Nancy Cedex, France
∗∗∗ Istituto Nazionale di Fisica Nucleare,
Sezione di Milano-Bicocca,
Piazza delle Scienze 3, 20126, Milano, Italia
lev@landau.ac.ru,
berche@lpm.u-nancy.fr,
paolo.butera@mib.infn.it
###### Abstract
Monte Carlo (MC) simulations and series expansion (SE) data for the energy,
specific heat, magnetization and susceptibility of the ferromagnetic 4-state
Potts model on the square lattice are analyzed in a vicinity of the critical
point in order to estimate universal combinations of critical amplitudes. The
quality of the fits is improved using predictions of the renormalization group
(RG) approach and of conformal invariance, and restricting the data within an
appropriate temperature window.
The RG predictions on the cancelation of the logarithmic corrections in the
universal amplitude ratios are tested. A direct calculation of the effective
ratio of the energy amplitudes using duality relations explicitly demonstrates
this cancelation of logarithms, thus supporting the predictions of RG.
We emphasize the role of corrections and of background terms on the
determnination of the amplitudes. The ratios of the critical amplitudes of the
susceptibilities obtained in our analysis differ significantly from those
predicted theoretically and supported by earlier SE and MC analysis. This
disagreement might signal that the two-kink approximation used in the
analytical estimates is not sufficient to describe with fair accuracy the
amplitudes of the 4-state model.
## 1 Introduction
In a first paper [1] (hereafter denoted as I), we studied the universal
combinations of critical amplitudes of the 3-state Potts model. The present
paper is devoted to a similar analysis in the 4-state case, which is much more
involved due to the presence of logarithmic corrections strongly influencing
the critical behavior.
We analyze numerical data obtained in Monte Carlo (MC) simulations using the
Wolff [2] single-cluster algorithm and also the series expansion (SE) data
available in the literature. In the following, we refer to the data type as MC
and SE data, respectively. For comparison with our own results, we shall also
reconsider the data obtained in MC simulations using the Swendsen-Wang cluster
algorithm [3] by Caselle, Tateo, and Vinci [4] and indicated as CTV data.
Our motivation in using different data sets is to achieve a better control of
the critical behavior, since one may expect, for the three sources, some
differences in the critical region due to the different interplay of the
finite size effects. In addition, one can apply different techniques to the
data analysis: the fits in the case of the MC data and the approximant
technique in the case of the SE data. The consistency of the final results
will increase our confidence. We care so much because the presence of
logarithmic corrections makes the numerical determination of the critical
behavior of the 4-state Potts model a rather delicate task.
To be even safer, two different approaches are used, which were successfully
applied also to the $q=3$ state Potts model in I. First we estimate the
critical amplitudes, which are then used to compute universal ratios. Second,
besides the direct determination of the amplitudes themselves, we estimate
ratios of critical amplitudes, constructing effective ratio functions, and
computing their limiting values at the critical point. This provides a direct
estimate of universal ratios. Analyzing the renormalization group equations,
we have shown in an Appendix (see also [5, 6]) that, in the absence of any
regular background term, the logarithmic corrections cancel in the effective
ratio functions.
The Hamiltonian of the ferromagnetic Potts model [7] reads as
$H=-\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\;,$ (1)
where $s_{i}$ is a “spin” variable taking integer values between $0$ and
$q-1$, and the sum is restricted to the nearest neighbor sites $\langle
ij\rangle$ on a lattice of $N$ sites with periodic boundary conditions. The
partition function $Z$ is defined by
$Z=\sum_{conf}{\rm e}^{-\beta H}.$ (2)
with $\beta=1/k_{B}T$ 111According to the usual terminology, the inverse
temperature and the critical exponent of the magnetization are denoted by the
same symbol $\beta$, since there is no risk of confusion in this context., and
$k_{B}$ the Boltzmann constant (fixed to unity). On the square lattice in zero
magnetic field, the model is self-dual. Denoting by $\beta^{*}$ the dual of
the inverse temperature $\beta$, the duality relation
$\left(e^{\beta}-1\right)\left(e^{\beta^{*}}-1\right)=q$ (3)
determines the critical value of the inverse temperature [7]
$\beta_{c}=\ln(1+\sqrt{q})\approx 1.09861$. Dual reduced temperatures $\tau$
and $\tau^{*}$ can be defined by
$\beta=\beta_{c}(1-\tau)\;\;\;{\rm and}\;\;\;\beta^{*}=\beta_{c}(1+\tau^{*}).$
(4)
Close to the critical point, $\tau$ and $\tau^{*}$ coincide through first
order, since
$\tau^{*}=\tau+\frac{\ln(1+\sqrt{q})}{\sqrt{q}}\tau^{2}+O(\tau^{3})$.
The critical amplitudes and the critical exponents describe the singular
behavior of the thermodynamic quantities close to the critical point. For
example the magnetization $M$, the (reduced) susceptibility $\chi$ and the
specific heat $C$ of a spin system in zero external field222In this paper we
only deal with the physical properties in zero magnetic field. behave as
333Note that for simplicity we have dropped for the moment the multiplicative
logarithmic corrections and allowed only for additive corrections.
$\displaystyle M(\tau)$ $\displaystyle=$ $\displaystyle
B(-\tau)^{\beta}\left(1+{\rm corr.\;terms}\right),\ \tau<0$ (5)
$\displaystyle\chi(\tau)_{\pm}$ $\displaystyle=$
$\displaystyle\Gamma_{\pm}|\tau|^{-\gamma}\left(1+{\rm corr.\;terms}\right),$
(6) $\displaystyle C(\tau)_{\pm}$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-\alpha}\left(1+{\rm
corr.\;terms}\right),$ (7)
Here $\tau$ is the reduced temperature $\tau=(T-T_{c})/T$ and the labels $\pm$
refer to the high-temperature (HT) and low-temperature (LT) sides of the
critical temperature $T_{c}$. In addition to the mentioned observables, for
the Potts models with $q>2$ a transverse susceptibility can be defined in the
low-temperature phase444In the following we shall use the notations
$\Gamma_{L}$ or $\Gamma_{T}$ for the longitudinal or transverse susceptibility
amplitudes in the low-temperature phase. When still used, $\Gamma_{-}$ is
identified with $\Gamma_{L}$.
$\chi_{T}(\tau)=\Gamma_{T}(-\tau)^{-\gamma}\left(1+{\rm corr.\;terms}\right).$
(8)
The critical exponents are known exactly for the 2D Potts model [8, 9, 10, 11,
12]:
$x_{\epsilon}=\frac{1+y}{2-y},\quad x_{\sigma}=\frac{1-y^{2}}{4(2-y)},$ (9)
where $y$ is related to the number of states $q$ of the Potts variable by
$\cos\frac{\pi y}{2}=\frac{1}{2}\sqrt{q}.$ (10)
The standard exponents follow from $x_{\epsilon}=(1-\alpha)/\nu$ and
$x_{\sigma}=\beta/\nu$. The central charge of the corresponding conformal
field theory is also simply expressed [11, 12] in terms of $y$
$c=1-\frac{3y^{2}}{2-y}.$ (11)
Analytical estimates of critical amplitude ratios for the $q$-state Potts
models with $q=1$, $2$, $3$, and $4$ were recently obtained by Delfino and
Cardy [13]. They used the exact 2D scattering field theory of Chim and
Zamolodchikov [14] and estimated the ratios using a two-kink approximation for
$1<q\leq 3$. For $3<q\leq 4$, they considered both the two-kink approximation
and the contribution from the bound state. For $q=4$ this approximation leads
to the value $c=0.985$ for the central charge, to be compared to the exact
value $c=1$. Using this approximate value, one can calculate the scaling
dimensions from (11) and (9) obtaining the values $x_{\sigma}=0.117$ and
$x_{\epsilon}=0.577$, to be compared with the exact values $1/8$ and $1/2$
respectively. So, the deviation from the exact values becomes as large as 6
and 15 per cent, emphasizing the difficulty of the $q=4$ case. In the 3-state
case the situation is much better (see I).
Let us recall that the existence of logarithmic corrections to scaling in the
4-state Potts model was pointed out in the pioneering works of Cardy,
Nauenberg and Scalapino [15, 16], where a set of non-linear RG equations was
solved. Their discussion was recently extended by Salas and Sokal [17].
The universal susceptibility amplitude ratio $\Gamma_{+}/\Gamma_{L}$ was
calculated by Delfino and Cardy in [13] for both the 3-state and 4-state Potts
models. Later Delfino, et al. [18] estimated analytically also the ratio of
the transverse to the longitudinal susceptibility amplitudes
$\Gamma_{T}/\Gamma_{L}$. The values obtained in the 4-state case are
$\displaystyle\Gamma_{+}/\Gamma_{L}=4.013,$
$\displaystyle\Gamma_{T}/\Gamma_{L}=0.129.$ (12)
In this latter paper [18], the results of MC simulations were also reported,
but they were considered inconclusive by the authors. More recently Delfino
and Grinza report compatible values in the case of the Ashkin-Teller model,
using the same technique at the same level of approximation [19].
Another contribution to the study of the amplitude ratios in the 2D 4-state
Potts model was reported by Caselle, et al [4]. These authors presented a MC
determination of various amplitudes. In particular, their estimate of the
susceptibility amplitude ratio $\Gamma_{+}/\Gamma_{L}=3.14(70)$ is in
reasonable agreement with the theoretical estimate of Delfino and Cardy, in
spite of a somewhat controversial [18] use of the logarithmic corrections in
the fitting procedure.
Enting and Guttmann [20] also analyzed SE data for the 4-state Potts model and
found
$\displaystyle\Gamma_{+}/\Gamma_{L}=3.5(4),$
$\displaystyle\Gamma_{T}/\Gamma_{L}=0.11(4),$ (13)
results which are compatible with the predictions of [13] and [18]. Their
series analysis does not rely on differential approximants, but, in the hope
to achieve better control of the log-corrections of the $q=4$ case, they
address directly the asymptotic behavior of the series coefficients.
In the present paper we present more accurate MC data supplemented by a
reanalysis of the extended series derived by Enting and Guttmann [20]. We
address the following question: Is it possible to estimate the influence of
the logarithmic corrections on the fit procedure? Is it possible to devise
some procedure in which the role of the logarithmic corrections is properly
taken into account?
In the rest of the paper, we shall be concerned with the following universal
combinations of critical amplitudes
$\frac{A_{+}}{A_{-}},\;\;\frac{\Gamma_{+}}{\Gamma_{L}},\;\;\frac{\Gamma_{T}}{\Gamma_{L}},\;\;R_{C}^{+}=\frac{A_{+}\Gamma_{+}}{B^{2}},\;\;R_{C}^{-}=\frac{A_{-}\Gamma_{L}}{B^{2}}$
(14)
where the last two are a consequence of the scaling relation555We refer the
reader to the review Ref. [21] for a detailed discussion of the universality
of the critical amplitudes ratios. $\alpha=2-2\beta-\gamma$. To the various
critical amplitudes of interest, $A_{\pm}$, $\Gamma_{\pm}$,…, we have
associated appropriately defined “effective amplitudes”, namely temperature-
dependent quantities $A_{\pm}(\tau)$, $\Gamma_{\pm}(\tau),\ldots$, which take
as limiting values, when $\tau\rightarrow 0$, the critical amplitudes
$A_{\pm}$, $\Gamma_{\pm},\dots$. In order to avoid any risk of confusion
between the critical amplitude and the corresponding effective temperature-
dependent amplitude, reference to these temperature-dependent quantities is
always made with their explicit $\tau-$dependence.
Also in this paper, we make use of the duality relation in order to improve
the estimates of the ratios between effective amplitudes measured at dual
temperatures. In the case of the 4-state Potts model, this procedure would
even eliminate all logarithmic corrections from the fit in absence of
background contributions, which unfortunately do exist for most quantities! We
again use the duality relation to estimate the correction-to-scaling
amplitudes in the behavior of the specific heat and of the susceptibility. For
this purpose, we compute ratios also on the duality line, e.g. the
susceptibility effective-amplitude ratio
$\Gamma_{+}(\tau)/\Gamma_{L}(\tau^{*})=\chi_{+}(\beta)/\chi_{L}(\beta^{*})$ as
the ratio of $\chi_{+}(\beta)$, the high-temperature susceptibility at inverse
temperature $\beta$, and of $\chi_{L}(\beta^{*})$, the low-temperature
susceptibility at the dual inverse temperature $\beta^{*}$. Furthermore, we
show analytically that the leading logarithmic corrections cancel on the
duality line for the ratio of the specific-heat amplitudes as extracted from
the energy at dual temperatures.
As a final result of our analysis666The figures given here are an average
between the MC and the SE determinations [5]., we propose estimates of the
susceptibility critical-amplitude ratios $\Gamma_{+}/\Gamma_{L}=6.49(44)$ and
$\Gamma_{T}/\Gamma_{L}=0.154(12)$ which are significantly different both from
the predictions eq.(12) of [13, 18], and from the numerical estimates of [4,
20]. The deviation from the numerical estimates of other authors might be
explained by the complicated logarithmic corrections used to fit the data in
[4, 20]. The difference from the theoretical predictions might be due to the
limited accuracy of the approximation scheme used for $q>3$.
In conclusion, obtaining an accurate (say, within a few per cent) and
generally accepted approximation of the critical amplitude combinations for
the four-state Potts model still remains an open issue both theoretically and
numerically.
## 2 Computational procedures
### 2.1 Monte Carlo simulations
We use the single-cluster Wolff algorithm [2] for studying square lattices of
linear size $L$ with periodic boundary conditions. Starting from an ordered
state, we let the system equilibrate in $10^{5}$ steps measured by the number
of flipped Wolff clusters. The averages are computed over $10^{6}$—$10^{7}$
steps. The random numbers are produced by an exclusive-XOR combination of two
shift-register generators with the taps (9689,471) and (4423,1393), which are
known [22] to be safe for the Wolff algorithm.
The order parameter of a microstate ${\tt M}({\tt t})$ is evaluated during the
simulations as
${\tt M}=\frac{qN_{m}/N-1}{q-1},$ (15)
where $N_{m}$ is the number of sites $i$ with $s_{i}=m$ at the time $\tt t$ of
the simulation [23], and $m\in[0,1,...,(q-1)]$ is the spin value of the
majority of the spins. $N=L^{2}$ is the total number of spins. The thermal
average is denoted $M=\langle{\tt M}\rangle$.
Thus, the longitudinal susceptibility in the low-temperature phase is measured
by the fluctuation of the majority spin orientation
$k_{B}T\;\chi_{L}=\frac{1}{N}(\langle N_{m}^{2}\rangle-\langle
N_{m}\rangle^{2})$ (16)
and the transverse susceptibility is defined in terms of the fluctuations of
the minority of the spins
$k_{B}T\;\chi_{T}=\frac{1}{(q-1)N}\sum_{\mu\neq m}(\langle
N_{\mu}^{2}\rangle-\langle N_{\mu}\rangle^{2}),$ (17)
while in the high-temperature phase $\chi_{+}$ is given by the fluctuations in
all $q$ states,
$k_{B}T\;\chi_{+}=\frac{1}{qN}\sum_{\mu=0}^{q-1}(\langle
N_{\mu}^{2}\rangle-\langle N_{\mu}\rangle^{2}),$ (18)
where $N_{\mu}$ is the number of sites with the spin in the state $\mu$.
Properly allowing for the finite-size effects, this definition of the
susceptibilities is, in both phases, completely consistent with the available
SE data [24].
The internal energy density of a microstate is calculated as
${\tt E}=-\frac{1}{N}\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\,$ (19)
its ensemble average is denoted by $E=\langle{\tt E}\rangle$ and the reduced
specific heat per spin measures the energy fluctuations,
$(k_{B}T)^{2}\;C=-\frac{\partial E}{\partial\beta}=\langle{\tt
E}^{2}\rangle-\langle{\tt E}\rangle^{2}.$ (20)
We have simulated the model on square lattices with linear sizes $L=20$, $40$,
$60$, $80$, $100$ and $200$. In each case, we have measured the physical
quantities within a range of reduced temperatures called the “critical window”
and defined as follows. Assuming a proportionality factor of order 1 in the
definition of the correlation length, the relation $L<\xi\propto|\tau|^{-\nu}$
yields the value of the reduced temperature at which the correlation length
becomes comparable with the system size $L$ and thus below which the finite-
size effects are not negligible. This value defines the lower end of the
critical window and avoids finite size effects which would make our analysis
more complex. The upper limit of the critical window is fixed for convenience
at $\tau=0.20-0.25$.
### 2.2 Series expansions
Our MC study of the critical amplitudes will be supplemented by an analysis of
the high-temperature and low-temperature expansions for $q=4$, recently
extended through remarkably high orders by Briggs, Enting and Guttmann [25,
20]. In terms of these series, we can compute the effective critical
amplitudes for the susceptibilities, the specific heat and the magnetization
and extrapolate them by the standard resummation techniques, namely simple
Padé approximants (PA) and differential approximants (DA), properly biased
with the exactly known critical temperatures and critical exponents.
The LT expansions are expressed in terms of the variable $z=\exp(-\beta)$. In
the $q=4$ case, the expansion of the energy extends through $z^{43}$. For the
longitudinal susceptibility the expansion extends through $z^{59}$, and for
the transverse susceptibility through $z^{47}$. In the case of the
magnetization, the expansion extends through $z^{43}$. The HT expansions are
computed in terms of the variable $v=(1-z)/(1+(q-1)z)$. They extend up to
$v^{43}$ in the case of the energy and up to $v^{24}$ for the susceptibility.
It is useful to point out that, for convenience, in Ref. [20] the product of
the susceptibility by the factor $q^{2}/(q-1)$, rather than the susceptibility
itself, is tabulated at HT, because it has integer expansion coefficients. For
the same reason, at LT the magnetization times $q/(q-1)$ is tabulated.
Therefore the appropriate normalization should be restored in order that the
series yield amplitudes consistent with the MC results.
As a general remark on our series analysis, we may point out that the accuracy
of the amplitude estimates given in Ref. I for the $q=3$ case is good due to
the relatively harmless nature of the power-like corrections to scaling, while
in the $q=4$ case the mentioned resummation methods cannot reproduce the
expected logarithmic corrections to scaling and therefore the extrapolations
to the critical point are more uncertain. In this case we have tested also a
somewhat unconventional use of DA’s: in computing the effective amplitudes, we
only retain DA estimates outside some small vicinity of the critical point,
where they appear to be stable and reliable. Finally, we perform the
extrapolations by fitting these data to an asymptotic form which includes
logarithmic corrections. We shall add further comments on the specific
analyses in the next sections.
## 3 Critical amplitudes of 4-state Potts model
### 3.1 Expected temperature-dependence of the observables
In the case of the 4-state Potts model, we have $y=0$ from (10) and the second
thermal exponent [26, 11, 12] $y_{\phi_{2}}=-4y/3(1-y)$ vanishes. Accordingly,
the leading power-behavior of the specific heat (and of other physical
quantities) is modified [15] by a logarithmic factor
${C}(\tau)=\frac{A_{\pm}}{\alpha}|\tau|^{-\alpha}(-\ln|\tau|)^{-1}{\cal
X}_{corr}(-\ln|\tau|)+{\cal Y}_{bt}(|\tau|).$ (21)
The exponent $\alpha$ takes the value $2/3$ and ${\cal X}_{corr}(-\ln|\tau|)$
contains correction terms with powers of $|\tau|$, $-\ln|\tau|$ and
$\ln(-\ln|\tau|)$. It may also contain non-integer power corrections due to
the higher thermal exponents [8, 9, 10, 11, 12] $y_{\phi_{n}}$ or to other
irrelevant fields, as well as power corrections due to the identity field.
${\cal Y}_{bt}(|\tau|)$ contains all regular contributions and is referred to
as the background term.
Extending the pioneering work of Cardy, Nauenberg and Scalapino [15, 16],
Salas and Sokal [17] obtained a set of non-linear RG equations. In the
appendix (see also Refs. [5, 6]), we derive from this set of equations a
closed expression for the leading logarithmic corrections, which is more
suitable to describe the temperature range accessible in a numerical study,
than the asymptotic form given by Salas and Sokal,
${C}(\tau)=\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}(-\ln|\tau|)^{-1}\left[1-\frac{3}{2}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{\ln|\tau|}\right)\right],$
(22)
which is the first term of a slowly convergent expansion of ${\cal
X}_{corr}(-\ln|\tau|)$ in logs. We have observed that the following expansion
(see Appendix) is better behaved in the temperature window near the critical
point accessible by MC and SEs
$\displaystyle{C}(\tau)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}{\cal G}^{-1}(-\ln|\tau|),$
(23) $\displaystyle{\cal G}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle(-\ln|\tau|)\times{\cal E}(-\ln|\tau|)\times{\cal
F}(-\ln|\tau|),$ (24) $\displaystyle{\cal E}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)$
(25) $\displaystyle\
\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right),$
$\displaystyle{\cal F}(-\ln|\tau|)$ $\displaystyle\simeq$
$\displaystyle\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{-1}.$
(26)
The function $\cal E(-\ln|\tau|)$ contains the exact form of the leading terms
with universal coefficients predicted by RG. The remaining part is made of log
terms, whose coefficients involve the non-universal dilution field $\psi_{0}$.
The multiplicative function ${\cal F}(-\ln|\tau|)$ mimics, in a given
temperature range close to the critical point, the higher-order terms of this
non-universal part, which is a slowly convergent series in powers of logs
starting with an $O\left(\frac{1}{\ln|\tau|}\right)$ term. The values of
$C_{1}$ and $C_{2}$ in (26) should not be considered as “real” amplitudes of
correction-to-scaling terms, but only as “effective” parameters. A “zeroth-
order” analysis may be performed taking $C_{1}=C_{2}=0$, i.e. ${\cal
F}(-\ln|\tau|)=1.$ A more refined estimate follows from the analysis of the
magnetization, as explained in the section 3.2 and in the Appendix. The
absence of a constant background term is a simplifying feature in the analysis
of the magnetization777The effective amplitudes are constructed by dividing
the corresponding physical quantity by the leading terms (main power
dependence with known logarithmic corrections), and, therefore the background
terms, if present, will be divided by the logarithmic terms. These terms may
seriously complicate the analysis of the limiting behavior of the effective
amplitudes.. The difference in behavior between the two types of expressions
(22) and (23) for the specific heat is illustrated by the fact that, for
example, at a typical value of temperature numerically accessible without
finite-size effects, $\tau=0.10$, we have
$1-\frac{3}{2}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\simeq 0.457$, while
$\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\times\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\simeq
0.786$ and the two functions are not simply proportional to each other in the
typical range $\tau=0.02-0.20$. Fitting the numerical data with one or the
other choice may thus spoil the outcome for the leading amplitude.
Similar expressions of the logarithmic corrections are obtained for the other
physical quantities in Eqs. (79-85) of the Appendix.
Further corrections to scaling may also be present in Eq. (23). They are
discussed in the Appendix and may be of the form $a_{2/3}|\tau|^{2/3}$,
$a_{h_{3}}|\tau|^{\Delta_{h_{3}}}$ or
$a_{\phi_{3}}|\tau|^{\Delta_{\phi_{3}}}$, as well as powers of these terms,
where $\Delta_{h_{3}}=3/4$ and $\Delta_{\phi_{3}}=5/3$. Pure power corrections
and background terms may also be needed. Here we also stress that the
inclusion of a leading correction in $a|\tau|^{2/3}$ and of analytic terms
seems to be necessary according to the papers by Joyce [27, 28], where the
magnetization of a model, expected to belong to the 4-state Potts model
universality class, is shown to have an expression of the form
$M(-|\tau|)=|\tau|^{1/12}(f_{0}(\tau)+|\tau|^{2/3}f_{1}(\tau)+|\tau|^{4/3}f_{2}(\tau))$
(27)
with $f_{i}(\tau)$ analytic functions when $\tau\to 0^{-}$. The correction
exponents are obtained from the table of the conformal scaling dimensions by
Dotsenko and Fateev [11, 12], but not all of them are necessarily present.
However, at least the presence of the exponents $2/3$, and possibly of $4/3$,
seems to be needed in order to account for the numerical results. Caselle et
al. [4] also considered an $a_{2/3}|\tau|^{2/3}$ term to fit the
magnetization. The parameter $a_{h_{3}}$ is a priori possibly needed only for
magnetic quantities, while the corrections in $a_{\phi_{3}}$ will
systematically be dropped in our fits, since they are sub-sub-dominant.
In conclusion, the most general expression that we will consider is the
following:
$\displaystyle{\rm Obs.}(\pm|\tau|)$ $\displaystyle\simeq$ $\displaystyle{\rm
Ampl.}\times|\tau|^{\blacktriangleleft}\times{\cal
G}^{\bigstar}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+$ (28)
$\displaystyle\qquad\qquad+{\rm\ backgr.\ terms},$ $\displaystyle{\rm corr.\
terms}$ $\displaystyle=$ $\displaystyle
a_{2/3}|\tau|^{2/3}+b_{\pm}|\tau|+a_{4/3}|\tau|^{4/3}+\dots,$ (29)
$\displaystyle{\rm backgr.\ terms}$ $\displaystyle=$ $\displaystyle
D_{0}+D_{1}|\tau|+\dots$ (30)
where ${\cal G}(-\ln|\tau|)$ is defined by Eqs. (24-26), while
${\blacktriangleleft}$ and $\bigstar$ stand for exponents which depend on the
observable considered. They are all given in the Appendix.
### 3.2 The magnetization amplitude
The amplitude $B$ of the magnetization is defined by the asymptotic behavior
(see the appendix for details)
$\displaystyle M(-|\tau|)$ $\displaystyle=$ $\displaystyle
B|\tau|^{1/12}(-\ln|\tau|)^{-1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)^{-1}\right.$
(31) $\displaystyle\left.\
\qquad\times\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right){\cal
F}(-\ln|\tau|)\right]^{-1/8}(1+a|\tau|^{2/3}+b|\tau|+\dots).$
We can extract an effective function ${\cal F}_{eff}(-\ln|\tau|)$ which mimics
the real one ${\cal F}(-\ln|\tau|)$ in the convenient temperature range
$|\tau|\simeq 0.01-0.10$. This is done by plotting an effective magnetization
amplitude
$\displaystyle B_{eff}(-|\tau|)$ $\displaystyle=$ $\displaystyle
M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)\right.$
(32)
$\displaystyle\left.\qquad\qquad\qquad\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right)\right]^{1/8}$
which is then fitted to the expression
$B_{eff}(-|\tau|)=B\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{1/8}\times(1+a|\tau|^{2/3}+b|\tau|+\dots)$
(33)
in which we have also included corrections to scaling. As we have already
noticed, the coefficients appearing in the function ${\cal F}$ are effective
parameters adapted to the temperature window considered, therefore the values
of $C_{1}$ and $C_{2}$ have no special meaning.
In order to analyze the numerical data and to extract the different
coefficients, one needs very accurate data. As an illustration, in Figure 1 we
compare MC and SE data, and MC data from Caselle et al. [4].
Figure 1: The magnetization $M$ in the critical window region. Our MC data are
represented by boxes, the MC data from Ref. [4] by stars and the SE data by a
solid line.
Figure 2: The effective amplitude of the magnetization $M$. Insert: The
magnetization $M$ as function of $|\tau|^{1/12}(-\ln|\tau|)^{-1/8}$. (Our MC
data are represented by open circles, the SE data by boxes and the fit by a
solid line).
The behavior of $B_{eff}(-|\tau|)$ is shown in Fig. 2. In table 1, we present
a selection of our fits of MC data to Eq. (33). The first column of the table
indicates the different choices of the function ${\cal F}(-\ln|\tau|)$. For
each line in the tables, several fits have been tried, varying the number of
points in the interval $|\tau|\in[0.005,0.25]$ (total number of points 50) and
calculating the $\chi^{2}/d.o.f.$ for each fit. A reasonable balance has to be
found between the distance of the points from the critical temperature and
their number. It appeared that limiting the fit window to 20 data points, i.e.
to the interval $[0.005,0.1]$, gives the best confidence level. This choice is
quite satisfactory, since it corresponds to a close vicinity of the critical
point. The criterion that we adopted in order to select a most convincing fit
among all possible fits is the stability of the correction-to-scaling
amplitudes $a$ and $b$ in equation (31) when the temperature window is varied,
typically in the range $|\tau|\in[0,0.06]$ to $|\tau|\in[0,0.30]$. As long as
these numbers have not converged to given values, we cannot pretend that it is
meaningful to include such corrections in the analysis. We have to mention
that stability of the correction amplitudes is never reached if we stop the
correction terms in equation (31) at the leading log, or even at the three
higher log terms (correction function ${\cal E}(-\ln|\tau|)$). Then even the
leading amplitude $B$ is questionable. The effective function ${\cal
F}_{eff}(-\ln|\tau|)$ is essential to reach convergence of all but the two
effective coefficients $C_{1}$ and $C_{2}$ which still strongly depend on the
temperature window.
For a given type of fit defined in the first column, Tables 1 and 2 show the
parameters of the fit minimizing $\chi^{2}$ and the minimization is performed
by varying the width and position of the temperature window.
Once our choice is made for $C_{1}$ and $C_{2}$, the results of the fit for
the amplitude $B$ show a remarkable stability. The entry called fit # 0
corresponds to a “zeroth-order” fit in which the function ${\cal
F}(-\ln|\tau|)$ is taken equal to unity. It is presented for comparison, in
order to emphasize the improvement occurring in the following lines. Fit #1
keeps all correction coefficients, while the lines which follow are obtained
decreasing the number of fit parameters. We favor fits # 2 and #3, where the
coefficient $b$ (the amplitude of the linear term $|\tau|$) is fixed to zero,
since from the first fit (#1) we see that leaving $b$ free leads to a value
close to zero and we consider more reliable a fit with fewer parameters. Fit #
4 corresponds to another extreme choice including no irrelevant correction at
all. It shows that the effective function ${\cal F}(-\ln|\tau|)$ is more
important than the irrelevant corrections to scaling in order to achieve a
stable amplitude $B$ (comparable to the outcomes of fits # 1 to # 3).
Table 1: Various fits of our MC data for the magnetization effective amplitude $M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}{\cal E}^{1/8}(-\ln|\tau|)$ to the expression Eq. (33). The stars in the first column indicate our favorite fits (the reasons for this choice are given in the text). Fit # | Amplitude | Correction terms
---|---|---
| |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
MC # 0 | $1.1355(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.41(1)$ | $\phantom{-}0.41(21)$
MC # 1 | $1.1566(5)$ | $-0.740(2)$ | $-0.630(51)$ | $-0.172(4)$ | $-0.018(6)$
MC # $2^{*}$ | $1.1570(1)$ | $-0.757(1)$ | $-0.522(11)$ | $-0.191(2)$ | $\phantom{-}-$
MC # $3^{*}$ | $1.1559(12)$ | $-0.88(5)$ | $\phantom{-}-$ | $-0.21(1)$ | $\phantom{-}-$
MC # 4 | $1.1593(1)$ | $-0.25(2)$ | $-3.04(5)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 2: Same as table 1 for the magnetization data obtained by the SE method. A star in the first column indicates our favorite fit with the coefficients $C_{1}$ and $C_{2}$ (now fixed and shown in bold face in this table and in the forthcoming tables) obtained from the fits of the MC data reported in table 1. Fit # | Amplitude | Correction terms
---|---|---
| |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
SE # 0 | $1.1364(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.435(5)$ | $\phantom{-}0.097(10)$
SE # 1 | $1.1597(1)$ | $-0.637(5)$ | $-1.417(28)$ | $-0.115(2)$ | $\phantom{-}0.013(2)$
SE # $2$ | $1.1589(1)$ | $-0.660(5)$ | $-1.246(25)$ | $-0.126(2)$ | $\phantom{-}-$
SE # $2^{*}$ | $1.1575(1)$ | $-{\bf 0.757}$ | ${\bf-0.522}$ | $-0.194(1)$ | $\phantom{-}-$
SE # $3$ | $1.1583(8)$ | $-0.981(34)$ | $\phantom{-}-$ | $-0.185(10)$ | $\phantom{-}-$
SE # $3^{*}$ | $1.1575(1)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $-0.225(1)$ | $\phantom{-}-$
SE # 4 | $1.1573(1)$ | $\phantom{-}0.164(7)$ | $-4.106(18)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 3: Same as table 1 but using the MC data of Caselle et al. [4] for the magnetization. The fit marked by a star is performed using the same coefficient $C_{1}$ (shown in bold) as in table 1. Fit # | Amplitude | Correction terms
---|---|---
| |
| $B$ | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
CTV # 0 | $1.1386(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $-0.518(6)$ | $\phantom{-}0.30(2)$
CTV # 1 | $1.196\pm 1.499$ | $-7.5\pm 163$ | $\phantom{-}19.6\pm 429$ | $-1.09\pm 17.3$ | $1.2\pm 29.2$
CTV # 2 | $1.148(6)$ | $\phantom{-}0.45(56)$ | $-3.6\pm 1.1$ | $-0.17(4)$ | $\phantom{-}-$
CTV # 3 | $1.162(9)$ | $-1.13(38)$ | $\phantom{-}-$ | $-0.14(12)$ | $\phantom{-}-$
CTV # $3^{*}$ | 1.1561(1) | ${\bf-0.88}$ | $\phantom{-}-$ | $-0.215(1)$ | $\phantom{-}-$
CTV # 4 | $1.153(2)$ | $\phantom{-}0.73(31)$ | $-5.5\pm 0.77$ | $\phantom{-}-$ | $\phantom{-}-$
The same procedure is now applied to the SE data. It is a non-conventional
approach to fit SE data, which are usually analyzed by approximant methods,
but the exercise is tempting. We thus apply exactly the same procedure as in
the case of MC data, varying the fitting interval and comparing the values of
the $\chi^{2}/d.o.f.$. The best results are collected in table 2. The
agreement between the results quoted in the two tables is amazing. Not only
the amplitudes, but also the correction coefficients are very close to each
other. In this table (and in the forthcoming tables), we also present the
results of this analysis of SE data with $C_{1}$ and $C_{2}$ fixed to their
best values extracted in table 1 from the MC data. They are indicated again
with a star (SE $\\#2^{*}$ and SE $\\#3^{*}$) and in order to emphasize the
fact that $C_{1}$ and $C_{2}$ in this case are not free parameters, they are
indicated in bold face. A reason for this approach is first to insist on the
consistency of the results and second to privilege a fit with less free
parameters (and in particular no free log term). For the magnetization
amplitude, a compromise between MC and SE data analysis provides our final
estimate
$B=1.157(1).$ (34)
As an additional test, we decided to fit also the MC data of Caselle et al.
[4], obtained using the Swendsen-Wang cluster algorithm, to our functional
expression Eq. (33). Notice that the various coefficients reported in table 3
(not only the critical amplitude, but also the correction terms) are very
close to those reported above in table 1. In particular this fit gives further
support to our choice in favor of fits # 2 and # 3.
In the following tables, we will refer to the best two fits by the labels #
$2^{*}$ and # $3^{*}$ (which means that the coefficients $C_{1}$ and/or
$C_{2}$ are fixed to their values indicated in table 1) when fitting other
quantities. Note that the MC data of Caselle et al. [4] are not fitted with
the choice # $2^{*}$, because they do not extend on a range of temperatures
wide enough to apply the corresponding functional expression.
### 3.3 The specific heat and the energy density
We now turn to the study of the specific-heat amplitudes. Figure 3 shows the
effective-amplitude ratio $A_{+}(\tau)/A_{-}(-|\tau|)$ computed at dual and
symmetric temperatures following the same prescription as in the case of the
3-state Potts model studied in paper [1]. The same ratio computed from SE data
is also shown for comparison. The linear fit of the most accurate MC data
(lattice size $L=100$ computed with $10^{7}$ measurement steps) yields for
this ratio the value $0.9999(1)$, which is remarkably close to the exact value
1 derived from duality.
Figure 3: The specific-heat effective-amplitude ratio computed from the energy
ratio
$\frac{(E(\beta)-E_{0})\tau^{\alpha-1}}{(E_{0}-E(\beta^{*}))(\tau^{*})^{\alpha-1}}$
as a function of the reduced temperature $\tau$ on the dual line. The SE data
are represented by boxes, our MC data ($L=100$) by circles. The solid line is
given by Eq. (36). The ratio of the effective amplitudes
$A_{+}(\tau)/A_{-}(-|\tau|)$ is shown for different sizes in the insert (for
lattice linear sizes $L=100$ (open circles), $L=200$ (squares), $L=300$ (up
triangles), and $L=400$ (down triangles). The simulations were performed with
$N_{th}=10^{5}$ thermalization steps and $N_{MC}=10^{6}$ MC steps. The results
of a simulation for $L=100$ with $N_{th}=10^{6}$ and $N_{MC}=10^{7}$, are
represented by closed circles. The SE data are represented by a solid line.
The dotted line represents a linear fit of the closed circles.
The ratio
$\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}=\frac{(E(\beta)-E_{0})\tau^{\alpha-1}}{(E_{0}-E(\beta^{*}))(\tau^{*})^{\alpha-1}},$
(35)
where the constant $E_{0}$ is the value of the energy at the transition
temperature, $E_{0}=E(\beta_{c})=-1-1/\sqrt{q}$, when expanded close to the
transition point leads to
$\displaystyle\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}$ $\displaystyle=$
$\displaystyle 1+\frac{7}{3}\alpha_{q}\tau+O(\tau^{1+\alpha})$ (36)
with
$\alpha_{q}=-E_{0}\beta_{c}e^{-\beta_{c}}=\frac{\ln(1+\sqrt{q})}{\sqrt{q}}\approx
0.5493$. Eq. (36) predicts an asymptotically linear $\tau-$dependence of the
effective-amplitude ratio. This linear dependence is observed in Fig. 3 which
also confirms that the leading logarithmic corrections in the scaling function
${\cal X}_{\it corr}(-\ln(|\tau|))$ asymptotically cancel in the ratio
$A_{+}(\tau)/A_{-}(\tau^{*})$. (Of course this is true also for the same
quantity computed at symmetric temperatures, since $\tau^{*}\approx|\tau|$ for
small values of $\tau$). Figure 3 compares the effective-amplitude ratio
obtained from MC simulation and SE data with Eq. (36). The slope of MC data is
1.25, very close to the predicted value 1.28.
Now, we make the natural conjecture (proven in the Appendix in the absence of
background corrections) that the cancelation of the leading logarithmic
corrections will also occur for the other ratios. For the leading log-
correction, $-\ln|\tau|$, and for the next correction in
$\ln(-\ln|\tau|)/(-\ln|\tau|)$, this can be shown analytically from the RG as
first indicated by Cardy, Nauenberg and Scalapino in Ref. [15] and by Salas
and Sokal in Ref. [17]. Our statement is stronger since it extends also to the
higher order log-terms, such as the next correction in $1/(-\ln|\tau|)$. We
believe that equations (79) to (85) in the appendix are exact, and since all
the log-terms come from Eq. (78), they should cancel in the appropriate ratios
(i.e. when the same powers of the dilution field appear in the numerator and
the denominator. This is always the case when one considers an effective
combination tending to a universal ratio as $\tau\to 0$).
According to a RG analysis (see Appendix A), we may write the energy in the
critical region as
$\displaystyle E_{\pm}(\pm|\tau|)$ $\displaystyle=$ $\displaystyle
E_{0}\pm\frac{A_{\pm}}{\alpha(1-\alpha)\beta_{c}}\frac{|\tau|^{1/3}}{(-\ln|\tau|)}\frac{1+a_{\pm}|\tau|^{2/3}}{{\cal
E}(-\ln|\tau|){\cal F}(-\ln|\tau|)}+D_{1,\pm}|\tau|.$ (37)
In a fixed range of values of the reduced temperature, the “correction
function” ${\cal F}(-\ln|\tau|)$ is now fixed and the only remaining freedom
is to include background terms (coeff. $D$) and possibly additive corrections
to scaling coming from irrelevant scaling fields (coeff. $a$). Therefore, once
the function ${\cal F}(-\ln|\tau|)$ is fixed after our study of the
magnetization, a reasonable fit of the energy data needs only three
parameters888Like in the case of the magnetization, a fourth parameter in
$b_{\pm}|\tau|$ appears to be unnecessary., $A_{\pm}$, $a_{\pm}$, and
$D_{\pm}$. The next step is the fit of the mean $\bar{A}(\tau)$ of the
effective amplitudes,
$\displaystyle\bar{A}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha(1-\alpha)\beta_{c}(E_{+}(\tau)-E_{-}(-|\tau|))\times{\cal
G}(-\ln|\tau|)/|\tau|^{1-\alpha}$ (38) $\displaystyle=$
$\displaystyle\frac{\beta_{c}}{9}(E_{+}(\tau)-E_{-}(-|\tau|))\times{\cal
G}(-\ln|\tau|)/|\tau|^{1/3}.$
We thus fit the MC data to the expression
$\bar{A}(\tau)=A(1+a|\tau|^{2/3})+D_{1}|\tau|\times{\cal
G}(-\ln|\tau|)/|\tau|^{1/3}.$ (39)
The log corrections, which indeed cancel in the singular part, unfortunately
reappear in the background term, albeit suppressed by the power
$|\tau|^{2/3}$.
Table 4: Fits of the energy difference (MC data computed at dual temperatures) to the expression $\bar{A}(\tau)\simeq{\rm Ampl.}\times(1+{\rm corr.\ terms})+{\rm backgrd.\ terms}\times|\tau|^{-1/3}{\cal G}(-\ln|\tau|)$. Fit # | Amplitude | correction terms | background term
---|---|---|---
| | |
| Ampl. | $\propto\frac{1}{-\ln|\tau|}$ | $\propto\frac{\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$
MC # $2^{*}$ | $1.338(3)$ | ${\bf-0.757}$ | ${\bf-0.522}$ | $-4.98(8)$ | $\phantom{-}0.920(13)$
MC # $3^{*}$ | $1.316(10)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $-4.88(38)$ | $\phantom{-}0.899(62)$
Figure 4: The effective amplitude $\bar{A}(\tau)$ computed from our MC data
using Eq. (38) (open circles) and from the fit to Eq. (39) (solid line).
Figure 5: The energy differences $\Delta E_{+}$ and $\Delta E_{-}$, calculated
from our MC data for lattice size $L=100$ (circles) and the fitted expressions
(solid lines).
We note that $\bar{A}(\tau)$ is constructed in Eq. (38) using the values of
energy density computed at symmetric temperatures. The same quantity
constructed from the energy densities at dual temperatures (with
$E_{-}(\tau^{*})$ instead of $E_{-}(-|\tau|)$ in Eq. (38)) can also be studied
and provides better results. In table 4, we show our results (the best fit is
obtained with the choices of $C_{i}$’s coefficients labeled $2^{*}$ in tables
1 and 2). Again, the agreement between the two fits is quite good, but this
time it is more trivial since the same data set is fitted. Taking the average
of the parameters from the two fits, we conclude
$A=1.327(12),$ (40)
and $a\simeq-4.93(23)$, and $D_{1}\simeq 0.910(38)$. Padé approximants of SE
data for the specific heat provide $A=1.35(1)$.
The expansion including corrections to scaling and background terms for the
specific heat follows from the expressions of the energy density. There is
some disagreement between our amplitude
$\frac{A}{\alpha(1-\alpha)\beta_{c}}\simeq 5.922(40)-6.021(13)$ and the result
reported by Caselle et al. [4], $6A\simeq 7.80(36)$999Notice that Caselle et
al. use a different definition of the energy.. We have to notice that the
amplitudes are very sensitive to the expression used for the fits. Our choice
of effective amplitude in Eq. (38) is supported by the quite regular behavior
shown in figure 4, and also by the natural choice of the fitting expression
(39). The comparison between the MC data and the resulting fit is shown in
figure 5. We agree on this point with Enting and Guttmann [20] who emphasized
that their estimates depend critically on the form assumed for the logarithmic
sub-dominant terms, and on the further assumption that the other sub-dominant
terms, including powers of logarithms, powers of logarithms of logarithms
etc., can all be neglected.
### 3.4 Susceptibilities amplitudes
#### 3.4.1 High temperature susceptibility amplitude
We proceed along the same lines as for the other physical quantities and fit
the high-temperature susceptibility to the expression
$\chi_{+}(\tau)=\Gamma_{+}\tau^{-7/6}{\cal
G}^{3/4}(-\ln\tau)(1+a_{+}\tau^{2/3}+b_{+}\tau)+D_{+}.$ (41)
Figure 6: The effective amplitude of the high-temperature susceptibility $\chi_{+}(\tau)$. We have shown SE data for $\tau\leq 0.05$, our MC data for $\tau>0.05$ and a fit to Eq. (41) (dotted line). Table 5: Fits of the high-temperature susceptibility to the expression $\chi_{+}(\tau)\simeq{\rm Ampl.}\times\tau^{-7/6}\times{\cal G}^{3/4}(-\ln\tau)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$. A star in the first column indicates that the coefficients $C_{i}$’s are those deduced from the MC fits of table 1. fit # | amplitude | correction terms | backgr.
---|---|---|---
| | |
| $\Gamma_{+}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.03144(15)$ | $\bf-0.757$ | $\bf-0.522$ | $\phantom{-}0.561(60)$ | $-0.053(17)$
MC # $3^{*}$ | $0.03178(30)$ | $\bf-0.88$ | $\phantom{-}-$ | $\phantom{-}0.53(23)$ | $\phantom{-}0.052(120)$
CTV # $3^{*}$ | $0.03051(29)$ | ${\bf-0.88}$ | $\phantom{-}-$ | $\phantom{-}1.48(34)$ | $-0.45(24)$
SE # $2^{*}$ | $0.03041(1)$ | ${\bf-0.757}$ | $\bf-0.522$ | $\phantom{-}1.30(1)$ | $-0.362(9)$
SE # $3^{*}$ | $0.03039(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $\phantom{-}1.67(1)$ | $-0.59(2)$
We can easily obtain the amplitude $\Gamma_{+}$ observing that a single
constant as a background term $D_{+}$ is sufficient for the fit (this will
also be the case at low temperature). The effective amplitude
$\Gamma_{eff}(\tau)=\chi_{+}(\tau)\tau^{7/6}{\cal G}^{-3/4}(-\ln\tau)$
is represented in Fig. 6 and table 5 collects the coefficients determined by
the fits. We finally obtain the high-temperature susceptibility amplitude
$\Gamma_{+}=0.0310(7).$ (42)
The value which follows from differential approximants to SE data, although
less accurate, is consistent with it, $\Gamma_{+}=0.033(2)$.
#### 3.4.2 Low temperature susceptibilities amplitudes
The behavior of the longitudinal susceptibility in the low-temperature phase
is less easy to analyze [5]. We use the expression
$\chi_{L}(-|\tau|)=\Gamma_{L}|\tau|^{-7/6}{\cal
G}(-\ln|\tau|)^{3/4}(1+a_{L}|\tau|^{2/3}+b_{L}|\tau|)+D_{L}.$ (43)
and the various coefficients are collected in table 6. For the transverse
susceptibility, the same procedure leads to the amplitudes also listed in the
table 6.
One may note that the values of the transverse susceptibility amplitude are
more stable than those of the longitudinal amplitude, while the estimates of
the corrections to scaling are less scattered in the latter case. Our final
estimates are
$\Gamma_{L}=0.00478(24)$ (44)
and
$\Gamma_{T}=0.00074(2).$ (45)
DA analysis of SE data gives approximately $\Gamma_{L}=0.005(1)$.
Table 6: Fits of the low-temperature longitudinal susceptibility to the expression $\chi_{L}(-|\tau|)\simeq{\rm Ampl.}\times|\tau|^{-7/6}\times{\cal G}^{3/4}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$ and of the low-temperature transverse susceptibility to the expression $\chi_{T}(-|\tau|)\simeq{\rm Ampl.}\times|\tau|^{-7/6}\times{\cal G}^{3/4}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backgr.\ terms}$. A star in the first column indicates our favorite fit with the coefficients deduced from the MC fits of table 1. fit # | amplitude | correction terms | backgr.
---|---|---|---
| | |
| $\Gamma_{L}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.00454(2)$ | $\bf-0.757$ | $\bf-0.522$ | $-2.83(3)$ | $\phantom{-}0.050(2)$
MC # $3^{*}$ | $0.00484(3)$ | $\bf-0.88$ | $\phantom{-}-$ | $-3.73(14)$ | $\phantom{-}0.13(1)$
CTV # $3^{*}$ | $0.00494(3)$ | $\bf-0.88$ | $\phantom{-}-$ | $-4.35(15)$ | $\phantom{-}0.210(19)$
SE # $2^{*}$ | $0.00483(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-3.77(3)$ | $\phantom{-}0.116(3)$
SE # $3^{*}$ | $0.00493(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-4.18(5)$ | $\phantom{-}0.178(6)$
| $\Gamma_{T}$ | $\propto\frac{1}{-\ln\tau}$ | $\propto\frac{\ln(-\ln\tau)}{(-\ln\tau)^{2}}$ | $\quad\propto\tau^{2/3}$ | $\quad\propto\tau^{0}$
MC # $2^{*}$ | $0.00076(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-0.805(34)$ | $-0.0028(2)$
MC # $3^{*}$ | $0.00073(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-0.25(13)$ | $-0.0050(14)$
SE # $2^{*}$ | $0.00073(1)$ | $\bf-0.757$ | $\bf-0.522$ | $-0.577(14)$ | $-0.00373(15)$
SE # $3^{*}$ | $0.00073(1)$ | $\bf-0.88$ | $\phantom{-}-$ | $-0.369(15)$ | $-0.00457(16)$
### 3.5 Universal amplitude ratio $R_{c}^{-}$.
Figure 7: The functions ${R}^{-}_{C}(-|\tau|)$ (open symbols) and
${R}^{-}_{C^{*}}(-\tau)$ (closed symbols) which approaches the universal ratio
$R_{C}^{-}$ as $\tau\to 0^{-}$. Our MC data are represented by circles and the
CTV data by stars.
We use the available MC data for $C$, $M$, and $\chi$ to estimate the
universal amplitude ratio $R_{C}^{-}$ in the low-temperature phase. To this
purpose, we form the function (compare with Eq. (69))
$R_{C}^{-}(-|\tau|)=\alpha\tau^{2}\frac{C(-|\tau|)\chi_{L}(-|\tau|)}{M^{2}(-|\tau|)}$
(46)
which is an estimator of the universal amplitude ratio in the limit
$|\tau|\rightarrow 0$. As discussed in the Appendix, we expect that all sets
of logarithmic corrections cancel in this ratio. Figure 7 shows with open
symbols the combination from Eq. (46) for two sets of MC data, those of
Caselle et al. (CTV) and our simulations. We may fit these data with
correction-to-scaling terms starting from $|\tau|^{2/3}$ or, assuming in plain
analogy with the energy ratio that such corrections cancel, with terms
starting with $|\tau|$. In table 7 we include these fits for our MC data set
varying the temperature window and the number of correction terms.
Table 7: Estimates of the critical amplitude ratio $R_{C}^{-}$ from our MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
| |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$ | $\quad\propto|\tau|^{5/3}$
$0.01-0.29$ | 0.00651(3) | $-3.04(12)$ | $-1.24(39)$ | $\phantom{-}4.18(32)$ | $\phantom{-}-$
| 0.00685(3) | $-4.60(3)$ | $\phantom{-}3.84(5)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00619(1) | $\phantom{-}-$ | $-16.13(11)$ | $\phantom{-}29.25(37)$ | $-14.34(31)$
| 0.00591(2) | $\phantom{-}-$ | $-11.01(10)$ | $\phantom{-}12.13(16)$ | $\phantom{-}-$
$0.01-0.10$ | 0.00628(7) | $-1.54(58)$ | $-6.78\pm 2.36$ | $\phantom{-}9.41\pm 2.53$ | $\phantom{-}-$
| 0.00654(3) | $-3.66(9)$ | $-1.99(18)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00627(3) | $-2.70(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00618(3) | $\phantom{-}-$ | $-15.90\pm 1.26$ | $\phantom{-}28.64\pm 5.51$ | $-14.09\pm 6.18$
| 0.00611(1) | $\phantom{-}-$ | $-13.05(17)$ | $\phantom{-}16.10(35)$ | $\phantom{-}-$
$0.01-0.046$ | 0.00642(6) | $-3.09(33)$ | $\phantom{-}0.62(80)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00637(1) | $-2.83(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00616(4) | $\phantom{-}-$ | $-14.20(94)$ | $\phantom{-}18.99\pm 2.44$ | $\phantom{-}-$
| 0.00588(4) | $\phantom{-}-$ | $-6.92(18)$ | $\phantom{-}-$ | $\phantom{-}-$
A similar analysis was also performed for the data from Caselle et al (not
shown here). The results of both analysis are consistent.
More traditionally, we may evaluate the ratio $R_{C}^{-}$ using the estimated
values of the amplitudes reported in sections 3.2, 3.3 and 3.4 (see Eq. (14)).
The results will be presented later, but anticipating the forthcoming
analysis, we can quote as a reliable estimate $R_{C}^{-}\simeq 0.0052\pm
0.0002$ (approximants of SE data lead to $0.0050(2)$). One can see from the
table 7 that the ratio obtained from the effective function Eq. (46) is
systematically larger. The difference may be explained by the fact that
fitting the effective ratio function Eq. (46) without any logarithmic
correction, we assume that the background corrections for the longitudinal
susceptibility $\chi_{L}$ and for the specific heat $C$ are small in the
critical temperature window. While this is indeed the case, these background
terms are not negligibly small and their presence leads to systematic
deviations of the estimates of $R_{C}^{-}$ presented in the table 7. The ratio
(46) may be written as
$\alpha\tau^{2}(C_{s}+C_{bt})(\chi_{s}+\chi_{bt})/M^{2}$, where $C_{s}$ and
$\chi_{s}$ are the singular parts of the specific heat and of the
susceptibility and $C_{bt}$ and $\chi_{bt}$ are the corresponding background
nonsingular terms. Eq. (46) may be rewritten as
$\alpha\tau^{2}C_{s}\chi_{s}(1+C_{bt}/C_{s})(1+\chi_{bt}/\chi_{s})/M^{2}$.
Thus, the background terms $C_{bt}$ and $\chi_{bt}$ contribute when divided by
the singular terms (or in other words multiplied by factors $\tau^{2/3}{\cal
G}$ and $\tau^{7/6}{\cal G}^{-3/4}$, respectively). Clearly, in the critical
region, the first factor has the dominant contribution. This “large” term may
be eliminated completely if we form a quantity equivalent to Eq. (46) from the
energy difference $E_{-}(|\tau|)-E_{0}$ instead of the specific heat,
$R^{-}_{C^{*}}(-|\tau|)=\alpha(\alpha-1)\beta_{c}\tau\frac{(E_{-}(-|\tau|)-E_{0})\chi_{L}(-|\tau|)}{M^{2}(-|\tau|)},$
(47)
which is shown in the Figure 7 with closed symbols. The extrapolation at
$\tau\to 0^{-}$ is obviously different. The results of the fit of MC data to
Eq. (47) are given in table 8. The outcome for the universal combination
$R_{C}^{-}$ is now fully consistent with the value $0.0052(2)$ and supports
our idea that the specific heat background term spoils the behavior of the
estimator (46).
Table 8: Estimates of the critical amplitude ratio $R_{C^{*}}^{-}$ (Expr. (47)) from our MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
| |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$ | $\quad\propto|\tau|^{5/3}$
$0.01-0.29$ | 0.00551(9) | $-2.53(5)$ | $\phantom{-}0.57(15)$ | $\phantom{-}1.11(12)$ | $\phantom{-}-$
| 0.00558(3) | $-2.97(3)$ | $\phantom{-}1.95(2)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00527(8) | $\phantom{-}-$ | $-11.44(16)$ | $\phantom{-}20.98(51)$ | $-11.25(42)$
| 0.00508(1) | $\phantom{-}-$ | $-7.29(9)$ | $\phantom{-}7.38(14)$ | $\phantom{-}-$
$0.01-0.10$ | 0.00558(4) | $-3.28(36)$ | $\phantom{-}3.66\pm 1.45$ | $\phantom{-}2.23\pm 1.55$ | $\phantom{-}-$
| 0.00553(1) | $-2.77(5)$ | $\phantom{-}1.58(9)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(2) | $-1.98(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00538(2) | $\phantom{-}-$ | $-16.11(99)$ | $\phantom{-}40.39\pm 4.30$ | $31.97\pm 4.81$
| 0.00525(2) | $\phantom{-}-$ | $-9.55(24)$ | $\phantom{-}11.82(50)$ | $\phantom{-}-$
$0.01-0.046$ | 0.00558(3) | $-3.09(20)$ | $\phantom{-}2.34(48)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00543(2) | $-2.11(3)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(3) | $\phantom{-}-$ | $-12.40(74)$ | $\phantom{-}18.95\pm 1.99$ | $\phantom{-}-$
| 0.00511(4) | $\phantom{-}-$ | $-5.04(19)$ | $\phantom{-}-$ | $\phantom{-}-$
Table 9: Estimates of the critical amplitude ratio $R_{C^{*}}^{-}$ from Caselle et al MC data using different fits and varying the temperature window. $\tau-$window | amplitude | correction terms
---|---|---
| |
| $R_{C}^{-}$ | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{4/3}$
$0.0058-0.029$ | 0.00548(2) | $-2.59(17)$ | $1.27(48)$ | $\phantom{-}-$
| 0.00543(1) | $-2.14(2)$ | $\phantom{-}-$ | $\phantom{-}-$
| 0.00535(1) | $\phantom{-}-$ | $-13.77(66)$ | $\phantom{-}21.79\pm 1.98$
| 0.00521(2) | $\phantom{-}-$ | $-5.93(23)$ | $\phantom{-}-$
Again, a similar analysis of CTV data (see table 9) leads to fully consistent
results.
## 4 Discussion
Our final goal is the determination of some universal combinations of
amplitudes. This can be done either directly from the values of the amplitudes
listed in the various tables of this paper, or also by extrapolating the
effective ratios to $\tau=0$. Let us start with an estimate obtained by the
second method, and let us concentrate on the most controversial amplitude
ratios, those of the susceptibilities. As we have just shown in the section on
$R_{C}^{-}$, this method may lead to systematic deviations if the background
terms are not handled with care. Later on we shall discuss the ratios of the
amplitudes listed in the previous tables.
The ratio $\Gamma_{+}/\Gamma_{L}$ can be estimated from the ratio of the SE of
the susceptibility $\chi_{+}$ at high-temperature and of the longitudinal
susceptibility $\chi_{L}$ in the low-temperature phase. We have again two
options to form this ratio, either from quantities computed at temperatures
symmetric with respect to the critical temperature $T_{c}\pm\tau$ or at
inverse temperatures $\beta$ and $\beta^{*}$ related by the duality relation
(3).
Figure 8: The ratio of the effective amplitudes
$\Gamma_{+}(\tau)/\Gamma_{L}(-|\tau|)$ obtained from SE data at symmetric
temperatures (boxes) and dual temperatures (circles) together with the
corresponding linear fits (solid lines).
Figure 8 shows both the ratios $\Gamma_{+}(\tau)/\Gamma_{L}(-|\tau|)$ and
$\Gamma_{+}(\tau)/\Gamma_{L}(\tau^{*})$, while the straight lines are drawn as
a guide for the eye. It would be naive to take the linear fit too seriously,
otherwise one should conclude that the background (non-singular) contribution
to the ratio is negligible. The value of the universal amplitude ratio
$\Gamma_{+}/\Gamma_{L}$ obtained from the SE data is approximately
$\Gamma_{+}/\Gamma_{L}\simeq 6.16(1)$ and $6.30(1)$, when using respectively
the fits MC $\\#2^{*}$ and MC $\\#3^{*}$. We have also analyzed the effective-
amplitude ratio from MC data obtained by dividing the high-temperature reduced
susceptibility by the longitudinal reduced susceptibility computed at
temperatures related by the duality relation. Neglecting the constant
background terms in the susceptibilities eliminates all logarithms and makes
the fit quite simple, leading to a ratio in the range
$\Gamma_{+}/\Gamma_{L}\simeq 6.30-6.60$. On the other hand, if we keep in the
fit the background terms, the logs reappear and we are lead to
$\Gamma_{+}/\Gamma_{L}\simeq 6.0-6.1$.
Thus we get values which are quite different from the analytical prediction
$\Gamma_{+}/\Gamma_{L}=4.013$ of Delfino and Cardy [13], as well as from the
value $3.5(4)$ estimated by Enting and Guttmann from an analysis of the SE
data for the susceptibility in both phases, and from the value $3.14(70)$
estimated by Caselle et al. [4].
Let us now estimate the effective-amplitude ratio
$\Gamma_{T}(-|\tau|)/\Gamma_{L}(-|\tau|)$. This ratio, shown in figure 9, has
been computed both by MC simulation, for various lattice sizes, and from SE
data. Due to the non-singular correction terms, its behavior is far from being
linear in $\tau$ as was the ratio $\chi_{+}(|\tau|)/\chi_{L}(-|\tau|)$
(compare with figure 8). A possible explanation is that there might be some
symmetry in the correction-to-scaling amplitudes occurring in the asymptotic
expansion of $\chi_{+}(|\tau|)$ and $\chi_{L}(-|\tau|)$, but not of
$\chi_{T}(-|\tau|)$, which introduces here a stronger background term still
containing the logs. Following again the same procedure, we arrive at
estimates close to $0.146$ when neglecting all logarithmic corrections, while
we get $0.152-0.153$ when allowing for these corrections and from the analysis
of SE data. Finally we obtain $\Gamma_{T}/\Gamma_{L}=0.151(3)$ and $0.148(3)$
from the fits MC $\\#2^{*}$ and $3^{*}$. All these values differ from the
analytical prediction $\Gamma_{T}/\Gamma_{L}=0.129$ of Delfino et al. [18] and
from the value $0.11(4)$ of Ref. [20].
Figure 9: 4-state Potts model. The ratio of the effective amplitudes
$\Gamma_{T}(-|\tau|)/\Gamma_{L}(-|\tau|)$ for the 4-state Potts model on
square lattices of linear sizes $L=20$ (boxes), $L=40$ (up triangles), $L=60$
(down triangles), $L=80$ (diamonds), $L=100$ (stars) computed with
$N_{MC}=10^{5}$ MC steps, and $L=100$ (closed circles) computed with
$N_{MC}=10^{6}$. The solid line represents the SE data.
Let us now extract what we believe are more reliable estimates for the
universal combinations of amplitudes by a direct evaluation of the ratios of
the numbers presented in this paper and collected again in table 10. The
universal combinations are presented in table 11, together with the
corresponding results available in the literature. Averaging our different
results, we quote the following final estimates:
$\displaystyle\Gamma_{+}/\Gamma_{L}$ $\displaystyle=$ $\displaystyle 6.49\pm
0.44,$ (48) $\displaystyle\Gamma_{T}/\Gamma_{L}$ $\displaystyle=$
$\displaystyle 0.154\pm 0.012,$ (49) $\displaystyle R_{C}^{+}$
$\displaystyle=$ $\displaystyle 0.0338\pm 0.0009,$ (50) $\displaystyle
R_{C}^{-}$ $\displaystyle=$ $\displaystyle 0.0052\pm 0.0002.$ (51)
These results clearly confirm the above-mentioned limits of effective ratios.
Table 10: Critical amplitudes and correction coefficients for the 4-state Potts model. They are written in the following format ${\rm Obs.}(\pm|\tau|)\simeq{\rm Ampl.}\times|\tau|^{\blacktriangleleft}\times{\cal G}^{\bigstar}(-\ln|\tau|)\times(1+{\rm corr.\ terms})+{\rm backg.\ terms}.$ The results of the MC analysis of Ref.[4] are compiled together with our results obtained by combining the MC and series expansion (SE) data analysis. observable | amplitude | correction terms, $\times{\cal G}^{\bigstar}(-\ln|\tau|)$ | background terms | source
---|---|---|---|---
| | | |
| | $\quad\propto|\tau|^{2/3}$ | $\quad\propto|\tau|$ | $\quad\propto|\tau|^{0}$ | $\quad\propto|\tau|$ |
$E_{\pm}(\tau)$ | $1.338(3)/\alpha(1-\alpha)\beta_{c}$ | $-4.98(8)$ | $\phantom{-}\alpha_{4}$ | $\phantom{--}E_{0}$ | $\phantom{-}3.77(6)$ | this paper MC# $2^{*}$
| $1.316(9)/\alpha(1-\alpha)\beta_{c}$ | $-4.88(38)$ | $\phantom{-}\alpha_{4}$ | $\phantom{--}E_{0}$ | $\phantom{-}3.68(30)$ | this paper MC# $3^{*}$
$\chi_{+}(\tau)$ | $\Gamma_{+}=0.0223(14)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.05(14)$ | $\phantom{-}-$ | [4]
| $\Gamma_{+}=0.031(5)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{+}=0.03144(15)$ | $\phantom{-}0.561(60)$ | $\phantom{-}-$ | $-0.053(17)$ | $\phantom{-}-$ | this paper MC# $2^{*}$
| $\Gamma_{+}=0.03178(30)$ | $\phantom{-}0.53(23)$ | $\phantom{-}-$ | $\phantom{-}0.052(120)$ | $\phantom{-}-$ | this paper MC# $3^{*}$
| $\Gamma_{+}=0.03041(1)$ | $\phantom{-}1.30(1)$ | $\phantom{-}-$ | $-0.362(9)$ | $\phantom{-}-$ | this paper SE# $2^{*}$
| $\Gamma_{+}=0.03039(1)$ | $\phantom{-}1.67(1)$ | $\phantom{-}-$ | $-0.59(2)$ | $\phantom{-}-$ | this paper SE# $3^{*}$
$\chi_{L}(-|\tau|)$ | $\Gamma_{L}=0.00711(10)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.02(1)$ | $\phantom{-}-$ | [4]
| $\Gamma_{L}=0.0088(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{L}=0.00454(2)$ | $-2.83(3)$ | $\phantom{-}-$ | $\phantom{-}0.050(2)$ | $\phantom{-}-$ | this paper MC# $2^{*}$
| $\Gamma_{L}=0.00484(3)$ | $-3.73(14)$ | $\phantom{-}-$ | $\phantom{-}0.13(1)$ | $\phantom{-}-$ | this paper MC# $3^{*}$
| $\Gamma_{L}=0.00483(1)$ | $-3.77(3)$ | $\phantom{-}-$ | $\phantom{-}0.116(3)$ | $\phantom{-}-$ | this paper SE# $2^{*}$
| $\Gamma_{L}=0.00493(1)$ | $-4.18(5)$ | $\phantom{-}-$ | $\phantom{-}0.178(6)$ | $\phantom{-}-$ | this paper SE# $3^{*}$
$\chi_{T}(\tau)$ | $\Gamma_{T}=0.0010(3)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
| $\Gamma_{T}=0.00076(1)$ | $-0.805(34)$ | $\phantom{-}-$ | $-0.0028(2)$ | $\phantom{-}-$ | this paper MC# $2^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.25(13)$ | $\phantom{-}-$ | $-0.0050(14)$ | $\phantom{-}-$ | this paper MC# $3^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.577(14)$ | $\phantom{-}-$ | $-0.00373(15)$ | $\phantom{-}-$ | this paper SE# $2^{*}$
| $\Gamma_{T}=0.00073(1)$ | $-0.369(15)$ | $\phantom{-}-$ | $-0.00457(16)$ | $\phantom{-}-$ | this paper SE# $3^{*}$
$M(-|\tau|)$ | $B=1.1621(11)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}0.05(14)$ | $\phantom{-}-$ | [4]
| $B=1.1570(1)$ | $-0.191(2)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | this paper MC# $2^{*}$
| $B=1.1559(12)$ | $-0.210(10)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | this paper MC# $3^{*}$
| $B=1.1575(1)$ | $-0.194(1)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | this paper SE# $2^{*}$
| $B=1.1575(1)$ | $-0.225(1)$ | $\phantom{-}-$ | $\phantom{-}-$ | $\phantom{-}-$ | this paper SE# $3^{*}$
Table 11: Universal combinations of the critical amplitudes in the 4-state Potts model. $A_{+}/A_{-}$ | $\Gamma_{+}/\Gamma_{L}$ | $\Gamma_{T}/\Gamma_{L}$ | $R_{C}^{+}$ | $R_{C}^{-}$ | source
---|---|---|---|---|---
$1.$ | $4.013$ | $0.129$ | $0.0204$ | $0.00508$ | [13, 18]
$\phantom{-}-$ | $4.02$ | $0.129$ | $\phantom{-}-$ | $\phantom{-}-$ | [19]
$\phantom{-}-$ | $3.14(70)$ | $\phantom{-}-$ | $0.021(5)$ | $0.0068(9)$ | [4]
$\phantom{-}-$ | $3.5(4)$ | $0.11(4)$ | $\phantom{-}-$ | $\phantom{-}-$ | [20]
$1.000(5)$ | $6.93(6)$ | $0.1674(30)$ | $0.03452(25)$ | $0.00499(3)$ | this paper MC# $2^{*}$
$1.000(13)$ | $6.57(10)$ | $0.1508(26)$ | $0.03439(63)$ | $0.00524(9)$ | this paper MC# $3^{*}$
$\phantom{-}-$ | $6.30(1)$ | $0.1511(24)$ | $0.03336(9)$ | $0.00530(3)$ | this paper SE# $2^{*}$
$\phantom{-}-$ | $6.16(1)$ | $0.1481(23)$ | $0.03279(24)$ | $0.00532(5)$ | this paper SE# $3^{*}$
The main outcome of this work are the surprisingly high values of the ratios
$\Gamma_{+}/\Gamma_{L}$, $\Gamma_{T}/\Gamma_{L}$ and $R_{C}^{+}$ (and the low
value for $R_{C}^{-}$), significantly deviating from the predictions of
Delfino and Cardy. We emphasize that our results are also supported by a
direct extrapolation of effective-amplitude ratios for which most corrections
to scaling disappear. We believe that our fitting procedure is reliable, and
since the disagreement with the theoretical calculations can hardly be
resolved, we suspect that it might be attributed to the approximations made in
Ref. [13] in order to predict the susceptibility ratios. Even more puzzling is
the fact that Delfino and Cardy argue in favor of a higher robustness of their
results for $\Gamma_{T}/\Gamma_{-}$ than for $\Gamma_{+}/\Gamma_{-}$, while
the disagreement is indisputable in both cases. Indeed, in the conclusion of
their paper, and in a footnote, Delfino et al [18] (p.533) explain that their
results are sensitive to the relative normalization of the order- and
disorder-operator form-factors, which could be the origin of some troubles for
$q=3$ and $4$ for the ratios $\Gamma_{+}/\Gamma_{L}$ and $R_{C}$ only.
As a final argument in favor of our results, we may mention a work of W. Janke
and one of us (LNS) on the amplitude ratios in the Baxter-Wu model (expected
to belong to the 4-state Potts model universality class), leading to the
estimate $\Gamma_{+}/\Gamma_{-}\simeq 6.9$ and $R_{C}^{-}\approx 0.005$ [30].
This result, obtained from an analysis of MC data shows a similar discrepancy
with Delfino and Cardy’s result and suggests that further analysis might still
be necessary.
## 5 Acknowledgements
Discussions with A. Zamolodchikov, V. Dotsenko, V. Plechko, W. Janke and M.
Henkel, and a correspondence with J. Cardy and J. Salas were very helpful.
LNS is grateful to the Statistical Physics group of the University Henri
Poincaré Nancy 1 for the kind hospitality. Both LNS and PB thank the
Theoretical group of the University Milano–Bicocca for hospitality and
support. Financial support from the Laboratoire Européen Associé “Physique
Théorique et Matière Condensée”, a common research program between the Landau
Institute, the Ecole Normale Supérieure de Paris, Paris Sud University and the
Russian Foundation for Basic Research is also gratefully acknowledged.
## Appendix A Solution of non-linear RG equations and cancelation of
logarithmic corrections in effective-amplitude ratios
For the 4-state Potts model, the non-linear RG equation for the relevant
thermal and magnetic fields $\phi$ and $h$, with the corresponding RG
eigenvalues $y_{\phi}$ and $y_{h}$, and the marginal dilution field $\psi$ are
given by
$\displaystyle\frac{d\phi}{d\ln b}$ $\displaystyle=$
$\displaystyle(y_{\phi}+y_{\phi\psi}\psi)\phi,$ (52)
$\displaystyle\frac{dh}{d\ln b}$ $\displaystyle=$
$\displaystyle(y_{h}+y_{h\psi}\psi)h,$ (53) $\displaystyle\frac{d\psi}{d\ln
b}$ $\displaystyle=$ $\displaystyle g(\psi)$ (54)
where $b$ is the length rescaling factor and $l=\ln b$. The function $g(\psi)$
may be Taylor expanded,
$g(\psi)=y_{\psi^{2}}\psi^{2}(1+\frac{y_{\psi^{3}}}{y_{\psi^{2}}}\psi+\dots)$.
Accounting for marginality of the dilution field, there is no linear term. The
first term has been considered by Nauenberg and Scalapino, and later by Cardy,
Nauenberg and Scalapino. The second term was introduced by Salas and Sokal. In
this appendix, we slightly change the notations of Salas and Sokal, keeping
however the notation $y_{ij}$ for all coupling coefficients between the
scaling fields $i$ and $j$. These parameters take the values
$y_{\phi\psi}=3/(4\pi)$, $y_{h\psi}=1/(16\pi)$, $y_{\psi^{2}}=1/\pi$ and
$y_{\psi^{3}}=-1/(2\pi^{2})$, while the relevant scaling dimensions are
$y_{\phi}=\nu^{-1}=3/2$ and $y_{h}=15/8$.
The fixed point is at $\phi=h=0$. Starting from initial conditions $\phi_{0}$,
$h_{0}$, the relevant fields grow exponentially with $l$. The field $\phi$ is
analytically related to the temperature, so the temperature behavior follows
from the renormalization flow from $\phi_{0}\sim|\tau|$ up to some $\phi=O(1)$
outside the critical region. Notice also that the marginal field $\psi$
remains of order $O(\psi_{0})$ and $\psi_{0}$ is negative, $\psi_{0}=O(-1)$.
In zero magnetic field, under a change of length scale, the singular part of
the free energy density transforms as
$f(\psi_{0},\phi_{0})=e^{-Dl}f(\psi,\phi),$ (55)
where $D=2$ is the space dimension. Solving Eq. (52) leads to
$\ln(\phi/\phi_{0})=y_{\phi}l+y_{\phi\psi}\int\psi dl$ where the last integral
is obtained from Eq. (54) rewritten as $\int_{0}^{l}\psi
dl=\frac{1}{y_{\psi^{2}}}\ln({\psi}/{\psi_{0}})+\frac{1}{y_{\psi^{2}}}\ln
G(\psi_{0},\psi)$. Note that $G(\psi_{0},\psi)$ takes the value 1 in Cardy,
Nauenberg and Scalapino and the value
$\frac{y_{\psi^{2}}+y_{\psi^{3}}\psi_{0}}{y_{\psi^{2}}+y_{\psi^{3}}\psi}$ in
Salas and Sokal. Since this term appears always in the same combination, we
write $z=\psi_{0}/\psi$, $\bar{z}=\frac{z}{G(\psi_{0},\psi)}$ and in the same
way we set $x=\phi_{0}/\phi$. One thus obtains
$l=-\frac{1}{y_{\phi}}\ln
x+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln\bar{z}.$ (56)
At the critical temperature $\phi=0$, the $l-$dependence on the magnetic field
obeys a similar expression and one is led to the equality
$l=-\nu\ln x+\mu\ln\bar{z}=-\nu_{h}\ln y+\mu_{h}\ln\bar{z},$ (57)
where $y=h_{0}/h$ and for brevity we will denote $\nu=1/y_{\phi}=\frac{2}{3}$,
$\mu=\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}=\frac{1}{2}$,
$\nu_{h}=1/y_{h}=\frac{8}{15}$ and
$\mu_{h}=\frac{y_{h\psi}}{y_{h}y_{\psi^{2}}}=\frac{1}{30}$. We can thus deduce
the following behavior for the free energy density in zero magnetic field in
terms of the thermal and dilution fields, or at the critical temperature in
terms of magnetic and dilution fields
$\displaystyle f(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu}\
\\!\bar{z}^{-D\mu}f(\phi,\psi)$ (58) $\displaystyle f(h_{0},\psi_{0})$
$\displaystyle=$ $\displaystyle y^{D\nu_{h}}\ \\!\bar{z}^{-D\mu_{h}}f(h,\psi)$
(59)
The other thermodynamic properties follow by derivation with respect to the
scaling fields, e.g.
$E(\phi_{0},\psi_{0})=\frac{\partial}{\partial\phi_{0}}f(\psi_{0},\phi_{0})$,
or
$\displaystyle E(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\frac{\partial\phi}{\partial\phi_{0}}E(\phi,\psi),$ (60) $\displaystyle
C(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\left(\frac{\partial\phi}{\partial\phi_{0}}\right)^{2}C(\phi,\psi),$
(61) $\displaystyle M(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\frac{\partial h}{\partial h_{0}}M(\phi,\psi),$ (62)
$\displaystyle\chi(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
e^{-Dl}\left(\frac{\partial h}{\partial h_{0}}\right)^{2}\chi(\phi,\psi).$
(63)
The derivatives $\frac{\partial\phi}{\partial\phi_{0}}$ and $\frac{\partial
h}{\partial h_{0}}$ will thus determine the scaling behavior of all the
thermodynamic quantities. The first one is obvious,
$\frac{\partial\phi}{\partial\phi_{0}}=x^{-1}$ and for the second one,
$\frac{\partial h}{\partial h_{0}}=y^{-1}$, we express $y$ in terms of $x$ and
$z$ using eq. (57) 101010It follows that
$y=x^{y_{h}/y_{\phi}}\bar{z}^{y_{h\psi}/y_{\psi^{2}}-y_{h}y_{\phi\psi}/y_{\phi}y_{\psi^{2}}}$..
Altogether, introducing the notations $\lambda=y_{h}/y_{\phi}=\frac{5}{4}$ and
$\kappa=y_{h}y_{\phi\psi}/y_{\phi}y_{\psi^{2}}-y_{h\psi}/y_{\psi^{2}}=\frac{7}{8}$
one has
$\displaystyle f(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu}\
\\!\bar{z}^{-D\mu}f(\phi,\psi)$ (64) $\displaystyle E(\phi_{0},\psi_{0})$
$\displaystyle=$ $\displaystyle x^{D\nu-1}\ \\!\bar{z}^{-D\mu}E(\phi,\psi)$
(65) $\displaystyle C(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
x^{D\nu-2}\ \\!\bar{z}^{-D\mu}C(\phi,\psi)$ (66) $\displaystyle
M(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle x^{D\nu-\lambda}\
\\!\bar{z}^{-D\mu+\kappa}M(\phi,\psi)$ (67)
$\displaystyle\chi(\phi_{0},\psi_{0})$ $\displaystyle=$ $\displaystyle
x^{D\nu-2\lambda}\ \\!\bar{z}^{-D\mu+2\kappa}\chi(\phi,\psi)$ (68)
What appears extremely useful in these expressions is that when defining
appropriate effective ratios111111i.e. effective ratios which eventually tend
to universal limits when $\tau\to 0$, the dependence on the quantity $\bar{z}$
cancels, due to the scaling relations among the critical exponents. This
quantity $\bar{z}$ is precisely the only one where the log terms are hidden,
and thus we may infer that not only the leading log terms, but all the log
terms hidden in the dependence on the marginal dilution field disappear in the
conveniently defined effective ratios. For example in an effective ratio like
$R_{C}(x)=x^{2}\frac{C(x,z)\chi(x,z)}{M^{2}(x,z)},$ (69)
all corrections to scaling coming from the variable $z$ disappear.
Now we proceed by iterations of $l=-\nu\ln x+\mu\ln\bar{z}$ and
$\bar{z}=z\frac{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}$.
The asymptotic solution of Eq. (54) is121212When $y_{\psi^{3}}=0$ the
asymptotic solution of Eq. (54) is simply
$\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}$. Thus one can try the ansatz
$\psi=\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}(1+X(l))$ with a small
correction $X(l)$ to solve asymptotically the full Eq. (54). Keeping only
terms of order $O(l^{-3})$ at most, we are led to the following expression
$X^{\prime}(l)=-(1/l)X(l)+y_{\psi^{3}}/((y_{\psi^{2}})^{2}l^{2})$ where we use
$X(l)=Y(l)/l$ to eventually obtain $Y(l)=(y_{\psi^{3}}/(y_{\psi^{2}})^{2})\ln
l$.
$\frac{\psi}{\psi_{0}}=\frac{1}{1-\psi_{0}y_{\psi^{2}}l}\left(1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}}\frac{\ln
l}{l}+O(1/l)\right).$ (70)
We get for the variable $z$
$\displaystyle
z=\frac{1-\psi_{0}y_{\psi^{2}}l}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}}\frac{\ln
l}{l}}\simeq-\psi_{0}y_{\psi^{2}}\nu(-\ln|\tau|)\frac{1+\frac{\mu}{\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{-\ln|\tau|}\right)}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}+O\left(\frac{1}{-\ln|\tau|}\right)}.$
(71)
Similarly, one has the combination
$\frac{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}\simeq\frac{1}{1+(y_{\psi^{3}}/y_{\psi^{2}})\psi_{0}}\left(1-\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{1}{(-\ln|\tau|)}+O\left(\frac{1}{(-\ln|\tau|)^{2}}\right)\right)$
(72)
and eventually one gets for the full correction-to-scaling variable the heavy
expression
$\displaystyle\bar{z}$ $\displaystyle=$ $\displaystyle{\rm
const}\times(-\ln|\tau|)\frac{1+\frac{\mu}{\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}{1+\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}\times\left(1-\frac{y_{\psi^{3}}}{(y_{\psi^{2}})^{2}\nu}\frac{1}{(-\ln|\tau|)}\right)\times{F}(-\ln|\tau|)$
(73) $\displaystyle=$ $\displaystyle{\rm
const}\times(-\ln|\tau|)\frac{1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}{1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}}\times\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right)\times{\cal
F}(-\ln|\tau|)$
where ${\cal F}(-\ln|\tau|)$ is a function of the variable $(-\ln|\tau|)$ only
where also appears the non-universal constant $\psi_{0}$. Using Eq. (67), we
deduce the behavior of the magnetization for example
$\displaystyle M(-|\tau|)$ $\displaystyle=$ $\displaystyle
B|\tau|^{1/12}(-\ln|\tau|)^{-1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\right.$
(74) $\displaystyle\left.\
\qquad\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right){\cal
F}(-\ln|\tau|)\right]^{-1/8}.$
Since all these log expressions are “lazy functions”, it is unsafe to expand
such terms, e.g.
$\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\simeq
1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}$, since the correction term is
not small enough in the accessible temperature range $|\tau|\simeq 0.05-0.10$.
We can thus only extract an effective function ${\cal F}_{eff}(-\ln|\tau|)$
which mimics the real one ${\cal F}(-\ln|\tau|)$ in the convenient temperature
range. This is done through a plot of an effective-magnetization amplitude
$\displaystyle B_{eff}(-|\tau|)$ $\displaystyle=$ $\displaystyle
M(-|\tau|)|\tau|^{-1/12}(-\ln|\tau|)^{1/8}\left[\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)\right.$
(75)
$\displaystyle\left.\qquad\qquad\qquad\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right)\right]^{1/8}$
which is found to behave as
$B_{eff}(-|\tau|)=B\left(1-\frac{C_{1}}{-\ln|\tau|}-\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{1/8}$
(76)
from which one deduces that the function ${\cal F}(-\ln|\tau|)$ takes the
approximate expression
${\cal
F}(-\ln|\tau|)\simeq\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{-1}.$
(77)
What is remarkable is the stability of the fit to Eq. (76). We obtain (see
table 1) $C_{1}=-0.757$ and $C_{2}=-0.522$ which yields an amplitude
$B=1.1570(1)$. It is also possible to try a simpler choice, fixing $C_{1}=0$
and approximating the whole series by the $C_{2}-$term only. We then find the
value $C_{2}^{\prime}=-0.88$ and this leads to a very close magnetization-
amplitude $B=1.1559(2)$.
For the following, we group all the terms coming from the variable $\bar{z}$
into a single function ${\cal G}(-\ln|\tau|)=(-\ln|\tau|)\times{\cal
E}(-\ln|\tau|)\times{\cal F}(-\ln|\tau|)$ where
$\displaystyle{\cal E}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)$
(78)
$\displaystyle\qquad\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{-\ln|\tau|}\right)$
in terms of which the singular parts of the physical quantities take a very
compact form,
$\displaystyle f(\tau)$ $\displaystyle=$ $\displaystyle
F_{\pm}|\tau|^{4/3}{\cal G}^{-1}(-\ln|\tau|)$ (79) $\displaystyle M(-|\tau|)$
$\displaystyle=$ $\displaystyle B|\tau|^{1/12}{\cal G}^{-1/8}(-\ln|\tau|)$
(80) $\displaystyle\chi_{+}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{+}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (81)
$\displaystyle\chi_{L}(-|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{L}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (82)
$\displaystyle\chi_{T}(-|\tau|)$ $\displaystyle=$
$\displaystyle\Gamma_{T}|\tau|^{-7/6}{\cal G}^{3/4}(-\ln|\tau|)$ (83)
$\displaystyle E_{\pm}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha(\alpha-1)}|\tau|^{1/3}{\cal
G}^{-1}(-\ln|\tau|)$ (84) $\displaystyle C_{\pm}(\pm|\tau|)$ $\displaystyle=$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-2/3}{\cal G}^{-1}(-\ln|\tau|).$
(85)
The function ${\cal E}$ is known exactly while the function ${\cal F}$ needs
to be fitted to the numerical data. In the same range of values of the reduced
temperature, the “correction function” ${\cal F}(-\ln|\tau|)$ is now fixed and
the only remaining freedom is to include background terms and possibly
additive corrections to scaling coming from irrelevant scaling fields 131313To
introduce corrections to scaling, let us consider the case of an irrelevant
scaling field, let say $g$, coupled to the temperature field through
$\frac{d\phi}{dl}=y_{\phi}\phi+y_{\phi\psi}\phi\psi+y_{\phi g}\phi
g\quad\hbox{and}\quad\frac{dg}{dl}=y_{g}g$ ($\Delta>0$ above plays the rôle of
$-y_{g}/y_{\phi}$, and is thus linked to the corresponding negative RG
eigenvalue $y_{g}$). Solving for $g$ gives $g=g_{0}e^{y_{g}l}$ (the irrelevant
scaling field decays exponentially when one approaches the fixed point).
Solving for $\psi$ gives $\psi=\frac{\psi_{0}}{1-\psi_{0}y_{\psi^{2}}l}$, and
for $\phi$,
$l=\frac{1}{y_{\phi}}\ln(\phi/\phi_{0})-\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln(\psi/\psi_{0})+\frac{y_{\phi
g}}{y_{\phi}y_{g}}g_{0}(e^{y_{g}l}-1).$ Iteration now leads to
$l=-\frac{1}{y_{\phi}}\ln|\tau|+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln(-\ln|\tau|)+\frac{y_{\phi\psi}}{y_{\phi}y_{\psi^{2}}}\ln\frac{|\psi_{0}|y_{\psi^{2}}}{y_{\phi}}+\frac{y_{\phi
g}g_{0}}{y_{\phi}|y_{g}|}(1-|\tau|^{|y_{g}|/y_{\phi}})$ and thus a free energy
density including the additive correction term $f\simeq e^{-Dl}={\rm
const}\times|\tau|^{D/y_{\phi}}(-\ln|\tau|)^{-Dy_{\phi\psi}/y_{\phi}y_{\psi^{2}}}\left(1+\frac{Dy_{\phi
g}g_{0}}{y_{\phi}|y_{g}|}|\tau|^{|y_{g}|/y_{\phi}}\right).$ In our case, the
$\ln|\tau|$ terms are due to the first scaling field (marginal) through the
complicated variable $\bar{z}$ and other correction terms could be added, e.g.
$f(\tau)=F_{\pm}|\tau|^{4/3}(-\ln|\tau|)^{-1}{\cal E}^{-1}(-\ln|\tau|){\cal
F}^{-1}(-\ln|\tau|)(1+D|\tau|^{\frac{2}{3}|y_{g}|}).$ . Among the additive
correction terms, we may have those coming from the thermal sector
$\Delta_{\phi_{n}}=-\nu y_{\phi_{n}}$, where the RG eigenvalues are
$y_{\phi_{n}}=D-\frac{1}{2}n^{2}$. The first dimension $y_{\phi_{1}}=y_{\phi}$
is the temperature RG eigenvalue, the next one, $y_{\phi_{2}}$, vanishes and
is responsible for the appearance of the logarithmic corrections, so the first
irrelevant correction to scaling in the thermal sector comes from
$\Delta_{\phi_{3}}=-\nu y_{\phi_{3}}=5/3$. One can also imagine a coupling of
the magnetic sector to irrelevant scaling fields. The magnetic scaling
dimensions $x_{\sigma_{n}}$ lead to RG eigenvalues
$y_{h_{n}}=D-x_{\sigma_{n}}$. The first dimension $y_{h_{1}}=y_{h}$ is the
magnetic field RG eigenvalue. The second one is still relevant,
$y_{h_{2}}=7/8$, and it could lead, if admissible by symmetry, to corrections
generically governed by the difference of relevant eigenvalues
$(y_{h_{1}}-y_{h_{2}})/y_{\phi}=2/3$. The next contribution comes from
$y_{h_{3}}=-9/8$ and leads to a Wegner correction-to-scaling exponent [29]
$\Delta_{h_{3}}=-\nu y_{h_{3}}=3/4$. Eventually, spatial inhomogeneities of
primary fields (higher order derivatives) bring the extra possibility of
integer correction exponents $y_{n}=-n$ in the conformal tower of the
identity. The first one of these irrelevant terms corresponds to a Wegner
exponent $\Delta_{1}=-\nu(-1)=2/3$ and it is always present. We may thus
possibly include the following corrections: $|\tau|^{2/3}$, $|\tau|^{3/4}$,
$|\tau|^{4/3}$, $|\tau|^{5/3}$, …, the first and third ones being always
present, while the other corrections depend on the symmetry properties of the
observables.
## References
* [1] L.N. Shchur, B. Berche and P. Butera, Phys. Rev. B 77, 144410 (2008).
* [2] U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
* [3] R.H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987).
* [4] M. Caselle, R. Tateo, and S. Vinci, Nucl. Phys. B 562, 549 (1999).
* [5] L.N. Shchur, B. Berche and P. Butera, Europhys. Lett. 81, 30008 (2008).
* [6] B. Berche, P. Butera and L.N. Shchur arXiv:0707.3317.
* [7] R.B. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952).
* [8] M.P.M. den Nijs, J. Phys. A 12, 1857 (1979).
* [9] R.B. Pearson, Phys. Rev. B 22, 2579 (1980).
* [10] B. Nienhuis, J. Stat. Phys. 34, 731 (1984); B. Nienhuis, in Phase Transitions and Critical Phenomena, Vol. 11, edited by C. Domb and J.L. Lebowitz (Academic Press, London, 1987).
* [11] Vl.S. Dotsenko, Nucl. Phys. B 225 [FS11], 54 (1984).
* [12] Vl.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [FS12], 312 (1984).
* [13] G. Delfino and J.L. Cardy, Nucl. Phys. B 519, 551 (1998).
* [14] L. Chim and A.B. Zamolodchikov, Int. J. Mod. Phys. A 7, 5317 (1992).
* [15] J. L. Cardy, N. Nauenberg and D.J. Scalapino, Phys. Rev. B 22, 2560 (1980).
* [16] M. Nauenberg and D.J. Scalapino, Phys. Rev. Lett. 44, 837 (1980).
* [17] J. Salas and A. Sokal, J. Stat. Phys. 88, 567 (1997).
* [18] G. Delfino, G.T. Barkema and J.L. Cardy, Nucl. Phys. B 565, 521 (2000).
* [19] G. Delfino and P. Grinza, Nucl. Phys. B 682, 521 (2004).
* [20] I.G. Enting and A.J. Guttmann, Physica A 321, 90 (2003).
* [21] V. Privman, P.C. Hohenberg, A. Aharony, in Phase Transitions and Critical Phenomena, Vol. 14, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1991).
* [22] L.N. Shchur, Comp. Phys. Comm. 121-122, 83 (1999).
* [23] K. Binder, J. Stat. Phys 24, 69 (1981).
* [24] L.N. Shchur, P. Butera, and B. Berche, Nucl. Phys. B 620, 579 (2002).
* [25] K.M. Briggs, I.G. Enting, and A.J. Guttmann, J. Phys. A 27, 1503 (1994).
* [26] B. Nienhuis, J. Phys. A 15, 199 (1982).
* [27] G.S. Joyce, Proc. R. Soc. Lond. A 343, 45 (1975).
* [28] G.S. Joyce, Proc. R. Soc. Lond. A 345, 277 (1975).
* [29] F.J. Wegner, Phys. Rev. B 5, 4529 (1972).
* [30] L. Shchur and W. Janke, to be published.
|
arxiv-papers
| 2008-09-26T07:53:07
|
2024-09-04T02:48:57.957901
|
{
"license": "Public Domain",
"authors": "Lev N. Shchur, Bertrand Berche, Paolo Butera",
"submitter": "Lev Shchur N",
"url": "https://arxiv.org/abs/0809.4553"
}
|
0809.4578
|
# Conductance of a disordered graphene superlattice
N. Abedpour Department of Physics, Sharif University of Technology,
11365-9161, Tehran, Iran Ayoub Esmailpour Department of physics, Shahid
Rajaei University , Lavizan, Tehran 16788, Iran School of Physics, Institute
for research in fundamental sciences, IPM 19395-5531 Tehran, Iran Reza Asgari
School of Physics, Institute for research in fundamental sciences, IPM
19395-5531 Tehran, Iran M. Reza Rahimi Tabar Department of Physics, Sharif
University of Technology, 11365-9161, Tehran, Iran Institute of Physics, Carl
von Ossietzky University, D-26111 Oldenburg, Germany
###### Abstract
We study the conductance of disordered graphene superlattices with short-range
structural correlations. The system consists of electron- and hole-doped
graphenes of various thicknesses, which fluctuate randomly around their mean
value. The effect of the randomness on the probability of transmission through
the system of various sizes is studied. We show that in a disordered
superlattice the quasiparticle that approaches the barrier interface almost
perpendicularly transmits through the system. The conductivity of the finite-
size system is computed and shown that the conductance vanishes when the
sample size becomes very large, whereas for some specific structures the
conductance tends to a nonzero value in the thermodynamics limit.
###### pacs:
68.65.Cd, 73.22.-f, 73.63.-b, 73.40.Lq
## I Introduction
Graphene, a single atomic layer of graphite, has been successfully produced in
experiment novoselov , which has resulted in intensive investigations on
graphene-based structures, due to the fundamental physics interests that is
involved and the promising applications geim . There are significant current
efforts devoted to growing graphene epitaxially berger by thermal
decomposition of silicon carbide (SiC), or by vapor deposition of hydrocarbons
on catalytic metallic surfaces, which could later be etched away, leaving
graphene on an insulating substrate. The low energy quasiparticle excitations
in graphene are linearly dispersing, and are described by Dirac cones at the
edges of the first Brillouin zone. The linear energy-momentum dispersion has
been confirmed by recent observations novoselov1 . The slope of the linear
relation corresponds the Fermi velocity of chiral Dirac electrons in graphene,
which plays an essential role in the Landau-Fermi liquid theory polini and
has a direct connection to the experimental measurement.
There are some unusual features of graphene, such as the effects of electron-
electron interactions on the ground-state properties yafis , anomalous
tunneling effect described by the Klein tunneling, the tunneling through a p-n
junctionklein ; katsnelson1 that follows from chiral band states, and the
energy-momentum linear dispersion relation. The Klein tunneling predicts that
the chiral massless carrier can pass through a high electrostatic potential
barrier with probability one, regardless of the height and width of the
barrier at normal incidence, which is in contrast with the conventional
nonrelativistic massive carrier tunneling where the transmission probability
decays exponentially with the increasing of the barrier hight and would depend
on the profile of the barrier. greiner ; su ; dombey ; krekora
An exciting experimental development is the ability to apply an electric field
effect or submicron gate voltage, in order to illustrate graphene p-n
junctions. williams By applying an external gate voltage, the system can be
switched from the n-type to the p-type carriers, thereby controlling the
electronic properties that give rise to graphene-based nanodevices. Recently,
strong evidence for Klein tunneling across potential steps which is steep
enough in graphene has been experimentally observed. stander
Clean graphene junctions were predicted to display a number of fascinating
physical phenomena, even in the absence of electron-electron interactions.
cheianov Interestingly, the Veselago lensing of electric current by a single
p-n junction in clean graphene cheianov1 and the Andreev reflection, and the
electron to hole conversion at the interface at normal incidence beenakker ,
have all been predicted theoretically. Such phenomena are predicted to change
both quantitatively and qualitatively when disorder is included in the model.
For instance, inhomogeneous graphene p-n junction systems were studied using
the Thomas-Fermi approximation, including disorder effects, by Fogler and
collaborators fogler . They showed that junction resistance is dominated by
either ballistic or diffusive contributions, depending on the density of
charged impurity and gradient of the carrier density.
In the semiconductor context there are basically a large number of works on
the tunneling, which have resulted in the ”obvious” declaration that the
electronic properties of semiconductor superlattice are different from those
calculated in a single-barrier junction. Moreover, the electronic properties
of semiconductor superlattices in the presence of disorder have been studied
by several groups diez1 ; adame ; diez2 ; bellani ; esmailpour . Importantly,
all the electronic states are localized in the thermodynamic limit for a
semiconductor superlattice in the presence of white-noise disorder. diez2
Graphene superlattices, on the other hand, may be fabricated by adsorbing
adatoms on graphene surface through similar techniques, by positioning and
aligning impurities with scanning tunneling microscopy eigler or by applying
a local top gate voltage to graphene. huard Recently, a periodic pattern in
the scanning tunneling microscope image has been demonstrated on a graphene on
top of a metallic substrate. marchini The transition of hitting massless
particles in graphene-based superlattice structure (GSLs) was first studied by
Bai and Zhang. bai They showed that the conductivity of the GSLs depends on
the superlattice structural parameters. Furthermore, the superlattice
structure of graphene nanoribbons has been recently studied by using first-
principles density functional theory calculations. sevincli These
calculations showed that the magnetic ground state of the constituent ribbons,
the symmetry of the junction, and their functionalization by adatoms represent
structural parameters to the electronic and magnetic properties of such
structures. Recently, novel physical properties of GSLs with one-dimensional
(1D) Kronig-Penney type and 2D muffin-tin type potentials were also studied.
cheol The results showed that a periodic potential applied by suitably
patterned modifications leads to further charge carrier behavior. The
propagation of charge carriers through such a superlattice is highly
anisotropic, and in extreme cases results in group velocities that are reduced
to zero in one direction but are unchanged in the other direction. Moreover,
they showed that the density and type of carrier are extremely sensitive to
the applied potential.
It would, therefore, be worthwhile to investigate how the conductance of
graphene superlattice junctions are affected by structural white noise, and
compare the conductances with those calculated for disordered semiconductor
superlattice. Due to the conservation of pseudospins in graphene,
backscattering process is suppressed at normal incidence, which makes the
disordered regions transparent. beenakker
The purpose of this paper is to study the electronic behavior of graphene
superlattices p-n junctions by using the transfer-matrix method. The system
that we study consists of a sequence of electron-doped graphene as wells, and
hole-doped graphene as barriers. We study the effect of the disorder imposed
on the size of the barriers in the transmission probability, $T$, through the
system as a function of the system size (number of the barriers), together
with various incident angles. The dc conductance of the finite-size system
takes on a nonzero value of the transmission in some special configurations.
Using the finite-size scaling of transmission, we show that the conductance,
in the thermodynamic limit, tends to a finite constant for spacial cases.
The rest of this paper is organized as follows. In Sec. II we introduce the
models and derive the related transfer matrix. We also explain how we
calculate the transmission probability and the dc conductivity. Section III
contains our numerical calculations. We conclude in Sec. IV with a brief
summary.
## II Model and Theory
Consider a system of superlattice p-n junctions in the independent carriers
model at zero temperature, and in the absence of carrier-phonon and spin-orbit
interactions. The low-energy massless Dirac-band Hamiltonian of graphene in
the continuum model can be written asslon ; haldane
${\cal H}_{0}=\hbar v\tau\left(\sigma_{1}\,k_{1}+\sigma_{2}\,k_{2}\right)$
where $\tau=\pm 1$ for the inequivalent $K$ and $K^{\prime}$ valleys at which
$\pi$ and $\pi^{*}$ bands touch, $k_{i}$ is an envelope function momentum
operator, $v$ is the Fermi velocity, and $\sigma_{i}$ is a Pauli matrix that
acts on the sublattice pseudospin degree of freedom. The total Hamiltonian of
a massless carrier in a special geometry is written as, ${\cal H}={\cal
H}_{0}+V(x)$ where $V(x)$ is the graphene-based superlattice potential which
is modeled as described below.
### II.1 Superlattice Model
We consider superlattice p-n junctions in a graphene-based structure. The
system consists of two kinds of graphene with different potentials, the first
being an electron-doped graphene with thickness $d_{W}$, while the second is a
hole-doped part with thickness $d_{B}$, standing alternately. The potential
for the electron- and hole-doped graphene are $V_{0}$ and zero, respectively.
The energy of the incident particle is $E_{0}=2\pi\hbar v/\lambda$ with the
wavelength $\lambda$ across the barriers, in such a way that the Fermi level
lies in the conduction band outside the barrier and the valence band inside
it, i.e., $(0<E_{0}<V_{0})$, as shown in Fig. 1. The growth direction is taken
to be the $x$ axis which is designed as the superlattice axis. In order to
neglect the strip edges, we assume that the width of the graphene strip is
much larger than $d_{B}$. We set disorder situations in which the value of
$d_{B}$ fluctuates around its mean value, given by $<d_{B}>=b$. In the model
the fluctuations are given by, $d_{B}|_{i}=b(1+\delta~{}\epsilon_{i})$, where
$\\{\epsilon_{i}\\}$ is a set of uncorrelated random variables or white noise
with the box distribution, $-1\leq\epsilon_{i}\leq 1$, and $i$ is the site
index. Here, the $\delta$ is the disorder strength.
Figure 1: Model of graphene superlattice p-n junctions.
We consider graphene-based superlattice potential in a simple model as
$\displaystyle V(x)=\left\\{\begin{array}[]{ll}V_{0}\hskip 113.81102pt&\hbox{
${\rm if}\hskip 8.5359pt|x-x_{2i}|<\frac{d_{B}|_{i}}{2}$}\\\ 0\hskip
113.81102pt&\hbox{${\rm otherwise}$}~{},\\\ \end{array}\right.$ (3)
where $x_{2i}$ is the position of barriers’ center. The model is similar to
the potential of semiconductor superlattices that has been used by other
groups. esmailpour
### II.2 DC conductivity
Let us now consider the case in which the incident massless electron in the
GSLs propagates at angle $\phi$ along the $x$ axis (see Fig. 1) and,
therefore, the Dirac spinor components, $\psi_{1}$ and $\psi_{2}$, which are
the solutions to the Dirac Hamiltonian, can be expressed bai as:
$\displaystyle\psi_{1}(x,y)$ $\displaystyle=$
$\displaystyle(a_{i}e^{iK_{i}x}+b_{i}e^{-iK_{i}x})e^{ik_{y}y}$
$\displaystyle\psi_{2}(x,y)$ $\displaystyle=$ $\displaystyle
s_{i}(a_{i}e^{iK_{i}x+i\phi_{i}}-b_{i}e^{-iK_{i}x-i\phi_{i}})e^{ik_{y}y}~{},$
(4)
where
$s_{i}=sgn(E_{0}-V(x)),\hskip 22.76228ptk_{y}=\frac{E_{0}}{\hbar
v}\sin(\phi)~{},$ (5)
and
$K_{i}=\left\\{\begin{array}[]{cc}k_{x}=E_{0}\cos(\phi)/\hbar v&for\,\,\,\,\
well\\\ q_{x}=\sqrt{(E_{0}-V_{0})^{2}/\hbar^{2}v^{2}-k_{y}^{2}}&\,\,\
for\,\,\,\,\ barrier\end{array}\right.$ (6)
In order to calculate the transmission coefficients, we use the transfer-
matrix method katsnelson . To this end, we apply the continuity of the wave
function at the boundaries, and construct the transfer matrices as follows
$\displaystyle\left(\begin{array}[]{c}1\\\ r\\\
\end{array}\right)=\frac{1}{2\cos\phi}\left(\begin{array}[]{cc}e^{-i\phi}-e^{i\theta}&e^{-i\phi}+e^{-i\theta}\\\
e^{i\phi}+e^{i\theta}&e^{i\phi}-e^{-i\theta}\\\ \end{array}\right)P(2N)\times$
(11)
$\displaystyle\left(\begin{array}[]{c}e^{ik_{x}l_{n}}(e^{-i\theta}-e^{i\phi})/[2e^{iq_{x}l_{n}}\cos\theta]\\\
e^{ik_{x}l_{n}}(e^{i\theta}+e^{i\phi})/[2e^{-iq_{x}l_{n}}\cos\theta]\\\
\end{array}\right)t_{2N}$ (14)
where $r$ and $t_{2N}$ are the reflection and transmission coefficients of the
system that consists of $N$ barriers, and $p(2N)$ is the transfer matrix given
by
$\displaystyle P(2N)=\prod_{i=3}^{2N}P_{i,i-1}$ (15) $\displaystyle
P_{i,i-1}=\left(\begin{array}[]{cc}M_{11}&M_{12}\\\ M_{21}&M_{22}\\\
\end{array}\right)$ (18)
where also $P_{i,i-1}$ is a transfer matrix from site $i$ to $i-1$ and
$M_{ij}$ are given by
$\displaystyle M_{11}$ $\displaystyle=$ $\displaystyle
e^{iK_{i}l_{(i-1)}}e^{-iK_{(i-1)}l_{(i-1)}}[e^{-i\varphi_{(i-1)}}-e^{i\varphi_{i}}]/2\cos(\varphi_{(i-1)})$
(19) $\displaystyle M_{12}$ $\displaystyle=$ $\displaystyle
e^{-iK_{i}l_{(i-1)}}e^{-iK_{(i-1)}l_{(i-1)}}[e^{-i\varphi_{(i-1)}}+e^{-i\varphi_{i}}]/2\cos(\varphi_{(i-1)})$
$\displaystyle M_{21}$ $\displaystyle=$ $\displaystyle
e^{iK_{i}l_{(i-1)}}e^{iK_{(i-1)}l_{(i-1)}}[e^{i\varphi_{(i-1)}}+e^{i\varphi_{i}}]/2\cos(\varphi_{(i-1)})$
$\displaystyle M_{22}$ $\displaystyle=$ $\displaystyle
e^{-iK_{i}l_{(i-1)}}e^{iK_{(i-1)}l_{(i-1)}}[e^{i\varphi_{(i-1)}}-e^{-i\varphi_{i}}]/2\cos(\varphi_{(i-1)})$
where $l_{i}=\sum_{j=1}^{j=int[i/2]}d_{B}|_{j}+int[(i-1)/2]d_{W}$ is the
length of system at $i$-th boundary and moreover,
$\varphi_{i}=\left\\{\begin{array}[]{cc}\phi&for\,\,\,\,well\\\
\theta=\tan^{-1}(k_{y}/q_{x})&\,\,\,for\,\,\,\,barrier\end{array}\right.$ (20)
It is evident that $T(E_{0},\phi)=|t_{2N}|^{2}$, and that it can be calculated
from Eq. (11) for a given $N$. When the transmission coefficients are
calculated, the conductivity of system is computed by means of the Büttiker
formula, datta taking the integral of $T(E_{0},\phi)$ over the angle
$G=G_{0}\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}T(E,\phi)\cos(\phi)d\phi$ (21)
where $G_{0}=e^{2}mvw/\hbar^{2}$ with $w$ being the width of the graphene
strip along the $y$ direction.
## III Results and Discussion
Let us first calculate the transmission probability and study the electronic
properties of disordered GSLs as a function of the strength of disorder
introduced in the system. We consider the width of barriers as a random
variable, so that the length of system in the numerical calculations will be
$L=N(b+d_{W})$. In all the numerical calculations, we assumed $b=<d_{B}>=50$
nm, while the wavelength of the incident particle is set by $\lambda=50$ nm,
or, equivalently, the energy of the carrier, $E_{0}=83$ meV. In all of the
calculations we used, $V_{0}=200$ meV, unless otherwise specified. The number
of realization of the random configurations is about 500.
Figure 2 shows the transmission probability, $T$, of the incident electrons
hitting a GSLs, as a function of the angle $\phi$ for several values of the
disorder strength, $\delta$. The number of the barriers in the figure is,
$N=100$, with $d_{W}=10$ nm. It is clear that, the transmission decreases by
increasing the disorder for all the angles, apart from the strictly normal
incidence case, $\phi=0$. This is physically understandable due to the Klein
tunneling process in graphene, where the backscattering process is suppressed.
Moreover, the massless carriers with incident angle close to normal incidence
can survive in the presence of disorder, while the width of the angles around
the normal incidence decreases with increasing strength of the disorder as
well.
In order to understand the finite-size effect and the effect of the width of
the wells, the transmissions probability of a massless particle through the
system were calculated as a function of the incident angle. The results are
shown in Fig. 3 for several sizes at $\delta=0.1$. In Fig. 3(a) we set
$d_{W}=10$ nm. The transmission decreases with increasing system size for all
the angels, except again at $\phi=0$. This behavior is in contrast with a
clean GSLs result, where the number of the peaks increases by increasing the
number of the barriers. bai In Fig. 3(b) the width of the wells is set to,
$d_{w}=30$ nm. Two sharp peaks in the transmission are obtained that disappear
in Fig. 3(a). Furthermore, in the case with $d_{W}=50$ nm there is only one
angle, $\phi\approx 60^{\circ}$, that the transmission survives in the
presence of the disorder, whereas under other angles the transmissions are
suppressed. Meanwhile, there is clearly a wide domain around $\phi=0$ for
which the transmission survives, and is larger than the one shown in Fig.
3(b). The width of the domain decreases with increasing strength of the
disorder. It is worthwhile to note qualitatively that for $\delta=0$, when we
have the condition that, $(q_{x}+k_{x})\times(b+d_{W})=2m\pi$ ( $m$ is an
integer), the transmission has finite values at angles different from
$\phi=0$. This is due to the resonance process in a system with $N$ barriers.
We also studied how disorder, introduced in the GSLs, affects the conductivity
of the system. Hence, we also calculated numerically the dc conductivity by
using Eq. (21), with the white-noise structural disorder imposed on the
system. Figure 4 shows the dc conductance of the GSLs as a function of $V_{0}$
for various strengths of the disorder, $\delta$. As shown in the figure, the
conductivity of the GSLs decreases by increasing the strength of the disorder.
However, the conductivity approaches a finite value, i.e., the existence of a
finite conductivity in finite-size disordered GSLs should be expected. In
general, the resonance condition is given by a function that yields,
$f(q_{x},d_{W},d_{B})=m\pi$. For instance, for the case $N=1$, the condition
yields, $q_{x}d_{B}=m\pi$, as a result of which $T(\phi)$ would be an
oscillating function of $q_{x}$. Note that $q_{x}$ is determined by $V_{0}$.
Consequently, this leads to a finite dc conductivity which is an oscillating
function of $V_{0}$. The observation of conductance oscillations in extremely
narrow graphene hetrostructures has been observed experimentally. young
In Fig. 5 the conductivity of the system is plotted as a function of $V_{0}$,
where the width of the wells is $d_{w}=10$ nm with $\delta=0.1$. The
conductivity decreases with increasing the size of system. The inset in the
figure shows the conductivity of a clean GSLs as a function of $V_{0}$ for
several system sizes. It indicates that the dc conductance of clean
superlattice behaves uniquely for different sizes, but in a disordered GSLs it
decreases by increasing the size of system, as shown in Fig. 5. At a constant
strength of the disorder, changing $d_{W}$ may also change the conductivity,
as depicted in Fig. 6. It demonstrates that the conductivity varies
periodically with increasing $d_{W}$. As a result, in the disordered GSLs, the
dc conductance of finite-size systems depends on the structural parameters,
especially $d_{W}$.
To compute all results that have been presented so far, we considered a system
of finite size. Next, we wish to calculate the finite-size scaling of
$G/G_{0}$. For this purpose, we calculated the conductivity as a function of
the system size. The results are summarized in Fig. 7. Importantly, the
conductivity vanishes by a simple power law, except for the case for which,
$\lambda=d_{W}=50$ nm. In general, for $\lambda=md_{W}$ the conductivity
approaches a finite value as $N$ becomes large.
In order to examine such results better, we also calculated the $G/G_{0}$ for
a case for which $\lambda=d_{W}=45$ nm. We found that the conductance tends to
a constant in the thermodynamic limit. the numerical data are fitted by using
$\frac{G}{G_{0}}=g_{\infty}+\frac{\gamma}{L^{\eta}}$ (22)
where $g_{\infty}$, $\gamma$, and $\eta$ are constants. $g_{\infty}$ is the
asymptotic value of $G/G_{0}$ in the thermodynamic limit, $N\to\infty$. As a
result, for the case $d_{W}=5$ nm we obtain $g_{\infty}=0$, $\gamma\simeq
1.0$, and $\eta\simeq 0.46$; for $d_{W}=10$ nm we obtained, $g_{\infty}=0$,
$\gamma\simeq 0.9$, and $\eta\simeq 0.42$, and $g_{\infty}\simeq 0.14$,
$\gamma\simeq 0.6$ and $\eta\simeq 0.2$ for $d_{W}=50$ nm. In all the case the
regression was with $r^{2}=0.99$, indicating very accurate fits. Note that
$g_{\infty}$ is zero for small $d_{W}$, but tends to a nonzero constant for
$d_{W}=50$ nm.
## IV Conclusion
We studied numerically the dc conductance of a discorded graphene superlattice
p-n junctions for various values of the strength of structural disorder
imposed on the material. It was shown that there exists a width around the
normal incidence angle for which the transmission becomes finite in the
presence of structural white-noise disorder. That is, the white-noise disorder
gives rise the largest number of the peaks in the transmission, suppressed in
the thermodynamic limit but quasiparticles which approach almost
perpendicularly to the barriers transmit through the material. We also
calculated the conductivity of a finite-size disordered system and showed that
the conductivity decreases by increasing the system size but that there are
cases for which the conductance approaches a nonzero value. This result is in
contrast with the case of a clean (ordered) GSLs. bai Furthermore, the
results of the finite-size scaling computations predict a zero conductance for
all the GSLs, except for some special $d_{W}$ values for which
$\lambda=md_{W}$, where $m$ is an integer, in which case the conductance tends
to a nonzero constant in the thermodynamic limit.
Apparently, such a feature is independent of the value of $b=<d_{B}>$.
Consequently, we predict a finite conductivity for a disordered GSLs when the
wavelength of incident particle is equal to $md_{W}$. These results are in
complete contrast with those calculated for disordered semiconductor
superlattice which become insulator. diez1 ; diez2 ; bellani Our finding for
the dc conductance of the GSLs should be important to the design of electronic
nano-devices based on graphene superlattices. It would probably worthwhile to
extend the present work to the case in which a correlated noise is used. In
this case one must replace the white-noise with a proper short- or long-range
correlated noise.
###### Acknowledgements.
R. A. would like to thank the International Center for Theoretical Physics,
Trieste for its hospitality during the period when part of this work was
carried out. A. E and N. A are supported by the IPM grant.
## References
* (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
* (2) A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007); A. K. Geim and A. H. MacDonald, Phys. Today 60, 35 (2007); A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
* (3) C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, Science 312, 1191 (2006).
* (4) K. S. Novoselov, E. McCann, S. V. Morosov, V. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jian, F. Scheden, and A. K. Geim, Nature 438, 197 (2005); Y. Zhang, J. W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005); Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E. Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, Phys. Rev. Lett. 98, 197403 (2007); A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nature Phys. 3, 36 (2007); New J. Phys. 9, 385 (2007); M. Mucha-Kruczyński, O. Tsyplyatyev, A. Grishin, E. McCann, V. I. Fal’ko, A. Bostwick, and E. Rotenberg, Phys. Rev. B 77, 195403 (2008); S. Y. Zhou, G.-H. Gweon, A. V. Fedorov, P. N. First, W. A. de Heer, D.-H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara, Nature Mat. 6, 770 (2007).
* (5) M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea and A. H. MacDonald, Phys. Rev. B 77, 081411(R) (2008); M. Polini, R. Asgari, Y. Barlas, T. Pereg-Barnea and A. H. MacDonald, Solid State Commun. 143, 58 (2007).
* (6) Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari and A. H. MacDonald, Phys. Rev. Lett. 98, 236601 (2007).
* (7) O. Klein, Z. Phys. 53, 157165 (1929).
* (8) M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Phys. 2 ,620 (2006).
* (9) W. Greiner, B. Mueller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, Berlin, 1985).
* (10) R. K. Su, G. C. Siu, and X. Chou, J. Phys. A 26, 1001 (1993).
* (11) N. Dombey and A. Calogeracos, Phys. Rep. 315, 41 (1999).
* (12) P. Krekora, Q. Su, and R. Grobe, Phys. Rev. Lett. 92, 040406 (2004).
* (13) J. R. Williams, L. DiCarlo and C. M. Marcus, Science 317, 638 (2007); B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, Phys. Rev. Lett. 98, 236803 (2007); B. Özyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov and P. Kim, Phys. Rev. Lett. 99, 166804 (2007).
* (14) N. Stander, B. Huard, and D. Goldhaber-Gordon, arXiv: 0806.2319.
* (15) V. V. Cheianov and V. I. Fal’ko, Physics. Rev. B 74, 041403(R) (2006); J. M. Pereira, Jr. V. Mlinar, F. M. Peeters, and P. Vasilopoulos, Phys. Rev. B 74, 045424 (2006).
* (16) V . V. Cheianov and V. I. Fal’ko, and B. L. Altshuler, Science 315, 1252 (2007).
* (17) C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008).
* (18) L. M. Zhang and M. M. Fogler, Phys. Rev. Lett. 100, 116804 (2008); M. M. Fogler, D. S. Novikov, L. I. Glazman, and B. I. Shklovskii, Phys. Rev. B 77, 075420 (2008).
* (19) E. Diez, A. Sánchez, and F. Domínguez-Adame, Phys. Rev. B 50, 14359 (1994).
* (20) F . Domínguez-Adame, A. Sánchez, and E. Diez, Phys. Rev. B 50, 17736 (1994).
* (21) E. Diez, A. Sánchez, and F. Domínguez-Adame, IEEE J. Quantum Electron. 31, 1919 (1995).
* (22) V. Bellani, et. al., Phys. Rev. Lett. 82, 2159 (1999).
* (23) A. Esmailpour, M. Esmaeilzadeh, E. Faizabadi, Pedro Carpena, and M. R. Rahimi Tabar, Phys. Rev. B 74, 024206 (2006).
* (24) H. Hiura, Appl. Surf. Sci. 222, 374 (2004); J. C. Meyer, C. O. Girit, M. F. Crommie, and A. Zettl, Appl. Phys. Lett. 92, 123110 (2008).
* (25) B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, Phys. Rev. Lett. 98, 236803 (2007).
* (26) S. Marchini, S. Gunther, J. Wintterlin, Phys. Rev. B76, 075429 (2007); A. L. Vázquez de Parga, F. Calleja, B. Borca, M. C. G. Passeggi, Jr., J. J. Hinarejos, F. Guinea, and R. Miranda, Phys. Rev. Lett. 100, 056807 (2008); I. Pletikosić, M. Kralj, P. Pervan, R. Brako, J. Coraux, A. T. ŃDiaye, C. Busse, and T. Michely, arXiv:0807.2770; Y. Pan, N. Jiang, J. T. Sun, D. X. Shi, S. X. Du, F. Liu, and H.-J. Gao, arXiv:0709.2858.
* (27) C. Bai and X. Zhang, Phys. Rev. B. 76, 075430 (2007).
* (28) H. Sevinçli, M. Topsakal, and S. Ciraci, arXiv: 0711.2414.
* (29) C.-H. Park, L. Yang, Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature Physics 4, 213 (2008).
* (30) J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109, 272 (1958).
* (31) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
* (32) M. I. Katsnelson, Eur. Phys. J. B 51, 157 (2006); J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, and C. W. Beebakker, Phys. Rev. lett. 96, 246802 (2006); M. Titov, Europhys. Lett. 79, 17004 (2007).
* (33) S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, London, 1995).
* (34) A. F. Young and P. Kim, arXiv: 0808.0855.
Figure 2: (Color online) Transmission probability $T$ of electrons through the
system as a function of the incident angle for several disorder strengths.
(a): $\delta=0$ and $0.1$, and (b): $\delta=0.0,0.05,0.1,0.15$ and $0.2$ for
$N=100$ and $d_{W}=10$nm.
Figure 3: (Color online) Transmission probability $T$ of the massless carriers
through the system as a function of the incident angle, the system size and
$\delta=0.1$ for $d_{W}=10$ nm. (a) $d_{W}=30$ nm. (b) $d_{W}=50$ nm. Figure
4: (Color online) DC conductivity as a function of the barrier potential,
$V_{0}$ (in units of meV) for various strengths of the disorder and $N=100$
and $d_{W}=10$ nm. Figure 5: (Color online) DC conductivity as a function of
the barrier potential $V_{0}$ (in units of meV) and system size for
$\delta=0.1$ and $d_{W}=10$ nm. Inset shows the same, but for clean (ordered)
GSLs. Figure 6: (Color online) DC conductivity as a function of $d_{W}$ and
system sizes for $\delta=0.1$ and $V_{0}=300$ meV. Figure 7: (Color online)
Finite-size scaling of the dc conductivity as a function of size $N$ for the
disorder strength $\delta=0.1$ and various values of $d_{W}$ and $\lambda$ (in
the inset).
|
arxiv-papers
| 2008-09-26T09:46:04
|
2024-09-04T02:48:57.965583
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Abedpour, Ayoub Esmailpour, Reza Asgari, M. Reza Rahimi Tabar",
"submitter": "Reza Asgari",
"url": "https://arxiv.org/abs/0809.4578"
}
|
0809.4587
|
English Translation of Chapter 9 of the book
Adams spectral sequence and stable homotopy groups of spheres
(In Chinese)
(Sciences Press, Beijing 2007)
By Jinkun Lin
September, 2007
Chapter 9
A sequence of new families in the stable homotopy groups of spheres
In this Chapter, we will state and prove the existence of a sequence of new
families in the stable homotopy groups of spheres which is in base on several
papers of the author (especially [7][8][9][24]).As a preminilaries, in §1 we
introduce some spectra which is closely related to Moore spectrum and Smith-
Toda spectrum $V(1)$ and state some of their properties. In §2 we state and
prove a general result on the convergence of $h_{0}\sigma$ and
$h_{0}\sigma^{\prime}$ for a pair of $a_{0}$-related elements $\sigma$ and
$\sigma^{\prime}$. ( the generalization of [8] Theorem A). §3 is devoted to
state and prove a general result on the convergence of
$(i^{\prime}i)_{*}(h_{0}\sigma)$ induces the convergence of
$(i^{\prime}i)_{*}(g_{0}\sigma)$ in the stable homotopy groups of Smith-Toda
spectrum $V(1)$ (the generalization of [7] Theorem II). In §4 we prove a pull
back Theorem in the Adam spectral sequence and as a corollary of the main
results in §2 and §4 , in §5 we obtain the convergence of a sequence of
$h_{0}h_{n},h_{0}b_{n},h_{0}h_{n}h_{m},h_{0}(h_{n}b_{m-1}-h_{m}b_{n-1})$ new
families in the stable homotopy groups of spheres. §6 concerns with the
convergence of a sequence of
$h_{0}\sigma\widetilde{\gamma}_{s},g_{0}\sigma\widetilde{\gamma}_{s}$-elements.
In §7, we first prove $h_{n}$ Theorem and then obtain the third periodicity
$\gamma_{p^{n}/s}$-families ([24] Theorem I and Theorem II)). At last , in §8,
the second periodicity $\beta_{tp^{n}/j,i+1}$-families in the stable homotopy
groups of spheres are detected.
§1. Some spectra closely related to the Moore spectrum and Smith-Toda spectrum
$V(1)$
Let $M$ be the Moore spectrum given by the cofibration
(9.1.1) $S\stackrel{{\scriptstyle
p}}{{\longrightarrow}}S\stackrel{{\scriptstyle
i}}{{\longrightarrow}}M\stackrel{{\scriptstyle j}}{{\longrightarrow}}\Sigma S$
Let $\alpha:\Sigma^{q}M\rightarrow M$ be Adams map and $K$ be its cofibre
given by the cofibration
(9.1.2)
$\Sigma^{q}M\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}M\stackrel{{\scriptstyle
i^{\prime}}}{{\longrightarrow}}K\stackrel{{\scriptstyle
j^{\prime}}}{{\longrightarrow}}\Sigma^{q+1}M$
The above spectrum $K$ which we briefly write as $K$ actually is the Smith-
Toda spectrum $V(1)$ in Chapter 6 §2.
Now we introduce some spectra closely related to $S,M$ or $K$ . Let $L$ be the
cofibre of $\alpha_{1}=j\alpha i:\Sigma^{q-1}S\rightarrow S$ given by the
cofibration
(9.1.3)
$\qquad\quad\Sigma^{q-1}S\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}S\stackrel{{\scriptstyle
i^{\prime\prime}}}{{\longrightarrow}}L\stackrel{{\scriptstyle
j^{\prime\prime}}}{{\longrightarrow}}\Sigma^{q}S$.
Let $Y$ be the cofibre of $i^{\prime}i:S\rightarrow K$ given by the
cofibration
(9.1.4) $\quad\qquad S\stackrel{{\scriptstyle
i^{\prime}i}}{{\longrightarrow}}K\stackrel{{\scriptstyle\overline{r}}}{{\longrightarrow}}Y\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\Sigma
S$.
$Y$ actually is the Toda spectrum $V(1\frac{1}{2})$, and it also is the
cofibre of $j\alpha:\Sigma^{q}M\rightarrow\Sigma S$ given by the cofibration
(9.1.5) $\quad\qquad\Sigma^{q}M\stackrel{{\scriptstyle
j\alpha}}{{\longrightarrow}}\Sigma
S\stackrel{{\scriptstyle\overline{w}}}{{\longrightarrow}}Y\stackrel{{\scriptstyle\overline{u}}}{{\longrightarrow}}\Sigma^{q+1}M$,
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma in the stable homotopy category (cf. Chapter 3 §7)
$\qquad\qquad S\quad\stackrel{{\scriptstyle
i^{\prime}i}}{{\longrightarrow}}\quad K\quad\stackrel{{\scriptstyle
j^{\prime}}}{{\longrightarrow}}\quad\Sigma^{q+1}M$
$\qquad\qquad\quad\searrow i\quad\nearrow
i^{\prime}\quad\searrow\overline{r}\quad\nearrow\overline{u}$
(9.1.6) $\qquad\qquad\qquad M\quad\qquad\qquad Y$
$\qquad\qquad\quad\nearrow\alpha\quad\searrow
j\quad\nearrow\overline{w}\quad\searrow\epsilon$
$\qquad\qquad\Sigma^{q}M\quad\stackrel{{\scriptstyle
j\alpha}}{{\longrightarrow}}\quad\Sigma S\quad\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\quad\Sigma S$
Note that $\alpha_{1}\cdot p=p\cdot\alpha_{1}=0$, then there exist
$\pi\in[\Sigma^{q}S,L]$ and $\xi\in[L,S]$ such that $p=j^{\prime\prime}\pi$
and $p=\xi i^{\prime\prime}$. since $\pi_{q}S$ = 0, then $\pi_{q}L\cong
Z_{(p)}\\{\pi\\}$. Moreover, $i^{\prime\prime}\xi
i^{\prime\prime}=i^{\prime\prime}\cdot p=(p\wedge 1_{L})i^{\prime\prime}$,
then $p\wedge 1_{L}=i^{\prime\prime}\xi+\lambda\pi j^{\prime\prime}$ for some
$\lambda\in Z_{(p)}$. By composing $j^{\prime\prime}$ on the above equation we
have $p\cdot j^{\prime\prime}=j^{\prime\prime}(p\wedge 1_{L})=\lambda
j^{\prime\prime}\pi\cdot j^{\prime\prime}=\lambda p\cdot j^{\prime\prime}$ so
that $\lambda=1$ and we have
(9.1.7) $p\wedge 1_{L}\quad=\quad i^{\prime\prime}\xi+\pi j^{\prime\prime}$.
By the following homotopy commutative diagram of $3\times 3$-Lemma
$\qquad\qquad\Sigma^{q}S\quad\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\quad\Sigma^{q}S\quad\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}\quad\Sigma
S$
$\qquad\qquad\quad\searrow\pi\quad\nearrow j^{\prime\prime}\quad\searrow
i\quad\nearrow j\alpha\quad\searrow i^{\prime\prime}$
(9.1.8) $\qquad\qquad\qquad
L\qquad\qquad\quad\Sigma^{q}M\qquad\qquad\Sigma^{q+1}L$
$\qquad\qquad\quad\nearrow
i^{\prime\prime}\quad\searrow\overline{h}\quad\nearrow\overline{u}\quad\searrow
j\quad\nearrow\pi$
$\qquad\qquad
S\quad\stackrel{{\scriptstyle\overline{w}}}{{\longrightarrow}}\quad\Sigma^{-1}Y\quad\stackrel{{\scriptstyle
j\overline{u}}}{{\longrightarrow}}\quad\Sigma^{q+1}S$
we obtain the following cofibration
(9.1.9)
$\quad\Sigma^{q}S\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}L\stackrel{{\scriptstyle\overline{h}}}{{\longrightarrow}}\Sigma^{-1}Y\stackrel{{\scriptstyle
j\overline{u}}}{{\longrightarrow}}\Sigma^{q+1}S$
and there are equations $\overline{u}\overline{h}=i\cdot
j^{\prime\prime},\quad\bar{h}i^{\prime\prime}=\overline{w},\quad\pi\cdot
j=i^{\prime\prime}j\alpha$ By $2\alpha ij\alpha=ij\alpha^{2}+\alpha^{2}ij$
(cf. (6.5.3)), then we have $\alpha_{1}\alpha_{1}$ = 0 and so there are
$\phi\in[\Sigma^{2q-1}S,L]$ and $(\alpha_{1})_{L}\in[\Sigma^{q-1}L,S]$ such
that
(9.1.10) $\quad
j^{\prime\prime}\phi\quad=\quad\alpha_{1}\quad=\quad(\alpha_{1})_{L}\cdot
i^{\prime\prime}$.
Let $W$ be the cofibre of $\phi:\Sigma^{2q-1}S\rightarrow L$, then $W$ also is
the cofibre of $(\alpha_{1})_{L}:\Sigma^{q-1}L\rightarrow S$ , This can be
seen by the following homotopy commutative diagram of $3\times 3$-Lemma
$\qquad\quad\Sigma^{2q-1}S\qquad\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}\quad\Sigma^{q}S\quad\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}\qquad\Sigma
S$
$\qquad\qquad\qquad\searrow\phi\quad\nearrow j^{\prime\prime}\quad\searrow
i^{\prime\prime}\quad\nearrow(\alpha_{1})_{L}$
(9.1.11) $\qquad\qquad\qquad L\qquad\qquad\qquad\Sigma^{q}L$
$\qquad\qquad\qquad\nearrow i^{\prime\prime}\quad\searrow w\quad\nearrow
u\quad\searrow j^{\prime\prime}$
$\qquad\qquad S\qquad\stackrel{{\scriptstyle
wi^{\prime\prime}}}{{\longrightarrow}}\qquad W\qquad\stackrel{{\scriptstyle
j^{\prime\prime}u}}{{\longrightarrow}}\qquad\Sigma^{2q}S$.
that is, we have two cofibrations
(9.1.12)
$\quad\Sigma^{2q-1}S\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}L\stackrel{{\scriptstyle
w}}{{\longrightarrow}}W\stackrel{{\scriptstyle
j^{\prime\prime}u}}{{\longrightarrow}}\Sigma^{2q}S$
(9.1.13)
$\quad\Sigma^{q-1}L\stackrel{{\scriptstyle(\alpha_{1})_{L}}}{{\longrightarrow}}S\stackrel{{\scriptstyle
wi^{\prime\prime}}}{{\longrightarrow}}W\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\Sigma^{q}L$.
We write the Toda spectrum $V(\frac{1}{2})$ as $K^{\prime}$, it is the cofibre
of $jj^{\prime}:\Sigma^{-1}K\rightarrow\Sigma^{q+1}S$ given by the cofibration
(9.1.14)$\quad\Sigma^{-1}K\stackrel{{\scriptstyle
jj^{\prime}}}{{\longrightarrow}}\Sigma^{q+1}S\stackrel{{\scriptstyle
z}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle
x}}{{\longrightarrow}}K$
$K^{\prime}$ also is the cobibre of $\alpha i:\Sigma^{q}S\rightarrow M$ given
by the cofibration
(9.1.15)$\quad\Sigma^{q}S\stackrel{{\scriptstyle\alpha
i}}{{\longrightarrow}}M\stackrel{{\scriptstyle
v}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle
y}}{{\longrightarrow}}\Sigma^{q+1}S$
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\qquad\Sigma^{q}S\quad\stackrel{{\scriptstyle\alpha
i}}{{\longrightarrow}}\quad M\quad\stackrel{{\scriptstyle
i^{\prime}}}{{\longrightarrow}}\quad K$
$\qquad\qquad\quad\searrow i\quad\nearrow\alpha\quad\searrow v\quad\nearrow x$
(9.1.16) $\Sigma^{q}M\qquad\qquad K^{\prime}$
$\qquad\qquad\quad\nearrow j^{\prime}\quad\searrow j\quad\nearrow
z\quad\searrow y$
$\qquad\quad\Sigma^{-1}K\stackrel{{\scriptstyle
jj^{\prime}}}{{\longrightarrow}}\quad\Sigma^{q+1}S\quad\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\Sigma^{q+1}S$
By $\alpha_{1}\wedge 1_{M}=ij\alpha-\alpha ij$, let
$\alpha^{\prime}=\alpha_{1}\wedge 1_{K}\in[\Sigma^{q-1}K,K]$, then
$j^{\prime}\alpha^{\prime}=-(\alpha_{1}\wedge 1_{M})j^{\prime}=\alpha
ijj^{\prime}\in[\Sigma^{-2}K,M]$. By (9.1.16) we have $y\cdot z=p$, then
$y\cdot z\cdot y=p\cdot y=y(1_{K^{\prime}}\wedge p)$, so that $z\cdot
y=1_{K^{\prime}}\wedge p$. This is because $[K^{\prime},M]=0$ which can be
seen by the following exact sequence induced by (9.1.15)
$0=[\Sigma^{q+1}S,M]\stackrel{{\scriptstyle
y^{*}}}{{\longrightarrow}}[K^{\prime},M]\stackrel{{\scriptstyle
v^{*}}}{{\longrightarrow}}[M,M]\stackrel{{\scriptstyle(\alpha
i)^{*}}}{{\longrightarrow}}$
where $[M,M]\cong Z_{p}\\{1_{M}\\}$ so that the above $(\alpha i)^{*}$ is
monic. Moreover, by the following homotopy commutative diagram of $3\times
3$-Lemma we know that the cofibre of $\alpha
ijj^{\prime}:\Sigma^{-1}K\rightarrow\Sigma M$ is $K^{\prime}\wedge M$ given by
the following cofibration
(9.1.17)$\quad\Sigma^{-1}K\stackrel{{\scriptstyle\alpha
ijj^{\prime}}}{{\longrightarrow}}\Sigma
M\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}K^{\prime}\wedge
M\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}K$
$\qquad\qquad\Sigma^{-1}K\quad\stackrel{{\scriptstyle\alpha
ijj^{\prime}}}{{\longrightarrow}}\Sigma M\quad\stackrel{{\scriptstyle
v}}{{\longrightarrow}}\quad\Sigma K^{\prime}$
$\qquad\qquad\quad\searrow jj^{\prime}\quad\nearrow\alpha
i\quad\searrow\psi\quad\nearrow_{1_{K^{\prime}}\wedge j}$
(9.1.18)$\qquad\qquad\quad\Sigma^{q+1}S\qquad\qquad K^{\prime}\wedge M$
$\qquad\qquad\quad\nearrow y\quad\searrow z\quad\nearrow_{1_{K^{\prime}}\wedge
i}\quad\searrow\rho$
$\qquad\qquad K^{\prime}\quad\stackrel{{\scriptstyle 1_{K^{\prime}}\wedge
p}}{{\longrightarrow}}\quad K^{\prime}\qquad\stackrel{{\scriptstyle
x}}{{\longrightarrow}}\qquad K$
From $(1_{K^{\prime}}\wedge j)(v\wedge 1_{M})\overline{m}_{M}=v(1_{M}\wedge
j)\overline{m}_{M}=v=(1_{K^{\prime}}\wedge j)\psi$ we have $(v\wedge
1_{M})\overline{m}_{M}=\psi$ and $d(\psi)\in[\Sigma^{2}M,K^{\prime}\wedge M]$
= 0. Similarly, from $m_{K}(x\wedge 1_{M})(1_{K^{\prime}}\wedge
i)=m_{K}(1_{K}\wedge i)x=x=\rho(1_{K^{\prime}}\wedge i)$ we have
$\rho=m_{K}(x\wedge 1_{M})$ and $d(\rho)\in[\Sigma K^{\prime}\wedge M,K]$ = 0.
Concludingly, up to sign we have
(9.1.19) $\rho=m_{K}(x\wedge 1_{M}),\quad\psi=(v\wedge
1_{M})\overline{m}_{M}$, $d(\rho)=0,d(\psi)=0$.
Let $\alpha^{\prime}=\alpha_{1}\wedge 1_{K}\in[\Sigma^{q-1}K,K]$, where
$\alpha_{1}=j\alpha i\in\pi_{q-1}S$, then
$j^{\prime}\alpha^{\prime}\alpha^{\prime}=0$ and so by (9.1.17), there exists
$\alpha_{K^{\prime}\wedge M}^{\prime}\in[\Sigma^{q-1}K,K^{\prime}\wedge M]$
such that $\rho\alpha_{K^{\prime}\wedge M}^{\prime}=\alpha^{\prime}$. and
$d(\alpha_{K^{\prime}\wedge M}^{\prime})\in[\Sigma^{q}K,K^{\prime}\wedge M]$ =
0. Hence $\rho\alpha_{K^{\prime}\wedge
M}^{\prime}i^{\prime}=\alpha^{\prime}i^{\prime}=i^{\prime}(\alpha_{1}\wedge
1_{M})=\rho(vi\wedge 1_{M})(\alpha_{1}\wedge 1_{M})$ and so we have
$\alpha_{K^{\prime}\wedge M}^{\prime}i^{\prime}=(vi\wedge
1_{M})(\alpha_{1}\wedge 1_{M})+\lambda\psi(ij\alpha ij)$ with $\lambda\in
Z_{p}$ ,this is because $[\Sigma^{q-2}M,M]\cong Z_{p}\\{ij\alpha ij\\}$. Since
the derivation $d(\alpha_{K^{\prime}\wedge
M}^{\prime})=0,d(i^{\prime})=0,d(vi\wedge 1_{M})=0,d(\alpha_{1}\wedge
1_{M})=0,d(\psi)=0$ and $d(ij\alpha ij)=-\alpha_{1}\wedge 1_{M}$, then by
applying $d$ to the above eqaution we have $\lambda\psi(\alpha_{1}\wedge
1_{M})$ = 0 so that $\lambda$ = 0. Concludingly we have
(9.1.20)$\quad\rho\alpha_{K^{\prime}\wedge
M}^{\prime}=\alpha^{\prime},\quad\alpha_{K^{\prime}\wedge
M}^{\prime}i^{\prime}=(vi\wedge 1_{M})(\alpha_{1}\wedge 1_{M})$,
$d(\alpha_{K^{\prime}\wedge M}^{\prime})=0$,
$\qquad\rho(1_{K^{\prime}}\wedge ij)\alpha^{\prime}_{K^{\prime}\wedge
M}=-\alpha^{\prime\prime}\in[\Sigma^{q-2}K,K]$
where we use $d(\alpha^{\prime\prime})=-\alpha^{\prime}$ (cf. (6.5.5) ).
Proposition 9.1.21 Let $p\geq 5$, $V$ be any spectrum and
$f:\Sigma^{t}K^{\prime}\rightarrow V\wedge K$ be any map, then $f\cdot
z=0\in[\Sigma^{t+q+1}S,V\wedge K]$.
Proof: By Theorem 6.5.16 and Theorem 6.5.19, there is a commutative
multiplication $\mu:K\wedge K\rightarrow K$ such that $\mu(i^{\prime}i\wedge
1_{K})=1_{K}=\mu(1_{K}\wedge i^{\prime}i)$ and there is an injection
$\nu:\Sigma^{q+2}K\rightarrow K\wedge K$ such that $(jj^{\prime}\wedge
1_{K})\nu=1_{K}$. Then by (9.1.14) we have $z\wedge 1_{K}=(z\wedge
1_{K})(jj^{\prime}\wedge 1_{K})\nu=0$ and $f\cdot
z=(1_{V}\wedge\mu)(1_{V\wedge K}\wedge i^{\prime}i)f\cdot
z=(1_{V}\wedge\mu)(f\cdot z\wedge 1_{K})i^{\prime}i$ = 0. Q.E.D.
By (9.1.6) we have $\epsilon\cdot\overline{w}=p$(up to sign), Then it is easy
to proof that $\overline{w}\cdot\epsilon=(1_{Y}\wedge p)$. By the following
homotopy commutative diagram of $3\times 3$-Lemma in the stable homotopy
category
$\qquad\quad\Sigma^{q}M\quad\stackrel{{\scriptstyle\alpha^{\prime}i^{\prime}}}{{\longrightarrow}}\qquad\Sigma
K\quad\stackrel{{\scriptstyle r}}{{\longrightarrow}}\qquad\Sigma Y$
$\qquad\qquad\quad\searrow j\alpha\quad\nearrow
i^{\prime}i\quad\searrow^{(r\wedge
1_{M})\overline{m}_{K}}\nearrow_{1_{Y}\wedge j}$
(9.1.22)$\quad\qquad\qquad\Sigma S\quad\qquad\qquad Y\wedge M$
$\qquad\qquad\quad\nearrow\epsilon\qquad\searrow\overline{w}\quad\nearrow_{1_{Y}\wedge
i}\quad\searrow^{m_{M}(\overline{u}\wedge 1_{M})}$
$\qquad\qquad Y\qquad\stackrel{{\scriptstyle 1_{Y}\wedge
p}}{{\longrightarrow}}\quad
Y\qquad\stackrel{{\scriptstyle\overline{u}}}{{\longrightarrow}}\quad\Sigma^{q+1}M$
we know that the cofibre of
$\alpha^{\prime}i^{\prime}:\Sigma^{q}M\rightarrow\Sigma K$ is $Y\wedge M$
given by the following cofibration
(9.1.23)
$\quad\Sigma^{q}M\stackrel{{\scriptstyle\alpha^{\prime}i^{\prime}}}{{\longrightarrow}}\Sigma
K\stackrel{{\scriptstyle(r\wedge
1_{M})\overline{m}_{K}}}{{\longrightarrow}}Y\wedge M\stackrel{{\scriptstyle
m_{M}(\overline{u}\wedge 1_{M})}}{{\longrightarrow}}\Sigma^{q+1}M$.
By (9.1.10), $\alpha_{1}=j^{\prime\prime}\cdot\phi$, where
$\phi\in\pi_{2q-1}L$. Then we have
(9.1.24) $\quad m_{M}(\overline{u}\bar{h}\wedge 1_{M})(\phi\wedge
1_{M})=m_{M}(ij^{\prime\prime}\wedge 1_{M})(\phi\wedge 1_{M})=\alpha_{1}\wedge
1_{M}$ .
Since $\alpha^{\prime}i^{\prime}\cdot\alpha=0$, then by the cofibration
(9.1.23), there is $\alpha_{Y\wedge M}\in[\Sigma^{2q+1}M,\\\ Y\wedge M]$ such
that $\alpha=m_{M}(\overline{u}\wedge 1_{M})\alpha_{Y\wedge M}$. In addition,
$m_{M}(\overline{u}\wedge 1_{M})\alpha_{Y\wedge M}m_{M}\\\ (\overline{u}\wedge
1_{M})=\alpha m_{M}(\overline{u}\wedge 1_{M})=m_{M}(\overline{u}\wedge
1_{M})(1_{Y}\wedge\alpha)$ so that by (9.1.23) we have $\alpha_{Y\wedge
M}m_{M}(\overline{u}\wedge 1_{M})=1_{Y}\wedge\alpha$ modulo $(r\wedge
1_{M})\overline{m}_{K}*[\Sigma^{q-1}Y\wedge M,K]$ = 0, this is because
$[\Sigma^{q}K,K]=0=[\Sigma^{2q}M,K]$ (cf. Theorem 6.5.9 and Theorem 6.2.11).
Note that $d(\alpha_{Y\wedge M})\in[\Sigma^{2q+2}M,Y\wedge M]$ = 0, this is
because $[\Sigma^{q+1}M,M]=0$ and $[\Sigma^{2q+1}M,K]$ = 0 (cf. Theorem 6.2.11
), then $(j\overline{u}\wedge 1_{M})\alpha_{Y\wedge
M}\in[\Sigma^{q-1}M,M]\cap(kerd)\cong Z_{p}\\{\alpha_{1}\wedge 1_{M}\\}$ and
so $(j\overline{u}\wedge 1_{M})\alpha_{Y\wedge M}=\alpha_{1}\wedge 1_{M}$ (up
to scalar). In addition, $m_{M}(\overline{u}\wedge 1_{M})\alpha_{Y\wedge
M}\cdot(\alpha_{1}\wedge 1_{M})=\alpha(\alpha_{1}\wedge
1_{M})=(\alpha_{1}\wedge 1_{M})\alpha=m_{M}(\overline{u}\wedge
1_{M})(\bar{h}\phi\wedge 1_{M})\alpha$, Then by (9.1.23) we have
$\alpha_{Y\wedge M}(\alpha_{1}\wedge 1_{M})=(\bar{h}\phi\wedge 1_{M})\alpha$,
this is because $[\Sigma^{3q-1}M,K]$ = 0\. Moreover we have,
$(1_{Y}\wedge\alpha)(\bar{h}\phi\wedge 1_{M})=\alpha_{Y\wedge
M}m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\phi\wedge 1_{M})=\alpha_{Y\wedge
M}(\alpha_{1}\wedge 1_{M})$. Hence, we have the following relations
(9.1.25) $m_{M}(\overline{u}\wedge 1_{M})\alpha_{Y\wedge M}=\alpha,\quad$
$\alpha_{Y\wedge M}m_{M}(\overline{u}\wedge 1_{M})=1_{Y}\wedge\alpha,\quad$
$\qquad(j\overline{u}\wedge 1_{M})\alpha_{Y\wedge M}=\alpha_{1}\wedge 1_{M}$
(up to scalar)
$\qquad\alpha_{Y\wedge M}(\alpha_{1}\wedge
1_{M})=(1_{Y}\wedge\alpha)(\bar{h}\phi\wedge 1_{M})=(\bar{h}\phi\wedge
1_{M})\alpha$
where $\alpha_{Y\wedge M}\in[\Sigma^{2q+1}M,Y\wedge M]\cap(kerd)$ and
$\phi\in\pi_{2q-1}L$.
Now recall the ring spectrum properties of the spetrum $K$. By Theorem 6.5.16
and (6.5.17), there is a homotopy equivalence $K\wedge K=K\vee\Sigma L\wedge
K\vee\Sigma^{q+2}K$ and there are projections and injections
(9.1.26) $\mu:K\wedge K\rightarrow K,\quad\mu_{2}:K\wedge K\rightarrow\Sigma
L\wedge K,\quad jj^{\prime}\wedge 1_{K}:K\wedge K\rightarrow\Sigma^{q+2}K$
$\quad i^{\prime}i\wedge 1_{K}:K\rightarrow K\wedge K,\quad\nu_{2}:\Sigma
L\wedge K\rightarrow K\wedge K,\quad\nu:\Sigma^{q+2}K\rightarrow K\wedge K$
such that (cf. Theorem 6.5.16, Theorem 6.5.19 )
$\quad\mu(i^{\prime}i\wedge 1_{K})=1_{K}=\mu(1_{K}\wedge i^{\prime}i)$,
$(jj^{\prime}\wedge 1_{K})\nu=1_{K}=(1_{K}\wedge jj^{\prime})\nu$ ,
$\quad(i^{\prime}i\wedge 1_{K})\mu+\nu_{2}\mu_{2}+(jj^{\prime}\wedge
1_{K})\nu=1_{K\wedge K}$ , $\mu_{2}(i^{\prime}i\wedge 1_{K})$ = 0.
Hence ,by (9.1.4), there exists $\overline{\mu}_{2}\in[Y\wedge K,\Sigma
L\wedge K]$ such that $\overline{\mu}_{2}(r\wedge 1_{K})=\mu_{2}$ and
$d(\overline{\mu}_{2})=0\in[Y\wedge K,L\wedge K]$, this can be obtained from
$d(\overline{\mu}_{2}(r\wedge 1_{K}))=d(\mu_{2})=0$ (cf. Theorem 6.5.19(H)).
By the first equation of (9.1.25) , (9.1.23)(9.1.3) and the following homotopy
commutative diagram (9.1.28) of $3\times 3$-Lemma we know that the cofibre of
$\alpha_{Y\wedge M}:\Sigma^{2q+1}M\rightarrow Y\wedge M$ is $\Sigma L\wedge K$
given by the following cofibration
(9.1.27) $\quad\Sigma^{2q+1}M\stackrel{{\scriptstyle\alpha_{Y\wedge
M}}}{{\longrightarrow}}Y\wedge
M\stackrel{{\scriptstyle\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime})}}{{\longrightarrow}}\Sigma L\wedge K\stackrel{{\scriptstyle
j^{\prime}(j^{\prime\prime}\wedge 1_{K})}}{{\longrightarrow}}\Sigma^{2q+2}M$
$\qquad\qquad\Sigma^{q}M\quad\stackrel{{\scriptstyle\alpha^{\prime}i^{\prime}}}{{\longrightarrow}}\quad\Sigma
K\quad\stackrel{{\scriptstyle i^{\prime\prime}\wedge
1_{K}}}{{\longrightarrow}}\Sigma L\wedge K$
$\qquad\qquad\quad\searrow
i^{\prime}\quad\nearrow\alpha^{\prime}\quad\searrow^{(r\wedge
1_{M})\overline{m}_{K}}\nearrow\overline{\mu}_{2}(1_{Y}\wedge i^{\prime})$
(9.1.28)$\qquad\qquad\quad\Sigma^{q}K\qquad\qquad\quad Y\wedge M$
$\qquad\qquad\quad\nearrow_{(j^{\prime\prime}\wedge 1_{K})}\searrow
j^{\prime}\quad\nearrow_{\alpha_{Y\wedge M}}\searrow^{m_{M}(\overline{u}\wedge
1_{M})}$
$\qquad\quad L\wedge K\stackrel{{\scriptstyle
j^{\prime}(j^{\prime\prime}\wedge
1_{K})}}{{\longrightarrow}}\Sigma^{2q+1}M\quad\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\quad\Sigma^{q+1}M$
Since $\epsilon\wedge 1_{K}=\mu(i^{\prime}i\wedge 1_{K})(\epsilon\wedge
1_{K})=0$, then the cofibration (9.1.4) induces a split cofibration
$K\stackrel{{\scriptstyle i^{\prime}i\wedge 1_{K}}}{{\longrightarrow}}K\wedge
K\stackrel{{\scriptstyle r\wedge 1_{K}}}{{\longrightarrow}}Y\wedge K$. that
is, there is a homotopy equivalence $K\wedge K=K\vee Y\wedge K$ so that
$Y\wedge K=\Sigma L\wedge K\vee\Sigma^{q+2}K$and there are projections
$\overline{\mu}_{2}:Y\wedge K\rightarrow\Sigma L\wedge K$ ,
$j\overline{u}\wedge 1_{K}:Y\wedge K\rightarrow\Sigma^{q+2}K$ and injections
$\nu_{Y}:\Sigma^{q+2}K\rightarrow Y\wedge K$, $\overline{\nu}_{2}:\Sigma
L\wedge K\rightarrow Y\wedge K$ such that $\nu_{Y}=(r\wedge 1_{K})\nu$ and
(9.1.29) $\quad(j\overline{u}\wedge
1_{K})\nu_{Y}=1_{K},\quad\overline{\mu}_{2}\overline{\nu}_{2}=1_{L\wedge
K},\quad\nu_{Y}(j\overline{u}\wedge
1_{K})+\overline{\nu}_{2}\overline{\mu}_{2}=1_{Y\wedge K}$.
By (9.1.1)(9.1.15)(9.1.3) and homotopy commutative diagram of $3\times
3$-Lemma we can easily know that the cofibre of $vi:S\rightarrow K^{\prime}$
is $\Sigma L$ given by the following cofibration
(9.1.30) $\quad S\stackrel{{\scriptstyle
vi}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle
k}}{{\longrightarrow}}\Sigma
L\stackrel{{\scriptstyle\xi}}{{\longrightarrow}}\Sigma S$
with relations $\xi\cdot i^{\prime\prime}=p$ so that $\xi
i^{\prime\prime}\wedge 1_{M}=p\wedge 1_{M}=0$ and so $\xi\wedge
1_{M}=\alpha(j^{\prime\prime}\wedge 1_{M})$. In addition, $\xi
i^{\prime\prime}\wedge 1_{K}=p\wedge 1_{K}=0$ so that $\xi\wedge
1_{K}\in(j^{\prime\prime}\wedge 1_{K})^{*}[\Sigma^{q}K,K]$ = 0. Then , the
cofibration (9.1.30) induces a split cofibration $K\stackrel{{\scriptstyle
vi\wedge 1_{K}}}{{\longrightarrow}}K^{\prime}\wedge K\stackrel{{\scriptstyle
k\wedge 1_{K}}}{{\longrightarrow}}\Sigma L\wedge K$. That is to say,
$K^{\prime}\wedge K$ splits into $K\vee\Sigma L\wedge K$ so that there is
$\nu^{\prime}_{2}:\Sigma L\wedge K\rightarrow K^{\prime}\wedge K$ such that
$(k\wedge 1_{K})\nu^{\prime}_{2}=1_{L\wedge K}$ and $\mu(x\wedge
1_{K})(vi\wedge 1_{K})=1_{K}$, $(vi\wedge 1_{K})\mu(x\wedge
1_{K})+\nu^{\prime}_{2}(k\wedge 1_{K})=1_{K^{\prime}\wedge K}$. Moreover,
$x(1_{K^{\prime}}\wedge\epsilon)=(1_{K}\wedge\epsilon)(x\wedge
1_{Y})=0\in[\Sigma^{-1}K^{\prime}\wedge Y,K]$, Hence , by (9.1.14) we have,
$1_{K^{\prime}}\wedge\epsilon=z\cdot\omega$, $\omega\in[K^{\prime}\wedge
Y,\Sigma^{q+2}S]$. We claim that $K^{\prime}\wedge Y$ splits into
$\Sigma^{q+2}S\vee\Sigma L\wedge K$, this can be seen by the following
homotopy commutative diagram of $3\times 3$-Lemma in the stable homotopy
category
$\qquad\qquad K^{\prime}\wedge Y\quad\stackrel{{\scriptstyle
1_{K^{\prime}}\wedge\epsilon}}{{\longrightarrow}}\quad\Sigma
K^{\prime}\quad\stackrel{{\scriptstyle x}}{{\longrightarrow}}\quad\Sigma K$
$\qquad\qquad\qquad\searrow\widetilde{\nu}\qquad\nearrow
z\quad\searrow^{1_{K^{\prime}}\wedge i^{\prime}i}\nearrow^{\mu(x\wedge
1_{K})}$
(9.1.31)$\quad\qquad\qquad\Sigma^{q+2}S\qquad\qquad\Sigma K^{\prime}\wedge K$
$\qquad\qquad\qquad\nearrow jj^{\prime}\quad\searrow
0\quad\nearrow\nu^{\prime}_{2}\qquad\searrow^{1_{K^{\prime}}\wedge r}$
$\quad\qquad\quad K\qquad\stackrel{{\scriptstyle
0}}{{\longrightarrow}}\quad\Sigma^{2}L\wedge
K\quad\stackrel{{\scriptstyle\tilde{\nu}_{2}}}{{\longrightarrow}}\quad\Sigma
K^{\prime}\wedge Y$.
That is, we have a split cofibration $\Sigma L\wedge
K\stackrel{{\scriptstyle\tilde{\nu}_{2}}}{{\longrightarrow}}K^{\prime}\wedge
Y\stackrel{{\scriptstyle\tilde{\nu}}}{{\longrightarrow}}\Sigma^{q+2}S$ so that
there are $\widetilde{\tau}:\Sigma^{q+2}S\rightarrow K^{\prime}\wedge
Y,\widetilde{\mu}_{2}:K^{\prime}\wedge Y\to\Sigma L\wedge K$ such that
(9.1.32) $\widetilde{\nu}\cdot\widetilde{\tau}=1_{S}$,
$\widetilde{\mu}_{2}\widetilde{\nu}_{2}=1_{L\wedge K}$,
$\widetilde{\tau}\widetilde{\nu}+\widetilde{\nu}_{2}\widetilde{\mu}_{2}=1_{K^{\prime}\wedge
Y}$.
Proposition 9.1.33 Let $V$ be any spectrum, then there is a direct sum
decomposition
$\qquad\qquad[\Sigma^{*}M,V\wedge K]=(kerd)i^{\prime}\oplus(kerd)i^{\prime}ij$
where $kerd=[\Sigma^{*}K,V\wedge K]\cap(kerd).$
Proof : For any $f\in[\Sigma^{*}M,V\wedge K]$ we have
$(1_{V}\wedge\mu)(fi\wedge 1_{K})i^{\prime}i=(1_{V}\wedge\mu(1_{K}\wedge
i^{\prime}i))fi=fi$, where $\mu:K\wedge K\rightarrow K$ is the multiplication
of the ring spectrum $K$ such that $\mu(i^{\prime}i\wedge
1_{K})=1_{K}=\mu(1_{K}\wedge i^{\prime}i)$ (cf. (9.1.26)). Then
$f=(1_{V}\wedge\mu)(fi\wedge 1_{K})i^{\prime}+f_{2}\cdot j$ for some
$f_{2}\in[\Sigma^{*+1}S,V\wedge K]$. It follows that
$f=(1_{V}\wedge\mu)(fi\wedge 1_{K})i^{\prime}+(1_{V}\wedge\mu)(f_{2}\wedge
1_{K})i^{\prime}ij$ which proves the result, where $d(fi\wedge 1_{K})=fi\wedge
d(1_{K})=0$, $d(1_{V}\wedge\mu)=1_{V}\wedge d(\mu)$ = 0 (cf. Theorem
6.5.19(G)). Q.E.D.
§2. A general result on convergence of $a_{0}$-related elements
From [12] p. 11 Theorem 1.2.14, there is a nontrivial secondary differential
in the Adams spectral sequence $d_{2}(h_{n})=a_{0}b_{n-1},n\geq 1$, where
$d_{2}:Ext_{A}^{1,p^{n}q}(Z_{p},Z_{p})\rightarrow
Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$ is the secondary diffenrential of the Adams
spectral sequence. We call $h_{n}\in Ext_{A}^{1,p^{n}q}(Z_{p},Z_{p})$ and
$b_{n-1}\in Ext_{A}^{2,p^{n}q}(Z_{p},Z_{p})$ is a pair of $a_{0}$-related
elements. In this section, we prove a general result on convergence of
$a_{0}$-related elements in the Adams spectral sequence of sphere spectrum and
Moore spectrum.
Definition 9.2.1 Let $p\geq 7,s\leq 4$, and there is a nontrivial secondary
differential of the Adams spectral sequence
$d_{2}(\sigma)=a_{0}\sigma^{\prime}$, we call $\sigma\in
Ext_{A}^{s,tq}(Z_{p},Z_{p})$ and $\sigma^{\prime}\in
Ext_{A}^{s+1,tq}(Z_{p},Z_{p})$ is a pair of $a_{0}$-related elements. We have
the following general result.
The main Theorem A (the generalization of [8] Theorem A) Let $p\geq 7,s\leq
4$, $\sigma$ be the unique generator of $Ext_{A}^{s,tq}(Z_{p},Z_{p})$ and
there is a nontrivial secondary differential
$d_{2}(\sigma)=a_{0}\sigma^{\prime}$ in the ASS, where $\sigma^{\prime}$ is
the unique generator (or the linear combination of the two generators) of
$Ext_{A}^{s+1,tq}(Z_{p},Z_{p})$. Moreover, suppose that
(I) $Ext_{A}^{s,tq+rq-u}(Z_{p},Z_{p})=0(r=2,3,4,u=1,2).$
$Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}\sigma\\},\quad
Ext_{A}^{s+1,tq+1}(Z_{p},Z_{p})\cong Z_{p}\\{a_{0}\sigma\\}$,
$Ext_{A}^{s+1,tq-q}(Z_{p},Z_{p})=0$
$Ext_{A}^{s+1,tq+kq+r-1}(Z_{p},Z_{p})=0(k=2,3,4,r=0,1)$,
$Ext_{A}^{s+1,tq+kq+r-2}(Z_{p},Z_{p})=0(k=1,2,3,r=0,1)$.
(II) $Ext_{A}^{s+2,tq+rq+u}(Z_{p},Z_{p})$ = 0,$r=2,3,4,u=-1,0$ or $r=3,4,u=1$,
$Ext_{A}^{s+2,tq}(Z_{p},Z_{p})$ = 0 or has unique generator $\iota$ such that
$a_{0}^{2}\iota\neq 0$,
$Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}\sigma^{\prime}\\}$ or
$Z_{p}\\{h_{0}\sigma^{\prime}_{1},h_{o}\sigma^{\prime}_{2}\\}$.
(III) $Ext_{A}^{s+3,tq+rq+1}(Z_{p},Z_{p})=0(r=1,3,4)$,
$Ext_{A}^{s+3,tq+rq}(Z_{p},Z_{p})=0(r=2,3)$
$Ext_{A}^{s+3,tq+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}\sigma^{\prime}\\}$ or
$Z_{p}\\{\widetilde{\alpha}_{2}\sigma^{\prime}_{1},\widetilde{\alpha}_{2}\sigma^{\prime}_{2}\\}$
$Ext_{A}^{s+3,tq+2}(Z_{p},Z_{p})\cong Z_{p}\\{a_{0}^{2}\sigma^{\prime}\\}$ or
$Z_{p}\\{a_{0}^{2}\sigma^{\prime}_{1},a_{0}^{2}\sigma^{\prime}_{2}\\}$
$Ext_{A}^{s+3,tq+1}(Z_{p},Z_{p})\cong Z_{p}\\{a_{0}\iota\\}$ or 0,
Then $h_{0}\sigma^{\prime}\in Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})$ and
$i_{*}(h_{0}\sigma)\in Ext_{A}^{s+1,tq+q}(H^{*}M,Z_{p})$ are permanent cycles
in the ASS.
To prove the main Theorem A, we need some preminilaries as follows.
For $(\alpha_{1})_{L}\in[\Sigma^{q-1}L,S]$ in (9.1.10) we have
$\alpha_{1}\cdot(\alpha_{1})_{L}\in[\Sigma^{2q-2}L,S]$ = 0 which is obtained
from $\pi_{rq-2}S$ = 0 $(r=2,3)$. Then there is
$\bar{\phi}\in[\Sigma^{2q-1}L,L]$ such that
$j^{\prime\prime}\bar{\phi}=(\alpha_{1})_{L}\in[\Sigma^{q-1}L,S]$ and
$\bar{\phi}\cdot i^{\prime\prime}\in\pi_{2q-1}L$. Since $\pi_{rq-1}S$ has
unique generator $\alpha_{1}=j\alpha i,\alpha_{2}=j\alpha^{2}i$ for $r=1,2$
respectively and $j^{\prime\prime}\phi\cdot p=\alpha_{1}\cdot p=0$, then
$\phi\cdot p=i^{\prime\prime}\alpha_{2}$ (up to scalar). That is,
$i^{\prime\prime}_{*}\pi_{2q-1}S$ also is generated by $\phi$ , so that
$\pi_{2q-1}L\cong Z_{p^{s}}\\{\phi\\}$, for some $s\geq 1$. Hence,
$\overline{\phi}i^{\prime\prime}=\lambda\phi$ with $\lambda\in Z_{(p)}$ and we
have $\lambda\alpha_{1}=\lambda
j^{\prime\prime}\phi=j^{\prime\prime}\overline{\phi}i^{\prime\prime}=(\alpha_{1})_{L}i^{\prime\prime}=\alpha_{1}$
so that $\lambda=1$ (mod $p$). Moreover,
$(\alpha_{1})_{L}\overline{\phi}\in[\Sigma^{3q-2}L,S]=0$, this is because
$\pi_{rq-2}S$ = 0 $(r=3,4)$, then, by (9.1.13), there is
$\bar{\phi}_{W}\in[\Sigma^{3q-1}L,W]$ such that
$u\overline{\phi}_{W}=\overline{\phi}$. Concludingly, we have elements
$\overline{\phi}\in[\Sigma^{2q-1}L,L],\overline{\phi}_{W}\in[\Sigma^{3q-1}L,W]$
such that
(9.2.2)
$j^{\prime\prime}\bar{\phi}=(\alpha_{1})_{L},\quad\overline{\phi}i^{\prime\prime}=\lambda\phi$,
$\lambda=1$ (mod $p$), $u\overline{\phi}_{W}=\overline{\phi}$.
Proposition 9.2.3 Let $p\geq 7$, then
(1) Up to nonzero scalar we have $\phi\cdot
p=i^{\prime\prime}\alpha_{2}=\pi\cdot\alpha_{1}\neq 0$ ,
$(\alpha_{1})_{L}\cdot\pi=\alpha_{2}$ ,
$p\cdot(\alpha_{1})_{L}=\alpha_{2}\cdot j^{\prime\prime}=(\alpha_{1})_{L}\pi
j^{\prime\prime}\neq 0$,
$[\Sigma^{2q-1}L,L]$ has unique generator $\overline{\phi}$ modulo some
elements of filtration $\geq 2$.
(2) $\bar{h}\overline{\phi}(p\wedge 1_{L})\neq 0\in[\Sigma^{2q}L,Y]$
(3) $\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge 1_{L})\neq
0\in[\Sigma^{3q}L,Y]$ , $j^{\prime\prime}\widetilde{\phi}(\pi\wedge
1_{L})\pi=j\alpha^{3}i\in\pi_{3q-1}S$ (up to mod $p$ nonzero scalar) , and
$\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi\neq 0\in\pi_{4q}Y$ , where
$\widetilde{\phi}\in[\Sigma^{2q-1}L\wedge L,L]$] such that
$\widetilde{\phi}(1_{L}\wedge i^{\prime\prime})=\bar{\phi}$.
(4) $\pi_{4q}Y$ has unique generator $\bar{h}\widetilde{\phi}(\pi\wedge
1_{L})\pi$ such that $\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi\cdot p$ = 0.
Proof: (1) Since $j^{\prime\prime}\phi\cdot p=\alpha_{1}\cdot
p=0=j^{\prime\prime}\pi\cdot\alpha_{1}$, and $\pi_{2q-1}S\cong
Z_{p}\\{\alpha_{2}\\}$, then $\phi\cdot
p=i^{\prime\prime}\alpha_{2}=\pi\cdot\alpha_{1}$ (up to scalar). We claim that
$\phi\cdot p\neq 0$, this can be proved as follows. Consider the following
exact sequence
$\quad Z_{p}\\{j\alpha^{2}\\}\cong[\Sigma^{2q-1}M,S]\stackrel{{\scriptstyle
i^{\prime\prime}_{*}}}{{\rightarrow}}[\Sigma^{2q-1}M,L]\stackrel{{\scriptstyle
j^{\prime\prime}_{*}}}{{\rightarrow}}[\Sigma^{q-1}M,S]\stackrel{{\scriptstyle(\alpha_{1})_{*}}}{{\rightarrow}}$
induced by (9.1.3). The right group has unique generator $j\alpha$ satisfying
$(\alpha_{1})_{*}j\alpha=j\alpha ij\alpha=\frac{1}{2}j\alpha\alpha ij\neq 0$,
then the above $(\alpha_{1})_{*}$ is monic, im$j^{\prime\prime}_{*}$ = 0 so
that $[\Sigma^{2q-1}M,L]\cong Z_{p}\\{i^{\prime\prime}j\alpha^{2}\\}$. Suppose
in contrast that $\phi\cdot p=0$, then $\phi\in i^{*}[\Sigma^{2q-1}M,L]$ and
so $\phi=i^{\prime\prime}j\alpha^{2}i$ ,
$\alpha_{1}=j^{\prime\prime}\phi=j^{\prime\prime}i^{\prime\prime}\alpha_{2}$ =
0 which is a contradiction. This shows that $\phi\cdot p\neq 0$ so that the
above scalar is nonzero (mod $p$).
The proof of the second result is similar. To prove the last result, let $x$
be any element of $[\Sigma^{2q-1}L,L]$, then
$j^{\prime\prime}x\in[\Sigma^{q-1}L,S]\cong Z_{p^{s}}\\{(\alpha_{1})_{L}\\}$
for some $s\geq 2$. Hence, $j^{\prime\prime}x=\lambda
j^{\prime\prime}\overline{\phi}$ with $\lambda\in Z_{p^{s}}$ so that
$x=\lambda\overline{\phi}+i^{\prime\prime}x^{\prime}$ ,where
$x^{\prime}\in[\Sigma^{2q-1}L,S]$. Since
$x^{\prime}i^{\prime\prime}\in\pi_{2q-1}S\cong Z_{p}\\{j\alpha^{2}i\\}$ and
$\pi_{3q-1}S\cong Z_{p}\\{j\alpha^{3}i\\}$ , then $x^{\prime}$ is an element
of filtration $\geq 2$ which shows the result.
(2) Suppose incontrast that $\bar{h}\bar{\phi}(p\wedge 1_{L})=0$, then by
(9.1.9) we have $\bar{\phi}(p\wedge
1_{L})=\lambda^{\prime}\pi\cdot(\alpha_{1})_{L}$ ,where $\lambda^{\prime}\in
Z_{(p)}$. Since $j^{\prime\prime}\pi\wedge 1_{M}=p\wedge 1_{M}=0$, then
$\pi\wedge 1_{M}=(i^{\prime\prime}\wedge 1_{M})\alpha$, and so
$\lambda^{\prime}(\pi\wedge 1_{M})i\cdot(\alpha_{1})_{L}$ =
$\lambda^{\prime}(1_{L}\wedge i)\pi(\alpha_{1})_{L}$ = 0. Moreover we have
$\lambda^{\prime}(i^{\prime\prime}\wedge 1_{M})\alpha
i(\alpha_{1})_{L}=\lambda^{\prime}(\pi\wedge 1_{M})i(\alpha_{1})_{L}$ = 0 ,
then $\lambda^{\prime}\alpha i(\alpha_{1})_{L}\in(\alpha_{1}\wedge
1_{M})_{*}[\Sigma^{q}L,M]$ and so $\lambda^{\prime}\alpha
i\alpha_{1}\in(\alpha_{1}\wedge 1_{M})(i^{\prime\prime})^{*}[\Sigma^{q}L,M]$ =
0 which can be obtained by the following exact sequence
$\quad[\Sigma^{2q}S,M]\stackrel{{\scriptstyle(j^{\prime\prime})^{*}}}{{\rightarrow}}[\Sigma^{q}L,M]\stackrel{{\scriptstyle(i^{\prime\prime})^{*}}}{{\rightarrow}}[\Sigma^{q}S,M]\stackrel{{\scriptstyle(\alpha_{1})^{*}}}{{\rightarrow}}$
induced by (9.1.3),where the right group has unique generaator $\alpha i$
satisfying $(\alpha_{1})^{*}\alpha i=\alpha ij\alpha i\neq 0$ so that
$(i^{\prime\prime})^{*}[\Sigma^{q}L,M]$ = 0. The above equation implies that
$\lambda^{\prime}$ = 0 so that we have $\bar{\phi}(p\wedge 1_{L})$ = 0, this
contradicts with the result in (1) on $j^{\prime\prime}\overline{\phi}(p\wedge
1_{L})=p\cdot(\alpha_{1})_{L}\neq 0$. This shows that
$\bar{h}\bar{\phi}(p\wedge 1_{L})\neq 0$.
(3) Since $\pi_{rq-2}S$ = 0( $r=2,3,4$), then
$\bar{\phi}(1_{L}\wedge\alpha_{1})\in[\Sigma^{3q-2}L,L]$ = 0, and so there is
$\widetilde{\phi}\in[\Sigma^{2q-1}L\wedge L,L]$ such that
$\widetilde{\phi}(1_{L}\wedge i^{\prime\prime})=\bar{\phi}$. We first prove
$\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge 1_{L})\neq 0$. For otherwise, if it
is zero, then $\overline{\phi}\pi\cdot p=\widetilde{\phi}(\pi\wedge
1_{L})(p\wedge 1_{L})i^{\prime\prime}$ = 0 so that $\overline{\phi}\pi\in
i^{*}[\Sigma^{3q-1}M,L]$. However,
$(j^{\prime\prime})_{*}[\Sigma^{3q-1}M,L]\subset[\Sigma^{2q-1}M,S]$ the last
of which has unique generator $j\alpha^{2}$ satisfying
$(\alpha_{1})_{*}(j\alpha^{2})=j\alpha ij\alpha^{2}\neq 0$, then
$(j^{\prime\prime})_{*}[\Sigma^{3q-1}M,L]$ = 0 and so we have
$(\alpha_{1})_{L}\pi=j^{\prime\prime}\overline{\phi}\pi\in
i^{*}(j^{\prime\prime})_{*}[\Sigma^{3q-1}M,\\\ L]$ = 0, this contradicts with
the result in (1).
Now suppose in contrast that $\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge
1_{L})=0$, then by (9.1.9) we have, $\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge
1_{L})=\pi\cdot\omega$ , where $\omega\in[\Sigma^{2q-1}L,S]$ satisfying
$\omega i^{\prime\prime}=\lambda_{1}\alpha_{2}$ for some $\lambda_{1}\in
Z_{p}$. It follows that $(i^{\prime\prime}\wedge 1_{M})\alpha
i\omega=(1_{L}\wedge i)\pi\cdot\omega$ = 0, then $\alpha
i\omega\in(\alpha_{1}\wedge 1_{M})_{*}[\Sigma^{2q}L,M]$ and so
$\lambda_{1}\alpha i\alpha_{2}=\alpha i\omega
i^{\prime\prime}\in(\alpha_{1}\wedge
1_{M})_{*}(i^{\prime\prime})^{*}[\Sigma^{2q}L,M]=(\alpha_{1})^{*}(i^{\prime\prime})^{*}[\Sigma^{2q}L.M]=0$.
This shows that $\lambda_{1}$ = 0 (since $\alpha i\alpha_{2}=\alpha
ij\alpha^{2}i\neq 0$). Hence, $\omega=\lambda_{2}j\alpha^{3}i\cdot
j^{\prime\prime}$ and $\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge
1_{L})=\lambda_{2}\pi\cdot j\alpha^{3}i\cdot j^{\prime\prime}$ for some
$\lambda_{2}\in Z_{(p)}$. It follows that $\overline{\phi}\pi\cdot
p=\widetilde{\phi}(\pi\wedge 1_{L})(p\wedge 1_{L})i^{\prime\prime}$ = 0 , then
$\overline{\phi}\pi\in i^{*}[\Sigma^{3q-1}M,L]$ and so
$(\alpha_{1})_{L}\pi=j^{\prime\prime}\overline{\phi}\pi\in
i^{*}(j^{\prime\prime})_{*}[\Sigma^{3q-1}M,L]$ = 0. This contradicts with the
result in (1) on $(\alpha_{1})_{L}\pi\neq 0$.
For the second result, by (9.1.9) we have $\pi\cdot j=i^{\prime\prime}j\alpha$
, then $j^{\prime\prime}\widetilde{\phi}(\pi\wedge 1_{L})\pi\cdot
j=j^{\prime\prime}\widetilde{\phi}(\pi\wedge
1_{L})i^{\prime\prime}j\alpha=j^{\prime\prime}\overline{\phi}\pi
j\alpha=(\alpha_{1})_{L}\pi j\alpha=\alpha_{2}j\alpha=j\alpha^{3}ij$ (up to
mod $p$ nonzero scalar) . Consequently we have
$j^{\prime\prime}\widetilde{\phi}(\pi\wedge 1_{L})\pi=j\alpha^{3}i$ (up to
nonzero scalar), This is because $\pi_{3q-1}S\cong Z_{p}\\{\alpha_{3}\\}$ so
that $p^{*}\pi_{3q-1}S$ = 0.
For the last result, we first prove $\widetilde{\phi}(\pi\wedge 1_{L})\pi\neq
0$. For otherwise , if it is zero, then $0=\widetilde{\phi}(\pi\wedge
1_{L})\pi\cdot j=\widetilde{\phi}(\pi\wedge
1_{L})i^{\prime\prime}j\alpha=\overline{\phi}\pi j\alpha$ and so
$\alpha_{2}j\alpha=(\alpha_{1})_{L}\pi
j\alpha==j^{\prime\prime}\overline{\phi}\pi j\alpha$ = 0 , this is a
contradiction (since $\alpha_{2}j\alpha=j\alpha^{2}ij\alpha\neq
0\in[\Sigma^{3q-2}M,S]$). Now suppose incontrast that
$\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi=0$, Then , by (9.1.9) and
$\pi_{3q-1}S\cong Z_{p}\\{\alpha_{3}\\}$ we have $\widetilde{\phi}(\pi\wedge
1_{L})\pi=\lambda\pi\cdot j\alpha^{3}i=\lambda i^{\prime\prime}j\alpha^{4}i$
for some $\lambda\in Z_{p}$ and so $j^{\prime\prime}\widetilde{\phi}(\pi\wedge
1_{L})\pi$ = 0, this contradicts with the second result.
(4) Since $(\overline{u})_{*}\pi_{4q}Y\subset\pi_{3q-1}M$ and the last of
which has unique generator
$ij\alpha^{3}i=ij^{\prime\prime}\widetilde{\phi}(\pi\wedge
1_{L})\pi=\overline{u}\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi$(up to
nonzero scalar) and $\pi_{4q-1}S\cong Z_{p}\\{j\alpha^{4}i\\}$ such that
$(\overline{w})_{*}\pi_{4q-1}S$ = 0, then $\pi_{4q}Y$ has unique generator
$\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi$. Moreover by (9.1.7) we have,
$\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi\cdot p=\bar{h}(p\wedge
1_{L})\widetilde{\phi}(\pi\wedge
1_{L})\pi=\bar{h}i^{\prime\prime}\xi\widetilde{\phi}(\pi\wedge
1_{L})\pi=\overline{w}j\alpha^{4}i$ = 0. This shows the result. Q.E.D.
Proposition 9.2.4 Let $p\geq 7$, then under the supposition of the main
Theorem A we have
$Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})$ = 0, $Ext_{A}^{s+1,tq}(H^{*}L,H^{*}L)\cong
Z_{p}\\{(\sigma^{\prime})_{L}\\}$ or
$Z_{p}\\{(\sigma^{\prime}_{1})_{L},(\sigma^{\prime}_{2})_{L}\\}$, where $L$ is
the spectrum in (9.1.3) and there are relations
$(i^{\prime\prime})^{*}(\sigma^{\prime})_{L}=(i^{\prime\prime})_{*}(\sigma^{\prime})$
or
$(i^{\prime\prime})^{*}(\sigma^{\prime}_{1})_{L}=(i^{\prime\prime})_{*}(\sigma^{\prime}_{1}),(i^{\prime\prime})^{*}(\sigma^{\prime}_{2})_{L}=(i^{\prime\prime})_{*}(\sigma^{\prime}_{2})$.
Proof: Consider the following exact sequence
$\quad Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\stackrel{{\scriptstyle
i^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle
j^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq}(Z_{p},Z_{p})\stackrel{{\scriptstyle(\alpha_{1})_{*}}}{{\longrightarrow}}$
induced by (9.1.3). The right group has unique generator $\sigma^{\prime}$ or
has two generators $\sigma^{\prime}_{1},\sigma^{\prime}_{2}$ satisfying
$(\alpha_{1})_{*}(\sigma^{\prime})=h_{0}\sigma^{\prime}\neq 0$ or
$(\alpha_{1})_{*}(\sigma^{\prime}_{1})=h_{0}\sigma^{\prime}_{1}\neq
0,(\alpha_{1})_{*}(\sigma^{\prime}_{2})=h_{0}\sigma^{\prime}_{2}\neq 0\in
Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})$ (cf. the supposition II), then the above
$(\alpha_{1})_{*}$ is monic so that im $j^{\prime\prime}_{*}$ = 0. Moreover,
the left group has unique generator $h_{0}\sigma=(\alpha_{1})_{*}(\sigma)$ ,
then im $i^{\prime\prime}_{*}$ = 0 so that $Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})$
= 0. Look at the following exact sequence
$\quad
0=Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(j^{\prime\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq}(H^{*}L,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(\alpha_{1})^{*}}}{{\longrightarrow}}$
induced by (9.1.3). Since $Ext_{A}^{s+1,tq}(Z_{p},Z_{p})\cong
Z_{p}\\{\sigma^{\prime}\\}$ or
$Z_{p}\\{\sigma^{\prime}_{1},\sigma^{\prime}_{2}\\}$ and
$Ext_{A}^{s+1,tq-q}(Z_{p},Z_{p})$ = 0, then the right group has unique
generator $(i^{\prime\prime})_{*}(\sigma^{\prime})$ or has two generators
$(i^{\prime\prime})_{*}(\sigma^{\prime}_{1})$,
$(i^{\prime\prime})_{*}(\sigma^{\prime}_{2})$, the image of which under
$(\alpha_{1})^{*}$ is zero. Then, the result on the middle group is proved.
Q.E.D.
Proposition 9.2.5 Let $p\geq 7$, then under the supposition of the main
Theorem A we have
(1) $Ext_{A}^{s+3,tq+3q+1}(H^{*}L,Z_{p})\cong
Z_{p}\\{\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{1}),\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{2})\\}$
or has unique generator $\overline{\phi}_{*}\pi_{*}\sigma^{\prime}$
(2) $Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)\cong
Z_{p}\\{\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{1})_{L},\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{2})_{L}\\}$ or has unique generator
$\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge 1_{L})_{*}(\sigma^{\prime})_{L}$,
where $\widetilde{\phi}\in[\Sigma^{2q-1}L\wedge L,L]$ such that
$\widetilde{\phi}(1_{L}\wedge
i^{\prime\prime})=\bar{\phi}\in[\Sigma^{2q-1}L,L]$ (cf. Prop. 9.2.3(3)).
Proof: (1) Consider the following exact sequence
$\quad Ext_{A}^{s+3,tq+3q+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
i^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+3q+1}(H^{*}L,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle
j^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+2q+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle(\alpha_{1})_{*}}}{{\longrightarrow}}$
induced by (9.1.3). The left group is zero and the right group has unique
generator $\widetilde{\alpha}_{2}\sigma^{\prime}$ or has two generators
$\widetilde{\alpha}_{2}\sigma^{\prime}_{1},\widetilde{\alpha}_{2}\sigma^{\prime}_{2}$(cf.
the supposition III). Note that $j\alpha\alpha
i=(\alpha_{1})_{L}\cdot\pi=j^{\prime\prime}\bar{\phi}\cdot\pi\in\pi_{2q-1}S$,
(cf. Prop. 9.2.3), then
$\widetilde{\alpha}_{2}\sigma^{\prime}_{1}=j_{*}\alpha_{*}\alpha_{*}i_{*}(\sigma^{\prime}_{1})=j^{\prime\prime}_{*}\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{1})$
and
$\widetilde{\alpha}_{2}(\sigma^{\prime}_{2})=j^{\prime\prime}_{*}\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{2})$
so that the result on the middle group follows.
(2) Consider the following exact sequence
$\quad
0=Ext_{A}^{s+3,tq+4q+1}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(j^{\prime\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+3q+1}(H^{*}L,H^{*}L)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+3q+1}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(\alpha_{1})^{*}}}{{\longrightarrow}}$
induced by (9.1.3). By the supposition III,
$Ext_{A}^{s+3,tq+rq+1}(Z_{p},Z_{p})$ = 0 ( $r=3,4$). By (1) and
$\bar{\phi}=\widetilde{\phi}(1_{L}\wedge i^{\prime\prime})$, the right group
has unique generator
$\bar{\phi}_{*}\pi_{*}(\sigma^{\prime})=(i^{\prime\prime})^{*}(\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime})_{L}$ or has two generators
$\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{1})=(i^{\prime\prime})^{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{1})_{L},\bar{\phi}_{*}\pi_{*}(\sigma^{\prime}_{2})=(i^{\prime\prime})^{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{2})_{L}$ the image of which under
$(\alpha_{1})^{*}$ is zero, then the middle group has unique generator
$\widetilde{\phi}_{*}(\pi\wedge 1_{L})_{*}(\sigma^{\prime})_{L}$ or has two
generators $\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{1})_{L},\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\sigma^{\prime}_{2})_{L}$. Moreover, by
$Ext_{A}^{s+3,tq+rq}(Z_{p},Z_{p})$ = 0( $r=2,3$) we know that
$Ext_{A}^{s+3,tq+2q}(Z_{p},\\\ H^{*}L)$ = 0, then by (9.1.9),
$Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)=\bar{h}_{*}Ext_{A}^{s+3,tq+3q+1}\\\
(H^{*}L,H^{*}L)$ and the result follows as desired. Q.E.D.
Proposition 9.2.6 Let $p\geq 7$, then under the supposition of the main
Theorem A we have
(1) $Ext_{A}^{s+2,tq+3q+1}(H^{*}Y,H^{*}L)=0$,
$Ext_{A}^{s+2,tq+4q+2}(H^{*}Y,Z_{p})$ = 0.
(2) $Ext_{A}^{s+1,tq+3q+r}(H^{*}Y,H^{*}L)=0$, $r=0,1$.
Proof: (1) Consider the following exact sequence
$Ext_{A}^{s+2,tq+3q}(H^{*}L,H^{*}L)\stackrel{{\scriptstyle(\bar{h})_{*}}}{{\longrightarrow}}Ext_{A}^{s+2,tq+3q+1}(H^{*}Y,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(j\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s+2,tq+2q-1}(Z_{p},H^{*}L)\stackrel{{\scriptstyle(\pi)_{*}}}{{\longrightarrow}}$
induced by (9.1.9). By the supposition II on
$Ext_{A}^{s+2,tq+rq}(Z_{p},Z_{p})$ = 0
( $r=2,3,4$) we know that the left group is zero. By the supposition II on
$Ext_{A}^{s+2,tq+rq-1}(Z_{p},Z_{p})$ = 0($r=2,3$) also know that the right
group is zero. Then the middle group is zero as desired.
For the second result, consider the following exact sequence
$Ext_{A}^{s+2,tq+4q+1}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(\bar{h})_{*}}}{{\longrightarrow}}Ext_{A}^{s+2,tq+4q+2}(H^{*}Y,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle(j\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s+2,tq+3q}(Z_{p},Z_{p})$
induced by (9.1.9). By the supposition II on
$Ext_{A}^{s+2,tq+rq+1}(Z_{p},Z_{p})$ = 0 ( $r=3,4$) we know that the left
group is zero. Similarly, the right group also is zero. Then the middle group
is zero as desired.
(2) Consider the following exact sequence ($r=0,1$)
$Ext_{A}^{s+1,tq+3q+r-1}(H^{*}L,H^{*}L)\stackrel{{\scriptstyle(\bar{h})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+3q+r}(H^{*}Y,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(j\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+2q+r-2}(Z_{p},H^{*}L)$
induced by (9.1.9). By the supposition I on
$Ext_{A}^{s+1,tq+kq+r-1}(Z_{p},Z_{p})$ = 0 ($k=2,3,4,r=0,1$) we know that the
left group is zero. By the supposition II on
$Ext_{A}^{s+1,tq+kq+r-2}(Z_{p},Z_{p})$ = 0 ($k=2,3,r=0,1$) also know that the
right group is zero, then the middle group is zero as desired. Q.E.D.
Proposision 9.2.7 Let $p\geq 7$, then under the supposition of the main
Theorem A we have
(1) $Ext_{A}^{s+1,tq+3q}(H^{*}W,H^{*}L)\cong
Z_{p}\\{(\overline{\phi}_{W})_{*}(\sigma)_{L}\\}$, where
$\overline{\phi}_{W}\in[\Sigma^{3q-1}L,\\\ W]$ satisfying
$u\overline{\phi}_{W}=\overline{\phi}\in[\Sigma^{2q-1}L,L]$ and
$(\sigma)_{L}\in Ext_{A}^{s,tq}(H^{*}L,H^{*}L)$ such that
$(i^{\prime\prime})^{*}(\sigma)_{L}=(i^{\prime\prime})_{*}(\sigma)\in
Ext_{A}^{s,tq}(H^{*}L,Z_{p})$.
(2) $Ext_{A}^{s,tq+3q}(H^{*}Y,H^{*}L)=0,\quad
Ext_{A}^{s,tq+q-1}(H^{*}M,H^{*}L)$ = 0
Proof: (1) Consider the following exact sequence
$Ext_{A}^{s+1,tq+3q}(H^{*}L,H^{*}L)\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+3q}(H^{*}W,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(j^{\prime\prime}u)_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(Z_{p},H^{*}L)\stackrel{{\scriptstyle\phi_{*}}}{{\longrightarrow}}$
induced by (9.1.12). By the supposition I on
$Ext_{A}^{s+1,tq+rq}(Z_{p},Z_{p})$ = 0 $(r=2,3,4$) we know that the left group
is zero. By $(i^{\prime\prime})^{*}\cdot
Ext_{A}^{s+1,tq+q}(Z_{p},H^{*}L)\subset Ext_{A}^{s+1,tq+q}(Z_{p}Z_{p})$ and
the last of which has unique generator
$h_{0}\sigma=(\alpha_{1})^{*}\cdot(\sigma)=(i^{\prime\prime})^{*}((\alpha_{1})_{L})^{*}(\sigma)$
and $Ext_{A}^{s+1,tq+2q}(Z_{p},Z_{p})$ = 0 , then the right group has unique
generator
$((\alpha_{1})_{L}))^{*}(\sigma)=((\alpha_{1})_{L})_{*}(\sigma)_{L}=(j^{\prime\prime}u)_{*}(\overline{\phi}_{W})_{*}(\sigma)_{L}$,
where $(\sigma)_{L}\in Ext_{A}^{s,tq}(H^{*}L,H^{*}L)$ satisfying
$(i^{\prime\prime})^{*}(\sigma)_{L}=(i^{\prime\prime})_{*}(\sigma)\in
Ext_{A}^{s,tq}(H^{*}L,Z_{p})$. Moreover,
$\phi_{*}((\alpha_{1})_{L})_{*}(\sigma)_{L}=0\in
Ext_{A}^{s+2,tq+3q}(H^{*}L,H^{*}L)$, then the middle group has unique
generator $(\overline{\phi}_{W})_{*}(\sigma)_{L}$.
(2) Consider the following exact sequences
$Ext_{A}^{s,tq+3q-1}(H^{*}L,H^{*}L)\stackrel{{\scriptstyle\bar{h}_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+3q}(H^{*}Y,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(j\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+2q-2}(Z_{p},H^{*}L)$
$Ext_{A}^{s,tq+q-1}(Z_{p},H^{*}L)\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+q-1}(H^{*}M,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+q-2}(Z_{p},H^{*}L)$
induced by (9.1.9) and (9.1.1) respectively. By the supposition I on
$Ext_{A}^{s,tq+rq-1}(Z_{p},Z_{p})$ = 0( $r=2,3,4$) we know that the upper left
group is zero. By the supposition I on $Ext_{A}^{s,tq+rq-2}(Z_{p},Z_{p})$ = 0
( $r=2,3$ ), the upper right group is zero. Then the upper middle group is
zero as desired. Similarly, the lower middle also is zero. Q.E.D.
Proposition 9.2.8 Let $p\geq 7$, then under the supposition (I)(III) of the
main Theorem A we have
$Ext_{A}^{s+3,tq+2}(H^{*}M,Z_{p})$ = 0,
$Ext_{A}^{s+1,tq+q+1}(H^{*}M\wedge L,Z_{p})\cong Z_{p}\\{(i\wedge
1_{L})_{*}\pi_{*}(\sigma)\\}$.
Proof: Consider the following exact sequence
$Ext_{A}^{s+3,tq+2}(Z_{p},Z_{p})\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+2}(H^{*}M,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}Ext_{A}^{s+3,tq+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p_{*}}}{{\longrightarrow}}$
induced by (9.1.1). By the supposition III, the right group is zero or has
unique generator $a_{0}\iota$ which satisfies
$p_{*}(a_{0}\iota)=a_{0}^{2}\iota\neq 0\in Ext_{A}^{s+4,tq+2}(Z_{p},Z_{p})$,
then im $j_{*}$ = 0. By the supposition III, the left group has unique
generator $a_{0}^{2}\sigma^{\prime}$or has two generators
$a_{0}^{2}\sigma^{\prime}_{1}=p_{*}(a_{0}\sigma^{\prime}_{1}),a_{0}^{2}\sigma^{\prime}_{2}=p_{*}(a_{0}\sigma^{\prime}_{2}$
so that we have im $i_{*}$ = 0. Then, the middle group is zero as desired.
For the second result, consider the following exact sequence
$Ext_{A}^{s+1,tq+q+1}(H^{*}L,Z_{p})\stackrel{{\scriptstyle(i\wedge
1_{L})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}M\wedge L,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle(j\wedge
1_{L})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})$
induced by (9.1.1). By Prop. 9.2.4, the right group is zero. Since
$(j^{\prime\prime})_{*}\\\ Ext_{A}^{s+1,tq+q+1}(H^{*}L,Z_{p})\subset
Ext_{A}^{s+1,tq+1}(Z_{p},Z_{p})\cong
Z_{p}\\{a_{0}\sigma=(j^{\prime\prime})_{*}\pi_{*}(\sigma)\\}$ and
$Ext_{A}^{s+1,tq+q+1}(Z_{p},Z_{p})$ = 0 then the left group has unique
generator $\pi_{*}(\sigma)$ and the result follows. Q.E.D.
Now we begin to prove the main Theorem A. The proof will be done by processing
an argument processing in the Adams resolution of some spectra related to $S$.
Let
(9.2.9)
$\quad\qquad\cdots\stackrel{{\scriptstyle\bar{a}_{2}}}{{\longrightarrow}}\quad\Sigma^{-2}E_{2}\quad\stackrel{{\scriptstyle\bar{a}_{1}}}{{\longrightarrow}}\quad\Sigma^{-1}E_{1}\quad\stackrel{{\scriptstyle\bar{a}_{0}}}{{\longrightarrow}}E_{0}=S$
$\qquad\qquad\qquad\qquad\qquad\qquad\big{\downarrow}\bar{b}_{2}\qquad\qquad\quad\big{\downarrow}\bar{b}_{1}\qquad\qquad\big{\downarrow}\bar{b}_{0}$
$\qquad\qquad\qquad\qquad\qquad\Sigma^{-2}KG_{2}\qquad\quad\Sigma^{-1}KG_{1}\qquad\quad
KG_{0}$
be the minimal Adams resolution of the sphere spectrum $S$ which satisfies
(1)
$E_{s}\stackrel{{\scriptstyle\bar{b}_{s}}}{{\rightarrow}}KG_{s}\stackrel{{\scriptstyle\bar{c}_{s}}}{{\rightarrow}}E_{s+1}\stackrel{{\scriptstyle\bar{a}_{s}}}{{\rightarrow}}\Sigma
E_{s}$ are cofibrations for all $s\geq 0$, which induce short exact sequences
in $Z_{p}$-cohomology $0\rightarrow
H^{*}E_{s+1}\stackrel{{\scriptstyle\bar{c}_{s}^{*}}}{{\rightarrow}}H^{*}KG_{s}\stackrel{{\scriptstyle\bar{b}_{s}^{*}}}{{\rightarrow}}H^{*}E_{s}\rightarrow
0$.
(2) $KG_{s}$ is a graded wedge sum of Eilenberg-Maclane spectra $KZ_{p}$ of
type $Z_{p}$.
(3) $\pi_{t}KG_{s}$ are the $E_{1}^{s,t}$-terms of the Adams spectral
sequence,
$(\bar{b}_{s}\bar{c}_{s-1})_{*}:\pi_{t}KG_{s-1}\rightarrow\pi_{t}KG_{s}$ is
the $d_{1}^{s-1,t}$-differentials of the Adams spectral sequence , and
$\pi_{t}KG_{s}\cong Ext_{A}^{s,t}(Z_{p},Z_{p})$ (cf. [3] p.180).
Then, an Adams resolution of an arbitrary spectrum $V$ can be obtained by
smashing $V$ to (9.2.9). We first prove some Lemmas.
Lemma 9.2.10 Let $p\geq 7$, then under the supposition of the main Theorem A
we have
$\bar{c}_{s+1}\cdot h_{0}\sigma=(1_{E_{s+2}}\wedge\alpha_{1})\kappa$ (up to
scalar)
where $\kappa\in\pi_{tq+1}E_{s+2}$ such that
$\bar{c}_{s+1}\cdot\sigma=\bar{a}_{s+1}\cdot\kappa$ and
$\bar{b}_{s+2}\cdot\kappa=a_{0}\sigma^{\prime}\in\pi_{tq+1}KG_{s+2}\cong
Ext_{A}^{s+2,tq+1}(Z_{p},Z_{p})$.
Proof: The $d_{1}$-cycle $(1_{KG_{s+1}}\wedge
i^{\prime\prime})h_{0}\sigma\in\pi_{tq+q}KG_{s+1}\wedge L$ represents an
element in $Ext_{A}^{s+2,tq+q}(H^{*}L,Z_{p})$ and this group is zero by Prop.
9.2.4 , then it is a $d_{1}$-boundary and so $(\bar{c}_{s+1}\wedge
1_{L})(1_{KG_{s+1}}\wedge i^{\prime\prime})h_{0}\sigma$ = 0 ,
$\bar{c}_{s+1}\cdot h_{0}\sigma=(1_{E_{s+2}}\wedge\alpha_{1})f^{\prime\prime}$
for some $f^{\prime\prime}\in\pi_{tq+1}E_{s+2}$. It follows that
$\bar{a}_{s+1}\cdot(1_{E_{s+2}}\wedge\alpha_{1})f^{\prime\prime}$ = 0 , then
$\bar{a}_{s+1}\cdot f^{\prime\prime}=(1_{E_{s+1}}\wedge
j^{\prime\prime})f^{\prime\prime}_{2}$ with
$f^{\prime\prime}_{2}\in\pi_{tq+q}(E_{s+1}\wedge L)$. The $d_{1}$-cycle
$(\bar{b}_{s+1}\wedge 1_{L})f^{\prime\prime}_{2}\in\pi_{tq+q}KG_{s+1}\wedge L$
represents an element in $Ext_{A}^{s+1,tq+q}(H^{*}L,Z_{p})$ and this group is
zero, then $(\bar{b}_{s+1}\wedge
1_{L})f^{\prime\prime}_{2}=(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{L})g^{\prime\prime}$ with $g^{\prime\prime}\in\pi_{tq+q}(KG_{s}\wedge L)$.
Hence, $f^{\prime\prime}_{2}=(\bar{c}_{s}\wedge
1_{L})g^{\prime\prime}+(\bar{a}_{s+1}\wedge 1_{L})f^{\prime\prime}_{3}$, for
some $f^{\prime\prime}_{3}\in\pi_{tq+q+1}E_{s+2}\wedge L$ and we have
$\bar{a}_{s+1}\cdot f^{\prime\prime}=\bar{a}_{s+1}(1_{E_{s+2}}\wedge
j^{\prime\prime})f^{\prime\prime}_{3}+\bar{c}_{s}(1_{KG_{s}}\wedge
j^{\prime\prime})g^{\prime\prime}=\bar{a}_{s+1}(1_{E_{s+2}}\wedge
j^{\prime\prime})f^{\prime\prime}_{3}+\lambda\bar{c}_{s}\cdot\sigma$
$=\bar{a}_{s+1}(1_{E_{s+2}}\wedge
j^{\prime\prime})f^{\prime\prime}_{3}+\lambda\bar{a}_{s+1}\cdot\kappa$ for
some $\lambda\in Z_{p}$, this is because $(1_{KG_{s}}\wedge
j^{\prime\prime})g^{\prime\prime}\in\pi_{tq}KG_{s}\cong
Ext_{A}^{s,tq}(Z_{p},Z_{p})\cong Z_{p}\\{\sigma\\}$. Then,
$f^{\prime\prime}=(1_{E_{s+2}}\wedge
j^{\prime\prime})f^{\prime\prime}_{3}+\lambda\kappa+\bar{c}_{s+1}\cdot
g^{\prime\prime}_{2}$ for some $g^{\prime\prime}_{2}\in\pi_{tq+1}KG_{s+1}$ and
so $\bar{c}_{s+1}\cdot h_{0}\sigma=(1_{E_{s+1}}\wedge\alpha_{1})\kappa$ (up to
scalar). Q.E.D.
Since $\bar{h}\phi\cdot p=\bar{h}i^{\prime\prime}j\alpha^{2}i$ = 0 (cf. Prop.
9.2.3(1) and (9.1.9)(9.1.5)), then $\bar{h}\phi=(1_{Y}\wedge j)\alpha_{Y\wedge
M}i$ , where $\alpha_{Y\wedge M}\in[\Sigma^{2q+1}M,Y\wedge M]$. Let $\Sigma U$
be the cofibre of $\bar{h}\phi=(1_{Y}\wedge j)\alpha_{Y\wedge
M}i:\Sigma^{2q}S\rightarrow Y$ given by the cofibration
(9.2.11)
$\Sigma^{2q}S\stackrel{{\scriptstyle\bar{h}\phi}}{{\longrightarrow}}Y\stackrel{{\scriptstyle
w_{2}}}{{\longrightarrow}}\Sigma U\stackrel{{\scriptstyle
u_{2}}}{{\longrightarrow}}\Sigma^{2q+1}S$
Moreover, $w_{2}(1_{Y}\wedge j)\alpha_{Y\wedge M}=\widetilde{w}\cdot j$, where
$\widetilde{w}:\Sigma^{2q}S\rightarrow U$ whose cofibre is $X$ given by the
cofibration
$\Sigma^{2q}S\stackrel{{\scriptstyle\tilde{w}}}{{\longrightarrow}}U\stackrel{{\scriptstyle\tilde{u}}}{{\longrightarrow}}X\stackrel{{\scriptstyle
j\tilde{\psi}}}{{\longrightarrow}}\Sigma^{2q+1}S$. Then, $\Sigma X$ also is
the cofibre of $\omega=(1_{Y}\wedge j)\alpha_{Y\wedge
M}:\Sigma^{2q}M\rightarrow Y$ given by the cofibration
(9.2.12) $\Sigma^{2q}M\stackrel{{\scriptstyle(1_{Y}\wedge j)\alpha_{Y\wedge
M}}}{{\longrightarrow}}Y\stackrel{{\scriptstyle\tilde{u}w_{2}}}{{\longrightarrow}}\Sigma
X\stackrel{{\scriptstyle\tilde{\psi}}}{{\longrightarrow}}\Sigma^{2q+1}M$
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\quad\Sigma^{2q}S\qquad\stackrel{{\scriptstyle\bar{h}\phi}}{{\longrightarrow}}\qquad
Y\qquad\stackrel{{\scriptstyle\tilde{u}w_{2}}}{{\longrightarrow}}\qquad\Sigma
X$
$\qquad\qquad\quad\searrow i\quad\nearrow_{(1_{Y}\wedge j)\alpha_{Y\wedge
M}}\searrow w_{2}\quad\nearrow\widetilde{u}\quad\searrow\widetilde{\psi}$
(9.2.13) $\Sigma^{2q}M\qquad\qquad\qquad\Sigma U\qquad\qquad\Sigma^{2q+1}M$
$\qquad\qquad\quad\nearrow\widetilde{\psi}\qquad\searrow
j\qquad\nearrow\widetilde{w}\qquad\searrow u_{2}\quad\nearrow i$
$\qquad\qquad X\quad\stackrel{{\scriptstyle
j\tilde{\psi}}}{{\longrightarrow}}\qquad\Sigma^{2q+1}S\quad\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\qquad\Sigma^{2q+1}S$
Since $j\overline{u}(\bar{h}\phi)$ = 0, then by (9.2.11) we have,
$j\overline{u}=u_{3}w_{2}$, for some $u_{3}\in[U,\Sigma^{q+1}S]$. Hence, the
spectrum $U$ in (9.2.11) also is the cofire of $w\pi:\Sigma^{q}S\rightarrow W$
given by the cofibration
(9.2.14) $\Sigma^{q}S\stackrel{{\scriptstyle
w\pi}}{{\longrightarrow}}W\stackrel{{\scriptstyle
w_{3}}}{{\longrightarrow}}U\stackrel{{\scriptstyle
u_{3}}}{{\longrightarrow}}\Sigma^{q+1}S$
This can be seen by the following homotopy commutative diagram of $3\times
3$\- Lemma in the stable homotopy category
$\qquad\quad\Sigma^{-1}Y\quad\stackrel{{\scriptstyle
j\overline{u}}}{{\longrightarrow}}\quad\Sigma^{q+1}S\quad\stackrel{{\scriptstyle
w\pi}}{{\longrightarrow}}\quad\Sigma W$
$\qquad\qquad\quad\searrow w_{2}\quad\nearrow
u_{3}\quad\searrow\pi\quad\nearrow w\quad\searrow w_{3}$
(9.2.15) $\quad\qquad\qquad U\qquad\qquad\qquad\Sigma L\qquad\qquad\Sigma U$
$\qquad\qquad\quad\nearrow w_{3}\quad\searrow
u_{2}\quad\nearrow\phi\quad\searrow\bar{h}\quad\nearrow w_{2}$
$\qquad\quad W\qquad\stackrel{{\scriptstyle
j^{\prime\prime}u}}{{\longrightarrow}}\quad\Sigma^{2q}S\qquad\stackrel{{\scriptstyle\bar{h}\phi}}{{\longrightarrow}}\qquad
Y$
Moreover, by $u_{3}\widetilde{w}=\alpha_{1}$, the cofibre of
$\widetilde{u}w_{3}:W\rightarrow X$ is $\Sigma^{q+1}L$ given by the
cofibration
(9.2.16)
$W\stackrel{{\scriptstyle\tilde{u}w_{3}}}{{\longrightarrow}}X\stackrel{{\scriptstyle
u^{\prime\prime}}}{{\longrightarrow}}\Sigma^{q+1}L\stackrel{{\scriptstyle
w^{\prime}(\pi\wedge 1_{L})}}{{\longrightarrow}}\Sigma W$
where $w^{\prime}\in[L\wedge L,W]$ such that $w^{\prime}(1_{L}\wedge
i^{\prime\prime})=w$. This can be seen by the following homotopy commutative
diagram of $3\times 3$-Lemma in the stable homotopy category
$\qquad\qquad
W\quad\stackrel{{\scriptstyle\tilde{u}w_{3}}}{{\longrightarrow}}\quad
X\quad\stackrel{{\scriptstyle
j\tilde{\psi}}}{{\longrightarrow}}\quad\Sigma^{2q+1}S$
$\qquad\qquad\quad\searrow w_{3}\quad\nearrow\widetilde{u}\quad\searrow
u^{\prime\prime}\quad\nearrow j^{\prime\prime}$
(9.2.17) $U\qquad\qquad\quad\Sigma^{q+1}L$
$\qquad\qquad\quad\nearrow\widetilde{w}\quad\searrow u_{3}\quad\nearrow
i^{\prime\prime}\quad\searrow^{w^{\prime}(\pi\wedge 1_{L})}$
$\qquad\qquad\Sigma^{2q}S\quad\stackrel{{\scriptstyle\alpha_{1}}}{{\longrightarrow}}\quad\Sigma^{q+1}S\quad\stackrel{{\scriptstyle
w\pi}}{{\longrightarrow}}\quad\Sigma W$
Lemma 9.2.18 (1) Let $\overline{\phi}_{W}\in[\Sigma^{3q-1}L,W]$ be the map in
Prop. 9.2.7 such that
$u\overline{\phi}_{W}=\overline{\phi}\in[\Sigma^{2q-1}L,L]$, then
(1) $\widetilde{u}w_{3}\overline{\phi}_{W}(p\wedge 1_{L})\neq
0\in[\Sigma^{3q-1}L,X]$.
(2) $Ext_{A}^{s,tq+3q-1}(H^{*}X,H^{*}L)$ = 0,
$Ext_{A}^{s+1,tq+3q}(H^{*}X,H^{*}L)=(\widetilde{u}w_{3})_{*}Ext_{A}^{s+1,tq+3q}(H^{*}W,H^{*}L)$.
Proof: (1) Suppose in contrast that
$\widetilde{u}w_{3}\overline{\phi}_{W}(p\wedge 1_{L})$ = 0, then by (9.2.16)
and the result of Prop. 9.2.3(1) on $[\Sigma^{2q-1}L,L]$ we have
(9.2.19) $\quad\overline{\phi}_{W}(p\wedge 1_{L})=\lambda w^{\prime}(\pi\wedge
1_{L})\overline{\phi}$ modulo $F_{3}[\Sigma^{3q-1}L,W]$
for some $\lambda\in Z_{(p)}$, where $F_{3}[\Sigma^{3q-1}L,W]$ denotes the
subgroup of $[\Sigma^{3q-1}L,W]$ consisting by all elements of filtration
$\geq 3$. Moreover, note that $uw^{\prime}(\pi\wedge 1_{L})\in[L,L]$ and this
group has two generators $(p\wedge 1_{L}),\pi j^{\prime\prime}$ which has
filtration one, then $uw^{\prime}(\pi\wedge 1_{L})=\lambda_{1}(p\wedge
1_{L})+\lambda_{2}\pi j^{\prime\prime}$ with $\lambda_{1},\lambda_{2}\in
Z_{(p)}$. By (9.1.13) we have
$\lambda_{1}p\cdot(\alpha_{1})_{L}+\lambda_{2}(\alpha_{1})_{L}\pi
j^{\prime\prime}$ = 0 so that $\lambda_{2}=\lambda_{0}\lambda_{1}$, here we
use the equation $(\alpha_{1})_{L}\pi
j^{\prime\prime}=-\lambda_{0}p\cdot(\alpha_{1})_{L}$, $\lambda_{0}\neq 0\in
Z_{p}$. Then , by composing $u$ on (9.2.19) we have$\overline{\phi}(p\wedge
1_{L})=u\overline{\phi}_{W}(p\wedge 1_{L})=\lambda uw^{\prime}(\pi\wedge
1_{L})\overline{\phi}=\lambda\lambda_{1}\overline{\phi}(p\wedge
1_{L})+\lambda\lambda_{0}\lambda_{1}\pi j^{\prime\prime}\overline{\phi}$ (mod
$F_{3}[\Sigma^{2q-1}L,L]$) and so by (9.1.9) $\bar{h}\overline{\phi}(p\wedge
1_{L})=\lambda\lambda_{1}\bar{h}\overline{\phi}(p\wedge 1_{L})$ (mod
$F_{3}[\Sigma^{2q}L,Y]$). This implies that $\lambda\lambda_{1}=1$ (mod $p$)
(cf. the following Remark 9.2.20).
Hence we have $\lambda\lambda_{1}\lambda_{0}\pi
j^{\prime\prime}\overline{\phi}$ = 0 (mod $F_{3}[\Sigma^{2q-1}L,L]$) and by
the same reason as shown in the following Remark 9.2.20, this implies that
$\lambda\lambda_{1}\lambda_{0}$ = 0 (mod $p$) which yields a contradiction.
(2) Consider the following exact sequence
$Ext_{A}^{s,tq+3q}(H^{*}Y,H^{*}L)\stackrel{{\scriptstyle(\tilde{u}w_{2})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+3q-1}(H^{*}X,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(\tilde{\psi})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+q-1}(H^{*}M,H^{*}L)$
induced by (9.2.12). By Prop. 9.2.7(2), both sides of groups are zero ,so tha
the middle group is zero as desired. Look at the following exact sequence
$Ext_{A}^{s+1,tq+3q}(H^{*}W,H^{*}L)\stackrel{{\scriptstyle(\tilde{u}w_{3})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+3q}(H^{*}X,H^{*}L)$
$\qquad\quad\stackrel{{\scriptstyle(u^{\prime\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+2q-1}(H^{*}L,H^{*}L)$
induced by (9.2.16). By the supposition on
$Ext_{A}^{s+1,tq+rq-1}(Z_{p},Z_{p})$ = 0 ( $r=1,2,3$) we know that the right
group is zero and so the result follows. Q.E.D.
Remark 9.2.20 Here we give an explanation on the reason why the coefficient in
the equation $(1-\lambda\lambda_{1})\bar{h}\overline{\phi}(p\wedge 1_{L})=0$
(mod $F_{3}[\Sigma^{2q}L,Y]$) must be zero (mod $p$). For otherwise , if
$1-\lambda\lambda_{1}\neq 0$ (mod $p$), then
$(1-\lambda\lambda_{1})\bar{h}\overline{\phi}(p\wedge 1_{L})$ must be
represented by some nonzero element $x\in Ext_{A}^{2,2q+2}(H^{*}Y,H^{*}L)$ in
the ASS. However , it also equals to an element of filtration $\geq 3$, then
$x$ must be a $d_{2}$-boundary, that is, $x=d_{2}(x^{\prime})\in
d_{2}Ext_{A}^{0,2q+1}(H^{*}Y,H^{*}L)$ = 0, this is because
$Ext_{A}^{0,2q+1}(H^{*}Y,H^{*}L)=Hom_{A}^{2q+1}(H^{*}Y,H^{*}L)$ = 0 which is
obtained by $H^{r}L\neq 0$ only for $r=0,q$. This is a contradiction so that
we have $1-\lambda\lambda_{1}=0$ (mod $p$).
Lemma 9.2.21 For the element $\kappa\in\pi_{tq+1}E_{s+2}$ in Lemma 9.2.19, it
is known that $\bar{a}_{s+1}\cdot\kappa=\bar{c}_{s}\cdot\sigma$ and
$\bar{b}_{s+2}\cdot\kappa=a_{0}\sigma^{\prime}\in\pi_{tq+1}KG_{s+2}\cong
Ext_{A}^{s+2,tq+1}(Z_{p},\\\ Z_{p})$, then there exists
$f\in\pi_{tq+3}E_{s+4}\wedge M$ and $g\in\pi_{tq+1}(KG_{s+1}\wedge M)$ such
that
(A) $(1_{E_{s+2}}\wedge i)\kappa=(\bar{c}_{s+1}\wedge
1_{M})g+(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{M})f$
and
(B) $(1_{E_{s+4}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge
M})f\cdot(\alpha_{1})_{L}$ = 0 $\in[\Sigma^{tq+4q+2}L,E_{s+4}\wedge Y]$,
where $\alpha_{Y\wedge M}\in[\Sigma^{2q+1}M,Y\wedge M]$ satisfying
$(1_{Y}\wedge j)\alpha_{Y\wedge M}i=\bar{h}\phi\in\pi_{2q}Y$.
Proof: Note that the $d_{1}$-cycle $(\bar{b}_{s+2}\wedge
1_{M})(1_{E_{s+2}}\wedge i)\kappa\in\pi_{tq+1}KG_{s+2}\wedge M$ represents an
element $i_{*}(a_{0}\sigma^{\prime})=i_{*}p_{*}(\sigma^{\prime})=0\in
Ext_{A}^{s+2,tq+1}(H^{*}M,Z_{p})$ so that it is a $d_{1}$-boundary. That is,
$(\bar{b}_{s+2}\wedge 1_{M})(1_{E_{s+2}}\wedge
i)\kappa=(\bar{b}_{s+2}\bar{c}_{s+1}\wedge 1_{M})g$ for some
$g\in\pi_{tq+1}KG_{s+1}\wedge M$ . Then, by $Ext_{A}^{s+3,tq+2}(H^{*}M,Z_{p})$
= 0 (cf. Prop. 9.2.8) we have $(1_{E_{s+2}}\wedge
i)\kappa=(\bar{c}_{s+1}\wedge 1_{M})g+(\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{M})f$ for some $f\in\pi_{tq+3}E_{s+4}\wedge M$. This shows (A). For (B), by
Prop. 9.2.3(1) we have $\phi\cdot p=i^{\prime\prime}j\alpha^{2}i$ (up to
nonzero scalar), then $\bar{h}\phi\cdot
p=\bar{h}i^{\prime\prime}j\alpha^{2}i=0$ and so $\bar{h}\phi=(1_{Y}\wedge
j)\alpha_{Y\wedge M}i$ for some $\alpha_{Y\wedge M}\in[\Sigma^{2q+1}M,Y\wedge
M]$. Then, by composing $1_{E_{s+2}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge M}$
on the equation (A) we have
(9.2.22) $(1_{E_{s+2}}\wedge\bar{h}\phi)\kappa=(1_{E_{s+2}}\wedge(1_{Y}\wedge
j)\alpha_{Y\wedge M}i)\kappa$
$\quad=(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{Y})(1_{E_{s+4}}\wedge(1_{Y}\wedge
j)\alpha_{Y\wedge M})f$
where $(1_{Y}\wedge j)\alpha_{Y\wedge M}$ induces zero homomorphism in
$Z_{p}$-cohomology so that $(\bar{c}_{s+1}\wedge
1_{Y})(1_{KG_{s+1}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge M})g$ = 0.
By composing $(\alpha_{1})_{L}$ on (9.2.22) we have
$(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{Y})(1_{E_{s+4}}\wedge(1_{Y}\wedge
j)\alpha_{Y\wedge
M})f\cdot(\alpha_{1})_{L}=(1_{E_{s+2}}\wedge\bar{h})(\kappa\wedge
1_{L})\phi\cdot(\alpha_{1})_{L}$ = 0, this is because
$\phi\cdot(\alpha_{1})_{L}\in[\Sigma^{3q-2}L,L]$ = 0 which is obtained by
$\pi_{rq-2}S$ = 0( $r=2,3,4$). Then we have
$(\bar{a}_{s+3}\wedge 1_{Y})(1_{E_{s+4}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge
M})f\cdot(\alpha_{1})_{L}=(\bar{c}_{s+2}\wedge 1_{Y})g_{1}$ = 0
where the $d_{1}$-cycle $g_{1}\in[\Sigma^{tq+3q+1}L,KG_{s+2}\wedge Y]$
represents an element in $Ext_{A}^{s+2,tq+3q+1}(H^{*}Y,H^{*}L)$ and this group
is zero (cf. Prop. 9.2.6(1)) so that it is a $d_{1}$-boundary and we have
$(\bar{c}_{s+2}\wedge 1_{Y})g_{1}$ = 0. Briefly write $(1_{Y}\wedge
j)\alpha_{Y\wedge M}=\omega$ and let $V$ be the cofibre of
$(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge
1_{L})=\omega\cdot(1_{M}\wedge(\alpha_{1})_{L}):\Sigma^{3q-1}M\wedge
L\rightarrow Y$ given by the cofibration
(9.2.23) $\Sigma^{3q-1}M\wedge
L\stackrel{{\scriptstyle(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge
1_{L})}}{{\longrightarrow}}Y\stackrel{{\scriptstyle
w_{4}}}{{\longrightarrow}}V\stackrel{{\scriptstyle
u_{4}}}{{\longrightarrow}}\Sigma^{3q}M\wedge L$
It follows that $(\bar{a}_{s+3}\wedge 1_{Y})(1_{E_{s+4}}\wedge
1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge 1_{L})(f\wedge
1_{L})=(\bar{a}_{s+3}\wedge 1_{Y})(1_{E_{s+4}}\wedge(1_{Y}\wedge
j)\alpha_{Y\wedge M})f\cdot(\alpha_{1})_{L}$ = 0, then by (9.2.23) we have
$(\bar{a}_{s+3}\wedge 1_{M\wedge L})(f\wedge 1_{L})=(1_{E_{s+3}}\wedge
u_{4})f_{2}$ for some $f_{2}\in[\Sigma^{tq+3q+2}L,E_{s+3}\wedge V]$.
Consequently, $(\bar{b}_{s+3}\wedge 1_{M\wedge L})(1_{E_{s+3}}\wedge
u_{4})f_{2}$ = 0 so that we have
(9.2.24) $(\bar{b}_{s+3}\wedge 1_{V})f_{2}=(1_{KG_{s+3}}\wedge w_{4})g_{2}$
with $g_{2}\in[\Sigma^{tq+3q+2}L,KG_{s+3}\wedge Y]$. Then,
$(\bar{b}_{s+4}\bar{c}_{s+3}\wedge 1_{V})(1_{KG_{s+3}}\wedge w_{4})g_{2}$ = 0
and so $(\bar{b}_{s+4}\bar{c}_{s+3}\wedge
1_{Y})g_{2}\in(1_{KG_{s+4}}\wedge(1_{Y}\wedge(\alpha_{1})_{L}(\omega\wedge
1_{L}))_{*}[\Sigma^{*}L,KG_{S+4}\wedge M\wedge L]$ = 0. That is, $g_{2}$ is a
$d_{1}$-cycle which represents an element $[g_{2}]\in
Ext_{A}^{s+3,tq+3q+2}(H^{*}Y\\\ ,H^{*}L)$ and this group has two generators as
shown in Prop. 9.2.5(2)), then we have
(9.2.25) $[g_{2}]=\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\lambda_{1}[\sigma^{\prime}_{1}\wedge
1_{L}]+\lambda_{2}[\sigma^{\prime}_{2}\wedge 1_{L}])$
with $\lambda_{1},\lambda_{2}\in Z_{p}$. By (9.2.24) we know that
$(w_{4})_{*}[g_{2}]\in
E_{2}^{s+3,tq+3q+2}(V)=Ext_{A}^{s+3,tq+3q+2}(H^{*}V,H^{*}L)$
is a permanent cycle in the ASS. However,
$(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge 1_{L})$ is a map of filtration 2,
then the cofibration (9.2.23) induces an exact sequence in $Z_{p}$-cohomology
which is split as $A$-module. That is , it induces a split exact sequence in
the $E_{1}$-term of the ASS :
$E_{1}^{s+3,*}(Y)\stackrel{{\scriptstyle(w_{4})_{*}}}{{\longrightarrow}}E_{1}^{s+3,*}(V)\stackrel{{\scriptstyle(u_{4})_{*}}}{{\longrightarrow}}E_{1}^{s+3,*-3q}(M\wedge
L)$. It follows that it induces a split exact sequence in the $E_{r}$-term of
the ASS for all $(r\geq 2$)
(9.2.26)
$E_{r}^{s+3,*}(Y)\stackrel{{\scriptstyle(w_{4})_{*}}}{{\longrightarrow}}E_{r}^{s+3,*}(V)\stackrel{{\scriptstyle(u_{4})_{*}}}{{\longrightarrow}}E_{r}^{s+3,*-3q}(M\wedge
L)$
Then , $d_{r}((w_{4})_{*}[g_{2}])=0$ implies that $d_{r}([g_{2}])=0$( $r\geq
2$). That is , (9.2.24) implies that $[g_{2}]$ also is a permanent cycle in
the ASS. Since the secondary differential $d_{2}[g_{2}]$ = 0 and
$d_{2}(\sigma)=a_{0}\sigma^{\prime}$ in which $\sigma^{\prime}$ is the linear
combination of $\sigma^{\prime}_{1},\sigma^{\prime}_{2}$, then
$\lambda_{1},\lambda_{2}$ linearly dependent. That is, (9.2.25) becomes
$[g_{2}]=\lambda_{1}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]$.
Now we consider the case $\lambda_{1}$ is nonzero or zero respectively.
If $\lambda_{1}\neq 0$ , (9.2.24) imp;ies $[g_{2}]$ and so
$\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge 1_{L})_{*}[\sigma^{\prime}\wedge
1_{L}]\in E_{2}^{s+3,tq+3q+2}\\\ (Y)=Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)$ is
a permanent cycle in the ASS. Moreover, by $(\bar{a}_{s+3}\wedge
1_{Y})(1_{E_{s+4}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge
M})f\cdot(\alpha_{1})_{L}$ = 0 we have
$(1_{E_{s+4}}\wedge(1_{Y}\wedge j)\alpha_{Y\wedge
M})f\cdot(\alpha_{1})_{L}=(\bar{c}_{s+3}\wedge 1_{Y})g_{3}$
for some $d_{1}$-cycle $g_{3}\in[\Sigma^{tq+3q+2}L,KG_{s+3}\wedge Y]$ and it
represents an element $[g_{3}]\in Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)$ so
that we have $[g_{3}]=\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}(\lambda_{3}[\sigma^{\prime}_{1}\wedge
1_{L}]+\lambda_{4}[\sigma^{\prime}_{2}\wedge 1_{L}])$ for some
$\lambda_{3},\lambda_{4}\in Z_{p}$. By the above equation and
$(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge 1_{L})$ has filtration 2 we know
that the secondary differential $d_{2}([g_{3}])$ = 0 so that by the similar
reason as above , $\lambda_{3},\lambda_{4}$ is linearly dependent . That is,
$[g_{3}]=\lambda_{3}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]$ so that we have
$(1_{E_{s+4}}\wedge(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge 1_{L}))(f\wedge
1_{L})=(\bar{c}_{s+3}\wedge 1_{Y})g_{3}=0$ and the result follows.
If $\lambda_{1}$ = 0, then $g_{2}=(\bar{b}_{s+3}\bar{c}_{s+2}\wedge
1_{Y})g_{4}$ for some $g_{4}\in[\Sigma^{tq+3q+2}L,KG_{s+2}\wedge Y]$ and
(9.2.24) becomes $(\bar{b}_{s+3}\wedge
1_{V})f_{2}=(\bar{b}_{s+3}\bar{c}_{s+2}\wedge 1_{V})(1_{KG_{s+2}}\wedge
w_{4})g_{4}$. Consequently we have $f_{2}=(\bar{c}_{s+2}\wedge
1_{V})(1_{KG_{s+2}}\wedge w_{4})g_{4}+(\bar{a}_{s+3}\wedge 1_{V})f_{3}$ with
$f_{3}\in[\Sigma^{tq+3q+3}L,E_{s+4}\wedge V]$ and so $(\bar{a}_{s+3}\wedge
1_{M\wedge L})(f\wedge 1_{L})=(1_{E_{s+3}}\wedge
u_{4})f_{2}=(\bar{a}_{s+3}\wedge 1_{M\wedge L})(1_{E_{s+4}}\wedge
u_{4})f_{3}$. Hence, $(f\wedge 1_{L})=(1_{E_{s+4}}\wedge
u_{4})f_{3}+(\bar{c}_{s+3}\wedge 1_{M\wedge L})g_{5}$ for some
$g_{5}\in[\Sigma^{tq+3q+3}L,KG_{s+3}\wedge M\wedge L]$ and so by (9.2.23) we
have $(1_{E_{s+4}}\wedge(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge
1_{L}))(f\wedge 1_{L})=(\bar{c}_{s+3}\wedge
1_{Y})(1_{KG_{s+3}}\wedge(1_{Y}\wedge(\alpha_{1})_{L})(\omega\wedge
1_{L}))g_{5}$ = 0 (this is because $(\alpha_{1})_{L}$ induces zero homomorphsm
is $Z_{p}$-cohomology). Q.E.D.
Proof of the main Theorem A: We will continue the argument in Lemma 9.2.21.
Note that the spectrum $V$ in (9.2.23) also is the cofibre of $(1_{M}\wedge
wi^{\prime\prime})\widetilde{\psi}:X\rightarrow\Sigma^{2q}M\wedge W$ given by
the cofibration
(9.2.27) $X\stackrel{{\scriptstyle(1_{M}\wedge
wi^{\prime\prime})\tilde{\psi}}}{{\longrightarrow}}\Sigma^{2q}M\wedge
W\stackrel{{\scriptstyle w_{5}}}{{\longrightarrow}}V\stackrel{{\scriptstyle
u_{5}}}{{\longrightarrow}}\Sigma X$
this can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\Sigma^{3q-1}M\wedge L\quad\longrightarrow\qquad
Y\qquad\stackrel{{\scriptstyle\tilde{u}w_{2}}}{{\longrightarrow}}\quad\Sigma
X$
$\qquad\qquad\quad\searrow^{1_{M}\wedge(\alpha_{1})_{L}}\quad\nearrow\omega\quad\searrow
w_{4}\quad\nearrow u_{5}\quad\searrow\widetilde{\psi}$
(9.2.28) $\Sigma^{2q}M\quad\qquad\qquad\qquad V\qquad\qquad\Sigma^{2q+1}M$
$\qquad\qquad\qquad\nearrow\widetilde{\psi}\quad\searrow^{1_{M}\wedge
wi^{\prime\prime}}\quad\nearrow w_{5}\quad\searrow
u_{4}\quad\nearrow_{1_{M}\wedge(\alpha_{1})_{L}}$
$\quad\qquad\quad X\quad\longrightarrow\quad\Sigma^{2q}M\wedge
W\quad\stackrel{{\scriptstyle 1_{M}\wedge
u}}{{\longrightarrow}}\quad\Sigma^{3q}M\wedge L$
By Lemma 9.2.21(B) and (9.2.23), $f\wedge 1_{L}=(1_{E_{s+4}}\wedge
u_{4})f_{5}$ for some $f_{5}\in[\Sigma^{tq+3q+3}L\\\ ,E_{s+4}\wedge V]$ and so
by Lemma 9.2.21(A) we have
(9.2.29) $(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{M\wedge L})(1_{E_{s+4}}\wedge
u_{4})f_{5}\quad=\quad(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{M\wedge L})(f\wedge
1_{L})$
$\qquad=\quad(1_{E_{s+2}}\wedge i\wedge 1_{L})(\kappa\wedge
1_{L})-(\bar{c}_{s+1}\wedge 1_{M\wedge L})(g\wedge 1_{L})$.
It follows that $(\bar{a}_{s}\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{M\wedge L})(1_{E_{s+4}}\wedge u_{4})f_{5}=0$ and so
$(\bar{a}_{s}\bar{a}_{s+1}\bar{a}_{s+2}\\\ \bar{a}_{s+3}\wedge
1_{V})f_{5}=(1_{E_{s}}\wedge w_{4})f_{6}$ with
$f_{6}\in[\Sigma^{tq+3q-1}L,E_{s}\wedge Y]$. Clearly we have
$(\bar{b}_{s}\wedge 1_{V})(1_{E_{s}}\wedge w_{4})f_{6}$ = 0, then
$(\bar{b}_{s}\wedge 1_{Y})f_{6}$ = 0 and by
$Ext_{A}^{s+1+r,tq+3q+r}(H^{*}Y,\\\ H^{*}L)$ = 0( $r=0,1$, cf. Prop. 9.2.6) we
have $(\bar{a}_{s}\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{V})f_{5}=(\bar{a}_{s}\bar{a}_{s+1}\\\ \bar{a}_{s+2}\wedge
1_{V})(1_{E_{s+3}}\wedge w_{4})f_{7}$, with
$f_{7}\in[\Sigma^{tq+3q+2}L,E_{s+3}\wedge Y]$. Consequently we have
(9.2.30) $(\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{V})f_{5}$
$=(\bar{a}_{s+1}\bar{a}_{s+2}\wedge 1_{V})(1_{E_{s+3}}\wedge
w_{4})f_{7}+(\bar{c}_{s}\wedge 1_{V})g_{6}$
with $d_{1}$-cycle $g_{6}\in[\Sigma^{tq+3q}L,KG_{s}\wedge V]$ which represents
an element $[g_{6}]\in Ext_{A}^{s,tq+3q}(H^{*}V,H^{*}L)$. Note that the
$d_{1}$-cycle $(\bar{b}_{s+3}\wedge 1_{Y})f_{7}\in[\Sigma^{tq+3q+2}L,\\\
KG_{s+3}\wedge Y]$ represents an element $[(\bar{b}_{s+3}\wedge
1_{Y})f_{7}]\in Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)$ which has two generators
(cf. Prop. 9.2.5)(2)), then $[(\bar{b}_{s+3}\wedge
1_{Y})f_{7}]=\lambda^{\prime}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}[\sigma^{\prime}_{1}\wedge
1_{L}]+\lambda^{\prime\prime}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}[\sigma^{\prime}_{2}\wedge 1_{L}]$ for some
$\lambda^{\prime},\lambda^{\prime\prime}\in Z_{p}$. By the vanishes of the
secondary differential : $0=d_{2}[(\bar{b}_{s+3}\wedge 1_{Y})f_{7}]$ we know
that $\lambda^{\prime},\lambda^{\prime\prime}$ is linearly dependent . Then we
have
(9.2.31) $[(\bar{b}_{s+3}\wedge
1_{Y})f_{7}]=\lambda^{\prime}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]$
$\qquad\qquad\qquad\in Ext_{A}^{s+3,tq+3q+2}(H^{*}Y,H^{*}L)$
We claim that the scalar $\lambda^{\prime}$ in (9.2.31) is zero. This can be
proved as follows.
The equation (9.2.30) means that the secondary differential of the ASS
$d_{2}[g_{6}]=0\in
E_{2}^{s+2,tq+3q+1}(L,V)=Ext_{A}^{s+2,tq+3q+1}(H^{*}V,H^{*}L)$, then
$[g_{6}]\in E_{3}^{s,tq+3q}(L,V)$ and
The third differential $d_{3}[g_{6}]=(w_{4})_{*}[(\bar{b}_{s+3}\wedge
1_{Y})f_{7}]\in E_{3}^{s+3,tq+3q+2}(L,V)$
Note that $(\omega\wedge 1_{L})(1_{M}\wedge(\alpha_{1})_{L})(i\wedge
1_{L})\pi=(1_{Y}\wedge j)\alpha_{Y\wedge
M}i(\alpha_{1})_{L}\pi=\bar{h}\phi(\alpha_{1})_{L}\pi$ = 0 , this is because
$\phi(\alpha_{1})_{L}\in[\Sigma^{3q-2}L,L]$ = 0 which is obtained by
$\pi_{rq-2}S$ = 0 ($r=2,3,4$). Hence , by (9.2.23), $(i\wedge
1_{L})\pi=u_{4}\tau$ with $\tau\in[\Sigma^{4q}S,V]$ which has filtration 1.
Moreover, $u_{4}\tau\cdot p=(i\wedge 1_{L})\pi\cdot p$ = 0 , then , by Prop.
9.2.3(4), $\tau\cdot
p=\widetilde{\lambda}w_{4}\bar{h}\widetilde{\phi}(\pi\wedge 1_{L})\pi$ for
some $\widetilde{\lambda}\in Z_{(p)}$. This scalar $\widetilde{\lambda}$ must
be zero (mod $p$),this is because the left hand side of the equation has
filtration 2 and the right hand side has filtration 3 (cf. Remark 9.2.20 and
$Ext_{A}^{0,4q+1}(H^{*}V,Z_{p})=0$ which is obtained by
$Ext_{A}^{0,4q+1}(H^{*}Y,Z_{p})=0=Ext_{A}^{0,q+1}(H^{*}M\wedge L,Z_{p})$).
Consequently, by Prop. 9.2.3(4) we have $\tau\cdot p$ = 0 and so
$\tau=\overline{\tau}i$ with $\overline{\tau}\in[\Sigma^{4q}M,V]$. Since
$(u_{4})_{*}(\pi)^{*}[g_{6}]\in Ext_{A}^{s+1,tq+q+1}(H^{*}M\wedge
L,Z_{p})\cong Z_{p}\\{(i\wedge 1_{L})_{*}(\pi)_{*}(\sigma)\\}$ ( cf. Prop.
9.2.8), then $(u_{4})_{*}\pi^{*}[g_{6}]=\lambda_{0}(i\wedge
1_{L})_{*}\pi_{*}(\sigma)=\lambda_{0}(u_{4})_{*}(\overline{\tau}i)_{*}(\sigma)$
for some $\lambda_{0}\in Z_{p}$ and so by (9.2.23) we have
$\pi^{*}[g_{6}]=\lambda_{0}\overline{\tau}_{*}i_{*}(\sigma)\in
Ext_{A}^{s+1,tq+3q+1}(H^{*}V,Z_{p})$, this is because
$Ext_{A}^{s+1,tq+3q+1}\\\ (H^{*}Y,H^{*}L)$ = 0 (cf. Prop. 9.2.6). By the
supposition on $d_{2}(\sigma)=a_{0}\sigma^{\prime}=p_{*}(\sigma^{\prime})\in
Ext_{A}^{s+2,tq+1}(Z_{p},Z_{p})$, we have $d_{2}i_{*}(\sigma)$ = 0 so that
$i_{*}(\sigma)\in E_{3}^{s+2,tq+1}(S,\\\ M)$. Moreover,
$E_{2}^{s+3,tq+2}(S,M)=Ext_{A}^{s+3,tq+2}(H^{*}M,Z_{p})$ = 0 (cf. Prop. 9.2.8)
then $E_{3}^{s+3,tq+2}(S,M)$ = 0 so that the third differential
$d_{3}i_{*}(\sigma)\in E_{3}^{s+3,tq+2}(S,M)$ = 0. Since
$\pi^{*}[g_{6}]=\lambda_{0}(\overline{\tau})_{*}i_{*}(\sigma)\in
E_{2}^{s+1,tq+4q+1}(S,V)$, then
$\pi^{*}[g_{6}]=\lambda_{0}\overline{\tau}_{*}(i_{*}(\sigma))\in
E_{3}^{s+1,tq+4q+1}(S,V)$ and so
$d_{3}\pi^{*}[g_{6}]=\lambda_{0}d_{3}(\overline{\tau})_{*}(i_{*}(\sigma))=\lambda_{0}(\overline{\tau})_{*}d_{3}(i_{*}(\sigma))=0\in
E_{3}^{s+4,tq+4q+3}(S,V)$
Hence, $(w_{4})_{*}\pi^{*}[(\bar{b}_{s+3}\wedge
1_{Y})f_{7}]=d_{3}\pi^{*}[g_{6}]$ = 0 $\in E_{3}^{s+4,tq+4q+2}(S,V)$. In
addition, by the split exact sequence (9.2.26) we have
$\pi^{*}[(\bar{b}_{s+3}\wedge 1_{Y})f_{7}]=0\in E_{3}^{s+4,tq+4q+3}(S,Y)$.
Then, in the $E_{2}$-term, $\pi^{*}[(\bar{b}_{5}\wedge 1_{Y})f_{7}]$ must be a
$d_{2}$-boundary, that is
$\pi^{*}[(\bar{b}_{s+3}\wedge 1_{Y})f_{7}]\in
d_{2}E_{2}^{s+2,tq+4q+2}(S,Y)=d_{2}Ext_{A}^{s+2,tq+4q+2}(H^{*}Y,Z_{p})$ = 0
(cf. Prop. 9.2.6(1)). Hence, by (9.2.31),
$\lambda^{\prime}\bar{h}_{*}\widetilde{\phi}_{*}(\pi\wedge
1_{L})_{*}\pi_{*}(\sigma^{\prime})$ = 0. This implies that the scalar
$\lambda^{\prime}$ is zero (cf. Prop. 9.2.9(3)) which shows the above claim.
So,(9.2.30) becomes $(\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{V})f_{5}=(\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{V})(1_{E_{s+4}}\wedge w_{4})f_{8}+(\bar{c}_{s}\wedge 1_{V})g_{6}$ for some
$f_{8}\in[\Sigma^{tq+3q+3}L,E_{s+4}\wedge Y]$. By composing $1_{E_{s+1}}\wedge
u_{5}$ on the above equation we have
$(\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{X})(1_{E_{s+4}}\wedge
u_{5})f_{5}=(\bar{a}_{s+1}\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{X})(1_{E_{s+4}}\wedge\widetilde{u}w_{2})f_{8}$ ( cf. (9.2.28)), this is
because $(1_{KG_{s}}\wedge u_{5})g_{6}\in[\Sigma^{tq+3q-1}L,KG_{s}\wedge X]$
represents an element in $Ext_{A}^{s,tq+3q-1}(H^{*}X,\\\ H^{*}L)$ = 0 (cf.
Lemma 9.2.18(2)) so that it is a $d_{1}$-boundary and $(\bar{c}_{s}\wedge
1_{X})(1_{KG_{s}}\wedge u_{5})g_{6}$ = 0. Consequently we have
(9.2.32) $(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{X})(1_{E_{s+4}}\wedge
u_{5})f_{5}$
$=(\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{X})(1_{E_{s+4}}\wedge\widetilde{u}w_{2})f_{8}+(\bar{c}_{s+1}\wedge
1_{X})g_{7}$
for some $d_{1}$-cycle $g_{7}\in[\Sigma^{tq+3q+1}L,KG_{s+1}\wedge X]$ such
that $[g_{7}]\in Ext_{A}^{s+1,tq+3q}\\\ (H^{*}X,H^{*}L)$.
Now we prove $(\bar{c}_{s+1}\wedge 1_{X})g_{7}$ = 0 as follows. By Lemma
9.2.18(2) and Prop. 9.2.7(1),
$[g_{7}]=\lambda_{3}(\widetilde{u}w_{3})_{*}(\overline{\phi}_{W})_{*}[\sigma\wedge
1_{L}]$ and the equation (9.2.32) means that the secondary differential
$d_{2}[g_{7}]=0$. Since
$d_{2}(\sigma)=a_{0}\sigma^{\prime}=p_{*}(\sigma^{\prime})\in
Ext_{A}^{s+2,tq+1}(Z_{p},Z_{p})$ , then
$\lambda_{3}(\widetilde{u}w_{3})_{*}(\overline{\phi}_{W})_{*}(p\wedge
1_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]=d_{2}[g_{7}]=0\in
Ext_{A}^{s+3,tq+3q+1}(H^{*}X,H^{*}L)$. By Lemma 9.2.18(1), this implies that
$\lambda_{3}=0$ so that $g_{7}$ is a $d_{1}$-boundary and
$(\bar{c}_{s+1}\wedge 1_{X})g_{7}$ = 0.
Hence, (9.2.32) becomes $(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{Y\wedge
W})(1_{E_{s+4}}\wedge u_{5})f_{5}=(\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{X})(1_{E_{s+4}}\wedge\widetilde{u}w_{2})f_{8}$ and so by (9.2.28)(9.2.12)
we have, $(\bar{a}_{s+2}\bar{a}_{s+3}\wedge 1_{M})(1_{E_{s+4}}\\\
\wedge(1_{M}\wedge(\alpha_{1})_{L})u_{4})f_{5}=(\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{M})(1_{E_{s+4}}\wedge\widetilde{\psi}u_{5})f_{5}=0$. On the other hand, by
composing $(1_{E_{s+2}}\wedge 1_{M}\wedge(\alpha_{1})_{L})$ on the equation
(9.2.29) we have $(1_{E_{s+2}}\wedge
i)\kappa\cdot(\alpha_{1})_{L}=(1_{E_{s+2}}\wedge
1_{M}\wedge(\alpha_{1})_{L})(1_{E_{s+2}}\wedge i\wedge 1_{L})(\kappa\wedge
1_{L})=(\bar{a}_{s+2}\bar{a}_{s+3}\wedge
1_{M})(1_{E_{s+4}}\wedge(1_{M}\wedge(\alpha_{1})_{L})u_{4})f_{5}=0$.
It follows that
(9.2.33) $\kappa\cdot(\alpha_{1})_{L}=(1_{E_{s+2}}\wedge p)f_{9}$
for some $f_{9}\in[\Sigma^{tq+q}L,E_{s+2}]$. Since
$\bar{b}_{s+2}\cdot\kappa=a_{0}\sigma^{\prime}=p_{*}(\sigma^{\prime})\in
Ext_{A}^{s+2,tq+1}(Z_{p},\\\ Z_{p})$, then $\kappa\cdot(\alpha_{1})_{L}$ lifts
to a map $\tilde{f}\in[\Sigma^{tq+q+1}L,E_{s+3}]$ such that
$\bar{b}_{s+3}\cdot\tilde{f}$ represents
$p_{*}((\alpha_{1})_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]\neq 0\in
Ext_{A}^{s+3,tq+q+1}(Z_{p},H^{*}L)$ ( cf. Prop. 9.2.3(1)). Then , by (9.2.33),
$p_{*}[\bar{b}_{s+2}\cdot
f_{9}]=p_{*}((\alpha_{1})_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]$ so that
$[\bar{b}_{s+2}\cdot f_{9}]\in Ext_{A}^{s+2,tq+q}(Z_{p},H^{*}L)$ must equal to
$((\alpha_{1})_{L})_{*}[\sigma^{\prime}\wedge 1_{L}]$ , this is because the
group has two generators $((\alpha_{1})_{L})_{*}[\sigma^{\prime}_{1}\wedge
1_{L}]$, $((\alpha_{1})_{L})_{*}[\sigma^{\prime}_{2}\wedge 1_{L}]$. Write
$\xi_{n,s+2}=f_{9}i^{\prime\prime}$, then
(9.2.34) $\kappa\cdot\alpha_{1}=(1_{E_{s}+2}\wedge p)\xi_{n,s+2}$
such that $\bar{b}_{s+2}\cdot\xi_{n,s+2}=h_{0}\sigma^{\prime}\in
Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})$ and by Lemma 9.2.10 we have
$(\bar{c}_{s+1}\wedge 1_{M})(1_{KG_{s+1}}\wedge
i)h_{0}\sigma=(1_{E_{s+2}}\wedge i)\kappa\cdot\alpha_{1}$ = 0. This shows the
second result of the main Theorem.
In addition, by (9.2.34) and Lemma 9.2.10(2),
$\bar{a}_{0}\bar{a}_{1}\cdots\bar{a}_{s+1}(1_{E_{s+2}}\wedge p)\xi_{n,s+2}\\\
=0$, this shows that
$\xi_{n}=\bar{a}_{0}\bar{a}_{1}\cdots\bar{a}_{s+1}\cdot\xi_{n,s+2}\in\pi_{tq+q-s-2}S$
is an element of order $p$ and it is represented by $h_{0}\sigma^{\prime}\in
Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})$ in the ASS. Q.E.D.
Remark 9.2.35 In the proof of the main Theorem A, we obtain a stronger result.
By (9.2.33), $\kappa\cdot(\alpha_{1})_{L}=(1_{E_{s+2}}\wedge p)f_{9}$, then
$(1_{E_{s+2}}\wedge i)\kappa\cdot(\alpha_{1})_{L}$ = 0, and so
$(1_{E_{s+2}}\wedge 1_{L}\wedge i)(\kappa\wedge 1_{L})\phi=(1_{E_{s+2}}\wedge
1_{L}\wedge i)(\kappa\wedge 1_{L})((\alpha_{1})_{L}\wedge
1_{L})\widetilde{i^{\prime\prime}}$ = 0, where
$\widetilde{i}^{\prime\prime}\in\pi_{q}L\wedge L$ such that
$((\alpha_{1})_{L}\wedge 1_{L})\widetilde{i}^{\prime\prime}=\phi$. It can be
easily proved that $(\kappa\wedge 1_{L})\phi=(\bar{c}_{s+1}\wedge
1_{L})\sigma\phi$, where $\sigma\phi\in\pi_{tq+2q}(KG_{s+1}\wedge L)$ is a
$d_{1}$-cycle which represents $(\phi)_{*}(\sigma)\in
Ext_{A}^{s+1,tq+2q}(H^{*}L,Z_{p})$. Then we obtain that $(\bar{c}_{s+1}\wedge
1_{L\wedge M})(1_{KG_{s+1}}\wedge i)\sigma\phi$ = 0. That is to say,
$(1_{L}\wedge i)_{*}(\phi)_{*}(\sigma)\in Ext_{A}^{s+1,tq+2q}(H^{*}L\wedge
M,Z_{p})$ is a permanent cycle in the ASS . Moreover, by (9.2.34) we have
$\xi_{n,s+2}=f_{9}i^{\prime\prime}$, then
$(1_{KG_{s+2}}\wedge\alpha_{1})\xi_{n,s+2}=(1_{KG_{s+2}}\wedge\alpha_{1})f_{9}i^{\prime\prime}=f_{9}i^{\prime\prime}\cdot\alpha_{1}=0$
and so $\xi_{n,s+2}=(1_{E_{s+2}}\wedge j^{\prime\prime})\tilde{f}_{9}$ with
$\tilde{f}_{9}\in\pi_{tq+2q}E_{s+2}\wedge L$. Since $\xi_{n,s+2}$ is
represented by
$h_{0}\sigma^{\prime}=(j^{\prime\prime})_{*}\phi_{*}(\sigma^{\prime})$ in the
ASS , then $\tilde{f}_{9}$ is represented by $(\phi)_{*}(\sigma^{\prime})\in
Ext_{A}^{s+2,tq+2q}(H^{*}L,Z_{p})$ in the ASS. That is to say
$(\phi)_{*}(\sigma^{\prime})\in Ext_{A}^{s+2,tq+2q}(H^{*}L,Z_{p})$ is a
permanent cycle in the ASS. This is a stronger result obtained in the main
Theorem A.
§3. A general result on convergence in the spectrum $V(1)$
In this section we will prove , under some suppositions, a general result on
the convergence of $i^{\prime}_{*}i^{\prime}*(h_{0}\sigma)\in
Ext_{A}^{s+1,tq+q}(H^{*}V(1),Z_{p})$ to the homotopy groups of the spectrum
$V(1)$ can implies the convergence of $i^{\prime}_{*}i_{*}(g_{0}\sigma)\in\\\
Ext_{A}^{s+2,tq+pq+2q}(H^{*}V(1),Z_{p})$ in the ASS. We have the following
main Theorem.
The main Thoerem B (generalization of [7] Theorem II) Let $p\geq 5,s\leq 4$,
$Ext_{A}^{s,tq}(Z_{p},Z_{p})\cong Z_{p}\\{\sigma\\}$,
$Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}\sigma\\}$,
$Ext_{A}^{s+2,tq+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}\sigma\\}$ and suppose that
(I) $Ext_{A}^{s+1,tq+rq+u}(Z_{p},Z_{p})$ = 0, for $r=1,u=-1,1,2,3$ or $r=2,u=$
$\qquad\quad-1,0,1,2,3$.
$Ext_{A}^{s+1,tq}(Z_{p},Z_{p})$ is zero or has (one or two) generator
$\sigma^{\prime}$ satisfying $a_{0}\sigma^{\prime}\neq 0$.
$Ext_{A}^{s+1,tq+r}(Z_{p},Z_{p})$ = 0 for $r=-2,-1,2,3$ and has unique
generator $a_{0}\sigma$ for $r=1$ satisfying $a_{0}^{2}\sigma\neq 0$.
$Ext_{A}^{s,tq+q}(Z_{p},Z_{p})$ = 0 or
$=Z_{p}\\{h_{0}\tau^{\prime}\\},Ext_{A}^{s,tq+1}(Z_{p},Z_{p})$ = 0 or
$Z_{p}\\{a_{0}\tau^{\prime}\\}$.
$Ext_{A}^{s,tq+rq+u}(Z_{p},Z_{p})$ = 0, $r=1,u=1,2$, $r=-1,u=-1,0$.
(II) $i^{\prime}_{*}i_{*}(h_{0}\sigma)\in Ext_{A}^{s+1,tq+q}(H^{*}K,Z_{p})$ is
a permanent cycle in the ASS,
then $i^{\prime}_{*}i_{*}(g_{0}\sigma)\in
Ext_{A}^{s+2,tq+pq+2q}(H^{*}K,Z_{p})$ also is a permanent cycle in the ASS and
it conveges to a nontrivial element in $\pi_{tq+pq+2q-s-2}K$.
To prove the main Theorem B, we need some knowledge on derivation of maps
between $M$\- module spectra and some lower dimensional Ext groups. These
preminilaries will be used in the proof of the main Theorem B and especially
in the proof of Theorem 9.3.9 below.
Prop. 9.3.0 Let $p\geq 5,s\leq 4$, then under the supposition of the main
Theorem B we have
(1) $Ext_{A}^{s+1,tq+r}(H^{*}M,H^{*}M)$ = 0 for $r=1,2$.
(2) $Ext_{A}^{s+1,tq+r}(Z_{p},H^{*}M)$ = 0 for $r=0,1$,
$Ext_{A}^{s,tq+r}(H^{*}M,Z_{p})$ = 0 for $r=1,2$,
Proof (1) Consider the following exact sequence ( $r=1,2,3$)
$Ext_{A}^{s+1,tq+r}(Z_{p},Z_{p})\stackrel{{\scriptstyle
i_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+r}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
j_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+r-1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p_{*}}}{{\rightarrow}}$
induced by (9.1.1). By the supposition, the left group is zero for $r=2,3$ and
has unique generator $a_{0}\sigma=p_{*}(\sigma)$ for $r=1$ so that im $i_{*}$
= 0. By the supposition, the right group is zero for $r=3$ and has unique
generator $a_{0}\sigma$ for $r=2$ which satisfies
$p_{*}(a_{0}\sigma)=a_{0}^{2}\sigma\neq 0$. By the supposition, the right
group is zero for $r=1$ and has (one or two) generator $\sigma^{\prime}$ for
$r=2$ (both) satisfying $p_{*}(\sigma^{\prime})=a_{0}\sigma^{\prime}\neq 0$.
Then , the above $p_{*}$ is monic so that im $j_{*}$ = 0. This shows that the
middle group is zero which shows the first result. The second result can be
obtained immediately by the first result.
(2) Consider the following exact sequence $(r=0,1)$
$Ext_{A}^{s+1,tq+r+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+r}(Z_{p},H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+r}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\longrightarrow}}$
induced by (9.1.1). By the supposition, the left group is zero for $r=1$ and
has unique generator $a_{0}\sigma=p^{*}(\sigma)$ for $r=0$ so that im $j^{*}$
= 0. The right group has unique generator $a_{0}\sigma$ for $r=1$ which
satisfies $p^{*}(a_{0}\sigma)=a_{0}^{2}\sigma\neq 0$. The right group is zero
for $r=0$ or has (one or two) generator $\sigma^{\prime}$ satisfying
$p^{*}(\sigma^{\prime})=a_{0}\sigma^{\prime}\neq 0$. Then im $i^{*}$ = 0 so
that the middle group is zero as desired. The proof of the second result is
similar. Q.E.D.
Proposition 9.3.1 Let $p\geq 5,s\leq 4$, then under the supposition of the
main Theorem B we have
(1) $Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\cong Z_{p}\\{\tilde{\sigma}\\}$ satisfying
$i^{*}(\tilde{\sigma})=i_{*}(\sigma)\in Ext_{A}^{s,tq}\\\ (H^{*}M,Z_{p})$,
$j_{*}(\tilde{\sigma})=j^{*}(\sigma)\in Ext_{A}^{s,tq-1}(Z_{p},H^{*}M)$.
(2) $Ext_{A}^{s+1,tq+q}(H^{*}M,H^{*}M)\cong
Z_{p}\\{(ij)_{*}\alpha_{*}(\tilde{\sigma}),$
$\alpha_{*}(ij)^{*}(\tilde{\sigma})\\}$ ,
(3) $Ext_{A}^{s+1,tq+q+1}(H^{*}M,H^{*}M)\cong
Z_{p}\\{\alpha_{*}(\tilde{\sigma})=\alpha^{*}(\tilde{\sigma})\\}$,
(4) $Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)\cong
Z_{p}\\{i^{\prime}_{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})=i^{\prime}_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{\sigma})\\}$,
where $\alpha_{1}=j\alpha i:\Sigma^{q-1}S\rightarrow S$ and
$\alpha_{*}:Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\to Ext_{A}^{s+1,tq+q+1}\\\
(H^{*}M,H^{*}M)$ is the connecting (or boundary) homomorphism induced by
$\alpha:\Sigma^{q}M\to M$.
Proof: (1) Consider the following exact sequence
$\quad 0=Ext_{A}^{s,tq+1}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\rightarrow}}Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle
i^{*}}}{{\rightarrow}}Ext_{A}^{s,tq}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\rightarrow}}$
induced by (9.1.1). The right group has unique generator $i_{*}(\sigma)$ ,
this is because $Ext_{A}^{s,tq-r}(Z_{p},Z_{p})$ = 0 for $r=1$) and has unique
generator $\sigma$ for $r=0$. Moreover,
$p^{*}i_{*}(\sigma)=i_{*}p^{*}(\sigma)=i_{*}(a_{0}\sigma)=i_{*}p_{*}(\sigma)$
= 0, then the middle has unique generator $\tilde{\sigma}$ such that
$i^{*}(\tilde{\sigma})=i_{*}(\sigma)$. This shows the result and the second
relation can be similarly proved.
(2) By the supposition, $Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})$ has unique generator
$h_{0}\sigma=j_{*}\alpha_{*}i_{*}(\sigma)$ =
$j_{*}\alpha_{*}i^{*}(\tilde{\sigma})$, Then the result follows by the
following exact sequence
$\stackrel{{\scriptstyle
p^{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}M,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle
i^{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\longrightarrow}}$
induced by (9.1.1), where the right group has unique generator
$i^{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})=(ij)_{*}\alpha_{*}i_{*}(\sigma)$
satisftying
$p^{*}(ij)_{*}\alpha_{*}i_{*}(\sigma)=(ij)_{*}\alpha_{*}i_{*}p_{*}(\sigma)$ =
0 and the left group has unique generator
$\alpha_{*}i_{*}(\sigma)=i^{*}\alpha_{*}(\tilde{\sigma})$.
(3) Consider the following exact sequence
$Ext_{A}^{s+1,tq+q+2}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}M,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\longrightarrow}}$
induced by (9.1.1). The left group is zero, this is because by the
supposition, $Ext_{A}^{s+1,tq+q+r}(Z_{p},Z_{p})$ = 0 for $r=1,2,3$. The right
group has unique generator $(\alpha
i)_{*}(\sigma)=i^{*}\alpha_{*}(\tilde{\sigma})$, this is because
$Ext_{A}^{s+1,tq+q+r}(Z_{p},Z_{p})$ is zero for $r=1$ and has unique generator
$h_{0}\sigma=j_{*}(\alpha i)_{*}(\sigma)$ for $r=0$. Since $p^{*}(\alpha
i)_{*}(\sigma)=(\alpha i)_{*}p_{*}(\sigma)$ = 0, then the middle group has
unique generator $\alpha_{*}(\tilde{\sigma})$ as desired. Moreover we have
$\alpha_{*}(\tilde{\sigma})=\alpha^{*}(\tilde{\sigma})$, this is because
$i^{*}j_{*}\alpha_{*}(\tilde{\sigma})=j_{*}\alpha_{*}i_{*}(\sigma)=h_{0}\sigma=(j\alpha
i)^{*}(\sigma)=i^{*}j_{*}\alpha^{*}(\tilde{\sigma})$.
(4) Consider the following exact sequence
$Ext_{A}^{s+1,tq+q}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle
i^{\prime}_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle
j^{\prime}_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq-1}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle\alpha_{*}}}{{\longrightarrow}}$
induced by (9.1.2). By the supposition, $Ext_{A}^{s+1,tq-r}(Z_{p},Z_{p})$ = 0
for $r=1,2$ and has unique generator $\sigma^{\prime}$ for $r=0$ , then the
right group has unique generator $(ij)^{*}(\tilde{\sigma^{\prime}})$
satisfying
$\alpha_{*}(ij)^{*}(\tilde{\sigma^{\prime}})=j^{*}\alpha_{*}i_{*}(\sigma^{\prime})\neq
0\in Ext_{A}^{s+2,tq+q}(H^{*}M,\\\ H^{*}M)$ . Hence,
$Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)=i^{\prime}_{*}Ext_{A}^{s+1,tq+q}\\\
(H^{*}M,H^{*}M)$ has unique generator
$(i^{\prime})_{*}(ij)_{*}\alpha_{*}\tilde{\sigma}$ =
$i^{\prime}_{*}(\alpha_{1}\wedge 1_{M})_{*}(\tilde{\sigma})$, this is because
$(\alpha_{1}\wedge 1_{M})_{*}(\tilde{\sigma})$ =
$(ij)_{*}\alpha_{*}(\tilde{\sigma})-\alpha_{*}(ij)_{*}(\tilde{\sigma})$ which
is obtained by $\alpha_{1}\wedge 1_{M}=ij\alpha-\alpha ij$. Q.E.D.
Proposition 9.3.2 Let $p\geq 5,s\leq 4$ , then under the supposition of the
main Theorem B we have
(1) $Ext_{A}^{s+1,tq+2q+r}(H^{*}K,H^{*}M)$ = 0, $r=0,1,2$,
$Ext_{A}^{s+1,tq+2q+1}(H^{*}K,Z_{p})$ = 0.
(2) $Ext_{A}^{s+1,tq+q+r}(H^{*}K,Z_{p})$ = 0 , $r=1,2,3$,
$Ext_{A}^{s+1,tq+q+r}(H^{*}K,H^{*}M)$ = 0 ,$r=1,2$.
(3) $Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}K)$ $\cong
Z_{p}\\{(h_{0}\sigma)^{\prime}\\}$ with
$(i^{\prime})^{*}(h_{0}\sigma)^{\prime}=(i^{\prime}ij\alpha)_{*}(\tilde{\sigma})$.
Proof: (1) Consider the following exact sequence
$\qquad Ext_{A}^{s+1,tq+2q+r}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle
i^{\prime}_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+2q+r}(H^{*}K,H^{*}M)$
$\qquad\qquad\stackrel{{\scriptstyle
j^{\prime}_{*}}}{{\rightarrow}}Ext_{A}^{s+1,tq+q+r-1}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle\alpha_{*}}}{{\rightarrow}}$
induced by (9.1.2). The left group is zero by the supposition on
$Ext_{A}^{s+1,tq+2q+u}\\\ (Z_{p},Z_{p})$ = 0 for $u=-1,0,1,2$ . The right
group has unique generator $(ij)^{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})$ for
$r=0$ and is generated by two generators $(ij)_{*}\alpha_{*}(\tilde{\sigma})$
, $(ij)^{*}\alpha_{*}(\tilde{\sigma})$ for $r=1$. Moreovre, the right group
has unique generator $\alpha_{*}(\tilde{\sigma})$ for $r=2$ (cf. Prop.
9.3.1(3)). We claim that (i)
$\alpha_{*}(ij)^{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})\neq 0$ . (ii)
$\alpha_{*}[\lambda_{1}(ij)_{*}\alpha_{*}(\tilde{\sigma})+\lambda_{2}\alpha_{*}(ij)^{*}(\tilde{\sigma})]\neq
0$. (iii) $\alpha_{*}\alpha_{*}(\tilde{\sigma})\neq 0$. Then the above
$\alpha_{*}$ is monic and so $imj^{\prime}_{*}$ = 0. This shows
$Ext_{A}^{s+1,tq+2q+r}(H^{*}K,H^{*}M)$ = 0 with $r=0,1,2$ and consequently we
have $Ext_{A}^{s+1,tq+2q+1}$ $(H^{*}K,Z_{p})$ = 0.
To prove the claim, recall from the supposition that
$\widetilde{\alpha}_{2}\sigma=j_{*}\alpha_{*}\alpha_{*}i_{*}(\sigma)\\\ \neq
0\in Ext_{A}^{s+2,tq+2q+1}(Z_{p},Z_{p})$, then
$i_{*}(\widetilde{\alpha}_{2}\sigma)\neq 0\in
Ext_{A}^{s+2,tq+2q+1}(H^{*}M,Z_{p})$ , this is because
$Ext_{A}^{s+1,tq+2q}(Z_{p},Z_{p})$ = 0 from the supposition. In addition, we
also have $j^{*}i_{*}(\alpha_{2}\sigma)\neq 0\in
Ext_{A}^{s+2,tq+2q}(H^{*}M,H^{*}M)$ ,this is because $Ext_{A}^{s+1,tq+2q}$
$(H^{*}M,Z_{p})$ = 0. Hence, by $2\alpha ij\alpha=ij\alpha^{2}+\alpha^{2}ij$
we have
(9.3.3)
$\qquad\alpha_{*}(ij)^{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})=j^{*}\alpha_{*}(ij)_{*}\alpha_{*}i_{*}(\sigma)$
$\qquad\quad=\frac{1}{2}j^{*}(ij)_{*}\alpha_{*}\alpha_{*}i_{*}(\sigma)=\frac{1}{2}j^{*}i_{*}(\alpha_{2}\sigma)\neq
0$
This shows the claim (i). For the claim (ii),
$\alpha_{*}[\lambda_{1}(ij)_{*}\alpha_{*}(\tilde{\sigma})+\lambda_{2}\alpha_{*}(ij)^{*}(\tilde{\sigma})]$
$\quad=\frac{1}{2}\lambda_{1}(ij)_{*}\alpha_{*}\alpha_{*}(\tilde{\sigma})+(\frac{1}{2}\lambda_{1}+\lambda_{2})\alpha_{*}\alpha_{*}(ij)^{*}(\tilde{\sigma})\neq
0$
, this is because this two terms is linearly independent which can be obtained
from $(ij)_{*}\alpha_{*}\alpha_{*}(ij)^{*}(\tilde{\sigma})\neq 0$ ( cf.
(9.3.3)). The claim (iii) is immediate , this is because
$i^{*}j_{*}\alpha_{*}\alpha_{*}(\tilde{\sigma})=j_{*}\alpha_{*}\alpha_{*}i_{*}(\sigma)=\widetilde{\alpha}_{2}\sigma\neq
0$.
(2) Consider the following exact sequence ($r=1,2,3)$
$Ext_{A}^{s+1,tq+q+r}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
i^{\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+r}(H^{*}K,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle
j^{\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+r-1}(H^{*}M,Z_{p})\stackrel{{\scriptstyle\alpha_{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero for $r=2,3$ which can be obtained
from the supposition of $Ext_{A}^{s+1,tq+q+u}(Z_{p},Z_{p})$ = 0( $u=1,2,3$).
The left group has unique generator $\alpha_{*}i_{*}(\sigma)$ for $r=1$, then
in any case we have im $i^{\prime}_{*}$ = 0. The right group is zero for
$r=2,3$ (cf. Prop. 9.3.0) and has unique generator $i_{*}(\sigma^{\prime})$
for $r=1$ which satisfies $\alpha_{*}i_{*}(\sigma^{\prime})\neq 0\in
Ext_{A}^{s+2,tq+q+1}(H^{*}M,Z_{p})$ so that $j^{\prime}_{*}$ = 0 and the
result follows.
(3) Consider the following exact sequence
$Ext_{A}^{s+1,tq+2q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$
induced by (9.1.2). The left group is zero by (1) and the right group has
unique generator $i^{\prime}_{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})$ (cf. Prop.
9.3.1(4)) which satisfies $\alpha^{*}i^{\prime}_{*}(ij)_{*}\\\
\alpha_{*}(\tilde{\sigma})=i^{\prime}_{*}(ij)_{*}\alpha_{*}\alpha^{*}(\tilde{\sigma})=i^{\prime}_{*}(ij)_{*}\alpha_{*}\alpha_{*}(\tilde{\sigma})$
= 0, this is because $i^{\prime}ij\alpha^{2}\\\ =2i^{\prime}\alpha
ij\alpha-i^{\prime}\alpha^{2}ij=0\in[\Sigma^{2q-1}M,K]$. Then the result
follows. Q.E.D.
Proposition 9.3.4 Let $p\geq 5,s\leq 4$, then under the supposition of the
main Theorem B we have
$\qquad\quad Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$ $\cong
Z_{p}\\{(h_{0}\sigma)^{\prime\prime}\\}$
satisfying
$(i^{\prime})^{*}(h_{0}\sigma)^{\prime\prime}=i^{\prime}_{*}(ij)_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{\sigma})$.
Proof: Consider the following exact sequence
$Ext_{A}^{s+1,tq+2q}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}M)$
induced by (9.1.2). The left group is zero by Prop. 9.3.2(1) and similar to
that in Prop. 9.3.1, the right group has unique generator
$(ij)^{*}i^{\prime}_{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})=i^{\prime}_{*}(ij)_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{\sigma})$ which satisfies
$\alpha^{*}i^{\prime}_{*}(ij)_{*}(\alpha_{1}\wedge 1_{M})_{*}(\tilde{\sigma})$
= $i^{\prime}_{*}(ij)_{*}(\alpha_{1}\wedge
1_{M})_{*}\alpha_{*}(\tilde{\sigma})$ = 0 $\in Ext_{A}^{s+2,tq+2q}(H^{*}K,$
$H^{*}M)$, this is because $i^{\prime}ij(\alpha_{1}\wedge 1_{M})\alpha$ = 0
$\in[\Sigma^{2q-2}M,K]$. Then the result follows. Q.E.D.
Proposition 9.3.5 Let $p\geq 5,s\leq 4,$ then under the supposition of the
main Theorem B we have
$Ext_{A}^{s+1,tq+q+1}(H^{*}K^{\prime}\wedge M,H^{*}M)\cong
Z_{p}\\{\psi_{*}(ij)_{*}\alpha_{*}(\tilde{\sigma}),\psi_{*}(ij)^{*}\alpha_{*}(\tilde{\sigma})\\}.$
where $\psi:\Sigma M\rightarrow K^{\prime}\wedge M$ is the map in (9.1.17).
Proof: Consider the following exact sequence
$Ext_{A}^{s+1,tq+q}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle\psi_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}K^{\prime}\wedge
M,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle\rho_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}K,H^{*}M)=0$
induced by (9.1.17). The result follows immediately form Prop. 9.3.1(2) and
Prop.9.3.2.(Note: By the supposition, similar to that given in Prop. 9.3.2(2),
we can prove that $Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)$ = 0 so that the above
$\psi_{*}$ is monic). Q.E.D.
Proposition 9.3.6 Let $p\geq 5,s\leq 4$, then under the supposition of the
main Theorem B we have
$Ext_{A}^{s,tq}(H^{*}K,H^{*}K)$ $\cong Z_{p}\\{(\sigma)^{\prime}\\}$
satisfying
$(i^{\prime})^{*}(\sigma)^{\prime}=(i^{\prime})_{*}(\tilde{\sigma})$.
Proof: Consider the following exact sequence
$Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}K,H^{*}K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}K,H^{*}M)$
induced by (9.1.2). Since $j^{\prime}_{*}Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)$
$\subset Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\\\ \cong Z_{p}\\{\tilde{\sigma}\\}$ and
$\alpha_{*}(\tilde{\sigma})\neq 0\in Ext_{A}^{s+1,tq+q+1}(H^{*}M,H^{*}M)$,
then im $(j^{\prime})_{*}$ = 0 and so
$Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)=i^{\prime}_{*}Ext_{A}^{s,tq+q+1}(H^{*}M,H^{*}M)$.
Moreover, by the supposition on $Ext_{A}^{s,tq+q+r}(Z_{p},Z_{p})$ = 0 ,
$r=1,2$, and $Ext_{A}^{s,tq+q}(Z_{p},Z_{p})$ is zero or $\cong
Z_{p}\\{h_{0}\sigma^{\prime\prime}\\}$ we have
$Ext_{A}^{s,tq+q+1}(H^{*}M,H^{*}M)\cong
Z_{p}\\{\alpha_{*}(\tilde{\sigma^{\prime\prime}})\\}$, then
$Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)=(i^{\prime})_{*}Ext_{A}^{s,tq+q+1}(H^{*}M,H^{*}M)$
= 0. On the other hand, it is easily seen that $Ext_{A}^{s,tq}(H^{*}K,H^{*}M)$
has unique generator $i^{\prime}_{*}(\tilde{\sigma})$ which satisfies
$\alpha^{*}i^{\prime}_{*}(\tilde{\sigma})=i^{\prime}_{*}\alpha_{*}(\tilde{\sigma})$
= 0. Then the result follows. Q.E.D.
By (6.5.5), there is $\alpha^{\prime\prime}\in[\Sigma^{q-2}K,K]$ such that
$\alpha^{\prime\prime}i^{\prime}=i^{\prime}ij\alpha ij$. Let $X$ be the
cofibre of $\alpha^{\prime\prime}:\Sigma^{q-2}K\rightarrow K$ given by the
cofibration
(9.3.7)
$\qquad\quad\Sigma^{q-2}K\stackrel{{\scriptstyle\alpha^{\prime\prime}}}{{\longrightarrow}}K\stackrel{{\scriptstyle
w}}{{\longrightarrow}}X\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\Sigma^{q-1}K$,
Then, $\alpha^{\prime\prime}$ induces a boundary homomorphism (or connecting
homomotphism) $(\alpha^{\prime\prime})^{*}$ :
$Ext_{A}^{s,tq}(H^{*}K,H^{*}K))\rightarrow Ext_{A}^{s+1,tq+q-1}$
$(H^{*}K,H^{*}K)$. Since $\alpha^{\prime\prime}i^{\prime}=i^{\prime}ij\alpha
ij=i^{\prime}ij(\alpha_{1}\wedge 1_{M})$ , then
$(i^{\prime})^{*}(\alpha^{\prime\prime})^{*}(\sigma)^{\prime}=(\alpha^{\prime\prime}i^{\prime})^{*}(\sigma)^{\prime}=(i^{\prime}ij(\alpha_{1}\wedge
1_{M}))^{*}(\sigma)^{\prime}=(\alpha_{1}\wedge
1_{M})^{*}(ij)^{*}(i^{\prime})^{*}(\sigma)^{\prime}=(i^{\prime}ij)_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{\sigma})=(i^{\prime})^{*}(h_{0}\sigma)^{\prime\prime}$ (cf.
Prop. 9.3.4) . Then we have
(9.3.8)
$\quad(h_{0}\sigma)^{\prime\prime}=(\alpha^{\prime\prime})^{*}(\sigma)^{\prime}\in
Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$
this is because the above $(i^{\prime})^{*}$ is monic which can be obtained by
$Ext_{A}^{s+1,tq+2q}\\\ (H^{*}K,H^{*}M)$ = 0 (cf. Prop. 9.3.2).
After finishing the above preminilaries , we now turn to prove the following
Theorem 9.3.9. It is proved by some argument processing in the Adams
resolution (9.2.9) of some spectra related to the sphere spectrum $S$.
Theorem 9.3.9 Let $p\geq 5,s\leq 4$ , then under the supposition of the main
Theorem B we have $(\bar{c}_{s+1}\wedge 1_{K})(h_{0}\sigma)^{\prime\prime}$ =
0, where $(h_{0}\sigma)^{\prime\prime}\in[\Sigma^{tq+q-1}K,\\\ KG_{s+1}\wedge
K]$ is a $d_{1}$-cycle which represents the unique generator
$(h_{0}\sigma)^{\prime\prime}$ of $Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$ (cf.
Prop. 9.3.4).
Before proving Theorem 9.3.9, we first prove the following Lemma.
Lemma 9.3.10 Let $p\geq 5,s\leq 4$, then under the supposition of the main
Theorem B we have
(1) $(\bar{c}_{s+1}\wedge 1_{K})(h_{0}\sigma)^{\prime\prime}$ =
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})$,
(2) $(\bar{c}_{s+1}\wedge 1_{K})(h_{0}\sigma\wedge
1_{K})=(1_{E_{s+2}}\wedge\alpha_{1}\wedge 1_{K})(\kappa\wedge 1_{K})$
where $\kappa\in\pi_{tq+1}E_{s+2}$ such that
$\bar{a}_{s+1}\kappa=\bar{c}_{s}\sigma$ with $\sigma\in\pi_{tq}KG_{s}\\\ \cong
Ext_{A}^{s,tq}(Z_{p},Z_{p})$.
Proof: Recall that $X$ is the cofibre of
$\alpha^{\prime\prime}:\Sigma^{q-2}K\rightarrow K$ given by the cofibration
(9.3.7). Since
$(h_{0}\sigma)^{\prime\prime}\in[\Sigma^{tq+q-1}K,KG_{s+1}\wedge K]$
represents
$(h_{0}\sigma)^{\prime\prime}=(\alpha^{\prime\prime})^{*}(\sigma)^{\prime}\in
Ext_{A}^{s+1,tq+q-1}(H^{*}K,$ $H^{*}K)$, then
$(h_{0}\sigma)^{\prime\prime}u\in[\Sigma^{tq}X,\\\ KG_{s+1}\wedge K]$ is a
$d_{1}$-boundary so that $(\bar{c}_{s+1}\wedge
1_{K})(h_{0}\sigma)^{\prime\prime}u$ = 0 and $(\bar{c}_{s+1}\wedge
1_{K})(h_{0}\sigma)^{\prime\prime}=f^{\prime}\alpha^{\prime\prime}$ for some
$f^{\prime}\in\Sigma^{tq+1}K,E_{s+2}\wedge K].$ It follows that
$(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}\alpha^{\prime\prime}=0$ and so
$(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}=f^{\prime}_{2}w$ with
$f^{\prime}_{2}\in[\Sigma^{tq}X,E_{s+1}\wedge K]$. Then, $(\bar{b}_{s+1}\wedge
1_{K})f^{\prime}_{2}w$ = 0 and $(\bar{b}_{s+1}\wedge
1_{K})f^{\prime}_{2}=g^{\prime}\cdot u$ for some
$g^{\prime}\in[\Sigma^{tq+q-1}K,KG_{s+1}\wedge K]$. $g^{\prime}$ is a
$d_{1}$-cycle, this is because $(\bar{b}_{s+2}\bar{c}_{s+1}\wedge
1_{K})g^{\prime}=g^{\prime}_{2}\alpha^{\prime\prime}$ (with
$g^{\prime}_{2}\in[\Sigma^{tq+1}K,KG_{s+2}\wedge K])$ = 0 since
$\alpha^{\prime\prime}$ induces zero homomorphism in $Z_{p}$-cohomology. Then,
by Prop. 9.3.4 and (9.3.8), $g^{\prime}$ represents
$(h_{0}\sigma)^{\prime\prime}=(\alpha^{\prime\prime})^{*}(\sigma)^{\prime}\in
Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$ and so $g^{\prime}\cdot u$ is a
$d_{1}$-boundary , that is $g^{\prime}\cdot u=(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{K})g^{\prime}_{3}$ with $g^{\prime}_{3}\in[\Sigma^{tq}X,KG_{s}\wedge K]$.
Then $(\bar{b}_{s+1}\wedge
1_{K})f^{\prime}_{2}=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{K})g^{\prime}_{3}$ and
so $f^{\prime}_{2}=(\bar{c}_{s}\wedge
1_{K})g^{\prime}_{3}+(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}_{3}$ for some
$f^{\prime}_{3}\in[\Sigma^{tq+1}X,E_{s+2}\wedge K]$ and we have
$(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}=f^{\prime}_{2}w=(\bar{c}_{s}\wedge
1_{K})g^{\prime}_{3}w+(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}_{3}w$. Clearly,
$g^{\prime}_{3}w\in[\Sigma^{tq}K,KG_{s}\wedge K]$ is a $d_{1}$-cycle which
represents $Ext_{A}^{s,tq}(H^{*}K,H^{*}K)\cong Z_{p}\\{(\sigma)^{\prime}\\}$
(cf. Prop. 9.3.6). Then $g^{\prime}_{3}w=\sigma\wedge 1_{K}$ (up to scalar and
modulo $d_{1}$-boundary), where $\sigma\in\pi_{tq}KG_{s}\cong
Ext_{A}^{s,tq}(Z_{p},Z_{p})$. So we have $(\bar{a}_{s+1}\wedge
1_{K})f^{\prime}=(\bar{c}_{s}\wedge 1_{K})(\sigma\wedge
1_{K})+(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}_{3}w$ = $(\bar{a}_{s+1}\wedge
1_{K})(\kappa\wedge 1_{K})+(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}_{3}w$ , where
$\kappa\in\pi_{tq+1}E_{s+2}$ satisfying
$\bar{a}_{s+1}\kappa=\bar{c}_{s}\sigma$. It follows that
$f^{\prime}=\kappa\wedge 1_{K}+f^{\prime}_{3}w+(\bar{c}_{s+1}\wedge
1_{K})g^{\prime}_{4}$ for some
$g^{\prime}_{4}\in[\Sigma^{tq+1}K,KG_{s+1}\wedge K]$ and we have
$(\bar{c}_{s+1}\wedge
1_{K})(h_{0}\sigma)^{\prime\prime}=f^{\prime}\alpha^{\prime\prime}=(\kappa\wedge
1_{K})\alpha^{\prime\prime}=(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K}).$ This shows (1). The proof of (2) is similar. Q.E.D.
Proof of Theorem 9.3.9 At first, by the supposition of the main Theorem B on
$i^{\prime}_{*}i_{*}(h_{0}\sigma)\in Ext_{A}^{s+1,tq+q}(H^{*}K,Z_{p})$ in a
permanent cycle in the ASS we have $(\bar{c}_{s+1}\wedge
1_{K})(h_{0}\sigma\wedge 1_{K})$ = 0. The there exists
$\eta^{\prime}_{n,s+1}\in[\Sigma^{tq+q}K,E_{s+1}\wedge K]$ such that
$(\bar{b}_{s+1}\wedge 1_{K})\eta^{\prime}_{n,s+1}=(h_{0}\sigma\wedge 1_{K})$.
By Lemma 9.3.10, it suffices to prove
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})$ = 0. Note that,
by $\bar{a}_{s+1}\kappa=\bar{c}_{s}\sigma$ we have
$\bar{a}_{s+1}(1_{E_{s+2}}\wedge\alpha_{1})\kappa=\bar{c}_{s}(1_{KG_{s}}\wedge\alpha_{1})\sigma$
= 0 and so $(1_{E_{s+2}}\wedge\alpha_{1})\kappa=\bar{c}_{s+1}(h_{0}\sigma)$
(up to scalar), this is because $\pi_{tq+q}KG_{s+1}\cong
Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}\sigma\\}$. Then, by Lemma
9.3.10 we have
(9.3.11) $(1_{E_{s+2}}\wedge\alpha_{1}\wedge 1_{K})(\kappa\wedge
1_{K})=(\bar{c}_{s+1}\wedge 1_{K})(h_{0}\sigma\wedge 1_{K})$ = 0.
Moreover, by (9.1.20) we have
$\quad(1_{E_{s+2}}\wedge\rho\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge
1_{K})i^{\prime}$ $\quad=\quad(1_{E_{s+2}}\wedge\alpha^{\prime})(\kappa\wedge
1_{K})i^{\prime}$ = 0, Then, by (9.1.17),
$(1_{E_{s+2}}\wedge\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge
1_{K})i^{\prime}=(1_{E_{s+2}}\wedge(v\wedge 1_{M})\overline{m}_{M})f$ for some
$f\in[\Sigma^{tq+q-1}M,E_{s+2}\wedge K^{\prime}\wedge M]$ and so
$(1_{E_{s+2}}\wedge i^{\prime})f=(1_{E_{s+2}}\wedge\rho(1_{K^{\prime}}\wedge
ij)\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge
1_{K})i^{\prime}=(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})i^{\prime}=(1_{E_{s+2}}\wedge\alpha^{\prime})(\kappa\wedge
1_{K})i^{\prime}ij$ = 0. Hence $f=(1_{E_{s+2}}\wedge\alpha)f_{2}$ for some
$f_{2}\in[\Sigma^{tq-1}M,E_{s+2}\wedge M]$ and we have
$(1_{E_{s+2}}\wedge(x\wedge 1_{M})\alpha^{\prime}_{K^{\prime}\wedge
M})(\kappa\wedge 1_{K})i^{\prime}=(1_{E_{s+2}}\wedge(i^{\prime}\wedge
1_{M})\overline{m}_{M}\alpha)f_{2}$ = 0 and $(1_{E_{s+2}}\wedge(x\wedge
1_{M})\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge 1_{K})\rho(v\wedge
1_{M})=(1_{E_{s+2}}\wedge(x\wedge 1_{M})\alpha^{\prime}_{K^{\prime}\wedge
M})(\kappa\wedge 1_{K})\rho(vi\wedge 1_{M})m_{M}++(1_{E_{s+2}}\wedge(x\wedge
1_{M})\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge 1_{K})\rho(v\wedge
1_{M})\overline{m}_{M}(j\wedge 1_{M})$ = 0, this is because $\rho(v\wedge
1_{M})\overline{m}_{M}$ = 0 , $\rho(vi\wedge 1_{M})=i^{\prime}$. Then we have
(9.3.12) $\qquad\quad(1_{E_{s+2}}\wedge(x\wedge
1_{M})\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge
1_{K})\rho=f_{3}(y\wedge 1_{M})$
with $f_{3}\in[\Sigma^{tq+2q+1}M,E_{s+2}\wedge K\wedge M]\cap(kerd)$ (cf.
(9.1.15) and Cor. 6.4.15). It follows that
(9.3.13) $\qquad\quad(\bar{a}_{s+1}\wedge 1_{K\wedge M})f_{3}=f_{4}(\alpha
i\wedge 1_{M})$
with $f_{4}\in[\Sigma^{tq+q}M\wedge M,E_{s+1}\wedge K\wedge M]\cap(kerd)$(cf.
(9.1.15) and Cor. 6.4.15). Note that the $d_{1}$-cycle $(\bar{b}_{s+1}\wedge
1_{M})(1_{E_{s+1}}\wedge jj^{\prime}\wedge 1_{M})f_{4}\in[\Sigma^{tq-2}M\wedge
M,KG_{s+1}\wedge M]\cong Z_{p}\\{(\sigma^{\prime}\wedge 1_{M})ij(j\wedge
1_{M})\\}$, then $(\bar{b}_{s+1}\wedge 1_{M})(1_{E_{s+1}}\wedge
jj^{\prime}\wedge 1_{M})f_{4}=\lambda\cdot(\sigma^{\prime}\wedge
1_{M})ij(j\wedge 1_{M})$ and by applying the derivation $d$ we have
$\lambda\cdot(\sigma^{\prime}\wedge 1_{M})(j\wedge 1_{M})$ = 0 and this
implies that $\lambda$ = 0. That is to say $(\bar{b}_{s+1}\wedge
1_{M})(1_{E_{s+1}}\wedge jj^{\prime}\wedge 1_{M})f_{4}$ = 0 , then
$(\bar{b}_{s+1}\wedge 1_{K\wedge M})f_{4}=(1_{KG_{s+1}}\wedge x\wedge 1_{M})g$
with $d_{1}$-cycle $g\in[\Sigma^{tq+q}M\wedge M,E_{s+1}\wedge K^{\prime}\wedge
M]\cap(kerd)$( cf. Cor. 6.4.15).
By Theorem 6.4.3, $g=g(i\wedge 1_{M})m_{M}+g\overline{m}_{M}(j\wedge 1_{M})$.
Now we claim that $g(i\wedge 1_{M})=\lambda_{1}(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(h_{0}\sigma\wedge 1_{M})$ and
$g\overline{m}_{M}=\lambda_{2}(1_{KG_{s+1}}\wedge(v\wedge
1_{M})\overline{m}_{M})(h_{0}\sigma\wedge 1_{M})$ (mod $d_{1}$-boundary),
where $\lambda_{1},\lambda_{2}\in Z_{p}$.
To prove the claim, note that the $d_{1}$-cycle $g(i\wedge 1_{M})$ represents
an element $[g(i\wedge 1_{M})]\in Ext_{A}^{s+1,tq+q}(H^{*}K^{\prime}\wedge
M,H^{*}M)$ and $[(1_{KG_{s+1}}\wedge\rho)g(i\wedge 1_{M})]\in
Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$ $\cong Z_{p}\\{[(1_{KG_{s+1}}\wedge
i^{\prime})(h_{0}\sigma\wedge 1_{M})]\\}$ (cf. Prop. 9.3.2(4)). Then
$(1_{KG_{s+1}}\wedge\rho)g(i\wedge
1_{M})=\lambda_{1}(1_{KG_{s+1}}\wedge\rho(vi\wedge 1_{M}))(h_{0}\sigma\wedge
1_{M})+(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{K})g_{2}$ for some
$g_{2}\in[\Sigma^{tq+q}M,KG_{s}\wedge K]$. Since $(1_{KG_{s}}\wedge
j^{\prime}\alpha^{\prime})g_{2}$ = 0, then $g_{2}=(1_{KG_{s}}\wedge\rho)g_{3}$
with $g_{3}\in[\Sigma^{tq+q}M,KG_{s}\wedge K^{\prime}\wedge M]$. Then
$g(i\wedge 1_{M})$ $=\lambda_{1}(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(h_{0}\sigma\wedge 1_{M})+(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{K^{\prime}\wedge M})g_{3}+(1_{KG_{s+1}}\wedge\psi)g_{4}$ with
$g_{4}\in[\Sigma^{tq+q-1}M,KG_{s+1}\wedge M]\cong Z_{p}\\{(h_{0}\sigma\wedge
1_{M})ij\\}$ and so $g_{4}=\lambda^{\prime}(h_{0}\sigma\wedge 1_{M})ij$ for
some $\lambda^{\prime}\in Z_{p}$. However, $d(i\wedge 1_{M})=0$ and $d(g)$ = 0
this implies that $d(g(i\wedge 1_{M}))$ = 0, then, by applying the derivation
$d$ to the above equation we have
$(1_{KG_{s+1}}\wedge\psi)d(g_{4})+(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{K^{\prime}\wedge M})d(g_{3})=0$, that is
$\lambda^{\prime}(1_{KG_{s+1}}\wedge\psi)(h_{0}\sigma\wedge
1_{M})=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{K^{\prime}\wedge M})d(g_{3})$ and
this means that the scalar $\lambda^{\prime}$ = 0, this is because
$\psi_{*}[h_{0}\sigma\wedge 1_{M}]\neq 0\in
Ext_{A}^{s+1,tq+q+1}(H^{*}K^{\prime}\wedge M,H^{*}M)$(cf. Prop. 9.3.5). This
shows that $g(i\wedge 1_{M})=\lambda_{1}(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(h_{0}\sigma\wedge 1_{M})$ (mod $d_{1}$-boundary). In addition, by
$d(\overline{m}_{M})$ $\in[\Sigma^{2}M,M\wedge M]\cong[\Sigma^{2}M,M]+[\Sigma
M,M]$ = 0, then similarly we have
$g\overline{m}_{M}=\lambda_{2}(1_{KG_{s+1}}\wedge\psi)(h_{0}\sigma\wedge
1_{M})$ (mod $d_{1}$-boundary). This proves the above claim.
Then, modulo $d_{1}$-boundary we have
(9.3.14) $\qquad g=g(i\wedge 1_{M})m_{M}+g\overline{m}_{M}(j\wedge 1_{M})$
$=\lambda_{1}(1_{KG_{s+1}}\wedge vi\wedge 1_{M})(h_{0}\sigma\wedge
1_{M})m_{M}+\lambda_{2}(1_{KG_{s+1}}\wedge\psi)(h_{0}\sigma\wedge
1_{M})(j\wedge 1_{M})$
$=\lambda_{1}(h_{0}\sigma\wedge 1_{K^{\prime}\wedge M})(vi\wedge
1_{M})m_{M}+\lambda_{2}(h_{0}\sigma\wedge 1_{K^{\prime}\wedge M})(v\wedge
1_{M})\overline{m}_{M}(j\wedge 1_{M})$
We claim that
(9.3.15) The scalar in (9.3.14) $\lambda_{1}=\lambda_{2}$.
This will be proved in the last. Then , $g=\lambda_{1}(1_{KG_{s+1}}\wedge
v\wedge 1_{M})(h_{0}\sigma\wedge 1_{M}\wedge 1_{M})$ and so we have
$(\bar{b}_{s+1}\wedge 1_{K\wedge M})f_{4}=(1_{KG_{s+1}}\wedge x\wedge
1_{M})g=\lambda_{1}(1_{KG_{s+1}}\wedge i^{\prime}\wedge
1_{M})(h_{0}\sigma\wedge 1_{M}\wedge 1_{M})=\lambda_{1}(h_{0}\sigma\wedge
1_{K}\wedge 1_{M})(i^{\prime}\wedge 1_{M})+(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{K\wedge M})g_{5}=\lambda_{1}(\bar{b}_{s+1}\wedge 1_{K\wedge
M})(\eta^{\prime}_{n,s+1}\wedge 1_{M})(i^{\prime}\wedge
1_{M})+(\bar{b}_{2}\bar{c}_{1}\wedge 1_{K\wedge M})g_{5}$ and
$f_{4}=\lambda_{1}(\eta^{\prime}_{n,s+1}i^{\prime}\wedge
1_{M})+(\bar{c}_{s}\wedge 1_{K\wedge M})g_{5}+(\bar{a}_{s+1}\wedge 1_{K\wedge
M})f_{5}$ with $f_{5}\in[\Sigma^{tq+q+1}M\wedge M,E_{s+2}\wedge K\wedge M]$.
It follows that $(\bar{a}_{s+1}\wedge 1_{K\wedge M})f_{3}=f_{4}(\alpha i\wedge
1_{M})=(\bar{a}_{s+1}\wedge 1_{K\wedge M})f_{5}(\alpha i\wedge 1_{M})$ and so
$f_{3}=f_{5}(\alpha i\wedge 1_{M})+(\bar{c}_{s+1}\wedge 1_{K\wedge M})g_{6}$
for some $g_{6}\in[\Sigma^{tq+2q+1}M,E_{s+2}\wedge K\wedge M]$. So
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})\rho=(1_{E_{s+2}}\wedge(1_{K}\wedge j)(x\wedge
1_{M})\alpha^{\prime}_{K^{\prime}\wedge M})(\kappa\wedge
1_{K})\rho=(1_{E_{s+2}}\wedge 1_{K}\wedge j)f_{3}(y\wedge
1_{M})=(\bar{c}_{s+1}\wedge 1_{K})(1_{KG_{s+1}}\wedge 1_{K}\wedge
j)g_{6}(y\wedge 1_{M})$ = 0, this is because the $d_{1}$-cycle
$(1_{KG_{s+1}}\wedge 1_{K}\wedge j)g_{6}\in[\Sigma^{tq+2q}M,KG_{s+1}\wedge K]$
represents an element in $Ext_{A}^{s+1,tq+2q}(H^{*}K,H^{*}M)$ = 0 (cf. Prop.
9.3.2(1)).
It follows from (9.1.11) that
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})=f_{6}\alpha
ijj^{\prime}$ for some $f_{6}\in[\Sigma^{tq+q+1}M,E_{s+2}\wedge K]$ and
$(\bar{a}_{s+1}\wedge 1_{K})f_{6}\alpha ijj^{\prime}=(\bar{a}_{s+1}\wedge
1_{K})(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})=(\bar{c}_{1}\wedge
1_{K})(1_{KG_{s}}\wedge\alpha^{\prime\prime})(\sigma\wedge 1_{K})$ = 0. Then,
by (9.1.14) we have $(\bar{a}_{s+1}\wedge 1_{K})f_{6}\alpha i=f_{7}z$ with
$f_{7}\in[\Sigma^{tq+q-1}K^{\prime},E_{s+1}\wedge K]$. Moreover, by Prop.
9.1.21, $f_{7}z$ = 0, then $f_{6}\alpha i=(\bar{c}_{s+1}\wedge 1_{K})g_{7}$
for some $g_{7}\in\pi_{tq+2q+1}(KG_{s+1}\wedge K)$ and so
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})=f_{6}\alpha
ijj^{\prime}=(\bar{c}_{s+1}\wedge 1_{K})g_{7}jj^{\prime}$ = 0, this is because
the $d_{1}$-cycle $g_{7}\in\pi_{tq+2q+1}(KG_{s+1}\wedge K)$ reprresents an
element in $Ext_{A}^{s+1,p^{n}q+2q+1}(H^{*}K,Z_{p})$ = 0. This shows the
result of the Theorem and the remaining work is to prove the claim (9.3.15).
To prove (9.3.15), Note that by Theorem 6.4.3 and (9.1.15) we have $(v\wedge
1_{M})\overline{m}_{M}(\alpha_{1}\wedge 1_{M})=(v\wedge
1_{M})\overline{m}_{M}(j\wedge 1_{M})(\alpha i\wedge 1_{M})=-(v\wedge
1_{M})(i\wedge 1_{M})m_{M}(\alpha i\wedge 1_{M})=-(vi\wedge 1_{M})\alpha$.
Similarly we have $\alpha(j\overline{u}\wedge 1_{M})=-(\alpha_{1}\wedge
1_{M})m_{M}(\overline{u}\wedge 1_{M})$, where
$\overline{u}:Y\rightarrow\Sigma^{q+1}M$ and $v:\Sigma M\rightarrow
K^{\prime}$ are the map (9.1.5)(9.1.15).
Then, modulo $d_{1}$-boundary we have
$\quad(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(\widetilde{h_{0}\sigma})=-(1_{KG_{s+1}}\wedge v\wedge
1_{M})\overline{m}_{M}(h_{0}\sigma\wedge 1_{M})$
$\quad(\widetilde{h_{0}\sigma})(j\overline{u}\wedge 1_{M})=-(h_{0}\sigma\wedge
1_{M})m_{M}(\overline{u}\wedge 1_{M})$
where $\widetilde{h_{0}\sigma}\in[\Sigma^{tq+q+1}M,KG_{s+1}\wedge M]$ is a
$d_{1}$-cycle which represents $\alpha_{*}(\tilde{\sigma})\in
Ext_{A}^{s+1,tq+q+1}(H^{*}M,H^{*}M)$. So, by (9.3.14), modulo $d_{1}$-boundary
we have
$\quad g(\overline{u}\wedge 1_{M})$ $=\lambda_{1}(1_{KG_{s+1}}\wedge v\wedge
1_{M})(h_{0}\sigma\wedge 1_{M}\wedge 1_{M})(\overline{u}\wedge
1_{M})+(\lambda_{2}-\lambda_{1})$
$\quad\qquad\qquad\qquad(1_{KG_{s+1}}\wedge v\wedge
1_{M})\overline{m}_{M}(h_{0}\sigma\wedge 1_{M})(j\overline{u}\wedge 1_{M})$
$\qquad\quad\qquad=(\lambda_{1}-\lambda_{2})(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(\widetilde{h_{0}\sigma})(j\overline{u}\wedge 1_{M})$
this is because $(1_{KG_{s+1}}\wedge v)(h_{0}\sigma\wedge
1_{M})\overline{u}=(1_{KG_{s+1}}\wedge
v)[\widetilde{h_{0}\sigma})ij+(1_{KG_{s+1}}\wedge
ij)(\widetilde{h_{0}\sigma})\overline{u}$ = 0 (mod $d_{1}$-boundary). On the
other hand, modulo $d_{1}$-boundary we have
$\quad g(\overline{u}\wedge 1_{M})=\lambda_{2}(1_{KG_{s+1}}\wedge v\wedge
1_{M})(h_{0}\sigma\wedge 1_{M}\wedge 1_{M})(\overline{u}\wedge
1_{M})+(\lambda_{1}-$
$\qquad\qquad\qquad\quad\lambda_{2})(1_{KG_{s+1}}\wedge vi\wedge
1_{M})(h_{0}\sigma\wedge 1_{M})m_{M}(\overline{u}\wedge 1_{M})$
$\quad\qquad\qquad=(\lambda_{2}-\lambda_{1})(1_{KG_{s+1}}\wedge vi\wedge
1_{M})\widetilde{h_{0}\sigma}(j\overline{u}\wedge 1_{M})$.
Moreover, $(1_{KG_{s+1}}\wedge vi\wedge
1_{M})\widetilde{h_{0}\sigma}(j\overline{u}\wedge 1_{M})$ represents an
nonzero element in the Exr group, this is because
$(1_{KG_{s+1}}\wedge(1_{K}\wedge i)(x\wedge 1_{M}))(1_{KG_{s+1}}\wedge
vi\wedge 1_{M})\widetilde{h_{0}\sigma}(j\overline{u}\wedge
1_{M})(\overline{r}\wedge 1_{M})(1_{K}\wedge i)=(1_{KG_{s+1}}\wedge
i^{\prime}ij)\widetilde{h_{0}\sigma}ijj^{\prime}=(1_{KG_{s+1}}\wedge
i^{\prime})(h_{0}\sigma\wedge 1_{M})ijj^{\prime}$ represents a nonzero element
in the Ext group. Then, by comparison to the above two equations we have
$\lambda_{1}-\lambda_{2}=\lambda_{2}-\lambda_{1}$ so that
$\lambda_{1}=\lambda_{2}$. This shows the claim (9.3.15). Q.E.D.
Remark In the last of section 4, we will also give another proof of Theorem
9.3.9.
Proof of the main Theorem B By Theorem 9.3.9, there exists
$(\eta_{n,s+1})^{\prime\prime}\in[\Sigma^{tq+q-1}K,E_{s+1}\wedge K]$ such that
$(\bar{b}_{s+1}\wedge
1_{K})(\eta_{n,s+1})^{\prime\prime}=(h_{0}\sigma)^{\prime\prime}\in[\Sigma^{tq+q-1}K,KG_{s+1}\wedge
K]$. Let $(\eta_{n})^{\prime\prime}=(\bar{a}_{0}\cdots\bar{a}_{s}\wedge
1_{K})(\eta_{n,s+1})^{\prime\prime}\in[\Sigma^{tq+q-s-2}K,\\\ K]$ and consider
the map
$(\eta_{n})^{\prime\prime}\beta i^{\prime}i\in\pi_{tq+pq+2q-s-2}K$
where $\beta\in[\Sigma^{(p+1)q}K,K]$ is the known second periodicity element
which has filtration 1. Since $(\eta_{n})^{\prime\prime}$ is represented by
$(h_{0}\sigma)^{\prime\prime}\in Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)$ in the
ASS, then $(\eta_{n})^{\prime\prime}\beta i^{\prime}i\in\pi_{tq+pq+2q-s-2}K$
is represented by $(\beta i^{\prime}i)^{*}(h_{0}\sigma)^{\prime\prime}\\\
=(\beta
i^{\prime}i)^{*}(\alpha^{\prime\prime})^{*}(\sigma)^{\prime}=\alpha^{\prime\prime}_{*}\beta_{*}(i^{\prime}i)_{*}(\sigma)\in
Ext_{A}^{s+2,tq+pq+2q}(H^{*}K,Z_{p})$. By [14] Theorem 3.2 and [15] Theorem
5.2 we know that $\alpha^{\prime\prime}\beta i^{\prime}i\in\pi_{pq+2q-2}K$ is
represented by
$\alpha^{\prime\prime}_{*}\beta_{*}(i^{\prime}i)_{*}(1)=(i^{\prime}i)_{*}(g_{0})\in
Ext_{A}^{2,pq+2q}(H^{*}K,Z_{p})$(up to nonzero scalar) in the ASS so that
$(\eta_{n})^{\prime\prime}\beta i^{\prime}i$ is represented by
$\alpha^{\prime\prime}_{*}\beta_{*}(i^{\prime}i)_{*}(\sigma)=(i^{\prime}i)_{*}(g_{0}\sigma)\in
Ext_{A}^{s+2,tq+pq+2q}(H^{*}K,Z_{p})$. Q.E.D.
Using the stronger result of the main Theorem A which is stated in the Remark
9.2.35, the result of the main Theorem B also can be obtained by the following
main Theorem B’.
The main Theorem B’ Let $\sigma\in
Ext_{A}^{s,tq}(Z_{p},Z_{p}),\sigma^{\prime}\in Ext_{A}^{s+1,tq}(Z_{p},\\\
Z_{p})$ be a pair of $a_{0}$-related elements, that is, there is a secondary
differential $d_{2}(\sigma)=a_{0}\sigma^{\prime}$. Suppose that all the
supposition of the main Theorem A hold, then
$(i^{\prime}i)_{*}(g_{0}\sigma)\in
Ext_{A}^{s+2,tq+pq+2q}(H^{*}K,Z_{p}),(i^{\prime}i)_{*}(g_{0}\sigma^{\prime})\in
Ext_{A}^{s+3,tq+pq+2q}\\\ (H^{*}K,Z_{p})$ are permanent cycles in the ASS.
Proof Let
$\phi\sigma\in\pi_{tq+2q}KG_{s+1},\phi\sigma^{\prime}\in\pi_{tq+2q}KG_{s+2}$
be $d_{1}$-cycles which represent $\phi_{*}(\sigma)\in
Ext_{A}^{s+1,tq+2q}(H^{*}L,Z_{p}),\phi_{*}(\sigma^{\prime})\in
Ext_{A}^{s+2,tq+2q}(H^{*}L,Z_{p})$ respectively. By the stronger result of the
main Theorem A (cf. Remark 9.2.35) we have $(\bar{c}_{s+2}\wedge
1_{L})\phi\sigma^{\prime}=0,(\bar{c}_{s+1}\wedge 1_{L\wedge
M})(1_{KG_{s+1}}\wedge 1_{L}\wedge i)\phi\sigma=0$. Then $(\bar{c}_{s+2}\wedge
1_{L\wedge K})(\phi\sigma^{\prime}\wedge 1_{K})$ = 0 and by using the
multiplication of the ring spectrum $K$ we have $(\bar{c}_{s+1}\wedge
1_{L\wedge K})(\phi\sigma)\wedge 1_{K})=0$. In addition,
$(h_{0}\sigma)^{\prime\prime}=(1_{KG_{s+1}}\wedge\overline{\Delta})(\phi\sigma\wedge
1_{K}),(h_{0}\sigma^{\prime})^{\prime\prime}=(1_{KG_{s+2}}\wedge\overline{\Delta})(\phi\sigma^{\prime}\wedge
1_{K})$, this is because $\overline{\Delta}(\phi\wedge
1_{K})=\alpha^{\prime\prime}\in[\Sigma^{q-2}K,K]$. Then we have
$(\bar{c}_{s+1}\wedge
1_{K})(h_{0}\sigma)^{\prime\prime}=0,(\bar{c}_{s+2}\wedge
1_{K})(h_{0}\sigma^{\prime})^{\prime\prime}=0$. The remaining steps is
similiar to that given in the proof of the main Theorem B. Q.E.D.
§4. A general result on pull back convergence of $h_{0}\sigma$
In this section, we will prove that, under some suppositions, the convergence
of the element $(1_{L}\wedge i)_{*}\phi_{*}(\sigma)\in
Ext_{A}^{s+1,tq+2q}(H^{*}L\wedge M,Z_{p})$ can be pull backed to obtain the
convergence of $h_{0}\sigma\in Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})$ in the stable
homotopy groups of spheres. We have the following main Theorem.
The main Theorem C ( generalization of [24] Theorem A) Let $p\geq 5,s\leq 4$
and suppose that
(I)(a) $Ext_{A}^{s,tq}(Z_{p},Z_{p})\cong Z_{p}\\{\sigma\\}$,
$Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}\sigma\\}$
$Ext_{A}^{s+2,tq+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}\sigma\\}$ satisfying $a_{0}^{2}\sigma\neq 0$.
(b) $Ext_{A}^{s+1,tq+u}(Z_{p},Z_{p})\cong Z_{p}\\{a_{0}\sigma\\}$ for $u=1$
and is zero for $u=2,3$.
$Ext_{A}^{s+1,tq}(Z_{p},Z_{p})$ is zero or has (one or two) generator
$\sigma^{\prime}$ such that (both) satisfies
$h_{0}\sigma^{\prime}\neq 0,a_{0}\sigma^{\prime}\neq 0$,
$Ext_{A}^{s+1,tq+rq+u}(Z_{p},Z_{p})$ = 0 for $r=-1,2,3,u=-2,-1,0,1,2,3$ or
for $r=1,u=-2,-1,1,2,3$
(c) $Ext_{A}^{s,tq+u}(Z_{p},Z_{p})$ = 0 for $u=-1,1,2,3$
$Ext_{A}^{s,tq+rq+u}(Z_{p},Z_{p})$ = 0 for $r=-2,-1,1,2,u=-2,-1,0,1,2,3$
(II) $(1_{L}\wedge i)_{*}(\phi)_{*}(\sigma)\in
Ext_{A}^{s+1,tq+2q}(H^{*}L\wedge M,Z_{p})$ is a permanent cycle in the ASS,
then $(\alpha i)_{*}(\sigma)\in Ext_{A}^{s+1,tq+q+1}(H^{*}M,Z_{p})$ also is a
permanent cycle in tha ASS so that $h_{0}\sigma=j_{*}(\alpha i)_{*}(\sigma)\in
Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})$ converges to an element in $\pi_{tq+q-s-1}S$
of order $p$.
Note that the supposition (I) of the main Thoerem C contains the supposition I
of the main Thoerem B, then some results on Ext groups in $\S 3$ also hold
under the supposition of the main Theorem C. Before proving the main Theorem
C, we first recall the properties of some spectra related to $K$ and $M$ and
prove some results on low dimensional Ext groups.
By (9.1.27), $((1_{Y}\wedge j)\alpha_{Y\wedge M}\wedge
1_{M})\overline{m}_{M}=\alpha_{Y\wedge M}$ , (9.2.12) and the following
homotopy commutative diagram of $3\times 3$-Lemma
$\qquad\quad X\wedge M\stackrel{{\scriptstyle m_{M}(\tilde{\psi}\wedge
1_{M})}}{{\longrightarrow}}\Sigma^{2q}M\qquad\stackrel{{\scriptstyle
0}}{{\longrightarrow}}\qquad\Sigma^{2q+2}M$
$\qquad\qquad\quad\searrow^{\tilde{\psi}\wedge 1_{M}}\quad\nearrow
m_{M}\qquad\searrow^{(\phi\wedge
1_{K})i^{\prime}}\nearrow_{j^{\prime}(j^{\prime\prime}\wedge
1_{K})}\quad\searrow\overline{m}_{M}$
(9.4.1) $\Sigma^{2q}M\quad\qquad\qquad\qquad\Sigma L\wedge
K\quad\qquad\qquad\Sigma^{2q+2}M\wedge M$
$\qquad\qquad\quad\nearrow\overline{m}_{M}\quad\searrow^{(1_{Y}\wedge
j)\alpha_{Y\wedge M}\wedge 1_{M}}\nearrow\quad\qquad\searrow
u^{\prime}\quad\nearrow\widetilde{\psi}\wedge 1_{M}$
$\qquad\quad\Sigma^{2q+1}M\quad\stackrel{{\scriptstyle\alpha_{Y\wedge
M}}}{{\longrightarrow}}\quad Y\wedge
M\quad\stackrel{{\scriptstyle\tilde{u}w_{2}\wedge
1_{M}}}{{\longrightarrow}}\quad\Sigma X\wedge M$
we know that the cofibre of $m_{M}(\widetilde{\psi}\wedge 1_{M}):X\wedge
M\rightarrow\Sigma^{2q}M$ is $\Sigma L\wedge K$ given by the following
cofibration
(9.4.2) $X\wedge M\stackrel{{\scriptstyle m_{M}(\tilde{\psi}\wedge
1_{M})}}{{\longrightarrow}}\Sigma^{2q}M\stackrel{{\scriptstyle(\phi\wedge
1_{K})i^{\prime}}}{{\longrightarrow}}\Sigma L\wedge K\stackrel{{\scriptstyle
u^{\prime}}}{{\longrightarrow}}\Sigma X\wedge M$
Since $(1_{L}\wedge i^{\prime})(\phi\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})$ = 0, then by $[\Sigma^{-q-1}X\wedge M,L\wedge M]\cap(kerd)\cong
Z_{p}\\{u^{\prime\prime}\wedge 1_{M}\\}$ and (9.1.2) we have $(\phi\wedge
1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})=(1_{L}\wedge\alpha)(u^{\prime\prime}\wedge 1_{M})$ (up to nonzero
scalar). Since $(\phi\wedge 1_{K})i^{\prime}\alpha$ = 0, then by (9.4.2),
there exists $\alpha_{X\wedge M}\in[\Sigma^{3q}M,X\wedge M]$ such that
$m_{M}(\widetilde{\psi}\wedge 1_{M})\alpha_{X\wedge M}=\alpha$. In addition,
$m_{M}(\widetilde{\psi}\wedge 1_{M})\alpha_{X\wedge
M}m_{M}(\widetilde{\psi}\wedge 1_{M})=\alpha m_{M}(\widetilde{\psi}\wedge
1_{M})=m_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge\alpha)$ so that by
(9.4.2) we have $\alpha_{X\wedge M}m_{M}(\widetilde{\psi}\wedge
1_{M})=1_{X}\wedge\alpha$ modulo $(u^{\prime})_{*}[\Sigma^{q}X\wedge M,L\wedge
K]$ = 0 , this is because $[\Sigma^{q}L\wedge K,L\wedge K]$ = 0 and
$[\Sigma^{3q}M,L\wedge K]$ = 0. Concludingly we have
(9.4.3) $(\phi\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})=(1_{L}\wedge\alpha)(u^{\prime\prime}\wedge 1_{M}),$
$\qquad\quad\alpha_{X\wedge M}m_{M}(\widetilde{\psi}\wedge
1_{M})=1_{X}\wedge\alpha$
The cofibre of the map $\alpha_{X\wedge M}:\Sigma^{3q}M\rightarrow X\wedge M$
is $W\wedge K$ given by the cofibration
(9.4.4) $\Sigma^{3q}M\stackrel{{\scriptstyle\alpha_{X\wedge
M}}}{{\longrightarrow}}X\wedge M\stackrel{{\scriptstyle\mu_{X\wedge
M}}}{{\longrightarrow}}W\wedge K\stackrel{{\scriptstyle
j^{\prime}(j^{\prime\prime}u\wedge 1_{K})}}{{\longrightarrow}}\Sigma^{3q+1}M$
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\qquad\Sigma^{3q}M\quad\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\quad\Sigma^{2q}M\quad\stackrel{{\scriptstyle(\phi\wedge
1_{K})i^{\prime}}}{{\longrightarrow}}\quad\Sigma L\wedge K$
$\qquad\qquad\quad\searrow\alpha_{X\wedge M}\nearrow_{m_{M}(\tilde{\psi}\wedge
1_{M})}\searrow i^{\prime}\quad\nearrow(\phi\wedge 1_{K})$
(9.4.5)$\qquad\qquad\quad X\wedge M\qquad\qquad\quad\Sigma^{2q}K$
$\qquad\qquad\quad\nearrow u^{\prime}\quad\searrow\mu_{X\wedge M}\quad\nearrow
j^{\prime\prime}u\wedge 1_{K}\searrow j^{\prime}$
$\qquad\qquad L\wedge K\quad\stackrel{{\scriptstyle w\wedge
1_{K}}}{{\longrightarrow}}\quad W\wedge K\stackrel{{\scriptstyle
j^{\prime}(j^{\prime\prime}u\wedge
1_{K})}}{{\longrightarrow}}\quad\Sigma^{3q+1}M$
By (9.2.13), $ijm_{M}(\widetilde{\psi}\cdot\widetilde{u}\wedge
1_{M})=ij(u_{2}\wedge 1_{M})=(u_{2}\wedge 1_{M})(1_{U}\wedge
ij)=m_{M}(\widetilde{\psi}\cdot\widetilde{u}\wedge 1_{M})(1_{U}\wedge
ij)=m_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge ij)(\widetilde{u}\wedge
1_{M})$, then we have $ijm_{M}(\widetilde{\psi}\wedge
1_{M})=m_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge
ij)+\lambda(j\widetilde{\psi}\wedge 1_{M})$ for some $\lambda\in Z_{p}$. It
follows that $\lambda j(j\widetilde{\psi}\wedge
1_{M})=-jm_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge
ij)=-j\widetilde{\psi}(1_{X}\wedge j)=j(j\widetilde{\psi}\wedge 1_{M})$ and so
$\lambda=1$. In addition, $i^{\prime}(\alpha_{1}\wedge
1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})=(j^{\prime\prime}\wedge
1_{K})(1_{L}\wedge i^{\prime})(\phi\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})$ = 0, then by (9.1.23) we have $m_{M}(\widetilde{\psi}\wedge
1_{M})=m_{M}(\overline{u}\wedge 1_{M})\psi_{X\wedge M}$ , where $\psi_{X\wedge
M}\in[\Sigma^{-q+1}X\wedge M,Y\wedge M]$. In addition, $[\Sigma^{-q+1}X\wedge
M,Y\wedge M]\cong Z_{p}\\{\psi_{X\wedge M}\\}$ , this can be obtained from
$[\Sigma^{-2q}X\wedge M,M]\cong Z_{p}\\{m_{M}(\widetilde{\psi}\wedge
1_{M})\\}$ , (9.1.23) and $[\Sigma^{-q}X\wedge M,K]$ = 0. Then, by
$j^{\prime}(j^{\prime\prime}u\wedge 1_{K})\cdot\mu_{X\wedge M}$ = 0 and
(9.1.27) we have $(u\wedge 1_{K})\mu_{X\wedge
M}=\overline{\mu}_{2}(1_{Y}\wedge i^{\prime})\psi_{X\wedge M}$ (up to nonzero
scalar). Concludingly we have
(9.4.6) $j\widetilde{\psi}\wedge 1_{M}=ijm_{M}(\widetilde{\psi}\wedge
1_{M})-m_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge ij)$,
$\qquad(u\wedge 1_{K})\mu_{X\wedge M}=\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime})\psi_{X\wedge M}$ (up to nonzero scalar)
$\qquad[\Sigma^{-q+1}X\wedge M,Y\wedge M]\cong Z_{p}\\{\psi_{X\wedge M}\\},$
$\qquad\quad m_{M}(\overline{u}\wedge 1_{M})\psi_{X\wedge
M}=m_{M}(\widetilde{\psi}\wedge 1_{M})$,
By the following homotopy commutative diagram of $3\times 3$-Lemma
$\qquad\qquad L\wedge K\quad\stackrel{{\scriptstyle(1_{X}\wedge
j)u^{\prime}}}{{\longrightarrow}}\quad\Sigma X\qquad\stackrel{{\scriptstyle
1_{X}\wedge p}}{{\longrightarrow}}\qquad\Sigma X$
$\qquad\qquad\quad\searrow u^{\prime}\qquad\nearrow 1_{X}\wedge
j\qquad\searrow\omega\qquad\nearrow\widetilde{u}w_{2}$
$X\wedge M\qquad\qquad\qquad\qquad Y$
$\qquad\qquad\quad\nearrow 1_{X}\wedge
i\quad\searrow^{m_{M}(\tilde{\psi}\wedge 1_{M})}\nearrow_{(1_{Y}\wedge
j)\alpha_{Y\wedge M}}\searrow^{\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}i)}$
$\qquad\qquad
X\qquad\stackrel{{\scriptstyle\tilde{\psi}}}{{\longrightarrow}}\qquad\Sigma^{2q}M\qquad\stackrel{{\scriptstyle(\phi\wedge
1_{K})i^{\prime}}}{{\longrightarrow}}\qquad\Sigma L\wedge K$
we know that the cofibre of $(1_{X}\wedge j)u^{\prime}:L\wedge
K\rightarrow\Sigma X$ is $Y$ given by the cofibration
(9.4.7) $L\wedge K\stackrel{{\scriptstyle(1_{X}\wedge
j)u^{\prime}}}{{\longrightarrow}}\Sigma
X\stackrel{{\scriptstyle\omega}}{{\longrightarrow}}Y\stackrel{{\scriptstyle\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime}i)}}{{\longrightarrow}}\Sigma L\wedge K$
In addition, by the commutativity of the above rectangle we have
(9.4.8) $\omega\wedge 1_{M}=\alpha_{Y\wedge M}m_{M}(\widetilde{\psi}\wedge
1_{M})$.
Proposition 9.4.9 Under the supposition (I) of the main Thoerem C we have
(1) $Ext_{A}^{s+1,tq+r}(H^{*}K,H^{*}M)$ = 0 for $r=1,2$,
(2) $Ext_{A}^{s+1,tq+rq+1}(H^{*}K,H^{*}K)$ = 0 for $r=-1,0,1,2$.
Proof: (1) By the supposition, $Ext_{A}^{s+1,tq-q+r}(Z_{p},Z_{p})$ = 0 for
$r=-1,0,1,\\\ 2,3$, then
$(j^{\prime})_{*}Ext_{A}^{s+1,tq+r}(H^{*}K,Z_{p})\subset Ext_{A}^{s+1,tq-
q-r-1}(H^{*}M,Z_{p})$ = 0 for $r=1,2,3$ and so
$Ext_{A}^{s+1,tq+r}(H^{*}K,Z_{p})=(i^{\prime})_{*}Ext_{A}^{s+1,tq+r}(H^{*}M,Z_{p})$
= 0 for $r=1,2,3$ (cf. Prop. 9.3.0(1)) and the result follows.
(2) Consider the following exact sequence ($r=-1,0,1,2$)
$0=Ext_{A}^{s+1,tq+(r+1)q+2}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+rq+1}(H^{*}K,H^{*}K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+rq+1}(H^{*}K,H^{*}M)$
induced by (9.1.2). The right group is zero for $r=0,1,2$( cf. (1) and Prop.
9.3.2(1)(2)) and also is zero for $r=-1$ which is obtained by the supposition
on $Ext_{A}^{s+1,tq-q+r}(Z_{p},Z_{p})$ = 0 for $r=-1,0,1,2$. The left group is
zero for $r=-1,0,1$ (cf. (1) and Prop. 9.3.2). The left group also is zero for
$r=2$, this is because $Ext_{A}^{s+1,tq+rq+u}(Z_{p},Z_{p})$ = 0 for
$r=2,3,u=0,1,2,3$ by the supposition. Then the middle group is zero as
desired. Q.E.D.
Proposition 9.4.10 Under the supposition (I) of the main Theorem C we have
(1) $Ext_{A}^{s+1,tq+q+1}(H^{*}W\wedge K,H^{*}X\wedge M)$ = 0.
(2) $Ext_{A}^{s+1,tq+2q+1}(H^{*}Y,H^{*}M)\cong Z_{p}\\{((1_{Y}\wedge
j)\alpha_{Y\wedge M})_{*}(\tilde{\sigma})\\}$,
$Ext_{A}^{s+1,tq+q}(H^{*}Y,H^{*}Y)\cong
Z_{p}\\{(\overline{u})^{*}((1_{Y}\wedge j)\alpha_{Y\wedge
M})_{*}(\tilde{\sigma})\\}$,
(3) $Ext_{A}^{s+1,tq+3q}(H^{*}X,H^{*}M)\cong Z_{p}\\{((1_{X}\wedge
j)\alpha_{X\wedge M})_{*}(\tilde{\sigma})\\}$
Proof: (1) Consider the following exact sequence
$0=Ext_{A}^{s+1,tq+3q+1}(H^{*}W\wedge K,H^{*}M)\stackrel{{\scriptstyle
m_{M}(\tilde{\psi}\wedge
1_{M})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}W\wedge
K,H^{*}X\wedge
M)\stackrel{{\scriptstyle(u^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}W\wedge
K,H^{*}L\wedge K)$
induced by (9.4.2). The right group is zero by Prop. 9.4.9(2) and
(9.1.12)(9.1.3). The left group also is zero by
$Ext_{A}^{s+1,tq+rq+1}(H^{*}K,H^{*}M)$ = 0(for $r=1,2,3$)(cf. the proof of
Prop. 9.4.9(2)) . Then the middle group is zero as desired.
(2) Since $\overline{u}(1_{Y}\wedge j)\alpha_{Y\wedge
M}\in[\Sigma^{q-1}M,M]\cong Z_{p}\\{ij\alpha,\alpha ij\\}$, then
$\overline{u}(1_{Y}\wedge j)\alpha_{Y\wedge
M}=\lambda_{1}ij\alpha+\lambda_{1}\alpha ij$ where the scalar
$\lambda_{1},\lambda_{2}\in Z_{p}$ satisfy $\lambda_{1}j\alpha
ij\alpha+\lambda_{2}j\alpha^{2}ij$ = 0. Consider the following exact sequence
$Ext_{A}^{s+1,tq+2q}(Z_{p},H^{*}M)\stackrel{{\scriptstyle(\overline{w})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+2q+1}(H^{*}Y,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle(\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle(j\alpha)_{*}}}{{\longrightarrow}}$
induced by (9.1.5). The left group is zero which can be obtained by the
supposition on $Ext_{A}^{s+1,tq+2q+k}(Z_{p},Z_{p})$ = 0 (for $k=0,1$). By
Prop. 9.3.1(2), the right group has two generators
$(ij)_{*}\alpha_{*}(\tilde{\sigma})$ and $\alpha_{*}(ij)_{*}(\tilde{\sigma})$
. Then $(\overline{u})_{*}\\\ Ext_{A}^{s+1,tq+2q+1}(H^{*}Y,H^{*}M)$ has unique
generator $(\overline{u})_{*}((1_{Y}\wedge j)\alpha_{Y\wedge
M})_{*}(\tilde{\sigma})$ so that the first result follows. For the second
result, consider the following exact sequence
$Ext_{A}^{s+1,tq+q}(Z_{p},Z_{p})\stackrel{{\scriptstyle(\overline{w})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq+q+1}(H^{*}Y,Z_{p})$
$\qquad\quad\stackrel{{\scriptstyle(\overline{u})_{*}}}{{\longrightarrow}}Ext_{A}^{s+1,tq}(H^{*}M,Z_{p})\stackrel{{\scriptstyle(j\alpha)_{*}}}{{\longrightarrow}}$
induced by (9.1.5). By the supposition, the left group has unique generator
$h_{0}\sigma=(j\alpha i)_{*}(\sigma)$ so that im $(\overline{w})_{*}$ = 0. The
right group is zero or has (one or two) generator $i_{*}(\sigma^{\prime})$
such that $(j\alpha)_{*}i_{*}(\sigma^{\prime})=h_{0}\sigma^{\prime}\neq 0$.
Then the middle group is zero and so the second result follows.
(3) Since $\widetilde{\psi}(1_{X}\wedge j)\alpha_{X\wedge
M}\in[\Sigma^{q-1}M,M]\cong Z_{p}\\{ij\alpha,\alpha ij\\}$, then
$\widetilde{\psi}(1_{X}\wedge j)\alpha_{X\wedge
M}=\lambda_{3}ij\alpha+\lambda_{4}\alpha ij$, where the scalar
$\lambda_{3},\lambda_{4}\in Z_{p}$ satisfy $\lambda_{3}(1_{Y}\wedge
j)\alpha_{Y\wedge M}ij\alpha+\lambda_{4}(1_{Y}\wedge j)\alpha_{Y\wedge
M}\alpha ij$ = 0. Then, similar to that in (2), $(\widetilde{\psi})_{*}\\\
Ext_{A}^{s+1,tq+3q}(H^{*}X,H^{*}M)$ has unique generator
$(\widetilde{\psi})_{*}((1_{X}\wedge j)\alpha_{X\wedge
M})_{*}(\tilde{\sigma})$ so that $Ext_{A}^{s+1,tq+3q}(H^{*}X,H^{*}M)$ has
unique generator $((1_{X}\wedge j)\alpha_{X\wedge M})_{*}(\tilde{\sigma})$ ,
this is because $Ext_{A}^{s+1,tq+3q+1}(H^{*}Y,H^{*}M)$ = 0 which can be
obtained by the supposition (I)(b) on $Ext_{A}^{s+1,tq+rq+k}(Z_{p},Z_{p})$ = 0
for $r=1,2,k=-1,0,1,2$). Q.E.D.
Proposition 9.4.11 Under the supposition (I) of the main Theorem C we have
(1) $Ext_{A}^{s,tq-2q}(H^{*}M,H^{*}X\wedge M)\cong
Z_{p}\\{m_{M}(\widetilde{\psi}\wedge 1_{M})^{*}(\tilde{\sigma})\\}$.
(2) $Ext_{A}^{s,tq+rq+u}(H^{*}K,H^{*}M)$ = 0 for $r=-1,1,2,3,u=0,1,2$
$Ext_{A}^{s,tq}(H^{*}K,H^{*}K)\cong Z_{p}\\{\sigma_{K}\\}$ satisfying
$(i^{\prime})^{*}(\sigma_{K})=(i^{\prime})_{*}(\tilde{\sigma})$,
(3) $Ext_{A}^{s,tq}(H^{*}L\wedge K,H^{*}L\wedge K)\cong
Z_{p}\\{\sigma_{L\wedge K}\\}$
satisfying $(j^{\prime\prime}\wedge 1_{K})_{*}(\sigma_{L\wedge
K})=(j^{\prime\prime}\wedge 1_{K})^{*}(\sigma_{K})$
$Ext_{A}^{s,tq+rq+u}(H^{*}L\wedge K,H^{*}M)$ = 0 for $r=1,2,3,u=0,1,2$,
(4) $Ext_{A}^{s,tq+rq+u}(H^{*}W\wedge K,H^{*}M)$ = 0 for $r=1,2,3,u=0,1,2$,
$Ext_{A}^{s,tq+q}(H^{*}W\wedge K,H^{*}X\wedge M)$ = 0
Proof: (1) Consider the following exact sequence
$Ext_{A}^{s,tq}(H^{*}M,H^{*}M\wedge
M)\stackrel{{\scriptstyle(\tilde{\psi}\wedge
1_{M})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-2q}(H^{*}M,H^{*}X\wedge M)$
$\qquad\quad\stackrel{{\scriptstyle(\tilde{u}w_{2}\wedge
1_{M})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-2q}(H^{*}M,H^{*}Y\wedge M)$
induced by (9.2.12). By the suppopsition on $Ext_{A}^{s,tq-rq+u}(Z_{p},Z_{p})$
= 0 with ($r=1,2$, $u=0,1,2$) and the degree of the top cell of $Y\wedge M$ is
$q+3$ we know that the right group is zero. Since
$(\overline{m}_{M})^{*}Ext_{A}^{s,tq}(H^{*}M,H^{*}M\wedge M)\subset
Ext_{A}^{s,tq+1}(H^{*}M,H^{*}M)$ = 0 ( cf. Prop. 9.3.0(2)), then the left
group has unique generator $(m_{M})^{*}(\tilde{\sigma})$ and so the result
follows.
(2) Consider the following exact sequence ($r=-1,1,2,3,u=0,1,2$)
$Ext_{A}^{s,tq+rq+u}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle(i^{\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+rq+u}(H^{*}K,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle(j^{\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+(r-1)q+u-1}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle\alpha_{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero for $r=-1,1,2,3,u=0,1,2$, this is
obtained from the supposition I(c) on $Ext_{A}^{s,tq+rq+k}(Z_{p},Z_{p})$ = 0
(for $r=-1,1,2,3,k=-1,0,1,2,3$). By the supposition and Prop. 9.3.0(2),the
right group is zero except for $r=1,u=0,1$ it has unique generator
$(ij)_{*}(\tilde{\sigma})$ or $\tilde{\sigma}$ respectively. However, it
satisfies $\alpha_{*}(ij)_{*}(\tilde{\sigma})\neq 0$,
$\alpha_{*}(\tilde{\sigma})\neq 0$ then, the middle group is zero as desired.
Consider the following exact sequence
$0=Ext_{A}^{s,tq+q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}K,H^{*}K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero as shown above. The right group has
unique generator $(i^{\prime})_{*}(\tilde{\sigma})$ , this is because
$(j^{\prime})_{*}Ext_{A}^{s,tq}\\\ (H^{*}K,H^{*}M)\subset
Ext_{A}^{s,tq-q-1}(H^{*}M,H^{*}M)$ = 0 and $Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\cong
Z_{p}\\{\tilde{\sigma}\\}$. Then the middle has unique generatot $\sigma_{K}$
as desired.
(3) Consider the following exact sequence ($r=-1,0$)
$Ext_{A}^{s,tq+(r+1)q}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle(j^{\prime\prime}\wedge
1_{K})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+rq}(H^{*}K,H^{*}L\wedge K)$
$\qquad\quad\stackrel{{\scriptstyle(i^{\prime\prime}\wedge
1_{K})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+rq}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle(\alpha_{1}\wedge
1_{K})^{*}}}{{\longrightarrow}}$
induced by (9.1.3). The left group is zero for $r=0$ , this is because by (2)
$(i^{\prime})^{*}Ext_{A}^{s,tq+q}(H^{*}K,H^{*}K)\subset
Ext_{A}^{s,tq+q}(H^{*}K,H^{*}M)$ = 0 and $Ext_{A}^{s,tq+2q+1}\\\
(H^{*}K,H^{*}M)$ = 0. Moreover, by (2), the left group has unique generator
$\sigma_{K}$ for $r=-1$. The right group is zero $r=-1$, this is because by
(2) $(i^{\prime})^{*}Ext_{A}^{s,tq-q}(H^{*}K,H^{*}K)\subset
Ext_{A}^{s,tq-q}(H^{*}K,H^{*}M)$ = 0 and $Ext_{A}^{s,tq+1}(H^{*}K,\\\ H^{*}M)$
= 0. The right group has unique generator $\sigma_{K}$ for $r=0$ which
satisfies $(\alpha_{1}\wedge 1_{K})^{*}(\sigma_{K})\neq 0\in
Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}K)$ , this is because
$(i^{\prime})^{*}(\alpha_{1}\wedge 1_{K})^{*}(\sigma_{K})=(\alpha_{1}\wedge
1_{M})^{*}(i^{\prime})^{*}(\sigma_{K})=(\alpha_{1}\wedge
1_{M})^{*}(i^{\prime})_{*}(\tilde{\sigma})=(i^{\prime})_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{\sigma})\neq 0\in Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$. Then
the middle group is zero for $r=0$ and has unique generator
$(j^{\prime\prime}\wedge 1_{K})^{*}(\sigma_{K})$ for $r=-1$ so that the first
result can be obtained by the following exact sequence
$0=Ext_{A}^{s,tq}(H^{*}K,H^{*}L\wedge
K)\stackrel{{\scriptstyle(i^{\prime\prime}\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}L\wedge K,H^{*}L\wedge K)$
$\qquad\quad\stackrel{{\scriptstyle(j^{\prime\prime}\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-q}(H^{*}K,H^{*}L\wedge
K)\stackrel{{\scriptstyle(\alpha_{1}\wedge 1_{K})_{*}}}{{\longrightarrow}}$
induced by (9.1.3). For the second result, look at the following exact
sequence ($r=1,2,3,u=0,1,2$)
$0=Ext_{A}^{s,tq+rq+u}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(i^{\prime\prime}\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+rq+u}(H^{*}L\wedge K,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle(j^{\prime\prime}\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+(r-1)q+u}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(\alpha_{1}\wedge
1_{K})_{*}}}{{\longrightarrow}}$
induced by (9.1.3). By (2), the left group is zero for $r=1,2,3,u=0,1,2$ and
the right group also is zero for $r=2,3,u=0,1,2$. By Prop. 9.3.0 and the
supposition , the right group also is zero for $r=1,u=1,2$. For $r=1,u=0$, The
right group has unique generator $(i^{\prime})_{*}(\tilde{\sigma})$ which
satisfies $(\alpha_{1}\wedge 1_{K})_{*}(i^{\prime})_{*}(\tilde{\sigma})\neq
0$. Then the middle group is zero for $r=1,2,3,u=0,1,2$.
(4) Consider the following exact sequence ($r=1,2,3,u=0,1,2$)
$0=Ext_{A}^{s,tq+rq+u}(H^{*}L\wedge K,H^{*}M)\stackrel{{\scriptstyle(w\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+rq+u}(H^{*}W\wedge K,H^{*}M)$
$\qquad\quad\stackrel{{\scriptstyle(j^{\prime\prime}u\wedge
1_{K})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+(r-2)q+u}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(\phi\wedge
1_{K})_{*}}}{{\longrightarrow}}$
induced by (9.1.12). By (3), the left group is zero for $r=1,2,3,u=0,1,2$. By
(2), the right group is zero for $r=1,3,u=0,1,2$ and by Prop. 9.3.0 and the
supposition, it also is zero for $r=2,u=1$. For $r=2,u=0$, the right group has
unique generator $(i^{\prime})_{*}(\tilde{\sigma})$ which satisfies
$(\phi\wedge 1_{K})_{*}(i^{\prime})_{*}(\tilde{\sigma})\neq 0\in
Ext_{A}^{s+1,tq+2q}(H^{*}L\wedge K,H^{*}M)$. Then the middle group is zero for
$r=1,2,3,u=0,1,2$ as desired.
Since $(\widetilde{u}w_{2}\overline{w}\wedge
1_{M})^{*}Ext_{A}^{s,tq+q}(H^{*}W\wedge K,H^{*}X\wedge M)\subset
Ext_{A}^{s,tq+q}(H^{*}W\wedge K,H^{*}M)$ = 0, then, by (9.1.5),
$(\widetilde{u}w_{2}\wedge 1_{M})^{*}Ext_{A}^{s,tq+q}(H^{*}W\wedge
K,H^{*}X\wedge M)=(\overline{u}\wedge
1_{M})^{*}Ext_{A}^{s,tq+2q+1}(H^{*}W\wedge K,H^{*}M\wedge M)$ = 0. and by
using (9.2.12) we know that $Ext_{A}^{s,tq+q}(H^{*}W\wedge K,H^{*}X\wedge
M)=(\widetilde{\psi}\wedge 1_{M})^{*}Ext_{A}^{tq+3q}(H^{*}W\wedge
K,H^{*}M\wedge M)$ = 0. Q.E.D.
The proof of the main Theorem C will be done by some argument processing in
the Adams resolution (cf. 9.2.9) of some spectra related to the sphere
spectrum $S$. Before proving the main Theorem C , we first prove the following
Lemmas.
Lemma 9.4.12 Under the supposition (I)(II) of the main Theorem C we have
(1) Let $\widetilde{h_{0}\sigma}\in[\Sigma^{tq+q+1}M,KG_{s+1}\wedge M]$ be a
$d_{1}$-cycle which represents $\alpha_{*}(\tilde{\sigma})\in
Ext_{A}^{s+1,tq+q+1}(H^{*}M,H^{*}M)$, then $(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(1_{E_{s+2}}\wedge\alpha)(\kappa\wedge 1_{M})$
(up to scalar), where $\kappa\in\pi_{tq+1}E_{s+2}$ such that
$\bar{a}_{s+1}\cdot\kappa=\bar{c}_{s}\cdot\sigma$ and
$\sigma\in\pi_{tq}KG_{s}\cong Ext_{A}^{s,tq}(Z_{p},Z_{p})$.
(2) $(1_{E_{s+2}}\wedge\phi\wedge 1_{M})(\kappa\wedge 1_{M})$ = 0,
$(1_{E_{s+2}}\wedge\alpha_{1}\wedge 1_{M})(\kappa\wedge 1_{M})$ = 0.
Proof: (1) Since $(1_{KG_{s+1}}\wedge i^{\prime})\widetilde{h_{0}\sigma}$ is a
$d_{1}$-boundary, then $(\bar{c}_{s+1}\wedge 1_{K})\\\ (1_{KG_{s+1}}\wedge
i^{\prime})(\widetilde{h_{0}\sigma})$ = 0 so that $(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(1_{E_{s+2}}\wedge\alpha)f^{\prime}$ for some
$f^{\prime}\in[\Sigma^{tq+1}M,E_{s+2}\wedge M]$. It follows that
$(\bar{a}_{s+1}\wedge 1_{M})(1_{E_{s+2}}\wedge\alpha)f^{\prime}$ = 0 and so
$(\bar{a}_{s+1}\wedge 1_{M})f^{\prime}=(1_{E_{s+1}}\wedge
j^{\prime})f^{\prime}_{2}$ with
$f^{\prime}_{2}\in[\Sigma^{tq+q+1}M,E_{s+1}\wedge K]$. The $d_{1}$-cycle
$(\bar{b}_{s+1}\wedge 1_{K})f^{\prime}_{2}$ represents an element in
$Ext_{A}^{s+1,tq+q+1}(H^{*}K,H^{*}M)$ and this group is zero by Prop.
9.3.2(2), then $(\bar{b}_{s+1}\wedge
1_{K})f^{\prime}_{2}=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{K})g^{\prime}_{0}$ for
some $g^{\prime}_{0}\in[\Sigma^{tq+q+1}M,KG_{s}\wedge K]$. Consequently we
have, $f^{\prime}_{2}=(\bar{c}_{s}\wedge
1_{K})g^{\prime}_{0}+(\bar{a}_{s+1}\wedge 1_{K})f^{\prime}_{3}$ for some
$f^{\prime}_{3}\in[\Sigma^{tq+q+2}M,E_{s+2}\wedge K]$ and so
$(\bar{a}_{s+1}\wedge 1_{M})f^{\prime}=(\bar{a}_{s+1}\wedge
1_{M})(1_{E_{s+2}}\wedge j^{\prime})f^{\prime}_{3}+(\bar{c}_{s}\wedge
1_{M})(1_{KG_{s}}\wedge j^{\prime})g^{\prime}_{0}=(\bar{a}_{2}\wedge
1_{M})(1_{E_{s+2}}\wedge j^{\prime})f^{\prime}_{3}+(\bar{c}_{s}\wedge
1_{M})(\sigma\wedge 1_{M})=(\bar{a}_{s+1}\wedge 1_{M})(1_{E_{s+2}}\wedge
j^{\prime})f^{\prime}_{3}+(\bar{a}_{s+1}\wedge 1_{M})(\kappa\wedge 1_{M})$,
where the $d_{1}$-cycle $(1_{KG_{s}}\wedge
j^{\prime})g^{\prime}_{0}\in[\Sigma^{tq}M,KG_{s}\wedge M]$ represents an
element in $Ext_{A}^{s,tq}(H^{*}M,H^{*}M)$ and this group has unique generator
$\tilde{\sigma}$ so that it equals to $\sigma\wedge 1_{M}$ (mod
$d_{1}$-boundary). Hence we have $f^{\prime}=(1_{E_{s+1}}\wedge
j^{\prime})f^{\prime}_{3}+(\kappa\wedge 1_{M})+(\bar{c}_{s+1}\wedge
1_{M})\tilde{g}_{1}$ for some $\tilde{g}_{1}\in[\Sigma^{tq+1}M,KG_{s+1}\wedge
M]$ and so $(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(1_{E_{s+2}}\wedge\alpha)f^{\prime}=(1_{E_{s+1}}\wedge\alpha)(\kappa\wedge
1_{M})$ which shows the result.
(2) Since $Ext_{A}^{s+1,tq+rq}(Z_{p},Z_{p})$ is zero for $r=2$ and has unique
generator $h_{0}\sigma=(j^{\prime\prime})_{*}(\phi)_{*}(\sigma)$ for$r=1$,
then $Ext_{A}^{s+1,tq+2q}(H^{*}L,Z_{p})\cong Z_{p}\\{(\phi)_{*}(\sigma)\\}$
and $Ext_{A}^{s+1,tq+2q}(H^{*}W,Z_{p})$ = 0. By this and a similar proof as
given in (1) we know that $(1_{E_{s+2}}\wedge\phi)\kappa=(\bar{c}_{s+1}\wedge
1_{L})\sigma\phi$ (up to scalar), where
$\sigma\phi\in\pi_{tq+2q}(KG_{s+1}\wedge L)$ is a $d_{1}$-cycle which
represents $(\phi)_{*}(\sigma)\in Ext_{A}^{s+1,tq+2q}(H^{*}L,\\\ Z_{p})$.
Then, by the supposition (II) of the main Theorem C we have
$(1_{E_{s+2}}\wedge\phi\wedge 1_{M})(\kappa\wedge 1_{M})=(\bar{c}_{s+1}\wedge
1_{L\wedge M})(\sigma\phi\wedge 1_{M})$ = 0 so that the result follows. Q.E.D.
Lemma 9.4.13 Under the supposition (I) of the main Theorem C we have
(1) $Ext_{A}^{s,tq}(H^{*}X\wedge M,H^{*}X\wedge M)\cong Z_{p}\\{[\sigma\wedge
1_{X\wedge M}]\\}$.
(2) For any $d_{1}$-cycle $g_{0}\in[\Sigma^{tq+q}X,KG_{s+1}\wedge X]$,
$g_{0}=\lambda^{\prime}(h_{0}\sigma\wedge 1_{X})$ (mod $d_{1}$-boundary) with
$\lambda^{\prime}\in Z_{p}$ and $(\psi_{X\wedge M})_{*}[h_{0}\sigma\wedge
1_{X\wedge M}]\neq 0\in Ext_{A}^{s+1,tq+1}(H^{*}Y\wedge M,H^{*}X\wedge M)$.
Proof (1) Consider the following exact sequence
$Ext_{A}^{s,tq+2q}(H^{*}L\wedge K,H^{*}M)\stackrel{{\scriptstyle
m_{M}(\tilde{\psi}\wedge 1_{M})^{*}}}{{\to}}\longrightarrow
Ext_{A}^{s,tq}(H^{*}L\wedge K,H^{*}X\wedge M)$
$\qquad\quad\stackrel{{\scriptstyle(u^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}L\wedge
K,H^{*}L\wedge K)\stackrel{{\scriptstyle((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))^{*}}}{{\longrightarrow}}$
induced by (9.4.2). By Prop. 9.4.11(3), the left group is zero and the right
group has unique generator $\sigma_{L\wedge K}$ which satisfies $((1_{L}\wedge
i^{\prime})(\phi\wedge 1_{M}))^{*}(\sigma_{L\wedge K})\neq 0\in
Ext_{A}^{s+1,tq+2q}(H^{*}L\wedge K,H^{*}M)$, this is because
$(j^{\prime\prime}\wedge 1_{K})_{*}((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))^{*}(\sigma_{L\wedge K})=((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))^{*}(j^{\prime\prime}\wedge 1_{K})_{*}(\sigma_{L\wedge
K})=((1_{L}\wedge i^{\prime})(\phi\wedge 1_{M})^{*}(j^{\prime\prime}\wedge
1_{K})^{*}(\sigma_{K})=((\alpha_{1}\wedge
1_{K})i^{\prime})^{*}(\sigma_{K})=(\alpha_{1}\wedge
1_{M})^{*}(i^{\prime})_{*}(\tilde{\sigma})=(i^{\prime}(\alpha_{1}\wedge
1_{M}))_{*}(\tilde{\sigma})\neq 0\in Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$. Then
the middle group is zero. Look at the following exact sequence
$Ext_{A}^{s,tq}(H^{*}L\wedge K,H^{*}X\wedge
M)\stackrel{{\scriptstyle(u^{\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq}(H^{*}X\wedge
M,H^{*}X\wedge M)$
$\qquad\quad\stackrel{{\scriptstyle m_{M}(\tilde{\psi}\wedge
1_{M})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-2q}(H^{*}M,H^{*}X\wedge
M)\stackrel{{\scriptstyle((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))_{*}}}{{\longrightarrow}}$
induced by (9.4.2). As shown above, the left group is zero. By Prop.
9.4.11(1), the right group has unique generator $m_{M}(\widetilde{\psi}\wedge
1_{M})^{*}(\tilde{\sigma})\\\ =m_{M}(\widetilde{\psi}\wedge
1_{M})^{*}[\sigma\wedge 1_{M}]=[(\sigma\wedge
1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})]=[(1_{KG_{s}}\wedge
m_{M}(\widetilde{\psi}\wedge 1_{M}))(\sigma\wedge 1_{X\wedge
M})]=m_{M}(\widetilde{\psi}\wedge 1_{M})_{*}[\sigma\wedge 1_{X\wedge M}]$ and
it satisfies $((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))_{*}m_{M}(\widetilde{\psi}\wedge
1_{M})^{*}(\tilde{\sigma})=((1_{L}\wedge i^{\prime})(\phi\wedge
1_{M}))_{*}m_{M}(\widetilde{\psi}\wedge 1_{M})_{*}[\sigma\wedge 1_{X\wedge
M}]$ = 0. Then the middle group has unique generator $[\sigma\wedge 1_{X\wedge
M}]$ as desired.
(2) Note that
$(\widetilde{\psi})_{*}(\widetilde{u}w_{2})^{*}Ext_{A}^{s+1,tq+q}(H^{*}X,H^{*}X)\subset
Ext_{A}^{s+1,tq-q-1}(H^{*}M,\\\ H^{*}Y)$. Similar to that in Prop. 9.3.0(1),
by the supposition we know that $Ext_{A}^{s+1,tq}(H^{*}M,H^{*}M)$ is zero or
has (one or two ) generator $\tilde{\sigma}^{\prime}$, then
$Ext_{A}^{s+1,tq-q-1}(H^{*}M,H^{*}Y)$ is zero or has (one or two) generator
$(\overline{u})^{*}(\tilde{\sigma}^{\prime})$ and it satisfies $((1_{Y}\wedge
j)\alpha_{Y\wedge
M})_{*}(\overline{u})^{*}(\tilde{\sigma}^{\prime})=((1_{Y}\wedge
j)\alpha_{Y\wedge M})_{*}(\overline{u})_{*}[\sigma^{\prime}\wedge
1_{Y}]=(1_{Y}\wedge\alpha_{1})_{*}[\sigma^{\prime}\wedge
1_{Y}]=[h_{0}\sigma^{\prime}\wedge 1_{Y}]\neq 0$, then
$(\widetilde{\psi})_{*}(\widetilde{u}w_{2})^{*}Ext_{A}^{s+1,tq+q}(H^{*}X,H^{*}X)$
= 0 and so we have
$(\widetilde{u}w_{2})^{*}Ext_{A}^{s+1,tq+q}(H^{*}X,H^{*}X)=(\widetilde{u}w_{2})_{*}Ext_{A}^{s+1,tq+q}(H^{*}Y,\\\
H^{*}Y)$ = 0 , this is because $Ext_{A}^{s+1,tq+q}(H^{*}Y,H^{*}Y)\cong
Z_{p}\\{((1_{Y}\wedge j)\alpha_{Y\wedge M})_{*}(\overline{u})^{*}\\\
(\tilde{\sigma})\\}$ ( cf. Prop. 9.4.10(2)). Then
$Ext_{A}^{s+1,tq+q}(H^{*}X,H^{*}X)=(\widetilde{\psi})^{*}Ext_{A}^{s+1,tq+3q}\\\
(H^{*}X,H^{*}M)$ and it has unique generator
$(\widetilde{\psi})^{*}((1_{X}\wedge j)\alpha_{X\wedge
M})_{*}(\tilde{\sigma})=((1_{X}\wedge j)\alpha_{X\wedge M})_{*}[(\sigma\wedge
1_{M})\widetilde{\psi}]=((1_{X}\wedge j)\alpha_{X\wedge
M})_{*}[(1_{KG_{s+1}}\wedge\widetilde{\psi})(\sigma\wedge
1_{X})]=((1_{X}\wedge j)\alpha_{X\wedge M})_{*}m_{M}(\widetilde{\psi}\wedge
1_{M})_{*}(1_{X}\wedge i)_{*}[\sigma\wedge 1_{X}]=(1_{X}\wedge j\alpha
i)_{*}[\sigma\wedge 1_{X}]=[h_{0}\sigma\wedge 1_{X}]$ (cf. Prop. 9.4.10(3))
Then the first result follows. For the second result , by (9.4.6), the
$d_{1}$-cycle $(1_{KG_{s+1}}\wedge m_{M}(\overline{u}\wedge
1_{M})\psi_{X\wedge M})(h_{0}\sigma\wedge 1_{X\wedge M})=(1_{KG_{s+1}}\wedge
m_{M}(\widetilde{\psi}\wedge 1_{M}))(h_{0}\sigma\wedge 1_{X\wedge
M})=(h_{0}\sigma\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})$ and it
represents an element $m_{M}(\widetilde{\psi}\wedge
1_{M})^{*}[h_{0}\sigma\wedge 1_{M}]=m_{M}(\widetilde{\psi}\wedge
1_{M})^{*}(\alpha_{1}\wedge 1_{M})_{*}(\tilde{\sigma})\neq 0$ so that the
second result follows. Q.E.D.
Proof the main Theorem C By Lemma 9.4.12(1), it suffices to prove
$(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(1_{E_{s+1}}\wedge\alpha)(\kappa\wedge
1_{M})=0$. The proof is divided into the following two steps.
Step 1 To prove $(\kappa\wedge 1_{X\wedge M})(1_{X}\wedge\alpha)$ = 0.
By (9.4.3), $(\phi\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})=(u^{\prime\prime}\wedge 1_{M})(1_{X}\wedge\alpha)$, then by Lemma
9.4.12(2) we have $(1_{E_{s+2}}\wedge u^{\prime\prime}\wedge
1_{M})(1_{E_{s+2}}\wedge 1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge
M})=(1_{E_{s+2}}\wedge\phi\wedge 1_{M})(\kappa\wedge
1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})$ = 0. Moreover, by (9.2.16) we have
$(1_{E_{s+2}}\wedge 1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge
M})=(1_{E_{s+2}}\wedge\widetilde{u}w_{3}\wedge 1_{M})f$ for some
$f\in[\Sigma^{tq+q+1}X\wedge M,E_{s+2}\wedge W\wedge M]\cap(kerd)$ (cf. Cor.
6.4.15). By composing $(1_{E_{s+2}}\wedge 1_{X}\wedge i^{\prime}i\wedge
1_{M})$ on the above equation we have
$(1_{E_{s+2}}\wedge\widetilde{u}w_{3}\wedge 1_{K\wedge M})(1_{E_{s+2}}\wedge
1_{W}\wedge i^{\prime}i\wedge
1_{M})f=(1_{E_{s+2}}\wedge(1_{X}\wedge(i^{\prime}i\wedge
1_{M})\alpha)(\kappa\wedge 1_{X\wedge M})=(1_{E_{s+2}}\wedge
1_{X}\wedge\overline{m}_{K}i^{\prime}(\alpha_{1}\wedge 1_{M})))(\kappa\wedge
1_{X\wedge M})$ = 0 ,where we use the result on
$(1_{E_{s+2}}\wedge\alpha_{1}\wedge 1_{M})(\kappa\wedge 1_{M})$ = 0 in Lemma
9.4.12(2). Consequently, by (9.2.16), $(1_{E_{s+2}}\wedge 1_{W}\wedge
i^{\prime}i\wedge 1_{M})f=(1_{E_{s+2}}\wedge w^{\prime}(\pi\wedge 1_{L})\wedge
1_{K\wedge M})f_{2}$ = 0 (for some $f_{2}\in[\Sigma^{tq+1}X\wedge
M,E_{s+2}\wedge L\wedge K\wedge M]$), this is because $\pi\wedge 1_{K}$ = 0.
Then by (9.1.4) we have $f=(1_{E_{s+2}}\wedge 1_{W}\wedge\epsilon\wedge
1_{M})f_{3}=(1_{E_{s+2}}\wedge 1_{W}\wedge\alpha m_{M}(\overline{u}\wedge
1_{M})f_{3}$ for some $f_{3}\in[\Sigma^{tq+q+2}X\wedge M,E_{s+2}\wedge W\wedge
Y\wedge M]\cap(kerd)$ ( cf. Cor. 6.4.15) and so
(9.4.14) $(1_{E_{s+2}}\wedge 1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge M})$
$\quad=(1_{E_{s+2}}\wedge\widetilde{u}w_{3}\wedge 1_{M})(1_{E_{s+2}}\wedge
1_{W}\wedge\alpha m_{M}(1_{M}\wedge\overline{u}))f_{3}$
$\quad=(1_{E_{s+2}}\wedge\alpha_{X\wedge M}(j^{\prime\prime}u\wedge
1_{M}))(1_{E_{s+2}}\wedge 1_{W}\wedge m_{M}(1_{M}\wedge\overline{u}))f_{3}$ (
cf. (9.4.3))
By (9.4.14), $(\bar{a}_{s+1}\wedge 1_{X\wedge
M})(1_{E_{s+2}}\wedge\widetilde{u}w_{3}\wedge
1_{M})(1_{E_{s+2}}\wedge(1_{W}\wedge\alpha m_{M}(\overline{u}\wedge
1_{M}))f_{3}$ = $(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge
1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge M})=(\bar{c}_{s}\wedge 1_{X\wedge
M})(1_{KG_{s}}\wedge 1_{X}\wedge\alpha)(\sigma\wedge 1_{X\wedge M})$ = 0 ,this
is because $\alpha$ induces zero homomorphism in $Z_{p}$-cohomology. Then, by
(9.2.16) and $w^{\prime}(\pi\wedge 1_{L})\wedge 1_{M}=(w\wedge
1_{M})(1_{L}\wedge\alpha)$ we have
(9.4.15) $(\bar{a}_{s+1}\wedge 1_{W\wedge M})(1_{E_{s+2}}\wedge
1_{W}\wedge\alpha m_{M}(\overline{u}\wedge 1_{M}))f_{3}$
$=(1_{E_{s+1}}\wedge(1_{W}\wedge\alpha)(w\wedge 1_{M}))f_{5}$
with $f_{5}\in[\Sigma^{tq}X\wedge M,E_{s+1}\wedge L\wedge M]\cap(kerd)$ ( cf.
Cor. 6.4.15 ).
By (9.4.15)(9.1.2) , $(\bar{a}_{s+1}\wedge 1_{W\wedge M})(1_{E_{s+2}}\wedge
1_{W}\wedge m_{M}(\overline{u}\wedge 1_{M}))f_{3}=(1_{E_{s+1}}\wedge w\wedge
1_{M})f_{5}+(1_{E_{s+1}}\wedge 1_{W}\wedge j^{\prime})f_{6}$ for some
$f_{6}\in[\Sigma^{tq+q+1}X\wedge M,E_{s+1}\wedge W\wedge K]\cap(kerd)$ ( cf.
Prop. 6.5.26). Since
$(1_{W}\wedge\alpha_{1})w=w(1_{L}\wedge\alpha_{1})=w\cdot\phi
j^{\prime\prime}$ = 0, then $w=(1_{W}\wedge j^{\prime\prime})\psi_{W}$ , where
$\psi_{W}\in[\Sigma^{q}L,W\wedge L]$. So we have $w\wedge 1_{M}=(1_{W}\wedge
j^{\prime\prime})\psi_{W}\wedge 1_{M}=(1_{W}\wedge m_{M}(\overline{u}\wedge
1_{M}))((1_{W}\wedge\bar{h})\psi_{W}\wedge 1_{M})$. Hence,
$-(\bar{a}_{s+1}\wedge 1_{W\wedge Y\wedge
M})f_{3}=(1_{E_{s+2}}\wedge(1_{W}\wedge h)\psi_{W}\wedge
1_{M})f_{5}+(1_{E_{s+1}}\wedge 1_{W}\wedge(1_{Y}\wedge
i)r)f_{6}+(1_{E_{s+1}}\wedge 1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f_{7}$
and by Prop. 6.5.26, $f_{7}=f_{8}(1_{X}\wedge i^{\prime})+f_{9}(1_{X}\wedge
i^{\prime}ij)$, where $f_{8}\in[\Sigma^{tq+q}X\wedge K,E_{s+1}\wedge W\wedge
K]\cap(kerd)$ and $f_{9}\in[\Sigma^{tq+q+1}X\wedge K,E_{s+1}\wedge W\wedge
K]\cap(kerd)$. Since $d((1_{Y}\wedge i)r)=((r\wedge 1_{M})d(1_{K}\wedge
i)=(r\wedge 1_{M})(1_{K}\wedge m_{M})(T_{K,M}\wedge 1_{M})(1_{M}\wedge
1_{K}\wedge i)\overline{m}_{K}=(r\wedge 1_{M})(1_{K}\wedge m_{M}(1_{M}\wedge
i)\overline{m}_{K}=(r\wedge 1_{M})\overline{m}_{K}$, by applying the
derivation $d$ using Theorem 6.4.8(1) we have $-(1_{E_{s+1}}\wedge
1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f_{6}-(1_{E_{s+1}}\wedge
1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f_{9}(1_{X}\wedge i^{\prime})$ = 0
(Note : $f_{6}$ has odd degree) and so
(9.4.16) $-(\bar{a}_{s+1}\wedge 1_{W\wedge Y\wedge
M})f_{3}=(1_{E_{s+1}}\wedge(1_{W}\wedge\bar{h})\psi_{W}\wedge 1_{M})f_{5}$
$\quad+(1_{E_{s+1}}\wedge 1_{W}\wedge(1_{Y}\wedge i)r)f_{6}+(1_{E_{s+1}}\wedge
1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f_{8}(1_{X}\wedge i^{\prime})$
$\quad-(1_{E_{s+1}}\wedge 1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})f_{6}(1_{X}\wedge ij)$
Note that the $d_{1}$-cycle $(\bar{b}_{s+1}\wedge 1_{W\wedge
K})f_{6}\in[\Sigma^{tq+q+1}X\wedge M,KG_{s+1}\wedge W\wedge K]\cap(kerd)$
represents an element in $Ext_{A}^{s+1,tq+q+1}(H^{*}W\wedge K,H^{*}X\wedge M)$
and by Prop. 9.4.10(1) this group is zero, then $(\bar{b}_{s+1}\wedge
1_{W\wedge K})f_{6}=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{W\wedge K})g$ for some
$g\in[\Sigma^{tq+q+1}X\wedge M,KG_{s}\wedge W\wedge K]\cap(kerd)$ (cf. Prop.
6.5.26) and so $f_{6}=(\bar{c}_{s}\wedge 1_{W\wedge K})g+(\bar{a}_{s+1}\wedge
1_{W\wedge K})f^{\prime}$ with $f^{\prime}\in[\Sigma^{tq+q+2}X\wedge
M,E_{s+2}\wedge W\wedge K]\cap(kerd)$ ( cf. Prop. 6.5.26). Then we have
(9.4.17) $-(\bar{a}_{s+1}\wedge 1_{W\wedge Y\wedge
M})f_{3}=(1_{E_{s+1}}\wedge(1_{W}\wedge\bar{h})\psi_{W}\wedge 1_{M})f_{5}$
$\qquad+(\bar{a}_{s+1}\wedge 1_{W\wedge Y\wedge M})(1_{E_{s+2}}\wedge
1_{W}\wedge(1_{Y}\wedge i)r)f^{\prime}$
$\qquad+(\bar{c}_{s}\wedge 1_{W\wedge Y\wedge M})(1_{KG_{s}}\wedge
1_{W}\wedge(1_{Y}\wedge i)r)g$
$\qquad+(\bar{a}_{s+1}\wedge 1_{W\wedge Y\wedge M})(1_{E_{s+2}}\wedge
1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f^{\prime}(1_{X}\wedge ij)$
$\qquad-(\bar{c}_{s}\wedge 1_{W\wedge Y\wedge M})(1_{KG_{s}}\wedge
1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})g(1_{X}\wedge ij)$
$\qquad+(1_{E_{s+1}}\wedge 1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})f_{8}(1_{X}\wedge i^{\prime})$
Let $P$ be the cofibre of
$(1_{W}\wedge\bar{h})\psi_{W}:\Sigma^{q+1}L\rightarrow W\wedge Y$ given by the
cofibration
(9.4.18)
$\Sigma^{q+1}L\stackrel{{\scriptstyle(1_{W}\wedge\bar{h})\psi_{W}}}{{\longrightarrow}}W\wedge
Y\stackrel{{\scriptstyle w_{5}}}{{\longrightarrow}}P\stackrel{{\scriptstyle
u_{5}}}{{\longrightarrow}}\Sigma^{q+2}L$
Then the cofibre of $w_{5}(1_{W}\wedge r):W\wedge K\rightarrow P$ is $\Sigma
X$ given by the cofibration
(9.4.19) $W\wedge K\stackrel{{\scriptstyle w_{5}(1_{W}\wedge
r)}}{{\longrightarrow}}V\stackrel{{\scriptstyle
w_{6}}}{{\longrightarrow}}\Sigma X\stackrel{{\scriptstyle
u_{6}}}{{\longrightarrow}}\Sigma W\wedge K$
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\qquad W\wedge K\quad\stackrel{{\scriptstyle w_{5}(1_{W}\wedge
r)}}{{\longrightarrow}}\quad P\quad\stackrel{{\scriptstyle
u_{5}}}{{\longrightarrow}}\quad\Sigma^{q+2}L$
$\qquad\qquad\quad\searrow 1_{W}\wedge r\quad\nearrow w_{5}\quad\searrow
w_{6}\quad\nearrow u^{\prime\prime}$
$\qquad\qquad\qquad\quad W\wedge Y\qquad\qquad\qquad\Sigma X$
$\qquad\qquad\quad\nearrow^{(1_{W}\wedge\bar{h})\psi_{W}}\searrow
1_{W}\wedge\epsilon\nearrow\widetilde{u}w_{3}\searrow u_{6}$
$\qquad\qquad\Sigma^{q+1}L\stackrel{{\scriptstyle w^{\prime}(\pi\wedge
1_{L})}}{{\longrightarrow}}\quad\Sigma W\quad\stackrel{{\scriptstyle
1_{W}\wedge i^{\prime}i}}{{\longrightarrow}}\quad\Sigma W\wedge K$
Note that $u_{6}=\mu_{X\wedge M}(1_{X}\wedge i)$, then by composing
$(\bar{b}_{s+1}\wedge 1_{P})(1_{E_{s+1}}\wedge w_{5}\wedge j)$ on the left
hand side of (9.4.17) and composing $(1_{X}\wedge i)$ on the right hand side
we have $(\bar{b}_{s+1}\wedge 1_{P})(1_{E_{s+1}}\wedge w_{5}(1_{W}\wedge
r))f_{8}(1_{X}\wedge i^{\prime}i)$ = 0 and so $(\bar{b}_{s+1}\wedge 1_{W\wedge
K})f_{8}(1_{X}\wedge i^{\prime}i)=(1_{KG_{s+1}}\wedge
u_{6})g_{0}=(1_{KG_{s+1}}\wedge\mu_{X\wedge M}(1_{X}\wedge
i))g_{0}=(1_{KG_{s+1}}\wedge\mu_{X\wedge M})(g_{0}\wedge 1_{M})(1_{X}\wedge
i)$ with $d_{1}$-cycle $g_{0}\in[\Sigma^{tq+q}X,KG_{s+1}\wedge X]$. Moreover ,
by Lemma 9.4.13(2), $g_{0}=\lambda_{1}(h_{0}\sigma\wedge 1_{X})$ (mod
$d_{1}$-boundary), where $\lambda_{1}\in Z_{p}$. On the other hand, by
applying the derivation $d$ to $(\bar{b}_{s+1}\wedge 1_{W\wedge
K})f_{8}(1_{X}\wedge i^{\prime}ij)=(1_{KG_{s+1}}\wedge\mu_{X\wedge
M})(g_{0}\wedge 1_{M})(1_{X}\wedge ij)$ we have
(9.4.20) $(\bar{b}_{s+1}\wedge 1_{W\wedge K})f_{8}(1_{X}\wedge
i^{\prime})==(1_{KG_{s+1}}\wedge\mu_{X\wedge M})(g_{0}\wedge 1_{M})$ ,
$\qquad\qquad g_{0}=\lambda_{1}(h_{0}\sigma\wedge
1_{X})\in[\Sigma^{tq+q}X,KG_{s+1}\wedge X]$ (mod $d_{1}$-boundary)
Consider the following commutative diagram of exact sequences
$\Sigma^{q+1}L\wedge M\quad\stackrel{{\scriptstyle w\wedge
1_{M}}}{{\longrightarrow}}\quad\Sigma^{q+1}W\wedge
M\quad\stackrel{{\scriptstyle j^{\prime\prime}u\wedge
1_{M}}}{{\longrightarrow}}\quad\Sigma^{3q+1}M\stackrel{{\scriptstyle\phi\wedge
1_{M}}}{{\longrightarrow}}\Sigma^{q+2}L\wedge M$
$\qquad\big{\uparrow}1_{L\wedge
M}\qquad\qquad\quad\qquad\big{\uparrow}1_{W}\wedge m_{M}(\overline{u}\wedge
1_{M})\qquad\big{\uparrow}u_{7}\qquad\qquad\big{\uparrow}1_{L\wedge M}$
$\Sigma^{q+1}L\wedge
M\stackrel{{\scriptstyle(1_{W}\wedge\bar{h})\psi_{W}\wedge
1_{M}}}{{\longrightarrow}}W\wedge Y\wedge M\quad\stackrel{{\scriptstyle
w_{5}\wedge 1_{M}}}{{\longrightarrow}}\quad P\wedge M\stackrel{{\scriptstyle
u_{5}\wedge 1_{M}}}{{\longrightarrow}}\Sigma^{q+2}L\wedge M$
of the cofibrations (9.1.12)(9.4.18). Since the left rectangle homotopy
commutes then there exists $u_{7}\in[\Sigma^{-3q-1}P\wedge M,M]$ such that all
the above rectangle homotopy commute. That is we have
(9.4.21) $u_{7}(w_{5}\wedge 1_{M})=(j^{\prime\prime}u\wedge 1_{M})(1_{W}\wedge
m_{M}(\overline{u}\wedge 1_{M})),\quad$
$\quad\qquad(\phi\wedge 1_{M})u_{7}=\pm\quad u_{5}\wedge 1_{M}$
where $u_{7}\in[\Sigma^{-3q-1}P\wedge M,M]$. By the above two equations , we
have the following homotopy commutative diagram of $3\times 3$-Lemma in which
we use the cofibrations (9.2.12)(9.4.18)(9.1.23)
$\qquad P\wedge M\quad\stackrel{{\scriptstyle u_{5}\wedge
1_{M}}}{{\longrightarrow}}\quad\Sigma^{q+2}L\wedge
M\quad\stackrel{{\scriptstyle w\wedge
1_{M}}}{{\longrightarrow}}\quad\Sigma^{q+2}W\wedge M$
$\qquad\qquad\searrow u_{7}\quad\nearrow\phi\wedge
1_{M}\quad\searrow^{((1_{W}\wedge\bar{h})\psi_{W}\wedge
1_{M}}\nearrow_{1_{W}\wedge m_{M}(\overline{u}\wedge
1_{M})}\searrow^{j^{\prime\prime}u\wedge 1_{M}}$
(9.4.22)$\qquad\qquad\Sigma^{3q+1}M\quad\qquad\qquad\qquad\Sigma W\wedge
Y\wedge M\qquad\qquad\quad\Sigma^{3q+2}M$
$\qquad\qquad\nearrow j^{\prime\prime}u\wedge 1_{M}\searrow^{(\phi_{W}\wedge
1_{K})i^{\prime}}\nearrow_{1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K}}\searrow
w_{5}\wedge 1_{M}\quad\nearrow u_{7}$
$\quad\Sigma^{q+1}W\wedge
M\stackrel{{\scriptstyle\tilde{\lambda}(1_{W}\wedge\alpha^{\prime}i^{\prime})}}{{\longrightarrow}}\Sigma^{2}W\wedge
K\qquad\quad\longrightarrow\qquad\quad\Sigma P\wedge M$
Then there is a cofibration
(9.4.23) $\Sigma^{3q-1}M\stackrel{{\scriptstyle(\phi_{W}\wedge
1_{K})i^{\prime}}}{{\longrightarrow}}W\wedge
K\stackrel{{\scriptstyle(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})}}{{\longrightarrow}}$
$\qquad\qquad\Sigma^{-1}P\wedge M\stackrel{{\scriptstyle
u_{7}}}{{\longrightarrow}}\Sigma^{3q}M$
in which $\phi_{W}\in[\Sigma^{3q-1}S,W]$ such that
$u\cdot\phi_{W}=\phi\in[\Sigma^{2q-1}S,L]$. Since $(\phi\wedge
1_{K})i^{\prime}\cdot u_{7}=(u\cdot\phi_{W}\wedge 1_{K})i^{\prime}\cdot u_{7}$
= 0, then by (9.4.2) we have
(9.4.24) $u_{7}=m_{M}(\widetilde{\psi}\wedge 1_{M})u_{8}$
where $u_{8}\in[\Sigma^{-q-1}P\wedge M,X\wedge M]$. On the other hand, by
(9.4.8), $(\omega\wedge 1_{M})u_{8}(w_{5}(1_{W}\wedge r)\wedge
1_{M})=\alpha_{Y\wedge M}m_{M}(\widetilde{\psi}\wedge
1_{M})u_{8}(w_{5}(1_{W}\wedge r)\wedge 1_{M})=\alpha_{Y\wedge M}u_{7}\\\
(w_{5}(1_{W}\wedge r)\wedge 1_{M})=\alpha_{Y\wedge
M}j^{\prime}(j^{\prime\prime}u\wedge 1_{K})(1_{W}\wedge m_{K})$ = 0 (cf.
(9.1.27)). Then, by (9.4.7), $u_{8}(w_{5}(1_{W}\wedge r)\wedge
1_{M})=((1_{X}\wedge j)u^{\prime}\wedge 1_{M})\Delta_{1}$ with
$\Delta_{1}\in[\Sigma^{-q}W\wedge K\wedge M,L\wedge K\wedge M]\cap(kerd)$. By
composing $\mu_{X\wedge M}(1_{X}\wedge i)\wedge 1_{M}$ on the above equation
and using (9.4.19) we have $((1_{X}\wedge j)u^{\prime}\wedge
1_{M})\Delta_{1}(\mu_{X\wedge M}(1_{X}\wedge i)\wedge 1_{M})$ = 0 and so by
(9.4.7)(9.4.6), $\Delta_{1}(\mu_{X\wedge M}(1_{X}\wedge i)\wedge
1_{M})=(\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}i)\wedge 1_{M})\psi_{X\wedge
M}$. Then $(j^{\prime\prime}\wedge 1_{K\wedge M})\Delta_{1}(\mu_{X\wedge
M}(1_{X}\wedge i)\wedge 1_{M})=((j^{\prime\prime}\wedge
1_{K})\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}i)\wedge 1_{M})\psi_{X\wedge
M}=(i^{\prime}\overline{u}\wedge 1_{M})\psi_{X\wedge M}=(i^{\prime}i\wedge
1_{M})m_{M}(\overline{u}\wedge 1_{M})\psi_{X\wedge M}+(i^{\prime}\wedge
1_{M})\overline{m}_{M}(j\overline{u}\wedge 1_{M})\psi_{X\wedge M}$ and so
$(j^{\prime\prime}\wedge 1_{K\wedge M})\Delta_{1}(\mu_{X\wedge M}(1_{X}\wedge
i)\cdot\widetilde{u}w_{2}\wedge 1_{M})$ = 0. Consequently we have
$(j^{\prime\prime}\wedge 1_{K\wedge M})\Delta_{1}(\mu_{X\wedge
M}(\widetilde{u}w_{2}\wedge 1_{M})\wedge 1_{M})\in(1_{Y}\wedge j\wedge
1_{M})^{*}[\Sigma^{-2q}Y\wedge M,K\wedge M]$ = 0, this is because the degree
of the top cell of $Y\wedge M$ is $q+3$. Then $(j^{\prime\prime}\wedge
1_{K\wedge M})\Delta_{1}(\mu_{X\wedge M}\wedge
1_{M})\in(\widetilde{\psi}\wedge 1_{M\wedge M})^{*}[M\wedge M\wedge M,K\wedge
M]$ and so $(j^{\prime}(j^{\prime\prime}\wedge 1_{K})\wedge
1_{M})\Delta_{1}(\mu_{X\wedge M}\wedge 1_{M})$ = 0 and by (9.4.4) we have
$(j^{\prime}(j^{\prime\prime}\wedge 1_{K})\wedge
1_{M})\Delta_{1}=\Delta_{2}(j^{\prime}(j^{\prime\prime}u\wedge 1_{K})\wedge
1_{M})=\lambda(j^{\prime}(j^{\prime\prime}u\wedge 1_{K})\wedge 1_{M})$ with
$\lambda\in Z_{p}$, this is because $\Delta_{2}\in[M\wedge M,M\wedge
M]\cap(kerd)\cong Z_{p}\\{1_{M\wedge M}\\}$. Hence, $(\widetilde{\psi}\wedge
1_{M})u_{8}(w_{5}(1_{W}\wedge r)\wedge 1_{M})=(\widetilde{\psi}(1_{X}\wedge
j)u^{\prime}\wedge 1_{M})\Delta_{1}=(j^{\prime}(j^{\prime\prime}\wedge
1_{K})\wedge 1_{M})\Delta_{1}=\lambda(j^{\prime}(j^{\prime\prime}u\wedge
1_{K})\wedge 1_{M})$ and by (9.4.21)(9.4.24) we know that $\lambda=1$ so that
(9.4.25) $m_{M}(\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)=j^{\prime}(j^{\prime\prime}u\wedge 1_{K})$
$\qquad\quad=(j\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})$ ,
$\qquad\quad(j\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)=ijj^{\prime}(j^{\prime\prime}u\wedge
1_{K})$
where we use $(jj^{\prime}\wedge 1_{M})\overline{m}_{K}=j^{\prime}$ in the
above equation. By composing $(1_{E_{s+1}}\wedge u_{8}(w_{5}\wedge 1_{M}))$
(it has odd degree) on (9.4.17) we have
(9.4.26) $(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge
u_{8}(w_{5}\wedge 1_{M}))f_{3}$
$=-(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)f^{\prime}$
$\qquad-\overline{\lambda}(\bar{a}_{s+1}\wedge 1_{X\wedge
M})(1_{E_{s+2}}\wedge u^{\prime}(u\wedge 1_{K}))f^{\prime}(1_{X}\wedge ij)$
$\qquad+(\bar{c}_{s}\wedge 1_{X\wedge M})(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)g$
$\qquad-(\bar{c}_{s}\wedge 1_{X\wedge M})(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K}))g(1_{X}\wedge ij)$
$\qquad+\overline{\lambda}(1_{E_{s+1}}\wedge u^{\prime}(u\wedge
1_{K}))f_{8}(1_{X}\wedge i^{\prime})$
where we use $u_{8}(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})=\overline{\lambda}u^{\prime}(u\wedge 1_{K})$, for some
nonzero $\overline{\lambda}\in Z_{p}$. Moreover, by (9.4.20)(9.4.6),
$(\bar{b}_{s+1}\wedge 1_{L\wedge K})(1_{E_{s+1}}\wedge u\wedge
1_{K})f_{8}(1_{X}\wedge i^{\prime})=(1_{KG_{s+1}}\wedge(u\wedge
1_{K})\mu_{X\wedge M})(g_{0}\wedge
1_{M})=(1_{KG_{s+1}}\wedge\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime})\psi_{X\wedge M})(g_{0}\wedge
1_{M})=\lambda_{1}(1_{KG_{s+1}}\wedge\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime})\psi_{X\wedge M})(h_{0}\sigma\wedge 1_{X\wedge
M})=\lambda_{1}(h_{0}\sigma\wedge 1_{L\wedge K})\overline{\mu}_{2}(1_{Y}\wedge
i^{\prime})\psi_{X\wedge M}$ (mod $d_{1}$-boundary). Then
$[(\bar{b}_{s+1}\wedge 1_{L\wedge K})(1_{E_{s+1}}\wedge u\wedge
1_{K})f_{8}(1_{X}\wedge i^{\prime})]=\lambda_{1}(\phi\wedge
1_{K})_{*}(j^{\prime\prime}\wedge 1_{K})_{*}[(\sigma\wedge 1_{L\wedge
K})\overline{\mu}_{2}(1_{Y}\wedge i^{\prime})\psi_{X\wedge
M}]=\lambda_{1}(\phi\wedge 1_{K})_{*}(j^{\prime\prime}\wedge
1_{K})_{*}(\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}))_{*}(\psi_{X\wedge
M})_{*}[\sigma\wedge 1_{X\wedge M}]=\lambda_{1}(\phi\wedge
1_{K})_{*}(i^{\prime})_{*}(m_{M}(\overline{u}\wedge 1_{M}))_{*}(\psi_{X\wedge
M})_{*}[\sigma\wedge 1_{X\wedge M}]=\lambda_{1}((1_{L}\wedge
i^{\prime})(\phi\wedge 1_{M}))_{*}(m_{M}(\widetilde{\psi}\wedge
1_{M})_{*}[\sigma\wedge 1_{X\wedge M}]$ = 0 $\in Ext_{A}^{s+1,tq}(H^{*}L\wedge
K,H^{*}X\wedge M)$. That is we have $(\bar{b}_{s+1}\wedge 1_{L\wedge
K})(1_{E_{s+1}}\wedge u\wedge 1_{K})f_{8}(1_{X}\wedge
i^{\prime})=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{L\wedge K})g_{3}$ with
$g_{3}\in[\Sigma^{tq}X\wedge M,KG_{s}\wedge L\wedge K]\cap(kerd)$ ( cf. Prop.
9.5.26) and so $(1_{E_{s+1}}\wedge u\wedge 1_{K})f_{8}(1_{X}\wedge
i^{\prime})=(\bar{c}_{s}\wedge 1_{L\wedge K})g_{3}+(\bar{a}_{s+1}\wedge
1_{L\wedge K})f^{\prime}_{2}$ with $f^{\prime}_{2}\in[\Sigma^{tq+1}X\wedge
M,E_{s+2}\wedge L\wedge K]\cap(kerd)$ (cf. Prop. 6.5.26). Hence, (9.4.26)
becomes
(9.4.27) $(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge
u_{8}(w_{5}\wedge 1_{M}))f_{3}$
$=-(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r))f^{\prime}$
$\quad-\overline{\lambda}(\bar{a}_{s+1}\wedge 1_{X\wedge M})(1_{E_{s+2}}\wedge
u^{\prime}(u\wedge 1_{K}))f^{\prime}(1_{X}\wedge ij)$
$\quad+(\bar{c}_{s}\wedge 1_{X\wedge M})(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r))g$
$\quad-(\bar{c}_{s}\wedge 1_{X\wedge M})(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K}))g(1_{X}\wedge ij)$
$\quad+\overline{\lambda}(\bar{c}_{s}\wedge 1_{X\wedge M})(1_{KG_{s}}\wedge
u^{\prime})g_{3}+\overline{\lambda}(\bar{a}_{s+1}\wedge 1_{X\wedge
M})(1_{E_{s+2}}\wedge u^{\prime})f^{\prime}_{2}$
By (9.4.27), $(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)g-(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})g(1_{X}\wedge
ij)+\overline{\lambda}(1_{KG_{s}}\wedge u^{\prime})g_{3}\in[\Sigma^{tq}X\wedge
M,KG_{s}\wedge X\wedge M]$ is a $d_{1}$-cycle which represents an element in
$Ext_{A}^{s,tq}(H^{*}X\wedge M,H^{*}X\wedge M)\cong Z_{p}\\{[\sigma\wedge
1_{X\wedge M}]\\}$ ( cf. Lemma 9.4.13). Then we have
(9.4.28) $(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge 1_{M})(1_{W}\wedge(1_{Y}\wedge
i)r)g+\overline{\lambda}(1_{KG_{s}}\wedge u^{\prime})g_{3}$
$\quad-(1_{KG_{s}}\wedge u_{8}(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K}))g(1_{X}\wedge ij)$
$\qquad=\bar{\lambda}_{0}(\sigma\wedge 1_{X\wedge M})$ (mod $d_{1}$-boundary).
Now we consider the cases of $\bar{\lambda}_{0}\neq 1$ or
$\bar{\lambda}_{0}=1$ separately .
If $\bar{\lambda}_{0}\neq 1$, then by (9.4.27) and
$\bar{c}_{s}\cdot\sigma=\bar{a}_{s+1}\cdot\kappa$ we have
$\qquad(1_{E_{s+2}}\wedge u_{8}(w_{5}\wedge 1_{M}))f_{3}=-(1_{E_{s+2}}\wedge
u_{8}(w_{5}\wedge 1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)f^{\prime}$
$\qquad\quad-\bar{\lambda}(1_{E_{s+2}}\wedge u^{\prime}(u\wedge
1_{K}))f^{\prime}(1_{X}\wedge ij)+\bar{\lambda}(1_{E_{s+2}}\wedge
u^{\prime})f^{\prime}_{2}$
$\qquad\quad+\bar{\lambda}_{0}(\kappa\wedge 1_{X\wedge
M})+(\bar{c}_{s+1}\wedge 1_{X\wedge M})g_{4}$
with $g_{4}\in[\Sigma^{tq+1}X\wedge M,KG_{s+1}\wedge X\wedge M]$ and by
composing $(1_{E_{s+2}}\wedge
1_{X}\wedge\alpha)=(1_{E_{s+2}}\wedge\alpha_{X\wedge
M}m_{M}(\widetilde{\psi}\wedge 1_{M}))$ we obtain that $(1_{E_{s+2}}\wedge
1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge
M})=(1_{E_{s+2}}\wedge\alpha_{X\wedge M}(j^{\prime\prime}u\wedge
1_{M})(1_{W}\wedge m_{M}(\overline{u}\wedge
1_{M}))f_{3}=(1_{E_{s+2}}\wedge\alpha_{X\wedge M}\cdot
m_{M}(\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M}))f_{3}=\bar{\lambda}_{0}(1_{E_{s+2}}\wedge
1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge M})$ so that the result of the step
1 follows.
If $\bar{\lambda}_{0}=1$, then by composing $(1_{KG_{s}}\wedge
m_{M}(\widetilde{\psi}\wedge 1_{M}))$ on (9.4.28) and using (9.4.25) we have
$(1_{KG_{s}}\wedge j^{\prime}(j^{\prime\prime}u\wedge
1_{K}))g=(1_{KG_{s}}\wedge m_{M}(\widetilde{\psi}\wedge
1_{M})u_{8}(w_{5}\wedge 1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)g=(\sigma\wedge
1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})$ (mod $d_{1}$-boundary). Moreover,
by composing $(1_{KG_{s}}\wedge j\widetilde{\psi}\wedge 1_{M})$ on (9.4.28)
and using (9.4.25) we have
$(1_{KG_{s}}\wedge j\widetilde{\psi}\wedge 1_{M})(\sigma\wedge 1_{X\wedge M})$
$=(1_{KG_{s}}\wedge(j\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)g$
$\quad-(1_{KG_{s}}\wedge(j\widetilde{\psi}\wedge 1_{M})u_{8}(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K}))g(1_{X}\wedge ij)$
$\quad+\overline{\lambda}(1_{KG_{s}}\wedge(j\widetilde{\psi}\wedge
1_{M})u^{\prime})g_{3}$
$=(1_{KG_{s}}\wedge ij(j^{\prime}(j^{\prime\prime}\wedge 1_{K})(u\wedge
1_{K}))g$
$\quad-(1_{KG_{s}}\wedge j^{\prime}(j^{\prime\prime}u\wedge
1_{K}))g(1_{X}\wedge ij)+\bar{\lambda}(1_{KG_{s}}\wedge
j^{\prime}(j^{\prime\prime}\wedge 1_{K}))g_{3}$ by (9.4.25)
$=(1_{KG_{s}}\wedge ij)(\sigma\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge
1_{M})-(\sigma\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})(1_{X}\wedge
ij)$
$\quad+\overline{\lambda}(1_{KG_{s}}\wedge j^{\prime}(j^{\prime\prime}\wedge
1_{K}))g_{3}$
$=(1_{KG_{s}}\wedge j\widetilde{\psi}\wedge 1_{M})(\sigma\wedge 1_{X\wedge
M})+\overline{\lambda}(1_{KG_{s}}\wedge j^{\prime}(j^{\prime\prime}\wedge
1_{K}))g_{3}$ by (9.4.6)
(mod $d_{1}$-boundary), then $(1_{KG_{s}}\wedge
j^{\prime}(j^{\prime\prime}\wedge 1_{K}))g_{3}$ = 0 and so
$g_{3}=(1_{KG_{s}}\wedge\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}))g_{5}$ (mod
$d_{1}$-boundary) for some $g_{5}\in[\Sigma^{tq+q+1}X\wedge M,KG_{s}\wedge
Y\wedge M]$. So, by (9.4.6)(9.4.20)
$(1_{KG_{s+1}}\wedge\overline{\mu}_{2}(1_{Y}\wedge i^{\prime})\psi_{X\wedge
M})(g_{0}\wedge 1_{M})=(1_{KG_{s+1}}\wedge(u\wedge 1_{K})\mu_{X\wedge
M})(g_{0}\wedge 1_{M})=(\bar{b}_{s+1}\wedge 1_{L\wedge K})(1_{E_{s+1}}\wedge
u\wedge 1_{K})f_{8}(1_{X}\wedge i^{\prime})=(\bar{b}_{s+1}\bar{c}_{s}\wedge
1_{L\wedge K})g_{3}=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{L\wedge
K})(1_{KG_{s}}\wedge\overline{\mu}_{2}(1_{Y}\wedge i^{\prime}))g_{5}$ so that
$(1_{KG_{s+1}}\wedge\psi_{X\wedge M})(g_{0}\wedge
1_{M})=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{Y\wedge M})g_{5}$ , this shows
$\lambda_{1}(\psi_{X\wedge M})_{*}[h_{0}\sigma\wedge 1_{X\wedge
M}]=(\psi_{X\wedge M})_{*}[g_{0}\wedge 1_{M}]=0\in
Ext_{A}^{s+1,tq+1}(H^{*}Y\wedge M,.H^{*}X\wedge M)$ and by Lemma 9.4.13(2) we
have $\lambda_{1}$ = 0 . Then $[g_{0}\wedge 1_{M}]$ = 0 and so
$(\bar{b}_{s+1}\wedge 1_{W\wedge K})f_{8}(1_{X}\wedge
i^{\prime})=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{W\wedge K})g_{6}$ for some
$g_{6}\in[\Sigma^{tq+q}X\wedge M,KG_{s}\wedge W\wedge K]$ and
$f_{8}(1_{X}\wedge i^{\prime})=(\bar{c}_{s}\wedge 1_{W\wedge
K})g_{6}+(\bar{a}_{s+1}\wedge 1_{W\wedge K})f^{\prime}_{3}$ with
$f^{\prime}_{3}\in[\Sigma^{tq+q+1}X\wedge M,E_{s+2}\wedge W\wedge K]$. Then,
by composing $(1_{E_{s+1}}\wedge w_{5}\wedge 1_{M})$ on (9.4.17) we have
$-(\bar{a}_{s+1}\wedge 1_{P\wedge M})(1_{E_{s+2}}\wedge w_{5}\wedge
1_{M})f_{3}$
$=(\bar{a}_{s+1}\wedge 1_{P\wedge M})(1_{E_{s+2}}\wedge(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)f^{\prime}$
$\quad+(\bar{a}_{s+1}\wedge 1_{P\wedge M})(1_{E_{s+2}}\wedge(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})f^{\prime}(1_{X}\wedge ij)$
$\quad+(\bar{a}_{s+1}\wedge 1_{P\wedge M})(1_{E_{s+2}}\wedge(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})f^{\prime}_{3}+(\bar{c}_{s}\wedge 1_{P\wedge M})g_{7}$
where the $d_{1}$-cycle $g_{7}=(1_{KG_{s}}\wedge(w_{5}\wedge
1_{M})(1_{W}\wedge(1_{Y}\wedge i)r)g-(1_{KG_{s}}\wedge(w_{5}\wedge
1_{M})(1_{W}\wedge(r\wedge 1_{M})\overline{m}_{K})g(1_{X}\wedge
ij)+(1_{KG_{s}}\wedge(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})g_{6}\in[\Sigma^{tq+q+1}X\wedge M,KG_{s}\wedge P\wedge
M]$ which represents an element in $Ext_{A}^{s,tq+q+1}(H^{*}P\wedge
M,H^{*}X\wedge M)$. However, this group is zero , this can be obtained by the
following exact sequence
$0=Ext_{A}^{s,tq+q}(H^{*}W\wedge K,H^{*}X\wedge
M)\stackrel{{\scriptstyle((w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})_{*}}}{{\longrightarrow}}$
$\qquad Ext_{A}^{s,tq+q+1}(H^{*}P\wedge M,H^{*}X\wedge
M)\stackrel{{\scriptstyle(u_{7})_{*}}}{{\longrightarrow}}$
$\qquad Ext_{A}^{s,tq-2q}(H^{*}M,H^{*}X\wedge
M)\stackrel{{\scriptstyle((1_{W}\wedge i^{\prime})(\phi_{W}\wedge
M))_{*}}}{{\longrightarrow}}$
induced by (9.4.23), where the left group is zero by Prop. 9.4.11(4) and by
Prop. 9.4.11(1) the right group has unique generator
$m_{M}(\widetilde{\psi}\wedge 1_{M})^{*}(\tilde{\sigma})$, which satisfies
$((1_{W}\wedge i^{\prime})(\phi_{W}\wedge
1_{M}))_{*}m_{M}(\widetilde{\psi}\wedge 1_{M})^{*}(\tilde{\sigma})\neq 0\in
Ext_{A}^{s+1,tq+q}(H^{*}W\wedge K,H^{*}X\wedge M)$.
Then, $(\bar{c}_{s}\wedge 1_{P\wedge M})g_{7}$ = 0 and so $-(1_{E_{s+2}}\wedge
w_{5}\wedge 1_{M})f_{3}=(1_{E_{s+2}}\wedge(w_{5}\wedge
1_{M})u_{8}(1_{W}\wedge(1_{Y}\wedge
i)r)f^{\prime}-(1_{E_{s+2}}\wedge(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K})f^{\prime}(1_{X}\wedge
ij)+(1_{E_{s+2}}\wedge(w_{5}\wedge 1_{M})(1_{W}\wedge(r\wedge
1_{M})\overline{m}_{K}))f^{\prime}_{3}+(\bar{c}_{s+1}\wedge 1_{P\wedge
M})g_{8}$ for some $g_{8}\in[\Sigma^{tq+q+2}X\wedge M,KG_{s+1}\wedge P\wedge
M]$. By composing $(1_{E_{s+2}}\wedge\alpha_{X\wedge M}\cdot u_{7})$ we have
$(1_{E_{s+2}}\wedge 1_{X}\wedge\alpha)(\kappa\wedge 1_{X\wedge
M})=(1_{E_{s+2}}\wedge\alpha_{X\wedge M}(j^{\prime\prime}u\wedge
1_{M})(1_{W}\wedge m_{M}(\overline{u}\wedge
1_{M}))f_{3}=(1_{E_{s+2}}\wedge\alpha_{X\wedge M}\cdot u_{7}(w_{5}\wedge
1_{M}))f_{3}$ = 0 . This shows the result of step 1.
Step 2 To prove $(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(\kappa\wedge 1_{M})\alpha$ = 0.
By (9.4.3)(9.4.4), $\mu_{X\wedge M}(1_{X}\wedge\alpha i)=\mu_{X\wedge
M}\alpha_{X\wedge M}\widetilde{\psi}$ = 0 and so by (9.1.15) $\mu_{X\wedge
M}=\mu_{X\wedge K^{\prime}}(1_{X}\wedge v)$ , where $\mu_{X\wedge
K^{\prime}}\in[X\wedge K^{\prime},W\wedge K]$. We claim that $X\wedge
K^{\prime}$ splits into $W\wedge K\vee\Sigma^{q}Y$, that is, there is a split
cofibration $\Sigma^{q}Y\rightarrow X\wedge K^{\prime}\rightarrow W\wedge K$,
this can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma and using $(1_{Y}\wedge j)\alpha_{Y\wedge
M}j^{\prime}=r(1_{K}\wedge\alpha_{1})$
$\qquad\qquad X\wedge M\quad\stackrel{{\scriptstyle\mu_{X\wedge
M}}}{{\longrightarrow}}\quad W\wedge K\quad\stackrel{{\scriptstyle
0}}{{\longrightarrow}}\quad\Sigma^{q+1}Y$
$\qquad\qquad\quad\searrow 1_{X}\wedge v\nearrow\mu_{X\wedge
K^{\prime}}\searrow^{j^{\prime}(j^{\prime\prime}u\wedge
1_{K})}\nearrow_{(1_{Y}\wedge j)\alpha_{Y\wedge M}}$
$\qquad\qquad\qquad X\wedge K^{\prime}\qquad\qquad\qquad\Sigma^{3q+1}M$
$\qquad\qquad\quad\nearrow\widetilde{\tau}_{X\wedge K^{\prime}}\searrow
1_{X}\wedge\quad\nearrow\widetilde{\psi}\quad\searrow\alpha_{X\wedge M}$
$\qquad\qquad\Sigma^{q}Y\quad\stackrel{{\scriptstyle\tilde{u}w_{2}}}{{\longrightarrow}}\qquad\Sigma^{q+1}X\quad\stackrel{{\scriptstyle
1_{X}\wedge\alpha i}}{{\longrightarrow}}\quad\Sigma X\wedge M$
Hence, there is a split cofibration
$\Sigma^{q}Y\stackrel{{\scriptstyle\tau_{X\wedge
K^{\prime}}}}{{\longrightarrow}}X\wedge
K^{\prime}\stackrel{{\scriptstyle\mu_{X\wedge
K^{\prime}}}}{{\longrightarrow}}W\wedge K$ and so there are $\nu_{X\wedge
K^{\prime}}:X\wedge K^{\prime}\rightarrow\Sigma^{q}Y$ and
$\widetilde{\nu}_{X\wedge K^{\prime}}:W\wedge K\rightarrow X\wedge K^{\prime}$
such that $\nu_{X\wedge K^{\prime}}\cdot\tau_{X\wedge
K^{\prime}}=1_{Y},\quad\mu_{X\wedge K^{\prime}}\cdot\widetilde{\nu}_{X\wedge
K^{\prime}}=1_{W\wedge K},$ $\widetilde{\tau}_{X\wedge
K^{\prime}}\cdot\nu_{X\wedge K^{\prime}}+\widetilde{\nu}_{X\wedge
K^{\prime}}\cdot\mu_{X\wedge K^{\prime}}=1_{X\wedge K^{\prime}}$.
By the result of step 1 we have $(\kappa\wedge 1_{M\wedge X\wedge
K^{\prime}})(\alpha\wedge 1_{X\wedge K^{\prime}})$ = 0, then $(\kappa\wedge
1_{M\wedge Y})(\alpha\wedge 1_{Y})=(1_{E_{s+2}}\wedge 1_{M}\wedge\nu_{X\wedge
K^{\prime}})(\kappa\wedge 1_{M\wedge X\wedge K^{\prime}})(\alpha\wedge
1_{X\wedge K^{\prime}})(1_{M}\wedge\tau_{X\wedge K^{\prime}})$ = 0. Moreover,
by using the splitness in (9.1.32) we have $(\bar{c}_{s+1}\wedge
1_{M})\widetilde{h_{0}\sigma}=(\kappa\wedge 1_{M})\alpha=(1_{E_{s+2}}\wedge
1_{M}\wedge\widetilde{\nu})(\kappa\wedge 1_{M\wedge Y\wedge
K^{\prime}})(\alpha\wedge 1_{Y\wedge
K^{\prime}})(1_{M}\wedge\widetilde{\tau})$ = 0 which shows the main Theorem C.
Q.E.D.
Remark. In the proof of the main Theorem C, We only use the supposition (II)
for our geometric input to obtain that $(1_{E_{s+2}}\wedge\phi\wedge
1_{M})(\kappa\wedge 1_{M})m_{M}(\widetilde{\psi}\wedge 1_{M})$ = 0. Then , the
geometric supposition (II) of the main Theorem C can be weakened to be the
supposition on $m_{M}(\widetilde{\psi}\wedge 1_{M})^{*}(\phi\wedge
1_{M})_{*}(\tilde{\sigma})\in Ext_{A}^{s+1,tq}(H^{*}L\wedge M,H^{*}X\wedge M)$
is a permanent cycle in the ASS.
Using some new cofibrations in this section , we also can give an alternative
proof of Theorem 9.3.9( and so the main Theorem B). We first do some
preminalaries.
Since $\alpha^{\prime}\alpha^{\prime}i^{\prime}=0$, then by (9.1.23), there
exists $\alpha^{\prime\prime}_{Y\wedge M}\in[\Sigma^{q-2}Y\wedge M,K]$ such
that $\alpha^{\prime\prime}_{Y\wedge M}(r\wedge
1_{M})\overline{m}_{K}=\alpha^{\prime}$. By applying the derivation $d$,
$d(\alpha^{\prime\prime}_{Y\wedge M})(r\wedge
1_{M})\overline{m}_{K}=-d(\alpha^{\prime})=0$ and so
$d(\alpha^{\prime\prime}_{Y\wedge M})\in(m_{M}(\overline{u}\wedge
1_{M}))^{*}[\Sigma^{2q}M,K]$ = 0. $\alpha_{Y\wedge M}(1_{Y}\wedge
i)r\in[\Sigma^{q-2}K,K]\cong Z_{p}\\{\alpha^{\prime\prime}\\}$ and so
$\alpha^{\prime\prime}_{Y\wedge M}(1_{Y}\wedge
i)r=\lambda\alpha^{\prime\prime}$ for some $\lambda\in Z_{p}$. Note that
$d((1_{Y}\wedge i)r)=(r\wedge 1_{M})d(1_{K}\wedge i)=(r\wedge
1_{M})\overline{m}_{K}$, then by applying the derivation $d$, we have
$\alpha^{\prime}=\alpha^{\prime\prime}_{Y\wedge M}(r\wedge
M)\overline{m}_{K}=\lambda d(\alpha^{\prime\prime})=-\lambda\alpha^{\prime}$
and so $\lambda=-1$. By (9.1.8), $\bar{h}i^{\prime\prime}=\overline{w}$,
$ri^{\prime}=\overline{w}\cdot j$( up to sign), then $(r\wedge
1_{M})\overline{m}_{K}i^{\prime}=-(ri^{\prime}\wedge
1_{M})\overline{m}_{M}=\pm(\overline{w}\wedge
1_{M})=\pm(\bar{h}i^{\prime\prime}\wedge 1_{M})$ and so
$\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}i^{\prime\prime}\wedge
1_{M})=\lambda_{0}\alpha^{\prime\prime}_{Y\wedge M}(r\wedge
1_{M})\overline{m}_{K}i^{\prime}=\lambda_{0}\alpha^{\prime}i^{\prime}=\lambda_{0}i^{\prime}((\alpha_{1})_{L}i^{\prime\prime}\wedge
1_{M})$ and we have $\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}\wedge
1_{M})=\lambda_{0}i^{\prime}((\alpha_{1})_{L}\wedge 1_{M})$, where
$\lambda_{0}=\pm 1$. On the other hand, $i^{\prime}((\alpha_{1})_{L}\wedge
1_{M})(1_{L}\wedge j^{\prime})(i^{\prime\prime}\wedge
1_{K})=i^{\prime}(\alpha_{1}\wedge 1_{M})j^{\prime}=i^{\prime}(ij\alpha-\alpha
ij)j^{\prime}$ = 0, then $i^{\prime}((\alpha_{1})_{L}\wedge 1_{M})(1_{L}\wedge
j^{\prime})=\lambda^{\prime}\alpha^{\prime\prime}(j^{\prime\prime}\wedge
1_{K})$ with $\lambda^{\prime}\in Z_{p}$. By composing the map
$\widetilde{\Delta}$ in Theorem 6.5.18 we have
$\lambda^{\prime}\alpha^{\prime}i^{\prime}ijj^{\prime}=\lambda^{\prime}\alpha^{\prime\prime}i^{\prime}j^{\prime}=\lambda^{\prime}\alpha^{\prime\prime}(j^{\prime\prime}\wedge
1_{K})\widetilde{\Delta}=i^{\prime}((\alpha_{1})_{L}\wedge 1_{M})(1_{L}\wedge
j^{\prime})\widetilde{\Delta}=-i^{\prime}((\alpha_{1})_{L}i^{\prime\prime}\wedge
1_{M})ijj^{\prime}=-\alpha^{\prime}i^{\prime}ijj^{\prime}$ so that
$\lambda^{\prime}=-1$. Concludingly , there is $\alpha^{\prime\prime}_{Y\wedge
M}\in[\Sigma^{q-2}Y\wedge M,K]$ such that
(9.4.29) $\alpha^{\prime\prime}_{Y\wedge M}(r\wedge
1_{M})\overline{m}_{K}=\alpha^{\prime},\quad\qquad\alpha^{\prime\prime}_{Y\wedge
M}(1_{Y}\wedge i)r=-\alpha^{\prime\prime},$
$\qquad\qquad d(\alpha^{\prime\prime}_{Y\wedge
M})=0,\quad\qquad\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}\wedge
1_{M})=\lambda_{0}i^{\prime}((\alpha_{1})_{L}\wedge 1_{M}),$
$\qquad\qquad i^{\prime}((\alpha_{1})_{L}\wedge 1_{M})(1_{L}\wedge
j^{\prime})=-\alpha^{\prime\prime}(j^{\prime\prime}\wedge 1_{K})$
where $\lambda_{0}=\pm 1$.
Note that the cofibre of $\alpha^{\prime\prime}_{Y\wedge
M}:\Sigma^{q-2}Y\wedge M\to K$ is $X\wedge M$ given by the cofibration
(9.4.30) $\Sigma^{q-2}Y\wedge
M\stackrel{{\scriptstyle\alpha^{\prime\prime}_{Y\wedge
M}}}{{\longrightarrow}}K\stackrel{{\scriptstyle
u^{\prime}(i^{\prime\prime}\wedge 1_{K})}}{{\longrightarrow}}X\wedge
M\stackrel{{\scriptstyle\psi_{X\wedge
M}}}{{\longrightarrow}}\Sigma^{q-1}Y\wedge M$
and the above map $\psi_{X\wedge M}\in[X\wedge M,\Sigma^{q-1}Y\wedge M]$ and
$u^{\prime}\in[L\wedge K,X\wedge M]$ is just the map in (9.4.2) and (9.4.6).
This can be seen by the equation $m_{M}(\overline{u}\wedge 1_{M})\psi_{X\wedge
M}=m_{M}(\widetilde{\psi}\wedge 1_{M})$ in (9.4.6),(9.4.2) and the following
homotopy commutative diagram of $3\times 3$-Lemma
$\qquad\quad X\wedge M\quad\stackrel{{\scriptstyle m_{M}(\tilde{\psi}\wedge
1_{M})}}{{\longrightarrow}}\quad\Sigma^{2q}M\qquad\stackrel{{\scriptstyle\alpha^{\prime}i^{\prime}}}{{\longrightarrow}}\qquad\Sigma^{q+1}K$
$\qquad\qquad\quad\searrow\psi_{X\wedge
M}\quad\nearrow^{m_{M}(\overline{u}\wedge 1_{M})}\quad\searrow^{(\phi\wedge
1_{K})i^{\prime}}\quad\nearrow j^{\prime\prime}\wedge 1_{K}\searrow^{(r\wedge
1_{M})\overline{m}_{K}}$
(9.4.31) $\Sigma^{q-1}Y\wedge M\qquad\qquad\qquad\quad\Sigma L\wedge
K\qquad\qquad\qquad\Sigma^{q}Y\wedge M$
$\qquad\qquad\nearrow^{(r\wedge
1_{M})\overline{m}_{K}}\quad\searrow^{\alpha^{\prime\prime}_{Y\wedge
M}}\quad\nearrow i^{\prime\prime}\wedge 1_{K}\quad\searrow
u^{\prime}\qquad\quad\nearrow\psi_{X\wedge M}$
$\qquad\quad\Sigma^{q}K\qquad\stackrel{{\scriptstyle\alpha^{\prime}}}{{\longrightarrow}}\qquad\quad\Sigma
K\qquad\stackrel{{\scriptstyle u^{\prime}(i^{\prime\prime}\wedge
1_{K})}}{{\longrightarrow}}\qquad\Sigma X\wedge M$
and by this we have the following relation
(9.4.32) $\psi_{X\wedge M}u^{\prime}=(r\wedge
1_{M})\overline{m}_{K}(j^{\prime\prime}\wedge 1_{K})$.
Proposition 9.4.33 Let $p\geq 5$ and $V$ be any spectrum, then for any map
$f\in[\Sigma^{*}K,V\wedge K]$ we have
$(1_{V}\wedge\alpha^{\prime})d(f)=d(f)\alpha^{\prime}=0$.
Proof: By (6.5.12), $\alpha\wedge
1_{K}=\overline{m}_{K}^{\prime}\alpha^{\prime}m_{K}^{\prime}$, where
$m_{K}^{\prime}=m_{K}T:M\wedge K\to K$,
$\overline{m}_{K}^{\prime}=T\overline{m}_{K}:\Sigma K\to M\wedge K$.
$d(f)\alpha^{\prime}m_{K}^{\prime}=(1_{V}\wedge
m_{K}^{\prime})(T^{\prime}\wedge 1_{K})(1_{M}\wedge
f)\overline{m}_{K}^{\prime}\alpha^{\prime}m_{K}^{\prime}=(1_{V}\wedge
m_{K}^{\prime})(T^{\prime}\wedge 1_{K})(1_{M}\wedge f)(\alpha\wedge
1_{K})=(1_{V}\wedge m_{K}^{\prime})(T^{\prime}\wedge 1_{K})(\alpha\wedge
1_{V}\wedge 1_{K})(1_{M}\wedge f)=(1_{V}\wedge m_{K}^{\prime}(\alpha\wedge
1_{K}))(T^{\prime}\wedge 1_{K})(1_{M}\wedge f)$ = 0, where $T^{\prime}:M\wedge
V\to V\wedge M$is the switching map. Q.E.D.
Proposition 9.4.34 Under the supposition (I) of the main Theorem B we have
(1) $Ext_{A}^{s,tq-1}(H^{*}K,H^{*}K)$ = 0.
(2) $Ext_{A}^{s,tq}(H^{*}Y\wedge M,H^{*}K)$ has unique generator $(1_{Y}\wedge
i)_{*}r_{*}[\sigma\wedge 1_{K}]$.
Proof: (1) Consider the following exact sequence
$\qquad
Ext_{A}^{s,tq+q}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle(i^{\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq+q}(H^{*}K,H^{*}M)$
$\qquad\qquad\stackrel{{\scriptstyle(j^{\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-1}(H^{*}M,H^{*}M)\stackrel{{\scriptstyle\alpha_{*}}}{{\longrightarrow}}$
induced by (9.1.2). By the supposition (I), the right group has unique
generator $j^{*}i_{*}(\sigma)$ which satisfies
$\alpha_{*}j^{*}i_{*}(\sigma)=(ij)_{*}\alpha_{*}(\tilde{\sigma})\neq 0$. Then
im $(j^{\prime})_{*}$ = 0. By the supposition (I), the left group is zero or
has two generators
$(ij)_{*}\alpha_{*}(\widetilde{\tau}^{\prime}),(ij)^{*}\alpha_{*}(\widetilde{\tau}^{\prime})$
(this can be obtained by a similar proof as given in Prop. 9.3.1(2)), then
$Ext_{A}^{s,tq+q}(H^{*}K,H^{*}M)=(i^{\prime})_{*}Ext_{A}^{s,tq+q}(H^{*}M,H^{*}M)$
is zero or has unique generator $(i^{\prime})_{*}(\alpha_{1}\wedge
1_{M})_{*}(\widetilde{\tau}^{\prime})$. Look at the following exact sequence
$\qquad
Ext_{A}^{s,tq+q}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-1}(H^{*}K,H^{*}K)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,tq-1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). By the supposition (I), the right group has unique
generator $j^{*}(i^{\prime}i)_{*}(\sigma)$ which satisfies
$\alpha^{*}j^{*}(i^{\prime}i)_{*}(\sigma)=(i^{\prime})_{*}(ij)_{*}\alpha_{*}(\tilde{\sigma})\neq
0\in Ext_{A}^{s+1,tq+q}(H^{*}K,H^{*}M)$ so that im $(i^{\prime})^{*}$ = 0. The
left group is zero or has unique generator $(i^{\prime})_{*}(\alpha_{1}\wedge
1_{M})_{*}(\widetilde{\tau}^{\prime})$ and so im $(j^{\prime})^{*}$ = 0 and
the middle group is zero as desired.
(2) For any $g\in Ext_{A}^{s,tq}(H^{*}Y\wedge
M,H^{*}K)$,$m_{M}(\overline{u}\wedge 1_{M}))_{*}(g)\in Ext_{A}^{s,tq-q-1}\\\
(H^{*}M,H^{*}K)\cong Z_{p}\\{(j^{\prime})^{*}(\tilde{\sigma})\\}$, this can be
obtained from $Ext_{A}^{s,tq}(H^{*}M,H^{*}M)\\\ \cong
Z_{p}\\{\tilde{\sigma}\\}$ in Prop. 9.3.0(2) and
$Ext_{A}^{s,tq-q-1}(H^{*}M,H^{*}M)$ = 0, where the last is obtained by the
supposition (I) on $Ext_{A}^{s,tq-q+u}\\\ (Z_{p},Z_{p})$ = 0(for $u=0,-1,1$).
Then $(m_{M}(\overline{u}\wedge
1_{M}))_{*}(g)=\lambda^{\prime}(j^{\prime})^{*}[\sigma\wedge
1_{M}]=\lambda^{\prime}[(\sigma\wedge
1_{M})j^{\prime}]=\lambda^{\prime}[(1_{KG_{s}}\wedge j^{\prime})(\sigma\wedge
1_{K})]=\lambda^{\prime}(j^{\prime})_{*}[\sigma\wedge
1_{K}]=\lambda^{\prime}(m_{M}(\overline{u}\wedge 1_{M}))_{*}(1_{Y}\wedge
i)_{*}r_{*}[\sigma\wedge 1_{K}]$ and so $g=\lambda^{\prime}(1_{Y}\wedge
i)_{*}r_{*}[\sigma\wedge 1_{K}]$ ( with $\lambda^{\prime}\in Z_{p}$) modulo
$((r\wedge 1_{M})\overline{m}_{K})_{*}Ext_{A}^{s,tq-1}(H^{*}K,H^{*}K)$ = 0.
Q.E.D.
An alternative proof of Theorem 9.3.9: By the supposition (II) of the main
Theorem B we have $(1_{E_{s+2}}\wedge\alpha^{\prime})(\kappa\wedge
1_{K})=(\bar{c}_{s+1}\wedge 1_{K})(h_{0}\sigma\wedge 1_{K})$ = 0, then
$(\kappa\wedge 1_{K})=(1_{E_{s+2}}\wedge j^{\prime\prime}\wedge 1_{K})f$ and
we have $d((1_{E_{s+2}}\wedge j^{\prime\prime}\wedge 1_{K})f)$ = 0. That is we
have
(9.4.35) $\kappa\wedge 1_{K}=(1_{E_{s+2}}\wedge j^{\prime\prime}\wedge
1_{K})f$ $d(f)=(1_{E_{s+2}}\wedge i^{\prime\prime}\wedge 1_{K})f^{\prime}$
for some $f\in[\Sigma^{tq+q+1}K,E_{s+2}\wedge L\wedge
K],f^{\prime}\in[\Sigma^{tq+q+2}K,E_{s+2}\wedge K]$.
By (9.4.29)(9.4.35),$(1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge
M}(\bar{h}\wedge 1_{M})(1_{L}\wedge
j^{\prime}))f=\lambda_{0}(1_{E_{s+2}}\wedge i^{\prime}((\alpha_{1})_{L}\wedge
1_{M})(1_{L}\wedge
j^{\prime}))f=-\lambda_{0}(1_{E_{s+2}}\wedge\alpha^{\prime\prime}(j^{\prime\prime}\wedge
1_{K}))f=-\lambda_{0}(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})$, where $\lambda_{0}=\pm 1$. That is we have
(9.4.36) $(1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}\wedge
1_{M})(1_{L}\wedge
j^{\prime}))f=-\lambda_{0}(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})$ where $\lambda_{0}=\pm 1$
It follows that $(\bar{a}_{s+1}\wedge
1_{K})(1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}\wedge
1_{M})(1_{L}\wedge j^{\prime}))f=-\lambda_{0}(\bar{a}_{s+1}\wedge
1_{K})(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})=-\lambda_{0}(\bar{c}_{s}\wedge
1_{K})(1_{KG_{s}}\wedge\alpha^{\prime\prime})(\sigma\wedge 1_{K})$ = 0 since
$\alpha^{\prime\prime}$ induces zero homomorphism in $Z_{p}$-cohomology. Then,
by (9.4.30),$(\bar{a}_{s+1}\wedge 1_{Y\wedge
M})(1_{E_{s+2}}\wedge(\bar{h}\wedge 1_{M})(1_{L}\wedge
j^{\prime}))f=(1_{E_{s+1}}\wedge\psi_{X\wedge M})f_{2}$ with
$f_{2}\in[\Sigma^{tq+q-1}K,\\\ E_{s+1}\wedge X\wedge M]$. Consequently,
$(\bar{b}_{s+1}\wedge 1_{Y\wedge M})(1_{E_{s+1}}\wedge\psi_{X\wedge M})f_{2}$
= 0 and so by (9.4.30) we have $(\bar{b}_{s+1}\wedge 1_{X\wedge
M})f_{2}=(1_{KG_{s+1}}\wedge u^{\prime}(i^{\prime\prime}\wedge 1_{K}))g$, with
$d_{1}$-cycle $g\in[\Sigma^{tq+q-1}K,KG_{s+1}\wedge K]$ and this $d_{1}$-cycle
represents an element in $Ext_{A}^{s+1,tq+q-1}(H^{*}K,H^{*}K)\cong
Z_{p}\\{(h_{0}\sigma)^{\prime\prime}\\}$. Then
$[g]=\lambda^{\prime}(h_{0}\sigma)^{\prime\prime}=\lambda^{\prime}(\alpha^{\prime\prime})_{*}[\sigma\wedge
1_{K}]$ for some $\lambda^{\prime}\in Z_{p}$ so that
$\quad[(\bar{b}_{s+1}\wedge 1_{X\wedge
M})f_{2}]=(u^{\prime}(i^{\prime\prime}\wedge
1_{K}))_{*}[g]=\lambda^{\prime}(u^{\prime}(i^{\prime\prime}\wedge
1_{K}))_{*}(\alpha^{\prime\prime})_{*}[\sigma\wedge 1_{K}]$
$=\lambda^{\prime}(u^{\prime}(i^{\prime\prime}\wedge
1_{K}))_{*}(\alpha^{\prime\prime}_{Y\wedge M})_{*}((1_{Y}\wedge
i)r)_{*}[\sigma\wedge 1_{K}]$ = 0
Hence, $(\bar{b}_{s+1}\wedge 1_{X\wedge
M})f_{2}=(\bar{b}_{s+1}\bar{c}_{s}\wedge 1_{X\wedge M})g_{2}$ for some
$g_{2}\in[\Sigma^{tq+q-1}K,KG_{s}\wedge X\wedge M]$ and so
$f_{2}=(\bar{c}_{s}\wedge 1_{X\wedge M})g_{2}+(\bar{a}_{s+1}\wedge 1_{X\wedge
M})f_{3}$ with $f_{3}\in[\Sigma^{tq+q-1}K,\\\ E_{s+2}\wedge X\wedge M]$ and we
have
$(\bar{a}_{s+1}\wedge 1_{Y\wedge M})(1_{E_{s+2}}\wedge(\bar{h}\wedge
1_{M})(1_{L}\wedge j^{\prime}))f$
$=(\bar{a}_{s+1}\wedge 1_{Y\wedge M})(1_{E_{s+2}}\wedge\psi_{X\wedge
M})f_{3}+(\bar{c}_{s}\wedge 1_{Y\wedge M})(1_{KG_{s}}\wedge\psi_{X\wedge
M})g_{2}$
$=(\bar{a}_{s+1}\wedge 1_{Y\wedge M})(1_{E_{s+2}}\wedge\psi_{X\wedge
M})f_{3}+\lambda(\bar{c}_{s}\wedge 1_{Y\wedge M})(1_{KG_{s}}\wedge(1_{Y}\wedge
i)r)(\sigma\wedge 1_{K})$
$=(\bar{a}_{s+1}\wedge 1_{Y\wedge M})(1_{E_{s+2}}\wedge\psi_{X\wedge M})f_{3}$
$\qquad+\lambda(\bar{a}_{s+1}\wedge 1_{Y\wedge
M})(1_{E_{s+2}}\wedge(1_{Y}\wedge i)r)(\kappa\wedge 1_{K})$
with $\lambda\in Z_{p}$, where the $d_{1}$-cycle
$(1_{KG_{s}}\wedge\psi_{X\wedge M})g_{2}\in[\Sigma^{tq}K,KG_{s}\wedge Y\wedge
M]$ represents an element $\lambda((1_{Y}\wedge i)r)_{*}[\sigma\wedge
1_{K}]\in Ext_{A}^{s,tq}(H^{*}Y\wedge M,H^{*}K)$( cf. Prop. 9.4.34(2)) and so
it equals to $\lambda(1_{KG_{s}}\wedge(1_{Y}\wedge i)r)(\sigma\wedge 1_{K})$ (
mod $d_{1}$-boundary). Then we have $(1_{E_{s+2}}\wedge(\bar{h}\wedge
1_{M})(1_{L}\wedge j^{\prime}))f=(1_{E_{s+2}}\wedge\psi_{X\wedge
M})f_{3}+\lambda(1_{E_{s+2}}\wedge(1_{Y}\wedge i)r)(\kappa\wedge
1_{K})+(\bar{c}_{s+1}\wedge 1_{Y\wedge M})g_{3}$ for some
$g_{3}\in[\Sigma^{tq+1}K,KG_{s+1}\wedge Y\wedge M]$. By composing
$1_{E_{s+2}}\wedge 1_{Y}\wedge\alpha$ and using(9.4.8) we have
$(1_{E_{s+2}}\wedge\omega\wedge 1_{M})f_{3}=(1_{E_{s+2}}\wedge\alpha_{Y\wedge
M}m_{M}(\widetilde{\psi}\wedge
1_{M}))f_{3}=(1_{E_{s+2}}\wedge(1_{Y}\wedge\alpha)\psi_{X\wedge
M})f_{3}=-\lambda(1_{E_{s+2}}\wedge(1_{Y}\wedge\alpha i)r)(\kappa\wedge
1_{K})=-\lambda(1_{E_{s+2}}\wedge(r\wedge
1_{M})\overline{m}_{K}\alpha^{\prime})(\kappa\wedge 1_{K})$ = 0 and by (9.4.7)
we have $f_{3}=(1_{E_{s+2}}\wedge(1_{x}\wedge j)u^{\prime}\wedge 1_{M})f_{4}$
with $f_{4}\in[\Sigma^{tq+q+1}K,E_{s+2}\wedge L\wedge K\wedge M]$. That is we
have
(9.4.37) $(1_{E_{s+2}}\wedge(\bar{h}\wedge 1_{M})(1_{L}\wedge
j^{\prime}))f=(1_{E_{s+2}}\wedge\psi_{X\wedge M}((1_{X}\wedge
j)u^{\prime}\wedge 1_{M}))f_{4}$
$\qquad+\lambda(1_{E_{s+2}}\wedge(1_{Y}\wedge i)r)(\kappa\wedge
1_{K})+(\bar{c}_{s+1}\wedge 1_{Y\wedge M})g_{3}$
$=(1_{E_{s+2}}\wedge\psi_{X\wedge
M}u^{\prime})f_{5}+(1_{E_{s+2}}\wedge\psi_{X\wedge M}(1_{X}\wedge
ij)u^{\prime}(1_{L}\wedge m_{K}))f_{4}$
$\qquad+\lambda(1_{E_{s+2}}\wedge(1_{Y}\wedge i)r)(\kappa\wedge
1_{K})+(\bar{c}_{s+1}\wedge 1_{Y\wedge M})g_{3}$
where we use $f_{4}=(1_{E_{s+2}}\wedge(1_{L}\wedge\overline{m}_{K})(1_{L\wedge
K}\wedge j))f_{4}+(1_{E_{s+2}}\wedge(1_{L\wedge K}\wedge i)(1_{L}\wedge
m_{K}))f_{4}$ and write $(1_{E_{s+2}}\wedge 1_{L\wedge K}\wedge
j)f_{4}=f_{5}$.
By composing $1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge M}$ on (9.4.37)
and using (9.4.36)(9.4.30) we have $-\lambda_{0}(1_{E_{s+2}}\\\
\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})=(1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge M}(\bar{h}\wedge
1_{M})(1_{L}\wedge
j^{\prime}))f=\lambda(1_{E_{s+2}}\wedge\alpha^{\prime\prime}_{Y\wedge
M}(1_{Y}\wedge i)r)(\kappa\wedge
1_{K})=-\lambda(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})$.
If $\lambda\neq\lambda_{0}$, then
$(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge 1_{K})$ = 0 and the
Theorem follows. So, we suppose that $\lambda=\lambda_{0}$.
By (9.1.8) we have $\overline{u}\bar{h}=i\cdot j^{\prime\prime}$ so that
$m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\wedge 1_{M})=j^{\prime\prime}\wedge
1_{M}$( up to sign). Then, what happen is either $m_{M}(\overline{u}\wedge
1_{M})(\bar{h}\wedge 1_{M})=\lambda_{0}(j^{\prime\prime}\wedge 1_{M})$ or
$m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\wedge
1_{M})=-\lambda_{0}(j^{\prime\prime}\wedge 1_{M})$. Now we consider this two
cases separately.
Case 1 $m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\wedge
1_{M})=\lambda_{0}(j^{\prime\prime}\wedge 1_{M})$.
In this case, by composing $1_{E_{s+2}}\wedge m_{M}(\overline{u}\wedge 1_{M})$
on (9.4.37) we have
(9.4.38) $\lambda_{0}(1_{E_{s+2}}\wedge j^{\prime})(\kappa\wedge
1_{K})=\lambda_{0}(1_{E_{s+2}}\wedge j^{\prime}(j^{\prime\prime}\wedge
1_{K}))f$
$=\lambda_{0}(1_{E_{s+2}}\wedge(j^{\prime\prime}\wedge 1_{M})(1_{L}\wedge
j^{\prime}))f=(1_{E_{s+2}}\wedge m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\wedge
1_{M})(1_{L}\wedge j^{\prime}))f$
$=(1_{E_{s+2}}\wedge m_{M}(\overline{u}\wedge 1_{M})\psi_{X\wedge
M}(1_{X}\wedge ij)u^{\prime}(1_{L}\wedge m_{K}))f_{4}$
$\qquad+\lambda_{0}(1_{E_{s+2}}\wedge j^{\prime})(\kappa\wedge
1_{K})+(\bar{c}_{s+1}\wedge 1_{M})(1_{KG_{s+1}}\wedge m_{M}(\overline{u}\wedge
1_{M}))g_{3}$
$=-(1_{E_{s+2}}\wedge j^{\prime}(j^{\prime\prime}\wedge 1_{K})(1_{L}\wedge
m_{K}))f_{4}$
$\qquad+\lambda_{0}(1_{E_{s+2}}\wedge j^{\prime})(\kappa\wedge
1_{K})+(\bar{c}_{s+1}\wedge 1_{M})(1_{KG_{s+1}}\wedge m_{M}(\overline{u}\wedge
1_{M}))g_{3}$
and so $(1_{E_{s+2}}\wedge j^{\prime}(j^{\prime\prime}\wedge
1_{K})(1_{L}\wedge m_{K}))f_{4}=(\bar{c}_{s+1}\wedge 1_{M})(1_{KG_{s+1}}\wedge
m_{M}(\overline{u}\wedge 1_{M}))g_{3}$, where we use $m_{M}(\overline{u}\wedge
1_{M})\psi_{X\wedge M}(1_{X}\wedge ij)u^{\prime}=m_{M}(\widetilde{\psi}\wedge
1_{M})(1_{X}\wedge ij)u^{\prime}=-(j\widetilde{\psi}\wedge
1_{M})u^{\prime}=-j^{\prime}(j^{\prime\prime}\wedge 1_{K})$ which is obtained
by (9.4.6) and the right rectangle of the diagram (9.4.1). Moreover, by
applying the derivation $d$ to the equation (9.4.37) we have
(9.4.39) $(1_{E_{s+2}}\wedge(\bar{h}\wedge 1_{M})(1_{L}\wedge
j^{\prime})(i^{\prime\prime}\wedge
1_{K}))f^{\prime}=(1_{E_{s+2}}\wedge\psi_{X\wedge M}u^{\prime})d(f_{5})$
$\quad+(1_{E_{s+2}}\wedge\psi_{X\wedge M}(1_{X}\wedge
ij)u^{\prime})d((1_{E_{s+2}}\wedge 1_{L}\wedge m_{K})f_{4})$
$\quad+(1_{E_{s+2}}\wedge\psi_{X\wedge M}u^{\prime}(1_{L}\wedge
m_{K}))f_{4}-\lambda_{0}(1_{E_{s+2}}\wedge(r\wedge
1_{M})\overline{m}_{K})(\kappa\wedge 1_{K})$
$\quad+(\bar{c}_{s+1}\wedge 1_{Y\wedge M})d(g_{3})$
By (9.4.32) we have $\psi_{X\wedge M}u^{\prime}=(r\wedge
1_{M})\overline{m}_{K}(j^{\prime\prime}\wedge 1_{K})$ so that
$(j\overline{u}\wedge 1_{M})\psi_{X\wedge
M}u^{\prime}=j^{\prime}(j^{\prime\prime}\wedge 1_{K})$. Then, by composing
$1_{E_{s+2}}\wedge\phi\cdot j\overline{u}\wedge 1_{M}$ on (9.4.39), it becomes
(9.4.40) $\lambda_{0}(1_{E_{s+2}}\wedge(\phi\wedge
1_{M})j^{\prime})(\kappa\wedge 1_{K})=(1_{E_{s+2}}\wedge(\phi\wedge
1_{M})j^{\prime}(j^{\prime\prime}\wedge 1_{K}))d(f_{5})$
$\quad+(1_{E_{s+2}}\wedge(\phi\wedge 1_{M})ijj^{\prime}(j^{\prime\prime}\wedge
1_{K}))d((1_{E_{s+2}}\wedge 1_{L}\wedge m_{K})f_{4})=0$
here we use $(1_{E_{s+2}}\wedge(\phi\cdot j\overline{u}\wedge
1_{M})\psi_{X\wedge M}u^{\prime}(1_{L}\wedge m_{K}))f_{4}=(1_{E_{s+2}}\wedge
j^{\prime}(j^{\prime\prime}\wedge 1_{K})(1_{L}\wedge
m_{K}))f_{4}=(\bar{c}_{s+1}\wedge 1_{L\wedge M})(1_{KG_{s+1}}\wedge(\phi\wedge
1_{M})m_{M}(\overline{u}\wedge 1_{M}))g_{3}$ = 0 and by
$1_{L}\wedge\alpha_{1}=\phi\cdot j^{\prime\prime}$(up to nonzero scalar) we
obtain that $(1_{E_{s+2}}\wedge(\phi\wedge
1_{M})j^{\prime}(j^{\prime\prime}\wedge
1_{K}))d(f_{5})=(1_{E_{s+2}}\wedge(1_{L}\wedge
j^{\prime}\alpha^{\prime}))d(f_{5})$ = 0 ( cf. Prop. 9.4.33) and so the first
term of the right hand side of (9.4.40) is zero. the second term of the right
hand side of (9.4.40) is zero by the same reason.
It follows from (9.4.40) that $(1_{E_{s+2}}\wedge\phi\wedge
1_{K})(\kappa\wedge 1_{K})=(1_{E_{s+2}}\wedge(1_{L}\wedge\mu(i^{\prime}i\wedge
1_{K}))(\phi\wedge 1_{K})(\kappa\wedge 1_{K})(jj^{\prime}\wedge 1_{K})\nu$ = 0
and so we have $(1_{E_{s+2}}\wedge\alpha^{\prime\prime})(\kappa\wedge
1_{K})=(1_{E_{s+2}}\wedge\overline{\Delta}(\phi\wedge 1_{K}))(\kappa\wedge
1_{K})$ = 0 and the Theorem follows.
Case 2 $m_{M}(\overline{u}\wedge 1_{M})(\bar{h}\wedge
1_{M})=-\lambda_{0}(j^{\prime\prime}\wedge 1_{M})$.
In this case, the left hand side of (9.4.38) changes sign, then
$-(1_{E_{s+2}}\wedge j^{\prime}(j^{\prime\prime}\wedge 1_{K})(1_{L}\wedge
m_{K}))f_{4}+(\bar{c}_{s+1}\wedge 1_{M})(1_{KG_{s+1}}\wedge
m_{M}(\overline{u}\wedge 1_{M}))g_{3}=-2\lambda_{0}(1_{E_{s+2}}\wedge
j^{\prime})(\kappa\wedge 1_{K})$ and by composing $1_{E_{s+2}}\wedge\phi\cdot
j\overline{u}\wedge 1_{M}$ on (9.4.39) we have
$\qquad(\lambda_{0}-2\lambda_{0})(1_{E_{s+2}}\wedge(\phi\wedge
1_{M})j^{\prime})(\kappa\wedge 1_{K})$
$=\lambda_{0}(1_{E_{s+2}}\wedge(\phi\wedge 1_{M})j^{\prime})(\kappa\wedge
1_{K})-(1_{E_{s+2}}\wedge(\phi\wedge 1_{M})j^{\prime}(j^{\prime\prime}\wedge
1_{K})(1_{L}\wedge m_{K}))f_{4}$
$=(1_{E_{s+2}}\wedge(\phi\wedge 1_{M})j^{\prime}(j^{\prime\prime}\wedge
1_{K}))d(f_{5})$
$\qquad+(1_{E_{s+2}}\wedge(\phi\wedge
1_{M})ijj^{\prime}(j^{\prime\prime}\wedge 1_{K}))d((1_{E_{s+2}}\wedge
1_{L}\wedge m_{K})f_{4})$ = 0
so that the Theorem follows by the same reason. Q.E.D.
§5. A sequence of $h_{0}\sigma$ new families in the stable homotopy groups of
spheres
In this section, the convergence of a sequence of $h_{0}\sigma$ and
$h_{0}\sigma^{\prime}$ new families will be derived by the main Theorem A in
§2 and the main Theorem C in §4, where $\sigma$ and $\sigma^{\prime}$ is a
pair of $a_{0}$-related elements.
Theorem 9.5.1 Let $p\geq 7,n\geq 2$, then
$h_{0}h_{n}\in Ext_{A}^{2,p^{n}q+q}(Z_{p},Z_{p}),\quad h_{0}b_{n-1}\in
Ext_{A}^{3,p^{n}q+q}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge in the ASS to homotopy
elements of order $p$ in $\pi_{p^{n}q+q-2}S,\pi_{p^{n}q+q-3}S$ respectively.
Proof : By [12] Theorem 1.2.14 we have $d_{2}(h_{n})=a_{0}b_{n-1}\in
Ext_{A}^{3,p^{n}q+1}\\\ (Z_{p},Z_{p}),n\geq 1$, where
$d_{2}:Ext_{A}^{1,p^{n}q}(Z_{p},Z_{p})\to Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$
is a secondary differential in the ASS. That is, $h_{n}$ and $b_{n-1}$ is a
pair of $a_{0}$-related elements so that the main Theorem A can apply to
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$. We only need to
check the supposition (I)(II)(III) in the main Theorem A hold. By knowledge on
the $Z_{p}$-base of $Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $(s\leq 3)$ we know that
the the supposition (I)(II) of the main Theorem A hold for
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$. On the other
hand, from some results on $Ext_{A}^{4,*}(Z_{p},Z_{p})$ in [17] we know that
the following hold.
$Ext_{A}^{4,p^{n}q+rq+1}(Z_{p},Z_{p})=0(r=1,3,4),$
$Ext_{A}^{4,p^{n}q+rq}(Z_{p},Z_{p})=0(r=2,3)$,
$Ext_{A}^{4,p^{n}q+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}b_{n-1}\\}$,
$Ext_{A}^{4,p^{n}q+2}(Z_{p},Z_{p})\cong
Z_{p}\\{a_{0}^{2}b_{n-1}\\},Ext_{A}^{4,p^{n}q+1}(Z_{p},Z_{p})=0$.
That is, the supposition )III) of the main Theorem A hold for
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$. Then, by the
main Theorem A we obtain that $h_{0}b_{n-1}\in
Ext_{A}^{3,p^{n}q+q}(Z_{p},Z_{p})$, $i_{*}(h_{0}h_{n})\in
Ext_{A}^{2,p^{n}q+q}\\\ (H^{*}M,Z_{p})$ are permanent cycles in the ASS. By
Remark 9.2.35, the main Theorem A also obtains that $(1_{L}\wedge
i)_{*}\phi_{*}(h_{n})\in Ext_{A}^{2,p^{n}q+2q}(H^{*}L,Z_{p})$ is a permanent
cycle in the ASS so that the main Theorem C can apply to obtain the result of
the Theorem, this is because by knowledge on the $A_{p}$-base of
$Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=1,2,3$ we can easy to see that the
supposition (I) of the main Theorem C hold for
$(\sigma,\sigma^{\prime},s,tq)=(h_{n},b_{n-1},1,p^{n}q)$. Q.E.D.
Now we apply the main Theorem A and the main Theorem C to
$(\sigma,\sigma^{\prime})=(h_{n}h_{m},h_{n}b_{m-1}-h_{m}b_{n-1}),(s,tq)=(2,p^{n}q+p^{m}q)$
to obtain another sequence of $h_{0}\sigma$ families in the stable homotopy
groups of spheres. For checking the supposition (I)(II)(III) of the main
Theorem A, we first prove the following Proposition.
Proposition 9.5.2 Let $p\geq 7,n\geq m+2\geq 4,tq=p^{n}q+p^{m}q$, then
(1) $Ext_{A}^{4,tq+rq+u}(Z_{p},Z_{p})$ = 0 for $r=2,3,4,u=-1,0$ or
$r=3,4,u=1$,
$Ext_{A}^{4,tq+q}(Z_{p},Z_{p})\cong
Z_{p}\\{h_{0}h_{n}b_{m-1},h_{0}h_{m}b_{n-1}\\},$
$Ext_{A}^{4,tq}(Z_{p},Z_{p})\cong Z_{p}\\{b_{n-1}b_{m-1}\\},$
$Ext_{A}^{4,tq+1}(Z_{p},Z_{p})\cong
Z_{p}\\{a_{0}h_{n}b_{m-1},a_{0}h_{m}b_{n-1}\\}$
(2) $Ext_{A}^{5,tq+rq+1}(Z_{p},Z_{p})=0$ for $r=1,3,4$,
$Ext_{A}^{5,tq+rq}(Z_{p},Z_{p})$ = 0 for $r=2,3$,
$Ext_{A}^{5,tq+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}h_{n}b_{m-1},\widetilde{\alpha}_{2}h_{m}b_{n-1}\\}$,
$Ext_{A}^{5,tq+2}(Z_{p},Z_{p})\cong
Z_{p}\\{a_{0}^{2}h_{n}b_{m-1},a_{0}^{2}h_{m}b_{n-1}\\}$,
$Ext_{A}^{5,tq+1}(Z_{p},Z_{p})\cong Z_{p}\\{a_{0}b_{n-1}b_{m-1}\\}$,
$a_{0}^{2}b_{n-1}b_{m-1}\neq 0\in Ext_{A}^{6,tq+2}(Z_{p},Z_{p})$
Proof : By Theorem 5.5.3, there is a May spectral sequence (MSS)
$\\{E_{r}^{s,t,*},d_{r}\\}$ which converges to $Ext_{A}^{s,t}(Z_{p},Z_{p})$
and whose $E_{1}$-term is
$\quad E_{1}^{*,*,*}=E(h_{i,j}\mid i>0,j\geq 0)\otimes P(b_{i,j}\mid i>0,j\geq
0)\otimes P(a_{i}\mid i\geq 0)$,
where $E$ denotes the exterior algebra and $P$ denotes a polynomial algebra, ,
$h_{i,j}\in$ $E_{1}^{1,2(p^{i}-1)p^{j},2i-1}$, $b_{i,j}\in\\\
E_{1}^{2,2(p^{i}-1)p^{j+1},p(2i-1)},a_{i}\in E_{1}^{1,2p^{i}-1,2i+1}$.
Consider the following second degrees (mod $p^{n}q$) of the generators in the
$E_{1}^{*,*,*}$-term, where $0\leq i\leq n,n\geq m+2\geq 4$
$\qquad\mid h_{s,i}\mid=(p^{s+i-1}+\cdots+p^{i})q$ (mod $p^{n}q),\quad 0\leq
i<s+i-1<n$
$\qquad\qquad\quad=(p^{n-1}+\cdots+p^{i})q$ (mod $p^{n}q$ ), $0\leq
i<s+i-1=n$,
$\qquad\mid b_{s,i-1}\mid=(p^{s+i-1}+\cdots+p^{i})q$ (mod $p^{n}q$), $1\leq
i<s+i-1<n$,
$\qquad\qquad\quad=(p^{n-1}+\cdots+p^{i})q$ (mod $p^{n}q$), $1\leq i<s+i-1=n$.
$\qquad\mid a_{i+1}\mid=(p^{i}+\cdots+1)q+1$ (mod $p^{n}q$ ), $1\leq i<n$
$\qquad\mid a_{i+1}\mid=(p^{n-1}+\cdots+1)q+1$ (mod $p^{n}q$ ), $i=n$.
For degree $k=tq+rq+u$ such that $0\leq r\leq 4,-1\leq u\leq 2$ we have
$k\equiv p^{m}q+rq+u$ (mod $p^{n}q$ ). Then, for $3\leq w\leq 5$,
$E_{1}^{w,tq+rq+u,*}$ has no such generators which have one of the above
elements as a factor, this is because such a generator will have second degree
$(c_{n}p^{n-1}+\cdots+c_{1}p+c_{0})q+d$ (mod $p^{n}q$ ), where $c_{i}\neq
0(1\leq i\leq m-1$ or $m<i<n$), $0\leq c_{l}<p,l=0,\cdots,n,0\leq d\leq 5$. In
addition, the second degree $\mid b_{1,i-1}\mid=p^{i}q$ (mod $p^{n}q$)( $1\leq
i\leq n$),$\mid h_{1,i}\mid=p^{i}q$ (mod $p^{n}q$)( $0\leq i\leq n$). Then ,
exclude the above factor and the factor which has second degree $\geq tq+pq$,
we know that the only possibility of the factor of the generators in
$E_{1}^{w,tq+rq+u,*}$ are $a_{1},a_{0},h_{1,0}$ , $h_{1,n},h_{1.m},$
$b_{1,n-1},b_{1,m-1}$ .
Then, by degree reasons we have
$E_{1}^{4,tq+rq+1,*}$ = 0 for $r=3,4$, $E_{1}^{4,tq+rq+u,*}$ = 0 for
$r=2,3,4,u=-1,0$
$E_{1}^{4,tq,*}=Z_{p}\\{b_{1,n-1}b_{1,m-1}\\},\quad E_{1}^{4,tq+1,*}\cong
Z_{p}\\{a_{0}h_{1,n}b_{1,m-1},a_{0}h_{1,m}b_{1,n-1}\\}$,
$E_{1}^{4,tq+2,*}=Z_{p}\\{a_{0}^{2}h_{1,n}h_{1,m}\\}$,
$E_{1}^{4,tq+2q+1,*}=Z_{p}\\{h_{1,0}a_{1}h_{1,n}h_{1,m}\\}$,
$E_{1}^{4,tq+q,*}=Z_{p}\\{h_{1,0}h_{1,n}b_{1,m-1},h_{1,0}h_{1,m}b_{1,n-1}\\}$,
$E_{1}^{3,tq+1,*}=Z_{p}\\{a_{0}h_{1,n}h_{1,m}\\}$,
$E_{1}^{3,tq.*}=Z_{p}\\{h_{1,n}b_{1,m-1},h_{1,m}b_{1,n-1}\\}$,
$E_{1}^{3,tq+q,*}=Z_{p}\\{h_{1,0}h_{1,n}h_{1,m}\\}$, $E_{1}^{3,tq+2q+1,*}=0$
Note that the differentials in the MSS is derivative, that is,
$d_{r}(xy)=d_{r}(x)y+(-1)^{s}xd_{r}(y)$ for $x\in E_{1}^{s,t,*},y\in
E_{1}^{s^{\prime},t^{\prime},*}$. In addition,
,$a_{0},h_{1,n},b_{1,n-1},h_{1,0}a_{1}$ are permanent cycles in the MSS which
converge to $a_{0},h_{n},b_{n-1},\widetilde{\alpha}_{2}\in Ext_{A}^{*,*}\\\
(Z_{p},Z_{p})$ respectively.
Then, the differential $d_{r}E_{r}^{3,tq+sq+u,*}$ = 0 for all $r\geq 1$ and
$s=u=0$ or $s=1,u=0$ or $s=0,u=1$ or $s=2,u=1$ so that
$b_{1,n-1}b_{1,m-1},a_{0}h_{1,n}b_{1,m-1},$
$a_{0}h_{1,m}b_{1,n-1},h_{1,0}h_{1,n}b_{1,m-1},h_{1,0}h_{1,m}b_{1,n-1}\in
E_{r}^{4,*,*}$ are not $d_{r}$-boundary in the MSS and so
$b_{n-1}b_{m-1},a_{0}h_{n}b_{m-1},a_{0}h_{m}b_{n-1},h_{0}h_{n}b_{m-1},\\\
h_{0}h_{m}b_{n-1}$ are all nontrivial in $Ext_{A}^{4,*}(Z_{p},Z_{p})$. This
shows (1).
Similarly, by degree reasons we have
$E_{1}^{5,tq+q+1,*}\cong
Z_{p}\\{a_{0}h_{1,0}h_{1,n}b_{1,m-1},a_{0}h_{1,0}h_{1,m}b_{1,n-1},a_{1}b_{1,n-1}b_{1,m-1}\\}$
$E_{1}^{5,tq+rq+1,*}$ = 0 for $r=3,4$, $E_{1}^{5,tq+rq,*}=0$ for $r=2,3$
$E_{1}^{5,tq+2q+1,*}\cong
Z_{p}\\{h_{1,0}a_{1}h_{1,n}b_{1,m-1},h_{1,0}a_{1}h_{1,m}b_{1,n-1}\\}$
$E_{1}^{5,tq+1,*}=Z_{p}\\{a_{0}b_{1,n-1}b_{1,m-1}\\},\quad
E_{1}^{4,tq+2q+1,*}\cong Z_{p}\\{h_{1,0}a_{1}h_{1,n}h_{1,m}\\}$
$E_{1}^{5,tq+2,*}=Z_{p}\\{a_{0}^{2}h_{1,m}b_{1,n-1},a_{0}^{2}h_{1,n}b_{1,n-1}\\}$
All the generators of $E_{1}^{5,tq+q+1,*}$ dy in the MSS, this is because
$a_{0}h_{1,0}h_{1,n}b_{1,m-1}\\\ =-d_{1}(a_{1}h_{1,n}b_{1,m-1})$,
$a_{0}h_{1,0}h_{1,m}b_{1,n-1}=-d_{1}(a_{1}h_{1,m}b_{1,n-1})$ and
$d_{1}(a_{1}b_{1,n-1}b_{1,m-1})=-a_{0}h_{1,0}b_{1,n-1}b_{1,m-1}\neq 0\in
E_{1}^{5,tq+q+1,*}$
Then, $Ext_{A}^{5,tq+q+1}(Z_{p},Z_{p})$ = 0. In addition, similar to that
given in (1) we have, $d_{r}E_{r}^{4,tq+u,*}=0$, $d_{r}E_{r}^{4,tq+2q+1,*}=0$
for all $r\geq 1$, $u=1,2$ . Then, the generators in $E_{1}^{5,*,*}$ converge
in the MSS to
$\widetilde{\alpha}_{2}h_{n}b_{m-1},\widetilde{\alpha}_{2}h_{m}b_{n-1},a_{0}b_{n-1}b_{m-1},\\\
a_{0}^{2}h_{m}b_{n-1},a_{0}^{2}h_{n}b_{m-1}$ respectively. For the last
result, note that $d_{r}E_{r}^{5,tq+2,*}$ = 0 for all $r\geq 1$ , then
$a_{0}^{2}b_{n-1}b_{m-1}\neq 0\in Ext_{A}^{6,tq+2}(Z_{p},Z_{p})$. This shows
(2). Q.E.D.
Theorem 9.5.3 Let $p\geq 7,n\geq m+2\geq 4$, then
$h_{0}h_{n}h_{m}\in Ext_{A}^{3,p^{n}q+p^{m}q+q}(Z_{p},Z_{p}),$
$h_{0}(h_{n}b_{m-1}-h_{m}b_{n-1})\in Ext_{A}^{4,p^{n}q+p^{m}q+q}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to homotopy elements of
order $p$ in $\pi_{p^{n}q+p^{m}q+q-3}S$ and $\pi_{p^{n}q+p^{m}q+q-4}S$
respectively.
Proof : By [12]p.11 Theorem 1.2.14, there is a nontrivial secondary
differential $d_{2}(h_{n})=a_{0}b_{n-1}(n\geq 1$ ) and it follows that
$d_{2}(h_{n}h_{m})=d_{1}(h_{n})h_{m}+(-1)^{1+p^{n}q}h_{n}d_{2}(h_{m})=a_{0}h_{m}b_{n-1}-a_{0}h_{n}b_{m-1}$.
That is, $(h_{n}h_{m},h_{m}b_{n-1}-h_{n}b_{m-1})$ is a pair of $a_{0}$-related
elements. By applying the main Theorem A to
$(\sigma,\sigma^{\prime})=(h_{n}h_{m},h_{m}b_{n-1}-h_{n}b_{m-1}),(s,tq)=(2,p^{n}q+p^{m}q)$
we have $h_{0}(h_{m}b_{n-1}-h_{n}b_{m-1})\\\ \in
Ext_{A}^{4,p^{n}q+p^{m}q+q}(Z_{p},Z_{p})$ and $i_{*}(h_{0}h_{n}h_{m})\in
Ext_{A}^{3,p^{n}q+p^{m}q+q}(Z_{p},Z_{p})$ are permanent cycles in the ASS,
this is because by knowledge of $Z_{p}$-base of $Ext_{A}^{s,*}(Z_{p},\\\
Z_{p})$ for $s\leq 3$ we know that the supposition (I)(II)(III) of the main
Theorem A hold. By Remark 9.2.35, the main Theorem A also obtains that
$(1_{L}\wedge i)_{*}\phi_{*}(h_{n}h_{m})\in
Ext_{A}^{3,p^{n}q+p^{m}q+2q}(H^{*}L,Z_{p})$ is a permanent cycle in the ASS so
that by the main Theorem C , the result of the Theorem follows. This is
because the supposition (I) of the main Theorem C hold by the knowledge of the
$Z_{p}$-base of $Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=1,2,3$. Q.E.D.
From Theorem 9.5.1 and Theorem 9.5.3 , we obtain four families of
$h_{0}\sigma$ new families. In fact, there are many pairs of $a_{0}$-related
elements so that we can expect to obtain some other sequence of $h_{0}\sigma$
new families in the stable homotopy groups of speheres. We have the following
conjectures.
Conjecture 9.5.4 Let $p\geq 7,n\geq 3$, then there is a secondary differential
$d_{2}(g_{n})=a_{0}l_{n}\in Ext_{A}^{4,p^{n+1}q+2p^{n}q+1}(Z_{p},Z_{p}),n\geq
3$ (up to nonzero scalar) and
$h_{0}g_{n}\in Ext_{A}^{3,p^{n+1}q+2p^{n}q+q}(Z_{p},Z_{p})$
$h_{0}l_{n}\in Ext_{A}^{4,p^{n+1}q+2p^{n}q+q}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to homotopy elements of
order $p$ in $\pi_{p^{n+1}q+2p^{n}q+q-3}S$ and $\pi_{p^{n+1}q+2p^{n}q+q-4}S$
respectively, where $g_{n}\in
Ext_{A}^{2,p^{n+1}q+2p^{n}q}(Z_{p},Z_{p}),l_{n}\in
Ext_{A}^{3,p^{n+1}q+2p^{n}q}(Z_{p},Z_{p})$.
Conjecture 9.5.5 Let $p\geq 7,n\geq 3$, then there is a secondary differential
$d_{2}(k_{n})=a_{0}l^{\prime}_{n}\in
Ext_{A}^{4,2p^{n+1}q+p^{n}q+1}(Z_{p},Z_{p})$ ( up to nonzero scalar), $n\geq
3$ and
$h_{0}k_{n}\in Ext_{A}^{3,2p^{n+1}q+p^{n}q+q}(Z_{p},Z_{p})$
$h_{0}l^{\prime}_{n}\in Ext_{A}^{4,2p^{n+1}q+p^{n}q+q}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to homotopy elementws of
order $p$ in $\pi_{2p^{n+1}+p^{n}q+q-3}S$ and $\pi_{2p^{n+1}q+p^{n}q+q-4}S$ ,
where $k_{n}\in Ext_{A}^{2,2p^{n+1}q+p^{n}q}(Z_{p},Z_{p}),l^{\prime}_{n}\in
Ext_{A}^{3,2p^{n+1}q+p^{n}q}(Z_{p},Z_{p})$.
Remark 9.5.6 By [10][25], there is Thom map
$\Phi:Ext_{BP_{*}BP}^{s,*}(BP_{*},BP_{*})\\\ \rightarrow
Ext_{A}^{s,*}(Z_{p},Z_{p})$( $s=2,3$) such that
$\Phi(\beta_{p^{n-1}/p^{n-1}-1})=h_{0}h_{n}$,
$\Phi(\beta_{p^{n-1}/p^{n-1}})=b_{n-1}$,
$\Phi(\beta_{p^{m-1}/p^{m-1}-1}\beta_{p^{n-1}/p^{n-1}}-\beta_{p^{n-1}/p^{n-1}-1}\beta_{p^{m-1}/p^{m-1}})=h_{0}(h_{m}b_{n-1}\\\
-h_{n}b_{m-1})$,
$\Phi(\gamma_{p^{n-2}/p^{n-2}-p^{m-1},p^{m-1}-1})=h_{0}h_{n}h_{m}$. Then, the
$h_{0}h_{n},h_{0}b_{n-1},\\\
h_{0}(h_{m}b_{n-1}-h_{n}b_{m-1}),h_{0}h_{n}h_{m}$-map obtained by Theorem
9.5.1 and Theorem 9.5.3 are represented by $\beta_{p^{n-1}/p^{n-1}-1}$ \+
other terms $\in Ext_{BP_{*}BP}^{2,p^{n}q+q}(BP_{*},BP_{*})$,
$\alpha_{1}\beta_{p^{n-1}/p^{n-1}}$ \+ other terms $\in
Ext_{BP_{*}BP}^{3,p^{n}q+q}(BP_{*},BP_{*})$,
$\beta_{p^{m-1}/p^{m-1}-1}\beta_{p^{n-1}/p^{n-1}}-\beta_{p^{n-1}/p^{n-1}-1}\cdot\beta_{p^{m-1}/p^{m-1}}$
\+ other terms $\in Ext_{BP_{*}BP}^{4,p^{n}q+p^{m}q+q}(BP_{*},BP_{*})$,
$\gamma_{p^{n-2}/p^{n-2}-p^{m-1},p^{m-1}-1}$ \+ other terms $\in
Ext_{BP_{*}BP}^{3,p^{n}q+p^{m}q+q}(BP_{*},BP_{*})$ respectively in the Adams-
Novikov spectral sequence.
§6. A sequence of
$h_{0}\sigma\widetilde{\gamma}_{s},g_{0}\sigma\widetilde{\gamma}_{s}$ new
families in the stable homotopy groups of spheres
In this section, we use the main Theorem B to obtain
$i^{\prime}_{*}i_{*}(g_{0}h_{n}),\\\
i^{\prime}_{*}i_{*}(g_{0}b_{n-1}),i^{\prime}_{*}i_{*}(g_{0}h_{n}h_{m}),i^{\prime}_{*}i_{*}(g_{0}(h_{n}b_{m-1}-b_{m}b_{n-1}))$
et al converge to the corresponding nontrivial homotopy elements in the
homotopy groups of Smith-Toda spectrum $V(1)$. In base of these results, we
obtain a sequence of
$g_{0}\sigma\widetilde{\gamma}_{s},h_{0}\sigma\widetilde{\gamma}_{s}$ new
families in the stable homotopy groups of spheres.
Theorem 9.6.1 Let $p\geq 5,n\geq 2$, then
$i^{\prime}_{*}i_{*}(g_{0}h_{n})\in Ext_{A}^{3,p^{n}q+pq+2q}(H^{*}K,Z_{p})$
$i^{\prime}_{*}i_{*}(g_{0}b_{n-1})\in Ext_{A}^{4,p^{n}q+pq+2q}(H^{*}K,Z_{p})$
are permanent cycles in the ASS and they converge to the corresponding
homotopy element in $\pi_{p^{n}q+pq+2q-3}K,\pi_{p^{n}q+pq+2q-4}K$
respectively.
Proof : We first apply the main Theorem B to $(\sigma,s,tq)=(h_{n},1,p^{n}q)$.
By Theorem 9.5.1, the supposition (II) of the main Theorem B holds. Moreover,
by knowledge on the $Z_{p}$-base of $Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=1,2,3$
we know that the supposition (I) of the main Theorem B holds, then the first
result of the Theorem follows by the main Theorem B.
For the second result, we apply the main Theorem B to
$(\sigma,s,tq)=(b_{n-1},2,p^{n}q)$. Similarly by Theorem 9.5.1, the
supposition (II) of the main Theorem B holds. Noreover, by knowledge on the
$Z_{p}$-base of $Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=2,3$ and some result in
[17] on $Ext_{A}^{4,*}(Z_{p},Z_{p})$ we know that the supposition (I) of the
main Theorem B holds. Then , the second result also follows by the main
Theorem B. Q.E.D.
alternative Proof : It is known from the proof of Theorem 9.5.1 that the
supposition (I)(II)(III) if the main Theorem A hold for
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$. Then , applying
the main Theorem B’ in $\S 3$ to
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$ we obtain the two
results of the Theorem. Q.E.D.
Theorem 9.6.2 Let $p\geq 7,n\geq m+2\geq 4$, then
$i^{\prime}_{*}i_{*}(g_{0}h_{n}h_{m})\in
Ext_{A}^{4,p^{n}q+p^{m}q+pq+2q}(H^{*}K,Z_{p})$
$i^{\prime}_{*}i_{*}(g_{0}(h_{n}b_{m-1}-h_{m}b_{n-1}))\in
Ext_{A}^{5,p^{n}q+p^{m}q+pq+2q}(H^{*}K,Z_{p})$
are permanent cycles in the ASS and they converge to nontrivial homotopy
elements in $\pi_{p^{n}q+p^{m}q+pq+2q-4}K,\pi_{p^{n}q+p^{m}q+pq+2q-5}K$
respectively.
Proof : We first apply the main Theorem B to
$(\sigma,s,tq)=(h_{n}h_{m},2,p^{n}q+p^{m}q)$. By Theorem 9.5.3, the
supposition (II) of the main Theorem B holds. By knowledge on the $Z_{p}$-base
of $Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=2,3$ and some result in [17] on
$Ext_{A}^{4,*}(Z_{p},Z_{p})$ we know that the supposition (I) of the main
Theorem B also holds. Then the first result follows by the main Theorem B.
Moreover, we apply the main Theorem B to
$(\sigma,s,tq)=(h_{n}b_{m-1}-h_{m}b_{n-1},3,p^{n}q+p^{m}q)$. Similarly by
Theorem 9.5.3, the supposition (II) of the main Theorem B holds. By knowledge
on the $Z_{p}$-base of $Ext_{A}^{3,*}(Z_{p},Z_{p})$ for $s=3,4$ and the result
on $Ext_{A}^{5,p^{n}q+p^{m}q+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}h_{n}b_{m-1},\widetilde{\alpha}_{2}h_{m}b_{n-1}\\}$
in Prop. 9.5.2 we know that the supposition (I) of the main Theorem B holds.
Then, the second result follows immediately by the main Theorem B. Q.E.D.
alternative Proof : It is known from the proof of Theorem 9.5.2 that the
supposition (I)(II)(III) of the main Theorem A hold for
$(\sigma,\sigma^{\prime})=(h_{n}h_{m},h_{n}b_{m-1}-h_{m}b_{n-1}),(s,tq)=(2,p^{n}q+p^{m}q)$.
Then by applying the main Theorem B’ in §3 to
$(\sigma,\sigma^{\prime})=(h_{n}h_{m},h_{n}b_{m-1}-h_{m}b_{n-1}),(s,tq)=(2,p^{n}q+p^{m}q)$,
we obtain the two result of the Theorem. Q.E.D.
Using the notation in the cofibration (6.2.7)–(6.2.10), we know that
$\widetilde{\gamma}_{s}=((j_{1}j_{2}j_{3})_{*}(\gamma)_{*}^{s}(i_{3}i_{2}i_{1})_{*}(1)\in
Ext_{A}^{s,sp^{2}q+(s-1)pq+(s-2)q+s-3}(Z_{p},Z_{p})$
converges to the following third periodicity element in the ASS
$\gamma_{s}=j_{1}j_{2}j_{3}\gamma^{s}i_{3}i_{2}i_{1}\in\pi_{sp^{2}q+(s-1)pq+(s-2)q-3}S$
where $3\leq s<p$ and $1\in Ext_{A}^{0,0}(Z_{p},Z_{p})$. Now we consider the
products
$g_{0}\sigma\widetilde{\gamma}_{s},h_{0}\sigma\widetilde{\gamma}_{s}$, in
$Ext_{A}^{*,*}(Z_{p},Z_{p})$ and we will prove that they converge to the
corresponding homotopy element of order $p$ in the stable homotopy groups of
spheres, where $\sigma=h_{n},b_{n-1},h_{n}h_{m},$ or
$h_{n}b_{m-1}-h_{m}b_{n-1}$.
Theorem 9.6.3 Let $p\geq 7,n\geq 3,3\leq s<p$, then the products
$g_{0}h_{n}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+3,p^{n}q+sp^{2}q+spq+sq+s-3}(Z_{p},Z_{p})$
$g_{0}b_{n-1}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+4,p^{n}q+sp^{2}q+spq+sq+s-3}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to the corresponding
homotopy elements of order $p$ in the stable homotopy groups of spheres.
Proof: By Theorem 9.6.1, there is a nontrivial $f\in\pi_{p^{n}q+pq+2q-3}K$
such that it is represented by $i^{\prime}_{*}i_{*}(g_{0}h_{n})\in
Ext_{A}^{3,p^{n}q+pq+2q}(H^{*}K,Z_{p})$ in the ASS. Let
$\tilde{f}=j_{1}j_{2}j_{3}\gamma^{s}i_{3}f$ be the following composition
($tq=p^{n}q+pq+2q-3$)
$\qquad\tilde{f}:\Sigma^{tq}S\stackrel{{\scriptstyle
f}}{{\longrightarrow}}V(1)\stackrel{{\scriptstyle
i_{3}}}{{\longrightarrow}}V(2)\stackrel{{\scriptstyle\gamma^{s}}}{{\longrightarrow}}\Sigma^{-s(p^{2}q+pq+q)}V(2)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
j_{1}j_{2}j_{3}}}{{\longrightarrow}}\Sigma^{-s(p^{2}+p+1)q+(p+2)q+q+3}S$
Since $f$ is represented by $(i_{2})_{*}(i_{1})_{*}(g_{0}h_{n})\in
Ext_{A}^{3,p^{n}q+pq+2q}(H^{*}K,Z_{p})$ in the ASS, then the above $\tilde{f}$
is represented by
$c=(j_{1}j_{2}j_{3})_{*}(\gamma_{*})^{s}(i_{3}i_{2}i_{1})_{*}(g_{0}h_{n})\in
Ext_{A}^{s+3,p^{n}q+s(p^{2}+p+1)q+s-3}(Z_{p},Z_{p})$
By knowledge of Yoneda products we know that the above element $c$ is just the
products $g_{0}h_{n}\widetilde{\gamma}_{s}\in
Ext_{A}^{s+3,p^{n}q+s(p^{2}+p+1)q+s-3}(Z_{p},Z_{p})$. Then, to obtain the
first result, it suffices to prove the product
$g_{0}h_{n}\widetilde{\gamma}_{s}$ is nonzero in the Ext group and it is not a
$d_{r}$-boundary in the ASS, that is, we still need to prove
$Ext_{A}^{s+3-r,p^{n}q+s(p^{2}+p+1)q+s-2-r}(Z_{p},Z_{p})$ is zero for $r\geq
2$. We may prove this two facts by an argument in the May spectral sequence.
By degree reasons, $h_{n},g_{0},\widetilde{\gamma}_{s}$ is represented by
$h_{1,n},h_{2,0}h_{1,0},h_{2,1}h_{1,2}h_{3,0}a_{3}^{s-3}\in E_{1}^{*,*,*}$ in
the MSS respectively. Then, the products $g_{0}h_{n}\widetilde{\gamma}_{s}$ is
represented by
$h_{1,n}h_{2,0}h_{1,0}h_{2,1}h_{1,2}h_{3,0}a_{3}^{s-3}\in
E_{1}^{s+3,p^{n}q+s(p^{2}+p+1)q+s-3,*}$
in the MSS and so we can do some computation in the degree to prove
$E_{1}^{s+2,p^{n}q+s(p^{2}+p+1)q+s-3,*}=0$ and
$E_{1}^{s+3-r,p^{n}q+s(p^{2}+p+1)q+s-2-r,*}=0(r\geq 2)$ so that the first
result follows. We leave this computation to the reader. The proof and
computation for the second result is similar. Q.E.D.
By using Theorem 9.6.2, Theorem 9.5.1 and Theorem 9.5.3, similar to that given
in the proof of Theorem 9.6.3, we can obtain the following Theorem
9.6.4–9.6.6.
Theorem 9.6.4 Let $p\geq 7,n\geq m+2\geq 5,3\leq s<p$, then
$g_{0}h_{n}h_{m}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+4,p^{n}q+p^{m}q+s(p^{2}+p+1)q+s-3}(Z_{p},Z_{p})$
$g_{0}(h_{n}b_{m-1}-h_{m}b_{n-1})\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+5,p^{n}q+p^{m}q+s(p^{2}+p+1)q+s-3}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to the corresponding
homotopy elements of order $p$ in the stable homotopy groups of spheres.
Theorem 9.6.5 Let $p\geq 7,n\geq 3,3\leq s<p$, then the products
$h_{0}h_{n}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+2,p^{n}q+sp^{2}q+(s-1)(p+1)q+s-3}(Z_{p},Z_{p})$
$h_{0}b_{n-1}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+3,p^{n}q+sp^{2}q+(s-1)(p+1)q+s-3}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to the corresponding
elements of order $p$ in the stable homotopy groups of spheres.
Theorem 9.6.6 Let $p\geq 7,n\geq m+2\geq 5,3\leq s<p$, then the products
$h_{0}h_{n}h_{m}\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+3,p^{n}q+p^{m}q+sp^{2}q+(s-1)(p+1)q+s-3}(Z_{p},Z_{p})$
$h_{0}(h_{n}b_{m-1}-h_{m}b_{n-1})\widetilde{\gamma}_{s}\neq 0\in
Ext_{A}^{s+4,p^{n}q+p^{m}q+sp^{2}q+(s-1)(p+1)q+s-3}(Z_{p},Z_{p})$
are permanent cycles in the ASS and they converge to the corresponding
homotopy elements of order $p$ in the stable homotopy groups of spheres.
Remark 9.6.7 The new families obtained in Theorem 9.6.5 and Theorem 9.6.6
are the composition products of $h_{0}h_{n}$-element , $h_{0}b_{n-1}$-element
in Theorem 9.5.1,
$h_{0}h_{n}h_{m}$-element,$h_{0}(h_{n}b_{m-1}-h_{m}b_{n-1})$\- element in
Theorem 9.5.3 and
$\gamma_{s}=j_{1}j_{2}j_{3}\gamma^{s}i_{3}i_{2}i_{1}\in\pi_{sp^{2}q+(s-1)pq+(s-2)q-3}S$.
However, the new families obtained in Theorem 9.6.3 and Theorem 9.6.4 are
indecomposable elements in the stable homotopy groups of spheres, that is,
they are not compositions of some other elements of lower degrees in the
stable homotopy groups of spheres. This is because $g_{0}\in
Ext_{A}^{2,pq+2q}(Z_{p},Z_{p})$ dies in the ASS, that is, it support a
nontrivial differential in the Adams spectral sequence :
$d_{2}(g_{0})=b_{0}\widetilde{\alpha}_{2}$ (up to nonzero scalar) $\in
Ext_{A}^{4,pq+2q+1}(Z_{p},Z_{p})$ which can be easily proved as follows. Since
$\widetilde{\alpha}_{2},b_{0}$ converge in the ASS to
$\alpha_{2}=j\alpha^{2}i,\beta_{1}=jj^{\prime}\beta
i^{\prime}i\in\pi_{*}S$,then the composition products of
$\beta_{1}\alpha_{2}\in\pi_{pq+2q-3}S$ must be represented by
$b_{0}\widetilde{\alpha}_{2}\in Ext_{A}^{4,pq+2q+1}(Z_{p},Z_{p})$ in the ASS.
However, it is easily seen that
$\beta_{1}\alpha_{2}=jj^{\prime}\beta\i^{\prime}ij\alpha^{2}i=0$ and
$b_{0}\widetilde{\alpha}_{2}\neq 0\in Ext_{A}^{4,pq+2q+1}(Z_{p},Z_{p})$, then
$b_{0}\widetilde{\alpha}_{2}$ must be a $d_{r}$-boundary. By degree reason,
the only possibility is $b_{0}\widetilde{\alpha}_{2}=d_{2}(g_{0})$(up to
nonzero scalar).
Conjecture 9.6.8 By the conjecture 9.5.4–9.5.5, we can conjecture that, for
$p\geq 7,n\geq 3,3\leq s<p$, the products
$h_{0}g_{n}\widetilde{\gamma}_{s},h_{0}l_{n}\widetilde{\gamma}_{s},g_{0}g_{n}\widetilde{\gamma}_{s},g_{0}l_{n}\widetilde{\gamma}_{s},\\\
h_{0}k_{n}\widetilde{\gamma}_{s},h_{0}l^{\prime}_{n}\widetilde{\gamma}_{s},g_{0}k_{n}\widetilde{\gamma}_{s},g_{0}l^{\prime}_{n}\widetilde{\gamma}_{s}$
are permanent cycles in the ASS and they converge to the corresponding
homotopy elements of order $p$ in the stable homotopy groups of spheres. In
addition, all results or conjectures in this section also hold when we replace
the products with $\widetilde{\gamma}_{s}$ to be the products with
$\widetilde{\beta}_{s}(2\leq s<p)$. That is, we can obtain a sequence of
$h_{0}\sigma\widetilde{\beta}_{s},g_{0}\sigma\widetilde{\beta}_{s}$-elements,
where
$\widetilde{\beta}_{s}=(j_{1}j_{2})_{*}\beta_{*}^{s}(i_{1}i_{1})_{*}(1)\in
Ext_{A}^{s,spq+(s-1)q+s-2}(Z_{p},Z_{p}),2\leq s<p$.
§7. Third periodicity families in the stable homotopy groups of spheres
In this section, we will first prove the convergence of $h_{n}$-elements in
the homotopy groups of Smith-Toda spectrum $V(1)$ and in base of this we
obtain the convergence of third periodicity $\gamma_{p^{n}/s}$ families
$(1\leq s\leq p^{n}-1)$ in the Adams-Novikov spectral sequence.
Theorem 9.7.1 ([9] Theorem II) Let $p\geq 5,n\geq 0$,
$h_{n}\in Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K)$
be the element represented by $[t_{1}^{p^{n}}]$ in the cobar complex. Then
this $h_{n}$ is a permanent cycle in the Adams-Novikov spectral sequence and
it converges to a nontrivial homotopy element in $\pi_{p^{n}q-1}K$.
The proof of the above $h_{n}$-Theorem will be the main content of this
section. By Theorem 8.1.6(b)(ii), there is a relation
(9.7.2) $h_{n}=c_{2}(p^{n-2})+v_{2}^{p^{n-2}}h_{n-2}\in
Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*}BP_{*}K)$
By [10].p.502 Cor. 7.8, the image of $v_{2}^{s}c_{2}(p^{n-2})(p^{n-2}>s\geq
1)$ under the boundary homomorphism (or connecting homomorphism)
$j^{\prime}_{*}:Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}K)\to
Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*}M)$
and
$j_{*}:Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*}M)\to
Ext_{BP_{*}BP}^{3,*}(BP_{*},BP_{*})$
is just the third periodicity family $\gamma_{p^{n-2}/p^{n-2}-s}\neq 0\in
Ext_{BP_{*}BP}^{3,*}(BP_{*},\\\ BP_{*})$. Then, by Theorem 9.7.1, the relation
(9.7.2) and Theorem 7.3.2, we can obtain the following convergence Theorem of
third periodicity families in the stable homotopy groups of spheres
immediately.
Theorem 9.7.3 ([9] Theorem I) Let $p\geq 5,n\geq 1$ and $1\leq s\leq p^{n}-1$,
then the following third periodicity family
$\gamma_{p^{n}/s}\in Ext_{BP_{*}BP}^{3,*}(BP_{*},BP_{*})$
is a permanent cycle in tha ASS and it converge to an element of order $p$ in
$\pi_{*}S$ which has degree $p^{n+2}q+(p^{n}-s)(p+1)q-q-3$.
To prove Theorem 9.7.1, we first prove the following weaker Theorem.
Theorem 9.7.4 ([9] Theorem 4.1) Let $p\geq 5,n\geq 0$, $h_{n}\in
Ext_{A}^{1,p^{n}q}(Z_{p},\\\ Z_{p})$ be the element represented by
$\xi^{p^{n}}$ in the cobar complex, then $i^{\prime}_{*}i_{*}(h_{n})\in
Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$ is a permanent cycle in the ASS and it
converge to a nontrivial homotopy element in $\pi_{p^{n}q-1}K$.
The proof of Theorem 9.7.4 will be the main content of the rest of this
section. The proof need some preminilaries on low dimensional Ext groups and
an argument processing in the Adams resolution of some spectra related to $S$.
We first prove some results on Ext groups.
Theorem 9.7.5 Let $p\geq 3,n\geq 2,$ , then
(1) $Ext_{A}^{s,p^{n}q+r}(H^{*}K,Z_{p})=0$ for $s=2,3,r=1,2$,
$Ext_{A}^{s,p^{n}q+1}(H^{*}K,H^{*}M)=0$,
$Ext_{A}^{3,p^{n}q+q}(H^{*}K,H^{*}K)\cong Z_{p}\\{(h_{0}b_{n-1})^{\prime}\\}$.
(2) $Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}Y,Z_{p})$ = 0 for s = 1,2,3.
(3) $Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}Y)$ = 0,
where $Y$ is the spectrum in the cofibration (9.1.4).
Proof : (1) Consider the following exact sequence ($s=1,2,3,r=1,2)$
$Ext_{A}^{s,p^{n}q+r}(Z_{p},Z_{p})\stackrel{{\scriptstyle
i_{*}}}{{\rightarrow}}Ext_{A}^{s,p^{n}q+r}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
j_{*}}}{{\rightarrow}}Ext_{A}^{s,p^{n}q+r-1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p_{*}}}{{\rightarrow}}$
induced by (9.1.1). By knowledge of $Z_{p}$-base of
$Ext_{A}^{s,*}(Z_{p},Z_{p})$ for $s=1,2,3$ we know that the right group is
zero except for $(s,r)=(1,1),(2,1),(2,2),(3,2)$ it has unique generator
$h_{n},b_{n-1},a_{0}h_{n},a_{0}b_{n-1}$. However,$p_{*}(h_{n})=a_{0}h_{n}\neq
0,p_{*}(b_{n-1})=a_{0}b_{n-1}\neq 0,p_{*}(a_{0}h_{n})=a_{0}^{2}h_{n}\neq
0,p_{*}(a_{0}b_{n-1})=a_{0}^{2}b_{n-1}\neq 0$, then the above $p_{*}$ is monic
and so im $j_{*}$ = 0. In addition, the left group is zero except for
$(s,r)=(2,1),(3,1),(3,2)$ it has unique generator
$a_{0}h_{n}=p_{*}(h_{n}),a_{0}b_{n-1}=p_{*}(b_{n-1}),a_{0}^{2}h_{n}=p_{*}(a_{0}h_{n})$
respectively. Then we have im $i_{*}$ = 0 and obtain that
$Ext_{A}^{s,p^{n}q+r}(H^{*}M,Z_{p})=0$ for $s=1,2,3,r=1,2$.
Look at the following exact sequence ($s=2,3,r=1,2)$
$0=Ext_{A}^{s,p^{n}q+r}(H^{*}M,Z_{p})\stackrel{{\scriptstyle
i^{\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q+r}(H^{*}K,Z_{p})$
$\qquad\qquad\quad\stackrel{{\scriptstyle
j^{\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q-q+r-1}(H^{*}M,Z_{p})$
induced by (9.1.2). The left group is zero as shown above. The right group
also is zero, this is because $Ext_{A}^{s,p^{n}q-q+r-1}(Z_{p},Z_{p})$ = 0 for
$s=2,3,r=1,2,3$(cf. Chap. 5). Then, the middle group is zero for $s=2,3,r=1,2$
and so $Ext_{A}^{s,p^{n}q+1}(H^{*}K,H^{*}M)$ = 0 $(s=2,3$).
For the last result, consider the following exact sequence
$Ext_{A}^{3,p^{n}q+2q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(H^{*}K,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero by Prop. 9.3.2(1) and the right
group has unique generator $i^{\prime}_{*}(\alpha_{1}\wedge
1_{M})_{*}(\tilde{b}_{n-1}))$ (cf. Prop. 9.3.1) such that
$\alpha^{*}(i^{\prime})_{*}(\alpha_{1}\wedge 1_{M})_{*}(\tilde{b}_{n-1})$ = 0.
Then, the middle group has unique generator $(h_{0}b_{n-1})^{\prime}$ such
that $(i^{\prime})^{*}(h_{0}b_{n-1})^{\prime}=i^{\prime}_{*}(\alpha_{1}\wedge
1_{M})(\tilde{b}_{n-1})$. Q.E.D.
(2) The result is obvious for $s=1$. For $s=2,3$, consider the following exact
sequence
$\stackrel{{\scriptstyle(i^{\prime}i)_{*}}}{{\longrightarrow}}Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}K,Z_{p})\stackrel{{\scriptstyle
r_{*}}}{{\longrightarrow}}Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}Y,Z_{p})$
$\qquad\qquad\stackrel{{\scriptstyle\epsilon_{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q+q+s-3}(Z_{p},Z_{p})\stackrel{{\scriptstyle(i^{\prime}i)_{*}}}{{\longrightarrow}}$
induced by (9.1.4). The left group is zero for $s=2$, this is because
$Ext_{A}^{1,t}(Z_{p},Z_{p})$ = 0 for $t=-1,-2$ (mod $q$). The left group has
unique genertor $(i^{\prime}i)_{*}(h_{0}h_{n})$ for $s=3$ so that im $r_{*}$ =
0. The right group is zero for $s=2$ and has unique generator $h_{0}b_{n-1}$
for $s=3$ which satisfies $(i^{\prime}i)_{*}(h_{0}b_{n-1})\neq 0$, then im
$\epsilon_{*}$ = 0 and so the middle group is zero for $s=1,2,3$.
(3) Observe the following exact sequence
$0=Ext_{A}^{0,p^{n}q}(H^{*}K,Z_{p})\stackrel{{\scriptstyle\epsilon^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}Y)$
$\qquad\qquad\stackrel{{\scriptstyle
r^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle(i^{\prime}i)^{*}}}{{\longrightarrow}}$
induced by (9.1.4). The left group clearly is zero and the right group has
unique generator $(h_{n})^{\prime}$ (cf. Prop. 9.3.6) which satisfies
$(i^{\prime}i)^{*}(h_{n})^{\prime}=(i^{\prime}i)_{*}(h_{n})\neq 0\in
Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$ , then the middle group is zero as desired.
Q.E.D.
Prop. 9.7.6 Let $p\geq 3,n\geq 2,$ then
(1) $\quad Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}M)$ = 0,
$Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}M)$ = 0.
(2) $\quad Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}K)$ = 0 ,
$Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}K)$ = 0.
(3) $\quad Ext_{A}^{2,p^{n}q+q-u}(Z_{p},H^{*}K)=0$ for $u=0,1$ ,
$Ext_{A}^{3,p^{n}q+q}(Z_{p},H^{*}K)$ = 0.
Proof: (1) Consider the following exact sequences
$Ext_{A}^{2,p^{n}q+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\rightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}M)\stackrel{{\scriptstyle
i^{*}}}{{\rightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\rightarrow}}$
$Ext_{A}^{3,p^{n}q+2}(Z_{p},Z_{p})\stackrel{{\scriptstyle
j^{*}}}{{\rightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}M)\stackrel{{\scriptstyle
i^{*}}}{{\rightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})\stackrel{{\scriptstyle
p^{*}}}{{\rightarrow}}$
induced by (9.1.1). The upper left group has unique generator $a_{0}h_{n}$
which satisfies $j^{*}(a_{0}h_{n})=j^{*}p^{*}(h_{n})$ = 0 and the upper right
group has unique generator $b_{n-1}$ satisfying
$p^{*}(b_{n-1})=a_{0}b_{n-1}\neq 0\in Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$ (cf.
Theorem 5.4.1), then we have $Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}M)$ = 0. The lower
left group has unique generator $a_{0}^{2}h_{n}$ satisfying
$j^{*}(a_{0}^{2}h_{n})=j^{*}p^{*}$ $(a_{0}h_{n})$ = 0 and the lower right
group has unique generator $a_{0}b_{n-1}$ such that
$p^{*}(a_{0}b_{n-1})=a_{0}^{2}b_{n-1}\neq 0\in
Ext_{A}^{4,p^{n}q+2}(Z_{p},Z_{p})$ (cf. Prop. 9.5.2(2)) , then
$Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}M)$ = 0.
(2) Consider the following exact sequences
$0=Ext_{A}^{2,p^{n}q+q+1}(Z_{p},H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}K)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}M)=0$
$0=Ext_{A}^{3,p^{n}q+q+2}(Z_{p},H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}K)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}M)=0$
induced by (9.1.2). Both two right groups are zero by (1) and both two left
groups are also zero , this is because $Ext_{A}^{2,p^{n}q+q+r}(Z_{p},Z_{p})$ =
0 for $r=1,2$ (cf. Chapter 5) and $Ext_{A}^{3,p^{n}q+q+r}(Z_{p},Z_{p})$ = 0
for $r=2,3$ (cf. Theorem 5.4.1), then the result follows.
(3) Consider the following exact sequence
$\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+2q}(Z_{p},H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+q-1}(Z_{p},H^{*}K)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+q-1}(Z_{p},H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero, this is because
$Ext_{A}^{2,p^{n}q+2q+r}(Z_{p},Z_{p})$ = 0 for $r=0,1$ (cf. Chapter 5). The
right group has unique generator $j^{*}(h_{0}h_{n})$ since
$Ext_{A}^{2,p^{n}q+q-1}(Z_{p},Z_{p})$ = 0 and
$Ext_{A}^{2,p^{n}q+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}h_{n}\\}$. In addition,
we claim that $\alpha^{*}j^{*}(h_{0}h_{n})$ = $\frac{1}{2}\cdot
j^{*}(\widetilde{\alpha}_{2}h_{n})\neq 0\in Ext_{A}^{3,p^{n}q+2q}(Z_{p},\\\
H^{*}M)$ . To prove this, it suffices to prove
$\alpha^{*}j^{*}(h_{0})=\frac{1}{2}j^{*}(\widetilde{\alpha}_{2})\in
Ext_{A}^{2,2q}(Z_{p},\\\ H^{*}M)$. Since
$i^{*}\alpha^{*}j^{*}(h_{0})=\alpha_{1}^{*}(h_{0})=h_{0}^{2}=0$,
then$\alpha^{*}j^{*}(h_{0})=\lambda j^{*}(\widetilde{\alpha}_{2})$ for some
scalar $\lambda\in Z_{p}$. Since both sides of the equation detect the
corresponding homotopy elements, then the relation
$\alpha_{1}j\alpha=\frac{1}{2}\alpha_{2}j$ implies $\lambda=\frac{1}{2}$. This
shows the above claim and so the above $\alpha^{*}$ is monic, im
$(i^{\prime})^{*}$ = 0 and we have $Ext_{A}^{2,p^{n}q+q-1}(Z_{p},H^{*}K)$ = 0.
The proof of the case for $u=0$ is similar.
For the second result, consider the following exact sequence
$Ext_{A}^{3,p^{n}q+2q+1}(Z_{p},H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(Z_{p},H^{*}K)$
$\qquad\qquad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(Z_{p},H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group has unique generator
$j_{*}\alpha_{*}\alpha_{*}(\tilde{h}_{n})=j_{*}\alpha_{*}\alpha^{*}(\tilde{h}_{n})$,
this is because $Ext_{A}^{3,p^{n}q+2q+1}(Z_{p},Z_{p})$ has unique generator
$\widetilde{\alpha}_{2}h_{n}\\\
=j_{*}\alpha_{*}\alpha_{*}i_{*}(h_{n})=i^{*}j_{*}\alpha_{*}\alpha_{*}(\tilde{h}_{n})$
and $Ext_{A}^{3,p^{n}q+2q+2}(Z_{p},Z_{p})$ = 0 (cf. Theorem 5.4.1), then im
$(j^{\prime})^{*}$ = 0. The right group has unique generator
$j_{*}\alpha_{*}(\tilde{b}_{n-1})$ since $Ext_{A}^{3,p^{n}q+q}(Z_{p},Z_{p})$
has unique generator
$h_{0}b_{n-1}=j_{*}\alpha_{*}i_{*}(b_{n-1})=j_{*}\alpha_{*}i^{*}(\tilde{b}_{n-1})$
and $Ext_{A}^{3,p^{n}q+q+1}(Z_{p},Z_{p})$ = 0 (cf. Theorem 5.4.1). In
addition,
$\alpha^{*}j_{*}\alpha_{*}\cdot(\tilde{b}_{n-1})=j_{*}\alpha_{*}\alpha_{*}(\tilde{b}_{n-1})\neq
0\in Ext_{A}^{4,p^{n}q+2q+1}(Z_{p},H^{*}M)$, this is because
$i^{*}j_{*}\alpha_{*}\alpha_{*}\cdot(\tilde{b}_{n-1})=\widetilde{\alpha}_{2}b_{n-1}\neq
0$ (cf. Prop. 9.5.2(2)). Then the above $\alpha^{*}$ is monic, im
$(i^{\prime})^{*}$ = 0 and so the middle group is zero as desired. Q.E.D.
Proposition 9.7.7 Let $p\geq 3,n\geq 2$, then
(1) $Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}X)$ = 0,
$Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}X)$ = 0.
(2) $Ext_{A}^{3,p^{n}q+q}(H^{*}X,H^{*}K)\cong
Z_{p}\\{w_{*}(h_{0}b_{n-1})^{\prime}\\}.$
(3) $Ext_{A}^{1,p^{n}q+q-1}(H^{*}X,Z_{p})\cong Z_{p}\\{\tau_{*}(h_{n})\\}$,
where $X$ is the spectrum in the cofibration (9.3.7) ,
$\tau:\Sigma^{q-1}S\rightarrow X$ is a map satisfying
$u\tau=i^{\prime}i:S\rightarrow K$ which is obtained by
$\alpha^{\prime\prime}i^{\prime}i=0$ and (9.3.7).
Proof : (1) Consider the following exact sequences
$0=Ext_{A}^{2,p^{n}q+q-1}(Z_{p},H^{*}K)\stackrel{{\scriptstyle
u^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
w^{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}K)=0$
$0=Ext_{A}^{3,p^{n}q+q}(Z_{p},H^{*}K)\stackrel{{\scriptstyle
u^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
w^{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}K)=0$
induced by (9.3.7). By Prop. 9.7.6(2)(3), Both sides four groups are zero so
that the result follows.
(2) We first claim that $Ext_{A}^{s,p^{n}q+1}(H^{*}K,H^{*}K)$ = 0( $s=2,3$),
then the result follows by the following exact sequence
$\stackrel{{\scriptstyle(\alpha^{\prime\prime})_{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+q}(H^{*}X,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(H^{*}K,H^{*}K)=0$
induced by (9.3.7), where the left group has unique generator
$(h_{0}b_{n-1})^{\prime}$(cf. Prop. 9.7.5(1)). To prove the above claim,
consider the following exact sequence
$\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q+q+2}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q+1}(H^{*}K,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{s,p^{n}q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2)¿ The right group is zero for $s=2,3$ (cf. Prop. 9.7.5(1))
and the left group is zero by Prop. 9.3.2(2). This shows the above claim.
(3) Since $\alpha^{\prime\prime}i^{\prime}i=0$, then, by (9.3.7), there is
$\tau\in\pi_{q-1}X$ such that $u\tau=i^{\prime}i:S\rightarrow K$. Consider the
following exact sequence
$0=Ext_{A}^{1,p^{n}q+q-1}(H^{*}K,Z_{p})\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q+q-1}(H^{*}X,Z_{p})$
$\qquad\qquad\quad\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})\stackrel{{\scriptstyle\alpha^{\prime\prime}_{*}}}{{\longrightarrow}}$
induced by (9.3.7). The left group is zero since $Ext_{A}^{1,t}(Z_{p},Z_{p})$
= 0 for $t\equiv-1,-2$ (mod $q$). The right group has unique generator
$(i^{\prime}i)_{*}(h_{n})$ which satisfies
$\alpha^{\prime\prime}_{*}(i^{\prime}i)_{*}(h_{n})$ = 0, then the middle group
has unique generator $\tau_{*}(h_{n})$ such that
$u_{*}\tau_{*}(h_{n})=(i^{\prime}i)_{*}(h_{n})$. Q.E.D.
Since $u\tau\cdot p=i^{\prime}i\cdot p=0$, then by (9.3.7) we have $\tau\cdot
p=wi^{\prime}i\alpha_{1}$ (uo to nonzero scalar), this is because
$\pi_{q-1}K\cong Z_{p}\\{i^{\prime}i(\alpha_{1})\\}.$ Then, by
$Ext_{A}^{2,p^{n}q+1}(H^{*}K,Z_{p})$ = 0(cf. Prop. 9.7.5(1)) and the Ext exact
sequence induced by (9.3.7) we have
(9.7.8)
$\tau_{*}(a_{0}b_{n-1})=\tau_{*}p_{*}(b_{n-1})=w_{*}(i^{\prime}i)_{*}(\alpha_{1})_{*}(b_{n-1})$
$\qquad=w_{*}(i^{\prime}i)_{*}(h_{0}b_{n-1})\neq 0\in
Ext_{A}^{3,p^{n}q+q}(H^{*}X,Z_{p})$.
Proposition 9.7.9 Let $p\geq 3,n\geq 2$, then
(1) $Ext_{A}^{1,p^{n}q+q-1}(H^{*}K,H^{*}K)$ = 0,
$Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}X)$ = 0.
(2) $Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}X)\cong
Z_{p}\\{u^{*}(h_{n})^{\prime}\\}$.
(3) $Ext_{A}^{2,p^{n}q+q}(H^{*}X,Z_{p})\cong
Z_{p}\\{w_{*}(i^{\prime}i)_{*}(h_{0}h_{n})\\}$.
(4) $Ext_{A}^{2,p^{n}q+1}(H^{*}X,H^{*}K)$ = 0.
Proof: (1) Consider the following exact sequence
$\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q+2q}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q+q-1}(H^{*}K,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q+q-1}(H^{*}K,H^{*}M)$
induced by (9.1.2). The right group is zero since
$Ext_{A}^{1,p^{n}q-2}(H^{*}M,H^{*}M)$ = 0,
$Ext_{A}^{1,p^{n}q+q-1}(H^{*}M,H^{*}M)$ = 0 which is obtained by
$Ext_{A}^{1,t}(Z_{p},Z_{p})$ = 0 for $t\equiv-1,-2$ (mod $q$) and
$Ext_{A}^{1,p^{n}q+q+t}(Z_{p},Z_{p})$ = 0 for $t=-1,0,1$. The left group also
is zero since $Ext_{A}^{1,p^{n}q+q-1}(H^{*}M,H^{*}M)$ = 0 and
$Ext_{A}^{1,p^{n}q+2q}(H^{*}M,\\\ H^{*}M)$ = 0 which is obtained by the same
reason as above. Then the middle group is zero.
The second result follows by the following exact sequence
$0=Ext_{A}^{1,p^{n}q+q-1}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle
u^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
w^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle(\alpha^{\prime\prime})^{*}}}{{\longrightarrow}}$
induced by (9.3.7), where the right group has unique generator
$(h_{n})^{\prime}$ which satisfies
$(\alpha^{\prime\prime})^{*}(h_{n})^{\prime}=(h_{0}h_{n})^{\prime\prime}\neq
0\in Ext_{A}^{2,p^{n}q+q-1}(H^{*}K,H^{*}K)$ ( cf. (9.3.8)).
(2) Consider the following exact sequence
$\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q+2}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle(j^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(i^{\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}M)\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}$
induced by (9.1.2). The left group is zero since
$Ext_{A}^{1,p^{n}q-q+1}(H^{*}M,H^{*}M)$ = 0 and
$Ext_{A}^{1,p^{n}q+2}(H^{*}M,H^{*}M)$ = 0 which is obtained by
$Ext_{A}^{1,p^{n}q-q+t}(Z_{p},Z_{p})$ = 0 for $t=0,1,2$ and
$Ext_{A}^{1,p^{n}q+t}(Z_{p},Z_{p})$ = 0 ($t=1,2$). The right group also is
zero since $Ext_{A}^{1,p^{n}q-2q}(H^{*}M,H^{*}M)$ = 0 and
$Ext_{A}^{1,p^{n}q-q+1}(H^{*}M,H^{*}M)$ = 0 . Then we have
$Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}K)$ = 0.
The desired result can be obtained by the following exact sequence
$\stackrel{{\scriptstyle(\alpha^{\prime\prime})^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle
u^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
w^{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}K)=0$
induced by (9.3.7), where the left group has unique generator
$(h_{n})^{\prime}$ (cf. Prop. 9.3.6).
(3) Consider the following exact sequence
$Ext_{A}^{1,p^{n}q+1}(H^{*}K,Z_{p})\stackrel{{\scriptstyle\alpha^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+q}(H^{*}K,Z_{p})\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}$
$\qquad\qquad Ext_{A}^{2,p^{n}q+q}(H^{*}X,Z_{p})\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+1}(H^{*}K,Z_{p})=0$
induced by (9.3.7). The right group is zero by Prop. 9.7.5(1) and the left
group has unique generator $(i^{\prime}i)_{*}(h_{0}h_{n})$ since
$Ext_{A}^{2,p^{n}q+q}(Z_{p},Z_{p})\cong Z_{p}\\{h_{0}h_{n}\\}$ and
$Ext_{A}^{2,t}(Z_{p},Z_{p})$ = 0 for $t\equiv-1,-2$ (mod $q$). In addition,
$Ext_{A}^{1,p^{n}q+1}(H^{*}K,\\\ Z_{p})$ = 0, this is because
$Ext_{A}^{1,p^{n}q+1}(H^{*}M,Z_{p})$ = 0 ( cf. the proof of Prop. 9.7.5(1))
and $Ext_{A}^{1,p^{n}q-q}(H^{*}M,Z_{p})$ = 0 ( cf Chapter 5), then $w_{*}$ is
monic so that the result follows.
(4) Consider the following exact sequence
$0=Ext_{A}^{2,p^{n}q+1}(H^{*}K,H^{*}K)\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+1}(H^{*}X,H^{*}K)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q-q+2}(H^{*}K,H^{*}K)=0$
induced by (9.3.7). The left group is zero as pointed out in the proof of
Prop. 9.7.7(2). The right group also is zero since
$Ext_{A}^{2,p^{n}q+3}(H^{*}K,H^{*}M)$ = 0 and
$Ext_{A}^{2,p^{n}q-q+2}(H^{*}K,H^{*}M)$ = 0 which is obtained by
$Ext_{A}^{2,p^{n}q+t}\\\ (Z_{p},Z_{p})$ = 0 for $t=2,3$ and
$Ext_{A}^{2,p^{n}q-rq+t}(Z_{p},Z_{p})$ = 0 for $r=1,2$ , $t=1,2,3$. Then the
middle group is zero as desired. Q.E.D.
Now we proceed to prove the main Theorem 9.7.4 in this section. The proof will
be done by some argument processing in the Adams resolution (9.2.9) . We first
prove the following Proposition and Lemmas.
Proposition 9.7.10 Let $p\geq 5,n\geq 2$,
$(h_{0}h_{n})^{\prime\prime}\in[\Sigma^{p^{n}q+q-1}K,KG_{2}\wedge K]$ be
$d_{1}$-cycle which represents the element
$(h_{0}h_{n})^{\prime\prime}=(\alpha^{\prime\prime})^{*}(h_{n})^{\prime}\in\\\
Ext_{A}^{2,p^{n}q+q-1}(H^{*}K,H^{*}K)$(cf. (9.3.8)), then there exist
$\eta^{\prime\prime}_{n,2}\in[\Sigma^{p^{n}q+q-1}K,\\\ E_{2}\wedge K]$ and
$(\eta^{\prime\prime}_{n,2})_{Y}\in[\Sigma^{p^{n}q+q-1}Y,E_{2}\wedge K]$ such
that $(\bar{b}_{2}\wedge 1_{K})\eta^{\prime\prime}_{n,2}=(\bar{b}_{2}\wedge
1_{K})(\eta^{\prime\prime}_{n,2})_{Y}\cdot
r=(h_{0}h_{n})^{\prime\prime}\in[\Sigma^{p^{n}q+q-1}K,KG_{2}\wedge K]$ where
$r:K\to Y$ is the map in (9.1.4).
Proof : Applying Theorem 9.3.9 to $(\sigma,s,tq)=(h_{n},1,p^{n}q)$, or
applying the mian Theorem B’ and its proof to
$(\sigma,\sigma^{\prime})=(h_{n},b_{n-1}),(s,tq)=(1,p^{n}q)$, we have
$(\bar{c}_{2}\wedge 1_{K})(h_{0}h_{n})^{\prime\prime}=0$,. Them there exists
$\eta^{\prime\prime}_{n,2}\in[\Sigma^{p^{n}q+q-1}K,KG_{2}\wedge K]$ such that
$(\bar{b}_{2}\wedge
1_{K})\eta^{\prime\prime}_{n,2}=(h_{0}h_{n})^{\prime\prime}\in[\Sigma^{p^{n}q+q-1}K,KG_{2}\wedge
K]$. For the second result, note that
$(h_{0}h_{n})^{\prime\prime}i^{\prime}i\in[\Sigma^{p^{n}q+q-1}S,KG_{2}\wedge
K]=0$, this is because $\pi_{p^{n}q+tq-u}KG_{2}\cong
Ext_{A}^{2,p^{n}q+tq-u}(Z_{p},Z_{p})$ = 0 for $t=0,1,u=1,2,3$. Then there is
$(h_{0}h_{n})^{\prime\prime}_{Y}\in[\Sigma^{p^{n}q+q-1}Y,KG_{2}\wedge K]$ such
that $(h_{0}h_{n})^{\prime\prime}=(h_{0}h_{n})^{\prime\prime}_{Y}\cdot r$,
where $r:K\to Y$ is the map in (9.1.4). Then by Theorem 9.3.9 we have
$(\bar{c}_{2}\wedge 1_{K})(h_{0}h_{n})^{\prime\prime}_{Y}\cdot r=0$ and so ,
by the cofibration (9.1.4), $(\bar{c}_{2}\wedge
1_{K})(h_{0}h_{n})^{\prime\prime}_{Y}=f^{\prime}_{0}\epsilon=(1_{E_{2}}\wedge\epsilon\wedge
1_{K})(1_{Y}\wedge f^{\prime}_{0})=0$, for some
$f^{\prime}_{0}\in[\Sigma^{p^{n}q+q}S,E_{3}\wedge K]$, where we use
$\epsilon\wedge 1_{K}=\mu(i^{\prime}i\wedge 1_{K})(\epsilon\wedge 1_{K})=0$
which is obtained by (9.1.26) and the cofibration (9.1.4). Hence, there is
$(\eta^{\prime\prime}_{n,2})_{Y}\in[\Sigma^{p^{n}q+q-1}Y,E_{2}\wedge K]$ such
that $(\bar{b}_{2}\wedge
1_{K})(\eta^{\prime\prime}_{n,2})_{Y}=(h_{0}h_{n})^{\prime\prime}_{Y}\in[\Sigma^{p^{n}q+q-1}Y,KG_{2}\wedge
K]$. Q.E.D.
Lemma 9.7.11 Let $p\geq 5,n\geq 2$ and
$(\eta^{\prime\prime}_{n})_{Y}=(\bar{a}_{0}\bar{a}_{1}\wedge
1_{K})(\eta^{\prime\prime}_{n,2})_{Y}\in[\Sigma^{p^{n}q+q-3}Y,K]$ be the map
obtained in Prop. 9.7.10, then $w(\eta^{\prime\prime}_{n})_{Y}\cdot
r=\lambda^{\prime}w(\zeta_{n-1}\wedge
1_{K})+(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}$ for some
$f^{\prime\prime}_{1}\in[\Sigma^{p^{n}q+q+1}K,E_{4}\wedge X]$ and nonzero
$\lambda^{\prime}\in Z_{p}$, where $\zeta_{n-1}\in\pi_{p^{n}q+q-3}S$ is the
element obtained in Theorem 9.5.1 which is represented by $h_{0}b_{n-1}\in
Ext_{A}^{3,p^{n}q+q}(Z_{p},Z_{p})$ in the ASS.
Proof : By Prop. 9.7.10 and (9.3.8)(9.3.7), $(\bar{b}_{2}\wedge
1_{X})(1_{E_{2}}\wedge w)(\eta^{\prime\prime}_{n,2})_{Y}\cdot r$ =
$(1_{KG_{2}}\wedge w)(h_{0}h_{n})^{\prime\prime}=(\bar{b}_{2}\bar{c}_{1}\wedge
1_{X})g^{\prime\prime}$ with
$g^{\prime\prime}\in[\Sigma^{p^{n}q+q-1}K,KG_{1}\wedge X]$, then
(9.7.12) $(1_{E_{2}}\wedge w)(\eta^{\prime\prime}_{n,2})_{Y}\cdot
r=(\bar{c}_{1}\wedge 1_{X})g^{\prime\prime}+(\bar{a}_{2}\wedge
1_{X})f^{\prime\prime}_{0}$
for some $f^{\prime\prime}_{0}\in[\Sigma^{p^{n}q+q}K,E_{3}\wedge X]$. The
$d_{1}$-cycle $(\bar{b}_{3}\wedge
1_{X})f^{\prime\prime}_{0}\in[\Sigma^{p^{n}q+q}K,KG_{3}\wedge X]$ represents
an element in $Ext_{A}^{3,p^{n}q+q}(H^{*}X,H^{*}K)$ and this group has unique
generator $w_{*}[h_{0}b_{n-1}\wedge 1_{K}]$ ( cf. Prop. 9.7.7(2)), then
$(\bar{b}_{3}\wedge
1_{X})f^{\prime\prime}_{0}=\lambda^{\prime}(1_{KG_{3}}\wedge
w)(h_{0}b_{n-1}\wedge 1_{K})+(\bar{b}_{3}\bar{c}_{2}\wedge
1_{X})\tilde{g}_{0}$
$\qquad=\lambda^{\prime}(\bar{b}_{3}\wedge 1_{X})(1_{E_{3}}\wedge
w)(\zeta_{n-1,3}\wedge 1_{K})+(\bar{b}_{3}\bar{c}_{2}\wedge
1_{X})\tilde{g}_{0}$
with $\lambda^{\prime}\in Z_{p}$ and
$\tilde{g}_{0}\in[\Sigma^{p^{n}q+q}K,KG_{2}\wedge X]$, where we use
$(\bar{b}_{3}\wedge 1_{K})(\zeta_{n-1,3}\wedge 1_{K})=h_{0}b_{n-1}\wedge
1_{K}$ (cf. Theorem 9.5.1) . Then
$f^{\prime\prime}_{0}=\lambda^{\prime}(1_{E_{3}}\wedge w)(\zeta_{n-1,3}\wedge
1_{K})+(\bar{c}_{2}\wedge 1_{X})\tilde{g}_{0}+(\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}$ for some
$f^{\prime\prime}_{1}\in[\Sigma^{p^{n}q+q+1}K,E_{4}\wedge X]$ and so we have
$(\bar{a}_{2}\wedge
1_{X})f^{\prime\prime}_{0}=\lambda^{\prime}(\bar{a}_{2}\wedge
1_{X})(1_{E_{3}}\wedge w)(\zeta_{n-1,3}\wedge
1_{K})+(\bar{a}_{2}\bar{a}_{3}\wedge 1_{X})f^{\prime\prime}_{1}$ and (9.7.12)
becomes
(9.7.13) $(1_{E_{2}}\wedge w)(\eta^{\prime\prime}_{n,2})_{Y}\cdot
r=(\bar{c}_{1}\wedge 1_{X})g^{\prime\prime}$
$\qquad\qquad+\lambda^{\prime}(\bar{a}_{2}\wedge 1_{X})(1_{E_{3}}\wedge
w)(\zeta_{n-1,3}\wedge 1_{K})+(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}$
with $g^{\prime\prime}\in[\Sigma^{p^{n}q+q-1}K,KG_{1}\wedge X]$,
$f^{\prime\prime}_{1}\in[\Sigma^{p^{n}q+q+1}K,E_{4}\wedge X]$ and
$\lambda^{\prime}\in Z_{p}$.
To prove the Lemma, it suffices to prove the scalar $\lambda^{\prime}$ in
(9.7.13) is nonzero. Suppose in contrast that $\lambda^{\prime}$ = 0, then by
(9.7.13)(9.1.4) we have
(9.7.14) $(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i=-(\bar{c}_{1}\wedge
1_{X})g^{\prime\prime}i^{\prime}i$
This will yield a contradiction as shown below.
Note that the $d_{1}$-cycle
$g^{\prime\prime}i^{\prime}i\in\pi_{p^{n}q+q-1}KG_{1}\wedge X$ represents an
element in $Ext_{A}^{1.p^{n}q+q-1}(H^{*}X,Z_{p})\cong
Z_{p}\\{\tau_{*}(h_{n})\\}$ ( cf. Prop. 9.7.7(3)). Then
$g^{\prime\prime}i^{\prime}i=\lambda_{0}(1_{KG_{1}}\wedge\tau)(h_{n})$ , where
$h_{n}\in\pi_{p^{n}q}KG_{1}\cong Ext_{A}^{1,p^{n}q}(Z_{p},Z_{p})$ and
$\lambda_{0}\in Z_{p}$. Consequently, (9.7.14) becomes
(9.7.15) $(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i=-\lambda_{0}(\bar{c}_{1}\wedge
1_{X})(1_{KG_{1}}\wedge\tau)(h_{n})$
The equation (9.7.15) means the secondary differential
$-\lambda_{0}d_{2}(\tau_{*}(h_{n}))$ = 0. However, by [12] p.11 Theorem
1.2.14, $d_{2}(h_{n})=a_{0}b_{n-1}\neq 0\in
Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$, where
$d_{2}:Ext_{A}^{1,p^{n}q}(Z_{p},Z_{p})\rightarrow
Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$ is the secondary differential in the ASS.
This implies that
$d_{2}(\tau_{*}(h_{n}))=\tau_{*}d_{2}(h_{n})=\tau_{*}(a_{0}b_{n-1})=w_{*}(i^{\prime}i)_{*}(h_{0}b_{n-1})\neq
0\in Ext_{A}^{3,p^{n}q+q}(H^{*}X,Z_{p})$ ( cf. (9.7.8)). This shows that
$\lambda_{0}$ = 0 and so by (9.7.15) we have $(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i$ = 0.
It follows that $(\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{2}\wedge
1_{X})g^{\prime\prime}_{2}$ = 0 , where the $d_{1}$-cycle
$g^{\prime\prime}_{2}\in\pi_{p^{n}q+q}KG_{2}\wedge X$ represents an element in
$Ext_{A}^{2,p^{n}q+q}(H^{*}X,Z_{p})\cong Z_{p}\\{w_{*}(i^{\prime}i)_{*}\\\
(h_{0}h_{n})\\}$ (cf. Prop. 9.7.9(3)) and the generator of this group is a
permanent cycle in the ASS (cf. Theorem 9.5.1) so that we have
$(\bar{c}_{2}\wedge 1_{X})g^{\prime\prime}_{2}=0$. Then
$f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{3}\wedge
1_{X})g^{\prime\prime}_{3}=(\bar{c}_{3}\wedge
1_{X})g^{\prime\prime}_{4}i^{\prime}i$ for some
$g^{\prime\prime}_{3}\in\pi_{p^{n}q+q+1}KG_{3}\wedge X$ and
$g^{\prime\prime}_{4}\in[\Sigma^{p^{n}q+q+1}K,KG_{3}\wedge X]$ , this is
because $g^{\prime\prime}_{3}\cdot\epsilon$ = 0 which is obtained by the fact
that $\epsilon:Y\rightarrow\Sigma S$ induces zero homomorphism in
$Z_{p}$-cohomology. Consequently we have
$f^{\prime\prime}_{1}=(\bar{c}_{3}\wedge
1_{X})g^{\prime\prime}_{4}+f^{\prime\prime}_{2}r$ with
$f^{\prime\prime}_{2}\in[\Sigma^{p^{n}q+q+1}Y,E_{4}\wedge X]$ and
$(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{2}r$. Hence, if $\lambda^{\prime}$ = 0, (9.7.13)
becomes
$(1_{E_{2}}\wedge w)(\eta^{\prime\prime}_{n,2})_{Y}\cdot
r=(\bar{a}_{2}\bar{a}_{3}\wedge 1_{X})f^{\prime\prime}_{2}r+(\bar{c}_{1}\wedge
1_{X})g^{\prime\prime}_{5}r$
where $g^{\prime\prime}_{5}\in[\Sigma^{p^{n}q+q-1}Y,KG_{2}\wedge X]$ such that
$g^{\prime\prime}_{5}r=g^{\prime\prime}$. Moreover, by the above equation we
have
(9.7.16) $(1_{E_{2}}\wedge
w)(\eta^{\prime\prime}_{n,2})_{Y}=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{2}+(\bar{c}_{1}\wedge
1_{X})g^{\prime\prime}_{5}+f^{\prime\prime}_{3}\epsilon$
with $f^{\prime\prime}_{3}\in\pi_{p^{n}q+q}E_{2}\wedge X$. Since
$\epsilon:Y\rightarrow\Sigma S$ induces zero homomorphism in
$Z_{p}$-cohomology , then the right hand side of (9.7.16) has filtration $\geq
3$. However, $(\eta^{\prime\prime}_{n})_{Y}=(\bar{a}_{0}\bar{a}_{1}\wedge
1_{K})(\eta^{\prime\prime}_{n,2})_{Y}$ has filtration 2, this is because it is
represented by $(h_{0}h_{n})^{\prime\prime}_{Y}\in
Ext_{A}^{2,p^{n}q+q-1}(H^{*}K,H^{*}Y)$ in the ASS. Moreover, by the following
exact sequence
$0=Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}Y)\stackrel{{\scriptstyle\alpha^{\prime\prime}_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+q-1}(H^{*}K,H^{*}Y)$
$\qquad\qquad\quad\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q+q-1}(H^{*}X,H^{*}Y)\stackrel{{\scriptstyle(\alpha^{\prime\prime})_{*}}}{{\longrightarrow}}$
induced by (9.3.7) we know that $w_{*}(h_{0}h_{n})^{\prime\prime}_{Y}\neq 0$,
where the left group is zero by Prop. 9.7.5(3). That is to say,
$(1_{E_{2}}\wedge w)(\eta^{\prime\prime}_{n})_{Y}$ has filtration 2 which is
represented by $w_{*}(h_{0}h_{n})^{\prime\prime}_{Y}$ in the ASS. This shows
that the equation (9.7.16) is a contradiction so that the scalar
$\lambda^{\prime}$ must be nonzero. Q.E.D.
Lemma 9.7.17 Let $w:K\rightarrow X$ be the map in the cofibration (9.3.7) and
$W$ is the cofibre of $wi^{\prime}i:S\rightarrow X$ given by the cofibration
$S\stackrel{{\scriptstyle
wi^{\prime}i}}{{\longrightarrow}}X\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}W\stackrel{{\scriptstyle
u_{1}}}{{\longrightarrow}}\Sigma S$ , then
(1) $Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}W,Z_{p})$ = 0 for $s=1,3$ and has unique
generator $(w_{1})_{*}\tau_{*}(h_{n})=(\tau)^{*}(w_{1})_{*}[h_{n}\wedge
1_{X}]$ for $s=2$.
(2) $Ext_{A}^{1,p^{n}q}(H^{*}W,H^{*}X)\cong
Z_{p}\\{(w_{1})_{*}(h_{n})^{\prime}_{X}=(w_{1})_{*}[h_{n}\wedge 1_{X}]\\}$.
(3) $(w_{1})_{*}[a_{0}b_{n-1}\wedge 1_{X}]\neq 0\in
Ext_{A}^{3,p^{n}q+1}(H^{*}W,H^{*}X)$.
Proof : (1) Note that $W$ also is the cofibre of
$r\alpha^{\prime\prime}:\Sigma^{q-2}K\rightarrow Y$, this can be seen by the
following homotopy commutative diagram of $3\times 3$-Lemma.
$\qquad\qquad\quad S\qquad\stackrel{{\scriptstyle
wi^{\prime}i}}{{\longrightarrow}}\qquad X\qquad\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\quad\Sigma^{q-1}K$
$\qquad\qquad\qquad\searrow i^{\prime}i\qquad\nearrow w\quad\searrow
w_{1}\qquad\nearrow u_{2}$
$\qquad\qquad\qquad\qquad K\quad\qquad\qquad\qquad W$
$\qquad\qquad\qquad\nearrow\alpha^{\prime\prime}\qquad\searrow r\quad\nearrow
w_{2}\qquad\searrow u_{1}$
$\qquad\qquad\Sigma^{q-2}K\quad\stackrel{{\scriptstyle
r\alpha^{\prime\prime}}}{{\longrightarrow}}\qquad
Y\qquad\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}\qquad\Sigma S$
That is , we have a cofibration
$\qquad\Sigma^{q-2}K\stackrel{{\scriptstyle
r\alpha^{\prime\prime}}}{{\longrightarrow}}Y\stackrel{{\scriptstyle
w_{2}}}{{\longrightarrow}}W\stackrel{{\scriptstyle
u_{2}}}{{\longrightarrow}}\Sigma^{q-1}K$ and it induces the following exact
sequence
$Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}Y,Z_{p})\stackrel{{\scriptstyle(w_{2})_{*}}}{{\longrightarrow}}Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}W,Z_{p})$
$\qquad\qquad\quad\stackrel{{\scriptstyle(u_{2})_{*}}}{{\longrightarrow}}Ext_{A}^{s-1,p^{n}q+s-2}(H^{*}K,Z_{p})\stackrel{{\scriptstyle(r\alpha^{\prime\prime})_{*}}}{{\longrightarrow}}$
The left group is zero for $s=1,2,3$ ( cf. Prop. 9.7.5(2)). The right group
also is zero for $s=1,3$ (cf. Prop. 9.7.5(1)) and has unique generator
$(i^{\prime}i)_{*}(h_{n})=u_{*}\tau_{*}(h_{n})=(u_{2})_{*}(w_{1})_{*}(\tau)_{*}(h_{n})$
for $s=2$. Then the result follows.
(2) Consider the following exact sequence
$Ext_{A}^{1,p^{n}q}(H^{*}X,H^{*}X)\stackrel{{\scriptstyle(w_{1})_{*}}}{{\longrightarrow}}Ext_{A}^{1,p^{n}q}(H^{*}W,H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(u_{1})_{*}}}{{\longrightarrow}}Ext_{A}^{2,p^{n}q}(Z_{p},H^{*}X)=0$
The right group is zero by Prop. 9.7.7(1) and by Prop. 9.7.9(2)(1) we know
that the left group has unique generator $(h_{n})^{\prime}_{X}=[h_{n}\wedge
1_{X}]$ which satisfies $u_{*}(h_{n})^{\prime}_{X}=u^{*}(h_{n})^{\prime}\in
Ext_{A}^{1,p^{n}q-q+1}(H^{*}K,H^{*}X)$ . Then the middle group has unique
generator $(w_{1})_{*}(h_{n})^{\prime}_{X}$.
(3) Since $\alpha^{\prime\prime}:\Sigma^{q-2}K\to K$ is not an $M$-module map,
then , as the cofibre of $\alpha^{\prime\prime}$, the spectrum $X$ is not an
$M$-moduld spectrum, that is, the map $p\wedge 1_{X}\neq 0\in[X,X]$. So
$[a_{0}\wedge 1_{X}]=(p\wedge 1_{X})_{*}[\tau\wedge 1_{X}]\neq 0\in
Ext_{A}^{1,1}(H^{*}X,X^{*}X)$ ( where $\tau$ is the unit in $\pi_{0}KG_{0}$),
and so $[a_{0}b_{n-1}\wedge 1_{X}]=(p\wedge 1_{X})_{*}[b_{n-1}\wedge
1_{X}]=[a_{0}\wedge 1_{X}][b_{n-1}\wedge 1_{X}]\neq 0\in
Ext_{A}^{3,p^{n}q+1}(H^{*}X,X^{*}X)$ which can be obtained by by knowledge of
Yoneda products and $a_{0}b_{n-1}\neq 0\in Ext_{A}^{3,p^{n}q+1}(Z_{p},Z_{p})$.
Note to the following exact sequence
$0=Ext_{A}^{3,p^{n}q+1}(Z_{p},H^{*}X)\stackrel{{\scriptstyle(wi^{\prime}i)_{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(H^{*}X,H^{*}X)$
$\qquad\qquad\quad\stackrel{{\scriptstyle(w_{1})_{*}}}{{\longrightarrow}}Ext_{A}^{3,p^{n}q+1}(H^{*}W,H^{*}X)$
where the left group is zero by Prop. 9.7.7(1), then $(w_{1})_{*}$ is monic
and so the reslt follows. Q.E.D.
Remark The result on $[a_{0}b_{n-1}\wedge 1_{X}]\neq 0$ in Lemma 9.7.17(3)
also can be proved by some computation in Ext groups as follows. Suppose in
contrast that $(p\wedge 1_{X})_{*}[b_{n-1}\wedge 1_{X}]=[a_{0}b_{n-1}\wedge
1_{X}]=0$, then by (9.1.1), $[b_{n-1}\wedge 1_{X}]=(j\wedge 1_{X})_{*}(x_{1})$
with $x_{1}\in Ext_{A}^{2,p^{n}q+1}(H^{*}M\wedge X,H^{*}X)$. Recall that $X$
is the spectrum in (9.3.7), then we have $w^{*}(1_{M}\wedge u)_{*}(x_{1})\in
Ext_{A}^{2,p^{n}q-q+1}(H^{*}M\wedge K,H^{*}K)$ = 0 which can be obtained by
$Ext_{A}^{2,p^{n}q-q+1}\\\ (H^{*}M\wedge K,H^{*}M)$ = 0 and
$Ext_{A}^{2,p^{n}q+2}(H^{*}M\wedge K,H^{*}M)$ = 0. By the Ext exact sequence
induced by (9.3.7) we have $(1_{M}\wedge u)_{*}(x_{1})\in
u^{*}Ext_{A}^{2,p^{n}q+1}(H^{*}M\wedge K,H^{*}K)$. However, this group has
unique generator $(\overline{m}_{K})_{*}(b_{n-1})^{\prime}$, then
$(1_{M}\wedge u)_{*}(x_{1})=\lambda
u^{*}(\overline{m}_{K})_{*}(b_{n-1})^{\prime}$ for some $\lambda\in Z_{p}$. By
applying $(1_{M}\wedge\alpha^{\prime\prime})_{*}$ we have $\lambda
u^{*}(1_{M}\wedge\alpha^{\prime\prime})_{*}(\overline{m}_{K})_{*}(b_{n-1})^{\prime}$
= 0 and so
$\lambda(1_{M}\wedge\alpha^{\prime\prime})_{*}(\overline{m}_{K})_{*}(b_{n-1})^{\prime}\in(\alpha^{\prime\prime})^{*}Ext_{A}^{2,p^{n}q+1}(H^{*}M\wedge
K,H^{*}K)$. Then $\lambda(\alpha_{1}\wedge
1_{K})_{*}(b_{n-1})^{\prime}=\lambda(m_{K})_{*}(1_{M}\wedge\alpha^{\prime\prime})_{*}(\overline{m}_{K})_{*}(b_{n-1})^{\prime}$
= 0 which shows that $\lambda=0$. So $x_{1}=(1_{M}\wedge w)_{*}(x_{2})$ for
some $x_{2}\in Ext_{A}^{2,p^{n}q+1}(H^{*}M\wedge K,H^{*}X)$. Similarly we can
prove that $w^{*}(x_{2})$ = 0. Then $x_{1}\in u^{*}(1_{M}\wedge
w)_{*}Ext_{A}^{2,p^{n}q+q}(H^{*}M\wedge K,H^{*}K)$. However,
$Ext_{A}^{2,p^{n}q+q}(H^{*}M\wedge K,H^{*}K)$ has two generators $(i\wedge
1_{K})_{*}(h_{0}h_{n})^{\prime}$ ,
$(\overline{m}_{K})_{*}(h_{0}h_{n})^{\prime\prime}$ and
$u^{*}(h_{0}h_{n})^{\prime\prime}=u^{*}(\alpha^{\prime\prime})^{*}(h_{n})^{\prime}$
= 0, then we have $x_{1}=u^{*}(1_{M}\wedge w)_{*}(i\wedge
1_{K})_{*}(h_{0}h_{n})^{\prime}$( up to scalar) and so $[b_{n-1}\wedge
1_{X}]=(j\wedge 1_{X})_{*}(x_{1})$ = 0. This is a contradiction and shows that
$[a_{0}b_{n-1}\wedge 1_{X}]\neq 0\in Ext_{A}^{3,p^{n}q+1}(H^{*}X,X^{*}X)$.
Proof of Theorem 9.7.4 : The result for $n=0,1$ is wellknown, then we assume
that $n\geq 2$. By Lemma 9.7.11 and (9.1.4) we have
(9.7.18)
$\lambda^{\prime}wi^{\prime}i\zeta_{n-1}=\lambda^{\prime}w(\zeta_{n-1}\wedge
1_{K})i^{\prime}i=-(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i.$
Moreover, by the cofibration in Lemma 9.7.17 we have
$(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})(1_{E_{4}}\wedge
w_{1})f^{\prime\prime}_{1}i^{\prime}i$ = 0 and so
$(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})(1_{E_{4}}\wedge
w_{1})f^{\prime\prime}_{1}u=k\tau^{\prime}$ for some
$k\in[\Sigma^{p^{n}q-2}X^{\prime},W]$ , where
$i^{\prime}i=u\tau:\Sigma^{q-1}S\stackrel{{\scriptstyle\tau}}{{\rightarrow}}X\stackrel{{\scriptstyle
u}}{{\rightarrow}}\Sigma^{q-1}K$ which is obtained by
$\alpha^{\prime\prime}i^{\prime}i$ = 0 and (9.3.7) and
$\tau^{\prime}:X\rightarrow X^{\prime}$ is the map in the following
cofibration
(9.7.19)
$\Sigma^{q-1}S\stackrel{{\scriptstyle\tau}}{{\rightarrow}}X\stackrel{{\scriptstyle\tau^{\prime}}}{{\rightarrow}}X^{\prime}\stackrel{{\scriptstyle\tau^{\prime\prime}}}{{\rightarrow}}\Sigma^{q}S.$
We claim that $k\in[\Sigma^{p^{n}q-2}X^{\prime},W]$ has filtration $\geq 4$,
this can be proved as follows.
By Lemma 9.7.17(1) and (9.7.18) we have
$(\tau^{\prime\prime})^{*}Ext_{A}^{s-1,p^{n}q+q+s-3}(H^{*}W,\\\
Z_{p})=0\subset Ext_{A}^{s,p^{n}q+s-2}(H^{*}W,H^{*}X^{\prime})$ and so
$(\tau^{\prime})^{*}:Ext_{A}^{s,p^{n}q+s-2}(H^{*}W,\\\
H^{*}X^{\prime})\rightarrow Ext_{A}^{s,p^{n}q+s-2}$ $(H^{*}W,H^{*}X)$ for
$s=1,2,3$ is monic. Then, the fact that $k\tau^{\prime}$ has filtration$\geq
4$ implies that $k\in[\Sigma^{p^{n}q-2}X^{\prime},W]$ also has filtration
$\geq 4$. This shows the above claim and so
$k=(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})k_{3}$ for some
$k_{3}\in[\Sigma^{p^{n}q+2}X^{\prime},E_{4}\wedge W]$ and
$(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})(1_{E_{4}}\wedge
w_{1})f^{\prime\prime}_{1}u=(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge
1_{W})k_{3}\tau^{\prime}$. It follows that
(9.7.20) $(\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})(1_{E_{4}}\wedge
w_{1})f^{\prime\prime}_{1}u=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{W})k_{3}\tau^{\prime}+(\bar{c}_{1}\wedge 1_{W})\bar{g}$
$\quad=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{W})k_{3}\tau^{\prime}+\lambda_{1}(\bar{c}_{1}\wedge 1_{W})(1_{KG_{1}}\wedge
w_{1})(h_{n}\wedge 1_{X})$
where the $d_{1}$-cycle $\bar{g}=\lambda_{1}(1_{KG_{1}}\wedge
w_{1})(h_{n}\wedge 1_{X})\in[\Sigma^{p^{n}q}X,KG_{1}\wedge W]$ with
$\lambda_{1}\in Z_{p}$ which is obtained by Lemma 9.7.17(2).
The equation (9.7.20) means that the differential
$d_{2}(\lambda_{1}(w_{1})_{*}[h_{n}\wedge 1_{X}])$ = 0. However,
$d_{2}((w_{1})_{*}[h_{n}\wedge 1_{X}])=(w_{1})_{*}[a_{0}b_{n-1}\wedge
1_{X}]\neq 0$ ( cf. Lemma 9.7.17(3)). Then the scalar $\lambda_{1}$ = 0 and we
have $\bar{g}$ = 0, $(\bar{a}_{2}\bar{a}_{3}\wedge 1_{W})(1_{E_{4}}\wedge
w_{1})f^{\prime\prime}_{1}u=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{W})k_{3}\tau^{\prime}$ and $(\bar{a}_{2}\bar{a}_{3}\wedge
1_{K})(1_{E_{4}}\wedge
u)f^{\prime\prime}_{1}i^{\prime}i=(\bar{a}_{2}\bar{a}_{3}\wedge
1_{K})(1_{E_{4}}\wedge u_{2}w_{1})f^{\prime\prime}_{1}u\tau$ = 0. Consequently
we have $(\bar{a}_{3}\wedge 1_{K})(1_{E_{4}}\wedge
u)f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{2}\wedge 1_{K})\bar{g}_{2}=0$ ,
this is because the $d_{1}$-cycle $\bar{g}_{2}\in\pi_{p^{n}q+1}KG_{2}\wedge K$
represents an element in $Ext_{A}^{2,p^{n}q+1}(H^{*}K,Z_{p})$ = 0 (cf.
9.7.5(1)) . Then, $(1_{E_{4}}\wedge
u)f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{3}\wedge 1_{K})\bar{g}_{3}$ for
some $\bar{g}_{3}\in[\Sigma^{p^{n}q+2}S,KG_{3}\wedge K]$. Since
$(1_{KG_{3}}\wedge\alpha^{\prime\prime})\bar{g}_{3}$ = 0, then
$\bar{g}_{3}=(1_{KG_{3}}\wedge u)\bar{g}_{4}$ for some
$\bar{g}_{4}\in[\Sigma^{p^{n}q+q+1}S,KG_{3}\wedge X]$ and so we have
$(1_{E_{4}}\wedge u)f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{3}\wedge
1_{K})(1_{KG_{3}}\wedge u)\bar{g}_{4}$ ,
$f^{\prime\prime}_{1}i^{\prime}i=(\bar{c}_{3}\wedge
1_{X})\bar{g}_{4}+(1_{E_{4}}\wedge w)\bar{f}_{2}$ with
$\bar{f}_{2}\in[\Sigma^{p^{n}q+q+1}S,E_{4}\wedge K]$.
Hence, by (9.7.18) we have
$\lambda^{\prime}wi^{\prime}i\zeta_{n-1}=-(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge
1_{X})f^{\prime\prime}_{1}i^{\prime}i=-(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\\\
\bar{a}_{3}\wedge 1_{X})(1_{E_{4}}\wedge w)\bar{f}_{2}$ and by (9.3.7),
$\lambda^{\prime}i^{\prime}i\zeta_{n-1}=-(\bar{a}_{0}\bar{a}_{1}\bar{a}_{2}\bar{a}_{3}\wedge
1_{K})\bar{f}_{2}+\alpha^{\prime\prime}\omega_{n}$ with
$\omega_{n}\in\pi_{p^{n}q-1}K$. Since $\lambda^{\prime}i^{\prime}i\zeta_{n-1}$
is a map of filtration 3 which is represented by
$\lambda^{\prime}(i^{\prime}i)_{*}(h_{0}b_{n-1})\in
Ext_{A}^{3,p^{n}q+q}(H^{*}K,Z_{p})$ in the ASS, then
$\alpha^{\prime\prime}\omega_{n}$ has filtration 3 and so
$\omega_{n}\in\pi_{p^{n}q-1}K$ has filtration $\leq 2$. However, by Prop.
9.7.5(1) we have $Ext_{A}^{2,p^{n}q+1}(H^{*}K,Z_{p})$ = 0 , then
$\omega_{n}\in\pi_{p^{n}q-1}K$ must be represented by the unique generator
$(i^{\prime}i)_{*}(h_{n})\in Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$ (up to nonzero
scalar). This shows the Theorem. Q.E.D.
Remark The element $\omega_{n}\in\pi_{p^{n}q-1}K$ obtained in Theorem 9.7.4
can be extended to $(\omega_{n})^{\prime}\in[\Sigma^{p^{n}q-1}K,K]$ such that
$(\omega_{n})^{\prime}i^{\prime}i=\omega_{n}$. Then, $(\omega_{n})^{\prime}$
is represented by $(h_{n})^{\prime}\in Ext_{A}^{1,p^{n}q}(H^{*}K,H^{*}K)$ in
the ASS and $\alpha^{\prime\prime}(\omega_{n})^{\prime}$,
$(\omega_{n})^{\prime}\alpha^{\prime\prime}\in[\Sigma^{p^{n}q+q-3}K,K]$ is
represented by
$\alpha^{\prime\prime}_{*}(h_{n})^{\prime}=(\alpha^{\prime\prime})^{*}(h_{n})^{\prime}=(h_{0}h_{n})^{\prime\prime}\\\
\in Ext_{A}^{2,p^{n}q+q-1}(H^{*}K,H^{*}K)$. By Theorem 9.7.4 and Lemma 9.7.11
we have
$\alpha^{\prime\prime}(\omega_{n})^{\prime}=(\omega_{n})^{\prime}\alpha^{\prime\prime}+\lambda^{\prime}\zeta_{n-1}\wedge
1_{K}$ (modulo higher filtration).
By [10] p.511, there is a map $\phi_{*}\phi:BP_{*}BP\rightarrow A_{*}$ such
that $t_{n}\mapsto$ the conjugate of $\xi_{n}$, where
$A_{*}=E[\tau_{0},\tau_{1},\tau_{2},...]\otimes P[\xi_{1},\xi_{2},...]$ is the
dual of the Steenrod algebra $A$. Then $\phi_{*}\phi$ induces the Thom map
$\Phi:Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},\\\ BP_{*}K)\rightarrow
Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$ such that the image of $h_{n}\in
Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},\\\ BP_{*}K)$ is
$\Phi(h_{n})=(i^{\prime}i)_{*}(h_{n})\in Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$.
Then, the element $\omega_{n}\in\pi_{p^{n}q-1}K$ obtained in Theorem 9.7.4 is
represented by $h_{n}$ \+ other terms $\in
Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},\\\ BP_{*}K)$ in the Adams-Novikov spectral
sequence. To know what the elements in the other terms, we first prove the
following Lemma.
Lemma 9.7.21 By degree reason, $Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K)$ is
generated (additively) by the following $v_{2}$-torsion elements
$c_{2}(p^{n-2})$ and $v_{2}$-torsion free elements
$h_{n},v_{2}^{p^{n-2}(p-1)}h_{n-2},v_{2}^{a_{i}p^{i}}h_{i}$ , where $i\geq
0,a_{i}=(p^{2k}-1)/(p+1)$ , $n-i=2k\geq 4$. In addition, there is a relation
$h_{n}=c_{2}(p^{n-2})+v_{2}^{p^{n-2}(p-1)}h_{n-2}\in
Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K)$.
Proof : By [19] Theorem 1.1 and 1.5, $Ext^{1,*}(BP_{*},BP_{*}K)$ is a
$Z_{p}[v_{2}]$-module which is generated by $v_{2}$-torsion elements
$c_{2}(ap^{s})$ and $v_{2}$-torsion free elements $w_{2},h_{i}$, where $a\neq
0$ (mod $p$), $s\geq 0$ and $i\geq 0$. Moreover, the internal degree $\mid
h_{i}\mid=p^{i}q$, $\mid c_{2}(ap^{s})\mid=ap^{s}(p^{2}+p+1)q-q(ap^{s})(p+1)q$
and $\mid w_{2}\mid=(p+1)^{2}q$.
Since $\mid v_{2}^{b}w_{2}\mid\equiv 0$ (mod $(p+1)q)$, then $\mid
v_{2}^{b}w_{2}\mid\neq p^{n}q$. If $\mid v_{2}^{b}h_{i}\mid=p^{n}q$, then
$b(p+1)q+p^{i}q=p^{n}q,b(p+1)=p^{i}(p^{n-i}-1)$ and so $(p^{n-i}-1)$ must be
divisible by $p+1$. Hence $b=0$ and $i=n$ or $b=a_{i}p^{i}$ with
$a_{i}=(p^{2k}-1)/(p+1)$ and $n-i=2k\geq 2$. Then
$h_{n},v_{2}^{p^{n-2}(p-1)}h_{n-2}$ and $v_{2}^{a_{i}p^{i}}h_{i}(0\leq i<n-2$)
are the only torsion free elements of $Ext^{1,p^{n}q}(BP_{*},BP_{*}K)$.
If $\mid v_{2}^{b}c_{2}(ap^{s})\mid=p^{n}q$, then
$p^{n}q=ap^{s}(p^{2}+p+1)q-(q(ap^{s})-b)(p+1)q$,
$ap^{s}(p^{2}+p+1)=p^{n}+(q(ap^{s})-b)(p+1)$ and so the right hand side must
be divisible by $p^{2}+p+1$. So we have
(9.7.22) $ap^{s}=p^{n-2}+\frac{(q(ap^{s})-b-p^{n-2})(p+1)}{p^{2}+p+1}.$
We claim that $s\leq n-2$ which will be proved below , then $q(ap^{s})-b$ must
be divisible by $p^{s}$. However, by [19] p.132, $q(ap^{s})=p^{s}$ for $a=1$
and $q(ap^{s})=p^{s}$ \+ other terms for $a\geq 2$. Then, the only possibility
is $q(ap^{s})=p^{s},a=1,b=0$ and $s=n-2$. That is to say, the only
$v_{2}$-torsion elements in $Ext^{1,p^{n}q}(BP_{*},BP_{*}K)$ is
$c_{2}(p^{n-2})$.
Now we prove the above claim. Suppose in contrast that $s\geq n-1$ , then, by
(9.7.22) we have
$\frac{(q(ap^{s})-b-p^{n-2})(p+1)}{p^{2}+p+1}=ap^{s}-p^{n-2}\geq
p^{s}-p^{n-2},$
$2p^{s}>q(ap^{s})-b-p^{n-2}\geq\frac{(p^{s}-p^{n-2})(p^{2}+p+1)}{p+1}>p^{s+1}-p^{n-1}$
and this is a contradiction which shows the above claim. Q.E.D.
Proof of Theorem 9.7.1 For the Thom
$\Phi:Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K)\\\ \rightarrow
Ext_{A}^{1,p^{n}q}(H^{*}K,Z_{p})$ we have
$\Phi(h_{n})=(i^{\prime}i)_{*}(h_{n})$. By this we know that the element
$\omega_{n}\in\pi_{p^{n}q-1}K$ obtained in Theorem 9.7.4 is represented by
$h_{n}$ \+ ( other terms) $\in Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K)$ in
the Adams-Novikov spectral sequence. By Lemma 9.7.21, the other terms are the
linear combination of $v_{2}^{p^{n-2}(p-1)}h_{n-2}$ and
$v_{2}^{a_{i}p^{i}}h_{i}$, where $i\geq 0,n-i=2k\geq 4$ and
$a_{i}=(p^{2k}-1)/(p+1)$. Let $\beta\in[\Sigma^{(p+1)q}K,K]$ be the known
$v_{2}$-map, then $i^{\prime}i\alpha_{1}\in\pi_{q-1}K$ and
$i^{\prime}j^{\prime}\beta i^{\prime}i\in\pi_{pq-1}K$ is represented by
$h_{0},h_{1}\in Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}K)$ respectively. That is,
$h_{0},h_{1}\in Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}K)$ are permanent cycles in
the Adams-Novikov spectral sequence. Suppose inductively that $h_{i}\in
Ext_{BP_{*}BP}^{1,p^{i}q}(BP_{*},BP_{*}K)$ for $i\leq n-1(n\geq 2)$ are
permanent cycles in the Adams-Novikov spectral sequence. Since
$\omega_{n}\in\pi_{p^{n}q-1}K,\omega_{n+1}\in\pi_{p^{n+1}q-1}K$ are
represented by the linear combination of
$h_{n}+v_{2}^{p^{n-2}(p-1)}h_{n-2}$ and $v_{2}^{a_{i}p^{i}}h_{i}\in
Ext_{BP_{*}BP}^{1,p^{n}q}(BP_{*},BP_{*}K),$
$h_{n+1}+v_{2}^{p^{n-1}(p-1)}h_{n-1}$ and $v_{2}^{a_{i}p^{i}}h_{i}\in
Ext_{BP_{*}BP}^{1,p^{n+1}q}(BP_{*},BP_{*}K)$
then $h_{n},h_{n+1}\in Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}K)$ also are
permanent cycles. This completes the induction and the result of the Theorem
follows. Q.E.D.
Conjecture 9.7.22 Theorem 9.7.4 can be generalzed to be the following general
result. Let $p\geq 5,s\leq 4$ , $Ext_{A}^{s,tq}(Z_{p},Z_{p})\cong
Z_{p}\\{x\\},Ext_{A}^{s+1,tq+q}\\\ (Z_{p},Z_{p})\cong
Z_{p}\\{h_{0}x\\},Ext_{A}^{s+2,tq+2q+1}(Z_{p},Z_{p})\cong
Z_{p}\\{\widetilde{\alpha}_{2}x\\}$ and some supposition on vanishes of some
Ext groups. If the secondary differential $d_{2}(x)=a_{0}x^{\prime}\in\\\
Ext_{A}^{s+2,tq+1}(Z_{p},Z_{p})$ with $x^{\prime}\in
Ext_{A}^{s+1,tq}(Z_{p},Z_{p})$, that is, $x$ and $x^{\prime}$ is a pair of
$a_{0}$-related elements, then there exists $\omega\in\pi_{tq-s}K$ such that
$i^{\prime}i\xi=\alpha^{\prime\prime}\cdot\omega$ (mod $F^{s+2}\pi_{*}K)$ and
$\omega\in\pi_{tq-s}K$ is represented by $(i^{\prime}i)_{*}(x)\in
Ext_{A}^{s,tq}(H^{*}K,Z_{p})$ in the ASS, where $\xi\in\pi_{tq+q-s-2}S$ is the
homotopy element which is represented by $h_{0}x^{\prime}\in
Ext_{A}^{s+2,tq+q}(Z_{p},Z_{p})$ in the ASS and $F^{s+2}\pi_{*}K$ denotes the
group consisting of all elements in $\pi_{*}K$ filtration $\geq s+2$.
§8. Second periodicity families in the stable homotopy groups of spheres
By Theorem 8.1.2 in chapter 8, $Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*})$ is
generated by $\alpha_{tp^{n}/n+1}(n\geq 0,p$ not divisible by $t\geq 1$) and
It was proved by Novikov that all these first periodicity families converge to
the im $J\subset\pi_{*}S$. In this section, using the $h_{0}h_{n+1}$-element
obtained in Theorem 9.5.1 and the elements $\beta_{p/r},1\leq r\leq p-1$ and
$\beta_{tp/r},t\geq 2,1\leq r\leq p$ as our geometric input, we prove the
following Theorem on the convergence of second periodicity families
$\beta_{tp^{n}/r}$ in the Adams-Novikov spectral sequence.
Theorem 9.8.1 Let $p\geq 5,n\geq 1,1\leq s\leq p^{n}-1$ if $t\geq 1$ is not
divisible by $p$ or $1\leq s\leq p^{n}$ if $t\geq 2$ is not divisible by $p$ ,
then The elements
$\beta_{tp^{n}/s}\in Ext_{BP_{*}BP}^{2,tp^{n}(p+1)q-sq}(BP_{*},BP_{*})$
in Theorem 8.1.3 are permanent cycles in the Adams-Novikov spectral sequence
and they converge to the corresponding homotopy elements of order $p$ in
$\pi_{tp^{n}(p+1)q-sq-2}S$.
We will prove Theorem 9.8.1 in case $t\geq 1$ or $t\geq 2$ separately. The
proof will be done by some arguments processing in the cannical Adams-Novikov
resolution. We first do some preminilaries as follows.
Let $M$ be the Moore spectrum whose $BP_{*}$-homology are
$BP_{*}(M)=BP_{*}/(p)$. Let $\alpha:\Sigma^{q}M\rightarrow M$ be the Adams map
which induces $BP_{*}$-homomorphisms are $v_{1}:BP_{*}/(p)\rightarrow
BP_{*}/(p)$. Let $K_{r}$ be the cofibre of $\alpha^{r}:\Sigma^{rq}M\rightarrow
M$ given by the cofibration
(9.8.2)$\qquad\qquad\Sigma^{rq}M\stackrel{{\scriptstyle\alpha^{r}}}{{\longrightarrow}}M\stackrel{{\scriptstyle
i^{\prime}_{r}}}{{\longrightarrow}}K_{r}\stackrel{{\scriptstyle
j^{\prime}_{r}}}{{\longrightarrow}}\Sigma^{rq+1}M$
The cofibration (9.8.2) induces a short exact sequence of $BP_{*}$-homology
$0\rightarrow BP_{*}/(p)\stackrel{{\scriptstyle
v_{1}^{r}}}{{\longrightarrow}}BP_{*}/(p)\longrightarrow
BP_{*}/(p,v_{1}^{r})\rightarrow 0$
Recall from §5 in chapter 6, $K_{r}$ is a $M$-module spectrum and we have the
following derivations
(9.8.3) $d(i^{\prime}_{r})$ = 0, $d(j^{\prime}_{r})$ = 0, $d(\alpha)=0,\quad
d(ij)=-1_{M}$.
Moreover, the cofibre of
$i^{\prime}_{s}j^{\prime}_{r}:K_{r}\rightarrow\Sigma^{rq+1}K_{s}$ is $\Sigma
K_{r+s}$ given by the cofibration
(9.8.4)
$\qquad\Sigma^{rq}K_{s}\stackrel{{\scriptstyle\psi_{s,s+r}}}{{\longrightarrow}}K_{s+r}\stackrel{{\scriptstyle\rho_{s+r,r}}}{{\longrightarrow}}K_{r}\stackrel{{\scriptstyle
i^{\prime}_{s}j^{\prime}_{r}}}{{\longrightarrow}}\Sigma^{rq+1}K_{s}$
This can be seen by the following homotopy commutative diagram of $3\times
3$-Lemma
$\qquad\qquad K_{r}\quad\stackrel{{\scriptstyle
i^{\prime}_{s}j^{\prime}_{r}}}{{\longrightarrow}}\quad\Sigma^{rq+1}K_{s}\quad\stackrel{{\scriptstyle
j^{\prime}_{s}}}{{\longrightarrow}}\quad\Sigma^{(r+s)q+2}M$
$\qquad\qquad\quad\searrow j^{\prime}_{r}\quad\nearrow
i^{\prime}_{s}\quad\searrow\psi_{s,r+s}\quad\nearrow
j^{\prime}_{r+s}\quad\searrow\alpha^{s}$
(9.8.5) $\Sigma^{rq+1}M\qquad\qquad\Sigma
K_{r+s}\qquad\qquad\qquad\Sigma^{rq+2}M$
$\qquad\qquad\quad\nearrow\alpha^{s}\quad\searrow\alpha^{r}\quad\nearrow
i^{\prime}_{r+s}\quad\searrow\rho_{r+s,r}\quad\nearrow j^{\prime}_{r}$
$\qquad\quad\Sigma^{(r+s)q+1}M\stackrel{{\scriptstyle\alpha^{r+s}}}{{\longrightarrow}}\Sigma
M\quad\quad\stackrel{{\scriptstyle
i^{\prime}_{r}}}{{\longrightarrow}}\qquad\Sigma K_{r}$
Moreover, the cofibration (9.8.4) induces a short exact sequence of
$BP_{*}$-homology
$0\rightarrow BP_{*}/(p,v_{1}^{s})\stackrel{{\scriptstyle
v_{1}^{r}}}{{\longrightarrow}}BP_{*}/(p,v_{1}^{s+r})\longrightarrow
BP_{*}/(p,v_{1}^{r})\rightarrow 0$
and by the homotopy commutative diagram (9.8.5) , we have the following
relations
(9.8.6) $\qquad\psi_{s,s+r}i^{\prime}_{s}=i^{\prime}_{s+r}\alpha^{r},\qquad
j^{\prime}_{r}\rho_{s+r,r}=\alpha^{s}j^{\prime}_{s+r}$
$\qquad\quad
j^{\prime}_{s+r}\psi_{s,s+r}=j^{\prime}_{s},\qquad\rho_{s+r,r}i^{\prime}_{s+r}=i^{\prime}_{r}$.
Proposition 9.8.7 Let $p\geq 5$ and $f\in[\Sigma^{t}K_{r},S]$ be any map, then
$f=jj^{\prime}_{r}\overline{f}$ for some
$\overline{f}\in[\Sigma^{t+rq+2}K_{r},K_{r}]$.
Proof : By Theorem 6.5.16(A) in chapter 6, there is
$\nu_{r}:\Sigma^{rq+2}K_{r}\rightarrow K_{r}\wedge K_{r}$ such that
$(jj^{\prime}_{r}\wedge 1_{K_{r}})\nu_{r}=1_{K_{r}}$. Let $K^{\prime}_{r}$ be
the cofibre of $jj^{\prime}_{r}:\Sigma^{-1}K_{r}\rightarrow\Sigma^{rq+1}S$
given by the cofibration $\Sigma^{-1}K_{r}\stackrel{{\scriptstyle
jj^{\prime}_{r}}}{{\longrightarrow}}\Sigma^{rq+1}S\stackrel{{\scriptstyle
z_{r}}}{{\longrightarrow}}K^{\prime}_{r}\rightarrow K_{r}$, then $z_{r}\wedge
1_{K_{r}}=(z_{r}jj^{\prime}_{r}\wedge 1_{K_{r}})\nu_{r}$ = 0
$\in[\Sigma^{rq+1}K_{r},K^{\prime}_{r}\wedge K_{r}]$. Consequently,
$z_{r}f=(1_{K^{\prime}_{r}}\wedge f)(z_{r}\wedge 1_{K_{r}})$ = 0 and so
$f=jj^{\prime}_{r}\overline{f}$ for some
$\overline{f}\in[\Sigma^{t+rq+1}K_{r},K_{r}]$. Q.E.D.
Let
(9.8.8)$\quad\qquad\cdots\stackrel{{\scriptstyle\tilde{a}_{2}}}{{\longrightarrow}}\quad\Sigma^{-2}\widetilde{E}_{2}\stackrel{{\scriptstyle\tilde{a}_{1}}}{{\longrightarrow}}\quad\Sigma^{-1}\widetilde{E}_{1}\quad\stackrel{{\scriptstyle\tilde{a}_{0}}}{{\longrightarrow}}\qquad\widetilde{E}_{0}=S$
$\quad\qquad\qquad\qquad\qquad\qquad\big{\downarrow}\tilde{b}_{2}\qquad\qquad\quad\big{\downarrow}\tilde{b}_{1}\qquad\qquad\qquad\big{\downarrow}\tilde{b}_{0}$
$\qquad\qquad\qquad\qquad\Sigma^{-2}BP\wedge\widetilde{E}_{2}\qquad\Sigma^{-1}BP\wedge\widetilde{E}_{1}\qquad
BP\wedge\widetilde{E}_{0}=BP$
be the canonnical Adams-Novikov resolution of the sphere spectrum $S$, where
$\widetilde{E}_{s}\stackrel{{\scriptstyle\tilde{b}_{s}}}{{\longrightarrow}}BP\wedge\widetilde{E}_{s}\stackrel{{\scriptstyle\tilde{c}_{s}}}{{\longrightarrow}}\widetilde{E}_{s+1}\stackrel{{\scriptstyle\tilde{a}_{s}}}{{\longrightarrow}}\Sigma\widetilde{E}_{s}$
are cofibrations for all $s\geq 0$ such that
$\widetilde{E}_{0}=S,\tilde{b}_{s}=\tau\wedge 1_{\widetilde{E}_{s}}$( $s>$0)
and $\tilde{b}_{0}=\tau:S\rightarrow BP$ is the injection of the bottom cell.
Then $\pi_{t}BP\wedge\widetilde{E}_{s}$ is the $E_{1}^{s,t}$-term of the
Adams-Novikov spectral sequence,
$(\tilde{b}_{s+1}\tilde{c}_{s})_{*}:\pi_{t}BP\wedge\widetilde{E}_{s}\rightarrow\pi_{t}BP\wedge\widetilde{E}_{s+1}$
are the $d_{1}^{s,t}$-differential and
$E_{2}^{s,t}=Ext_{BP_{*}BP}^{s,t}(BP_{*},BP_{*})\Longrightarrow(\pi_{t-s}S)_{p}$
Proposition 9.8.9 Let $p\geq 3,r\geq 1,s\geq 0$ and $\widetilde{E}_{s}$ be the
spectrum in the Adams-Novikov resolution (9.8.8),
$\wedge^{s}BP=BP\wedge\cdots\wedge BP$ be the smash products of $s$ copies of
$BP$, then
$(BP\wedge\widetilde{E}_{s})^{*},(BP\wedge\widetilde{E}_{s})^{*}(M),(BP\wedge\widetilde{E}_{s})^{*}(K_{r})$
are the direct summand of
$(\wedge^{s+1}BP)^{*},(\wedge^{s+1}BP)^{*}(M),(\wedge^{s+1}BP)^{*}(K_{r})$ and
we have
$[\Sigma^{t}M,BP\wedge\widetilde{E}_{s}]=(BP\wedge\widetilde{E}_{s})^{-t}(M)$
= 0 for $t\neq-1$ (mod q),
$[\Sigma^{t}K_{r},BP\wedge\widetilde{E}_{s}]=(BP\wedge\widetilde{E}_{s})^{-t}(K_{r})$
= 0 for $t\neq-2$ (mod q).
Proof : We first consider the $BP^{*}$-cohomology. It is known that
$\pi_{t}BP=BP_{t}=BP^{-t}$, then $BP^{*}=Z_{(p)}[v_{1},v_{2},\cdots]$ , where
$\mid v_{i}\mid=-2(p^{i}-1)$ and
$I_{n}=(p,v_{1},\cdots,v_{n-1}),(p,v_{1}^{r})$ is the invariant ideal of
$BP^{*}$. Clearly, there are two exact sequences on $BP^{*}$-cohomology as
follows
$\qquad 0\rightarrow BP^{*}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}BP^{*}\stackrel{{\scriptstyle\rho_{0}}}{{\longrightarrow}}BP^{*}/(p)\rightarrow
0$ $\qquad 0\rightarrow\Sigma^{-rq}BP^{*}/(p)\stackrel{{\scriptstyle
v_{1}^{r}}}{{\longrightarrow}}BP^{*}/(p)\stackrel{{\scriptstyle\rho_{1}}}{{\longrightarrow}}BP^{*}/(p,v_{1}^{r})\rightarrow
0$
where $\rho_{0},\rho_{1}$ are the projections.
Note that
$(\wedge^{s}BP)^{*}=\pi_{*}(\wedge^{s}BP)=BP_{*}(\wedge^{s-1}BP)=BP_{*}BP\otimes\cdots\otimes
BP_{*}BP$ with $s-1$ copies of $BP_{*}BP$ and $s\geq 2$. Then we have the
follwing short exact sequences ($s\geq 1$)
(9.8.10) $\qquad 0\rightarrow(\wedge^{s}BP)^{*}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}(\wedge^{s}BP)^{*}\longrightarrow(\wedge^{s}BP)^{*}/(p)\rightarrow
0$
$\quad 0\rightarrow\Sigma^{-rq}(\wedge^{s}BP)^{*}/(p)\stackrel{{\scriptstyle
v_{1}^{r}}}{{\longrightarrow}}(\wedge^{s}BP)^{*}/(p)\longrightarrow(\wedge^{s}BP)^{*}/(p,v_{1}^{r})\rightarrow
0$
For any $f\in[\Sigma^{t}M,\wedge^{s}BP]=(\wedge^{s}BP)^{-t}(M)$, if $t\neq
0$(mod q) , then by the sparseness of
$(\wedge^{s}BP)^{*}=BP_{*}BP\otimes\cdots\otimes BP_{*}BP$ ( that is,
$BP_{r}BP$ = 0 for $r\neq 0$ (mod $q$)) we have $fi\in(\wedge^{s}BP)^{-t}$ = 0
; if $t=0$ (mod q), then $fi$ is an element of order $p$ in $Z_{(p)}$-module
$(\wedge^{s}BP)^{-t}$ so that we have $fi$ = 0. This shows that
$i^{*}=0:(\wedge^{s}BP)^{*}(M)\rightarrow(\wedge^{s}BP)^{*}$. Similarly we
have
$(i^{\prime}_{r})^{*}=0:(\wedge^{s}BP)^{*}(K_{r})\rightarrow(\wedge^{s}BP)^{*}(M)$
( $r\geq 1$). Then, the cofibration (9.1.1) (9.8.2) induces respectively the
following short exact sequences for all $(r\geq 1)$
$\qquad 0\rightarrow(\wedge^{s}BP)^{*}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}(\wedge^{s}BP)^{*}\stackrel{{\scriptstyle
j^{*}}}{{\longrightarrow}}(\wedge^{s}BP)^{*}(M)\rightarrow 0$ $\qquad
0\rightarrow(\wedge^{s}BP)^{*}(M)\stackrel{{\scriptstyle(\alpha^{r})^{*}}}{{\longrightarrow}}(\wedge^{s}BP)^{*}(M)\stackrel{{\scriptstyle(j^{\prime}_{r})^{*}}}{{\longrightarrow}}(\wedge^{s}BP)^{*}(K_{r})\rightarrow
0$
where the degrees $\mid j^{*}\mid=-1$, $\mid(j^{\prime}_{r})^{*}\mid=-(rq+1)$.
By comparison to the above two short exact sequences with (9.8.10) we have
(9.8.11) $\qquad(\wedge^{s}BP)^{*}(M)\cong\Sigma(\wedge^{s}BP)^{*}/(p),$
$\qquad\quad(\wedge^{s}BP)^{*}(K_{r})\cong\Sigma^{rq+2}(\wedge^{s}BP)^{*}/(p,v_{1}^{r})$.
Let $\mu:BP\wedge BP\rightarrow BP$ be the multiplication of the ring spectrum
$BP$ and $\tau:S\rightarrow BP$ be the injection of the bottom cell, , then we
have $\mu(1_{BP}\wedge\tau)=1_{BP}=\mu(\tau\wedge 1_{BP})$ so that the
cofibration
$\widetilde{E}_{s-1}\stackrel{{\scriptstyle\tilde{b}_{s-1}}}{{\longrightarrow}}BP\wedge\widetilde{E}_{s-1}\stackrel{{\scriptstyle\tilde{c}_{s-1}}}{{\longrightarrow}}\widetilde{E}_{s}\stackrel{{\scriptstyle\tilde{a}_{s-1}}}{{\longrightarrow}}\Sigma\widetilde{E}_{s-1}$
induces a split short exact sequence
$BP\wedge\widetilde{E}_{s-1}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{b}_{s-1}}}{{\longrightarrow}}BP\wedge
BP\wedge\widetilde{E}_{s-1}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{c}_{s-1}}}{{\longrightarrow}}BP\wedge\widetilde{E}_{s}$
this is because $(\mu\wedge
1_{\widetilde{E}_{s-1}})(1_{BP}\wedge\tilde{b}_{s-1})=(\mu\wedge
1_{\widetilde{E}_{s-1}})(1_{BP}\wedge\tau\wedge
1_{\widetilde{E}_{s-1}})=1_{BP\wedge\widetilde{E}_{s-1}}$. That is to say,
$BP\wedge\widetilde{E}_{s}$ is the direct summand of $BP\wedge
BP\wedge\widetilde{E}_{s-1}$ and by induction we have
$BP\wedge\widetilde{E}_{s}$ is the direct summand of $\wedge^{s+1}BP$. Hence,
$(BP\wedge\widetilde{E}_{s})^{*},(BP\wedge\widetilde{E}_{s})^{*}(M),(BP\wedge\widetilde{E}_{s})^{*}(K_{r})$
are the direct summand of
$(\wedge^{s+1}BP)^{*},(\wedge^{s+1}BP)^{*}(M),(\wedge^{s+1}BP)^{*}(K_{r})$
respectively and the last result can be obtained by (9.8.11). Q.E.D.
Proposition 9.8.12 Let $p\geq 3,n\geq 1$, then
$\qquad\qquad
Ext_{BP_{*}BP}^{0,p^{n}(p+1)q}(BP_{*},BP_{*}/(p,v_{1}^{p^{n}-1}))$
is generated additively by the generators
$v_{2}^{p^{n}},v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(t_{r}p^{n-2r})$ ( $r\geq
1$), where $t_{r}=(p^{2r+1}+1)/(p+1)$ and $\tilde{c}_{1}(ap^{s})$ is the
generator in Theorem 8.1.7 in chapter 8 which has degree $sp^{s}(p+1)q$.
Proof : By Theorem 8.1.7, the desired generators are of the form
$v_{1}^{b}\tilde{c}_{1}(ap^{s})$ with degrees $bq+ap^{s}(p+1)q=p^{n}(p+1)q$,
$a\geq 1$ is not divisible by $p$, $0\leq b<p^{n}-1$ and $b\geq$ max
$\\{0,p^{n}-1-q_{1}(ap^{s})\\}$, where $q_{1}(ap^{s})=p^{s}$ if $a=1$,
$q_{1}(ap^{s})=p^{s}+p^{s-1}-1$ if $a\geq 2$ is not divisible by $p$.
If $b=0$, then the generator is $v_{2}^{p^{n}}$. Since $b<p^{n}-1$, then $s<n$
and $b\equiv 0$ ( mod $p^{s}$) and so $b\geq p^{n}-1-q_{1}(ap^{s})\geq
p^{n}-p^{n-1}$ if $b\geq 1$. Let
$b=(p-1)p^{n-1}+c_{n-2}p^{n-2}+\cdots+c_{s}p^{s}$ be the p-adic expasion of
$b$ such that $0\leq c_{i}\leq p-1$. By $b\geq(p^{n}-1)-(p^{s}+p^{s-1}-1)$ or
$b\geq(p^{n}-1)-p^{s}$ we have $c_{n-2}p^{n-2}+\cdots+c_{s}p^{s}\geq
p^{n-1}-p^{s}-p^{s-1}$ or $p^{n-1}-p^{s}-1$. Consequently we have
$c_{n-2}=\cdots=c_{s}=p-1$. On the other hand, $b$ is divisible by $p+1$, then
$(p-1)-c_{n-2}+\cdots+(-1)^{n-1-s}c_{s}$ = 0 so that $n-1-s$ must be odd. Let
$n-1-s=2r-1$, then we have $s=n-2r,b=p^{n}-p^{n-2r}$ as desired and
$a=(p^{2r+1}+1)/(p+1)$. Q.E.D.
Proposition 9.8.13 Let $p\geq 3,n\geq 1$, then
(1) $Ext_{BP_{*}BP}^{2,p^{n+1}q+q}(BP_{*},BP_{*})$ is generated additively by
the generators
$\beta_{p^{n}/p^{n}-1},\beta_{t_{r}p^{n-2r}/p^{n-2r}-1}$ for all $r\geq 1$),
where $t_{r}=(p^{2r+1}+1)/(p+1)$.
(2) $Ext_{BP_{*}BP}^{1,p^{n+1}q+q}(BP_{*},BP_{*}M)$ is generated additively by
the generators
$\beta^{\prime}_{p^{n}/p^{n}-1},\beta^{\prime}_{t_{r}p^{n-2r}/p^{n-2r}-1}$ for
all $r\geq 1$), where $t_{r}=(p^{2r+1}+1)/(p+1)$ and
$\beta^{\prime}_{tp^{n}/s}$ is the generator in
$Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}M)$ such that
$j_{*}(\beta^{\prime}_{tp^{n}/s})=\beta_{tp^{n}/s}\in
Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})$.
Proof : By Theorem 8.1.3 in chapter 8, $Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})$
is generated additively by the generators $\beta_{ap^{s}/b,c+1}\in
Ext_{BP_{*}BP}^{2,ap^{s}(p+1)q-bq}(BP_{*},BP_{*})$ , where $s\geq 0$, $a\geq
1$ is not divisible by $p$, $b\geq 1,c\geq 0$ and subject to
(i) $b\leq s$ if $a$ = 1.
(ii) $p^{c}\mid b\leq p^{s-c}+p^{s-c-1}-1$
(iii) $p^{s-c-1}+p^{s-c-2}-1<b$ if $p^{c+1}\mid b$,
and $\beta_{ap^{s}/b,1}=\beta_{ap^{s}/b}$. Then, for $\beta_{ap^{s}/b,c+1}\in
Ext_{BP_{*}BP}^{2,p^{n+1}q+q}(BP_{*},BP_{*})$ we have
$ap^{s}(p+1)q-bq=p^{n+1}q+q=p^{n}(p+1)q-(p^{n}-1)q$ so that
$ap^{s}(p+1)q+(p^{n}-1-b)q=p^{n}(p+1)q$. Similar to that in the proof of Prop.
9.8.12 we have $a=1,s=n,b=p^{n}-1$ or
$a=(p^{2r+1}+1)/(p+1),b=p^{n-2r}-1,s=p^{n-2r}$( $r\geq 1$) and consequently
$c=0$. This shows (1) and the proof of (2) is similar. Q.E.D.
After finishing the proof of the above Proposition, we proceed to prove
Theorem 9.8.1 in case $t\geq 1$. The proof will be done by some argument
processing in the Adams-Novikov resolution of some spectra and using the
$h_{0}h_{n+1}$-map in Theorem 9.5.1 as our geometric input. We first prove the
following Lemma.
Lemma 9.8.14 If $g^{\prime\prime}$ is the element in $\pi_{p^{n}(p+1)q}BP$
such that $\tilde{b}_{1}\tilde{c}_{0}g^{\prime\prime}j\\\
j^{\prime}_{p^{n}-1}=0\in[\Sigma^{p^{n+1}q+q-2}K_{p^{n}-1},BP\wedge E_{1}]$,
then there exists $\bar{g}=px_{1}+v_{1}^{p^{n}-1}x_{2}\in\pi_{p^{n}(p+1)q}BP$
such that
$\tilde{b}_{1}\tilde{c}_{0}(g^{\prime\prime}-\bar{g})=0\in\pi_{p^{n}(p+1)q}BP\wedge\widetilde{E}_{1}$,
where $x_{1},x_{2}$ is some elements in $\pi_{*}BP$.
Proof : Let $\mu:BP\wedge BP\rightarrow BP$ be the multiplication of the ring
spectrum $BP$, then $\mu(\tilde{b}_{0}\wedge
1_{BP})=1_{BP}=\mu(1_{BP}\wedge\tilde{b}_{0})$, where
$\tilde{b}_{0}=\tau:S\rightarrow BP$ is the injection of the bottom cell as
stated above. Then we have the following split cofibration
$\quad BP\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{b}_{0}}}{{\longrightarrow}}BP\wedge
BP\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{c}_{0}}}{{\longrightarrow}}BP\wedge\widetilde{E}_{1}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{a}_{0}=0}}{{\longrightarrow}}\Sigma BP$
$BP\wedge\widetilde{E}_{1}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{b}_{0}\wedge
1_{\widetilde{E}_{1}}}}{{\longrightarrow}}BP\wedge
BP\wedge\widetilde{E}_{1}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{c}_{1}}}{{\longrightarrow}}BP\wedge\widetilde{E}_{2}\stackrel{{\scriptstyle
1_{BP}\wedge\tilde{a}_{0}=0}}{{\longrightarrow}}\Sigma
BP\wedge\widetilde{E}_{1}$
and there is $\mu^{\prime}:BP\wedge\widetilde{E}_{1}\rightarrow BP\wedge BP$
such that
$(1_{BP}\wedge\tilde{b}_{0})\mu+\mu^{\prime}(1_{BP}\wedge\tilde{c}_{0})=1_{BP\wedge
BP}$.
By $\tilde{b}_{1}\tilde{c}_{0}g^{\prime\prime}jj^{\prime}_{p^{n}-1}$ = 0 we
have
$\tilde{c}_{0}g^{\prime\prime}jj^{\prime}_{p^{n}-1}=\tilde{a}_{1}g^{\prime}$
with $g^{\prime}\in$
$[\Sigma^{p^{n+1}q+q-1}K_{p^{n}-1},\widetilde{E}_{2}]$. Then
$(1_{BP}\wedge\tilde{c}_{0}g^{\prime\prime}jj^{\prime}_{p^{n}-1})=(1_{BP}\wedge\tilde{a}_{1})(1_{BP}\wedge
g^{\prime})$ = 0 so that$1_{BP}\wedge
g^{\prime\prime}jj^{\prime}_{p^{n}-1}=[(1_{BP}\wedge\tilde{b}_{0})\mu+\mu^{\prime}(1_{BP}\wedge\tilde{c}_{0})](1_{BP}\wedge
g^{\prime\prime}jj^{\prime}_{p^{n}-1})$ =
$(1_{BP}\wedge\tilde{b}_{0})\mu(1_{BP}\wedge
g^{\prime\prime}jj^{\prime}_{p^{n}-1})$ and we have
(9.8.15) $\qquad(\tilde{b}_{0}\wedge
1_{BP})g^{\prime\prime}jj^{\prime}_{p^{n}-1}=(1_{BP}\wedge
g^{\prime\prime}jj^{\prime}_{p^{n}-1})(\tilde{b}_{0}\wedge 1_{K_{p^{n}-1}})$
$\qquad\qquad\quad=(1_{BP}\wedge\tilde{b}_{0})\mu(1_{BP}\wedge
g^{\prime\prime}jj^{\prime}_{p^{n}-1})(\tilde{b}_{0}\wedge 1_{K_{p^{n}-1}})$
$\qquad\qquad\quad=(1_{BP}\wedge\tilde{b}_{0})g^{\prime\prime}jj^{\prime}_{p^{n}-1}$.
Note that $(\tilde{b}_{0}\wedge
1_{BP})_{*},(1_{BP}\wedge\tilde{b}_{0})_{*}:BP_{*}\rightarrow BP_{*}BP$ are
the right and left unit $\eta_{R},\eta_{L}:BP_{*}\rightarrow BP_{*}BP$
respectively, then by (9.8.15) we have
$\eta_{R}(g^{\prime\prime})=\eta_{L}(g^{\prime\prime})$ mod
$(p,v_{1}^{p^{n}-1})$. This means that $(p,v_{1}^{p^{n}-1},g^{\prime\prime})$
is a $BP_{*}$ invariant ideal , or equivalently, $g^{\prime\prime}\in
Ext_{BP_{*}BP}^{0.p^{n}(p+1)q}(BP_{*},BP_{*}/(p,v_{1}^{p^{n}-1}))$. Then by
Prop. 9.8.12 we have
(9.8.16)$\qquad g^{\prime\prime}=\lambda
v_{2}^{p^{n}}+\Sigma\lambda_{r}v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(t_{r}p^{n-2r})+px_{1}+v_{1}^{p^{n}-1}x_{2}\in
BP_{*}$
where $1\leq\lambda,\lambda_{r}\leq p-1,t_{r}=(p^{2r+1}+1)/(p+1)$ and
$x_{1},x_{2}$ are some elements in $BP_{*}$.
Let $\bar{g}=px_{1}+v_{1}^{p^{n}-1}x_{2}$, then $(\tilde{b}_{0}\wedge
1_{BP})(g^{\prime\prime}-\bar{g})=\eta_{R}(g^{\prime\prime}-\bar{g})=\eta_{L}(g^{\prime\prime}-\bar{g})=(1_{BP}\wedge\tilde{b}_{0})(g^{\prime\prime}-\bar{g})$
so that
$\tilde{b}_{1}\tilde{c}_{0}(g^{\prime\prime}-\bar{g})=(1_{BP}\wedge\tilde{c}_{0})(\tilde{b}_{0}\wedge
1_{BP})(g^{\prime\prime}-\bar{g})$ = 0. Q.E.D.
Proof of Theorem 9.8.1 in case $t\geq 1$ By Theorem 9.5.1, there is
$\tilde{\eta}_{n+1}\in\pi_{p^{n+1}q+q-1}M$ such that
$\eta_{n+1}=j\tilde{\eta}_{n+1}\in\pi_{p^{n+1}q+q-2}S$ is represented in the
ASS by $h_{0}h_{n+1}\in Ext_{A}^{2,p^{n+1}q+q}(Z_{p},Z_{p})$. By Theorem 8.1.5
in chapter 8, $\Phi(\beta_{p^{n}/p^{n}-1})=h_{0}h_{n+1}$, where
$\Phi:Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})\rightarrow
Ext_{A}^{2,*}(Z_{p},Z_{p})$ is the Thom map. Then
$\eta_{n+1}=j\tilde{\eta}_{n+1}$ is represented by $\beta_{p^{n}/p^{n}-1}+x\in
Ext_{BP_{*}BP}^{2,p^{n+1}q+q}(BP_{*},BP_{*})$ in the Adams-Novikov spectral
sequence, where $x=\Sigma_{r\geq
1}\lambda_{r}\beta_{t_{r}p^{n-2r}/p^{n-2r}-1}$ with $\lambda_{r}\in Z_{(p)}$ (
cf. Prop. 9.8.13). Moreover, $\tilde{\eta}_{n+1}\in\pi_{p^{n+1}q+q-1}M$ is
represented by $\beta^{\prime}_{p^{n}/p^{n}-1}+x^{\prime}+i_{*}(y)$ in the
Adams spectral sequence, where $y\in Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*})$, and
$x^{\prime}=\Sigma_{r\geq
1}\lambda_{r}\beta^{\prime}_{t_{r}p^{n-2r}/p^{n-2r}-1}$ ,
$\beta^{\prime}_{tp^{n}/s}$ are the elements in
$Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}(M))$ such that
$j_{*}\beta^{\prime}_{tp^{n}/s}=\beta_{tp^{n}/s}\in
Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})$ . It is known that all the generators in
$Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*})$ are permanent cycles in the Adams-
Novikov spectral sequence , then there exists
$\tilde{f}\in\pi_{p^{n+1}q+q-1}M$ such that it is represented by
$\beta^{\prime}_{p^{n}/P^{n}-1}+x^{\prime}$. In addition, $\tilde{f}$ can be
extended by $f\in[\Sigma^{p^{n+1}q+q-1}M,M]\cap kerd$ such that
$\tilde{f}=fi$. Recall from (9.8.8)
$\qquad\cdots\stackrel{{\scriptstyle\tilde{a}_{2}\wedge
1_{M}}}{{\longrightarrow}}\Sigma^{-2}\widetilde{E}_{2}\wedge
M\stackrel{{\scriptstyle\tilde{a}_{1}\wedge
1_{M}}}{{\longrightarrow}}\Sigma^{-1}\widetilde{E}_{1}\wedge
M\stackrel{{\scriptstyle\tilde{a}_{0}\wedge
1_{M}}}{{\longrightarrow}}\widetilde{E}_{0}\wedge M=M$
$\qquad\quad\qquad\qquad\qquad\Big{\downarrow}\tilde{b}_{2}\wedge
1_{M}\qquad\qquad\Big{\downarrow}\tilde{b}_{1}\wedge
1_{M}\quad\qquad\Big{\downarrow}\tilde{b}_{0}\wedge 1_{M}$
$\qquad\qquad\Sigma^{-2}BP\wedge\widetilde{E}_{2}\wedge
M\qquad\Sigma^{-1}BP\wedge\widetilde{E}_{1}\wedge M\qquad BP\wedge M$
is the Adams-Novikov resolution of the Moore spectrum $M$. Then $fi$ can be
lifted to $f_{1}i\in\pi_{p^{n+1}q+q}(\widetilde{E}_{1}\wedge M)$ with
$f_{1}\in[\Sigma^{p^{n+1}q+q}M,E_{1}\wedge M]\cap kerd$ such that
$\tilde{a}_{0}\wedge 1_{M})f_{1}i=fi$ and the $d_{1}$-cycle
$(\tilde{b}_{1}\wedge
1_{M})f_{1}i\in\pi_{p^{n+1}q+q}BP\wedge\widetilde{E}_{1}\wedge M$ represents
$\beta^{\prime}_{p^{n}/p^{n}-1}+x^{\prime}\in
Ext_{BP_{*}BP}^{1,p^{n+1}q+q}(BP_{*},BP_{*}(M))$. By applying $d$ to the
equation $(\tilde{a}_{0}\wedge 1_{M})f_{1}ij=fij$ we have
$(\tilde{a}_{0}\wedge 1_{M})f_{1}=f$.
Since $(\tilde{a}_{0}\wedge 1_{M})f_{1}i=fi\in\pi_{p^{n+1}q+q-1}M$ is
represented by $\beta^{\prime}_{p^{n}/p^{n}-1}+x^{\prime}\in
Ext_{BP_{*}BP}^{1,p^{n+1}q+q}(BP_{*},BP_{*}(M))$ in the Adams-Novikov spectral
sequence and $v_{1}^{p^{n}-1}(\beta^{\prime}_{p^{n}/p^{n}-1}+x^{\prime})$ = 0,
then $(\tilde{a}_{0}\wedge
1_{M})f_{1}\alpha^{p^{n}-1}i=f\alpha^{p^{n}-1}i=\alpha^{p^{n}-1}fi$ has $BP$\-
filtration $>1$ so that $(\tilde{b}_{1}\wedge 1_{M})f_{1}\alpha^{p^{n}-1}i$ is
a $d_{1}$-boundary and it equals to $(\tilde{b}_{1}\tilde{c}_{0}\wedge
1_{M})gi$ for some $g\in[\Sigma^{p^{n}(p+1)q}M,BP\wedge M]$. Hence,
$(\tilde{b}_{1}\wedge
1_{M})f_{1}\alpha^{p^{n}-1}=(\tilde{b}_{1}\tilde{c}_{0}\wedge 1_{M})g$ , this
is because $\pi_{p^{n}(p+1)q+1}BP\wedge\widetilde{E}_{1}\wedge M$ = 0 which is
obtained by the sparseness fo $BP_{*}(\widetilde{E}_{1}\wedge M)$.
Consequently we have
$f_{1}\alpha^{p^{n}-1}=(\tilde{c}_{0}\wedge 1_{M})g+(\tilde{a}_{1}\wedge
1_{M})f_{2}$ with $f_{2}\in[\Sigma^{p^{n}(p+1)q+1}M,\widetilde{E}_{2}\wedge
M]$
and
$(1_{\widetilde{E}_{1}}\wedge
j)f_{1}\alpha^{p^{n}-1}=\tilde{c}_{0}g^{\prime\prime}j+\tilde{a}_{1}(1_{\widetilde{E}_{2}}\wedge
j)f_{2}$ with $g^{\prime\prime}\in\pi_{p^{n}(p+1)q}BP$,
where $g^{\prime\prime}j=(1_{BP}\wedge j)g$, this is because $(1_{BP}\wedge
j)gi\in\pi_{p^{n}(p+1)q-1}BP$ = 0. In addition, by
$\tilde{b}_{2}(1_{\widetilde{E}_{2}}\wedge
j)f_{2}\in[\Sigma^{p^{n}(p+1)q}M,BP\wedge\widetilde{E}_{2}]$= 0 and
$[\Sigma^{p^{n}(p+1)q+1}M,BP\\\ \wedge\widetilde{E}_{3}]$ = 0 ( cf. Prop.
9.8.9), then $(1_{\widetilde{E}_{2}}\wedge
j)f_{2}=\tilde{a}_{2}\tilde{a}_{3}f_{3}$ for some
$f_{3}\in[\Sigma^{p^{n}(p+1)q+2}M,E_{4}]$ and we have
(9.8.17) $\qquad(1_{E_{1}}\wedge
j)f_{1}\alpha^{p^{n}-1}=\bar{c}_{0}g^{\prime\prime}j+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{3}$.
By (9.8.17) we have
$\tilde{b}_{1}\tilde{c}_{0}g^{\prime\prime}jj^{\prime}_{p^{n}-1}$ = 0, then by
Lemma 9.8.14, there is
$\bar{g}=px_{1}+v_{1}^{p^{n}-1}x_{2}\in\pi_{p^{n}(p+1)q}BP$ with
$x_{1},x_{2}\in\pi_{*}BP$ such that
$\tilde{b}_{1}\tilde{c}_{0}(g^{\prime\prime}-\bar{g})$ = 0. Consequently,
$\tilde{c}_{0}(g^{\prime\prime}-\bar{g})=\tilde{a}_{1}f_{4}$ for some
$f_{4}\in\pi_{p^{n}(p+1)q+1}\widetilde{E}_{2}$ and
$f_{4}=\tilde{a}_{2}\tilde{a}_{3}f_{5}$ with
$f_{5}\in\pi_{p^{n}(p+1)q+3}\widetilde{E}_{4}$ which is obtained by the
sparseness of $\pi_{*}BP\wedge\widetilde{E}_{s}$. So, (9.8.17) becomes
(9.8.18) $\qquad(1_{\widetilde{E}_{1}}\wedge
j)f_{1}\alpha^{p^{n}-1}=\tilde{c}_{0}\bar{g}j+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{5}j+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{3}$.
Note that $\bar{g}=px_{1}+v_{1}^{p^{n}-1}x_{2}$, then ,
$\bar{g}jj^{\prime}_{p^{n}-1}$ = 0 $\in
BP^{*}(K_{p^{n}-1})\cong\Sigma^{-(p^{n}-1)q-2}\\\ BP^{*}/(p,v_{1}^{p^{n}-1})$
so that $\bar{g}j=\tilde{g}\alpha^{p^{n}-1}$ for some
$\tilde{g}\in[\Sigma^{p^{n+1}q+q-1}M,BP]$. Consequently, by (9.8.4), the
equation (9.8.18) becomes
(9.8.19) $\qquad(1_{\widetilde{E}_{1}}\wedge
j)f_{1}j^{\prime}_{1}\rho_{p^{n},1}=(1_{\widetilde{E}_{1}}\wedge
j)f_{1}\alpha^{p^{n}-1}j^{\prime}_{p^{n}}$
$\qquad\qquad=\tilde{c}_{0}\tilde{g}\alpha^{p^{n}-1}j^{\prime}_{p^{n}}+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}(f_{5}j+f_{3})j^{\prime}_{p^{n}}$
$\qquad\qquad=\tilde{c}_{0}\tilde{g}j^{\prime}_{1}\rho_{p^{n},1}+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}(f_{5}j+f_{3})j^{\prime}_{p^{n}}$
Moreover, by (9.8.4)(9.8.6) we have
$\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}(f_{5}j+f_{3})j^{\prime}_{p^{n}-1}=\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}(f_{5}j+f_{3})j^{\prime}_{p^{n}}\psi_{p^{n}-1,p^{n}}$
= 0 and so $(f_{5}j+f_{3})j^{\prime}_{p^{n}-1}$ = 0 , this is because
$[\Sigma^{p^{n+1}q+q+r}\\\ K_{p^{n}-1},BP\wedge\widetilde{E}_{2+r}]$ = 0 for
$r=-1,0,1$( cf. Prop. 9.8.9). This shows that
$(f_{5}j+f_{3})=f_{6}\alpha^{p^{n}-1}$ with
$f_{6}\in[\Sigma^{p^{n+1}q+q+2}M,\widetilde{E}_{4}]$. Hence, the equation
(9.8.19) becomes
(9.8.20) $\qquad(1_{\widetilde{E}_{1}}\wedge
j)f_{1}j^{\prime}_{1}\rho_{p^{n},1}=\tilde{c}_{0}\tilde{g}j^{\prime}_{1}\rho_{p^{n},1}+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{2}f_{6}\alpha^{p^{n}-1}j^{\prime}_{p^{n}}$
$\qquad\qquad\qquad=\tilde{c}_{0}\tilde{g}j^{\prime}_{1}\rho_{p^{n},1}+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{6}j^{\prime}_{1}\rho_{p^{n},1}$
and by (9.8.6) we have
(9.8.21) $\qquad(1_{\widetilde{E}_{1}}\wedge
j)f_{1}j^{\prime}_{1}=\tilde{c}_{0}\tilde{g}j^{\prime}_{1}+\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{6}j^{\prime}_{1}+\epsilon
i^{\prime}_{p^{n}-1}j^{\prime}_{1}$
for some $\epsilon\in[\Sigma^{p^{n+1}q+q-1}K_{p^{n}-1},\widetilde{E}_{1}]$. By
composing $\tilde{a}_{0}$ to (9.8.21) we have
$jfj^{\prime}_{1}=\tilde{a}_{0}\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{6}j^{\prime}_{1}+\tilde{a}_{0}\epsilon
i^{\prime}_{p^{n}-1}j^{\prime}_{1}=\tilde{a}_{0}\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{6}j^{\prime}_{1}+jj^{\prime}_{p^{n}-1}\overline{\epsilon}i^{\prime}_{p^{n}-1}j^{\prime}_{1}$
, this is because
$\tilde{a}_{0}\epsilon=jj^{\prime}_{p^{n}-1}\overline{\epsilon}$ for some
$\overline{\epsilon}\in[\Sigma^{p^{n}(p+1)q}K_{p^{n}-1},K_{p^{n}-1}]$ (cf.
Prop. 9.8.2). Consequently we have
(9.8.22) $\qquad
jfi=\tilde{a}_{0}\tilde{a}_{1}\tilde{a}_{2}\tilde{a}_{3}f_{6}i+jj^{\prime}_{p^{n}-1}\overline{\epsilon}i^{\prime}_{p^{n}-1}i+f_{7}\alpha
i$
with $f_{7}\in[\Sigma^{p^{n+1}q-2}M,S]$.
We claim that the map $f_{7}\alpha i$ in (9.8.22) has filtration $\geq 3$ so
that by (9.8.22) we obtain that
$jj^{\prime}_{p^{n}-1}\tilde{\epsilon}i^{\prime}_{p^{n}-1}i\in\pi_{p^{n+1}q+q-2}S$
is represented by $h_{0}h_{n+1}\in$ $Ext_{A}^{2,p^{n+1}q+q}(Z_{p},Z_{p})$ in
the Adams spectral sequence. This claim will be proved in the last.
Then, $jj^{\prime}_{p^{n}-1}\tilde{\epsilon}i^{\prime}_{p^{n}-1}i$ is
represented by $\lambda_{0}\beta_{p^{n}/p^{n}-1}+\Sigma_{r\geq
1}\lambda_{r}\beta_{t_{r}p^{n-2r}/p^{n-2r}-1}\\\ \in
Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})$ in the Adams-Novikov spectral sequence so
that by Prop. 9.8.12 we know that
$\tilde{\epsilon}i^{\prime}_{p^{n}-1}i\in\pi_{p^{n}(p+1)q}K_{p^{n}-1}$ is
represented by $v_{2}^{p^{n}}+\Sigma_{r\geq
1}\lambda_{r}v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(t_{r}p^{n-2r})\in
Ext_{BP_{*}BP}^{0,p^{n}(p+1)q}(BP_{*},BP_{*}K_{p^{n}-1})$, where
$\lambda_{r}\in Z_{p}$ and $t_{r}=(p^{2r+2}+1)/(p+1)$.
By [22] Theorem C,D, it is known that $v_{2}^{tp}\in
Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{r})$ , for $t\geq 1,1\leq r\leq p-1$, is
a permanent cyce in the Adams-Novikov spectral sequence. Suppose inductively
that $v_{2}^{tp^{s}}\in Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{r})$ (for $t\geq
1,1\leq r\leq p^{s}-1$ and $s\leq n-1$) is a permanent cycle in the Adams-
Novikov spectral sequence, then we know that
$v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(t_{r}p^{n-2r})\in
Ext_{BP_{*}BP}^{0,p^{n}(p+1)q}\\\ (BP_{*},BP_{*}K_{p^{n}-1})$ also is a
permanent cycle for all $r\geq 1$. Moreover, by the representation of the
above $\tilde{\epsilon}i^{\prime}_{p^{n}-1}i$ we obtain that $v_{2}^{p^{n}}\in
Ext_{BP_{*}BP}^{0,p^{n}(p+1)q}(BP_{*},\\\ BP_{*}K_{p^{n}-1})$ is a permanent
cycle. Hence, by (9.8.6), there exists
$k\in[\Sigma^{p^{n}(p+1)q}K_{p^{n}-1},K_{p^{n}-1}]$ such that the induced
$BP_{*}$-homomorphism $k_{*}=v_{2}^{p^{n}}$. In addition, the map
$\rho_{p^{n}-1,r}:K_{p^{n}-1}\rightarrow K_{r}$ in (9.8.4) for all $r\leq
p^{n}-1$ is a projection, then
$\rho_{p^{n}-1,r}k^{t}i^{\prime}_{p^{n}-1}i\in\pi_{tp^{n}(p+1)q}K_{r}$ is
represented by $v_{2}^{tp^{n}}\in Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{r})$ in
the Adams-Novikov spectral sequence. This completes the induction and
$jj^{\prime}_{r}\rho_{p^{n}-1,r}k^{t}i^{\prime}_{p^{n}-1}i\in\pi_{*}S$ is just
the $\beta_{tp^{n}/r}$-element of the Theorem.
Now our remaining work is to prove the above claim. We turn to an argument in
the ASS and let $A$ be the mod $p$ Steenrod algebra. By
$Ext_{A}^{1,p^{n+1}q-1}(Z_{p},H^{*}M)\cong Z_{p}\\{j^{*}(h_{n+1})\\}$ and the
result on $\beta_{p^{n}/p^{n}}\in Ext_{BP_{*}BP}^{2,p^{n+1}q}\\\
(BP_{*},BP_{*})$ support a nontrivial differential in the Adams-Novikov
spectral sequence in [12] p.106 Thoerem 5.4.8(i), we know that
(9.8.23) $\qquad j^{*}(h_{n+1})\in Ext_{A}^{1,p^{n+1}q-1}(Z_{p},H^{*}M)$ dies
in the ASS
Then, the map $f_{7}\in[\Sigma^{p^{n+1}q-2}M,S]$ in (9.8.22) has filtration
$\geq 2$ in the ASS and so $f_{7}\alpha i$ has filtration $\geq 3$. Moreover,
by (9.8.21) we know that
$jj^{\prime}_{p^{n}-1}\overline{\epsilon}i^{\prime}_{p^{n}-1}i$ and
$jfi\in\pi_{p^{n+1}q+q-2}S$ must have the same filtration so that it is
repesented by $h_{0}h_{n+1}\in Ext_{A}^{2,p^{n+1}q+q}(Z_{p},Z_{p})$ in the
ASS. This shows the above claim and the Theorem is proved. Q.E.D.
Remark 9.8.24 We give a detail proof of the result in (9.8.23) as follows. It
will be done by some argument processing in the Adams resolution (9.2.9).
Suppose in contrast that the map $j^{*}h_{n+1}\in
Ext_{A}^{1,p^{n+1}q-1}(Z_{p},H^{*}M)$ is a permanent cycle in the ASS, then we
have $\bar{c}_{1}h_{n+1}\cdot j$ = 0 , where
$h_{n+1}\in\pi_{p^{n+1}q}KG_{1}\cong Ext_{A}^{1,p^{n+1}q}(Z_{p},Z_{p})$.
Consequently $\bar{c}_{1}h_{n+1}=\bar{f}\cdot p$ for some
$\bar{f}\in\pi_{p^{n+1}q}E_{2}$. On the other hand,
$\bar{b}_{2}\bar{f}\in\pi_{p^{n+1}q}KG_{2}\cong
Ext_{A}^{2,p^{n+1}q}(Z_{p},Z_{p})\cong Z_{p}\\{b_{n}\\}$ so that we have
$\bar{b}_{2}\bar{f}=\lambda\cdot b_{n}$ with $\lambda\in Z_{p}$. However, the
scalar $\lambda$ must be zero, this is because $b_{n}$ support a nontrivial
differential $d_{2p-1}(b_{n})=d_{2p-1}\Phi(\beta_{p^{n}/p^{n}})=\Phi
d_{2p-1}(\beta_{p^{n}/p^{n}})=\Phi(\alpha_{1}\beta_{p^{n-1}/p^{n-1}}^{p})=h_{0}b_{n-1}^{p}\neq
0$ (cf. [12] p.206 Theorem 5.4.8(i) ). Hence $\bar{f}=\bar{a}_{2}\bar{f}_{1}$
for some $\bar{f}_{1}\in\pi_{p^{n+1}q+1}E_{3}$ and we have
$\bar{c}_{1}h_{n+1}=\bar{a}_{2}\bar{f}_{1}\cdot
p=\bar{a}_{2}\bar{a}_{3}\bar{f}_{2}$ with
$\bar{f}_{2}\in\pi_{p^{n+1}q+2}E_{4}$. This means that the secondary
differential $d_{2}(h_{n+1})$ = 0 which contradicts with the following known
nontrivial differential $d_{2}(h_{n+1})=a_{0}b_{n}\neq 0\in
Ext_{A}^{3,p^{n+1}q+1}(Z_{p},Z_{p})$ ( cf. [12] p.11 Theorem 1.2.14). So we
have $\tilde{c}_{1}h_{n+1}\cdot j\neq 0$ and so (9.8.23) holds.
Now we proceed to prove Theorem 9.8.1 in case $t\geq 2$. We first prove the
following Lemmas and Propositions.
Lemma 9.8.25 Let $p\geq 3$ and $v_{1}x\in
Ext_{BP_{*}BP}^{0,tp^{n}(p+1)q}(BP_{*},BP_{*}K_{p^{n}})$, then
$v_{1}x=\Sigma_{r=1}^{[n/2]}\lambda_{r}v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(a_{r}p^{n-2r})$,
where $\lambda_{r}\in Z_{p},a_{r}=(tp^{2r+1}+tp^{2r}-p^{2r}+1)/(p+1)$ and
$\tilde{c}_{1}(ap^{s})$ is the generator in Theorem 8.1.7 in chapter 6 which
has degree $ap^{s}(p+1)q$.
Proof : By Theorem 8.1.7 in chapter 8, $v_{1}x$ is a linear combination of the
following generators $v_{1}^{b}\tilde{c}_{1}(ap^{s})$, where $a\geq 1$ is not
divisible by $p$, $1\leq b<p^{n},b\geq max\\{0,p^{n}-q_{1}(ap^{s}\\}$ and
$q_{1}(ap^{s})=p^{s}$ if $a=1,q_{1}(ap^{s})=p^{s}+p^{s-1}-1$ if $a\geq 2$.
By degree reasons we have $bq+ap^{s}(p+1)q=tp^{n}(p+1)q$, then $s<n,b\geq
p^{n}-(p^{s}+p^{s-1}-1)>0$ and so $b\geq p^{n}-p^{n-1}$ if $s\leq n-2$. If
$s=n-1$, then $b$ is divisible by $p^{n-1}(p+1)$ so that $b\geq
p^{n}+p^{n-1}$. So, in any case we have $b\geq p^{n-1}(p-1)$ and the remaining
steps is similar to that given in the proof of Prop. 9.8.12. Q.E.D.
Proposition 9.8.26 Let $r>s$ and $\rho_{r,s}:K_{r}\rightarrow K_{s}$ be the
map in (9.8.4), then $d(\rho_{r,s})=i^{\prime}_{s}\xi j^{\prime}_{r}$ with
$\xi\in[\Sigma^{rq+1}M,M]\cap kerd$.
Proof : By (9.8.6)(9.8.3) we have
$j^{\prime}_{s}d(\rho_{r,s})=d(j^{\prime}_{s}\rho_{r,s})=d(\alpha^{r-s}j^{\prime}_{r})$
= 0 , then $d(\rho_{r,s})=i^{\prime}_{s}\overline{\xi}$ for some
$\overline{\xi}\in[\Sigma K_{r},M]$ and $\overline{\xi}=\xi j^{\prime}_{r}$
with $\xi\in[\Sigma^{rq+1}M,M]$ , this is because
$\overline{\xi}i^{\prime}_{r}\in[\Sigma M,M]$ = 0. By Theorem 6.4.14 in
chapter 6, we may assume $\xi=\xi_{1}+\xi_{2}ij$ with $\xi_{1},\xi_{2}\in
kerd\cap[\Sigma^{*}M,M]$. Then
$d(\rho_{r,s})=i^{\prime}_{s}\xi_{1}j^{\prime}_{r}+i^{\prime}_{s}\xi_{2}ijj^{\prime}_{r}$
and by applying the derivation $d$ using (9.8.3) we have
$i^{\prime}_{s}\xi_{2}j^{\prime}_{r}$ = 0. Consequently we have
$i^{\prime}_{s}\xi_{2}=\xi_{3}\alpha^{r}=0$, this is because
$\xi_{3}\in[\Sigma^{2}M,K_{s}]$ = 0. Then
$d(\rho_{r,s})=i^{\prime}_{s}\xi_{1}j^{\prime}_{r}$ with $\xi_{1}\in
kerd\cap[\Sigma^{rq+1}M,M]$. Q.E.D.
Proof of Thoerem 9.8.1 in case $t\geq 2$: By Theorem 9.8.1 in case $t\geq 1$,
there exists $f^{\prime}\in[\Sigma^{p^{n}(p+1)q}K_{a},K_{a}]$ such that the
induced $BP_{*}$-homomorphism $f^{\prime}_{*}=v_{2}^{p^{n}}$, where we briefly
write $p^{n}-1$ as $a$. By Theorem 6.5.22 in chapter 6, we may assume $f\in
Mod_{*}$ , this is because the components of $f^{\prime}$ in $Der_{*}$ and
$Mod_{*}\delta_{0}$ induce zero $BP_{*}$-homomorphism.
Write
$\delta^{\prime}=i^{\prime}_{s}j^{\prime}_{s}\in[\Sigma^{-sq-1}K_{s},K_{s}]$.
By Theorem 6.5.23 in chapter 6,
$\delta^{\prime}f^{\prime}-f^{\prime}\delta^{\prime}\in Mod_{*}$ and this
group is a commutative subring of $[\Sigma^{*}K_{s},K_{s}]$. Then we have
$f^{\prime}(\delta^{\prime}f^{\prime}-f^{\prime}\delta^{\prime})=(\delta^{\prime}f^{\prime}-f^{\prime}\delta^{\prime}f^{\prime})f^{\prime}$
or equivalently,
$(f^{\prime})^{2}\delta^{\prime}-\delta^{\prime}(f^{\prime})^{2}=2((f^{\prime})^{2}\delta^{\prime}-f^{\prime}\delta^{\prime}f^{\prime})$.
By induction we have
$(f^{\prime})^{s}\delta^{\prime}-\delta^{\prime}(f^{\prime})^{s}=s((f^{\prime})^{s}\delta^{\prime}-(f^{\prime})^{s-1}\delta^{\prime}f^{\prime})$
, $s\geq 1$. That is
(9.8.27) $\qquad
s\cdot(f^{\prime})^{s-1}\delta^{\prime}f^{\prime}=\delta^{\prime}(f^{\prime})^{s}+(s-1)(f^{\prime})^{s}\delta^{\prime}$,
$s\geq 1$
Let $\rho_{a,1}:K_{a}\rightarrow K_{1}$ be the projection in (9.8.4), then by
Theorem in chapter 6,
$\rho_{a,1}(f^{\prime})^{s}i^{\prime}_{a}i\in\pi_{*}K_{1}$ can be extended to
$k_{s}\in Mod_{*}\subset[\Sigma^{sp^{n}(p+1)q}K_{1},\\\ K_{1}]$ such that
$\rho_{a,1}(f^{\prime})^{s}i^{\prime}_{a}i=k_{s}i^{\prime}_{1}i$ and
$(k_{s})_{*}=v_{2}^{sp^{n}}$. Since
$j^{\prime}_{1}k_{s}i^{\prime}_{1}i=\alpha^{a-1}j^{\prime}_{a}(f^{\prime})^{s}i^{\prime}_{a}i$
and $(i^{\prime}_{1}j^{\prime}_{1}k_{s}-k_{s}i^{\prime}_{1}j^{\prime}_{1})\in
Mod_{*}$, then
$(i^{\prime}_{1}j^{\prime}_{1}k_{s}-k_{s}i^{\prime}_{1}j^{\prime}_{1})i^{\prime}_{1}i$
= 0 and so
$i^{\prime}_{1}j^{\prime}_{1}k_{s}=k_{s}i^{\prime}_{1}j^{\prime}_{1}$. By
applying the derivation $d$ to the equation
$\rho_{a,1}(f^{\prime})^{s}i^{\prime}_{a}\delta=k_{s}i^{\prime}_{1}\delta$
(where we write $\delta=ij$) we have
(9.8.28)
$\qquad\rho_{a,1}(f^{\prime})^{s}i^{\prime}_{a}=k_{s}i^{\prime}_{1}-d(\rho_{a,1})(f^{\prime})^{s}i^{\prime}_{a}\delta=k_{s}i^{\prime}_{1}-i^{\prime}_{1}\xi
j^{\prime}_{a}(f^{\prime})^{s}i^{\prime}_{a}\delta$
$\qquad=k_{s}i^{\prime}_{1}-i^{\prime}_{1}j^{\prime}_{a}(f^{\prime})^{s}i^{\prime}_{a}\xi\delta$,
$s\geq 1$,
where $\xi\in[\Sigma^{aq+1}M,M]\cap kerd$( cf. Prop. 9.8.26). Let $t\geq 2$ is
not divisible by $p$, then by (9.8.27)(9.8.28) we have
$i^{\prime}_{1}j^{\prime}_{a}(f^{\prime})^{t}i^{\prime}_{a}=\rho_{a,1}i^{\prime}_{a}j^{\prime}_{a}(f^{\prime})^{t}i^{\prime}_{a}=t\cdot\rho_{a,1}(f^{\prime})^{t-1}i^{\prime}_{a}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}$
= $t\cdot k^{t-1}i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}-t\cdot
i^{\prime}_{1}j^{\prime}_{a}(f^{\prime})^{t-1}i^{\prime}_{a}\xi\delta
j^{\prime}_{a}f^{\prime}i^{\prime}_{a}$ and
(9.8.29) $\qquad i^{\prime}_{1}j^{\prime}_{a}\phi=t\cdot
k^{t-1}i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}$,
where we write
$\phi=(f^{\prime})^{t}i^{\prime}_{a}+t\cdot(f^{\prime})^{t-1}i^{\prime}_{a}\xi\delta
j^{\prime}_{a}f^{\prime}i^{\prime}_{a}$.
Let $X$ be the cofibre of
$i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}:\Sigma^{p^{n+1}q+q-1}M\rightarrow
K_{1}$ given by the cofibration in the upper row of the following homnotopy
commutative diagram ($m=(t-1)p^{n}(p+1)q+(p^{n}-1)q$)
$\quad\Sigma^{-1}X\quad\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\quad\Sigma^{p^{n+1}q+q-1}M\stackrel{{\scriptstyle
i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}}}{{\longrightarrow}}\quad
K_{1}\qquad\stackrel{{\scriptstyle w}}{{\longrightarrow}}\qquad X$
(9.8.30)$\quad\Big{\downarrow}\bar{\phi}\quad\qquad\qquad\qquad\Big{\downarrow}\phi\qquad\qquad\qquad\quad\Big{\downarrow}t\cdot
k^{t-1}\qquad\quad\Big{\downarrow}\bar{\phi}$
$\quad\Sigma^{-m-1}K_{a+1}\stackrel{{\scriptstyle\rho_{a+1,a}}}{{\longrightarrow}}\Sigma^{-m-1}K_{a}\stackrel{{\scriptstyle
i^{\prime}_{1}j^{\prime}_{a}}}{{\longrightarrow}}\quad\Sigma^{-m+aq}K_{1}\stackrel{{\scriptstyle\psi_{1,a+1}}}{{\longrightarrow}}\Sigma^{-m}K_{a+1}$
Note that the above middle rectangle is homotopy commutative by (9.8.29), then
there exists $\bar{\phi}$ such that all the above rectangles commute up to
homotopy.
By $wi^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}$ = 0 we have
$wi^{\prime}_{1}j^{\prime}_{a}f^{\prime}=yj^{\prime}_{a}$ with
$y\in[\Sigma^{p^{n}(p+1)q}M,X]$ so that $uyj^{\prime}_{a}$ = 0 and
$uy=\lambda\cdot\alpha^{a}$ for some $\lambda\in[M,M]\cong Z_{p}\\{1_{M}\\}$,
that is we have
(9.8.31)$\qquad wi^{\prime}_{1}j^{\prime}_{a}f^{\prime}=yj^{\prime}_{a}$,
$uy=\lambda\cdot\alpha^{a}$ for some $\lambda\in Z_{p}$.
On the other hand,
$j^{\prime}_{a}(f^{\prime})^{s}\psi_{1,a}i^{\prime}_{1}=j^{\prime}_{a}(f^{\prime})^{s}i^{\prime}_{a}\alpha^{a-1}=\alpha^{a-1}j^{\prime}_{a}(f^{\prime})^{s}i^{\prime}_{a}=j^{\prime}_{1}\rho_{a,1}\\\
(f^{\prime})^{s}i^{\prime}_{a}=j^{\prime}_{1}k_{s}i^{\prime}_{1}$, then
$j^{\prime}_{a}f^{\prime}\psi_{1,a}=j^{\prime}_{1}k_{1}+\eta j^{\prime}_{1}$
with $\eta\in[\Sigma^{*}M,M]$ and so
$yj^{\prime}_{1}=wi^{\prime}_{1}j^{\prime}_{a}f^{\prime}\psi_{1,a}=wi^{\prime}_{1}j^{\prime}_{1}k_{1}+wi^{\prime}_{1}\eta
j^{\prime}_{1}=wk_{1}i^{\prime}_{1}j^{\prime}_{1}+wi^{\prime}_{1}\eta
j^{\prime}_{1}$ and we have
(9.8.32) $\qquad y=wk_{1}i^{\prime}_{1}+wi^{\prime}_{1}\eta+z\alpha$ with
$z\in[\Sigma^{*}M,X]$,
$\qquad\quad\bar{\phi}y=t\cdot\psi_{1,a+1}k_{t-1}k_{1}i^{\prime}_{1}+t\cdot\psi_{1,a+1}k_{t-1}i^{\prime}_{1}\eta+\bar{\phi}z\alpha$
which is obtained by (9.8.31).
We claim that
(9.8.33) $\bar{\phi}z\alpha i\in\pi_{tp^{n}(p+1)q+aq}K_{a+1}$ has $BP$
filtration $>0$
This will be proved in the last. Then
$\bar{\phi}yi=t\cdot\psi_{1,a+1}k_{t-1}k_{1}i^{\prime}_{1}i$ (modulo higher
filtration) is represented by $t\cdot v_{1}^{a}v_{2}^{tp^{n}}\in
Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{a+1})$ in the Adams-Novikov spectral
sequence.
Hence , by (9.8.31)(9.8.30)(9.8.28)(9.8.27) we have
$\bar{\phi}yj^{\prime}_{a}=\bar{\phi}wi^{\prime}_{1}j^{\prime}_{a}f^{\prime}=t\cdot\psi_{1,a+1}k_{t-1}i^{\prime}_{1}j^{\prime}_{a}f^{\prime}=t\cdot\psi_{1,a+1}\rho_{a,1}(f^{\prime})^{t-1}i^{\prime}_{a}j^{\prime}_{a}f^{\prime}=\psi_{1,a+1}\rho_{a,1}(i^{\prime}_{a}j^{\prime}_{a}(f^{\prime})^{t}+(t-1)(f^{\prime})^{t}i^{\prime}_{a}j^{\prime}_{a})$
= $(t-1)\psi_{1,a+1}\rho_{a,1}(f^{\prime})^{t}i^{\prime}_{a}j^{\prime}_{a}$
and so
$\bar{\phi}y=(t-1)\psi_{1,a+1}\rho_{a,1}(f^{\prime})^{t}i^{\prime}_{a}+\tilde{f}\alpha^{a}$
with $\tilde{f}\in[\Sigma^{tp^{n}(p+1)q}M,K_{a+1}]$.
By the claim (9.8.33), $\bar{\phi}yi$ is represented by $t\cdot
v_{1}^{a}v_{2}^{tp^{n}}$ in the Adams-Novikov spectral sequence, then
$\tilde{f}\alpha^{a}i$ is represented by $v_{1}^{a}v_{2}^{tp^{n}}$ and so
$\tilde{f}i\in\pi_{tp^{n}(p+1)q}K_{a+1}$ is represented by
$v_{2}^{tp^{n}}+v_{1}x\in
Ext_{BP_{*}BP}^{0,tp^{n}(p+1)q}(BP_{*},BP_{*}K_{a+1})$, where
$v_{1}x=\Sigma_{r=1}^{[n/2]}\lambda_{r}v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(a_{r}p^{n-2r})$
which is obtained by Lemma 9.8.25.
By [20], if $t\geq 2$ is not divisible by $p$ and $1\leq r\leq p$,
$v_{2}^{tp}\in Ext_{BP_{*}BP}^{0,*}(BP_{*},\\\ BP_{*}K_{r})$ is a permanent
cycle in the Adams-Novikov spectral sequence. Suppose inductively that
$v_{2}^{tp^{s}}\in Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{r})$ are permanent
cycles for all $t\geq 2$ is not divisible by $p$, $1\leq r\leq p^{s}$ and
$s\leq n-1$. Then, it is easily seen that
$v_{1}^{p^{n}-p^{n-2r}}\tilde{c}_{1}(a_{r}p^{n-2r})$ is realizable in
$[\Sigma^{tp^{n}(p+1)q}K_{a+1},K_{a+1}]$ so that the above by the induction
hypothesis we know that $v_{1}x$ also is a permanent cycle. . So,
$v_{2}^{tp^{n}}\in Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{a+1})$ is a permanent
cycle in the Adams-Novikov spectral sequence and there exists
$h\in\pi_{tp^{n}(p+1)q}K_{a+1}$ such that the induced $BP_{*}$-homomorphism
$h_{*}=v_{2}^{tp^{n}}$. Hence, for $1\leq r\leq a+1=p^{n}$,
$jj^{\prime}_{r}\rho_{a+1,r}h\in\pi_{tp^{n}(p+1)q-rq-2}S$ is just the
$\beta_{tp^{n}/s}$-element of the Theorem.
Now our remaining work is to prove the claim (9.8.33). Recall as known above
that $j^{\prime}_{a}f^{\prime}i^{\prime}_{a}i\in\pi_{*}M$ is represented by
$\beta^{\prime}_{p^{n}/p^{n}-1}\in Ext_{BP_{*}BP}^{1,*}(BP_{*},\\\ BP_{*}M)$
in the Adams-Novikov spectral sequence and
$\beta^{\prime}_{p^{n}/p^{n}-1}=v_{1}\beta^{\prime}_{p^{n}/p^{n}}$, then
$(i^{\prime}_{1})_{*}(\beta^{\prime}_{p^{n}/p^{n}-1})=0\in
Ext_{BP_{*}BP}^{1,*}(BP_{*},BP_{*}K_{1})$ and so
$i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}i\in\pi_{*}K_{1}$ has
$BP$-filtration $\geq q+1$. Then, in the Adams-Novikov resolution of the
spectrum $K_{1}$ , $i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}i$ can
be lifted to $\kappa\in\pi_{*}\widetilde{E}_{q+1}\wedge K_{1}$ such that
$(\tilde{a}_{0}\wedge 1_{K_{1}})\cdots(\tilde{a}_{q}\wedge
1_{K_{1}})\kappa=i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}i$. Since
$K_{1}$ is an $M$-module spectrum , then $\kappa=\kappa^{\prime}\cdot i$ with
$\kappa^{\prime}\in[\Sigma^{*}M,\widetilde{E}_{q+1}\wedge K_{1}]$.
Consequently we have
$i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}=(\tilde{a}_{0}\wedge
1_{K_{1}})\cdots(\tilde{a}_{q}\wedge 1_{K_{1}})\kappa^{\prime}+\sigma j$ with
$\sigma\in[\Sigma^{p^{n+1}q+q}S,K_{1}]$. Note that $(\tilde{b}_{0}\wedge
1_{K_{1}})\sigma\in\pi_{p^{n+1}q+q}BP\wedge K_{1}\cong
Hom_{BP_{*}BP}^{p^{n+1}q+q}(BP_{*},BP_{*}(BP\wedge K_{1}))$ is a $d_{1}$-cycle
in the Adams-Novikov resolution of $K_{1}$ and it represents an element in
$Ext_{BP_{*}BP}^{0,p^{n+1}q+q}(BP_{*},BP_{*}K_{1})$. However, this group is
zero by degree reason , this is because
$Ext_{BP_{*}BP}^{0,*}(BP_{*},BP_{*}K_{1})\cong Z_{p}[v_{2}]$). Then we have
$(\tilde{b}_{0}\wedge 1_{K_{1}})\sigma$ = 0 so that $\sigma$ can be lifted to
$\sigma^{\prime}\in\pi_{*}\widetilde{E}_{q+1}\wedge K_{1}$ such that
$(\tilde{a}_{0}\wedge 1_{K_{1}})\cdots(\tilde{a}_{q}\wedge
1_{K_{1}})\sigma^{\prime}=\sigma$. So we have
$i^{\prime}_{1}j^{\prime}_{a}f^{\prime}i^{\prime}_{a}=(\tilde{a}_{0}\wedge
1_{K_{1}})\cdots(\tilde{a}_{q}\wedge
1_{K_{1}})(\kappa^{\prime}+\sigma^{\prime}j)$. By this we know that the
following short exact sequence induced by the cofibration in the top row of
(9.8.30) is a split exact sequence of $BP_{*}BP$-comodule:
$\quad 0\rightarrow BP_{*}K_{1}\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}BP_{*}X\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}BP_{*}M\rightarrow 0$
where $\mid w_{*}\mid=-(p^{n+1}+1)q$
Moreover, this splitness also hold in the following
$Ext_{BP_{*}BP}^{0,*}$-stage :
$\quad 0\rightarrow Ext^{0}K_{1}\stackrel{{\scriptstyle
u_{*}}}{{\longrightarrow}}Ext^{0}X\stackrel{{\scriptstyle
w_{*}}}{{\longrightarrow}}Ext^{0}M\rightarrow 0$
That is to say, there is an invariant $BP_{*}$-homomorphism
$u^{\prime}:Ext^{0}X\rightarrow Ext^{0}K_{1}$ and
$w^{\prime}:Ext^{0}M\rightarrow Ext^{0}X$ such that
$u^{\prime}u_{*}=1_{Ext^{0}K_{1}},w_{*}w^{\prime}=1_{Ext^{0}M}$ and
$u_{*}u^{\prime}+w^{\prime}w_{*}=1_{Ext^{0}X}$, where we briefly write
$Ext_{BP_{*}BP}^{0,*}(BP_{*},\\\ BP_{*}X)$ as $Ext^{0}X$.
To prove the claim (9.8.33), suppose in contrast that $\bar{\phi}z\alpha
i\in\pi_{*}K_{a+1}$ has $BP$-filtration 0, then, by (9.8.32), it is
represented by $\lambda v_{1}^{a}v_{2}^{tp^{n}}\in Ext^{0}K_{a+1}$ in the
Adams-Novikov spectral sequence, where $\lambda\neq 0\in Z_{p}$. Then
$zi\in\pi_{tp^{n}(p+1)q+(a-1)q}X$ must have $BP$ filtration 0 and it is
represented by some $x\in Ext^{0,*}X$ and
$(\bar{\phi})_{*}(v_{1}x)=\lambda\cdot v_{1}^{a}v_{2}^{tp^{n}}$. However,
$\qquad x=u_{*}u^{\prime}(x)+w^{\prime}w_{*}(x)=w^{\prime}w_{*}(x)$
, this is because by degree reason we have $u^{\prime}(x)\in
Ext^{0,tp^{n}(p+1)q+(a-1)q}K_{1}$ = 0, Then
$x=\lambda^{\prime}w^{\prime}(v_{1}^{r})$, for some $\lambda^{\prime}\in
Z_{p},rq=tp^{n}(p+1)q+(a-1)q-p^{n+1}q-q$
since $w_{*}(x)\in Ext^{0,r}M\cong Z_{p}\\{v_{1}^{r}\\}$, Then
$\lambda
v_{1}^{a}v_{2}^{tp^{n}}=(\bar{\phi})_{*}(v_{1}x)=\lambda^{\prime}(\bar{\phi})_{*}w^{\prime}(v_{1}^{r+1})$
Moreover, since $w^{\prime}(v_{1}^{r+1})$ is belong to $Ext^{0,*}X$ the
summand which is isomorphic to $Ext^{0,*}M$ and $Ext^{0,*}M$ is a trivial
$Z_{p}[v_{2}]$-module, then we have
$\qquad
0=v_{2}^{p^{n}}\cdot(\bar{\phi})_{*}w^{\prime}(v_{1}^{r+1})=\lambda\cdot
v_{1}^{a}v_{2}^{(t+1)p^{n}}\in Ext^{0,*}K_{a+1}$
This is a contradiction and then shows the claim(9.8.33). Q.E.D.
After finishing the proof of Thoerem 9.8.1 on second periodicity elements in
the stable homotopy groups of spheres, we state the following Theorem on
further result on second periodicity families in the stable homotopy groups of
spheres without proof. The proof is done in base on the result of Theorem
9.8.1 and using some properties of the spectrum $M(p^{r},v_{1}^{ap^{s}})$
which is the geometric realization of $BP_{*}/(p^{r},v_{1}^{ap^{s}})$. The
details of the proof can be seen in [23] §3.
Theorem 9.8.34 Let $p\geq 5$. $j=cp^{i}\leq p^{n-i}-1$ if $t\geq 1$
($cp^{i}\leq p^{n-i}$ if $t\geq 2$), then the element
$\beta_{tp^{n}/j,i+1}\in Ext_{BP_{*}BP}^{2,*}(BP_{*},BP_{*})$
is a permanent cycle in the Adams-Novikov spectral sequence and it converges
to the corresponding homotopy element of order $p^{i+1}$ in $\pi_{*}S$.
REFERENCES
[1] Aikawa T. , 3-Dimensional cohomology of the mod p Steenrod algebra Math.
Scanfd. 47(1980), 91–115.
[2] Cohen R. , Odd primary families in stable homotopy theory. Memoirs of
Amer. Math. Soc. No. 242(1981).
[3] Cohen R. and Goerss P. , Secondary cohomology operations that detect
homotopy classes. Topology 22(1984), 177–194.
[4] Hoffman P. , Relations in the stable homotopy of Moore spaces. Proc.
London Math. Soc. 18(1968), 621–634.
[5] Jinkun Lin and Qibing Zheng , A new family of filtration seven in the
stable homotopy of spheres. Hiroshima Math. J. 28(1998) 183–205.
[6] Jinkun Lin , A new family of filtration three in the stable homotopy of
spheres. Hiroshima Math. J. 31(2001) 477–492.
[7] Jinkun Lin , Some new families in the stable homotopy of spheres
revisited. Acta Math. Sinica 18(2002) 95–106.
[8] Jinkun Lin , Two new families in the stable homotopy groups of sphere and
Moore spectrum. Chin. Ann. of Math. 27B(2006) 311-328.
[9] Jinkun Lin, Third periodicity families in the stable homotopy of spheres.
JP Journal of Geometry and Topology 3(2003), 179-219.
[10] Miller H. R. , Ravenel D.C. and Wilson W.S. . Periodic phenomena in the
Adams-Novikov spectra sequence. Ann. of Math. 106(1977) 469-516.
[11] Oka S. , Multilpicative structure of finite spectra and stable homotopy
of spheres. Lecture Notes in Math. v.1051 Springer-verlag (1984).
[12] Ravenel D.C. , Complex cobordism and stable homotopy groups of spheres .
Academic Press Inc. (1986)
[13] Thomas E. and Zahler R. , Generalized higher order cohomology operations
and stable homotopy groups of spheres. Advances in Math. 20(1976) 287–328.
[14] Toda H. , Algebra of stable homotopy of $Z_{p}$-spaces and applications.
J. Math. Kyoto Univ. 11(1971),197–251.
[15] Toda H. , On spectra realizing exterior part of the Steenrod algebra.
Topology 10(1971), 53–65.
[16] Wang X. and Zheng Q. , The convergence of
$\widetilde{\alpha}^{(n)}h_{0}h_{k}$ Science in China 41(1998), 622–636.
[17] Wang X., On the 4-dimensional cohomology of the Steenrod algebra. Beijing
Mathematics 1(1995), 80-99.
[18] Zhou X., Higher cohomology operations that detect homology class. Lecture
Notes in Math. v.1370 Springer-verlag 1980, 416–436.
[19] Miller H.R. and Wilson W.S. , On Novikov’s $Ext^{1}$ modulo an invariant
prime ideal. Topology 15(1976), 131-141.
[20] Oka S., Realizing some cyclic $BP_{*}$ modules and applications to stable
homotopy of spheres. Hiroshima Math. J. 7(1977), 427-447.
[21] Oka S., Small ring spectra and p-rank of the stable homotopy of spheres.
Contemp. Math. 49(1983), 267-308.
[22] Oka S., A new familiy in the stable homotopy of spheres. Hiroshima Math.
J. 5(1975),87-114.
[23] Jinkun Lin , Detection of second periodicity families in stable homotopy
of spheres. American J. of Math. 112(1990), 595-210.
[24] Jinkun Lin, A pull back Theorem in the Adams spectral sequence, Acta
Math. Sinica. v.34(2008) no.3,471-490
[25] Hirofumi Nakai, The Chrometic $E_{1}$-term $H^{0}M_{1}^{2}$ for p ¿ 3.
New York Journal of Math. 6(2000), 21-54.
|
arxiv-papers
| 2008-09-26T10:10:28
|
2024-09-04T02:48:57.976875
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinkun Lin",
"submitter": "Jinkun Lin",
"url": "https://arxiv.org/abs/0809.4587"
}
|
0809.4664
|
Physics and Astrophysics of Planetary Systems
# The Atmospheres of Extrasolar Planets
Mark S. Marley Mail Stop 245-3; NASA Ames Research Center; Moffett Field,
California 94035; USA
###### Abstract
The characteristics of irradiated solar system planetary atmospheres have been
studied for decades, consequently modern planetary science benefits from an
exhaustive body of ground- and space-based data. The study of extrasolar
planetary atmospheres, by contrast, is still in its infancy and currently
rests on a few score of datapoints, mostly of the transiting planets. This
short survey aims not to review this dynamic field but rather stresses the
importance of a few theoretical concepts and processes for our understanding
of exoplanet atmospheres. Topics covered include atmospheric structure and
dynamics, cloud processes and photochemistry of planetary atmospheres.
Influences on the albedos, spectra, and colors of extrasolar planets are
reviewed and caution is urged in the interpretation of exoplanet colors.
## 1 Introduction
The atmosphere controls a planet’s evolution through time and provides a
window into the chemical and physical conditions under which the planet
formed. Except for airless worlds, like Mercury, the atmosphere also mediates
the flow of information we receive about the nature of the planet. In this
contribution I aim to discuss a sampling of a few of the more important
concepts which relate to our understanding of extrasolar planetary
atmospheres. I will focus on topics that are important for understanding
spectral and photometric data from extrasolar planets, with an eye towards
providing examples from the solar system.
In some sense the trajectory of extrasolar planet science is recapitulating
the history of solar system planetary science. Three decades ago in the
University of Arizona Space Science series book “Jupiter” Wallace (1976)
reviewed the sparse data then available on Jupiter’s thermal emission spectrum
and presented models of the planet’s atmospheric structure. His effort to
piece together a consistent picture of the atmosphere from a handful of
thermal infrared photometric measurements, reflection spectra, and
transmission data from stellar occultations bears a striking resemblance to
current efforts to understand the transiting extrasolar planets.
Today, with our eyes now fixed on exoplanets, we are fortunate to have at hand
the insights gained from such work over the past half century of planetary
exploration. For a few transiting planets we again find ourselves in the
situation where we have a few broad band brightness temperature measurements
from which we are attempting to infer global atmospheric processes. As we
carry out our science we would do well to remember that this ground has been
crossed before: irradiated planetary atmospheres have been encountered before
the $21^{\rm st}$ century (a fact sometimes overlooked in the modern
literature). The excitement of our time, of course, is that the range of
parameter space, from planetary masses to the dizzying range of incident
radiation is far larger than previously encountered. Nevertheless we should
not be surprised that well understood processes familiar from solar system
planetary science, including photochemistry, hot stratospheres, and cloud
processes also play important roles in extrasolar planet atmospheres (Marley
1997).
To aid those attempting to traverse this new landscape of theory and data of
exoplanet atmospheres for the first time, I discuss in this chapter a few
elementary themes that are helpful for understanding the conceptual scenery.
While I will illustrate concepts by drawing from work in the field, I am not
in any sense attempting to provide a comprehensive review of exoplanet
science, as many such recent reviews are available (e.g., Charbonneau (2008),
Deming (2008), and Seager et al. (2008) review recent progress; Burrows et al.
(2001, 2006) and Barman et al. (2005) present useful surveys of much of the
theory of these objects; and Marley et al. (2007) review the pre-2007 field).
Instead I present a brief, illustrated (if somewhat idiosyncratic) guidebook
to a few of the important processes encountered at these new worlds. Hence I
will touch on atmospheric structure, cloud formation, the interpretation of
brightness temperature and the importance of photochemistry. I will mention
some of the processes contributing to stratospheric heating and conclude with
some brief comments about albedos and colors of planets.
## 2 Atmospheric Structure
The basic relationships governing a static atmosphere in hydrostatic
equilibrium are well known and presented in introductory textbooks, such as
Chamberlain & Hunten (1987). Such texts make for excellent introductory
reading as many of the fundamental relationships and processes governing
extrasolar planets are of course familiar from the solar system planets. One
particularly useful relation can be derived for an ideal gas atmosphere in
hydrostatic equilibrium. The number density of molecules per unit area in a
gaseous column above a specified height, $\cal N$, can be related to the local
number density of molecules, $n$, by
${\cal N}\approx{p(z)\over{g(z)m}}=n(z)H.$ $None$
Here the mean molecular weight is $m$, altitude is $z$ and $p,\,g$ and $H$ are
the pressure, gravity and the scale height; the equality is exact for an
isothermal atmosphere.
Figure 1: Atmospheric temperature-pressure profiles for Uranus, Jupiter, and
Earth. In all three planets the temperature increases with depth below a few
hundred millibars. In each planet this atmospheric region–the
troposphere–transports heat by convection from the deep interior, in the case
of the giants, or the surface in the case of Earth. In each atmosphere the
temperature also rises at low pressure–the stratosphere–owing to the
absorption of a fraction of the incident ultraviolet light by photochemical
products (ozone in the case of Earth, various hydrocarbon products in the
giants’ atmospheres). Uranus and Jupiter data are from the Voyager Radio
Science occultation experiments and the Earth profile is from the 1976
Standard Atmosphere. All data are available on-line
(http://atmos.nmsu.edu/planetary_datasets/indextemppres.html).
Equation (1) is useful for understanding a variety of basic characteristics of
atmospheres. For example if a molecular absorber has a cross section for
interaction with radiation of $\sigma\,$($\rm cm^{2}$), then the column
optical depth above a given pressure surface is $\tau_{\rm col}\sim{\cal
N}\sigma$. All else being equal, $\tau_{\rm col}$ falls linearly with
pressure. In a typical planetary atmosphere energy is transported upwards by
convection until the atmosphere becomes optically thin to thermal radiation,
when $\tau_{\rm col}\sim 1$. Above this level the outgoing energy is
transported by radiation. The pressure level in the atmosphere of this
radiative-convective boundary clearly depends upon the composition of the
atmosphere, the opacity of the major atmospheric constituents, gravity and the
temperature. For example, in the Earth’s atmosphere the surface temperature is
about $290\,\rm K$. Only at a temperature of about $220\,\rm K$ near $100\,\rm
mb$ (at the tropopause) is the optical depth at the peak of the Planck
function low enough that the air can radiate efficiently to space (Figure 1).
At lower pressures still a gray atmosphere would reach a constant temperature,
known as the skin temperature $T_{0}=(1/2)^{1/4}T_{\rm eff}$, where $T_{\rm
eff}$ is the effective temperature.
Figure 2: Pressure-temperature profiles for $\sim 4.5\,\rm Gyr$-old Jupiter-
like planets from 0.02 to 10 AU (left to right) from a solar-type star. Thick
lines are convective regions while thin lines are radiative regions. The
profiles at 5 and 10 AU show deviations that arise from numerical noise in the
chemical equilibrium table near condensation points, but this has a negligible
effect on planetary evolution. Figure adapted from Fortney et al. (2007).
The column abundance of molecules above a given pressure level in the
atmosphere controls the level at which the atmosphere becomes optically thin
to outgoing thermal radiation, and hence the location of the tropopause. The
tropopause pressure will thus vary with gravity as well as with atmospheric
structure and composition. In a lower gravity atmosphere each molecule
“weighs” less, so the column number of molecules, $\cal N$ above a given
pressure level must be larger to compress the gas than in a higher gravity
atmosphere. Indeed Equation (1) tells us that the column optical depth is
inversely proportional to $g$, so the column number density above the 1 bar
surface on a planet with half Earth’s gravity and a similar atmosphere would
be twice that of our atmosphere. All else being equal, the tropopause would be
at a lower pressure. For gas giant planets, which tend to have roughly
constant radii regardless of mass, higher mass generally corresponds to higher
gravity and thus more transparent atmospheres to a given pressure level. Given
the actual complexity of varying depths of absorption of incident radiation
with variations in gravity and atmospheric composition, theoretical models,
which relate atmospheric temperature to pressure, are required to fully
recognize the subtleties of atmospheric structure. Such models are shown in
Figure 2.
## 3 Atmospheric Dynamics
The overall structure of an irradiated atmosphere depends both upon the depth
at which incident energy is absorbed and internal sources of energy. For a
giant planet, convection transports energy outwards from the planetary
interior, as the planet cools slowly over time. For planets more distant than
a few AU from their primary star most incident energy is absorbed fairly deep
in the atmosphere, below the depth at which the atmosphere becomes optically
thick in the thermal infrared (Fortney et al. 2007). This is because most
gasses are more transparent in the optical than in the infrared. As a result
the absorbed incident energy simply adds to the internal energy being
transported outwards by convection and the temperature profile resembles that
of Jupiter shown in Figure 1. Another consequence of this relatively deep
deposition of solar radiation is that the internal heatflow is preferentially
transported by convection to the poles of Jupiter yielding a relatively
isothermal planet at the radiative-convective boundary (Ingersoll & Porco
1978).
However, for those giant planets found closer to their primary stars, the
radiative-convective boundary is deeper but absorption of incident flux still
occurs at a similar altitude (to the extent that composition is unchanged).
Thus the large incident flux upon a hot Jupiter is absorbed above the
radiative-convective boundary (Figure 2). As a consequence an isothermal layer
appears between the top of the deep convective zone and the region of the
atmosphere in which incident flux is absorbed. In this case the deep internal
heatflow is distinct from the thermalized incident radiation and the global
temperature distribution is no longer relatively homogeneous and equator to
pole temperature gradients can be large.
The nature of the atmospheric circulation of giant planets, which is
ultimately driven by the various energy fluxes, depends as well on a variety
of influences, particularly including rotation rate and atmospheric scale
height. Showman et al. (2007) present a very useful introduction and review of
this topic. The atmospheric redistribution of energy by winds is
unquestionably of paramount importance for the hot Jupiters and the efficiency
of redistribution controls the global temperature map and consequently the
phase variation of thermal emission, which has been successfully measured for
multiple planets by Spitzer (notably by Knutson et al. 2007; Cowan & Agol give
a complete summary of observations through mid-2008). Since a planet’s thermal
emission can arise from different depths in the atmosphere at different
wavelengths (§5), any variation in redistribution efficiency with altitude
will manifest itself as differing thermal emission maps as a function of
wavelength (Burrows et al. 2008a; Fortney et al. 2008). Ultimately coupled
models of radiative transfer and dynamics, similar to terrestrial global
circulation models, will be required to understand all of the contributing
factors (Showman et al. 2008 present one such model and review recent progress
in the field).
## 4 Clouds
Clouds play an important role in planetary atmospheres. They scatter incident
light back to space and sequester condensed species from the overlying
atmosphere. Jupiter’s atmosphere provides a point of departure for
understanding the diversity of giant planet atmospheres that we expect to
encounter outside of the solar system (see Lodders 2006 for a full
discussion).
Figure 3: Cloud structure expected to be found in, from left to right,
Jupiter’s atmosphere, a Jupiter-like planet too warm for water clouds to form,
and a hot Jupiter. Note that the overall cloud structure is similar in each
panel, but cloud levels move upward as the atmosphere is warmed. Important
labeled condensates include perovskite ($\rm CaTiO_{3}$) which removes TiO
from the gas phase, the salts $\rm Na_{2}S$ and KCl which remove two important
alkali absorbers (Figure 7) from the gas, and water clouds which appear near
the top of the Jupiter-like structure on the left. Figure modified from
Lodders (2004).
To understand Jupiter’s cloud structure we might imagine an air parcel moving
upwards from the deep interior (Figure 3). We start at a temperature of about
$2000\rm\,K$ and slowly raise the gas parcel up; as the gas rises, it cools
adiabatically. In the deep atmosphere all gasses are well mixed. At the point
in the atmosphere where a condensible species’ saturation mixing ratio first
equals the vapor mixing ratio, a cloud base is found. The first constituents
to condense are refractory oxides such as perovskite and corundum, followed by
various magnesium silicates including enstatite and forsterite. Iron is
predicted to condense as a native metal. Although we cannot see this region of
Jupiter’s atmosphere, we know that iron clouds are there because hydrogen
sulfide gas is detected in Jupiter’s visible atmosphere111If iron grains were
distributed uniformly above the iron condensation layer, $\rm H_{2}S$ gas
would react to form iron sulfide, FeS, thus removing sulphur-bearing gasses
from the atmosphere (Lodders 2004). (Niemann et al. 1998). As we move upwards
in the atmosphere the temperature continues to fall and eventually water
clouds form, removing $\rm H_{2}O$ from the gas phase. Above the water clouds,
the atmosphere continues to cool until ammonia clouds form. It is the ammonia
clouds of Jupiter, dusted by various photochemical pollutants, that we see
reflecting sunlight back from the planet. In §9 we’ll consider how giant
planets somewhat different than Jupiter might appear.
Since clouds scatter and absorb incident stellar radiation as well as emergent
thermal radiation, they play a very important role in controlling the
appearance and thermal structure of a planet. However clouds are intrinsically
difficult to model from a priori physical considerations. The detailed
behavior of terrestrial cloud cover as a function of atmospheric temperature
is the leading source of uncertainty in global climate models, for example.
Although the chemistry is thought to be well understood, predicting cloud
behavior for extrasolar planets, including such issues as particle sizes,
vertical distribution, and any horizontal patchiness is difficult.
Nevertheless some efforts at cloud modeling have been made (see the recent
review by Helling et al. (2008)). The hot L-type ultracool dwarfs have
atmospheres with thick silicate and iron clouds. Accounting for the effects of
these clouds has proven challenging. Given the difficulty of cloud modeling in
general and our experience with brown dwarfs, it seems that model predictions
for the spectra of extrasolar planets must be regarded skeptically, at least
when clouds are expected.
## 5 Brightness Temperature
The spectrum of any planet is composed of two components: scattered incident
radiation from the planet’s star and thermal emitted flux from the planet. The
thermal flux represents both energy arising from processes interior to the
planet and re-radiated absorbed incident radiation. For solar system planets
these two components of the spectrum are usually well separated in wavelength,
but for the hottest exoplanets there can be substantial overlap between
thermal radiation and scattered incident light. For a planet, such as a
transiting planet, with a known radius, the thermal emission spectrum is often
equated for convenience, wavelength by wavelength, to the thermal emission
from a blackbody. For an isothermal solid sphere with emissivity unity the
observed spectrum would equal that of a blackbody with a fixed temperature.
However for a real planet the flux will differ from that of a blackbody with
the same radius and at each wavelength a “brightness” temperature
$T_{B}(\lambda)$ may be defined.
Figure 4: Contribution functions calculated for a cloudless hot-Jupiter
lacking a stratosphere. Contribution functions are calculated for various
Spitzer broadband filters (the unlabeled curves correspond to various IRAC
filters), K band, and the Kepler band at 450–900 nm (black solid curve). For
clarity some of the curves have been normalized to 0.5 or 0 .75 rather than
1\. Figure from Showman et al. (2008).
Although thermal emission data for transiting planets are often reported in
terms of of $T_{B}(\lambda)$, such data must be regarded with some care.
Except in special cases (e.g., an isothermal atmosphere) brightness
temperature is not a measure of physical temperature or effective temperature.
Rather it gives a weighted measure of atmospheric temperatures over a range of
pressures from which flux emerges from the planet. To see this it is useful to
consider the expression for the upwards or outgoing intensity measured from an
atmosphere as a function of frequency, $I_{\nu}(0,\mu)$, where $\mu$ is the
cosine of the angle from the vertical. Making use of Eq. (1):
$I_{\nu}(0,\mu)={{\sigma_{\nu}n(z_{0})}\over\mu}\int\limits_{0}^{\infty}B_{\nu}(T)\exp{\biggl{(}}-{{z-z_{0}}\over
H}-{\tau_{\nu}\over\mu}{\biggr{)}}dz.$ $None$
The exponential function in this equation, known as a weighting function (see
Chamberlain & Hunten 1987), describes the relative contribution to the
outgoing flux from each altitude $z$, defined here as the distance from the
peak of the function, $z_{0}$. Crudely the brightness temperature can indeed
be equated to the temperature at the peak of the contribution function with
height. More importantly, the emergent flux, expressed as a single brightness
temperature is actually measuring thermal emission from a range of altitudes
in the atmosphere around $z_{0}$ with varying physical temperatures. Since
$\tau_{\nu}$ can vary dramatically with frequency $\nu$, the contribution
function—and thus the brightness temperature—can be quite different at
different wavelengths. For example Figure 4 illustrates theoretical
contribution functions for several different commonly used Spitzer filters
applied to the atmosphere of a cloud-free hot Jupiter (Showman et al. 2008).
Note that for each bandpass the measured emitted flux emerges over different
vertical regions of the atmosphere with different temperatures. In regions of
low opacity one sees deeper into the atmosphere which, for a monotonically
increasing temperature profile with depth, means higher brightness
temperature. High opacity spectral regions correspond to lower pressures and
lower temperatures. However if there is an inverted temperature profile then a
situation could emerge where $T_{B}(\lambda_{1})>T_{B}(\lambda_{2})$ and
$T_{B}(\lambda_{2})<T_{B}(\lambda_{3})$ where
$\tau(\lambda_{1})>\tau(\lambda_{2})>\tau(\lambda_{3})$. This is a commonplace
occurrence in the atmospheres of solar system giants and the chromospheres of
stars.
Furthermore, because of limb effects, this range in pressures from which flux
emerges also varies over the disk. Thus even for a gray atmosphere, with
constant optical depth as a function of wavelength, the brightness temperature
is in general not equal to the effective temperature at all wavelengths.
For these reasons, while brightness temperatures are useful shorthands to
convey information about planetary spectra, they must be regarded with some
caution. For example $8\,\rm\mu m$ Spitzer observation of the hot Neptune GJ
436b (Deming et al. 2007, Demory et al. 2007) yield a brightness temperature
of $712\pm 36\,\rm K$ which is modestly above the predicted effective
temperature (Deming et al. 2007). Since we do not expect, in general for
$T_{B}=T_{\rm eff}$ the information content of this single datapoint is
limited. With atmosphere models and additional data points the value of each
brightness temperature measurement increases, as was the case with Wallace’s
1976 study of Jupiter cited in the introduction.
## 6 Photochemistry
Figure 5: Incident flux at the top of the atmospheres of two transiting
planets compared to that received by Jupiter. Vertical lines denote the
approximate maximum wavelengths at which various molecules can be dissociated
(after Marley et al. 2007). Given the large incident fluxes received by these
and other hot Jupiters, many of the most abundant atmospheric species will be
easily photolyzed, which will likely lead to a rich photochemistry.
Atmospheric molecules can be dissociated by the absorption of ultraviolet
light, a process that happens high in the atmosphere before most incident UV
light is scattered back to space. Photochemical products can then participate
in complex reaction chains, producing various molecular products. A familiar
example is atmospheric ozone in Earth’s stratosphere, which ultimately results
from the photodissociation of molecular oxygen. Yung & DeMore (1999) provide a
useful guide to these topics on many planets. Photochemical products can
themselves become important players in the atmospheric radiative transfer of
giant planets, as discussed in the next section.
Photochemistry has long been expected (Marley 1998, Liang et al. 2004) to be
important for hot Jupiter atmospheres and will likely be far more complex than
in the solar system. This is because molecular species that are condensed
below the jovian clouds (e.g., $\rm H_{2}O,\ H_{2}S,\ NH_{3}$) and thus
protected from photodissociation will be gaseous in such hot atmospheres. Some
of these species, such as $\rm H_{2}S$, are easily photodissociated (Figure
5), and will likely produce new or unexpected species. Sulfur and nitrogen
compounds, in particular, may be important players in hot Jupiter
photochemistry and perhaps haze production (Marley et al. 2007). While the
carbon photochemistry has been studied (Liang et al. 2004), preliminary work
on photochemistry in water-bearing $\rm H_{2}-He$ atmospheres suggests that
compounds including $\rm CO_{2}$, HCN, and $\rm C_{2}H_{6}$ will be present
well in excess of the abundance predicted by equilibrium chemistry (Troyer et
al. 2007). Photochemical products may play a role in the formation of hot
stratospheres (Marley 1998; Marley et al. 2007; Burrows et al. 2008a). This
area is certainly rich for further study.
Photochemistry can also be very important in terrestrial atmospheres as is the
case with Venus and Titan. An earthlike planet with a greater abundance of
methane could well be enveloped in a photochemical haze and not appear
anything like a ‘pale blue dot’, even if there are indeed underlying oceans
(Zahnle 2008). Such obscuration by hazes is a major concern if planets are to
be characterized solely by their colors (§9).
## 7 Temperature Inversions (Stratospheres)
A gray atmosphere becomes asymptotically isothermal at small optical depths,
reaching the skin temperature $T_{0}$ (§2). It is often the case that an
atmosphere is optically thick to incident radiation at some wavelengths at low
pressures where the atmosphere is simultaneously relatively transparent at
infrared wavelengths. In this case more incident energy may be absorbed than
can be emitted by isothermal atmosphere with temperature $T_{0}$. As a result
the atmospheric layer with strong absorption must, in the absence of other
energy transport mechanisms, heat up until the thermal emission from the layer
equals the absorbed incident flux. An inverted temperature structure with a
warm, radiative upper atmosphere overlying a cooler tropopause, such as that
shown for several planets’ atmospheres in Figure 1, is known as a
stratosphere. Almost all solar system planets with an atmosphere exhibit a
stratosphere. In Earth’s atmosphere ozone absorbs ultraviolet light, which
warms the stratosphere to $270\,\rm K$, about 50 Kelvin warmer than the
temperature at the top of the troposphere. Solar system giant planet
atmospheres are heated by UV absorption by a combination of methane and
hydrocarbon photochemical products, including $\rm C_{2}H_{2}$ and $\rm
C_{2}H_{6}$ and photochemically produced hazes (e.g., see the review for
Neptune by Bishop et al. 1995).
The atmosphere of Jupiter provides a specific example. Without an energy
source the planet’s middle atmosphere would be close to 104 K, (the skin
temperature for Jupiter with $T_{\rm eff}=124\,\rm K$), as seen above the
tropopause in Figure 1. In the region where most of the incident UV flux is
absorbed (near 10 mbar) there is little overlap between a 100 K Planck
function and the important thermal opacity sources, so little flux can be
emitted. Since the absorbed energy cannot be radiated away by a 100 K
atmosphere, the atmosphere warms and the Planck function moves to shorter
wavelengths. Eventually the blue side of the Planck function overlaps the
strong $\nu_{4}$ methane fundamental and the $\nu_{9}$ ethane bands at 7.7 and
$12.2\,\rm\mu m$, allowing the atmosphere to radiatively cool, balancing the
absorbed incident flux (Chamberlain & Hunten 1987). As in other solar system
giant planet atmospheres, these strong mid-infrared bands of ethane and
methane act as a thermostat, regulating the stratospheric temperatures.
Likewise in exoplanet atmospheres a balance must be struck between the
absorption of incident radiation and thermal emission. For hot Jupiters which
are so warm that even the most refractory Ti- and V-bearing compounds do not
condense, TiO and VO gas may be exceptionally important absorbers (Hubeny et
al. 2003; Burrows et al. 2007; Fortney et al. 2008). These gasses, while not
abundant, have extraordinarily large absorption cross sections across the
entire optical spectrum. When present these molecules can absorb much of the
remarkably high incident flux above $1\,\rm mbar$ where the atmosphere is
optically thin in the thermal infrared. The atmosphere therefore becomes very
hot, perhaps as hot as $1900\,\rm K$ (Knutson et al. 2008), hot enough for
emission by the near-infrared and optical bands of water, CO, and even TiO to
balance this influx of energy (Fortney et al. 2008). There is a hint from
transit spectra of HD209458b that TiO and VO are indeed present in the
atmosphere (Desert et al. 2008).
As in solar system atmospheres, photochemistry may also play an important role
in exoplanet stratospheres. Photochemical hazes or photochemical gaseous
products that absorb well in the optical and UV could also provide prodigious
energy sources for exoplanet stratospheric heating. Photochemical pathways
have not yet been fully explored for these planets (but see the work on the
carbon chemistry by Liang et al. 2004 and speculation by Burrows et al. 2007)
and the opacity spectra for many potential molecules are not well known. This
area remains ripe for further study.
## 8 Albedos
Albedo (from the Latin word for ‘white’) is a measure of the reflectivity of
an object. In planetary science one encounters a variety of albedos (from
single-scattering to geometric to Bond to spherical, to name a few) and care
must be taken to carefully define the term in use. From a planet-wide
perspective the albedo of most importance is the Bond albedo, $A$, the ratio
of incident energy reflected into all angles by a planet to the total incident
energy. The Bond albedo appears in the equation for the equilibrium
temperature of a rapidly rotating planet with radius $R$ receiving an incident
flux ${\cal F}$:
$4\pi R^{2}\sigma T_{\rm eq}^{4}=(1-A)\pi R^{2}{\cal F}$ $None$
The geometric albedo, $p_{\lambda}$, is defined as the ratio of a planet’s
reflectivity measured at zero phase angle (opposition) to that of a Lambert
disk of the same radius. Unlike the Bond albedo, the geometric albedo is a
function of wavelength and, because it is measured at opposition (when the
phase angle $\Phi=0$), does not require information on the dependence of
scattering with phase to measure. For a perfectly reflecting Lambert sphere
the geometric albedo is $2\over 3$; for a semi-infinite purely Rayleigh
scattering atmosphere it is $3\over 4$. Both such idealized, perfectly
scattering objects would have a Bond albedo of 1, but the Rayleigh atmosphere
sends more light directly back to the observer at zero phase angle and thus
has a higher geometric albedo. Observed geometric albedo spectra for Uranus
and Jupiter as well as a model spectrum for HD 209458b are shown in Figure 6.
Note that the hot Jupiter is quite dark beyond about $0.4\,\rm\mu m$ since
gaseous Na and K absorb most incident photons before they can be scattered
back to space (see also Burrows et al. 2008b). Uranus and Jupiter would
likewise be quite dark in the red if not for their cloud layers that scatter
red photons before they can be absorbed by methane (Figure 7; Marley et al.
1999).
Figure 6: Observed geometric albedo of Uranus and Jupiter (Karkoschka 1994),
compared to one possible model of HD 209458b which ignores thermal emission,
and the $1\sigma$ upper limit from MOST (Rowe et al. 2007). The geometric
albedo of a deep Rayleigh scattering atmosphere and a Lambertian sphere are
shown as well. Figure from Fortney et al. (in prep.).
The Bond albedo is related to the monochromatic geometric albedo $q_{\lambda}$
by
$A=\int\limits_{0}^{\infty}p_{\lambda}q_{\lambda}f_{\lambda}d\lambda\Bigr{/}\int\limits_{0}^{\infty}f_{\lambda}d\lambda$
$None$
where $q_{\lambda}$ is the monochromatic phase integral and $f_{\lambda}$ is
the incident monochromatic flux. The phase integral is a measure of the
angular distribution of scattered light. For a Lambert sphere and a Rayleigh
sphere $q_{\lambda}={3\over 2}$ and $4\over 3$ respectively. In the general
case, however, a planet is neither a perfectly Lamberian nor a Rayleigh
scatterer and the scattered light must be measured as a function of phase
angle $\Phi$ and $\lambda$. This is why the Bond albedos for solar system
giants could only be accurately determined after the Voyager spacecraft had
measured their brightness at many different phase angles so that $q_{\lambda}$
could be measured (e.g. Pollack et al. 1986 for Uranus, see also the review by
Conrath et al. 1989).
Without a measurement of $q_{\lambda}$ it is common to assume that a planet
scatters light isotropically following a Lambertian phase function. The ratio
$C$ of the brightness of a planet at a distance $a$ from its star as seen in
reflected light at an arbitrary phase angle $\Phi$ to the brightness of the
star can then be written as
$C_{\lambda}(\Phi)=q_{\lambda}(R/a)^{2}\Bigl{[}{{\sin(\Phi)+(\pi-\Phi)\cos(\Phi)}\over\pi}\Bigr{]}.$
$None$
At quadrature $\Phi={\pi\over 2}$ and $C_{\lambda}=q_{\lambda}(R/a)^{2}/\pi$.
Note that for the giant planets in Figure 6 the contrast would be far more
favorable in the visible than in the red or infrared bands.
Figure 7: Model geometric albedo spectra for a Jupiter-mass planet at various
distances from a solar type star. Dotted lines are models for the 3 and 2 AU
planets but without cloud opacity. While these spectra are cast as a function
of orbital radius, the same sequence would result for a giant planet kept at a
fixed distance and modeled with progressively younger ages, higher masses, or
earlier stellar spectral types, all of which would produce warmer effective
temperatures, all else being equal. For example an $8\,\rm M_{J}$ planet at an
age of about 1 Gyr would have a similar spectrum to the planet at 1.0 AU. A
$4\,\rm M_{J}$ at the same age would be similar to a cloudless planet (dotted
line) at 2 AU. Figure adapted from Fortney et al. (in prep.).
Because molecular bands tend to be stronger at longer wavelengths and because
Rayleigh scattering is more efficient in the blue, most solar system giant
planets have larger geometric albedos at shorter wavelengths than at longer
wavelengths. Given that the Bond albedo is weighted by the incident flux,
which varies with stellar type, the same planet with the same geometric albedo
spectrum will have a variety of different Bond albedos as the color of the
incident flux is changed. Around an early type star, which is brightest in the
blue, a Jupiter-like planet might have a large Bond albedo of 0.45. But around
an M star the same planet’s Bond albedo might be less than 0.1 (Marley et al.
1999). For this reason it is usually best to consider the geometric albedo
which does not overtly depend on the incident flux (other than through its
dependence on the temperature profile).
## 9 Colors of Planets
Early efforts to directly image extrasolar planets in reflected light, such as
by small or intermediate-sized coronagraphs, will likely first produce images
in a few broad passbands. Given such limited available data it is worthwhile
to consider what might be learned about the nature of the detected planets
from such data.
Figure 8: Observed and model broad-band colors of giant planets. J, S, U, & N
are the observed colors of the corresponding solar system giants. Labeled
crosses show model colors of giant planets at the given distances from a
solar-type primary star and a simple cloud model. Unlabeled crosses are
Jupiter and Saturn models with a different set of assumptions about the cloud
properties. Depending on the assumed effective temperature and other model
details (cloud structure, etc.), the color of a $1\,\rm M_{J}$ planet can vary
widely. Color alone is thus not a useful discriminant of mass.
Given an orbital separation from the primary star, a single photometric
detection, combined with an assumed phase function and bounds placed on the
geometric albedo, would allow a crude estimate of the planet’s size. Assuming
an upper limit of $p<0.75$ (the pure Rayleigh scattering limit) and a lower
limit $p>0.06$ (typical of low albedo asteroids (Dotto et al. 2002)), for
example, would result in an uncertainty in the radius inferred for a directly
imaged planet of a factor of 3.5. A bright planet with a radius slightly
larger than Earth’s could not be distinguished from a dark planet with
Neptune’s radius on the basis of brightness alone. If the planet were also
detected by other means, for example radial velocity or astrometric methods,
then the known mass would discriminate between these two extremes. Without
such a detection, however, the nature of the planet would have to be discerned
by spectroscopic or photometric methods. Even low resolution spectroscopy
likely will be beyond the reach of modest aperture space-based coronagraphic
telescopes. This means that planets will have to be characterized, at least
initially, by their broadband colors.
Indeed based on our experience in the solar system, broadband colors of giant
planets at first seem to be promising markers for discerning planet type
(Figure 8 for the giants). Uranus and Neptune are blue while Jupiter and
Saturn are red. Among the terrestrial planets the Moon, Mercury, and Mars are
red while the Earth is slightly blue. It has been suggested (e.g., Traub 2003)
that such color trends, when applied to exoplanets, may help identify the type
of planets detected by direct imaging. A planet might be imaged in a few broad
spectral filters and characterized by comparison to solar system planetary
colors. There are substantial difficulties with this approach, however, since
a single planet can have very different colors just depending on its
temperature and the range of plausible colors is certainly much larger than
that sampled by solar system planets. The reflectivity of giant planets in the
blue is strongly influenced by stratospheric hazes (e.g., see Baines &
Bergstralh (1986) for Uranus) while the brightness in the red depends upon
cloud properties. As a result both photochemical hazes and clouds (Figure 7)
can substantially alter the broadband color of a planet and are difficult to
model on an a priori basis. Also there will certainly be exoplanets with
characteristics quite different from solar system objects (warm Neptunes,
super-Earths, water worlds, etc.) and their colors may be surprising.
For example Figure 7 plots the geometric albedo spectra of a Jupiter mass
planet at various distances from its primary star and thus warmer (see Marley
et al. (1999), Sudarsky et al. (2003), and Burrows (2005) for more complete
discussions). An extrasolar planet in the same orbit could also be warmer than
Jupiter if it were younger, more massive,222Since giants more massive than
Jupiter take longer to cool, a four Jupiter mass planet that is 2 billion
years old, would have $T_{\rm eff}\sim 400\,\rm K$ and exhibit a spectrum
similar to one of the lower curves of Figure 7. or orbiting a brighter and
hotter star than our sun. As we imagine warming Jupiter (Figure 3), first the
ammonia clouds would evaporate, allowing ammonia to be well mixed throughout
the atmosphere and allowing us to see the water clouds, which underlie the
ammonia clouds. Since the water clouds are optically thick and good scatterers
in the optical, such a planet would likely be much brighter and whiter than
our current Jupiter. As we continue to warm the planet the atmosphere would
heat up and the cloud base would move progressively higher in the atmosphere.
Eventually, even the massive water clouds would evaporate, and suddenly we
would have a clear atmosphere. Jupiter would then appear very dark blue
(Figure 7), because red photons, which do not efficiently Rayleigh scatter,
would burrow down into the atmosphere, never to return. As the atmosphere
continues to warm, alkali metals, which at low temperatures are found as
chlorides such as KCl, would also evaporate, again altering the optical
spectrum of the planet (Figures 6 & 7) as they absorb strongly in the red
(Burrows et al. 2000). The important lesson from this thought experiment is
that the same planet can have vastly different spectra–and by extension colors
(Figure 8)–depending on its effective temperature. It would be a mistake to
presume that one could recognize a Jupiter-mass planet simply by its broadband
color. In this example Jupiter varies from red to white to blue over the span
of a few hundred Kelvin. Spectra, such as those shown in Figure 7, along with
inferences on age of the primary star (hence placing limits on the age of the
planet) would be the best means of discriminating the nature of the planet.
## 10 Conclusions
As we enter the age of the direct detection and characterization of extrasolar
planets it is important not to overlook the lessons learned from the past half
century of planetary exploration. Solar system planets give ample
demonstration of the importance of atmospheric dynamics, cloud and
condensation processes, and photochemistry in controlling the face of planets.
The great diversity of atmospheres seen in our own solar system, from the hazy
skies of Titan to the turbulent atmosphere of Jupiter to the blue vistas of
Earth, are emblematic of the diversity of processes that can affect properties
of planets. Exoplanets will exhibit even larger ranges of properties and we
should not be surprised by hazy yellow Earths, red, white, or blue Jupiters,
or other unexpected worlds. Certainly extrasolar planet atmospheres will be
influenced by some of the processes mentioned here. (Other yet-to-be-
discovered influences will undoubtedly be important as well.) Ultimately,
however, such discoveries will extend the journey of planetary exploration
begun by the Mariners and Voyagers of the last century out into the galaxy.
Acknowledgements: The author thanks Jonathan Fortney and Kevin Zahnle for
helpful comments on the manuscript and Thierry Montmerle and the entire Les
Houches staff for organizing an exceptionally productive and informative
program.
## References
* [Baines & Bergstralh(1986)] Baines, K. H., & Bergstralh, J. T. 1986, Icarus, 65, 406
* [Barman et al.(2005)] Barman, T. S., Hauschildt, P. H., & Allard, F. 2005, Ap. J., 632, 1132
* [Bishop et al.(1995)] Bishop, J., Atreya, S. K., Romani, P. N., Orton, G. S., Sandel, B. R., & Yelle, R. V. 1995, Neptune and Triton, 427
* [Burrows(2005)] Burrows, A. 2005, Nature, 433, 261
* [Burrows et al.(2000)] Burrows, A., Marley, M. S., & Sharp, C. M. 2000, Ap. J., 531, 438
* [Burrows et al.(2001)] Burrows, A., Hubbard, W. B., Lunine, J. I., & Liebert, J. 2001, Reviews of Modern Physics, 73, 719
* [Burrows et al.(2006)] Burrows, A., Sudarsky, D., & Hubeny, I. 2006, Ap. J., 650, 1140
* [Burrows et al.(2007)] Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., & Charbonneau, D. 2007, Ap. J. Lett., 668, L171
* [Burrows et al.(2008)] Burrows, A., Budaj, J., & Hubeny, I. 2008a, Ap. J., 678, 1436
* [Burrows et al.(2008)] Burrows, A., Ibgui, L., & Hubeny, I. 2008b, Ap. J., 682, 1277
* [1987] Chamberlain, J. & Hunten, D. 1987, Theory of Planetary Atmospheres (Orlando: Academic Press)
* [Charbonneau(2008)] Charbonneau, D. 2008, ArXiv e-prints, 808, arXiv:0808.3007
* [Conrath et al.(1989)] Conrath, B. J., Hanel, R. A., & Samuelson, R. E. 1989, Origin and Evolution of Planetary and Satellite Atmospheres, 513
* [Cowan & Agol(2008)] Cowan, N. B., & Agol, E. 2008, ArXiv e-prints, 806, arXiv:0806.4606
* [Deming(2008)] Deming, D. 2008, ArXiv e-prints, 808, arXiv:0808.1289
* [Demory et al.(2007)] Demory, B.-O., et al. 2007, Astron. & Astrophys., 475, 1125
* [Desert et al.(2008)] Desert, J. -., Vidal-Madjar, A., Lecavelier des Etangs, A., Sing, D., Ehrenreich, D., Hebrard, G., & Ferlet, R. 2008, ArXiv e-prints, 809, arXiv:0809.1865
* [Dotto et al.(2002)] Dotto, E., Barucci, M. A., Müller, T. G., Brucato, J. R., Fulchignoni, M., Mennella, V., & Colangeli, L. 2002, A & A, 393, 1065
* [Fortney et al.(2007)] Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, Ap. J., 659, 1661
* [Fortney et al.(2008)] Fortney, J. J., Lodders, K., Marley, M. S., & Freedman, R. S. 2008, Ap. J., 678, 1419
* [Helling et al.(2008)] Helling, C., et al. 2008, MNRAS in press, arXiv:0809.3657
* [Hubeny et al.(2003)] Hubeny, I., Burrows, A., & Sudarsky, D. 2003, Ap. J., 594, 1011
* [Ingersoll & Porco(1978)] Ingersoll, A. P., & Porco, C. C. 1978, Icarus, 35, 27
* [Karkoschka(1994)] Karkoschka, E. 1994, Icarus, 111, 174
* [Knutson et al.(2007)] Knutson, H. A., et al. 2007, Nature, 447, 183
* [Knutson et al.(2008)] Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., & Megeath, S. T. 2008, Ap. J., 673, 526
* [Liang et al.(2004)] Liang, M.-C., Seager, S., Parkinson, C. D., Lee, A. Y.-T., & Yung, Y. L. 2004, Ap. J. Lett., 605, L61
* [Lodders(2004)] Lodders, K. 2004, Science, 303, 323
* [Marley(1998)] Marley, M. S. 1998, Brown Dwarfs and Extrasolar Planets, 134, 383
* [Marley et al.(2007)] Marley, M. S., Fortney, J., Seager, S., & Barman, T. 2007, Protostars and Planets V, 733
* [Niemann et al.(1998)] Niemann, H. B., et al. 1998, J. Geophys. Res., 103, 22831
* [Pollack et al.(1986)] Pollack, J. B., Rages, K., Baines, K. H., Bergstralh, J. T., Wenkert, D., & Danielson, G. E. 1986, Icarus, 65, 442
* [Rowe et al.(2007)] Rowe, J. F., et al. 2007, ArXiv e-prints, 711, arXiv:0711.4111
* [Seager et al.(2008)] Seager, S., Deming, D., & Valenti, J. A. 2008, ArXiv e-prints, 808, arXiv:0808.1913
* [Showman et al.(2007)] Showman, A. P., Menou, K., & Y-K. Cho, J. 2007, ArXiv e-prints, 710, arXiv:0710.2930
* [Showman et al.(2008)] Showman, A. P., Fortney, J. J., Lian, Y., Marley, M. S., Freedman, R. S., Knutson, H. A., & Charbonneau, D. 2008, ArXiv e-prints, 809, arXiv:0809.2089
* [Sudarsky et al.(2003)] Sudarsky, D., Burrows, A., & Hubeny, I. 2003, Ap. J., 588, 1121
* [Traub(2003)] Traub, W. A. 2003, Scientific Frontiers in Research on Extrasolar Planets, 294, 595
* [Troyer et al.(2007)] Troyer, J., Moses, J. I., Fegley, B., Lodders, K., Marley, M. S., & Fortney, J. J. 2007, Bulletin of the American Astronomical Society, 38, 450
* [Wallace(1976)] Wallace, L. 1976, Jupiter, 284
* [Yung & Demore(1999)] Yung, Y. L., & Demore, W. B. 1999, Photochemistry of planetary atmospheres / Yuk L. Yung, William B. DeMore. New York : Oxford University Press, 1999.
* [Zahnle(2008)] Zahnle, K. 2008, Nature, 454, 41
|
arxiv-papers
| 2008-09-26T16:31:56
|
2024-09-04T02:48:57.995225
|
{
"license": "Public Domain",
"authors": "Mark S. Marley",
"submitter": "Mark S. Marley",
"url": "https://arxiv.org/abs/0809.4664"
}
|
0809.4736
|
# Entanglement generation in double-$\Lambda$ system
Ling Zhou, Yong Hong Ma, Xin Yu Zhao School of physics and optoelectronic
technology, Dalian University of Technology, Dalian 116024, P.R.China
###### Abstract
In this paper, we study the generation of entanglement in a double-$\Lambda$
system. Employing standard method of laser theory, we deduce the dynamic
evolution equation of the two-mode field. We analyze the available
entanglement criterion for double-$\Lambda$ system and the condition of
entanglement existence. Our results show that under proper parameters, the
two-mode field can entangled and amlified.
###### pacs:
42.50.Dv, 03.67.Mn
## I
Introduction
Continuous variables entanglement (CVE), as entanglement resource, has
attracted lots of attention because CVE not only has advantages in quantum-
information science 1 but also can be prepared unconditionally, whereas the
preparation of discrete entanglement usually relies on an event selection via
coincidence measurements. Conventionally, continuous variables entanglement
has been produced by nondegenerate parametric down- conversion (NPD) pan . In
order to improve the strength of the NPD, engineering the NPD Hamiltonian
within cavity QED has also attracted much attention strength ; zhou ; guzman .
Besides parametric down- conversion pan ; zhang ; simon1 , Xiong et. al. han
had shown that two-photon correlated spontaneous emission laser can work as a
continuous variables entanglement producer and amplifier, which open a new
attracting research domain. And then, a number of different schemes have been
proposed [9-14]. Different from the gain medium atoms in [8-14], Ref. LU has
studied a single-molecular-magnets system to produce CVE where physics process
is similar to kiffner . All of these works deal with the similar physics
process where both of the two mode will be created (annihilated) a photon in
one loop respectively ( similar to down-conversion system ).
In this paper, we proposed a scheme to generate CVE where the one mode is
created a photon and the other is annihilated, which is different from [8-16].
The system consists of atoms in double-$\Lambda$ configuration interacting
with two modes cavity fields. The atoms are driven into a coherent state of
the upper two levels by two classical field. We obtain the master equation of
the two mode fields. Through analysis of entanglement, we find that the
criterion proposed in zubairy can be used to judge entanglement. We show that
in double-$\Lambda$ system, entanglement exist on the condition that the two-
mode quantum field is tuned away from the atomic transition, and the initial
field is in a quantum state. Our study is helpful to understand the
entanglement charateristic within a system where quantum field is in ”$V$”
configuration.
## II The model and theory calculation
We consider a system of atoms in double-$\Lambda$ configuration shown in
Fig.1. Two cavity fields interact with atomic transition
$|a\rangle\leftrightarrow|c\rangle$ and $|b\rangle\leftrightarrow|c\rangle$
with detuning $\Delta_{a}$ and $\Delta_{b},$ respectively. The two classical
pumping fields with Rabi frequency $\Omega_{2}$ and $\Omega_{1}$ drive the
atomic level between $|a\rangle\leftrightarrow|d\rangle$ and
$|b\rangle\leftrightarrow|d\rangle$ with detuning $\Delta_{1}$ and
$\Delta_{2}$ respectively. Our double-$\Lambda$ system can be sodium atoms in
a vapor cell EA where the lower states are the two hyperfine levels
$|$$F=1\rangle$ and $F=2\rangle$ of 3${}^{2}S_{1/2}$, and the upper state are
$|$$F=1\rangle$ and $F=2\rangle$ of 3${}^{2}P_{1/2}$. The double-$\Lambda$
system also can be atomic Pb vapor harris . The phase-dependent
electromagnetically induced transparency EA and efficient nonlinear frequency
conversion harris have been investigated experimentally in double-$\Lambda$
system. Ref. chong studied dark-state polaritons in double-$\Lambda$ system.
Here, we are interested in producing two-mode entangled laser via the
double-$\Lambda$ system. In interaction picture, the Hamiltonian of the system
can be written as
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\nu_{1}a_{1}^{\dagger}a_{1}+\nu_{2}a_{2}^{\dagger}a_{2}$
$\displaystyle+\omega_{a}|a\rangle\langle a|+\omega_{b}|b\rangle\langle
b|+\omega_{c}|c\rangle\langle c|+\omega_{d}|d\rangle\langle d|$
Figure 1: The level configuration of atoms. Two cavity modes interact with
atomic transition $|b\rangle\leftrightarrow|c\rangle$ and
$|a\rangle\leftrightarrow|c\rangle$ with detuning $\Delta_{b}$ and
$\Delta_{a}$ respectively while the two classical fields drive the atomic
level between $|b\rangle\leftrightarrow|d\rangle$ and
$|a\rangle\leftrightarrow|d\rangle$ with detuning $\Delta_{2}$ and
$\Delta_{1}$. For simplicity, we assume the spontaneous-emission rate of four
level are the same.
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
g_{1}a_{1}|b\rangle\langle c|+g_{2}a_{2}|a\rangle\langle c|$
$\displaystyle+\Omega_{1}|b\rangle\langle
d|e^{-i\omega_{1}t}+\Omega_{2}|a\rangle\langle d|e^{-i\omega_{2}t}+H.c.$
We hope that the Hamiltonian do not contain time $t$ so as to simplify the
density matrix deduction of the field. In order to do that, we assume that the
classical fields detuning $\Delta_{1}=$ $\Delta a-\Delta$ and
$\Delta_{2}=\Delta_{b}$ $-\Delta$. Now we goes into a frame by performing a
unitary transformation $U=exp\\{i[H_{0}+\Delta_{a}|a\rangle\langle
a|+\Delta_{b}|b\rangle\langle b|+\Delta|d\rangle\langle d|]t\\}$. In the new
frame, the Hamiltonian is as
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle-\Delta a|a\rangle\langle
a|-\Delta b|b\rangle\langle b|-\Delta|d\rangle\langle d|$
$\displaystyle+[g_{1}a_{1}|b\rangle\langle c|+g_{2}a_{2}|a\rangle\langle c|$
$\displaystyle+\Omega_{1}|b\rangle\langle
d|e^{-i\omega_{1}t}+\Omega_{2}|a\rangle\langle d|e^{-i\omega_{2}t}+H.c.].$
In order to see the entanglement of the two-mode field, we need to obtain the
equation of motion of the two-mode field. Using the standard procedure in
laser theory developed by Scully and Zubairyscully ; scully2 ; walls , we
obtain the following master equation governing the dynamics of the two-mode
cavity fields as
$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-\kappa_{1}(a_{1}^{\dagger}a_{1}\rho-a_{1}\rho
a_{1}^{\dagger})-\kappa_{2}(a_{2}^{\dagger}a_{2}\rho-a_{2}\rho
a_{2}^{\dagger})$ $\displaystyle-\alpha_{1}(\rho
a_{1}a_{1}^{\dagger}-a_{1}^{\dagger}\rho a_{1})-\alpha_{2}(\rho
a_{2}a_{2}^{\dagger}-a_{2}^{\dagger}\rho a_{2})$
$\displaystyle-\alpha_{12}(\rho a_{2}a_{1}^{\dagger}-a_{1}^{\dagger}\rho
a_{2})-\alpha_{21}(\rho a_{1}a_{2}^{\dagger}-a_{2}^{\dagger}\rho a_{1})+h.c..$
We can see that the master equation has the term $\rho
a_{2}a_{1}^{\dagger}-a_{1}^{\dagger}\rho a_{2}$ which means that the one mode
is created a photon and the other mode is annihilated a photon. The detail
deduction of the equation is given in appendix A. In Eq.(4), we have include
the loss of the two-mode cavity with loss rate $\kappa_{1}$ and $\kappa_{2}.$
The coefficients are
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{g_{1}^{2}}{D}[A_{11}L_{bb}+A_{21}L_{ab}+A_{31}L_{db}],$
(5) $\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{g_{2}^{2}}{D}[A_{12}L_{ba}+A_{22}L_{aa}+A_{32}L_{da}],$
$\displaystyle\alpha_{12}$ $\displaystyle=$
$\displaystyle\frac{g_{1}g_{2}}{D}[A_{11}L_{ba}+A_{21}L_{aa}+A_{31}L_{da}],$
$\displaystyle\alpha_{21}$ $\displaystyle=$
$\displaystyle\frac{g_{1}g_{2}}{D}[A_{12}L_{bb}+A_{22}L_{ab}+A_{32}L_{db}],$
where
$D=(\gamma-i\Delta_{a})(\gamma-i\Delta_{b})[\gamma-i(\Delta+\frac{\Delta_{a}}{2}+\frac{\Delta_{b}}{2}]+\Omega_{1}^{2}(\gamma-i\Delta_{a})+\Omega_{2}^{2}(\gamma-i\Delta_{b})$,
and
$\displaystyle L_{aa}$ $\displaystyle=$
$\displaystyle\frac{-i\Omega_{2}}{\gamma}y_{3},L_{bb}=\frac{-i\Omega_{1}}{\gamma}y_{2},$
(6) $\displaystyle L_{ab}$ $\displaystyle=$
$\displaystyle\frac{\gamma+i(\Delta_{2}-\Delta_{1})}{2\gamma}y_{1}-\frac{i\Omega_{2}}{2\gamma}y_{2}-\frac{-i\Omega_{1}}{2\gamma}y_{3},$
$\displaystyle L_{db}$ $\displaystyle=$
$\displaystyle\frac{\gamma-i\Delta_{2}}{2\gamma}y_{2}-\frac{i\Omega_{2}}{2\gamma}y_{1},$
$\displaystyle L_{da}$ $\displaystyle=$
$\displaystyle\frac{\gamma-i\Delta_{1}}{2\gamma}y_{3}+\frac{i\Omega_{1}}{2\gamma}y_{1},$
and
$\displaystyle A_{11}$ $\displaystyle=$
$\displaystyle(\gamma-i\Delta_{a})[\gamma-i(\Delta+\frac{\Delta_{a}}{2}+\frac{\Delta_{b}}{2}],$
(7) $\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle
A_{21}=-\Omega_{1}\Omega_{2},A_{31}=-i\Omega_{1}(\gamma-i\Delta_{a}),$
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle(\gamma-i\Delta_{b})[\gamma-i(\Delta+\frac{\Delta_{a}}{2}+\frac{\Delta_{b}}{2}],$
$\displaystyle A_{32}$ $\displaystyle=$
$\displaystyle-i\Omega_{2}(\gamma-i\Delta_{b})$
with
$\displaystyle y_{2}$ $\displaystyle=$
$\displaystyle\frac{2ir_{in}s(a_{1}\Omega_{2}-b\Omega_{1})}{a_{1}a_{2}-b^{2}},y_{3}=\frac{2ir_{in}s(a_{2}\Omega_{1}-b\Omega_{2})}{a_{1}a_{2}-b^{2}}$
(8) $\displaystyle y_{1}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{2}(\Delta_{1}-2\Delta_{2})}{s}y_{2}+\frac{\Omega_{1}(2\Delta_{1}-\Delta_{2})}{s}y_{3},$
in which
$\displaystyle s$ $\displaystyle=$
$\displaystyle\gamma^{2}+\Omega_{1}^{2}+\Omega_{2}^{2}+(\Delta_{2}-\Delta_{1})^{2},$
(9) $\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
M_{1}s-\Omega_{2}^{2}(2\Delta_{2}-\Delta_{1})^{2},$ $\displaystyle a_{2}$
$\displaystyle=$ $\displaystyle
M_{2}s-\Omega_{1}^{2}(2\Delta_{1}-\Delta_{2})^{2},$ $\displaystyle b$
$\displaystyle=$
$\displaystyle\Omega_{1}\Omega_{2}[3s-(\Delta_{1}-2\Delta_{2})(2\Delta_{1}-\Delta_{2})]$
$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\gamma^{2}+4\Omega_{1}^{2}+\Omega_{2}^{2}+\Delta_{2}^{2},$
$\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle\gamma^{2}+4\Omega_{2}^{2}+\Omega_{1}^{2}+\Delta_{1}^{2}.$
Although our four-level atom is similar to kiffner ; LU , the physical process
of the two-mode quantum fields is different because the two quantum fields
work in different atomic level. In [8-15], both of the two mode will be
created or annihilated a photon in one loop. So the master equation is of the
form $\rho a_{2}^{\dagger}a_{1}^{\dagger}-a_{1}^{\dagger}\rho a_{2}^{\dagger}$
($\rho a_{2}a_{1}-a_{1}\rho a_{2}$). In our system, the two quantum fields are
in a ”V” form levels if we do not see the two classical pumping fields. The
simplified ”V” form levels is similar to ”Hanle effect” laser hanle where the
master equation is with the term $\rho
a_{2}a_{1}^{\dagger}-a_{1}^{\dagger}\rho a_{2}$. In our system, the two
classical fields make the atoms with the coherence of the two up-level
$|a\rangle$ and $|b\rangle$ [see (A10)]. When the spontaneous emissions from
$|a\rangle$ and $|b\rangle$ to $|c\rangle$ take place, entangled photons will
be produced.
## III Entanglement criterion choice and the discussion of the entanglement
condition
How to determine the entanglement is a key problem. In Ref.[8-15], employing
the criterion $(\Delta u)^{2}+(\Delta v)^{2}<2$duan , a inequality of the sum
of the quantum fluctuations of two operators $u$ and $v$ for some entangled
state, they find the entanglement between the two mode fields. However, the
criterion inequality of the sum of the quantum fluctuations can not be applied
to measure coherent stateshchu . Although the entanglement criterion on
measure continuous variable have been developed[22-25], we still can not find
a criterion to judge all kind of continuous variable entanglement. In order to
make clear the kind of entanglement existing in our model, we now discuss the
analytic solution in our system so as to choose a appropriate entanglement
criterion as well as to know the condition of entanglement.
Now we analyze the entanglement condition . If $g_{1}=g_{2}$,
$\Omega_{1}=\Omega_{2}$, and $\Delta=\Delta_{a}=\Delta_{b}\gg$
$\Omega_{1},\Omega_{2},\gamma$, through Eq.(6) to (10), one can obtain the
relation $\alpha_{1}=\alpha_{2}=\alpha_{12}=\alpha_{21}=i\alpha$ ($\alpha$ is
a real number). Usually, the loss of the cavity do not change the entanglement
structure of the state. It just destroy or sometimes enhance the entanglement
a little. So, in our choice entanglement criterion, we omit the loss of the
cavity. Therefore, the master equation of our system Eq.(4) can be simplified
as
$\dot{\rho}=i\alpha[a_{1}a_{1}^{\dagger}+a_{2}a_{2}^{\dagger}+a_{2}a_{1}^{\dagger}+a_{1}a_{2}^{\dagger},\rho].$
(10)
The effective Hamiltonian
$H_{eq}=-\alpha(a_{1}a_{1}^{\dagger}+a_{2}a_{2}^{\dagger}+a_{2}a_{1}^{\dagger}+a_{1}a_{2}^{\dagger})$.
Due to
$[a_{1}a_{1}^{\dagger}+a_{2}a_{2}^{\dagger}$,$a_{2}a_{1}^{\dagger}+a_{1}a_{2}^{\dagger}]=0$,in
interaction picture $H_{eqI}=$
$-\alpha(a_{2}a_{1}^{\dagger}+a_{1}a_{2}^{\dagger})$. One can easy check that
the system state, evolved by $H_{eqI}=$
$-\alpha(a_{2}a_{1}^{\dagger}+a_{1}a_{2}^{\dagger}),$ never meet with the
criterion $(\Delta u)^{2}+(\Delta v)^{2}<r^{2}+\frac{1}{r^{2}}$ for the
initial field number $|n_{1,}n_{2}\rangle$. We recognize the field Hamiltonian
is the generator of the $SU(2)$ coherent state gerry . The evolution of the
state $|\Psi(0)\rangle$ is
$|\Psi(t)\rangle=e^{-iH_{eqI}t}|\Psi(0)\rangle=e^{x_{+}K_{+}}e^{K_{0}\ln{x_{0}}}e^{x_{-}K_{-}}|\Psi(0)\rangle,$
where $K_{+}=a_{1}^{\dagger}a_{2}$, $K_{-}=a_{1}a_{2}^{\dagger}$. These
operators satisfy the $SU(2)$ commutation relations, i.e.,
$[K_{-},K_{+}]=-2K_{0}$, $[K_{0},K_{+}]=K_{+}$, $[K_{0},K_{-}]=-K_{-}$, with
$K_{0}=\frac{1}{2}(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})$; and in which
$\displaystyle x_{0}$ $\displaystyle=$ $\displaystyle\\{\cosh i\alpha
t\\}^{-\frac{1}{2}},$ $\displaystyle x_{+}$ $\displaystyle=$ $\displaystyle
x_{-}=\tanh i\alpha t.$
If the initial field state is two-mode Fock state $|0,N\rangle$, the evolution
of the state is
$|\Psi(t)\rangle=(\cos\alpha
t)^{N/2}\sum_{n=0}^{N}\left(\begin{array}[]{c}N\\\
n\end{array}\right)^{1/2}(i\tan\alpha t)^{n}|n,N-n\rangle.$ (11)
From the entanglement definition of pure state, we know that the state
$|\Psi(t)\rangle$ is a entangled one.
Unfortunately, the Hamiltonian $H_{eqI}$ can not entangle initial coherent
state, because the evolution of the system as
$|\Psi(t)\rangle=e^{x_{+}K_{+}}e^{K_{0}\ln{x_{0}}}e^{x_{-}K_{-}}|\beta_{1},\beta_{2}\rangle=|\tilde{\beta}_{1},\tilde{\beta}_{2}\rangle,$
(12)
with
$\displaystyle\tilde{\beta}_{1}$ $\displaystyle=$
$\displaystyle\beta_{1}\cos\alpha t+i\beta_{2}\sin\alpha t,$
$\displaystyle\tilde{\beta}_{2}$ $\displaystyle=$
$\displaystyle\beta_{2}\cos\alpha t+i\beta_{1}\sin\alpha t.$
So, it is not entangled.
The two-mode $SU(2)$ cat state is sub-Poissonian distribution. We recall the
criterion, proposed by Hillery and Zubairyzubairy can be used for non-
Gaussionian state. The criterion say that if
$\langle N_{1}N_{2}\rangle<|\langle a_{1}a_{2}^{\dagger}\rangle|^{2}$ (13)
the two-mode field is entangled. If the field initially is in number state
$|n_{1},n_{2}\rangle$, using the differential equation Eq.(B1-B13)(let
$\kappa=0$ and $\alpha_{1}=\alpha_{2}=\alpha_{12}=\alpha_{21}=i\alpha$), we
finally obtain
$\langle N_{1}N_{2}\rangle-|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}=n_{1}n_{2}-\frac{1}{4}(n_{1}+n_{2}+2n_{1}n_{2})\sin^{2}2\alpha
t.$ (14)
The maximum value of $\sin^{2}2\alpha t$ is 1; therefore if
$2n_{1}n_{2}<n_{1}+n_{2},$ (15)
the two mode field will be entangled. Because $n_{1}$ and $n_{2}$ are integer,
in order to meet with $2n_{1}n_{2}<n_{1}+n_{2}$, the number $n_{1}$ and
$n_{2}$ should be not equal. If either $n_{1}$ or $n_{2}$ is zero (the state
is standard $SU(2)$ coherent state), we can see that $\langle
N_{a}N_{b}\rangle-|\langle ab^{\dagger}\rangle|^{2}$ is always less than zero;
thus we say the state is entangled. Therefore, the criterion Eq.(12) can be
used for judge entanglement within our system.
However, for resonant case ($\Delta_{b}=\Delta_{a}=\Delta=0$), if
$\gamma_{b}=\gamma_{a}$ , $g_{1}=g_{2}$ and $\Omega_{1}=\Omega_{2}$, the
coefficients $\alpha_{1}=\alpha_{2}=\alpha_{12}=\alpha_{21}=\beta$ (real
number). For the initial state $|n_{1},n_{2}\rangle$, after complicated
calculation employing Eqs. B1-B13 for
$\alpha_{1}=\alpha_{2}=\alpha_{12}=\alpha_{21}=\beta$, we have
$\displaystyle\langle N_{1}N_{2}\rangle-|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{n_{1}+n_{2}}{16}(3e^{8\beta t}+2e^{4\beta t}-5)$
$\displaystyle+\frac{n_{1}n_{2}}{8}(1+6e^{4\beta t}+e^{8\beta t})$
$\displaystyle+\frac{1}{4}(1+e^{8\beta t}-2e^{4\beta t})\succeq 0$
If $n_{1}$ or $n_{2}$ is zero, $\langle N_{1}N_{2}\rangle-|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}$ equal to zero at initial time. Except that
the $\langle N_{1}N_{2}\rangle-|\langle a_{1}a_{2}^{\dagger}\rangle|^{2}$ is
larger than zero. It is obvious that we can not obtain entanglement in
resonant case. This conclusion is consistent with the work in Ref.manzoor ,
where author show that for two-level quantum beat laser, entanglement can be
created only when the strong driving field should be tuned away from the
atomic transition.
Figure 2: (a): The time evolution of the entanglement.(b): The time evolution
of the two-mode fields where red line is for $N_{1}$ and blue line is for
$N_{2.}$ Initially, the atom is in number state $|10,0\rangle$ The parameters
are $g_{1}=g_{2}=1;$ $\Delta_{a}=\Delta_{b}=50$, $\Delta=4$, $r_{in}=20$,
$\gamma=1$, $\kappa_{1}=\kappa_{2}=0.010$. $\Omega_{2}=\Omega_{1}=5$.
## IV
The entanglement of the cavity field
In above section, we discuss a special case so as to choose entanglement
criterion and make clear the condition of entanglement. Although above
analysis is for pure state (approximation of master equation Eq.(4)), But the
criterion $\langle N_{1}N_{2}\rangle<|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}$ should be available in judging entanglement
for general case. Now, considering the loss of the cavity and the decay of the
atomic levels, we numerical solve the differential Eqs. (B1) to (B13) and plot
the entanglement criterion $\langle N_{1}N_{2}\rangle-|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}$ and the $N_{1}(\langle
a_{1}^{\dagger}a_{1}\rangle)$,$N_{2}$($\langle a_{2}^{\dagger}a_{2}\rangle$).
In Fig.2, we plot the case that the initial field state is a number state
$|10,0\rangle$ where $\Delta_{b}=\Delta_{a}\gg\gamma_{b}=\gamma_{a}$ which
means that the classical field resonantly drive the atom
($\Delta_{1}=\Delta_{2}=0)$ and the quantum field interact with the atoms with
equal detunings. We see that due to the loss of the cavity, the entanglement
gradually disappear and photon number of the two-mode field also decrease
under large detuning case. Of course, if the cavity is ideal, one will observe
the entanglement oscillation.
Figure 3: The time evolution of entanglement. Initially, the atom is in number
state $|1,0\rangle$ The parameters are $g_{1}=g_{2}=1;$
$\Delta_{a}=50,\Delta_{b}=20$, $\Delta=10$, $r_{in}=20$, $\gamma=1$,
$\kappa_{1}=\kappa_{2}=0.01$. $\Omega_{2}=\Omega_{1}=4,5,6$ for dotted, dashed
and solid line, respectively.
However, with the same detuning $\Delta_{b}=\Delta_{a}$ (when $g_{1}=g_{2})$,
we can not have amplified entangled laser shown in Fig.2. The quantum fields
are in ”V” form. If $\Delta_{b}=\Delta_{a}$, the photon number in two mode
only oscillate because of the symmetry. In our numerical simulation, we find
that in order to have amplified entangled laser, $\Delta_{a}$ and $\Delta_{b}$
should be different. For initial field state in number state $|1,0\rangle$, we
plot entanglement and average photon numbers in Fig.3 and 4 for several values
of $\Omega_{1}(\Omega_{2})$. One can see clearly that entanglement can be
obtained without preparation atomic coherence before (here, atoms are injected
in state $|d\rangle$). But the photon number in two mode has large difference.
By adjusting the values of $\Omega_{1}(\Omega_{2})$, we can adjust the time
region of entanglement. Because we inject the atom in atomic state
$|d\rangle$, it will need time to evolve into a coherence among the atomic
level $|a\rangle$, $|b\rangle$ and $|d\rangle$. So, we have no entanglement
during a initial short time . With large value of $\Omega_{1}(\Omega_{2})$,
the atoms will acquire their coherence quickly so that the entanglement appear
quickly. However, with large value of $\Omega_{1}(\Omega_{2})$, the photon
number also will be amplified quickly shown in Fig. 4. As in our analytic
calculation, we have known that the photon number in two mode differ
(Eq.(15)). Here, in order to amplify the photon number, the photon number not
only should have difference but also can not put up with very large photon
number. With the increasing of photon number, the entanglement disappear. But
the disentanglement is not resulted from loss of the cavity because we find
even for $\kappa=0$, entanglement also disappear. We conclude that the
disentanglement result from the increase of photon number rather than from the
loss of the cavity. As it is pointed out in Ref.[21],in the high-gain limit
the condition in Eq.(9) is no longer able to detect whether there is
entanglement in the state.
Figure 4: The time evolution of average photon number.
Now, we show another function of the classical fields,i.e., the ability to
overcome the loss of the cavity which is shown in Fig.4. Let us compare dotted
line and solid line. The two lines correspond to the loss rate of the cavity
$\kappa_{1}=\kappa_{2}=0.01$ and $0.1$, respectively; and all the other
parameters are the same. Due to the increasing of $\kappa_{1}$($\kappa_{2}$),
the values of $\langle N_{1}N_{2}\rangle-|\langle
a_{1}a_{2}^{\dagger}\rangle|^{2}$ move up. If $\kappa_{1}$($\kappa_{2}$) keep
increasing, we will loss entanglement. However, with the help of classical
fields, we still can obtain entanglement even through
$\kappa_{1}$($\kappa_{2}$) is large, which can be observed by comparing dashed
line and solid one. Although the loss rate $\kappa_{1}=\kappa_{2}=0.1$,
through increasing $\Omega_{1}(\Omega_{2})$ to $6$, we still can have
entanglement. Of course, because of the increasing of
$\Omega_{1}(\Omega_{2})$, the time region move left, which we have analyze it
in Fig.3.
Figure 5: The time evolution of entanglement. Initially, the atom is in
number state $|10,0\rangle$ The parameters are $g_{1}=g_{2}=1;$
$\Delta_{a}=50,\Delta_{b}=20$, $\Delta=10$, $r_{in}=30$, $\gamma=1$. Dotted
line: $\kappa_{1}=\kappa_{2}=0.010$. $\Omega_{2}=\Omega_{1}=4;$solid line:
$\kappa_{1}=\kappa_{2}=0.1$, $\Omega_{2}=\Omega_{1}=4$; dashed
line:$\kappa_{1}=\kappa_{2}=0.1$, $\Omega_{2}=\Omega_{1}=6.$
## V
Conclusion
In conclusion, we have studied the generation of entanglement in a
double-$\Lambda$ system. We derive the theory of this system and analyze the
available entanglement criterion for double-$\Lambda$ system. When the atoms
are injected in the ground state $|d\rangle$, the entangled laser can be
achieved under the condition of suitable parameters. Due to the classical
pumping field introduction, we do not need to prepare atomic coherence, and
the intensity of the quantum fields will be amplified. The classical pumping
can overcome the loss of the cavity. Our results show that the time for which
the two modes remain entangled depends upon the strength of the Rabi frequency
of the classical driving field.
Our results is helpful in understanding the entanglement characteristic when
the master equation contain the term $\rho
a_{2}a_{1}^{\dagger}-a_{1}^{\dagger}\rho a_{2}$ such as quantum beats laser
and Hanle effect laser system. Our studies is limited to the initial state
$|1,0\rangle$. One can research other initial field state. Our initial field
should be easy to obtain. Let excited two-level atom with transition frequency
$\nu 1($ or $\nu 2)$ passing through the vacuum two-mode cavity, when we
detect the output atom in ground state, we will have the field state
$|1,0\rangle$.
Acknowledgments: Authors thank Professor M. S. Zubairy and M. Ikram for their
critical reading. The project was supported by NSFC under Grant No.10774020,
and also supported by SRF for ROCS, SEM.
## Appendix A Calculation details of density matrix of two-mode fields
The classical fields will be treated to all orders in the Rabi frequency. The
transitions $|a\rangle$ \- $|c\rangle$ and $|b\rangle$ \- $|c\rangle$ are
treated fully quantum mechanically but only up to second order in the
corresponding coupling constants. By partially tracing the global state of
Schrödinger equation over the atomic variables, we have the formal reduced
fields
$\dot{\rho}_{f}=-i([H_{cb,}\rho_{bc}]+[H_{ca},\rho_{ac}]+[H_{ac},\rho_{ca}]+[H_{bc},\rho_{cb}]).$
(17)
Eq.(17) reveals that we need to get the density matrix elements
$\rho_{ca},\rho_{bc,}$ etc.. Inserting the Hamiltonian Eq.(1) into A.1, from
the schrödinger equation, we have
$\displaystyle\dot{\rho}_{bc}$ $\displaystyle=$
$\displaystyle-(\gamma-i\Delta_{b})\rho_{bc}-i\Omega_{1}\rho_{dc}$
$\displaystyle+(-ig_{1}a_{1}\rho_{cc}+ig_{1}\rho_{bb}a_{1}+ig_{2}\rho_{ba}a_{2}),$
$\displaystyle\dot{\rho}_{ac}$ $\displaystyle=$
$\displaystyle-(\gamma-i\Delta_{a})\rho_{ac}-i\Omega_{2}\rho_{dc}$
$\displaystyle+(ig_{1}\rho_{ab}a_{1}+ig_{2}\rho_{aa}a_{2}-ig_{2}a_{2}\rho_{cc}),$
$\displaystyle\dot{\rho}_{dc}$ $\displaystyle=$
$\displaystyle-[\gamma-i\Delta]\rho_{dc}-i\Omega_{1}\rho_{bc}-i\Omega_{2}\rho_{ac}$
$\displaystyle+(ig_{1}\rho_{db}a_{1}+ig_{2}\rho_{da}a_{2}).$
In the last equations Eq.(A), we have consider the spontaneous-emission of the
atomic level. We rewrite it in a matrix form as
$\dot{\rho}=-M\rho+A$ (19)
where
$\rho=\left(\begin{array}[]{c}\rho_{bc}\\\ \rho_{ac}\\\
\rho_{dc}\end{array}\right),$ (20)
$M=\left(\begin{array}[]{ccc}\gamma-i\Delta_{b}&0&i\Omega_{1}\\\
0&\gamma-i\Delta_{a}&i\Omega_{2}\\\
i\Omega_{1}&i\Omega_{2}&\gamma-i\Delta\end{array}\right),$ (21)
$A=\left(\begin{array}[]{c}ig_{1}\rho_{bb}a_{1}+ig_{2}\rho_{ba}a_{2}\\\
ig_{1}\rho_{ab}a_{1}+ig_{2}\rho_{aa}a_{2}\\\
ig_{1}\rho_{db}a_{1}+ig_{2}\rho_{da}a_{2}\end{array}\right).$ (22)
When we write matrix $A$, we let $\rho_{cc}=0$ and will explain the reason
later. A solution of Eq.(19) which is a linear in the coupling constant
$g_{1(2)}$ can be obtained scully ; scully2 ; walls . Here we only care for
the matrix elements $\rho_{bc}$ and $\rho_{ac}$, so we just write the solution
of the two terms as
$\displaystyle\rho_{bc}$ $\displaystyle=$
$\displaystyle\frac{i}{D}[(A_{11}\rho_{bb}^{0}+A_{21}\rho_{ab}^{0}+A_{31}\rho_{db}^{0})g_{1}a_{1}$
(23)
$\displaystyle+(A_{11}\rho_{ba}^{0}+A_{21}\rho_{aa}^{0}+A_{31}\rho_{da}^{0})g_{2}a_{2}],$
$\displaystyle\rho_{ac}$ $\displaystyle=$
$\displaystyle\frac{i}{D}[(A_{12}\rho_{bb}^{0}+A_{22}\rho_{ab}^{0}+A_{32}\rho_{db}^{0})g_{1}a_{1}$
(24)
$\displaystyle+(A_{12}\rho_{ba}^{0}+A_{22}\rho_{aa}^{0}+A_{32}\rho_{da}^{0})g_{2}a_{2}]$
with
$\displaystyle A_{11}$ $\displaystyle=$
$\displaystyle(\gamma-i\Delta_{a})(\gamma-i\Delta)+\Omega_{2}^{2},$ (25)
$\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle
A_{21}=-\Omega_{1}\Omega_{2},A_{31}=-i\Omega_{1}(\gamma-i\Delta_{a}),$
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle(\gamma-i\Delta_{b})(\gamma-i\Delta)+\Omega_{1}^{2},$
$\displaystyle A_{32}$ $\displaystyle=$
$\displaystyle-i\Omega_{2}(\gamma-i\Delta_{b}),$
where
$D=(\gamma-i\Delta_{a})(\gamma-i\Delta_{b})(\gamma-i\Delta)+\Omega_{1}^{2}(\gamma-i\Delta_{a})+\Omega_{2}^{2}(\gamma-i\Delta_{b})$
in Eqs.(23)(24). As a approximation, the density matrix elements in right side
of Eqs.(23)(24) such as $\rho_{bb}^{0}$, $\rho_{ba}^{0}$, etc. will be
determined by steady state of classical fields. In other words, the density
matrix elements $\rho_{bb}^{0}$, $\rho_{ba}^{0}$, etc. of classical fields, as
a zero order approximation, are substituted into right side of Eqs.(23) and
(24) . And then, we can obtain a first order approximation of density matrix
elements $\rho_{ca}$ , $\rho_{ab}$ in terms of couplings $g_{1}(g_{2)}$.
Now, we just consider classical fields to determine the zero order
approximation of the density matrix elements $\rho_{bb}^{0}$, $\rho_{ba}^{0}$,
etc.. The differential equations of density matrix elements only with
classical fields and atomic decay are
$\displaystyle\dot{\rho}_{bb}^{0}$ $\displaystyle=$
$\displaystyle-\gamma\rho_{bb}^{0}-i\Omega_{1}(\rho_{db}^{0}-\rho_{bd}^{0}),$
(26) $\displaystyle\dot{\rho}_{aa}^{0}$ $\displaystyle=$
$\displaystyle-\gamma\rho_{aa}^{0}-i\Omega_{2}(\rho_{da}^{0}-\rho_{ad}^{0}),$
$\displaystyle\dot{\rho}_{ba}^{0}$ $\displaystyle=$
$\displaystyle-[\gamma-i(\Delta_{2}-\Delta_{1})]\rho_{ba}^{0}+i\Omega_{2}\rho_{bd}^{0}-i\Omega_{1}\rho_{da}^{0},$
$\displaystyle\dot{\rho}_{da}^{0}$ $\displaystyle=$
$\displaystyle-[\gamma+i\Delta_{1})]\rho_{da}^{0}-i\Omega_{2}(\rho_{aa}^{0}-\rho_{dd}^{0})-i\Omega_{1}\rho_{ba}^{0},$
$\displaystyle\dot{\rho}_{db}^{0}$ $\displaystyle=$
$\displaystyle-[\gamma+i\Delta_{2})]\rho_{db}^{0}-i\Omega_{2}\rho_{ab}^{0}-i\Omega_{1}(\rho_{bb}^{0}-\rho_{dd}^{0}),$
$\displaystyle\dot{\rho}_{dd}^{0}$ $\displaystyle=$
$\displaystyle-\gamma\rho_{db}^{0}-i\Omega_{1}(\rho_{bd}^{0}-\rho_{db}^{0})-i\Omega_{2}(\rho_{ad}^{0}-\rho_{da}^{0})+r_{in}\rho$
with $\Delta_{1}=\Delta_{a}-\Delta,\Delta_{2}=\Delta_{b}-\Delta$. For
$\dot{\rho}_{cc}$, we have $\dot{\rho}_{cc}^{0}=-\gamma\rho_{cc}^{0}$. The
steady state solution $\rho_{cc}^{0}=0$ (It is the reason why we let
$\rho_{cc}=0$ in Eq.(A6)). Substituting the steady state solution of (A10)
into (A7) and (A8), we obtain $\rho_{bc}$ , $\rho_{ac}$. And then, inserting
$\rho_{bc}$ and $\rho_{ac}$ back into (A1), one can have the master equation
Eq.(4) with coefficients Eq.(5) to (9).
## Appendix B Calculation details of density matrix of two-mode fields
In order to numerical calculate the entanglement criterion $\langle
N_{1}N_{2}\rangle-|\langle a_{1}a_{2}^{\dagger}\rangle|^{2}$ and the
$N_{1}(N_{2})$, we need to deduce a series differential equations from master
equations (4) which are listed below.
$\displaystyle\frac{d\langle a_{1}^{\dagger}a_{1}\rangle}{dt}$
$\displaystyle=$
$\displaystyle(\alpha_{1}+\alpha_{1}^{\ast}-2\kappa_{1})\langle
a_{1}^{\dagger}a_{1}\rangle$ $\displaystyle+\alpha_{12}^{\ast}\langle
a_{2}^{\dagger}a_{1}\rangle+\alpha_{12}\langle
a_{2}a_{1}^{\dagger}\rangle+\alpha_{1}+\alpha_{1}^{\ast},$
$\displaystyle\frac{d\langle a_{1}a_{2}^{\dagger}\rangle}{dt}$
$\displaystyle=$
$\displaystyle(\alpha_{1}+\alpha_{2}^{\ast}-\kappa_{1}-\kappa_{2})\langle
a_{1}a_{2}^{\dagger}\rangle$ $\displaystyle+\alpha_{21}^{\ast}\langle
a_{1}^{\dagger}a_{1}\rangle+\alpha_{12}\langle
a_{2}^{\dagger}a_{2}\rangle+\alpha_{12}+\alpha_{21}^{\ast},$
$\displaystyle\frac{d\langle a_{2}^{\dagger}a_{2}a_{2}^{+}a_{1}\rangle}{dt}$
$\displaystyle=$
$\displaystyle(\alpha_{1}+\alpha_{2}+2\alpha_{2}^{\ast}-\kappa_{1}-3\kappa_{2})\langle
a_{2}^{\dagger}a_{2}a_{2}^{+}a_{1}\rangle$ (29)
$\displaystyle+(\alpha_{2}+\alpha_{2}^{\ast}+2\kappa_{2})\langle
a_{2}^{+}a_{1}\rangle+2\alpha_{21}^{\ast}\langle
a_{2}^{\dagger}a_{2}a_{1}^{+}a_{1}\rangle$
$\displaystyle+2\alpha_{21}^{\ast}\langle
a_{2}^{\dagger}a_{2}\rangle+\alpha_{21}\langle a_{2}^{\dagger
2}a_{1}^{2}\rangle+\alpha_{12}\langle
a_{2}a_{2}^{\dagger}a_{2}a_{2}^{+}\rangle$
$\displaystyle+\alpha_{21}^{\ast}\langle
a_{1}^{+}a_{1}\rangle+\alpha_{21}^{\ast},$ $\displaystyle\frac{d\langle
a_{2}^{2}a_{1}^{+2}\rangle}{dt}$ $\displaystyle=$ $\displaystyle
2(\alpha_{1}^{\ast}+\alpha_{2}-\kappa_{2}-\kappa_{1})\langle
a_{2}^{2}a_{1}^{+2}\rangle$ $\displaystyle+2\alpha_{21}\langle
a_{1}^{\dagger}a_{1}a_{1}^{+}a_{2}\rangle+2(\alpha_{21}+\alpha_{12}^{\ast})\langle
a_{1}^{+}a_{2}\rangle$ $\displaystyle+2\alpha_{12}^{\ast}\langle
a_{2}a_{2}^{\dagger}a_{2}a_{1}^{+}\rangle,$ $\displaystyle\frac{d\langle
a_{1}a_{1}^{+}a_{1}a_{1}^{+}\rangle}{dt}$ $\displaystyle=$ $\displaystyle
2(\alpha_{1}+\alpha_{1}^{\ast}-2\kappa_{1})\langle
a_{1}a_{1}^{+}a_{1}a_{1}^{+}\rangle$
$\displaystyle+(\alpha_{1}+\alpha_{1}^{\ast}+6\kappa_{1})\langle
a_{1}^{+}a_{1}\rangle$ $\displaystyle+2\alpha_{12}\langle
a_{1}^{\dagger}a_{1}a_{1}^{+}a_{2}\rangle+2\alpha_{12}^{\ast}\langle
a_{1}a_{1}^{+}a_{1}a_{2}^{\dagger}\rangle$ $\displaystyle+\alpha_{12}\langle
a_{2}a_{1}^{+}\rangle+\alpha_{12}^{\ast}\langle a_{1}a_{2}^{\dagger}\rangle$
$\displaystyle+4\kappa_{1}+\alpha_{1}+\alpha_{1}^{\ast},$
$\displaystyle\frac{d\langle a_{1}^{+}a_{1}a_{2}^{\dagger}a_{2}\rangle}{dt}$
$\displaystyle=$
$\displaystyle(\alpha_{1}+\alpha_{1}^{\ast}+\alpha_{2}+\alpha_{2}^{\ast}-2\kappa_{1}-2\kappa_{2})\langle
a_{1}^{+}a_{1}a_{2}^{\dagger}a_{2}\rangle$ (32)
$\displaystyle+(\alpha_{1}+\alpha_{1}^{\ast})\langle
a_{2}^{+}a_{2}\rangle+(\alpha_{2}+\alpha_{2}^{\ast})\langle
a_{1}^{\dagger}a_{1}\rangle$ $\displaystyle+\alpha_{21}^{\ast}\langle
a_{1}^{+}a_{1}a_{1}^{\dagger}a_{2}\rangle+\alpha_{21}\langle
a_{1}a_{1}^{+}a_{1}a_{2}^{\dagger}\rangle$ $\displaystyle+\alpha_{12}\langle
a_{2}a_{2}^{\dagger}a_{2}a_{1}^{+}\rangle\alpha_{12}^{\ast}\langle
a_{2}^{\dagger}a_{2}a_{2}^{+}a_{1}\rangle.$
Substituting the subscript 1 (2) with 2 (1) and then making their Hermitian
conjugate through (B1) to (B5), we can obtain the other seven differential
equations. The totall thirteen differenttial equations will be a closed set.
We can numerical solve it.
## References
* (1) S. L. Braunstein and P. van Look, Rev. Mod. Phys. 77, 513(2005).
* (2) D. Bouwmeester, J. W. Pan, K. Mattle, et al, Nature 390, 575(1997).
* (3) R. M. Serra, C. J. Villas-Boas, N. G. de Almeida, and M. H. Y. Moussa, Phys. Rev. A 71, 045802 R (2005); R. M. Serra, C.J. Villas-Boas, N. G. de Almeida, and M. H. Y. Moussa, ibid 71, 045802(2005).
* (4) L. Zhou, H. Xiong,and M. S. Zubairy, Phys. Rev. A 74, 022321 2006 .
* (5) R. Guzmán, J.C. Retamal, E. Solano, and N. Zagury, Phys. Rev. Lett. 96, 010502(2006).
* (6) Y. Zhang, H. Wang, X. Y. Li, J. T. Jing, C. D. Xie, and K. C. Peng, Phys. Rev. A 62, 023813(2000); W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, Phys. Rev. Lett. 89, 253601(2002).
* (7) C. Simon and D. Bouwmeester, Phys. Rev. Lett. 91,053601(2003).
* (8) H.Xiong, M.O. Scully, and M. S. Zubairy, Phys.Rev.Lett. 94, 023601(2005).
* (9) H.T.Tan, S.Y.Zhu,M. S. Zubairy, Phys. Rev. A 72,022305(2005).
* (10) E. Alebachew, Phys. Rev.A 76,023808(2007)
* (11) M. Kiffner, M. S. Zubairy, J. Evers, and C.H. Keitel, Phys.Rev.A 75,033816(2007).
* (12) M. Ikram, G. X. Li and M. S. Zubairy, Phys. Rev. A 76, 042317 (2007).
* (13) C. H. Raymond Ooi, Phys. Rev.A 76,013809(2007).
* (14) G. X. Li, H. T. Tan, and M. Macovei, Phys. Rev. A 76,053827(2007).
* (15) X. Y. Lü, J. B. Liu, Y. Tian, P. J. Song and Z. M. Zhan, Euro. Phys. Lett. 82, 6403 (2008).
* (16) M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press(1997).
* (17) M. O. Scully,Phys. Rev. Lett. 55, 2802(1985).
* (18) M. Hillery and M. S. Zubairy, Phys. Rev. Lett. 96, 050503(2006).
* (19) A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G.Y. Yin, and S. E. Harris,Phys. Rev. Lett. 84,5308(2000).
* (20) Y. D. Chong and M. Soljačić, Phys. Rev. A 77, 013823(2008).
* (21) M. O. Scully and M. S. Zubairy, Phys. Rev. A, 35, 752(1987).
* (22) C. A. Blockley and D. F. Walls, Phys. Rev. A 43, 5049(1991).
* (23) L. M. Duan, G.Giedke, J.I.Cirac, and P. Zoller, Phys. Rev. Lett.84, 2722(2000).
* (24) E. Shchukin and W. Vogel, Phys. Rev. Lett 95, 230502(2005).
* (25) R. Simon, Phys. Rev. Lett. 84, 2726(2000).
* (26) A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, quant-ph/06050001.
* (27) C. C. Gerry and R. Grobe, J. Mod. Optics 44, 41(1997).
* (28) E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev,* and L. Windholz, Phys. Rev. A 59, 2302(1999)
|
arxiv-papers
| 2008-09-27T03:25:23
|
2024-09-04T02:48:58.003131
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ling Zhou, Yong Hong Ma, Xin Yu Zhao",
"submitter": "Ling Zhou",
"url": "https://arxiv.org/abs/0809.4736"
}
|
0809.4762
|
Ultra-high energy cosmic rays threshold in Randers-Finsler space
Zhe Chang 111changz@mail.ihep.ac.cn and Xin Li 222lixin@mail.ihep.ac.cn
Institute of High Energy Physics
Chinese Academy of Sciences
P. O. Box 918(4), 100049 Beijing, China
###### Abstract
Kinematics in Finsler space is used to study the propagation of ultra high
energy cosmic rays particles through the cosmic microwave background
radiation. We find that the GZK threshold is lifted dramatically in Randers-
Finsler space. A tiny deformation of spacetime from Minkowskian to Finslerian
allows more ultra-high energy cosmic rays particles arrive at the earth. It is
suggested that the lower bound of particle mass is related with the negative
second invariant speed in Randers-Finsler space.
PACS numbers: 03.30.+p, 11.30.Cp, 98.70.Sa
Decades ago, Greisen, Zatsepin and Kuz’min (GZK) [1] discussed the propagation
of the ultra-high energy cosmic rays (UHECR) particles through the cosmic
microwave background radiation (CMBR) [2]. Due to photopion production process
by the CMBR, the UHECR particles will lose their energies drastically down to
a theoretical threshold (about $5\times 10^{19}$eV). That is to say, the UHECR
particles which their energy beyond the threshold can not be observed[3]. This
strong suppression is called GZK cutoff. However hundreds of events with
energies above $10^{19}$eV and about 20 events above $10^{20}$eV have been
observed[4].
To explain this puzzle, one general accepted hypothesis is that the Lorentz
Invariance (LI) is violated[5]. The violation of the LI and the Planck scale
physics have long been suggested as possible solutions of the cosmic rays
threshold anomalies[5]. LI is one of the foundations of the Standard model of
particle physics. Coleman and Glashow have set up a perturbative framework for
investigating possible departures of local quantum field theory from LI[6, 7].
In a different approach, Cohen and Glashow suggested [8] that the exact
symmetry group of nature may be isomorphic to a subgroup SIM$(2)$ of the
Poincare group. The mere observation of ultra-high energy cosmic rays and
analysis of neutrino data give an upper bound of $10^{-25}$ on the Lorentz
violation[9].
In fact, Gibbons, Gomis and Pope[10] showed that the Finslerian line element
$ds=(\eta_{\mu\nu}dx^{\mu}dx^{\nu})^{(1-b)/2}(n_{\rho}dx^{\rho})^{b}$ is
invariant under the transformations of the group DISIM${}_{b}(2)$. The very
special relativity is a Finsler geometry.
Recently, we proposed a gravitational field equation in Berwald-Finsler
space[11]. The asymmetric term in field equation violated LI naturally. A
modified Newton’s gravity is obtained as the weak field approximation of the
Einstein’s equation in Berwald-Finsler space[12]. The flat rotation curves of
spiral galaxies can be deduced naturally without invoking dark matter in the
framework of Finsler geometry.
In this Letter, we use the kinematics in Randers-Finsler space to study the
propagation of the UHECR particles through CMBR. We obtain a deformed GZK
threshold for the UHECR particles interacting with soft photons, which depends
on an intrinsic parameter of the Randers-Finsler space[13].
Denote by $T_{x}M$ the tangent space at $x\in M$, and by $TM$ the tangent
bundle of $M$. Each element of $TM$ has the form $(x,y)$, where $x\in M$ and
$y\in T_{x}M$. The natural projection $\pi:TM\rightarrow M$ is given by
$\pi(x,y)\equiv x$. A Finsler structure[14] of $M$ is a function
$\displaystyle F:TM\rightarrow[0,\infty).$
The Finsler structure $F$ is regularity (F is $C^{\infty}$ on the entire slit
tangent bundle $TM\backslash 0$), positive homogeneity ($F(x,\lambda
y)=\lambda F(x,y)$, for all $\lambda>0$) and strong convexity (the $n\times n$
Hessian matrix $g_{ij}\equiv\frac{\partial^{2}}{\partial y^{i}\partial
y^{j}}\left(\frac{1}{2}F^{2}\right)$ is positive-definite at every point of
$TM\backslash 0$).
It is convenient to take $y\equiv\frac{dx}{d\tau}$ being the intrinsic speed
on Finsler space.
In 1941, G. Randers[15] studied a very interesting class of Finsler manifolds.
The Randers metric is a Finsler structure $F$ on $TM$ with the form
$\displaystyle
F(x,y)\equiv\sqrt{\eta_{ij}\frac{dx^{i}}{d\tau}\frac{dx^{j}}{d\tau}}+\frac{\eta_{ij}\kappa^{i}}{2m}\frac{dx^{j}}{d\tau}~{}.$
(1)
The action of a free moving particle on Randers space is given as
$\displaystyle
I=\int^{r}_{s}\mathcal{L}d\tau=m\int^{r}_{s}F\left(\frac{dx}{d\tau}\right)d\tau.$
(2)
Define the canonical momentum $p_{i}$ as
$\displaystyle p_{i}=m\frac{\partial
F}{\partial\left(\frac{dx^{i}}{d\tau}\right)}~{}.$ (3)
Using Euler’s theorm on homogeneous functions, we can write the mass–shell
condition as
$\displaystyle{\cal M}(p)=g^{ij}p_{i}p_{j}=m^{2}~{}.$ (4)
The modified dispersion relation in Randers spaces is of the form
$\displaystyle
m^{2}=\eta^{ij}p_{i}p_{j}-\eta^{ij}\kappa_{i}(\mu,M_{p})p_{j}~{},$ (5)
where we have used the notation
$\displaystyle\eta_{ij}={\rm diag}\\{1,-1,-1,-1\\}~{},$ (6)
$\displaystyle\kappa_{i}=\kappa\\{1,-1,-1,-1\\}~{},$ (7)
and $\eta^{ij}$ is the inverse matrix of $\eta_{ij}$. Here $\kappa$ can be
regarded as a measurement of LI violation. We consider the head-on collision
between a soft photon of energy $\epsilon$, momentum q and a high energy
particle $m_{1}$ of energy $E_{1}$, momentum $p_{1}$, which leads to the
production of two particles $m_{2}$, $m_{3}$ with energies $E_{2}$, $E_{3}$
and momentums $p_{2}$, $p_{3}$, respectively. By making use of the energy and
momentum conservation law and the modified dispersion relation (5), we obtain
the deformed GZK threshold in Randers-Finsler space
$\displaystyle
E_{th}=\frac{(m_{2}+m_{3})^{2}-m_{1}^{2}}{4(\epsilon-\kappa/2)}.$ (8)
Taking roughly the energy of soft photon to be $10^{-3}$eV, we give a plot for
the dependence of the threshold $E^{N}_{th}$ on the deformation parameter
$\kappa$ in FIG. 1.
FIG.1
We can see clearly that a tiny deformation of spacetime ($\kappa$ with the
order of the CMBR) can provide sufficient correction to the primary predicted
threshold for the propagation of UHECR particles through the CMBR[1]. If the
nature of our universe is Finslerian, more UHECR particles should be detected
than Greisen, Zatsepin and Kuzmin expected.
Another invariant speed in Randers-Finsler space is expressed as[13]
$\displaystyle C_{2}=\frac{\kappa-4m}{\kappa+4m}~{}.$ (9)
From the above discussion, we know that the deformation parameter $\kappa$ may
be the same order with CMBR. So far as we know that there is no observational
evidence for the existence of the second invariant speed $C_{2}$. Thus, we
suppose that the $C_{2}$ is negative or $C_{2}$ is beyond the speed of light.
The negative condition of the invariant speed $C_{2}$ deduces that
$m\geq\kappa/4$. This gives particle mass a lower bound for massive particle.
The condition that $C_{2}$ is beyond the speed of light deduces that the mass
of particle is negative. In such a case, $C_{2}$ may be corresponded to the
speed of Goldstone boson.
Recently, there is a renewed interest in experimental tests of LI and CPT
symmetry. Kostelecky[16] has tabulated experimental results for LI and CPT
violation in the minimal Standard-Model Extension. Our result would not
violate the minimal Standard-Model Extension, since $\kappa$ can be eliminated
by a redefinition of the energy and momentum. $\kappa$ is very small, the
minor change in energy and momentum can be neglected except for soft photon.
Acknowledgements
We would like to thank T. Chen, J. X. Lu, N. Wu, M. L. Yan and Y. Yu for
useful discussion. One of us (X. Li) indebt W. Bietenholz for useful
discussion on UHECR. The work was supported by the NSF of China under Grant
NOs. 10575106 and 10875129.
## References
* [1] K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 (1966).
* [2] P. G. Roll and D. T. Wilkinson, Phys. Revs. Lett. 16, 405 (1966).
* [3] F. W. Stecker, Phys. Rev. Lett. 21, 1016 (1968).
* [4] M. Takeda et al., Phys. Rev. Lett. 81, 1163 (1998);M. Takeda et al., Astrophys. J. 522, 225 (1999); N. Hayashida et al., Phys. Rev. Lett. 73, 3491 (1994); D. J. Bird et al., Astrophys. J. 441, 144 (1995); D. J. Bird et al., Phys. Rev. Lett. 71, 3401 (1993); D. J. Bird et al., Astrophys. J. 424, 491 (1994); M. A. Lawrence, R. J. O. Reid, and A. A. Watson, J. Phys. G 17, 733 (1991).
* [5] V. A. Kostelecky, Phys. Rev. D. 69, 105009 (2004); R. Aloisio, P. Blasi, P. L. Ghia, and A. F. Grillo, Phys. Rev. D 62, 053010 (2000); O. Bertolami and C. S. Carvalho, Phys. Rev. D 61, 103002 (2000); H. Sato, arXiv: astro-ph/0005218; T. Kifune, Astrophys. J. 518, L21 (1999). W. Kluzniak, arXiv: astro-ph/9905308; R. J. Protheroe and H. Meyer, Phys. Lett. B 493, 1 (2000).
* [6] S.R. Coleman and S.L. Glashow, Phys. Lett. B405, 249 (1997).
* [7] S.R. Coleman and S.L. Glashow, Phys. Rev. D59, 116008 (1999).
* [8] A.G. Cohen and S.L. Glashow, Phys. Rev. Lett. 97 021601 (2006).
* [9] G. Battistoni et al., Phys. Lett. B615 14 (2005).
* [10] G.W. Gibbons, J. Gomis and C.N. Pope, ”General Very Special Relativity is Finsler Geometry”, hep-th/0707.2174.
* [11] X. Li and Z. Chang, arXiv: gr-qc/0711.1934.
* [12] Z. Chang and X. Li, arXiv: gr-qc/0806.2184, to be published in Phys. Lett. B.
* [13] Z. Chang and X. Li, Phys. Lett. B. 663, 103 (2008).
* [14] D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer, New York, 2000.
* [15] G. Randers, Phys. Rev. 59, 195 (1941).
* [16] V. A. Kostelecky, arXiv: hep-ph/0801.0287v1.
|
arxiv-papers
| 2008-09-27T11:07:25
|
2024-09-04T02:48:58.007867
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang and Xin Li",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/0809.4762"
}
|
0809.4829
|
# Optimisation of future long baseline neutrino experiments
Olga Mena
IEEC/CSIC, Bellaterra, Spain and
IFIC/CSIC, Valencia, Spain
E-mail It is a pleasure to thank all my long-baseline collaborators and to
Enrique Fernández Martínez who has produced the sensitivity plots shown in
this talk. O. M is supported by a _Ramón y Cajal_ contract from MEC, Spain.
mena@ieec.uab.es
###### Abstract:
The aim of this _talk_ is to review near and far future long baseline neutrino
experiments as superbeams, $\beta$-Beams and neutrino factories, comparing
their sensitivities to the unknown parameters in the neutrino oscillation
sector. We focus on the extraction of the neutrino mass hierarchy, exploring
alternatives to the commonly used neutrino-antineutrino comparison. Special
attention to a new concept of neutrino factory design, the _low energy
neutrino factory_ , is given.
## 1 Introduction
During the last several years the physics of neutrinos has achieved remarkable
progress. The present data require two large ($\theta_{12}$ and $\theta_{23}$)
and one small ($\theta_{13}$) angles in the neutrino mixing matrix, and at
least two mass squared differences, $\Delta m_{ij}^{2}\equiv
m_{j}^{2}-m_{i}^{2}$ (where $m_{j}$’s are the neutrino masses), one driving
the atmospheric ($\Delta m_{23}^{2}$) and the other one the solar ($\Delta
m_{12}^{2}$) neutrino oscillations. The mixing angles $\theta_{12}$ and
$\theta_{23}$ control the solar and the atmospheric neutrino oscillations,
while $\theta_{13}$ is the angle which connects the atmospheric and solar
neutrino realms.
A recent global fit [1] provides the following $3\sigma$ allowed ranges for
the atmospheric mixing parameters $|\mbox{$\Delta m_{23}^{2}$}|=(2-3.2)\times
10^{-3}$ eV2 and $0.32<\sin^{2}\theta_{23}<0.64$. The sign of $\Delta
m_{23}^{2}$, sign$(\mbox{$\Delta m_{23}^{2}$})$, cannot be determined with the
existing data. The two possibilities, $\mbox{$\Delta m_{23}^{2}$}>0$ or
$\mbox{$\Delta m_{23}^{2}$}<0$, correspond to two different types of neutrino
mass ordering: normal hierarchy and inverted hierarchy. In addition,
information on the octant in which $\theta_{23}$ lies, if
$\sin^{2}2\theta_{23}\neq 1$, is beyond the reach of present experiments. The
solar neutrino oscillation parameters lie in the low-LMA (Large Mixing Angle)
region, with best fit values [1] $\mbox{$\Delta m_{12}^{2}$}=7.9\times
10^{-5}~{}{\rm eV^{2}}$ and $\sin^{2}\theta_{12}=0.30$. A combined 3-neutrino
oscillation analysis of the solar, atmospheric, reactor and long-baseline
neutrino data [1] constrains the third mixing angle to be
$\sin^{2}\theta_{13}<0.04$ at the $3\sigma$ C.L.
The future goals for the study of neutrino properties is to precisely
determine the already measured oscillation parameters and to obtain
information on the unknown ones: namely $\theta_{13}$, the CP–violating phase
$\delta$ and the type of neutrino mass hierarchy (or equivalently
sign$(\mbox{$\Delta m_{23}^{2}$})$).
## 2 The golden channels
The most promising way to determine the unknown parameters $\delta$ and
$\theta_{13}$ is through the detection of the subleading transitions
$\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})\leftrightarrow\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$.
These channels have been named as _golden channels_ [2]. Defining
$\Delta_{ij}\equiv\frac{\Delta m^{2}_{ij}}{2E}$, a convenient and precise
approximation is obtained by expanding to second order in the following small
parameters: $\theta_{13}$, $\Delta_{12}/\Delta_{23}$, $\Delta_{12}/A$ and
$\Delta_{12}\,L$ [2, 3]
$\displaystyle P_{\nu_{e}\nu_{\mu}(\bar{\nu}_{e}\bar{\nu}_{\mu})}$
$\displaystyle=$ $\displaystyle
s_{23}^{2}\sin^{2}2\mbox{$\theta_{13}$}\,\left(\frac{\mbox{$\Delta_{23}$}}{\tilde{B}_{\mp}}\right)^{2}\,\sin^{2}\left(\frac{\tilde{B}_{\mp}\,L}{2}\right)\,+\,c_{23}^{2}\sin^{2}2\theta_{12}\,\left(\frac{\Delta_{12}}{A}\right)^{2}\,\sin^{2}\left(\frac{A\
L}{2}\right)$ (1) $\displaystyle+$
$\displaystyle\tilde{J}\;\frac{\Delta_{12}}{A}\,\frac{\mbox{$\Delta_{23}$}}{\tilde{B}_{\mp}}\,\sin\left(\frac{AL}{2}\right)\,\sin\left(\frac{\tilde{B}_{\mp}L}{2}\right)\,\cos\left(\pm\delta-\frac{\mbox{$\Delta_{23}$}\,L}{2}\right)\,,$
where $L$ is the baseline, $\tilde{B}_{\mp}\equiv|A\mp\mbox{$\Delta_{23}$}|$
and the matter parameter $A$ is defined in terms of the average electron
number density, $n_{e}(L)$, as $A\equiv\sqrt{2}\,G_{F}\,n_{e}(L)$, and
$\tilde{J}$ is defined as
$\displaystyle\tilde{J}\equiv\cos\theta_{13}\;\sin 2\theta_{13}\;\sin
2\theta_{23}\;\sin 2\theta_{12}~{}.$ (2)
The golden transitions Eqs. (1) are sensitive to all the unknown parameters
quoted in the introductory section. They are clearly sensitive to the mixing
angle $\theta_{13}$. These channels are also sensitive to the CP–violating
phase (via the third term or the _interference_ term, the only term which
differs for neutrinos and antineutrinos). The golden transitions allow us also
to extract the sign of the atmospheric mass difference exploiting matter
effects. In the presence of matter effects, the neutrino (antineutrino)
oscillation probability gets enhanced [4] for the normal (inverted) hierarchy.
Making use of the different matter effects for neutrinos and antineutrinos
seems, in principle, the most promising way to distinguish among the two
possibilities: normal versus inverted hierarchy.
## 3 Degenerate solutions
We can ask ourselves whether it is possible to unambiguously determine
$\delta$ and $\theta_{13}$ by measuring the golden transition probabilities,
$\mbox{$\nu_{e}$}\to\mbox{$\nu_{\mu}$}$ and
$\mbox{$\overline{\nu}_{e}$}\to\mbox{$\overline{\nu}_{\mu}$}$, Eqs. (1) (or
equivalently, $\mbox{$\nu_{\mu}$}\to\mbox{$\nu_{e}$}$ and
$\mbox{$\overline{\nu}_{\mu}$}\to\mbox{$\overline{\nu}_{e}$}$) at fixed
neutrino energy $E$ and at just one baseline $L$. The answer is no. By
exploring the full (allowed) range of the $\delta$ and $\theta_{13}$
parameters, that is, $-180^{\circ}<\delta<180^{\circ}$ and
$0^{\circ}<\mbox{$\theta_{13}$}<10^{\circ}$, one finds, at fixed neutrino
energy and at fixed baseline, the existence of degenerate solutions
($\theta^{{}^{\prime}}_{13}$, $\delta^{{}^{\prime}}$), that we label
_intrinsic degeneracies_ , which give the same oscillation probabilities than
the set ($\theta_{13}$, $\delta$) chosen by nature [5]. It has also been
pointed out that other fake solutions might appear from unresolved
degeneracies in two other oscillation parameters:
1. 1.
The sign of the atmospheric mass difference $\Delta m_{23}^{2}$ may remain
unknown. In this particular case,
$P(\theta^{{}^{\prime}}_{13},\delta^{{}^{\prime}},-\Delta
m_{23}^{2})=P(\theta_{13},\delta,\Delta m_{23}^{2})$ [6, 7].
2. 2.
Disappearance experiments only give us information on $\sin^{2}2\theta_{23}$:
is $\theta_{23}$ in the first octant, or is it in the second one,
$(\pi/2-\theta_{23}$)? . In terms of the probabilities,
$P(\theta^{{}^{\prime}}_{13},\delta^{{}^{\prime}},\frac{\pi}{2}-\mbox{$\theta_{23}$})=P(\theta_{13},\delta,\mbox{$\theta_{23}$})$
[7, 8].
All these ambiguities complicate the experimental determination of $\delta$
and $\theta_{13}$. The situation has been dubbed the _eight-fold degeneracy_.
A lot of work has been devoted to resolve the degeneracies by exploiting the
different neutrino energy and baseline dependence of two (or more) LBL
experiments. A complete list of references is beyond the scope of this talk. I
suggest to see Ref. [9] and references therein.
## 4 The tree level approach: superbeams
The next generation of long baseline $\nu_{e}$ neutrino appearance experiments
will be the so-called superbeam experiments. The major goal of superbeam
experiments is to set a non-zero value of the small mixing angle $\theta_{13}$
(or, in the absence of a positive signal, to improve the current upper bound
on this mixing angle). A superbeam experiment consists, basically, of a higher
intensity version of a conventional neutrino (antineutrino) beam. Superbeams
represent the logical next step in accelerator-based neutrino physics. There
are two possible strategies regarding the neutrino beam. The _off-axis_
technique produces a neutrino spectrum very narrow in energy (nearly
monochromatic, $\Delta E/E\sim 15-25\%$), which peaks at lower energies with
respect to the on-axis one. The off-axis technique allows a discrimination
between the peaked $\nu_{e}$ oscillation signal and the intrinsic $\nu_{e}$
background which has a broad energy spectrum. In addition, the off-axis
technique reduces significantly the background resulting from neutral current
interactions of higher energy neutrinos with a $\pi^{0}$ in the final state.
Unfortunately, off axis experiments are counting experiments in which one has
only two measurements (the number of neutrinos and the number of antineutrino
events) and resolving degeneracies becomes an impossible task. This is the
technique exploited by the $\nu_{e}$ appearance experiments T2K [10] and
NO$\nu$A [11].
The _wide band beam (WBB)_ technique exploits the spectral information of the
signal, being sensitive to many $E/L$’s at the same time. The neutrino beam is
on-axis and therefore the fluxes and the beam energies are higher than the
ones exploited in the off-axis case. Higher beam energies imply longer
distances, and therefore larger detectors. The WBB technique requires Mton
class detectors with extremely good energy resolution and optimal neutral-
current background rejection. See Refs. [12] for the physics opportunities
with a WBB at a Deep Underground Science and Engineering Laboratory (DUSEL).
The authors of [13] have studied carefully the two possible techniques,
finding (for the same exposure) the WBB option better for the mass hierarchy
extraction, and the NO$\nu$A off-axis experiment better for CP–violation
searches.
## 5 The race for the hierarchy
Typically, the proposed near term LBL neutrino oscillation experiments
(superbeams) have a single far detector and plan to run with the beam in two
different modes, muon neutrinos and muon antineutrinos. Suppose we compute the
oscillation probabilities $P(\nu_{\mu}\to\nu_{e})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ for a given set of oscillation parameters
and the CP–phase $\delta$ is varied between $0$ and $2\pi$: we obtain a closed
CP trajectory (an ellipse) in the bi–probability space of neutrino and
antineutrino conversion [14]. Matter effects are responsible for the departure
of the center of the ellipses from the diagonal line in the bi–probability
plane for normal and inverted hierarchy. In Fig. 1, we have illustrated the
case for $E=2.0$ GeV and $L=810$ km, which roughly correspond to those of the
NO$\nu$A experiment. The distance between the center of the ellipse for the
normal hierarchy (lower blue) and that for the inverted hierarchy (upper red)
is governed by the size of the matter effects. Notice that the ellipses
overlap for a significant fraction of values of the CP–phase $\delta$ for
every allowed value of $\sin^{2}2\theta_{13}$. This makes the determination of
sign$(\mbox{$\Delta m_{23}^{2}$})$ extremely difficult, i. e., the
sign$(\mbox{$\Delta m_{23}^{2}$})$-extraction is not free of degeneracies and
it is highly dependent on the value of $\delta$.
| |
---|---|---
Figure 1: Left panel: bi–probability plot for $P(\nu_{\mu}\to\nu_{e})$ versus
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ at a baseline of 810 km and an energy of
2.0 GeV for the normal (blue) and the inverted (red) hierarchies. The smaller,
lower (larger, upper) ellipses are for $\sin^{2}2\theta_{13}=0.02$
($~{}0.10$). Medium panel: bi–probability plot for $P(\nu_{\mu}\to\nu_{e})$
versus $P(\nu_{\mu}\to\nu_{e})$ with baselines 295 km and 810 km. The mean
neutrino energies are chosen such that the $\langle E\rangle/L$ for the two
experiments are approximately identical. The right panel is the bi–probability
plot for $P(\nu_{\mu}\to\nu_{e})$ versus $P(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$
for the normal (blue) and the inverted (red) hierarchies. The baseline and
mean neutrino energy for both experiments are 810 km and $\sim$ 2 GeV.
Following the line of thought developed by Minakata, Nunokawa and Parke [15],
we exploited [16] the neutrino data only from two experiments at different
distances and at different off-axis locations, such that the $\langle
E\rangle/L$ is the same for the two experiments (see also Refs. [17, 18, 19,
20]). In the case of bi–probability plots for neutrino–neutrino modes at
different distances (which will be referred as near (N) and far (F)), the
CP–trajectory is also elliptical. In Fig. 1 (medium panel) we present the
bi–probability plot for the mean energies and baselines of the $\nu_{e}$
appearance experiments T2K and NO$\nu$A. The overlap of the two ellipses,
which implies the presence of a degeneracy of the type of hierarchy with other
parameters, is determined by their width and the difference in the slopes. The
ratio of the slopes, at first order in the matter parameter, and assuming that
the $\langle E\rangle/L$ of the near and far experiments is the same, reads
$\displaystyle\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+2\left(A_{\rm N}L_{\rm
N}-A_{\rm F}L_{\rm
F}\right)\left(\frac{1}{\Delta_{13}L/2}-\frac{1}{\tan(\Delta_{13}L/2)}\right)~{},$
(3)
where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the
ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies,
$A_{\rm F}$ and $A_{\rm N}$ are the matter parameters, and $L_{\rm F}$ and
$L_{\rm N}$ are the baselines for the two experiments. The separation between
the center of the ellipses for the two hierarchies increases as the difference
in the matter parameter times the path length, ($AL$), for the two experiments
increases. However the width of the ellipses is crucial: even when the
separation between the central axes of the two regions is substantial, if the
ellipses for the normal and inverted hierarchy overlap, the hierarchy cannot
be resolved for values of the CP–phase, $\delta$, for which there is overlap.
The width of the ellipses is determined by the difference in the $\langle
E\rangle/L$ of the two experiments.
In the case of bi–probability plots for the $\nu_{\mu}\to\nu_{e}$ and its CPT
conjugated channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ at the same energy
divided by baseline,$\langle E\rangle/L$, the CPT–trajectory collapses to a
line (see Fig. 1, right panel). As for the neutrino-neutrino case, assuming
that the $\langle E\rangle/L$ of the CPT conjugated channels is the same (to
minimize the ellipses width), at first order, the ratio of the slopes reads
[15]
$\displaystyle\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+2\left(AL+A_{\rm
CPT}L_{\rm
CPT}\right)\left(\frac{1}{\Delta_{13}L/2}-\frac{1}{\tan(\Delta_{13}L/2)}\right)~{},$
(4)
where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the
ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies, $A$
and $A_{\rm CPT}$ are the matter parameters and $L$ and $L_{\rm CPT}$ are the
baselines for the two experiments which exploit the $\nu_{\mu}\to\nu_{e}$ and
its CPT conjugated channel ($\bar{\nu}_{e}\to\bar{\nu}_{\mu}$). Notice that,
compared to the neutrino–neutrino case given by Eq. (3), the separation
between the center of the ellipses for the two hierarchies increases as the
sum of the matter parameter times the baseline, $AL$, for both experiments
does. Here the ratio of the slopes is enhanced by matter effects for both
$\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel
$\bar{\nu}_{e}\to\bar{\nu}_{\mu}$. Figure 1 (right panel) shows the
bi–probability curves for the combination of these two flavor transitions,
assuming that the two experiments are performed at the same mean energy and
baseline. If the $\langle E\rangle/L$ of both experiments is the same, the
ellipses will become lines with a negligible width. The separation of the
lines for the normal and inverted hierarchy grows as the matter effects for
both experiments increase. Consequently, the comparison of CPT conjugated
channels is more sensitive to the neutrino mass hierarchy than the
neutrino–neutrino one, see Ref. [21].
## 6 Higher order corrections: $\beta$-beams and neutrino factories
Precision lepton flavor physics requires powerful machines and extremely pure
neutrino beams. Future LBL experiments which exploits pure $\nu_{e}$
($\bar{\nu}_{e}$) neutrino beams are $\beta$-beams and neutrino factories. A
$\beta$-beam experiment [22] exploit ions which are accelerated to high
Lorentz factors, stored and then $\beta$-decay, producing a collimated
electron neutrino beam. The typical neutrino energies are in the $200$ MeV-GeV
range, requiring detectors with hundred-of-MeV thresholds and good energy
resolution. The only requirement is good muon identification in order to
detect the appearance of muon neutrinos (or muon antineutrinos) from the
initial electron neutrino (or antineutrino) beam. No magnetisation is required
and therefore several detectors technologies (water Cherenkov, totally active
scintillator (TASD), liquid argon and non-magnetised iron calorimeter) could
be used, depending on the peak energy.
The initial $\beta$-beam setup [22] considers a _low_ -$\gamma$ machine which
accelerates ${}^{6}He$ ($\bar{\nu}_{e}$ emitter) and ${}^{18}Ne$ ($\nu_{e}$
emitter) up to $\gamma\sim 100$. In order to tune the $E/L$ at the vacuum
oscillation maximum, a large water Cherenkov detector is located at a distance
$\mathcal{O}(100)$ km. The first exciting option to improve this initial
$\beta$-beam scenario was presented in Ref. [23], where the possibility of
using higher $\gamma$ factors was first suggested. The second exciting option,
see Ref. [24], proposes to accelerate alternative ions, as ${}^{8}Li$
($\bar{\nu}_{e}$ emitter) and ${}^{8}B$ ($\nu_{e}$ emitter), with higher
Q-values. A plethora of setups have been proposed in the literature (see Ref.
[9] for a complete list of references).
A neutrino factory (NF) [25, 26] consists, essentially, of a muon storage ring
with long straight sections along which the muons decay. These muons provide
high intensity and extremely pure neutrino beams. Hence, the NF provides
$\nu_{e}$ and $\overline{\nu}_{e}$ beams in addition to $\nu_{\mu}$ and
$\overline{\nu}_{\mu}$ beams, with minimal systematic uncertainties on the
neutrino flux and spectrum. One of the most important advantages of the NF,
compared to the $\beta$-beam, is its ability to measure with high precision
the atmospheric mixing parameters via the disappearance channels
($\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})\to\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$)
The NF exploits the golden signature of the _wrong-sign muon_ events [2]. What
is a “wrong sign muon” event? Suppose, for example, that positive charged
muons have been stored in the ring. These muons will decay as $\mu^{+}\to
e^{+}+\mbox{$\nu_{e}$}+\mbox{$\overline{\nu}_{\mu}$}$. The muon antineutrinos
will interact in the detector to produce positive muons. Then, any _wrong-sign
muons_ (negatively-charged muons) detected are an unambiguous proof of
electron neutrino oscillations in the $\mbox{$\nu_{e}$}\to\mbox{$\nu_{\mu}$}$
channel. A magnetized detector with good muon charge identification is
mandatory.
In Ref. [9] a complete study of possible near and far future LBL facilities
has been performed, including superbeams, $\beta$-beams and NF. The optimal
setup is found to be a $20$ GeV NF delivering $5\cdot 10^{20}$ muon decays per
year, baseline and polarity. The running time assumed is 5 years per polarity.
Two iron calorimeter detectors of $50$ kton are placed at two different
baselines, at $\mathcal{O}(4000)$ km and at $\mathcal{O}(7000)$ km (the so-
called _magic baseline_ [27]). The oscillated data from detector at the
largest baseline helps enormously in resolving the mass hierarchy degeneracy.
Once that the sign($\Delta m_{23}^{2}$) degeneracy is resolved, leptonic
CP–violation can be measured unambiguously using the data from the detector
located at $\mathcal{O}(4000)$ km.
### 6.1 The low energy neutrino factory
The optimal $20$ GeV plus two detectors NF setup described in the previous
section outperforms any other planned scenario so far, as we will shortly
show. However, such an aggressive setup could be extremely challenging to
construct (a $\mathcal{O}(7000)$ km baseline would require the construction of
a decay tunnel with an inclination of $\sim 30^{\circ}$ 111I would like to
thank C. Quigg for making this observation). More important, such an
aggressive scenario might not be needed if
$\sin^{2}2\theta_{13}>10^{-4}-10^{-3}$. The reason for that is simple: if the
mixing angle $\theta_{13}$ is not so small, there is no need to go to very
long baselines to amplify it. Shorter baselines require lower energies. Lower
energy Neutrino Factories (LENF), which store muons with energies $<10$ GeV,
require a detector technology that can detect lower energy muons. In previous
studies [28, 29], we have considered a LENF with an energy of about $4$ GeV
providing $5\cdot 10^{20}$ muon decays per year. The detector exploited was a
magnetized TASD [29] of $20$ kton, with a muon energy detection threshold of
$500$ MeV, located at a distance of $1480$ km (Fermilab to Henderson mine).
The results are similar for a baseline of $1280$ km (Fermilab to Homestake).
The intrinsic background fraction is $10^{-3}$. Here, we improve the LENF
setup in two ways. First, the detector energy resolution would be $dE/E\sim
10\%$ 222Based on NO$\nu$A results, we expect the TASD dE/E to be better than
$6\%$ at 2 GeV.. Second, and more interesting, since it seems possible to
measure in a magnetized TASD the electron charge [30], apart of exploiting the
$\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})\to\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$
channels, their _T_ -conjugate channels
$\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})\to\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})$
channels are also added to the analysis. These extra _T_ -conjugate channels
will help enormously in resolving degeneracies. We assume here that the
electron charge identification is constant in energy and equal to $50\%$ (a
detailed analysis will be presented elsewhere [30]).
Figure 2 shows the 3-$\sigma$ $\theta_{13}$ discovery potential and the
sensitivities to the mass hierarchy to CP–violation expected from data at a
future LENF with the characteristics quoted above (the exposure is $10^{23}$
kton-decays). As a comparison, we show as well the expected sensitivities
exploiting data from the future LBL facilities presented in Ref. [9]. Notice
that the high-$\gamma$ $\beta$-beam [23], labelled as BB$350$, provides a
slightly better sensitivity than the LENF to both CP violation and to
$\theta_{13}$ due to its lower energy and its huge statistics. The $20$ GeV NF
with two baselines ($4000$ km+$7000$ km) is unbeatable, but we might only need
such an aggressive scenario if $\sin^{2}2\theta_{13}<10^{-4}-10^{-3}$. To
conclude, the low energy neutrino factory (LENF) [28, 29] provides a
compromise between super precision machines and feasible setups, and it could
provide an ideal and realistic scenario for precision lepton physics.
| |
---|---|---
Figure 2: The left, medium and right panels show the 3-$\sigma$ $\theta_{13}$
discovery potential, the mass hierarchy sensitivity and the CP-violation
sensitivity, respectively, expected from future data at a LENF. We present as
well the expected sensitivities from future data at the different LBL
experiments presented in Ref. [9]. Figure produced using the GLOBES software
[31].
## References
* [1] M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rept. 460, 1 (2008).
* [2] A. Cervera et al., Nucl. Phys. B 579, 17 (2000) [Erratum-ibid. B 593, 731 (2001)].
* [3] E. K. Akhmedov et al., JHEP 0404, 078 (2004).
* [4] L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); V. D. Barger et al., Phys. Rev. D 22, 2718 (1980); S. P. Mikheev and A. Y. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985); S. J. Parke, Phys. Rev. Lett. 57, 1275 (1986).
* [5] J. Burguet-Castell et al., Nucl. Phys. B 608, 301 (2001).
* [6] H. Minakata and H. Nunokawa, JHEP 0110, 001 (2001).
* [7] V. Barger, D. Marfatia and K. Whisnant, Phys. Rev. D 65, 073023 (2002).
* [8] G. L. Fogli and E. Lisi, Phys. Rev. D 54, 3667 (1996).
* [9] A. Bandyopadhyay et al. [ISS Physics Working Group], arXiv:0710.4947 [hep-ph].
* [10] Y. Hayato et al., Letter of Intent, available at _http://neutrino.kek.jp/jhfnu/_
* [11] D. S. Ayres et al. [NOvA Collaboration], hep-ex/0503053.
* [12] V. Barger et al., Phys. Rev. D 74, 073004 (2006); V. Barger et al., arXiv:0705.4396 [hep-ph].
* [13] V. Barger et al., Phys. Rev. D 76, 031301 (2007).
* [14] H. Minakata and H. Nunokawa, JHEP 0110, 001 (2001).
* [15] H. Minakata, H. Nunokawa and S. J. Parke, Phys. Rev. D 68, 013010 (2003).
* [16] O. Mena, H. Nunokawa and S. J. Parke, Phys. Rev. D 75, 033002 (2007)
* [17] P. Huber, M. Lindner and W. Winter, Nucl. Phys. B 654, 3 (2003).
* [18] V. Barger, D. Marfatia and K. Whisnant, Phys. Lett. B 560, 75 (2003).
* [19] O. Mena Requejo, S. Palomares-Ruiz and S. Pascoli, Phys. Rev. D 72, 053002 (2005); O. Mena, S. Palomares-Ruiz and S. Pascoli, Phys. Rev. D 73, 073007 (2006).
* [20] M. Ishitsuka et al., Phys. Rev. D 72, 033003 (2005); K. Hagiwara, N. Okamura and K. i. Senda, Phys. Lett. B 637, 266 (2006).
* [21] A. Jansson et al., arXiv:0711.1075 [hep-ph].
* [22] P. Zucchelli, Phys. Lett. B 532, 166 (2002).
* [23] J. Burguet-Castell et al., Nucl. Phys. B 695, 217 (2004).
* [24] C. Rubbia et al., Nucl. Instrum. Meth. A 568, 475 (2006).
* [25] S. Geer, Phys. Rev. D 57, 6989 (1998) [Erratum-ibid. D 59, 039903 (1999)].
* [26] A. De Rujula, M. B. Gavela and P. Hernandez, Nucl. Phys. B 547, 21 (1999).
* [27] P. Huber and W. Winter, Phys. Rev. D 68, 037301 (2003).
* [28] S. Geer, O. Mena and S. Pascoli, Phys. Rev. D 75, 093001 (2007).
* [29] A. D. Bross et al, Phys. Rev. D 77, 093012 (2008).
* [30] A. D .Bross, in these _proceedings_ ; A. D .Bross et al, in preparation.
* [31] P. Huber, M. Lindner and W. Winter, Comput. Phys. Commun. 167, 195 (2005).
|
arxiv-papers
| 2008-09-28T08:36:38
|
2024-09-04T02:48:58.012006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Olga Mena",
"submitter": "Olga Mena Requejo",
"url": "https://arxiv.org/abs/0809.4829"
}
|
0809.4957
|
# Probability density functions of work and heat near the stochastic resonance
of a colloidal particle
Alberto Imparato∗, Pierre Jop+, Artyom Petrosyan +& Sergio Ciliberto + ∗
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C,
Denmark
+Université de Lyon, Laboratoire de Physique de l’École Normale Supérieure de
Lyon, CNRS UMR 5276, , 46 allée d’Italie, 69364 Lyon cedex 7, France.
###### Abstract
We study experimentally and theoretically the probability density functions of
the injected and dissipated energy in a system of a colloidal particle trapped
in a double well potential periodically modulated by an external perturbation.
The work done by the external force and the dissipated energy are measured
close to the stochastic resonance where the injected power is maximum. We show
a good agreement between the probability density functions exactly computed
from a Langevin dynamics and the measured ones. The probability density
function of the work done on the particle satisfies the fluctuation theorem.
###### pacs:
82.70.Dd
## I Introduction
The study of fluctuations of the injected and dissipated power in a system
driven out of equilibrium by an external force is nowadays a widely studied
problem which is not yet completely understood. This is a very important and
general issue within the context of Fluctuation Theorems (FT) which constitute
extremely useful relations for characterizing the probabilities of observing
entropy production or consumption in out of equilibrium systems. These
relations were first observed in the simulations of a sheared fluidevans and
later proved both for chaotic dynamical systems gallavotti95 and for
stochastic dynamicskurchan98 . These works lead to different formulations
which find powerful applications for measuring free-energy difference in
biology (see e.g. ritort06 for a review). The hypothesis and the extensions
of fluctuation theoremsCohen have been tested in various experimental systems
such as colloidal particlesblickle06 ; wang02 ; al_pre07 , mechanical
oscillatorsjoubaud07 , electric circuitsgarnier and optically driven single
two-level systemsschuler05 . The effect of anharmonic potential on the motion
of a colloidal particle has been tested by Blickle et al blickle06 ; seifert07
; schuler05 . In a recent experiment jop it has been shown that FT holds for
a colloidal particle confined in a double well potential and driven out of
equilibrium near the stochastic resonance (see next section). The purpose of
this article is to compare the probability density function (PDF) for work and
heat measured in this experiment with those analytically computed from a non-
linear Langevin dynamics. We also discuss the difference between the PDF
obtained from a phase average of the driving and those obtained from a fixed
phase only. Such a precise comparison between theory and experiment in a
double well potential driven out of equilibrium has never been done before.
There is only a numerical study, which has explored the distributions of the
dissipated heat and of the work in a Langevin dynamics near the stochastic
resonance saikia07 ; sahoo07 . The paper is organized as follow. The
properties of the stochastic resonance are recalled in the next section. The
experimental set-up is described in section 3. The analytical PDF are derived
in section 4. The comparison between the theoretical predictions and the
experimental measurements is done in section 5. Finally we conclude in section
6.
## II Stochastic resonance
A colloidal particle, confined in a double well potential, hops between the
two wells at a rate $r_{k}$, named the Kramers’ rate, which is determined by
the height $\delta U$ of the energy barrier between the two wells,
specifically $r_{k}=\tau_{0}^{-1}\exp({-{\delta U\over k_{B}T}})$, where
$\tau_{0}$ is a characteristic time, $k_{B}$ the Boltzmann constant and $T$
the heat bath temperature libchaber . When the double well potential $U$ is
modulated by an external periodic perturbation whose frequency is close to
$r_{k}$ the system presents the stochastic resonance phenomenonBenzi , i.e.
the hops of the particle between the two wells synchronize with the external
forcing. The stochastic resonance has been widely studied in many different
systems and it has been shown to be a bona fide resonance looking at the
resident timeBenzi ; gammaitoni95 , the Fourier transform of the signal for
different noise intensitybabic04 . Numerically, the stochastic resonance has
been characterized by computing the injected work done by the external agent
as a function of noise and frequency iwai01 ; dan05 . In a recent experiment
jop some of us have studied experimentally the Steady State Fluctuation
Theorem (SSFT) in a system composed by a Brownian particle trapped in a double
well potential periodically modulated by an external driving force. We have
measured the energy injected into the system by the sinusoidal perturbation
and we have analyzed the distributions of work and heat fluctuations. We find
that although the dynamics of the system is strongly non-linear the SSFT holds
for the work integrated on time intervals which are only a few periods of the
driving force. In this paper we will compare this measured PDF with those
derived analytically from a Langevin dynamics in which the exact potential of
the experiment has been used.
## III Experimental Set-up
The experimental setup is composed by a custom built inverted optical tweezers
made of an oil-immersion objective (63$\times$, N.A.=1.3) which focuses a
laser beam (wavelength $\lambda=1064$ nm) to the diffraction limit for
trapping glass beads ($2~{}\mu$m in diameter). The silica beads are dispersed
in bidistilled water in very small concentration. The suspension is introduced
in the sample chamber of dimensions $0.25\times 10\times 10$ mm3, then a
single bead is trapped and moved away from others.
The position of the bead is tracked using a fast-camera with the resolution of
108 nm/pixels which gives after treatment the position of the bead with an
accuracy better than 20 nm. The trajectories of the bead are sampled at 50 Hz.
The position of the trap can be easily displaced on the focal plane of the
objective by deflecting the laser beam using an acousto-optic deflector (AOD).
To construct the double well potential the laser is focused alternatively at
two different positions at a rate of 5 kHz. The residence times $\tau_{i}$
(with $i=1,2$) of the laser in each of the two positions determine the mean
trapping strength felt by the trapped particle. Indeed if
$\tau_{1}=\tau_{2}=100\mu s$ the typical diffusion length of the bead during
this period is only 5 nm. As a consequence the bead feels an average double-
well potential:
$U_{0}(x)=ax^{4}-bx^{2}-dx\ ,$ (1)
where $a$, $b$ and $d$ are determined by the laser intensity and by the
distance of the two focal points. In our experiment the distance between the
two spots is $1.45~{}\mu$m, which produces a trap whose minima are at
$x_{min}=\pm$0.45 $\mu$m. The total intensity of the laser is $58$ mW on the
focal plane which corresponds to an inter-well barrier energy $\delta
U_{0}=1.8~{}k_{B}T$. Starting from the static symmetric double-trap,
($\tau_{1}=\tau_{2}$) we modulate the depth of the wells at low frequency by
modulating the residence times ($\tau_{i}$) during which the spot remains in
each position. We keep the total intensity of the laser constant in order to
produce a more stable potential. The modulation of the average intensity is
harmonic at frequency $f$ and its amplitude
$(\tau_{2}-\tau_{1})/(\tau_{2}+\tau_{1})$, is $0.7~{}\%$ of the average
intensity in the static symmetric case. Thus the potential felt by the bead
has the following profile in the axial direction:
$U(x,t)=U_{0}(x)+U_{p}(x,t)=U_{0}+c\ x\ \sin(2\pi ft),$ (2)
with $ax_{min}^{4}=1.8\ k_{B}T$, $bx_{min}^{2}=3.6\ k_{B}T$, $d|x_{min}|=0.44\
k_{B}T$ and $c|x_{min}|=0.81~{}k_{B}T$. The amplitude of the time dependent
perturbation is synchronously acquired with the bead trajectory. The
parameters given here are average parameters since the coefficients $a$, $b$
and $c$ ,obtained from fitted steady distributions at given phases, vary with
the phase ($\delta a/a\approx 10\%$, $\delta b/b\approx\delta c/c\approx
5\%$).
An example of the measured potential at $t=\frac{1}{4f}$ and at
$t=\frac{3}{4f}$ is shown on the Fig. 1a). The figure is obtained by measuring
the two steady state probability distribution function $P(x)$ of $x$,
(corresponding to $U_{0}+c$ and $U_{o}-c$ respectively) and by taking
$U(x)=-\ln\left[P(x)\right]$.
Figure 1: a) The perturbed potential at $t=\frac{1}{4f}$ and half a forcing
period later. b) Example of trajectory of the glass bead and the corresponding
perturbation at $f=0.1$ Hz.
### III.1 The equation of motion.
The $x$ position of the particle can be described by a Langevin equation:
$\gamma\dot{x}=-\frac{\partial U}{\partial x}+\xi,$ (3)
with $\gamma=1.61\ 10^{-8}$ N s m-1 the friction coefficient and $\xi$ the
stochastic force. The natural Kramers’ rate ($c=0$) for the particle is
$r_{k}=0.3$ Hz at $T=300$ K. When $c\neq 0$ the particle can experience a
stochastic resonance when the forcing frequency is close to the Kramers’ rate.
An example of the sinusoidal force with the corresponding position are shown
on the figure 1b).
### III.2 The work and the heat.
Since the synchronization is not perfect, sometimes the particle receives
energy from the perturbation, sometimes the bead moves against the
perturbation leading to a negative work on the system.
In the following, all energies are normalized by $k_{B}T$. From the
trajectories, we compute the stochastic $W_{s}$ and the classical $W_{cl}$
works done by the perturbation on the system and the heat $Q$ exchanged with
the bath. These three quantities are defined by the following equations as in
ref.sekimoto96 :
$\displaystyle W_{s}(t_{0},t_{f})=\int^{t_{0}+t_{f}}_{t_{0}}{dt\frac{\partial
U(x,t)}{\partial t}}$ $\displaystyle
W_{cl}(t_{0},t_{f})=-\int^{t_{0}+t_{f}}_{t_{0}}{dt\dot{x}\frac{\partial
U_{p}(x,t)}{\partial x}}$ (4) $\displaystyle
Q(t_{0},t_{f})=-\int^{t_{0}+t_{f}}_{t_{0}}{dx\frac{\partial U(x,t)}{\partial
x}}$
where in this case $t_{f}={n\over f}$ is a multiple of the forcing period. We
use both $W_{s}$ and $W_{cl}$ because they give complementary information on
the fluctuations of the energy injected by the external perturbation into the
system (see ref. Taniguchi and reference therein for a discussion on this
point). For example, as discussed in ref.jop $W_{s}/T$ is the total entropy
production rate in this specific case seifert07b . The heat and the work,
defined in eq.4, are related through the first principle of thermodynamics:
$Q=-\Delta U+W_{s}$, where $\Delta
U=U(x(t_{f}+t_{0}),t_{0}+t_{f})-U(x(t_{0}),t_{0})$, whereas the two works are
related by a boundary term $W_{cl}=-\Delta U_{p}+W_{s}$, where $\Delta
U_{p}=U_{p}(x(t_{f}+t_{0}),t_{f}+t_{0})-U_{p}(x(t_{0}),t_{0})$. Since the
characteristic time evolution of the perturbation is small compared to the
fluctuation of position and due to the harmonic form of the perturbation, the
integrals are computed as follows:
$\displaystyle W_{s}(t_{0},t_{f})$ $\displaystyle=$ $\displaystyle\omega\ c\
\delta t\ \sum_{i=1}^{t_{f}/\delta t}x(i)\cos(\omega(t_{0}+t_{i}))$
$\displaystyle W_{cl}(t_{0},t_{f})$ $\displaystyle=$ $\displaystyle-\Delta
U_{p}+W_{s}$ (5) $\displaystyle Q(t_{0},t_{f})$ $\displaystyle=$
$\displaystyle-\Delta U+W_{s}$
where $\delta t$ is the sampling time. We checked that the direct computing of
integrals of $Q$ and $W_{cl}$ gives the same results. It is important to
stress that $t_{0}$ can either take any value (as it has been done in ref.jop
) or be a multiple of $1/f$: the fluctuations in the two cases exhibit
different PDFs, as we will see in the next sections. To compute the works and
heat from experimental data for a given duration $t_{f}$, we thus divide a
single trajectory into different segments starting either with a fixed phase,
or with different phases, before averaging the results over the whole
trajectory, and then over different runs. In ref.jop the average work
received over one period has been measured for different frequencies
($t_{f}={1\over f}$ in eq. 4). Each trajectory is here recorded during 3200 s
in different consecutive runs, which corresponds to 160 up to 7500 forcing
periods, for the range of frequencies explored. It has been found that the
maximum injected energy is around the frequency $f\approx 0.1$ Hz, which is
comparable with half of the Kramers’ rate of the fixed potential $r_{K}=0.3$
Hz. This maximum of transferred energy shows that the stochastic resonance for
a Brownian particle is a bona fide resonance, as it was previously shown in
experiments using resident time distributions gammaitoni95 ; schmitt06 or
directly in simulations iwai01 ; dan05 . It is worth noting that the average
values of work in the case of a periodic forcing do not dependent on their
definitions: only the boundary terms, which vanish in average with time, are
different jop . As the difference between $W_{cl}$ and $W_{s}$ has been
discussed in ref.jop we will focus here only on $W_{s}$ that, in order to
simplify the notation, will be indicated by $W$.
## IV Equations for the W and Q PDFs
In this section we discuss the equation governing the time evolution of the
work and heat PDFs. For a stochastic process described by eq. (3) the Fokker-
Planck equation reads
$\partial_{t}p(x,t)=\Gamma\frac{\partial}{\partial
x}\left[{U^{\prime}(x,t)p}\right]+k_{B}T\Gamma\frac{\partial^{2}p}{\partial
x^{2}},$ (6)
where $p(x,t)$ is the PDF associated to the coordinate $x$, and
$\Gamma=1/\gamma$. Here and in the following, the prime denotes derivative
with respect to $x$.
Let us consider the joint probability distribution function of the position
and of the stochastic work $\phi(x,W,t)$: in ref. al_w it has been shown that
the time evolution of such a function is governed by the partial differential
equation
$\partial_{t}\phi(x,W,t)=\Gamma\frac{\partial}{\partial
x}\left[{U^{\prime}(x,t)\phi}\right]+k_{B}T\Gamma\frac{\partial^{2}\phi}{\partial
x^{2}}-\frac{\partial U}{\partial t}\frac{\partial\phi}{\partial W},$ (7)
with the starting condition $\phi(x,W,t_{0})=p(x,t_{0})\delta(W)$. The
unconstrained probability distribution of the work is given by
$\Phi(W,t)=\int dx\phi(x,W,t).$ (8)
By introducing the Fourier transform
$\psi(x,\lambda,t)=\int dW\mathrm{e}^{-\lambda W}\phi(x,W,t),$ (9)
eq. (7) becomes
$\partial_{t}\psi(x,\lambda,t)=\Gamma\frac{\partial}{\partial
x}\left[{U^{\prime}(x,t)\psi}\right]+k_{B}T\Gamma\frac{\partial^{2}\psi}{\partial
x^{2}}-\lambda\frac{\partial U}{\partial t}\psi,$ (10)
with the starting condition $\psi(x,\lambda,t_{0})=p(x,t_{0})$. Note that, by
using the Fourier transform definition eq. (9), the wavenumber $\lambda$
associated to $W$ is a purely imaginary number, $\lambda=i|\lambda|$.
Let us now consider the joint probability distribution $\varphi(x,Q,t)$ of the
position $x$ and the heat $Q$ exchanged by the brownian particle whose motion
is described by eq. (3). The Fokker-Planck-like equation, governing the time
evolution of such a function, reads al_pre07 ; SpSe
$\displaystyle\partial_{t}\varphi(x,Q,t)$ $\displaystyle=$
$\displaystyle\partial_{x}\left({\Gamma
U^{\prime}\varphi}\right)-\partial_{Q}(\Gamma U^{\prime
2}\varphi)-\partial_{x}\left[{(\Gamma
U^{\prime}k_{B}T)\partial_{Q}\varphi}\right]-\partial_{Q}\left[{(\Gamma
U^{\prime}k_{B}T)\partial_{x}\varphi}\right]$ (11) $\displaystyle+\Gamma
k_{B}T\frac{\partial^{2}{\varphi}}{\partial x^{2}}+\Gamma k_{B}TU^{\prime
2}\frac{\partial^{2}{\varphi}}{\partial Q^{2}},$
with the starting condition $\varphi(x,Q,t_{0})=p(x,t_{0})\delta(Q)$. Note
that we use here the opposite sign convention for $Q$, eq. (5), with respect
to that adopted in ref. al_pre07 .
The unconstrained probability distribution functions of the heat reads
$\mathit{\Phi}(Q,t)=\int{\mathrm{d}}x\,\varphi(x,Q,t).$ (12)
Let $\chi(x,\lambda,t)$ be the Fourier transform of $\varphi(x,Q,t)$, as given
by
$\chi(x,\lambda,t)=\int{\mathrm{d}}Q\,\mathrm{e}^{-\lambda Q}\varphi(x,Q,t),$
(13)
eq. (11) becomes
$\partial_{t}\chi(x,\lambda,t)=\Gamma k_{B}T\frac{\partial^{2}{\chi}}{\partial
x^{2}}+\partial_{x}\left({\Gamma U^{\prime}\chi}\right)-\lambda\Gamma
U^{\prime 2}\chi-\lambda\partial_{x}\left[{(\Gamma
U^{\prime}k_{B}T)\chi}\right]-\lambda(\Gamma
U^{\prime}k_{B}T)\partial_{x}\chi+\lambda^{2}\Gamma k_{B}TU^{\prime 2}\chi,$
(14)
with the starting condition $\chi(x,\lambda,t_{0})=p(x,t_{0})$. In ref.
al_pre07 , it has been shown that by introducing the function $g(x,\lambda,t)$
defined by
$\chi(x,\lambda,t)=g(x,\lambda,t)\exp\left[{-\frac{\beta-2\lambda}{2}U(x,t)}\right],$
(15)
equation (14) simplifies, as one gets rid of the first order derivatives with
respect to $x$, and it becomes
$\partial_{t}g=\Gamma k_{B}T\frac{\partial^{2}{g}}{\partial
x^{2}}-\Gamma\beta\frac{U^{\prime
2}}{4}g+\frac{\Gamma}{2}U^{\prime\prime}g+\frac{\beta-2\lambda}{2}g\,\partial_{t}U,$
(16)
with the starting condition as given by
$g(x,\lambda,t_{0})=p(x,t_{0})\exp[(\beta-2\lambda)U(x,t_{0})/2]$.
We now show that a similar simplification can be obtained for eq. (10): let
$h(x,\lambda,t)$ be defined as
$\psi(x,\lambda,t)=h(x,\lambda,t)\exp[-\beta U(x,t)/2],$ (17)
substituting $h(x,\lambda,t)$ into eq. (10), we obtain the following equation
for $h(x,\lambda,t)$:
$\partial_{t}h=\Gamma k_{B}T\frac{\partial^{2}{h}}{\partial
x^{2}}-\Gamma\beta\frac{U^{\prime
2}}{4}h+\frac{\Gamma}{2}U^{\prime\prime}h+\frac{\beta-2\lambda}{2}h\,\partial_{t}U.$
(18)
The last equation is identical to (16), however its starting condition reads
$h(x,\lambda,t_{0})=p(x,t_{0})\exp[\beta U(x,t_{0})/2]$, which is different
from the starting condition of eq. (16).
As $\lambda$ is an imaginary number, one can split eq. (16) and eq. (18), in a
set of two equations for the real and imaginary part. For example, eq. (18)
becomes
$\displaystyle\partial_{t}h_{\mathrm{R}}$ $\displaystyle=$
$\displaystyle\Gamma k_{B}T\frac{\partial^{2}{h_{\mathrm{R}}}}{\partial
x^{2}}-\Gamma\beta\frac{U^{\prime
2}}{4}h_{\mathrm{R}}+\frac{\Gamma}{2}U^{\prime\prime}h_{\mathrm{R}}+\left({\frac{\beta}{2}h_{\mathrm{R}}+|\lambda|h_{\mathrm{I}}}\right)\partial_{t}U,$
(19) $\displaystyle\partial_{t}h_{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\Gamma k_{B}T\frac{\partial^{2}{h_{\mathrm{I}}}}{\partial
x^{2}}-\Gamma\beta\frac{U^{\prime
2}}{4}h_{\mathrm{I}}+\frac{\Gamma}{2}U^{\prime\prime}h_{\mathrm{I}}+\left({\frac{\beta}{2}h_{\mathrm{I}}-|\lambda|h_{\mathrm{R}}}\right)\partial_{t}U,$
(20)
where $h_{\mathrm{R}}$ and $h_{\mathrm{I}}$ are the real and the imaginary
part of $h$, respectively.
Equations (19) and (20), and the analogous equations for $g_{\mathrm{R}}$ and
$g_{\mathrm{I}}$, can be solved numerically for any value of $|\lambda|$, and
for any choice of the initial condition $p(x,t_{0})$ using, e.g., MATHEMATICA
mat_book . The functions $\psi(x,\lambda,t)$ and $\chi(x,\lambda,t)$, can be
thus obtained by using equations (17) and (15). The target functions
$\phi(x,W,t)$ and $\varphi(x,Q,t)$ can then be obtained by taking the Fourier
inverse transform, i.e. by inverting eqs. (9) and (13), respectively. Also in
this case the computation can be performed numerically.
## V Comparison with experiments
In this section, we compare the results for the work and heat PDFs, as
obtained by solving the differential equations introduced in the previous
section, with the experimental outcomes. We take the external driving
frequency to be equal to $f=0.25$ Hz, to have a good statistic, by allowing
the observation of the system over a sufficient number of periods. Such a
value is close to the natural Kramers’ rate, which ensures that the system is
in the stochastic resonance regime.
We first consider the distribution of the work over a single period of the
external potential (2). The unconstrained probability distribution of the work
$\Phi(W,t)$, is obtained as follows. In order to solve eq. (10), we solve
numerically eq. (6) up to the time $t_{0}=50/f\gg 1/f$, so as to ensure that
the solution $p(x,t_{0})$ of eq. (6) represents the steady state distribution
of the position of the particle. Such a solution is used as a starting
condition for eq. (10), which is solved numerically up to time
$t_{1}=t_{0}+1/f$, i.e. along a single period, and for different values of
$\lambda$, so as to obtain $\psi(x,\lambda,t_{1})$. The resulting
unconstrained Fourier transform of the work PDF, defined as
$\Psi(\lambda,t_{1})=\int{\mathrm{d}}x\,\psi(x,\lambda,t_{1})$, is plotted in
fig. 2 .
Figure 2: Unconstrained Fourier transform of the work PDF as a function of
$\lambda$, as obtained by numerical solution of eq. (10), with $t_{0}=50/f$
and $t_{1}=t_{0}+1/f$. Full line $\Psi_{\mathrm{R}}(\lambda,t_{1})$, dashed
line $\Psi_{\mathrm{I}}(\lambda,t_{1})$.
As discussed in the previous section, by inverting eq. (9) and exploiting eq.
(8), we finally obtain $\Phi(W,t_{1})$. In fig. 3, the experimental histogram
of the work done on the particle along a single period $1/f$ is plotted; in
the same figure the unconstrained probability distribution of the work
$\Phi(W,t_{1})$ is also plotted. We find a good agreement between the
experimental distribution of the work and the expected PDF. It is worth noting
first that we start sampling $W$ only few periods $1/f$ after the beginning of
each experiment and second that the experimental results are averaged over
different values of $t_{0}$ in order to improve the statistics (either with a
fixed phase or over different phases).
Figure 3: Diamonds: experimental probability distribution of the work done on
the particle, along a single period, and with a single value of the initial
phase. Full line: PDF of the work as obtained by numerical solution of eqs.
(7)-(10), with $t_{0}=50/f$ and $t_{1}=t_{0}+1/f$.
We now check that the work PDF satisfies the SSFT, which we expect to be true
for any time $t_{1}$ which is an integer multiple of the period $1/f$ as
discussed in seifert07 ; seifert05 . Given the periodicity of the external
driving, the stochastic work here considered corresponds to the total entropy
production, as defined by Seifert in seifert07 ; seifert05 . In figure 4, we
plot the symmetry function for the work PDF, defined as
$S(W,t_{1})=\log\left[{\Phi(W,t_{1})/\Phi(-W,t_{1})}\right],$ (21)
together with the corresponding work PDF, for two values of the final time
$t_{1}$. The fluctuation theorem states that the simmetry function has to
satisfy the equality $S(W,t_{1})=W$, for any value of $W$. Inspection of the
figure indicates that the fluctuation theorem is satisfied even at short times
$t_{1}$, the small deviations for the largest value of $W$ being due to
numerical and experimental difficulties in the sampling of the tails of the
work PDF. More details on SSFT have been already discussed in ref. jop .
Figure 4: Right panel: Symmetry function for the work PDF $S(W,t_{1})$, eq.
(21), for two different values of $t_{1}$. The symbols corresponds to the
experimental data, the lines to the numerical solution of eqs. (7)-(10) with
$t_{0}=50/f$ . Red boxes and red dashed-dotted line: symmetry function
obtained with $t_{1}=t_{0}+2/f$. Blue diamonds and blue dashed line: symmetry
function obtained with $t_{1}=t_{0}+4/f$. The full line corresponds to the
expected behaviour $S(W,t_{1})=W$. Left panel: PDF of the work for
$t_{1}=t_{0}+2/f$ (red dashed-dotted line), and $t_{1}=t_{0}+4/f$ (blue dashed
line).
We now consider the work distribution averaged over several phases. In order
to obtain the probability distribution of the work, we solve eq. (6) up to the
time $t_{0}(k)=(50+k/N)/f$, where $N=20$ and $k=0,\dots N-1$, i.e. we consider
$N$ initial conditions with a phase difference $1/(Nf)$. For each value of $k$
we solve eq. (10), up to time $t_{1}(k)=t_{0}(k)+1/f$, so as to obtain
$\psi(x,\lambda,t_{1}(k)|p(x,t_{0}(k))$, i.e. the Fourier transform of the
work PDF with the starting condition $p(x,t_{0}(k))$, and by antitransforming,
we obtain $\phi(x,W,t_{1}(k)|p(x,t_{0}(k))$. Finally the average work PDF is
obtained as $\overline{\Phi}(W)=1/N\sum_{k}\int
dx\phi(x,\lambda,t_{1}(k)|p(x,t_{0}(k))$. In fig. 5, this distribution is
compared with the experimental outcomes. We find a good agreement between the
expected PDF and the histogram of the work.
Figure 5: Diamonds: experimental probability distribution of the work done on
the particle averaged over different initial phases. Full line, PDF of the
work $\overline{\Phi}(W)$ as obtained by numerical solution of eqs. (7)-(10),
averaged over $N=20$ different phases, see text.
Similarly to what has been done for the work, we now consider the PDF of the
heat $\mathit{\Phi}(Q)$, both with a single value of the initial phase, and
averaged over several initial conditions. Also in this case, we use the
function $p(x,t_{0})$, solution of eq. (6), as a starting condition for the
numerical integration of eq. (14), then by antitransforming the function
$\chi(x,\lambda,t_{1})$, we obtain $\varphi(x,Q,t_{1})$ and finally
$\mathit{\Phi}(Q,t_{1})$, as discussed in the previous section. The results
for the case of a single value of the initial phase is plotted in fig. 6,
where we show a nice agreement with the experimental data.
Figure 6: Diamonds: experimental probability distribution of the heat
exchanged by the particle, along a single period, and with a single value of
the initial phase. Full line, $\mathit{\Phi}(Q,t_{1})$ as obtained by
numerical solution of eqs. (11)-(14), with $t_{0}=50/f$ and $t_{1}=t_{0}+1/f$.
In order to obtain the heat PDF averaged over several initial phases, we solve
numerically eq. (14) in the time intervals $[t_{0}(k),t_{1}(k)]$, where
$t_{0}(k)$, and $t_{1}(k)$ have been defined above. Then by antitransforming
the function $\chi(x,\lambda,t_{1}(k))$, we obtain
$\varphi(x,Q,t_{1}|p(x,t_{0}(k))$, and finally the unconstrained PDF of $Q$
defined as $\mathit{\overline{\Phi}}(Q)=1/N\sum_{k}\int
dx\varphi(x,Q,t_{1}|p(x,t_{0}(k))$. In fig. 7 we compare this expected PDF
with the experimental outcomes, the agreement between the numerical PDF and
the experimental data is good also in this case.
Figure 7: Diamonds: experimental probability distribution of the heat
exchanged by the particle with the bath averaged over different initial
phases. Full line, heat PDF $\mathit{\overline{\Phi}}(Q)$ as obtained by
numerical solution of eqs. (11)-(14), averaged over $N=20$ different phases,
see text.
## VI Conclusions
In conclusion, we have experimentally and theoretically investigated the power
injected in a bistable colloidal system by an external oscillating force.
We have compared the PDFs of the stochastic work and of the heat measured in
the experiment with those obtained theoretically. They exhibit large tails
toward negative values and their shape are non-gaussian close to the resonance
when averaging over a single period.
In refs. al_pre07 , it has been shown that if the external potential is
quadratic, an analytical gaussian solution can be obtained for equation (7) at
any value of $t$ and for eq. (11) in the long time limit. Here we have shown
that the same equations can be solved numerically, also at finite time and for
a more complex potentials, and they lead predictions on the system energy
exchange that can be verified experimentally. We have shown that the Fourier
transform of the heat and work PDFs obey the same differential equations (16)
and (18), and the only difference lies in the initial conditions. This ensures
$\Phi(W)$ and $\mathit{\Phi}(Q)$ to be different at short time, while one has
$\Phi(W)=\mathit{\Phi}(Q)$ in the long time regime, as one would expect, since
$\left<{U}\right>=0$ for long times.
Thus, our results suggest that the set of equations (7) and (11) represents an
useful and general tool to investigate the energy balance of microscopic
systems in arbitrary potentials.
Finally, we find that the work PDF satisfy the fluctuation theorem, even at
short times.
###### Acknowledgements.
AI is grateful to L. Peliti for long and interesting discussions.
## References
* (1) A. Simon, A. Libchaber, Phys. Rev. Lett., 68, 3375 (1992).
* (2) R.Benzi,G. Parisi, A. Sutera,A. Vulpiani, SIAM J. Appl. Math., 43, 565 (1983).
* (3) L. Gammaitoni, F. Marchesoni and S. Santucci. Phys. Rev. Lett. 74 (7), 1052-1055 (1995).
* (4) D. Babic, C. Schmitt, I. Poberaj and C. Bechinger. Europhys. Lett. 67 (2), p. 158 (2004).
* (5) T. Iwai. Physica A 300, pp. 350-358 (2001)
* (6) D. Dan and A. M. Jayannavar. Physica A 345, pp. 404-410 (2005)
* (7) D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993); D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994).
* (8) G. Gallavotti, E. G. D. Cohen. Phys. Rev. Lett. 74, pp. 2694 - 2697 (1995).
* (9) Jorge Kurchan. J. Phys. A: Math. Gen. 31, pp. 3719?3729 (1998). J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 (1999).
* (10) F. Ritort J. Phys.: Condens. Matter 18, R531 (2006).
* (11) R. van Zon and E.G.D. Cohen, Phys. Rev. Lett. 91 (11) 110601 (2003); Phys. Rev. E 67 046102 (2003); Phys. Rev. E 69 056121 (2004). R. van Zon, S. Ciliberto, E.G.D. Cohen, Phys. Rev. Lett. 92 (13) 130601 (2004).
* (12) G.M. Wang, E.M. Sevick, E. Mittag, D. J. Searles and D. J. Evans. Phys. Rev. Lett. 89, 050601 (2002)
* (13) V. Blickle, T. Speck, L. Helden, U. Seifert and C. Bechinger. Phys. Rev. Lett. 96, 070603 (2006).
* (14) F. Douarche, S. Joubaud, N. B. Garnier, A. Petrosyan, and S. Ciliberto Phys. Rev. Lett. 97 (14), 140603 (2006).
* (15) N. Garnier, S. Ciliberto, Phys. Rev. E 71 060101(R) (2005).
* (16) S. Schuler, T. Speck, C. Tietz, J. Wrachtrup and U. Seifert. Phys. Rev. Lett. 94, 180602 (2005).
* (17) T. Speck, V. Blickle, C. Bechinger and U. Seifert, Europhys.Lett, 79, 30002 (2007).
* (18) S. Saikia, R. Roy and A. M. Jayannavar. Phys. Lett. A, 369 pp. 367 371 (2007).
* (19) M. Sahoo, S. Saikia, M. C. Mahato and A. M. Jayannavar in press Physica A (2008).
* (20) K. Sekimoto J. Phys. Soc. Jpn. 66 (5), pp. 1234-1237 (1997).
* (21) C. Schmitt, B. Dybiec, P. Hänggi and C. Bechinger. Europhys. Lett. 74 (6), p. 937 (2006).
* (22) U. Seifert Eur. Phys. J. B, 64 3-4, pp. 423-431 (2008).
* (23) U. Seifert Phys. Rev. Lett. 95, 040602 (2005).
* (24) T. Taniguchi and E. G. D. Cohen J. Stat. Phys. 126 1 (2007)
* (25) T. Mai and A. Dhar. Phys. Rev. E 75, 061101 (2007)
* (26) P. Jop, A. Petrossyan, S. Ciliberto, Europhys. Lett., 81 50005 (2008).
* (27) A. Imparato, L. Peliti, G. Pesce, G. Rusciano, A. Sasso, Phys. Rev. E, 76 050101R (2007).
* (28) T. Speck and U. Seifert, J. Phys. A 38, L581 (2005).
* (29) A. Imparato and L. Peliti, Phys. Rev. E 72, 046114 (2005); A. Imparato and L. Peliti, Europhys. Lett. 70, 740 (2005).
* (30) S. Wolfram, The Mathematica Book (Cambridge University Press, Cambridge, 1999).
|
arxiv-papers
| 2008-09-29T12:41:28
|
2024-09-04T02:48:58.018083
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alberto Imparato, Pierre Jop, Artyom Petrosyan, Sergio Ciliberto",
"submitter": "Alberto Imparato",
"url": "https://arxiv.org/abs/0809.4957"
}
|
0809.4995
|
# Discrete coherent states for $n$ qubits
Carlos Muñoz Departamento de Física, Universidad de Guadalajara, 44420
Guadalajara, Jalisco, Mexico Andrei B. Klimov Departamento de Física,
Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
klimov@cencar.udg.mx Luis L. Sánchez-Soto Departamento de Óptica, Facultad
de Física, Universidad Complutense, 28040 Madrid, Spain
lsanchez@fis.ucm.es Gunnar Björk School of Information and Communication
Technology, Royal Institute of Technology (KTH), Electrum 229, SE-164 40
Kista, Sweden
gbjork@kth.se
(Day Month Year; Day Month Year)
###### Abstract
Discrete coherent states for a system of $n$ qubits are introduced in terms of
eigenstates of the finite Fourier transform. The properties of these states
are pictured in phase space by resorting to the discrete Wigner function.
###### keywords:
Discrete phase space; coherent states; Wigner function.
## 1 Introduction
Discrete quantum systems were studied originally by Weyl[1] and Schwinger,[2]
and later by many authors.[3, 4] However, many concepts that appear sharp for
continuous systems become fuzzy when one tries to apply them to discrete ones.
The reason is that the in the continuum we have only one harmonic oscillator,
while for finite systems, there are a number of candidates for that role, each
one with its own virtues and drawbacks.[5]
Coherent states constitute an archetypical example of the situation: for the
standard harmonic oscillator they are well understood, and sensible
generalizations have been devised to deal with systems with more general
dynamical groups.[6] However, in spite of interesting advances,[7, 8] the
discrete counterparts are still under heated discussion.
Number states are eigenstates of the Fourier transform, but they are not the
only ones: coherent states, with their associated Gaussian wave functions,
also are. In our opinion, this subtle, yet obvious, observation has not been
taken in due consideration in this field. From this perspective, a decisive
step was given by Mehta,[9] who obtained the eigenstates of the finite Fourier
transform. The purpose of this paper is to further explore this path, showing
how we can define physically feasible coherent states for $n$ qubits that are
also Fourier eigenstates.
Since the natural arena of these discrete coherent states is the phase space,
we use the discrete Wigner function to picture the corresponding results. The
problem of generalizing the Wigner function to finite systems has a long
story. Using the notions presented in the comprehensive review Bjork2008, we
construct a Wigner function for these coherent states and discuss some of
their properties.
## 2 Coherent states for $n$ qubits
We consider $n$ identical qubits (i.e., $n$ noninteracting spin 1/2 systems).
We recall that the Dicke states, belonging to the symmetric subspace of the
representation of SU(2)⊗n, are given by
$|n,k\rangle=\sqrt{\frac{k!(n-k)!}{n!}}\sum_{k}P_{k}(|1_{1},1_{2},\ldots,1_{k},0_{k+1},\ldots,0_{n}\rangle)\,,$
(1)
where $\\{P_{k}\\}$ denotes the complete set of all the possible permutations
of the qubits. These states can be expressed in terms of the elements of the
Galois field $\mathrm{GF}(2^{n})$ using the standard decomposition in the
self-dual basis (a short review of the concepts of finite fields needed in
this paper is presented in the appendix)
$|1_{1},1_{2},\ldots,1_{k},0_{k+1},\ldots,0_{n}\rangle\mapsto|1\sigma_{1}+1\sigma_{2}+\ldots+1\sigma_{k}+0\sigma_{k+1}+\ldots+0\sigma_{n}\rangle\,.$
(2)
Now, consider a SU(2) coherent state[6]
$|\xi\rangle=\frac{1}{(1+|\xi|^{2})^{n/2}}\sum_{k=0}^{n}\sqrt{\frac{n!}{k!(n-k)!}}\;\xi^{k}|n,k\rangle\,,$
(3)
where the complex number $\xi$ is related with the angular coordinates
$(\vartheta,\varphi)$ on the Bloch sphere by
$\xi=\cot(\vartheta/2)\,e^{-i\varphi}\,.$ (4)
Using the previous correspondence, $|\xi\rangle$ can be recast as
$|\xi\rangle=\frac{1}{(1+|\xi|^{2})^{n/2}}\sum_{\gamma\in\mathrm{GF}(2^{n})}\xi^{h(\gamma)}\,|\gamma\rangle\,,$
(5)
where the function $h(\gamma)$, when applied to the field element
$\gamma=\sum_{k=1}^{n}\gamma_{k}\sigma_{k}$ indicates the number of nonzero
coefficients $\gamma_{k}$.
In this case, the Fourier operator is
$F=\frac{1}{2^{n/2}}\sum_{\mu,\nu\in\mathrm{GF}(2^{n})}\chi(\mu\nu)\,|\mu\rangle\langle\nu|\,,$
(6)
$\chi$ being an additive character defined in (36). As it is well known,
$F^{2}=\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}$, so if we impose
that the states $|\xi\rangle$ are also eigenstates of $F$ we are lead to
$F|\xi\rangle=\pm|\xi\rangle\,.$ (7)
This immediately implies (all the spins are pointing in the same direction)
that there are two SU(2) coherent states (with $\xi_{\pm}=\pm\sqrt{2}-1$) that
simultaneously are eigenstates of the Fourier operator: they are precisely our
candidates to be coherent states for $n$ qubits. In particular,
$|\xi_{\pm}\rangle$ satisfy the following condition
$\frac{1}{2^{n/2}}\sum_{\mu,\gamma\in\mathrm{GF}(2^{n})}\xi^{h(\gamma)}\chi(\mu\gamma)|\mu\rangle=\pm\sum_{\mu\in\mathrm{GF}(2^{n})}\xi^{h(\mu)}|\mu\rangle\,,$
(8)
or, equivalently,
$\frac{1}{2^{n/2}}\sum_{\gamma\in\mathrm{GF}(2^{n})}\xi^{h(\gamma)}\chi(\mu\gamma)=\pm\xi^{h(\mu)}\,,$
(9)
and the minus sign may appear only for odd number of qubits.
Equation (5) is the abstract form of the SU(2) coherent state. It factorizes
in a product of single-qubit states when represented in the self-dual basis,
i.e.,
$|\xi\rangle=\frac{1}{(1+|\xi|^{2})^{n/2}}\sum_{c_{1},\ldots,c_{n}\in\mathbb{Z}_{2}}\xi^{h\left(\sum_{k=1}^{n}c_{k}\sigma_{k}\right)}|c_{1}\rangle\ldots|c_{n}\rangle=\prod_{j=1}^{n}\frac{(|0\rangle+\xi|1\rangle)_{j}}{(1+|\xi|^{2})^{1/2}}\,,$
(10)
$c_{k}$ being the expansion coefficients of $\gamma$ in that basis. The
operator transforming from the arbitrary basis
$\\{\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{n}\\}$ into a
factorized form is always a permutation given by
$P=\sum_{\mu\in\mathrm{GF}(2^{n})}(|\mu_{1}^{\prime}\rangle\ldots|\mu_{n}^{\prime}\rangle)\,(\langle\mu_{1}|\ldots\langle\mu_{n}|)\,,$
(11)
where
$\mu=\sum_{i=1}^{n}\mu_{i}\,\varepsilon_{i}=\sum_{i=1}^{n}\mu_{i}^{\prime}\,\sigma_{i}\,,\qquad\mu\in\mathrm{GF}(2^{n}),\quad\mu_{i},\mu_{i}^{\prime}\in\mathbb{Z}_{2}\,.$
(12)
Let us examine the simple yet illustrative example of a two-qubit coherent
state. In its abstract form it reads as
$|\xi\rangle=\frac{1}{1+\xi^{2}}(|0\rangle+\xi|\sigma\rangle+\xi|\sigma^{2}\rangle+\xi^{2}|\sigma^{3}\rangle).$
(13)
In the self-dual basis ($\sigma,\sigma^{2}$) we have the representation
$|0\rangle=|00\rangle=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
1\end{array}\right),\,|\sigma\rangle=|10\rangle=\left(\begin{array}[]{c}0\\\
0\\\ 1\\\
0\end{array}\right),\,|\sigma^{2}\rangle=|01\rangle=\left(\begin{array}[]{c}0\\\
1\\\ 0\\\
0\end{array}\right),\,|\sigma^{3}\rangle=|11\rangle=\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right),$ (14)
in such a way that
$|\xi\rangle=\frac{1}{1+\xi^{2}}\left(\begin{array}[]{c}\xi^{2}\\\ \xi\\\
\xi\\\
1\end{array}\right)=\frac{1}{\sqrt{1+\xi^{2}}}\left(\begin{array}[]{c}\xi\\\
1\end{array}\right)\otimes\frac{1}{\sqrt{1+\xi^{2}}}\left(\begin{array}[]{c}\xi\\\
1\end{array}\right)\,.$ (15)
In a non self-dual basis ($\sigma,\sigma^{3}$) we have
$|0\rangle=|00\rangle=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
1\end{array}\right),\,|\sigma\rangle=|10\rangle=\left(\begin{array}[]{c}0\\\
0\\\ 1\\\
0\end{array}\right),\,|\sigma^{3}\rangle=|01\rangle=\left(\begin{array}[]{c}0\\\
1\\\ 0\\\
0\end{array}\right),\,|\sigma^{2}\rangle=|11\rangle=\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)\,,$ (16)
and
$|\xi\rangle=\frac{1}{1+\xi^{2}}\left(\begin{array}[]{c}\xi\\\ \xi^{2}\\\
\xi\\\ 1\end{array}\right)\,,$ (17)
which cannot be factorized. The transition operator for this case is
$P=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\,,$ (18)
and it is nothing but a CNOT gate performing the operation
$|00\rangle+|01\rangle\rightarrow|00\rangle+|11\rangle\,.$ (19)
## 3 Discrete Wigner function
To gain further insights into the coherent states $|\xi_{\pm}\rangle$ we
proceed to picture them in phase space. To this end, we first note that, while
in the continuous case it is possible to translate a state by an infinite
distance, this is clearly not possible if the space is finite. To “prevent” a
state from “escaping” the finite phase space it is natural and convenient to
use the field $\mathrm{GF}(2^{n})$ in the representation of the states.
In consequence, we denote by $|\alpha\rangle$, with
$\alpha\in\mathrm{GF}(2^{n})$, an orthonormal basis in the Hilbert space of
the system. Operationally, the elements of the basis can be labeled by powers
of a primitive element, and the basis reads
$\\{|0\rangle,\,|\sigma\rangle,\ldots,\,|\sigma^{2^{n}-1}=1\rangle\\}\,.$ (20)
These vectors are eigenvectors of the operators $Z_{\beta}$ belonging to the
generalized Pauli group, whose generators are now defined as
$Z_{\beta}=\sum_{\alpha\in\mathrm{GF}(2^{n})}\chi(\alpha\beta)\,|\alpha\rangle\langle\alpha|\,,\qquad\qquad
X_{\beta}=\sum_{\alpha\in\mathrm{GF}(2^{n})}|\alpha+\beta\rangle\langle\alpha|\,,\\\
$ (21)
so that
$Z_{\alpha}X_{\beta}=\chi(\alpha\beta)\,X_{\beta}Z_{\alpha}\,.$ (22)
The operators (21) can be factorized into tensor products of powers of single-
particle Pauli operators $\sigma_{z}$ and $\sigma_{x}$, whose expression in
the standard basis of the two-dimensional Hilbert space is
$\hat{\sigma}_{z}=|1\rangle\langle 1|-|0\rangle\langle
0|\,,\qquad\qquad\hat{\sigma}_{x}=|0\rangle\langle 1|+|1\rangle\langle 0|\,.$
(23)
This factorization can be carried out by mapping each element of
$\mathrm{GF}(2^{n})$ onto an ordered set of natural numbers. As we have
already seen, a convenient choice for this is the self-dual basis, since the
finite Fourier transform factorizes then into a product of single-particle
Fourier operators, which leads to
$Z_{\alpha}=\hat{\sigma}_{z}^{a_{1}}\otimes\ldots\otimes\hat{\sigma}_{z}^{a_{n}}\,,\qquad\qquad
X_{\beta}=\hat{\sigma}_{x}^{b_{1}}\otimes\ldots\otimes\hat{\sigma}_{x}^{b_{n}}\,,$
(24)
where $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ are the expansion
coefficients of $\alpha$ and $\beta$, respectively, in the self-dual basis.
It was shown that the operators
$D(\alpha,\beta)=\phi(\alpha,\beta)\,Z_{\alpha}X_{\beta}\,,$ (25)
where $\phi(\alpha,\beta)$ is a phase, form an operational basis in the
discrete phase space.[10] The unitarity condition imposes the condition
$\phi^{2}(\alpha,\beta)=\chi(-\alpha\beta)$. These displacement operators (or
phase-point operators in the notation of Wootters[11]) allows us to introduce
a Hermitian kernel
$\Delta(\alpha,\beta)=\frac{1}{2^{n}}\sum_{\mu,\nu\in\mathrm{GF}(2^{n})}\chi(\alpha\nu-\beta\mu)\,D(\mu,\nu)\,,$
(26)
in terms of which we can define a well-behaved Wigner function as
$W_{\varrho}(\alpha,\beta)=\mathop{\mathrm{Tr}}\nolimits[\varrho\,\Delta(\alpha,\beta)]\,,$
(27)
where $\varrho$ is the density matrix of the system.
Figure 1: Wigner function for a coherent state of a system of three identical
qubits.
After some calculations, the Wigner function for our coherent states turns out
to be
$W_{|\xi_{+}\rangle}(\alpha,\beta)=\frac{1}{2^{n}}\frac{1}{(1+|\xi|^{2})^{n}}\sum_{\mu,\nu,\gamma\in\mathrm{GF}(2^{n})}\xi^{h(\gamma)}\overline{\xi}^{h(\gamma+\nu)}\chi(\alpha\nu+\beta\mu+\mu\nu+\mu\gamma)\phi(\mu,\nu)\,.$
(28)
A plot of this function for the case of three qubits is shown in Fig. 1. We
also note that the marginal distributions take a very simple form:
$\sum_{\alpha}W_{|\xi_{+}\rangle}(\alpha,\beta)=\frac{|\xi^{h(\beta)}|^{2}}{(1+|\xi|^{2})^{n}},\qquad\qquad\sum_{\beta}W_{|\xi_{+}\rangle}(\alpha,\beta)=\frac{|\xi^{h(\alpha)}|^{2}}{(1+|\xi|^{2})^{n}}\,.$
(29)
To conclude we wish to mention that it is also possible to introduce the
notion of squeezing for these states[12]. In the basis of the eigenstates of
$Z_{\alpha}$, such an operator has the following form
$S_{\lambda}=\sum_{\kappa\in\mathrm{GF}(2^{n})}|\kappa\rangle\langle\lambda\kappa|\,.$
(30)
The following relations hold
$S_{\lambda}^{\dagger}Z_{\alpha}S_{\lambda}=Z_{\alpha\lambda^{-1}}\,,\qquad\qquad
S_{\lambda}^{\dagger}X_{\alpha}S_{\lambda}=X_{\alpha\lambda}\,,$ (31)
so that
$\langle\xi_{\pm}|S_{\lambda}^{\dagger}X_{\alpha}S_{\lambda}|\xi_{\pm}\rangle=\langle\xi_{\pm}|S_{\lambda}^{\dagger}Z_{\alpha\lambda^{2}}S_{\lambda}|\xi_{\pm}\rangle\,.$
(32)
## 4 Conclusions
In summary, we have formulated a new sensible approach to deal with coherent
states for a system of $n$ qubits. The associated discrete Wigner function has
also been worked out. Some related problems, as the behavior under time
evolution or the extension to systems of qudits, will be addressed elsewhere.
## Appendix A Galois fields
We briefly recall the minimum background of finite fields needed to proceed
through this paper. The reader interested in more mathematical details is
referred, e.g., to the excellent monograph by Lidl and Niederreiter.[13]
A commutative ring is a set $R$ equipped with two binary operations, called
addition and multiplication, such that it is an Abelian group with respect the
addition, and the multiplication is associative. Perhaps, the motivating
example is the ring of integers $\mathbb{Z}$ with the standard sum and
multiplication. On the other hand, the simplest example of a finite ring is
the set $\mathbb{Z}_{n}$ of integers modulo $n$, which has exactly $n$
elements.
A field $F$ is a commutative ring with division, that is, such that 0 does not
equal 1 and all elements of $F$ except 0 have a multiplicative inverse (note
that 0 and 1 here stand for the identity elements for the addition and
multiplication, respectively, which may differ from the familiar real numbers
0 and 1). Elements of a field form Abelian groups with respect to addition and
multiplication (in this latter case, the zero element is excluded).
The characteristic of a finite field is the smallest integer $p$ such that
$p\,1=\underbrace{1+1+\ldots+1}_{\mbox{\scriptsize$p$ times}}=0$ (33)
and it is always a prime number. Any finite field contains a prime subfield
$\mathbb{Z}_{p}$ and has $d=p^{n}$ elements, where $n$ is a natural number.
Moreover, the finite field containing $p^{n}$ elements is unique and is called
the Galois field $\mathrm{GF}(p^{n})$.
Let us denote as $\mathbb{Z}_{p}[x]$ the ring of polynomials with coefficients
in $\mathbb{Z}_{p}$. Let $P(x)$ be an irreducible polynomial of degree $n$
(i.e., one that cannot be factorized over $\mathbb{Z}_{p}$). Then, the
quotient space $\mathbb{Z}_{p}[X]/P(x)$ provides an adequate representation of
$\mathrm{GF}(p^{n})$. Its elements can be written as polynomials that are
defined modulo the irreducible polynomial $P(x)$. The multiplicative group of
$\mathrm{GF}(p^{n})$ is cyclic and its generator is called a primitive element
of the field.
As a simple example of a nonprime field, we consider the polynomial
$x^{2}+x+1=0$, which is irreducible in $\mathbb{Z}_{2}$. If $\sigma$ is a root
of this polynomial, the elements
$\\{0,1,\sigma,\sigma^{2}=\sigma+1=\sigma^{-1}\\}$ form the finite field
$\mathrm{GF}(2^{2})$ and $\sigma$ is a primitive element.
A basic map is the trace
$\mathop{\mathrm{tr}}\nolimits(\alpha)=\alpha+\alpha^{2}+\ldots+\alpha^{p^{n-1}}\,,$
(34)
which satisfies
$\mathop{\mathrm{tr}}\nolimits(\alpha+\beta)=\mathop{\mathrm{tr}}\nolimits(\alpha)+\mathop{\mathrm{tr}}\nolimits(\beta),$
(35)
and leaves the prime field invariant. In terms of it we define the additive
characters as
$\chi(\alpha)=\exp\left[\frac{2\pi
i}{p}\mathop{\mathrm{tr}}\nolimits(\alpha)\right],$ (36)
and posses two important properties:
$\chi(\alpha+\beta)=\chi(\alpha)\chi(\beta),\qquad\qquad\sum_{\alpha\in\mathrm{GF}(p^{n})}\chi(\alpha\beta)=p^{n}\delta_{0,\beta}.$
(37)
Any finite field $\mathrm{GF}(p^{n})$ can be also considered as an
$n$-dimensional linear vector space. Given a basis $\\{\theta_{k}\\}$,
($k=1,\ldots,n$) in this vector space, any field element can be represented as
$\alpha=\sum_{k=1}^{n}a_{k}\,\theta_{k},$ (38)
with $a_{k}\in\mathbb{Z}_{p}$. In this way, we map each element of
$\mathrm{GF}(p^{n})$ onto an ordered set of natural numbers
$\alpha\Leftrightarrow(a_{1},\ldots,a_{n})$.
Two bases $\\{\theta_{1},\ldots,\theta_{n}\\}$ and
$\\{\theta_{1}^{\prime},\ldots,\theta_{n}^{\prime}\\}$ are dual when
$\mathop{\mathrm{tr}}\nolimits(\theta_{k}\theta_{l}^{\prime})=\delta_{k,l}.$
(39)
A basis that is dual to itself is called self-dual.
There are several natural bases in $\mathrm{GF}(p^{n})$. One is the polynomial
basis, defined as
$\\{1,\sigma,\sigma^{2},\ldots,\sigma^{n-1}\\},$ (40)
where $\sigma$ is a primitive element. An alternative is the normal basis,
constituted of
$\\{\sigma,\sigma^{p},\ldots,\sigma^{p^{n-1}}\\}.$ (41)
The choice of the appropriate basis depends on the specific problem at hand.
For example, in $\mathrm{GF}(2^{2})$ the elements $\\{\sigma,\sigma^{2}\\}$
are both roots of the irreducible polynomial. The polynomial basis is
$\\{1,\sigma\\}$ and its dual is $\\{\sigma^{2},1\\}$, while the normal basis
$\\{\sigma,\sigma^{2}\\}$ is self-dual.
## References
* [1] H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950).
* [2] J. Schwinger, Unitary operator basis, Proc. Natl. Acad. Sci. USA 46 (1960) 570–576.
* [3] A. Vourdas, Quantum systems with finite Hilbert space, Rep. Prog. Phys. 67 (2004) 267–320.
* [4] A. Vourdas, Quantum systems with finite Hilbert space: Galois fields in quantum mechanics, J. Phys. A 40 (2007) R285–R331.
* [5] M. Ruzzi, Jacobi $\vartheta$ functions and discrete Fourier transforms, J. Math. Phys. 47 (2006) 063507.
* [6] A. Perelomov, Generalized Coherent States and their Applications (Springer, Berlin, 1986).
* [7] D. Galetti and M. A. Marchiolli, Discrete coherent states and probability distributions in finite-dimensional spaces, Ann. Phys. (N Y) 249 (1996) 454–480.
* [8] M. Ruzzi, M. Marchiolli and D. Galetti, Extended Cahill-Glauber formalism for finite-dimensional spaces: I. fundamentals, J. Phys. A 38 (2005) 6239–6245.
* [9] M. L. Mehta, Eigenvalues and eigenvectors of the finite Fourier transform, J. Math. Phys. 28 (1987) 781–785.
* [10] G. Björk, A. B. Klimov and L. L. Sánchez-Soto, The discrete Wigner function, Prog. Opt. 51 (2008) 470–514
* [11] W. K. Wootters, A Wigner-function formulation of finite-state quantum mechanics, Ann. Phys. (N Y) 176 (1987) 1–21.
* [12] M. A. Marchiolli, M. Ruzzi and D. Galetti, Discrete squeezed states for finite-dimensional spaces, Phys. Rev. A 76 (2007) 032102.
* [13] R. Lidl, H. Niederreiter, Introduction to finite fields and their applications (Cambridge University Press, Cambridge, 1986).
|
arxiv-papers
| 2008-09-29T15:30:50
|
2024-09-04T02:48:58.023633
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Munoz, A. B. Klimov, L. L. Sanchez-Soto and G. Bjork",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0809.4995"
}
|
0809.5045
|
# Universal ratios of critical amplitudes in the Potts model universality
class
B. Berche P. Butera W. Janke L. Shchur Laboratoire de Physique des
Matériaux, Université Henri Poincaré Nancy 1, BP 239, F-54506 Vandœuvre les
Nancy Cedex, France Istituto Nazionale di Fisica Nucleare, Sezione di Milano-
Bicocca, Piazza delle Scienze 3, 20126, Milano, Italia Institut für
Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig,
Germany Landau Institute for Theoretical Physics, 142432 Chernogolovka,
Russia lev@landau.ac.ru
###### Abstract
Monte Carlo (MC) simulations and series expansions (SE) data for the energy,
specific heat, magnetization, and susceptibility of the three-state and four-
state Potts model and Baxter-Wu model on the square lattice are analyzed in
the vicinity of the critical point in order to estimate universal combinations
of critical amplitudes. We also form effective ratios of the observables close
to the critical point and analyze how they approach the universal critical-
amplitude ratios. In particular, using the duality relation, we show
analytically that for the Potts model with a number of states $q\leq 4$, the
effective ratio of the energy critical amplitudes always approaches unity
linearly with respect to the reduced temperature. This fact leads to the
prediction of relations among the amplitudes of correction-to-scaling terms of
the specific heat in the low- and high-temperature phases. It is a common
belief that the four-state Potts and the Baxter-Wu model belong to the same
universality class. At the same time, the critical behavior of the four-state
Potts model is modified by logarithmic corrections while that of the Baxter-Wu
model is not. Numerical analysis shows that critical amplitude ratios are very
close for both models and, therefore, gives support to the hypothesis that the
critical behavior of both systems is described by the same renormalization
group fixed point.
###### keywords:
Potts model; Baxter-Wu model; Critical exponents; Critical amplitudes;
Universality; Monte Carlo simulations; Series Expansions; Renormalization
Group
###### PACS:
0.50.+q, 75.10.-b
††journal: Computer Physics Communications
The fixed points of the renormalization group define the universal behavior of
a system through a set of critical exponents and universal combinations of
critical amplitudes [1]. The universality concept divides all systems at
criticality into a number of universality classes. It is instructive to know
the set of values of the critical exponents and of the universal combinations
of critical amplitudes for a given universality class.
The two-dimensional Potts model [2] is the simplest model which exhibits a
phase transition. It is solved exactly at the critical point for any number of
spin components $q$ and it is known that for $q\leq 4$ it has a continuous
phase transition while for $q>4$ the phase transition is of the first order.
The model has a great theoretical interest as new theories may be tested in
this model.
At the same time, these models may have some practical interest as they may be
realized in an adsorbate lattice placed onto a clean crystalline surface. The
full classification of such systems with continuous transitions is known
theoretically [3]. There are experiments in which some of them realize the
3-state and 4-state Potts models [4] and the critical exponents can be
experimentally estimated.
Critical exponents for the Potts model with $q\leq 4$ can be computed exactly
by different theoretical techniques [5, 6]. The values of the thermal critical
exponents and of the magnetic critical exponents follow from the
identification of the dimensions of the conformal algebra operators [6].
Nowadays, there is no doubt on the values of the leading critical exponents
whereas the values of the correction-to-scaling exponents are still under
discussion, as well as the values of the universal ratios of the critical
amplitudes. Our presentation is intended to give a short review of the
research on the subject.
The Potts model Hamiltonian [2] (see review [7] for details) can written as
$H=-\sum_{\langle ij\rangle}\delta_{s_{i}s_{j}}\;,$ where $s_{i}$ is a spin
variable taking integer values between $0$ and $q{-}1$, and the sum is
restricted to the nearest neighbor sites $\langle ij\rangle$ on the square
lattice.
Close to the critical temperature $T_{c}$ at which the continuous phase
transition occurs, the residual magnetization $M$ and the singular part of the
reduced susceptibility $\chi$ and of the specific heat $C$ of the system in
zero external field are characterized by the critical exponents $\beta$,
$\gamma$, and $\alpha$ and by the critical amplitudes $B$, $\Gamma_{\pm}$, and
$A_{\pm}$
$\displaystyle M(\tau)$ $\displaystyle\approx$ $\displaystyle
B(-\tau)^{\beta},\ \tau<0$ (1) $\displaystyle\chi_{\pm}(\tau)$
$\displaystyle\approx$ $\displaystyle\Gamma_{\pm}|\tau|^{-\gamma},$ (2)
$\displaystyle C_{\pm}(\tau)$ $\displaystyle\approx$
$\displaystyle\frac{A_{\pm}}{\alpha}|\tau|^{-\alpha},$ (3)
where $\tau$ is the reduced temperature $\tau=(T-T_{c})/T$ and the labels
$\pm$ refer to the high-temperature and low-temperature sides of the critical
temperature $T_{c}$. The critical amplitudes are not universal by themselves
but some combinations of them, f.e., $A_{+}/A_{-}$, $\Gamma_{+}/\Gamma_{-}$,
and $\Gamma_{+}A_{+}/B^{2}$, are universal [1] due to the scaling laws.
On the square lattice, in zero field, the model is self-dual. The duality
relation
$\left(e^{\beta}-1\right)\left(e^{\beta^{*}}-1\right)=q$ (4)
fixes the inverse critical temperature to $\beta_{c}{=}\ln(1{+}\sqrt{q})$. The
values $E(\beta)$ and $E(\beta^{*})$ of the internal energy at dual
temperatures are simply related through
$\left(1-e^{-\beta}\right)E(\beta)+\left(1-e^{-\beta^{*}}\right)E(\beta^{*})=2.$
(5)
Dual reduced temperatures $\tau$ and $\tau^{*}$ can be defined by
$\beta{=}\beta_{c}(1{-}\tau)$ and $\beta^{*}{=}\beta_{c}(1{+}\tau^{*})$. Close
to the critical point, $\tau$ and $\tau^{*}$ coincide through linear order,
since
$\tau^{*}{=}\tau{+}\frac{\ln(1{+}\sqrt{q})}{\sqrt{q}}\tau^{2}{+}O(\tau^{3})$.
The ratio of the free energy critical amplitudes $A_{+}/A_{-}$ is equal to
unity due to duality. Moreover, duality relations may be used to understand
the dependence on temperature of the effective amplitude functions which may
be constructed from the energy in the symmetric phase $E_{+}(\tau)$ and in the
ordered phase $E_{-}(\tau^{*})$
$\displaystyle A_{+}(\tau)$ $\displaystyle=$
$\displaystyle\alpha(1-\alpha)\beta_{c}(E_{+}(\tau)-E_{0})\tau^{\alpha-1},$
(6) $\displaystyle A_{-}(\tau^{*})$ $\displaystyle=$
$\displaystyle\alpha(1-\alpha)\beta_{c}(E_{0}-E_{-}(\tau^{*}))(\tau^{*})^{\alpha-1}$
(7)
as an effective amplitude ratio
$\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}=\frac{(E_{+}(\tau)-E_{0})\tau^{\alpha-1}}{(E_{0}-E_{-}(\tau^{*}))(\tau^{*})^{\alpha-1}},$
(8)
where the constant $E_{0}$ is the value of the energy at the transition
temperature, $E_{0}{=}E(\beta_{c}){=}{-}1{-}1/\sqrt{q}$.
Evaluating expression (8) for small $\tau$ and denoting
$\alpha_{q}=-E_{0}\beta_{c}e^{-\beta_{c}}=\frac{\ln(1+\sqrt{q})}{\sqrt{q}}$,
we obtain
$\displaystyle\frac{A_{+}(\tau)}{A_{-}(\tau^{*})}$ $\displaystyle=$
$\displaystyle 1+(3-\alpha)\alpha_{q}\tau+O(\tau^{1+\alpha}).$ (9)
Note the linear dependence on $\tau$ of the effective amplitude ratio.
The 2-state Potts model is equivalent to the Ising model which was solved
exactly [8] (see Ref. [9] for details). The susceptibility behavior was
understood in the paper by Wu, McCoy, Tracy and Barouch [10]. It turns out
that there exist only integer corrections to scaling (for a recent and
detailed discussions we refer readers to Refs. [11, 12, 13]). Values of the
critical exponents and of some amplitude ratios are presented in Table 1.
Table 1: Exact values of critical exponents and ratios of critical amplitudes for the Ising model (2-state Potts model). $\nu$ | $\alpha$ | $\beta$ | $\gamma$ | $A_{+}/A_{-}$ | $\Gamma_{+}/\Gamma_{L}$ | $R_{C}^{+}=\Gamma_{+}A_{+}/B^{2}$
---|---|---|---|---|---|---
1 | 0 | $1/8$ | $7/4$ | 1 | 37.69365… | 0.318569…
The critical behavior of the susceptibility reads as
$\chi(\tau)=\Gamma_{\pm}|\tau|^{-\gamma}{\cal X}_{corr}(|\tau|^{\Delta})+{\cal
Y}_{bt}(\tau),$ (10)
where ${\cal X}_{corr}(|\tau|^{\Delta})$ is the correction-to-scaling function
and ${\cal Y}_{bt}(\tau)$ represents an analytic expression (“background
term”) which accounts for non-singular contributions to susceptibility.
Set of values of the thermal critical exponents $x_{\epsilon_{n}}$ and of the
magnetic critical exponents $x_{\sigma_{n}}$ are known analytically [5, 6]
$x_{\epsilon_{n}}=\frac{n^{2}+ny}{2-y},\;\;\;x_{\sigma_{n}}=\frac{\left(2n-1\right)^{2}-y^{2}}{4(2-y)}$
(11)
in terms of the variable $y$ linked to the number of states $q$ by
$\cos\frac{\pi y}{2}=\frac{1}{2}\sqrt{q}$.
For the 3-state Potts model there is a finite number of correction terms [6],
$x_{\epsilon_{2}}=14/5$, $x_{\epsilon_{2}}=6$, and $x_{\sigma_{2}}=4/3$. The
leading correction-to-scaling contribution is
$\Delta=-(2-x_{\epsilon_{2}})/(2-x_{\epsilon_{1}})=2/3$ and it was first
supported in numerical simulation [14].
Clear evidence for these leading correction to the scaling behavior may be
seen in Figure 1, where we plot the difference of the longitudinal and
transverse susceptibilities $(\chi_{L}-\chi_{T})$ (note the cancelation of
background terms (10) in the difference) multiplied by the leading behavior
factor $|\tau|^{\gamma}$, as a function of $|\tau|^{2/3}$. The ratio of the
susceptibilities $(\chi_{T}/\chi_{L})$ is shown in Figure 2 and may be used to
estimate the universal ratio of associated amplitudes $\Gamma_{T}/\Gamma_{L}$.
Analytical predictions for the amplitude ratios in Potts model for $q=2,3$,
and 4 were given in the papers [15, 16]. The values are shown in the Table 2
together with numerical estimations from Monte Carlo (MC) and series
expansions (SE) analyses [17, 18, 19]. The coincidence of the data is a good
indication for the validity of both two-kink approximation [15, 16] to the
exact scattering theory [20] for 3-state Potts model and of the analysis of MC
data and series expansions.
Table 2: Exact values of critical exponents and estimates of the ratios of critical amplitudes for the 3-state Potts model. $\nu$ | $\alpha$ | $\beta$ | $\gamma$ | $\Gamma_{+}/\Gamma_{L}$ | $\Gamma_{T}/\Gamma_{L}$ | $R_{C}^{+}{=}\Gamma_{+}A_{+}/B^{2}$ | Remark
---|---|---|---|---|---|---|---
5/6 | 1/3 | $1/9$ | $13/9$ | - | - | - | Exact result
| | | | 13.848 | 0.327 | 0.1041 | [15, 16]
| | | | 13.83(9) | 0.325(2) | | [19] \- SE
| | | | 13.83(9) | 0.3272(7) | 0.1044(8) | [18] \- SE
| | | | 13.86(12) | 0.322(3) | 0.1049(29) | [18] \- MC
The analysis of the 4-state Potts model is much more complicated because in
addition to the corrections to scaling there are confluent logarithmic
corrections [21, 22]. The result of the analysis of MC data and SE [23, 24,
19] is shown in the Table 3 together with the analytical estimates [15, 16].
Conformal field theory predicts the set of renormalization group (RG)
exponents $y_{\epsilon_{n}}{=}2{-}x_{\epsilon_{n}}{=}$3/2, 0, -5/2, … . The
leading correction-to-scaling exponent $y_{\epsilon_{2}}$ vanishes and gives
rise to a logarithmic behavior [21]. In our recent publication [24, 25] we
revised the renormalization group equations and included in our analysis the
known form of the logarithmic corrections and of the next-to-leading
corrections, taking into account the width of the temperature region window
examined. The set of magnetic exponents for the Potts model
$x_{\sigma_{n}}$=1/8, 9/8, 25/8, … translates into the magnetization exponent
$\beta{=}1/12$ and the leading correction-to-scaling exponent
$\Delta_{\sigma}{=}2/3$. Finally, the following behavior of the susceptibility
is assumed
$\chi_{+}(\tau)=\Gamma_{+}\tau^{-7/6}{\cal
G}^{3/4}(-\ln\tau)(1+a_{+}\tau^{2/3}+b_{+}\tau)+D_{+},$ (12)
where the function $\cal G$ contains a universal correction function $\cal E$
[22, 24, 25] and the leading nonuniversal correction function $\cal F$
$\displaystyle{\cal G}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle(-\ln|\tau|)\times{\cal E}(-\ln|\tau|)\times{\cal
F}(-\ln|\tau|),$ (13) $\displaystyle{\cal E}(-\ln|\tau|)$ $\displaystyle=$
$\displaystyle\left(1+\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)$
(14) $\displaystyle\
\times\left(1-\frac{3}{4}\frac{\ln(-\ln|\tau|)}{-\ln|\tau|}\right)^{-1}\left(1+\frac{3}{4}\frac{1}{(-\ln|\tau|)}\right),$
$\displaystyle{\cal F}(-\ln|\tau|)$ $\displaystyle\simeq$
$\displaystyle\left(1+\frac{C_{1}}{-\ln|\tau|}+\frac{C_{2}\ln(-\ln|\tau|)}{(-\ln|\tau|)^{2}}\right)^{-1}.$
(15)
We fit our data to estimate the amplitude $\Gamma_{+}$, the coefficient of the
leading correction to scaling $a_{+}$ in Eq. (12), and coefficients $C_{1}$
and $C_{2}$ in Eq. (15).
It is obvious that the logarithmic corrections (the whole function $\cal
G(-\ln|\tau|)$) cancels in simple ratios, like $A_{+}/A_{-}$,
$\Gamma_{+}/\Gamma_{L}$, $\Gamma_{T}/\Gamma_{L}$, etc. This has been
demonstrated analytically for the effective ratio $A_{+}/A_{-}$ (see Eq. 9).
We note also that the RG analysis predicts [24, 25] powers of logarithmic
corrections to specific heat $\alpha^{\prime}{=}{-}1$, susceptibility
$\gamma^{\prime}{=}3/4$, and magnetization $\beta^{\prime}{=}{-}1/8$ such that
they cancel in all universal ratios. For example, the universal amplitude
ratio $R_{C}^{-}$ may be calculated as the limit of the ratio of functions
$R_{C}^{-}=\lim_{\tau\rightarrow
0}\tau\frac{(E_{-}(|\tau|)-E_{0})\chi_{-}(|\tau|)}{M(|\tau|)^{2}}\alpha(\alpha-1)\beta_{c}$
(16)
where $E_{0}=E(0)=\sqrt{2}$. One can check that in the ratio (16) not only
powers of $|\tau|$ cancel but also powers of $\cal E$. In the ratio, the
magnetization $M$ and the energy difference $E_{-}(|\tau|)-E_{0}$ have only
singular contributions and the only systematic deviation may come from the
background correction to susceptibility $\chi_{-}(|\tau|)$. It was shown in
[18] that the contribution from this background correction is negligible in
the critical region window, and the estimator (16) tends to the value
$0.0055(1)$ as $\tau\rightarrow 0$.
Figure 1: 3-state Potts model. Difference of susceptibilities
$(\chi_{L}-\chi_{T})|\tau|^{\gamma}$ as function of $|\tau|^{2/3}$. The almost
linear dependence supports the value 2/3 for the power of the leading
correction to scaling.
The Baxter-Wu [26] model is defined on a triangular lattice, with spins
$\sigma_{i}$ located at vertices. The three spins forming a triangular face
are coupled with a strength $J$, and the Hamiltonian reads
${\cal H}=-J\sum_{\rm faces}\sigma_{i}\sigma_{j}\sigma_{k},$ (17)
where the summation extends over all triangular faces of the lattice, both
pointing up and down. The ground state is four-fold degenerate and the
critical exponents are found to be the same as for the 4-state Potts model.
The exact behavior of the magnetization, energy, and specific heat are known
analytically [26, 27, 28]. An analysis of Monte Carlo data was performed by
two of us [29] and preliminary estimates shows that the values of the
susceptibility amplitude-ratio $\Gamma_{+}/\Gamma_{L}\approx 6.9$ and of the
ratio $R_{C}^{-}\approx 0.005$ are very near to those obtained from our
analysis of MC and SE data for the 4-state Potts model (see Table 3). We have
to note that logarithmic corrections to scaling are absent in the critical
behavior of Baxter-Wu model and this gives us more confidence in our analysis.
Delfino and Grinza [30] use the same analytical approach as in [15] to study
the Ashkin-Teller model which also belongs to the 4-state Potts model
universality class with some particular choice of coupling constants. This
leads to the estimatation $\Gamma_{+}/\Gamma_{L}\approx 4.02$ and
$\Gamma_{T}/\Gamma_{L}\approx 0.129$. This result is also very near to those
for 4-state Potts model (see second entry in Table 3.)
A possible explanation of the deviation of our results from the analytical
predictions may be explained as follows: the two-kink approximation is exact
for the 2-state Potts model, it gives good accuracy for the 3-state Potts
model, but it may be insufficient to produce accurate values for 4-state Potts
model. Further analyses have to be done to resolve the contradiction among
these results.
Table 3: Exact values of critical exponents and estimations of the ratios of critical amplitudes for the 4-state Potts model. $\nu$ | $\alpha$ | $\beta$ | $\gamma$ | $\Gamma_{+}/\Gamma_{L}$ | $\Gamma_{T}/\Gamma_{L}$ | $R_{C}^{-}{=}\Gamma_{-}A_{-}/B^{2}$ | Remark
---|---|---|---|---|---|---|---
2/3 | 2/3 | $1/12$ | $7/6$ | - | - | - | Exact result
| | | | 4.013 | 0.129 | 0.00508 | [15, 16]
| | | | 3.5(4) | 0.11(4) | | [19] \- SE
| | | | 3.14(70) | | 0.0068(9) | [23] \- MC
| | | | 6.93(6) | 0.1674(30) | 0.00512(13) | [24, 25] \- MC
| | | | 6.30(1) | 0.1511(24) | 0.00531(5) | [24, 25] \- SE
BB and WJ acknowledge partial support within the Graduate School “Statistical
Physics of Complex Systems” of DFH-UFA under Contract No. CDFA-02-07.
Financial support within a common research program between the Landau
Institute and the Ecole Normale Supérieure de Paris, Paris Sud University is
also gratefully acknowledged.
## References
* [1] V. Privman, P.C. Hohenberg, A. Aharony, in Phase Transitions and Critical Phenomena, Vol. 14, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1991).
* [2] R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106.
* [3] E. Domany, M. Schick, J. Walker, and R.B. Griffiths, Phys. Rev. B 18 (1978) 2209; E. Domany and M. Schick, Phys. Rev. B 20 (1979) 3828; C. Rottman, Phys. Rev. B 24 (1981) 1482.
* [4] A. Aharony, K. A. Muller and W. Berlinger, Phys. Rev. Lett. 38 (1977) 33;H. Pfnür and P. Piercy, Phys. Rev. B 41 (1990) 582; M. Sokolowski and H. Pfnür, Phys. Rev. Lett. 49 (1994) 7716; Y. Nakajima, et al, Phys. Rev. B 55(1997) 8129; C. Voges and H. Pfnür, Phys. Rev. B 57 (1998) 3345.
* [5] M.P.M. den Nijs, J. Phys. A 12 (1979)1857; R.B. Pearson, Phys. Rev. B 22 (1980) 2579; B. Nienhuis, J. Phys. A 15 (1982) 199; B. Nienhuis, J. Stat. Phys. 34 (1984) 731; B. Nienhuis, in Phase Transitions and Critical Phenomena,Vol. 11, edited by C. Domb and J.L. Lebowitz(Academic Press, London, 1987).
* [6] Vl.S. Dotsenko, Nucl. Phys. B 235 (1984) 54; Vl.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 (1984) 312.
* [7] F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235.
* [8] L. Onsager, Phys. Rev. ]65 (1944) 117.
* [9] B. M. McCoy and T.T. Wu, The two-dimensional Ising model (Harvard Uni. Press, Camdridge, Massachusetts, 1973)
* [10] E. Barouch, B.M. McCoy, and T.T. Wu, Phys. Rev. Lett. 31 (1973) 409; C.A. Tracy and B.M. McCoy, Phys. Rev. Lett. 31 (1973) 1500; T.T. Wu, B.M. McCoy, C.A. Tracy, and E. Barouch, Phys. Rev. B 13 (1976) 316.
* [11] Orrick, W. P., Nickel, B. G., Guttmann, A. J., and Perk, J. H. H., Phys. Rev. Lett. 86 (2001) 4120; J. Stat. Phys. 102 (2001) 795.
* [12] M. Caselle, M. Hasenbusch, A. Pelissetto, E. Vicari, J.Phys. A 35 (2002) 4861.
* [13] G. Delfino, Phys.Lett. B450 (1999) 196
* [14] G. von Gehlen, V. Rittenberg, and H. Ruegg, J. Phys. A 19 (1985) 107.
* [15] G. Delfino and J.L. Cardy, Nucl. Phys. B 519 (1998) 551.
* [16] G. Delfino, G.T. Barkema and J.L. Cardy, Nucl. Phys. B 565 (2000) 521.
* [17] L.N. Shchur, P. Butera, and B. Berche, Nucl. Phys. B 620 (2002) 579.
* [18] L.N. Shchur, L.N. Shchur, and P. Butera, Phys. Rev. B 77 (2008) 144410.
* [19] I.G. Enting and A.J. Guttmann, Physica A 321 (2003) 90.
* [20] L. Chim and A.B. Zamolodchikov, Int. J. Mod. Phys. A 7 (1992) 5317.
* [21] M. Nauenberg and D.J. Scalapino, Phys. Rev. Lett. 44 (1980) 837; J. L. Cardy, N. Nauenberg and D.J. Scalapino, Phys. Rev. B 22 (1980) 2560.
* [22] J. Salas and A. Sokal, J. Stat. Phys. 88 (1997) 567.
* [23] M. Caselle, R. Tateo, and S. Vinci, Nucl. Phys. B 562 (1999) 549.
* [24] L.N. Shchur, B. Berche and P. Butera, Europhys. Lett. bf 81 (2008) 30008.
* [25] L.N. Shchur, B. Berche and P. Butera, unpublished.
* [26] R. J. Baxter and F.Y. Wu, Phys. Rev. Lett. 31 (1973) 1294; Aust. J. Phys. 27 (1974) 357; R.J. Baxter, Aust. J. Phys. 27 (1974) 369.
* [27] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (New York, Academic Press, 1982).
* [28] G.S. Joyce, Proc. R. Soc. Lond. A 343 (1975); ibid 345 (1975) 277.
* [29] L.N. Shchur and W. Janke, unpublished.
* [30] G. Delfino and P. Grinza, Nucl. Phys. B 682 (2004) 521.
Figure 2: 3-state Potts model. Ratio of transverse to longitudinal
susceptibilities.
|
arxiv-papers
| 2008-09-29T19:04:08
|
2024-09-04T02:48:58.028034
|
{
"license": "Public Domain",
"authors": "Bertrand Berche, Paolo Butera, Wolfhard Janke, and Lev Shchur",
"submitter": "Lev Shchur N",
"url": "https://arxiv.org/abs/0809.5045"
}
|
0809.5067
|
# New Relativistic Particle-In-Cell Simulation Studies of Prompt and Early
Afterglows from GRBs
K.-I. Nishikawa J. Niemiec H. Sol M. Medvedev B. Zhang Å. Nordlund J.
Frederiksen P. Hardee Y. Mizuno D. H. Hartmann G. J. Fishman
###### Abstract
Nonthermal radiation observed from astrophysical systems containing
relativistic jets and shocks, e.g., gamma-ray bursts (GRBs), active galactic
nuclei (AGNs), and microquasars commonly exhibit power-law emission spectra.
Recent PIC simulations of relativistic electron-ion (or electron-positron)
jets injected into a stationary medium show that particle acceleration occurs
within the downstream jet. In collisionless, relativistic shocks, particle
(electron, positron, and ion) acceleration is due to plasma waves and their
associated instabilities (e.g., the Weibel (filamentation) instability)
created in the shock region. The simulations show that the Weibel instability
is responsible for generating and amplifying highly non-uniform, small-scale
magnetic fields. These fields contribute to the electron’s transverse
deflection behind the jet head. The resulting “jitter” radiation from
deflected electrons has different properties compared to synchrotron
radiation, which assumes a uniform magnetic field. Jitter radiation may be
important for understanding the complex time evolution and/or spectra in
gamma-ray bursts, relativistic jets in general, and supernova remnants.
###### Keywords:
Weibel instability, magnetic field generation, radiation
###### :
Relativistic Particle-In-Cell Simulation, Studies of Prompt and Early
Afterglows from GRBs
## 1 Introduction
Shocks are believed to be responsible for prompt emission from gamma-ray
bursts (GRBs) and their afterglows, for variable emission from blazars, and
for particle acceleration processes in jets from active galactic nuclei (AGN)
and supernova remnants (SNRs). The predominant contribution to the observed
emission spectra is often assumed to be synchrotron- and inverse Compton
radiation from these accelerated particles Piran (2005a); Zhang (2007). It is
assumed that turbulent magnetic fields in the shock region lead to Fermi
acceleration, producing higher energy particles Fermi (1949); Blandford &
Eichler (1987). To make progress in understanding emission from these object
classes, it is essential to place modeling efforts on a firm physical basis.
This requires studies of the microphysics of the shock process in a self-
consistent manner Piran (2005b); Waxman (2006).
### 1.1 New Numerical Method for Calculating Emission
The retarded electric field from a charged particle moving with instantaneous
velocity $\boldsymbol{\beta}$ under acceleration $\boldsymbol{\dot{\beta}}$ is
obtained Jackson (1999); Nishikawa et al. (2008a, b, c). After some
calculation and simplifying assumptions the total energy $W$ radiated per unit
solid angle per unit frequency can be expressed as
$\displaystyle\frac{d^{2}W}{d\Omega d\omega}=$ (1)
$\displaystyle\frac{\mu_{0}cq^{2}}{16\pi^{3}}\left|\int^{\infty}_{\infty}\frac{\bf{n}\times[(\bf{n}-\boldsymbol{\beta})\times\boldsymbol{\dot{\beta}}]}{(1-\boldsymbol{\beta}\cdot\bf{n})^{2}}e^{i\omega(t^{{}^{\prime}}-\bf{n}\cdot\bf{r}_{0}({\rm
t}^{{}^{\prime}})/{\rm c})}dt^{{}^{\prime}}\right|^{2}$
Here,
$\bf{n}\equiv\bf{R}(\rm{t}^{{}^{\prime}})/|\bf{R}(\rm{t}^{{}^{\prime}})|$ is a
unit vector that points from the particle’s retarded position towards the
observer. The first term on the right hand side, containing the velocity
field, is the Coulomb field from a charge moving without influence from
external forces in eq. 2.4 Hededal (2005). The second term is a correction
term that arises when the charge is subject to acceleration. Since the
velocity-dependent field falls off in distance as $R^{-2}$, while the
acceleration-dependent field scales as $R^{-1}$, the latter becomes dominant
when observing the charge at large distances ($R\gg 1$). The choice of unit
vector $\bf{n}$ along the direction of propagation of the jet (hereafter taken
to be the $x$-axis) corresponds to head-on emission. For any other choice of
$\bf{n}$ (e.g., $\theta=1/\gamma$), off-axis emission is seen by the observer.
The observer’s viewing angle is set by the choice of $\bf{n}$ ($n_{\rm
x}^{2}+n_{\rm y}^{2}+n_{\rm z}^{2}=1$).
| $B_{\rm x}$ | $V_{\rm j1,2}$ | $V_{\perp,1}$ | $V_{\perp,2}$ | $\gamma_{\max}$ | $\theta_{\Gamma}$ | Remarks
---|---|---|---|---|---|---|---
P | 3.70 ($B_{\rm z}$) | 0.0c | 0.998c | 0.9997c | 40.08 | 4.491 | gyrating
A | 3.70 | 0.99c | 0.1c | 0.12c | 13.48 | 13.35 | jet
B | 3.70 | 0.9924c | 0.1c | 0.12c | 36.70 | 4.905 | jet
C | 3.70 | 0.99c | 0.01c | 0.012c | 7.114 | 25.30 | jet
D | 0.370 | 0.99c | 0.01c | 0.012c | 7.114 | 25.30 | jet
E | 0.370 | 0.99c | 0.1c | 0.12c | 13.48 | 13.35 | $\Delta t=0.005$
F | 0.370 | 0.99c | 0.1c | 0.12c | 13.48 | 13.35 | $\Delta t=0.025$
Table 1: Seven cases of radiation
Figure 1: Summary of six cases with jet velocity (Cases A - F) (for Case P,
see Nishikawa et al. Nishikawa et al. (2008c))
## 2 Radiation from two electrons
In the previous section we discussed how to obtain the retarded electric field
from relativistically moving particles (electrons) observed at large distance.
Using eq. 1 we calculated the time evolution of the retarded electric field
and the spectrum from a gyrating electron in a uniform magnetic field to
verify the technique used in this calculation. We have calculated the
radiation from two electrons gyrating in the $x-y$ plane in the uniform
magnetic field $B_{\rm z}$ with Lorentz factors ($\gamma=15.8,40.8$) (Case P
in Table 1) Nishikawa et al. (2008a, b, c). We have very good agreement
between the spectrum obtained from the simulation and the theoretical
synchrotron spectrum expectation from eq. 7.10 Hededal & Nishikawa (2005).
In order to calculate more realistic radiation from relativistic jets we
included a parallel magnetic field ($B_{\rm x}$). Relativistic jets are
propagating along the $x$ direction. Table 1 shows six cases including the
previous case P (first row) Nishikawa et al. (2008c). The jet velocity is
0.99c (except Case B). Two different magnetic field strengths are used. Two
electrons are injected with two different perpendicular velocities (Cases A -
F). The maximum Lorenz factors, $\gamma_{\max}=\\{(1-(V_{\rm j2}^{2}+V_{\perp
2}^{2})/c^{2}\\}^{-1/2}$ are calculated for larger perpendicular velocity. The
critical angles for the off-axis radiation is calculated with
$\theta_{\Gamma}=\Gamma^{-1}$.
Figure 1 shows the summary of the six cases. Trajectories of the two electrons
are shown in the left column (red: larger perpendicular velocity, blue:
smaller perpendicular velocity). The two electrons propagate from left to
right with gyration in the $y-z$ plane (not shown). The gyroradius is about
$0.44\Delta$ ($\Delta=1$: the simulation grid length) for the electron with a
larger perpendicular velocity (Case A). The radiation electric field from the
two electrons is shown in the middle column. The spectra were calculated at
the point $(x,y,z)=(64,000,000.0,43.0,43.0)$ shown in he right column. The
seven curves show the spectrum at the viewing angles 0∘ (red), 1∘ (orange), 2∘
(yellow), 3∘ (moss green), 4∘ (green), 5∘ (light blue), and 6∘ (blue) ($n_{\rm
y}\neq 0$). The higher frequencies become stronger with the increasing viewing
angle. For Case A the power spectrum is scaled as $P\sim\omega^{1}$ as
proposed for jitter radiation Medvedev (2006). For all Cases the spectra are
much steeper than the slope $1/3$ for the synchrotron radiation.
The second row in Fig. 1 shows Case B with a larger jet velocity $V_{\rm
j1,2}=0.9924c$ with the other parameters kept the same as Case A. The spectra
with larger viewing angles are similar to those of Case A. The spectrum slope
is smaller than that in Case A. However, due to the large jet velocity the
higher frequencies at larger viewing angles (0∘, 1∘, 2∘) become stronger. On
the other hand, with the smaller perpendicular velocities (Cases C and D), the
gyroradius becomes very small. The spectra become weaker than those in Case A.
As shown in the third and fourth rows in Fig. 1, the viewing angle dependence
becomes very small. It should be noted that the slope of spectra is very steep
for Case C. Spectral leakage is found.
Cases D - F have a weaker magnetic fields ($B_{\rm x}=0.370$) than Cases A -
C. Case D has a small perpendicular velocity. The trajectories are almost
straight. The spectra look very similar to that for Bremsstrahlung Hededal &
Nishikawa (2005). The spectra become flat at lower frequencies. The peak
spectral power is the weakest of all the cases. With larger perpendicular
velocities the spectra become stronger than those with the smaller
perpendicular velocities. Case F shows the case with a larger time step (5
times) with the same parameters as Case E. The spectrum slope is very steep.
The spectra in this case show two differences with those in Case E. First
there exist positive slopes in the lower frequency. Second, due to the gyro-
motion the spectra split due to the viewing angles. In particular the spectrum
with larger viewing angle becomes stronger at high frequencies.
As shown in Table 1, the critical angles for the off-axis radiation
$\theta_{\Gamma}=\Gamma^{-1}$ are different. In this study we have obtained
the off-axis radiation for the angles 0∘, 1∘, 2∘, 3∘, 4∘, 5∘, and 6∘ ($n_{\rm
y}\neq 0$). For cases D, E, and F the variation among different viewing angles
is small since the angles are much smaller than the critical angle (25.3∘ and
13.35∘). However, for case B ($\theta_{\Gamma}=5^{\circ}$) the radiation shows
larger differences for different viewing angles due to the small critical
angle. For Case F (a longer time ($340/\omega_{\rm pe}$) the spectra at high
frequencies become stronger with larger viewing angles.
These results validate the technique used in our code. It should be noted that
the method based on the integration of the retarded electric fields calculated
by tracing many electrons described in the previous section can provide a
proper spectrum in turbulent electromagnetic fields. On the other hand, if the
formula for the frequency spectrum of radiation emitted by a relativistic
charged particle in instantaneous circular motion is used Jackson (1999);
Rybicki & Lightman (1979), the complex particle accelerations and trajectories
are not properly accounted for and the jitter radiation spectrum is not
properly obtained.
## 3 Discussion
The procedure used to calculate jitter radiation using the technique described
in the previous section has been implemented in our code.
In order to obtain the spectrum of synchrotron (jitter) emission Medvedev
(2000, 2006); Fleishman (2006), we consider an ensemble of electrons randomly
selected in the region where the filamentation (Weibel) instability Weibel
(1959) has fully developed, and electrons are accelerated in the generated
magnetic fields. We calculate emission from about 20,000 electrons during the
sampling time, $t_{\rm s}=t_{\rm 2}-t_{\rm 1}$ with Nyquist frequency
$\omega_{\rm N}=1/2\Delta t$ where $\Delta t$ is the simulation time step and
the frequency resolution $\Delta\omega=1/t_{\rm s}$. However, since the
emission coordinate frame for each particle is different, we accumulate
radiation at fixed angles in simulation system coordinates after transforming
from the individual particle emission coordinate frame. This provides an
intensity spectrum as a function of angle relative to the simulation frame
$x$-axis (this can be any angle by changing the unit vector ${\bf n}$ in eq.
(1)). A hypothetical observer in the ambient medium (viewing the external GRB
shock) views emission along the system $x$-axis. This computation is carried
out in the reference frame of the ambient medium in the numerical simulation.
For an observer located outside the direction of bulk motion of the ambient
medium, e.g., internal jet shocks in an ambient medium moving with respect to
the observer, an additional Lorentz transformation would be needed along the
line of sight to the observer. Spectra obtained from simulations can be
rescaled to physical time scales.
Emission obtained by the method described above is self-consistent, and
automatically accounts for magnetic field structures on the small scales
responsible for jitter emission. By performing such calculations for
simulations with different parameters, we can then investigate and compare the
quite contrasted regimes of jitter- and synchrotron-type emission Medvedev
(2000, 2006); Fleishman (2006) for prompt and afterglow emission. The
feasibility of this approach has been demonstrated and implemented Hededal &
Nordlund (2005); Hededal & Nishikawa (2005). Thus, we will be able to address
the issue of low frequency GRB spectral index violation of the synchrotron
line of death Medvedev (2006).
Simulations incorporating jitter radiation are in progress using an MPI code
Niemiec et al. (2008) which speeds up considerably from the previous OpenMP
code Ramirezet al. (2007). New results of jitter radiation will be presented
separately.
We have benefited from many useful discussions with A. J. van der Horst. This
work is supported by AST-0506719, AST-0506666, NASA-NNG05GK73G and NNX07AJ88G.
JN was supported by MNiSW research projects 1 P03D 003 29 and N N203 393034,
and The Foundation for Polish Science through the HOMING program, which is
supported through the EEA Financial Mechanism.Simulations were performed at
the Columbia facility at the NASA Advanced Supercomputing (NAS). Part of this
work was done while K.-I. N. was visiting the Niels Bohr Institute. He thanks
the director of the institution for generous hospitality.
## References
* Blandford & Eichler (1987) Blandford, R. & Eichler, D. 1987, Phys. Rep., 154, 1
* Fermi (1949) Fermi, E. 1949, Phys. Rev. 75, 1169
* Fleishman (2006) Fleishman, G. D. 2006, ApJ, 638, 348
* Hededal (2005) Hededal, C.B., 2005, PhD thesis (arXiv:astro-ph/0506559)
* Hededal & Nishikawa (2005) Hededal, C. B. and Nishikawa, K.-I. 2005, ApJ, 623, L89
* Hededal & Nordlund (2005) Hededal, C.B., & Nordlund, Å. 2005, submitted to ApJL (arXiv:astro-ph/0511662)
* Jackson (1999) Jackson, J. D. 1999, Classical Electrodynamics, Interscience
* Medvedev (2006) Medvedev, M. V. 2006, ApJ, 637, 869
* Medvedev (2000) Medvedev, M. V. 2000, ApJ, 540, 704
* Medvedev & Loeb (1999) Medvedev, M.V. & Loeb, A. 1999, ApJ, 526, 697
* Niemiec et al. (2008) Niemiec, J., Pohl, M., Stroman, T. & Nishikawa, K.-I. 2008, ApJ, 684, 1174
* Nishikawa et al. (2008a) Nishikawa, K.-I., Mizuno, Y., Fishman, G. J., & Hardee, P., 2008a, (arXiv:0801.4390)
* Nishikawa et al. (2008b) Nishikawa, K.-I.. Hardee, P., Mizuno, Y., Medvedev, M., Zhang, B., Hartmann, D. H., & Fishman, G. J. 2008b, (arXiv:0802.2558)
* Nishikawa et al. (2008c) Nishikawa, K. -I., Mizuno, Y., Hardee, P., Sol, H., Medvedev, M., Zhang, B., Nordlund, A., Frederiksen, J. T., Fishman, G. J. & Preece, R. 2008c, published in for the Proceedings of Science of the Workshop on Blazar Variability across the Electromagnetic Spectrum, April 22 to 25, 2008 (arXiv:0808.3781)
* Piran (2005a) Piran, T. 2005a, Rev. Mod. Phys. 76, 1143
* Piran (2005b) Piran, T. 2005b, in the proceedings of Magnetic Fields in the Universe, Angra dos Reis, Brazil, Nov. 29-Dec 3, 2004, Ed. E. de Gouveia del Pino, (arXiV:astro-ph/0503060)
* Ramirezet al. (2007) Ramirez-Ruiz, E., Nishikawa, K.-I., & Hededal, C. B., 2007, ApJ, 671, 1877
* Rybicki & Lightman (1979) Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics, John Wiley & Sons, New York
* Waxman (2006) Waxman, E. 2006, Plasma Phys. Control. Fusion, 48 B137
* Weibel (1959) Weibel, E. S. 1959, Phys. Rev. Lett., 2, 83
* Zhang (2007) Zhang, B. 2007, Chin. J. Astron. Astrophys. 7, 1
|
arxiv-papers
| 2008-09-30T15:19:15
|
2024-09-04T02:48:58.032312
|
{
"license": "Public Domain",
"authors": "K.-I. Nishikawa, J. Niemiec, H. Sol, M. Medvedev, B. Zhang, A.\n Nordlund, J. Frederiksen, P. Hardee, Y. Mizuno, D. H. Hartmann and G. J.\n Fishman",
"submitter": "Ken-Ichi Nishikawa",
"url": "https://arxiv.org/abs/0809.5067"
}
|
0809.5216
|
# Macrostates thermodynamics and its stable classical limit in Global
One–Dimensional Quantum General Relativity
L. A. Glinka111E-mail to: glinka@theor.jinr.ru , laglinka@gmail.com
_Nicolai N. Bogoliubov Laboratory of Theoretical Physics_ ,
_Joint Institute for Nuclear Research_ ,
_141980 Dubna, Moscow Region, Russian Federation_
###### Abstract
Global One–Dimensional Quantum General Relativity is the toy model with
nontrivial field theoretical content, describing classical one-dimensional
massive bosonic fields related to any $3+1$ metric, where the dimension is a
volume of three-dimensional embedding. In fact it constitutes the
midisuperspatial Quantum Gravity model.
We use one-particle density operator method in order to building macrostates
thermodynamics related with any $3+1$ metric. Taking the Boltzmann gas limit,
which is given by the energy equipartition law for the Bose–Einstein gas of
space quantum states generated from the Bogoliubov vacuum, we receive
consistent with General Relativity thermodynamical degrees of freedom number.
It confirm that the proposed Quantum Gravity toy model has well-defined
classical limit in accordance with classical gravity theory.
## 1 Introduction
The toy model – Global One–Dimensionality proposal in Quantum General
Relativity – considered in my last topical papers [1, 2, 3, 4, 5, 6], is a
nontrivial quantum field theoretical model describing one dimensional
classical massive bosonic fields related immediately to $3+1$ decomposed
metrics according to standard the Dirac–ADM approach in General Relativity.
For construction of the model elementary quantum field theory methods, as
field quantization by the Fock second quantization method and the
Bogoliubov–Heisenberg diagonalization procedure, are used. In fact this simple
type divagations constitutes a new and nontrivial midisuperspatial Quantum
Gravity model, which results in space quantum states conception and unique
connection between quantum correlations and physical scales of the system.
This paper is devoted to consider an application of one-particle density
operator method in order to building thermodynamics of quantum macrostates
related with any $3+1$ decomposed metric of General Relativity. Macrostates in
the Quantum Gravity model are given by the Bose–Einstein gas of space quantum
states. The self-consistence with General Relativity is achieved by the
classical limit – the Boltzmann gas limit of the macrostates thermodynamics,
which is given by the energy equipartition law for classically stable phase of
the Bose–Einstein gas of space quantum states generated from the Bogoliubov
vacuum. The classically stable phase is defined by appropriate limit of
quantum correlations for infinite number of vacuum space quantum states. In
result we obtain classical degrees of freedom number which equal to number od
space-time coordinates used in General Relativity.
The content of this paper is as follows. In the section 2 I recall the crucial
elements of the Global One–Dimensional model of Quantum Gravity. There is
present shortly a way from the Einstein–Hilbert General Relativity, by $3+1$
Arnowitt–Deser–Misner decomposition of metric and the Dirac primary
quantization of the Hamiltonian constraint which lead to the Wheeler–DeWitt
theory of quantum geometrodynamics, till my supposition about Global
One–Dimensionality and field theoretical content of the Wheeler–DeWitt model.
The section 3 is devoted to presentation of the main point of this article,
that is macrostates thermodynamics and its classically stable limit. We use
one-particle approximation. It is shown that the Boltzmann gas limit for
classically stable configuration of the Bose–Einstein gas of macrostates
generated from the stable Bogoliubov vacuum, leads to thermodynamical degrees
of freedom number which is consistent with General Relativity.
## 2 Global 1D Quantum Gravity
The classical gravity theory – General Relativity – describes 4-dimensional
pseudo–Riemannian [7] differentiable manifold $(M,g)$ defined by metric
$g_{\mu\nu}$ and coordinate system $x^{\mu}=(x^{0},x^{1},x^{2},x^{3})$, and
characterized by the Christoffel connections $\Gamma^{\rho}_{\mu\nu}$, the
Riemann curvature tensor $R^{\lambda}_{\mu\alpha\nu}$, the Ricci curvature
tensor $R_{\mu\nu}$, and the scalar curvature $R$ [8, 9]
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},\qquad\Gamma^{\rho}_{\mu\nu}=\dfrac{1}{2}g^{\rho\sigma}\left(g_{\mu\sigma,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma}\right)$
(1)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!R^{\lambda}_{\mu\alpha\nu}=\Gamma^{\lambda}_{\mu\nu,\alpha}-\Gamma^{\lambda}_{\mu\alpha,\nu}+\Gamma^{\lambda}_{\sigma\alpha}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\alpha},\quad
R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu},\quad
R=g^{\kappa\lambda}R_{\kappa\lambda}.$ (2)
According to Einstein [10], evolution of $(M,g)$ is given by the field
equations222The units $8\pi G/3=c=\hbar=k_{B}=1$ are used.
$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=3T_{\mu\nu},$ (3)
where $\Lambda$ is cosmological constant, and $T_{\mu\nu}$ is Matter stress-
energy tensor. The Einstein equations (3) can be received from the Hilbert
dynamical action [11] modified by the Hartle–Hawking boundary $(\partial M,h)$
action [12]
$S[g]=\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}R+\dfrac{\Lambda}{3}+\mathcal{L}\right\\}-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K,$ (4)
where $K$ is extrinsic curvature of $(\partial M,h)$, by the Palatini
principle [13] $\delta S[g]=0$ which relates the Matter Lagrangian
$\mathcal{L}$ with $T_{\mu\nu}$
$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta
g^{\mu\nu}}.$ (5)
### 2.1 3+1 Dirac–ADM approach
By employing of the $3+1$ Dirac–ADM decomposition [14, 15, 16]
$\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N^{i}N_{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right],\qquad
g^{\mu\nu}=\left[\begin{array}[]{cc}-1/N^{2}&N^{j}/N^{2}\vspace*{5pt}\\\
N^{i}/N^{2}&h^{ij}-N^{i}N^{j}/N^{2}\end{array}\right],$ (10)
where $h_{ij}$, $N$, $N_{i}$ are embedding metric, lapse, shift functions,
$h_{ik}h^{kj}=\delta_{i}^{j}$, $N^{i}=h^{ij}N_{j}$, the action (4) takes the
Hamiltonian form
$\displaystyle S[g]=\int dt\int_{\partial
M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}-NH-
N_{i}H^{i}\right\\},$ (11)
where dot means $t$-differentiation, non vanishing conjugate momenta $\pi$’s
are
$\pi^{ij}=-\sqrt{h}\left(K^{ij}-h^{ij}K\right),$ (12)
and $H$, $H^{i}$ are defined as
$\displaystyle
H=\sqrt{h}\left\\{K^{2}-K_{ij}K^{ij}+{{}^{(3)}R}-2\Lambda-6\varrho\right\\},\qquad
H^{i}=-2\pi^{ij}_{\leavevmode\nobreak\ ;j}\leavevmode\nobreak\ ,$ (13)
where ${{}^{(3)}R}=h^{ij}R_{ij}$ is scalar curvature of embedding and
$\varrho=n^{\mu}n^{\nu}T_{\mu\nu}$ is energy density related to normal vector
field $n^{\mu}=[1/N,-N^{i}/N]$ to a spacelike hypersurface. The Gauss–Codazzi
equations [17, 18, 19] determine the extrinsic curvature tensor $K_{ij}$ and
extrinsic scalar curvature $K$ as
$K_{ij}=\dfrac{1}{2N}\left[N_{i|j}+N_{j|i}-\dot{h}_{ij}\right],\qquad
K=\mathrm{Tr}K_{ij},$ (14)
where stroke means intrinsic covariant differentiation. $H^{i}$ are
diffeomorphisms $\widetilde{x}^{i}=x^{i}+\delta x^{i}$ generators
$\displaystyle i\left[h_{ij},\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-h_{ij,k}\delta x^{k}-h_{kj}\delta
x^{k}_{\leavevmode\nobreak\ ,i}-h_{ik}\delta x^{k}_{\leavevmode\nobreak\
,j}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (15) $\displaystyle
i\left[\pi_{ij},\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-\left(\pi_{ij}\delta
x^{k}\right)_{,k}+\pi_{kj}\delta x^{i}_{\leavevmode\nobreak\
,k}+\pi_{ik}\delta x^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (16)
where $H_{i}=h_{ij}H^{j}$, and the DeWitt algebra [20]
$\displaystyle i\left[\int_{\partial M}H\delta x_{1}d^{3}x,\int_{\partial
M}H\delta x_{2}d^{3}x\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{\partial M}H^{a}\left(\delta x_{1,a}\delta
x_{2}-\delta x_{1}\delta x_{2,a}\right)d^{3}x,$ (17) $\displaystyle
i\left[H_{i}(x),H_{j}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{\partial M}H_{a}c^{a}_{ij}d^{3}z,$ (18)
$\displaystyle i\left[H(x),H_{i}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!H\delta^{(3)}_{,i}(x,y),$ (19)
where $c^{a}_{ij}$ are structure constants of diffeomorphism group
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-\delta^{a}_{j}\delta^{b}_{i}\delta^{(3)}_{,b}(y,z)\delta^{(3)}(x,z)\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (20)
is first-class type. Dirac’s primary constraints time-preservation [20, 21]
leads to the secondary constraints (scalar and vector)
$\displaystyle\pi\approx 0\rightarrow H\approx 0,\qquad\pi^{i}\approx
0\rightarrow H^{i}\approx 0.$ (21)
Vector constraint merely reflects spatial diffeoinvariance, scalar constraint
gives dynamical information. Employing conjugate momenta (12) the scalar
constraint becomes the Einstein–Hamilton–Jacobi equation [22]–[71]
$G_{ijkl}\pi^{ij}\pi^{kl}+\sqrt{h}\left({}^{(3)}R-2\Lambda-6\varrho\right)=0,$
(22)
where
$G_{ijkl}=\dfrac{1}{2}h^{-1/2}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)$
is superspace metric.
### 2.2 Quantum Geometrodynamics
Canonical quantization [14, 72] of the Hamiltonian constraint (22)
$\displaystyle i\left[\pi^{ij}(x),h_{kl}(y)\right]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(23) $\displaystyle i\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y),\qquad
i\left[\pi(x),N(y)\right]=\delta^{(3)}(x,y),$ (24)
leads to the Wheeler–DeWitt equation [73, 20]
$\left\\{-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}-h^{1/2}\left(-{{}^{(3)}R}+2\Lambda+6\varrho\right)\right\\}\Psi[h_{ij},\phi]=0,$
(25)
where $\phi$ are Matter fields. Other first class constraints
$\pi\Psi[h_{ij},\phi]=0,\qquad\pi^{i}\Psi[h_{ij},\phi]=0,\qquad
H^{i}\Psi[h_{ij},\phi]=0,$ (26)
merely reflects diffeoinvariance. The canonical commutation relations hold
$\left[\pi(x),\pi^{i}(y)\right]=\left[\pi(x),H^{i}(y)\right]=\left[\pi^{i}(x),H^{j}(y)\right]=\left[\pi^{i}(x),H(y)\right]=0.$
(27)
### 2.3 Global One–Dimensionality
Supposing that Matter fields and the wave function $\Psi[h_{ij},\phi]$ are
functionals of embedding’s volume
$\Psi[h_{ij},\phi]\rightarrow\Psi[h],\leavevmode\nobreak\ \leavevmode\nobreak\
h=\det h_{ij},$ (28)
and apply change of variables $h_{ij}\rightarrow\det h_{ij}$ in the
Wheeler–DeWitt operator we obtain the Klein–Gordon–Fock type field equation
for massive field $\Psi$
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+m^{2}\right)\Psi=0,\qquad
m^{2}=\dfrac{2}{3h}\left({}^{(3)}R-2\Lambda-6\varrho\right),$ (29)
where $m^{2}$ is the mass square of $\Psi$. Elementary dimensional reduction
of (29) leads to the Clifford algebra and the Dirac type equation
$\left\\{\mathbf{\Gamma}^{a},\mathbf{\Gamma}^{b}\right\\}=2\eta^{ab}\mathbb{I},\qquad\eta^{ab}=\left[\begin{array}[]{cc}-1&0\\\
0&0\end{array}\right],\qquad\left(i\mathbf{\Gamma}\vec{\partial}-\mathbb{M}\right)\Phi=0.$
(30)
Here $\mathbf{\Gamma}=\left[-i\mathbb{I},\mathbb{O}\right]$ and we introduced
notation
$\Phi=\left[\begin{array}[]{c}\Psi\\\
\Pi_{\Psi}\end{array}\right],\qquad\vec{\partial}=\left[\begin{array}[]{c}\dfrac{\delta}{\delta
h}\\\ 0\end{array}\right],\qquad\mathbb{M}=\left[\begin{array}[]{cc}0&1\\\
-m^{2}&0\end{array}\right]\geq 0,$ (31)
where $\Pi_{\Psi}$ is conjugate momentum to $\Psi$ obtained from action
$S[\Psi]$
$\Pi_{\Psi}=\dfrac{\delta S[\Psi]}{\delta\Psi},\qquad
S[\Psi]=-\dfrac{1}{2}\int\delta
h\Psi^{\dagger}\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+m^{2}\right)\Psi.$ (32)
### 2.4 Field quantization
Field quantization of (30) according to bosonic relations [74, 75, 76]
$\displaystyle
i\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]=\delta(h^{\prime}-h),\quad
i\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Pi}_{\Psi}[h]\right]=0,\quad
i\left[\mathbf{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]=0,$ (33)
and the second quantization method [77, 78, 79] leads to the solution
$\mathbf{\Phi}=\mathbb{Q}\mathfrak{B},\qquad\mathbb{Q}=\dfrac{1}{\sqrt{2}}\left[\begin{array}[]{cc}|m|^{-1/2}&|m|^{-1/2}\\\
-i|m|^{1/2}&i|m|^{1/2}\end{array}\right].$ (34)
Here $\mathfrak{B}$ is a basis of creators $\mathsf{G}^{\dagger}[h]$ and
annihilators $\mathsf{G}[h]$
$\mathfrak{B}=\left\\{\left[\begin{array}[]{c}\mathsf{G}[h]\\\
\mathsf{G}^{\dagger}[h]\end{array}\right]:\left[\mathsf{G}[h^{\prime}],\mathsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\leavevmode\nobreak\
\leavevmode\nobreak\
\left[\mathsf{G}[h^{\prime}],\mathsf{G}[h]\right]=0\right\\}.$ (35)
with dynamics determined by the system of equations
$\dfrac{\delta\mathfrak{B}}{\delta
h}=\mathbb{L}\mathfrak{B},\qquad\mathbb{L}=\left[\begin{array}[]{cc}-im&\dfrac{\delta}{\delta
h}\ln\sqrt{|m|}\\\ \dfrac{\delta}{\delta
h}\ln\sqrt{|m|}&im\end{array}\right].$ (36)
Assuming new basis $\mathfrak{B}^{\prime}$ as compilation of the Bogoliubov
transformation and the Heisenberg equations
$\displaystyle\mathfrak{B}^{\prime}=\left[\begin{array}[]{cc}u&v\\\
v^{\ast}&u^{\ast}\end{array}\right]\mathfrak{B},\qquad|u|^{2}-|v|^{2}=1,\qquad\dfrac{\delta\mathfrak{B}^{\prime}}{\delta
h}=\left[\begin{array}[]{cc}-i\omega&0\\\
0&i\omega\end{array}\right]\mathfrak{B}^{\prime},$ (41)
where coefficients $u$, $v$ and frequency $\omega$ are functionals of $h$,
gives the Bogoliubov coefficients dynamics
$\dfrac{\delta\mathbf{b}}{\delta
h}=\mathbb{L}\mathbf{b},\qquad\mathbf{b}=\left[\begin{array}[]{c}u\\\
v\end{array}\right],\qquad|u|^{2}-|v|^{2}=1,$ (42)
and the new static basis $\mathfrak{B}^{\prime}=\mathfrak{B}_{I}$ with stable
vacuum $|0\rangle_{I}$
$\mathfrak{B}_{I}=\left\\{\left[\begin{array}[]{c}\mathsf{G}_{I}\\\
\mathsf{G}^{\dagger}_{I}\end{array}\right]:\left[\mathsf{G}_{I},\mathsf{G}^{\dagger}_{I}\right]=1,\leavevmode\nobreak\
\leavevmode\nobreak\
\left[\mathsf{G}_{I},\mathsf{G}_{I}\right]=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \mathsf{G}_{I}|0\rangle_{I}=0\right\\}.$ (43)
Integration of (42) can be done in the superfluid parametrization
$\displaystyle u=e^{i\theta}\cosh\phi,\quad
v=e^{-i\theta}\sinh\phi,\quad\theta=m_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}},\quad\phi=-\ln{\sqrt{\left|\lambda\right|}},$
(44)
where $\lambda=\dfrac{m_{I}}{m}=\dfrac{l}{l_{I}}$ scales sizes. By this reason
we obtain finally
$\mathbf{\Phi}=\mathbb{Q}\mathbb{G}\mathfrak{B}_{I},\qquad\mathbb{G}=\left[\begin{array}[]{cc}u^{\ast}&-v\\\
-v^{\ast}&u\end{array}\right],$ (45)
where $\mathbb{G}$ is the inverted Bogoliubov transformation matrix.
### 2.5 Quantum correlations
After quantization the equation (29) can be rewritten in the form
$\left(\dfrac{\delta^{2}}{\delta
h^{2}}+\dfrac{m^{2}_{I}}{\lambda^{2}}\right)\mathbf{\Psi}=0,$ (46)
and its solution can be red from (45)
$\displaystyle\mathbf{\Psi}=\frac{\lambda}{2\sqrt{2m_{I}}}\left(\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\mathsf{G}_{I}+\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\mathsf{G}_{I}^{\dagger}\right).$ (47)
It is sensible to consider the many-field states acting on the vacuum
$\displaystyle|h,n\rangle\equiv\mathbf{\Psi}^{n}|0\rangle_{I}=\left(\frac{\lambda}{2\sqrt{2m_{I}}}e^{i\theta}\right)^{n}\mathsf{G}^{\dagger
n}_{I}|0\rangle_{I},$ (48)
and determine the two-point quantum correlator $\langle
n^{\prime},h^{\prime}|h,n\rangle$. In the normalization $\langle
1,h_{I}|h_{I},1\rangle\equiv 1$ the one-point correlator is fundamental
$\displaystyle\langle 1,h|h,1\rangle=\lambda^{2}.$ (49)
## 3 Macrostates thermodynamics
### 3.1 The Bose–Einstein gas
The main point of this paper is macrostates thermodynamics and its classical
stable limit. Field quantization with the stable Bogoliubov vacuum presented
in the previous section allows to formulate formal thermodynamics of
macrostates. We will use here one-particle approximation only.
In the one-particle approximation the density operator is number of states
operator, which in static basis has a matrix $\mathbb{D}$ obtained in the
Heisenberg–Von Neumann picture
$\mathsf{D}={\mathsf{G}}^{\dagger}{\mathsf{G}}=\mathfrak{B}^{\dagger}\left[\begin{array}[]{cc}1&0\\\
0&0\end{array}\right]\mathfrak{B}^{\dagger}=\mathfrak{B}_{I}^{\dagger}\left[\begin{array}[]{cc}|u|^{2}&-uv\\\
-u^{\ast}v^{\ast}&|v|^{2}\end{array}\right]\mathfrak{B}_{I}\equiv{\mathfrak{B}}_{I}^{\dagger}\mathbb{D}{\mathfrak{B}}_{I}.$
(50)
The number of states generated from the stable Bogoliubov vacuum is
$\xi=\dfrac{{{}_{I}}\langle
0\left|{\mathsf{G}}^{\dagger}{\mathsf{G}}\right|0\rangle_{I}}{{{}_{I}}\langle
0|0\rangle_{I}}=|v|^{2},$ (51)
and using of elementary linear algebra methods allows to compute formal
entropy
$S=-\dfrac{\mathrm{Tr}\left(\mathbb{D}\ln\mathbb{D}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{8\xi(\xi+1)}{(2\xi+1)^{2}}-\ln\left(2\xi+1\right).$
(52)
Comparison of (52) with the Bose–Einstein gas entropy [80] leads to the
identification
$\displaystyle 2\xi+1\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\exp\dfrac{U-\mu N}{T}-1,$ (53)
$\displaystyle\dfrac{8\xi(\xi+1)}{(2\xi+1)^{2}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left(\dfrac{U-\mu N}{T}\right)\dfrac{\exp\dfrac{U-\mu
N}{T}}{\exp\dfrac{U-\mu N}{T}-1},$ (54)
which fix averaged number of states as
$N=\dfrac{1}{2\xi+1}.$ (55)
Taking the correct Hamiltonian matrix $\mathbb{H}$ of the Bose–Einstein gas
$\mathsf{H}=\mathfrak{B}_{I}^{\dagger}\left[\\!\\!\begin{array}[]{cc}\dfrac{m}{2}\left(|v|^{2}+|u|^{2}\right)&-muv\\\
-mu^{\ast}v^{\ast}&\dfrac{m}{2}\left(|v|^{2}+|u|^{2}\right)\end{array}\\!\\!\right]\mathfrak{B}_{I}\equiv{\mathfrak{B}}_{I}^{\dagger}\mathbb{H}{\mathfrak{B}}_{I},$
(56)
One can compute internal energy and chemical potential according to standard
rules
$U=\dfrac{\mathrm{Tr}\mathbb{D}\mathbb{H}}{\mathrm{Tr}\mathbb{D}},\qquad\mu=\dfrac{\delta
U}{\delta N}.$ (57)
### 3.2 Classically stable Boltzmann gas limit
The second formula of (44) and the number of states (51) allow to establish
the relations for mass and size scales as well as for quantum correlations
$\displaystyle\dfrac{m}{m_{I}}$ $\displaystyle=$
$\displaystyle\left(\sqrt{\xi}\pm\sqrt{\xi+1}\right)^{2},$ (58)
$\displaystyle\dfrac{l}{l_{I}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left(\sqrt{\xi}\pm\sqrt{\xi+1}\right)^{2}},$ (59)
$\displaystyle\langle 1h|h1\rangle$ $\displaystyle=$
$\displaystyle\dfrac{1}{\left(\sqrt{\xi}\pm\sqrt{\xi+1}\right)^{4}}.$ (60)
These formulas for the classical Boltzmann gas limit $\xi\rightarrow\infty$
becomes
$\displaystyle\lim_{\xi\rightarrow\infty}\dfrac{m}{m_{I}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\infty&\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\leavevmode\nobreak\ $+$\\\ 0&,\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ \leavevmode\nobreak\ $--$\end{array}\right.$
(63) $\displaystyle\lim_{\xi\rightarrow\infty}\dfrac{l}{l_{I}}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}0&\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\leavevmode\nobreak\ $+$\\\ \infty&,\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ \leavevmode\nobreak\ $--$\end{array}\right.$
(66) $\displaystyle\lim_{\xi\rightarrow\infty}\langle 1h|h1\rangle$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}0&\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\leavevmode\nobreak\ $+$\\\ \infty&,\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ \leavevmode\nobreak\ $--$\end{array}\right.$
(69)
So it is clear the the classically stable physical object is obtained for the
sign $-$. Computing for this case internal energy and temperature
$\displaystyle U$ $\displaystyle=$ $\displaystyle
m_{I}\dfrac{3\xi^{2}+3\xi+1}{2\xi+1}\left(\sqrt{\xi}-\sqrt{\xi+1}\right)^{2},$
(70) $\displaystyle T$ $\displaystyle=$ $\displaystyle
m_{I}\left[4\xi^{2}+4\xi+1-\dfrac{3\xi^{2}+3\xi+1}{\sqrt{\xi(\xi+1)}}(2\xi+1)\right]\dfrac{3\left(\sqrt{\xi}-\sqrt{\xi+1}\right)^{2}}{8\xi},$
(71)
and using of the equipartition law according to the Boltzmann gas limit
$\displaystyle\dfrac{U}{T}$ $\displaystyle=$
$\displaystyle\dfrac{\dfrac{8}{3}\dfrac{\xi}{2\xi+1}}{\dfrac{4\xi^{2}+4\xi+1}{3\xi^{2}+3\xi+1}-\dfrac{2\xi+1}{3\sqrt{\xi(\xi+1)}}},$
(72) $\displaystyle\lim_{\xi\rightarrow\infty}\dfrac{U}{T}$ $\displaystyle=$
$\displaystyle\dfrac{f}{2},$ (73)
leads to the number of thermodynamical degrees of freedom consistent with
General Relativity
$f=4.$ (74)
## 4 Conclusion
This article was devoted to presentation of the next result of the Global
One–Dimensionality model of Quantum General Relativity. There was recalled the
idea of the model that is global change of variables $h_{ij}\rightarrow\det
h_{ij}$ in the Wheeler–DeWitt equation and demanding that the Matter fields as
well as effectively the Wheeler–DeWitt wave function are functionals of the
global dimension. The model reduces 6 wave functions connected to 6
independent components of an embedding metric to 1 global wave function
related to an embedding volume.
There was presented macrostates thermodynamics and its classically stable
limit. The Bose–Einstein gas model was employed for computation of internal
energy and temperature, and the Boltzmann gas limit was applied for the case
of classically stable object, that is $l=\infty$ in the size scales. In result
we have obtained the consistence with General Relativity - thermodynamical
degrees of freedom number for the object is $f=4$, that lies in full agreement
with the fact that space-time coordinates $x^{\mu}=(x^{0},x^{1},x^{2},x^{3})$
are considered as the degrees of freedom.
By this reason the presented model expresses nontrivial relation between the
Einstein–Hilbert theory of the pseudo–Riemannian differentiable manifold and
thermodynamics of macrostates generated from the stable Bogoliubov vacuuum,
obtained by using the $3+1$ ADM decomposition of space-time metric and the
Dirac–ADM canonical approach to General Relativity.
## Acknowledgements
The author benefited valuable discussions and excellent critical remarks from
Profs. A.B. Arbuzov, I.Ya. Aref’eva, B.M. Barbashov, K.A. Bronnikov, I.L.
Buchdinder, D.I. Kazakov, V.N. Pervushin, V.B. Priezzhev, and D.V. Shirkov.
## References
* [1] L. A. Glinka, On Global One-Dimensionality proposal in Quantum General Relativity, submitted to Pisma v EChAYa 0808.1035[gr-qc]
* [2] L. A. Glinka, Quantum gravity as the way from spacetime to space quantum states thermodynamics, New Advances in Physics, Vol. 2, No. 1, 1 - 62, 2008 0803.1533[gr-qc]
* [3] L. A. Glinka, in Frontiers of Fundamental and Computational Physics. 9th International Symposium, Udine and Trieste, Italy 7–9 January 2008, p.94, eds. B. G. Sidharth, F. Honsell, O. Mansutti, K. Sreenivasan, and A. De Angelis. AIP Conf. Proc. 1018, American Institute of Physics, Melville, New York (2008). 0801.4157[gr-qc]
* [4] L. A. Glinka, Quantum Information from Graviton-Matter Gas. SIGMA 3, 087, (2007). 0707.3341[gr-qc]
* [5] L. A. Glinka, 1D Global Bosonization of Quantum Gravity, to appear in New Advances in Physics 0804.3516[gr-qc]
* [6] L. A. Glinka, On quantum cosmology as field theory of bosonic string mass groundstate, to appear in New Advances in Physics 0712.1674[gr-qc]
* [7] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen. Abh. Königl. Gesell. der Wissen. zu Göttingen, Band 13, 133 (1920).
* [8] M. Kriele, Spacetime. Foundations of General Relativity and Differential Geometry. Lect. Notes Phys. Monogr. 59, Springer-Verlag, Berlin Heidelberg New York (1999).
* [9] P. Petersen, Riemannian Geometry (2nd ed.). Grad. Texts Math. 171. Springer-Verlag, Berlin (2006).
* [10] A. Einstein, Die formale Grundlage der allgemeinen Relativitätstheorie. Sitzungsber. Preuss. Akad. Wiss. Berlin 2, 1030 (1914);
Prinzipielles zur verallgemeinerten Relativitätstheorie und
Gravitationstheorie. Phys. Z. 15, 176 (1914);
Zür allgemeinen Relativitätstheorie. Sitzungsber. Preuss. Akad. Wiss. Berlin
44, 778 (1915);
Zür allgemeinen Relativitätstheorie (Nachtrag). Sitzungsber. Preuss. Akad.
Wiss. Berlin 46, 799 (1915);
Die Feldgleichungen der Gravitation. Sitzungsber. Preuss. Akad. Wiss. Berlin
48, 844 (1915);
Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49, 769 (1916).
* [11] D. Hilbert, Die Grundlagen der Physik. Konigl. Gesell. d. Wiss. Göttingen, Nachr., Math.-Phys. Kl. 27, 395 (1915);
Die Grundlagen der Physik (Zweite Mitteilung). Konigl. Gesell. d. Wiss.
Göttingen, Nachr., Math.-Phys. Kl. 61, 53 (1917).
* [12] J. B. Hartle and S. W. Hawking, Wave function of the Universe. Phys. Rev. D 28, 2960 (1983).
* [13] A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Mat. Palermo 43, 203 (1919).
* [14] P. A. M. Dirac, The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. Lond. A 246, 333 (1958);
Fixation of coordinates in the Hamiltonian theory of gravitation. Phys. Rev.
114, 924 (1959);
Energy of the Gravitational Field. Phys. Rev. Lett. 2, 368 (1959);
Generalized Hamiltonian dynamics. Proc. Roy. Soc. Lond. A 246, 326 (1958);
Generalized Hamiltonian dynamics. Can. J. Math. 2, 129 (1950).
* [15] R. Arnowitt, S. Deser and Ch.W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, ed. by L. Witten, p. 227. Wiley, New York (1962).
* [16] B. DeWitt, The Global Approach to Quantum Field Theory, Vol. 1,2. Int. Ser. Monogr. Phys. 114, Clarendon Press, Oxford (2003).
* [17] K. F. Gauss, Disquisitiones generales circa superficies curvas. Gottingae: Typis Di eterichiansis, (1828).
* [18] D. Codazzi, Sulle coordinate curvilinee d’una superficie dello spazio. Ann. math. pura applicata 2, 101, (1868-1869).
* [19] A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems. Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e loro Applicazioni, n. 22, Accademia Nazionale dei Lincei, Roma (1976).
* [20] B. S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev. 160, 1113 (1967);
Quantum Theory of Gravity. II. The Manifestly Covariant Theory. Phys. Rev.
162, 1195 (1967);
Quantum Theory of Gravity. III. Applications of the Covariant Theory. Phys.
Rev. 162, 1239 (1967).
* [21] P. A. M. Dirac, Lectures on Quantum Field Theory, Belfer Graduate School of Science, Yeshiva University, New York (1966).
* [22] F. A. E. Pirani and A. Schild, On the Quantization of Einstein’s Gravitational Field Equations. Phys. Rev. 79, 986 (1950).
* [23] P. G. Bergmann, Introduction of ”true observables” into the quantum field equations. Nuovo Cim. 3, 1177 (1956);
Summary of the Chapel Hill Conference. Rev. Mod. Phys. 29, 352 (1957);
Observables in General Relativity. Rev. Mod. Phys. 33, 510 (1961);
Hamilton–Jacobi and Schrödinger Theory in Theories with First-Class
Hamiltonian Constraints. Phys. Rev. 144, 1078 (1966).
* [24] P. G. Bergmann and A. B. Komar, in Recent Developments in General Relativity, p. 31, Pergamon Press, Inc., New York, (1962).
* [25] J. A. Wheeler, in Battelle Rencontres: 1967 Lectures in Mathematics and Physics, eds. C. M. DeWitt and J. A. Wheeler, pp. 242. W.A. Benjamin, New York (1968).
* [26] P. W. Higgs, Integration of Secondary Constraints in Quantized General Relativity. Phys. Rev. Lett. 1, 373 (1958);
Integration of Secondary Constraints in Quantized General Relativity. Phys.
Rev. Lett. 3, 66 (1959).
* [27] J. L. Anderson, Factor Sequences in Quantized General Relativity. Phys. Rev. 114, 1182 (1959).
* [28] R. Arnowitt, S. Deser, and Ch. W. Misner, Dynamical Structure and Definition of Energy in General Relativity. Phys. Rev. 116, 1322 (1959);
Canonical Variables for General Relativity. Phys. Rev. 117, 1595 (1960);
Energy and the Criteria for Radiation in General Relativity. Phys. Rev. 118,
1100 (1960);
Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem.
Phys. Rev. 120, 313 (1960);
Canonical Variables, Expression for Energy, and the Criteria for Radiation in
General Relativity. Nuovo Cim. 15, 487 (1960);
Finite Self-Energy of Classical Point Particles Phys. Rev. Lett. 4, 375
(1960);
Consistency of the Canonical Reduction of General Relativity. J. Math. Phys.
1, 434 (1960);
Note on positive-definiteness of the energy of the gravitational field. Ann.
Phys. 11, 116, (1960);
Wave Zone in General Relativity. Phys. Rev. 121, 1556 (1961);
Coordinate Invariance and Energy Expressions in General Relativity. Phys. Rev.
122, 997 (1961).
* [29] A. Peres, On the Cauchy problem in general relativity. Nuovo Cim. 26, 53 (1962).
* [30] R. F. Beierlein, D. H. Sharp, and J. A. Wheeler, Three–Dimensional Geometry as Carrier of Information about Time. Phys. Rev. 126, 1864 (1962).
* [31] H. Leutwyler, Gravitational Field: Equivalence of Feynman Quantization and Canonical Quantization. Phys. Rev. 134, B1155 (1964).
* [32] A. B. Komar, Hamilton–Jacobi Quantization of General Relativity. Phys. Rev. 153, 1385 (1967);
Gravitational Superenergy as a Generator of Canonical Transformation. Phys.
Rev. 164, 1595 (1967).
* [33] B. S. DeWitt, Quantum theories of gravity. Gen. Rel. Grav. 1, 181 (1970).
* [34] D. R. Brill and R. H. Gowdy, Quantization of general relativity. Rep. Prog. Phys. 33, 413 (1970).
* [35] V. Moncrief and C. Teitelboim, Momentum Constraints as Integrability Conditions for the Hamiltonian Constraint in General Relativity. Phys. Rev. D 6, 966 (1972).
* [36] A. E. Fischer and J. E. Marsden, The Einstein equations of evolution - A geometric approach. J. Math. Phys. 13, 546 (1972).
* [37] C. Teitelboim, How commutators of constraints reflect the spacetime structure. Ann. Phys. NY 80, 542 (1973).
* [38] A. Ashtekar and R. Geroch, Quantum theory of gravitation. Rep. Progr. Phys. 37, 1211 (1974).
* [39] T. Regge and C. Teitelboim, Improved Hamiltonian for general relativity. Phys. Lett. B 53, 101 (1974);
Role of surface integrals in the Hamiltonian Formulation of General
Relativity. Ann. Phys. NY 88, 286, (1974).
* [40] R. Geroch, Structure of the Gravitational Field at Spatial Infinity. J. Math. Phys. 13, 956 (1972).
* [41] K. Kucha$\mathrm{\check{r}}$, Ground State Functional of the Linearized Gravitational Field. J. Math. Phys. 11, 3322 (1970);
Canonical Quantization of Cylindrical Gravitational Waves. Phys. Rev. D 4, 955
(1971);
A Bubble-Time Canonical Formalism for Geometrodynamics. J. Math. Phys. 13, 768
(1972);
Geometrodynamics regained: A Lagrangian approach. J. Math. Phys. 15, 708
(1974);
General relativity: Dynamics without symmetry. J. Math. Phys. 22, 2640 (1981);
Dirac constraint quantization of a parametrized field theory by anomaly-free
operator representations of spacetime diffeomorphisms. Phys. Rev. D 39, 2263
(1989).
* [42] M. A. H. MacCallum, in Quantum Gravity, Oxford Symposium, eds. C. J. Isham, R. Penrose, and D. W. Sciama. Clarendon Press, Oxford (1975);
* [43] C. J. Isham, in Quantum Gravity, Oxford Symposium, eds. C. J. Isham, R. Penrose, and D. W. Sciama. Clarendon Press, Oxford (1975);
Canonical groups and the quantization of general relativity. Nucl. Phys. B
Proc. Suppl. 6, 349, (1989).
* [44] C. J. Isham and A. C. Kakas, A group theoretical approach to the canonical quantisation of gravity: I. Construction of the canonical group. Class. Quantum Grav. 1, 621 (1984);
A group theoretical approach to the canonical quantisation of gravity. II.
Unitary representations of the canonical group. Class. Quantum Grav. 1, 633
(1984).
* [45] C. J. Isham and K. V. Kucha$\mathrm{\check{r}}$, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories. Ann. Phys. 164, 288 (1985);
Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics.
Ann. Phys. 164, 316 (1985).
* [46] S. A. Hojman, K. Kucha$\mathrm{\check{r}}$, and C. Teitelboim, Geometrodynamics regained. Ann. Phys. NY 96, 88 (1976).
* [47] G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752, (1977).
* [48] D. Christodoulou, M. Francaviglia, and W. M. Tulczyjew, General relativity as a generalized Hamiltonian system. Gen. Rel. Grav. 10, 567 (1979).
* [49] M. Francaviglia, Applications of infinite-dimensional differential geometry to general relativity. Riv. Nuovo Cim. 1, 1303 (1978).
* [50] J. A. Isenberg, in Geometrical and topological methods in gauge theories. Lect. Notes Phys. 129, eds. J. P. Harnad and S. Shnider, Springer–Verlag Berlin Heidelberg New York, New York (1980).
* [51] J. A. Isenberg and J. M. Nester, in General Relativity and Gravitation. One Hundred Years After the Birth of Albert Einstein., p.23, ed. A. Held, Plenum Press, NewYork and London (1980).
* [52] Z. Bern, S. K. Blau, and E. Mottola, General covariance of the path integral for quantum gravity. Phys. Rev. D 33, 1212 (1991).
* [53] P. O. Mazur, Quantum gravitational measure for three-geometries. Phys. Lett. B 262, 405 (1991).
* [54] C. Kiefer and T. P. Singh, Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067 (1991).
* [55] M. Ferraris, M. Francaviglia, and I. Sinicco, Covariant ADM formulation applied to general relativity. Nuovo Cim. B 107, 11 (1992).
* [56] N. Pinto-Neto and A. F. Velasco, The search for new representations of the Wheeler–DeWitt equation using the first order formalism. Gen. Rel. Grav. 25, 10, 991 (1993).
* [57] C. Kiefer, in Canonical Gravity: From Classical to Quantum, eds. J. Ehlers and H. Friedrich. Springer, Berlin (1994), arXiv:gr-qc/9312015
* [58] D. Giulini and C. Kiefer, Consistency of semiclassical gravity. Class. Quantum Grav. 12, 403 (1995).
* [59] V. N. Pervushin, V. V. Papoian, G. A. Gogilidze, A. M. Khvedelidze, Yu. G. Palii, and V. I. Smirichinsky, The Time surface term in quantum gravity. Phys. Lett. B 365, 35 (1996).
* [60] V. V. Papoian, V. N. Pervushin, and V. I. Smirichinsky, Conformal quantum cosmology: Integrable models and Friedmann observables. Phys. Atom. Nucl. 61, 1908 (1998), Yad. Fiz. 61, 2020 (1998).
* [61] V. N. Pervushin and V. I. Smirichinski, Bogolyubov Quasiparticles in Constrained Systems. J. Phys. A 32, 6191 (1999).
* [62] M. Pawlowski, V. N. Pervushin, and V. I. Smirichinski, Invariant Hamiltonian Quantization of General Relativity. JINR-E2-99232
* [63] N. Pinto-Neto and E. S. Santini, Must quantum spacetimes be Euclidean? Phys. Rev. D 59, 123517 (1999).
* [64] N. Pinto-Neto and E. S. Santini, The Consistency of Causal Quantum Geometrodynamics and Quantum Field Theory. Gen. Rel. Grav. 34, 505 (2002).
* [65] M. J. W. Hall, K. Kumar, and M. Reginatto, Bosonic field equations from an exact uncertainty principle. J. Phys A: Math. Gen. 36, 9779 (2003).
* [66] C. Rovelli, Quantum gravity. Cambridge University Press, Cambridge (2004).
* [67] N. Pinto-Neto, The Bohm Interpretation of Quantum Cosmology. Found. Phys. 35, 577 (2005).
* [68] M. J. W. Hall, Exact uncertainty approach in quantum mechanics and quantum gravity. Gen. Rel. Grav. 37, 1505 (2005).
* [69] B. M. Barbashov, V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk, Quantum gravity as theory of superfluidity. AIP Conf. Proc. 841, 362 (2006).
* [70] V. N. Pervushin and V. A. Zinchuk, Bogoliubov’s integrals of motion in quantum cosmology and gravity. Phys. Atom. Nucl. 70, 593 (2007).
* [71] R. Carroll, Metric fluctuations, entropy, and the Wheeler–DeWitt equation. Theor. Math. Phys. 152, 904 (2007).
* [72] L. D. Faddeev, The energy problem in Einstein’s theory of gravitation (Dedicated to the memory of V. A. Fock). Usp. Fiz. Nauk 136, 435 (1982).
* [73] J. A. Wheeler, On the Nature of Quantum Geometrodynamics. Ann. Physics 2, 604 (1957).
* [74] J. von Neumann, Die Eindeutigket der Schrodingerschen Operatoren. Math. Ann. 104, 570 (1931).
* [75] H. Araki and E. J. Woods, Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637 (1963).
* [76] J. -P. Blaizot and G. Ripka, Quantum theory of finite systems. Massachusetts Institute of Technology Press, Cambridge (1986).
* [77] F. A. Berezin, The Method of Second Quantization (2nd ed.). Nauka, Moscow (1987).
* [78] N. N. Bogoliubov and D. V. Shirkov, Introduction to the theory of quantized fields (3rd ed.). John Wiley and Sons (1980).
* [79] N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov, General Principles of Quantum Field Theory. Nauka, Moscow (1991).
* [80] K. Huang, Statistical Mechanics (2nd ed.). Wiley, New York (1987).
|
arxiv-papers
| 2008-09-30T17:00:26
|
2024-09-04T02:48:58.038554
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. A. Glinka",
"submitter": "L. A. Glinka",
"url": "https://arxiv.org/abs/0809.5216"
}
|
0809.5283
|
# Star Clusters in M31: I. A Catalog and a Study of the Young Clusters
Nelson Caldwell Smithsonian Astrophysical Observatory, 60 Garden Street,
Cambridge, MA 02138, USA
electronic mail: caldwell@cfa.harvard.edu Paul Harding Department of
Astronomy, Case Western Reserve University, Cleveland OH 44106-7215
electronic mail: paul.harding@case.edu Heather Morrison Department of
Astronomy, Case Western Reserve University, Cleveland OH 44106-7215
electronic mail: heather@vegemite.case.edu James A. Rose Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599,
USA
electronic mail: jim@physics.unc.edu Ricardo Schiavon Gemini Observatory,
670 N. A’ohoku Place , Hilo, HI 96720, USA
electronic mail: rschiavo@gemini.edu Jeff Kriessler Department of Astronomy,
Case Western Reserve University, Cleveland OH 44106-7215
electronic mail: jeffrey.kriessler@case.edu
###### Abstract
We present an updated catalog of 1300 objects in the field of M31, including
670 likely star clusters of various types, the rest being stars or background
galaxies once thought to be clusters. The coordinates in the catalog are
accurate to 0.2˝, and are based on images from the Local Group Survey (LGS,
Massey et al., 2006) or from the DSS. Archival HST images and the LGS were
inspected to confirm cluster classifications where possible, but most of the
classifications are based on spectra taken of $\sim 1000$ objects with the
Hectospec fiber positioner and spectrograph on the 6.5m MMT. The spectra and
images of young clusters are analyzed in detail in this paper; analysis of
older clusters will appear in a later paper. Ages and reddenings of 140 young
clusters are derived by comparing the observed spectra with model spectra.
Seven of these clusters also have ages derived from HST color-magnitude
diagrams (two of which we present here); these agree well with the
spectroscopically determined ages. Combining new V band photometry with the
M/L values that correspond to the derived cluster ages, we derive masses for
the young clusters, finding them to have masses as great as $10^{5}$ with a
median of $10^{4}{\rm M}_{\sun}$, and a median age of 0.25 Gyr. In comparison
therefore, Milky Way open clusters have the lowest median mass, the Milky Way
and M31 globulars the highest, and the LMC young massive clusters and the M31
young clusters are in between. The young clusters in M31 show a range of
structure. Most have the low concentration typical of Milky Way open clusters,
but there are a few which have high concentrations. We expect that most of
these young clusters will be disrupted in the next Gyr or so, however, some of
the more massive and concentrated of the young clusters will likely survive
for longer. The spatial distribution of the young clusters is well correlated
with the star-forming regions as mapped out by mid-IR emission. A kinematic
analysis likewise confirms the spatial association of the young clusters with
the star forming young disk in M31.
###### Subject headings:
catalogs – galaxies: individual (M31) – galaxies: star clusters – globular
clusters: general – star clusters: general
## 1\. Introduction
In the Milky Way, there is a clear separation between known open clusters
(which have diffuse structure, generally have low masses and ages, and belong
to the disk) and globular clusters (which have a more concentrated structure,
higher masses and ages, and in many cases belong to the halo). When the only
well-studied globular cluster system was that of the Milky Way, it was
generally thought that this separation was because globular clusters were
fundamentally different from other star clusters, perhaps because of
conditions in the early universe (Peebles & Dicke, 1968; Fall & Rees, 1985).
However, it is possible to produce this apparent bimodality from clusters
formed in a single process, with the same cluster initial mass function. In
this picture, cluster disruption mechanisms, which are more effective at
destroying low-mass clusters in particular because of two-body relaxation
(Spitzer, 1958; Spitzer & Harm, 1958), would remove almost all of the low-mass
older clusters. If all clusters were born with similar cluster mass functions,
then we would expect to see the occasional high-mass young cluster. In fact,
we do see these in other galaxies. The “populous blue clusters” of the LMC
(Freeman, 1980; Hodge, 1981; Mateo, 1993) have been suggested as examples of
young objects which will evolve into globular clusters. M33 also has a few
likely massive young clusters (Chandar et al., 1999) , and such clusters have
been found in a number of normal isolated spirals (Larsen, 2004). It is
possible that the seeming absence of such objects in the Milky Way is merely
an observational selection effect; recently, there have been discoveries of
heavily reddened open clusters such as Westerlund 1, which likely has mass in
excess of $10^{5}$M⊙ (Clark et al., 2005).
What of M31’s clusters? While its clusters have been studied since the 1960s
(eg Kinman, 1963; Vetešnik, 1962; van den Bergh, 1969) and it was noted even
then that some of these clusters had colors indicating young populations,
their nature is still not entirely clear. van den Bergh (1969) called them
open clusters, while Krienke & Hodge (2007) adopted the simple convention of
calling any cluster projected on M31’s disk a “disk cluster” until proved
otherwise by kinematics. This avoids the question of whether young clusters
are fundamentally different from globular clusters in structure, formation,
etc. Our detailed study of M31 young clusters, incorporating kinematics,
should cast some more light on these questions.
It is only recently that detailed constraints on the mass, kinematics, age and
structure of cluster populations in M31 have been obtained, particularly for
clusters projected on the inner disk and bulge. Multi-fiber spectroscopy and
HST imaging have played an important role here. A number of M31 globular
cluster catalogs have been created over the years, giving a very heterogeneous
result, with significant contamination by both background galaxies, foreground
objects and even non-clusters in M31 itself. While the work of Perrett et al.
(2002), Barmby et al. (2000), Galleti et al. (2007), Kim et al. (2007), and
Lee et al. (2008) has gone a long way towards cleaning up the catalogs and
winnowing out the non-clusters, still more work is needed for both young and
old clusters. Here we present a new catalog of M31 star clusters which were
originally classified as globular clusters, all with updated high-quality
coordinates. We have observed a large number of these clusters with the MMT
and the Hectospec multi-fiber system. In this paper we study more than 100
young M31 clusters in detail. In subsequent papers we will address the
kinematics, ages and metallicities of the older clusters.
The M31 young clusters have been studied both by authors aiming to study its
open clusters (eg Hodge, 1979; Hodge et al., 1987; Williams & Hodge, 2001a;
Krienke & Hodge, 2007), and also by others who have aimed to study its
globular clusters. For example, Elson & Walterbos (1988) pointed out the
existence of young clusters in M31, estimated masses between $10^{4}$ and
$10^{5}M_{\odot}$, and drew attention to their similarity to the populous blue
clusters in the LMC. Barmby et al. (2000) noted the existence of 8 clusters
with strong Balmer lines in their spectra, which they tentatively classified
as young globular clusters. Beasley et al. (2004) (confirmed with a later
sample by Puzia et al., 2005) added more clusters, and commented that HST
imaging (then unavailable) was needed to distinguish between structure typical
of open and globular clusters. Burstein et al. (2004) added more new young
clusters, bringing the total to 19, and Fusi Pecci et al. (2005) increased the
sample to 67. In general, authors have associated these young clusters with
M31’s disk, although Burstein et al. (2004) invoke an accretion of an LMC-
sized galaxy by M31.
Observations are particularly challenging for clusters projected on M31’s
disk: many of the early velocities had large errors, and there were issues
with background subtraction. Here we discuss high-quality spectroscopic
measurements of kinematics and age for the young clusters, supplemented with
HST imaging to delineate the structural, spatial and kinematical properties of
these young clusters. We find that while they are almost all kinematically
associated with M31’s young disk, and their age distribution will allow us to
test suggestions that M32 has had a recent passage through M31’s disk (Gordon
et al., 2005; Block et al., 2006). The young clusters have structure and
masses which range all the way from the low mass Milky Way open clusters to
higher mass, more concentrated globular clusters, although they are dominated
by the lower concentration clusters. We will also discuss the likelihood that
these clusters will survive.
## 2\. Revised Catalog of M31 Clusters
The starting point for our cluster catalog was the Revised Bologna Catalog
(RBC) (Galleti et al., 2004, 2007), itself a compilation of many previous
catalogs. Coordinates from this catalog were used to inspect images from the
Local Group Survey (LGS) images (Massey et al., 2006), which cover a region
out to 18 kpc radius on the major axis and 2.8 kpc away from the major axis,
HST archival WFPC2 and ACS images, and the DSS for the outermost clusters. The
LGS images have median seeing $\sim$ 1 ˝ . We also added in some new clusters,
found visually by ourselves on the LGS images and on HST images (discussed
below). Even a casual inspection of the LGS images, particularly the I band
images, reveals the presence of a vast number of uncataloged faint disk
clusters, presumably similar to the galactic open clusters (Krienke & Hodge,
2007, estimate 10000 such clusters from HST images) . We have elected not to
take on the enormous task of cataloging and measuring those clusters in this
paper; rather we choose to deal with the more massive clusters for which some
information, however fragmentary, already exists. At a later stage in the
project, the catalogs of Kim et al. (2007), extracted from KPNO 0.9m images,
and Krienke & Hodge (2007), from HST images became available, from which we
collected objects not already in our own catalog and subjected them to the
same editing as we now describe. The archival images were thus used to answer
the following questions from the catalog we created from the RBC and our own
additions.
First, did the catalog coordinates correspond to a unique object? In cases
where the identification of the cluster on the Local Group Survey was
ambiguous or unclear, we consulted the original papers and finding charts
where these were available (Vetešnik, 1962; Baade & Arp, 1964; Sargent et al.,
1977; Battistini et al., 1980, 1987, 1993; Crampton et al., 1985; Racine,
1991; Auriere et al., 1992; Mochejska et al., 1998; Barmby & Huchra, 2001). In
some cases, there is no clear object that can be associated with the published
coordinates, or the nearest object in fact already had a different ID. The
large number of Hectospec fibers meant that we were able to verify
classifications of many low-probability candidates. The cataloging and
observation parts of this program occurred in a feedback fashion, allowing
some target names and/or coordinates to be changed for the succeeding
observations. As a result, we had some objects whose identifications were
incorrect; to these we add an “x” to their original name in our tables below.
Second, were the existing coordinates accurate? In general, the answer was no.
Our final catalog contains 1200 objects from the RBC, without considering the
newer candidates of Kim et al. (2007) and Krienke & Hodge (2007). 830 of those
required coordinate corrections larger than 0.5˝ to place them on the FK5
system used in the DSS and the LGS images. The median error in coordinates is
0.8˝ , with the largest error being of order 10˝ ; at which point the
identification of the actual object becomes uncertain. Similarly, there are
379 objects in the Kim et al. (2007) catalog found within 2˝ of an LGS object.
For these, the median error is 0.8˝ offset, where the largest error is 1.9˝ .
Many of the discrepant velocities between us and the RBC or Kim et al. (2007)
tabulations reported here are likely due to inaccurate coordinates used in
previous spectroscopic work. The coordinates newly derived from the LGS images
are accurate to 0.2˝ rms.
Third, were the targets really clusters? The LGS V and I band images, and
WFPC2 or ACS images taken with non-UV filters were used to confirm the cluster
nature of the objects, by visual inspection as well as the automated image
classifier contained in the SExtractor code (Bertin & Arnouts, 1996). A number
of cluster candidates were stellar on the LGS images; all of these were later
confirmed as stars in our spectra, from either the spectral characteristics or
the velocities, in regions of M31 where there is no confusion between the
local M31 velocity field and the velocity distribution of foreground galactic
stars. We found that about 90 of the Kim et al. (2007) candidates listed as
new, probable and possible (indicating that they appeared non-stellar in their
KPNO 0.9m images) appeared stellar on the LGS or HST images. Objects for which
we have no new classification data are kept in the catalog, but noted as still
in question in our table.
The large majority of the misclassified objects are stars (foreground galactic
or M31 members); more than 130 objects considered to be clusters as recently
as 2007 by Galleti et al. (2007) are in fact stars. Quite a number of
cataloged objects are background galaxies, and a few are either
unidentifiable, or are accidental clumpings of galactic or M31 member stars.
Figure 1.— The disputed cluster B314-G037. The LGS I band image is shown on
the left, next to the Cohen et al. (2005) AO image, taken in the K´ band. The
I band reveals the star cluster clearly (arrow), though the magnitude measured
for the cluster previously using a large aperture was certainly too bright.
Figure 2.— Locations of the spectra taken with Hectospec of M31 cluster
candidates. Confirmed clusters, stars, possible stars and background galaxies
are shown in blue, green, yellow and red, respectively. Figure 3.—
Hectospectra of young clusters in M31. Shown are spectra of clusters disputed
as real by Cohen et al. (2005), the average of three young clusters verified
by ACS images, and the average of single, blue supergiant stars as a
comparison. If the disputed clusters were in fact merely a few stars mistaken
for a cluster, their absorption line widths, particularly the Balmer lines,
would be narrow as seen in the blue supergiant spectra. These spectra have
been gaussian smoothed and have had their continuum shapes removed for ease of
display.
Cohen et al. (2005) have recently stated that four clusters that were
classified as disk clusters by Morrison et al. (2004) are “asterisms”. They
note that in their Adaptive Optics (AO) images there was no cluster visible –
generally there were only a few bright stars. However, for young clusters, red
supergiants would dominate the light at infrared wavelengths and the hotter
mainsequence stars would appear much fainter. We would need to use multi-
wavelength data to classify these objects correctly. We show below that the
optical spectra of those four clusters are indeed consistent with clusters of
massive, main-sequence stars, and although the magnitudes, and hence masses,
of these few objects were certainly overestimated, the objects will still be
considered as clusters in our catalog, at least until HST images show
otherwise. Figure 1 shows this complication for one cluster, by comparing the
high resolution, but long wavelength AO image with the LGS I band image. A
star cluster is clearly visible in the I band, and even more prominent at
bluer wavelengths - the object is indeed a young star cluster, though
certainly not a globular and not very massive. Our own HST images do reveal
two cataloged clusters as asterisms: these are comprised of a small number of
OB stars and late supergiants, resulting in a distinctive integrated spectrum
with strong Balmer and He I lines in the blue, and TiO bands in the red. Even
if these two are real clusters, the derived masses are small enough to exclude
them from a list of massive clusters.
Some cataloged objects have no real object even within a generous radius. In a
few cases, a nearby background galaxy had also led to confusion in previous
papers (though not in the most recent version of the RBC), whereby an actual
cluster was labeled as background. Thus, while for the most part we have
removed objects from the list of clusters, we have also restored a few objects
to the cluster list.
Table Star Clusters in M31: I. A Catalog and a Study of the Young Clusters
lists all objects believed to be clusters. For object names, we use the naming
convention of Barmby et al. (2000) where possible, where the name consists of
a prefix with the RBC number followed by the number of the object from the
next most significant catalog. Objects for which we have no new information
other than improved coordinates, and which have not been convincingly shown to
be clusters by previous workers, are italicized. Objects in the RBC which we
did not observe and for which our coordinates are within 0.5˝ of the RBC
coordinates are not listed here, nor are the Williams & Hodge (2001a); Kim et
al. (2007) or Krienke & Hodge (2007) cluster candidates that we did not
observe. Some objects that we did not observe could of course still be
background even though they have non-stellar profiles, but these, few in
number, are still listed here.
A rough classification based on the spectra is included in this table, for
objects with good quality spectra. “Young” clusters are those with ages less
than 1 Gyr, “interm” refers to those with ages between 1 and 2 Gyr, and “old”
refers to clusters older than that. A subsequent paper will provide a detailed
analysis and evidence that few if any of these “old” clusters have ages less
than 10 Gyr. “HII” indicates the spectrum is emission-line dominated. “na”
appears for objects known to be clusters from an HST image, but for which we
have no spectrum, or cases where the spectrum is too poor to determine the
age, even in a coarse manner. The V magnitudes come from this paper, using the
aperture size listed (see 5.3) or otherwise as indicated. Column C describes
what information was used to classify the target as a cluster. The
possibilities are “S”, where our spectrum clearly indicates a star cluster,
“L”, where the LGS image is non-stellar, and/or “H”, where an HST image
indicates a star cluster.
Table Star Clusters in M31: I. A Catalog and a Study of the Young Clusters
gives a list of those clusters that have ages less than 2 Gyr. (Sections 4, 5
and 6.4 of this paper will discuss the measurement of ages and masses for
these clusters.) Table Star Clusters in M31: I. A Catalog and a Study of the
Young Clusters lists objects from previous cluster catalogs that are in fact
stars. Many of these had also been classified as stars by previous workers.
Asterisms are also listed here. Table Star Clusters in M31: I. A Catalog and a
Study of the Young Clusters gives a list of possible stars. In these cases,
the Local Group Survey imaging indicates a stellar profile, but either we have
no spectrum, or the spectrum is ambiguous. Note that some of the stars in
Tables Star Clusters in M31: I. A Catalog and a Study of the Young Clusters
and Star Clusters in M31: I. A Catalog and a Study of the Young Clusters are
certainly members of M31. Objects thought to be clusters in the Kim et al.
(2007) catalog but which have stellar profiles in the LGS images are listed in
Star Clusters in M31: I. A Catalog and a Study of the Young Clusters, with
coordinates derived from the LGS. Table Star Clusters in M31: I. A Catalog and
a Study of the Young Clusters lists background galaxies. Table Star Clusters
in M31: I. A Catalog and a Study of the Young Clusters lists cataloged objects
where there was no obvious object within a reasonable distance of the
previously published coordinates, which are listed here again.
As a commentary on the difficulty experienced by all of those who have
endeavored to collect true M31 star clusters (including us), here is a brief
summary of the contents of the most recent version of the RBC, excluding the
additions of Kim et al. (2007); Williams & Hodge (2001a); Huxor et al. (2005)
and Krienke & Hodge (2007), but including the lists compiled by other
astronomers starting with Edwin Hubble. The RBC, restricted as just mentioned,
contains 1170 entries. We here, and others (particularly Barmby et al., 2000),
have provided classifications for 991 of these. Only 620 entries are actually
star clusters, while 20 more could be considered clusters though the large
amount of ionized gas present indicates the clusters may still be forming. 270
entries are stars, mostly foreground, and 224 entries are background galaxies
or AGN. At least 2 objects are chance arrangements of luminous M31 stars,
together which appear as clusters from the ground.
In the Kim et al. (2007) catalogs, there 113, 258 and 234 “new”, “probable”
and “possible” clusters, respectively. The LGS survey contains images of 94,
152 and 105 members of those catalogs, respectively. Of those subsets, 79,
106, and 129, respectively have non-stellar profiles in the LGS, some of which
were observed with Hectospec.
## 3\. Hectospec Observations
The Hectospec multi-fiber positioner and spectrograph is ideally suited for
this project, in that its usable field is 1 degree in diameter, and the
instrument itself is mounted on the 6.5m MMT telescope. We obtained data in
observing runs in the years 2004 to 2007, and now have high-quality spectral
observations of over 400 confirmed clusters in M31, and lower quality spectra
of 50 more. We used the 270 gpm grating (except for a small number of objects
taken with a 600 gpm grating), which gave spectral coverage from 3650–9200Å
and a resolution of $\sim$5Å. The normal operating procedure with Hectospec,
and other multi-fiber spectrographs, is to assign a number of fibers to blank
sky areas in the focal plane, and then combine those in some fashion to allow
sky subtraction of the target spectra. For instance, the 4-5 fibers nearest on
the sky to the target may be combined. These methods are satisfactory for our
outer M31 fields, but not for the central areas where the local background is
high relative to the cluster targets. For those fields, we alternated
exposures on-target and off-target to allow local background subtraction to be
performed. Many of the discrepancies between our bulge cluster velocities and
those of previous workers might be explained by their lack of proper
background subtraction, and/or inaccurate target coordinates.
We obtained exposures for 25 fields, with total exposure times varying between
1800s and 4800s. The signal/noise ratios for the 500 objects we classified as
clusters have a median of 60 at 5200Å and 30 at 4000Å, with more than 100
clusters having a ratio at 5200Å better than 100. Figure 2 shows the locations
and types of objects observed in all of these fields.
The multifiber spectra were reduced in a uniform manner. For each field, or
configuration, the separate exposures were debiased and flat fielded, and then
compared before extraction to allow identification and elimination of cosmic
rays through interpolation. Spectra were then extracted, combined and
wavelength calibrated. Each fiber has a distinct wavelength dependence in
throughput, which can be estimated using exposures of a continuum source or
the twilight sky. The object spectra were thus corrected for this dependence
next, followed by a correction to put all the spectra on the same exposure
level. The latter correction was estimated by the strength of several night
sky emission lines. Sky subtraction was performed, using object-free spectra
as near as possible to each target. For the targets where local M31 background
was high, the method was reversed, such that only sky spectra far from the
disk of M31 were used. An offset exposure for such fields, taken concurrently,
was reduced in a similar way (thus contemporaneous sky subtraction was
performed for on- and off-target exposures), and then the off-target spectra
were subtracted from the on-target. This process increased the resultant noise
of course, but we deemed it essential for targets in the bulge and disk of
M31. The off-target exposures have the additional advantage of giving
measurements of the unresolved light in over 800 locations over the entire
disk of M31.
Velocities were measured using the SAO xcsao software. Given the wide variety
of spectra in this study, it was deemed necessary to develop new velocity
templates, from the spectra themselves. The procedure was to derive an initial
velocity of all spectra using library templates (typically a K giant star).
The spectra were shifted to zero velocity, and sorted into three different
spectral types, A, F and G type spectra. The best spectra in each group were
combined to make new templates, and the procedure was repeated now using the
new templates. By using these templates we have assured that all the M31
targets are on the same velocity system. They are tied to an external velocity
in the initial step, whose accuracy depended on the accuracy of the initial
set of templates used. A good test of the internal accuracy was provided by
repeat measurements of clusters. We have 386 repeat measurements (on different
nights) for 224 clusters. The median difference in velocity for these repeats
was 0.5 km s-1 , with an implied median single measurement error of 11 km s-1
(smaller than our formal errors listed in the tables). We will present
external comparisons in a subsequent paper, but note that the cluster
velocities agree very well with the HI rotation curve (see 6.3). Velocities
for the young clusters, stars and galaxies are presented in Tables Star
Clusters in M31: I. A Catalog and a Study of the Young Clusters, Star Clusters
in M31: I. A Catalog and a Study of the Young Clusters, Star Clusters in M31:
I. A Catalog and a Study of the Young Clusters, and Star Clusters in M31: I. A
Catalog and a Study of the Young Clusters. Velocities for the old clusters
will presented in a subsequent paper.
The spectra were corrected to relative flux values, using observations of
standard stars (the MMT F/5 optics system employs an atmospheric dispersion
compensator, ADC). The flux correction has been determined to be very stable
over several years. Thus, observations of the same targets taken in different
seasons can be combined where available.
## 4\. Using HST images to Determine Cluster Classification
Cohen et al. (2005) highlighted the heterogeneous quality of the M31 cluster
catalogs when they claimed, using AO techniques at K´ , that four out of six
observed young clusters were in fact asterisms. Figure 3 shows, from top to
bottom, spectra of the four disputed clusters, the average of three genuine
young clusters verified by ACS images, and the average spectra of single
supergiant stars (these were verified to be stars from the LGS images, and
members of M31 from their velocities). If the disputed clusters were in fact
merely a few stars, those stars would have to be supergiant stars, whose
absorption line widths would be as narrow as seen in our blue supergiant
spectra. This is not the case for the four disputed clusters (note in
particular the H$\beta$ and H$\delta$ widths, narrow in the stars and wide in
the other spectra), and we conclude that those objects are true clusters and
not asterisms. To be sure, these particular clusters are not globular clusters
either, and, additionally, are perhaps not massive enough to be considered
young, populous clusters.
High spatial resolution imaging can both check for asterisms and also explore
the clusters’ spatial structure: is their concentration low, like typical
Milky Way open clusters, or high, like globular clusters?
There are ACS or WFPC2 images available for 25 of the clusters with ages less
than 2 Gyr. Two of these show no evidence of an underlying cluster, but the
remaining 23 are clearly not asterisms. Figure 4 shows the range of structure
seen in these young clusters. While many of them show the low-concentration
structure typical of Milky Way open clusters, a number of them, such as
B374-G306 and B018-G071, are quite centrally concentrated, like the majority
of the Milky Way globular clusters.
Figure 4.— HST ACS or WFPC2 images of a selection of clusters with ages less
than 2 Gyr, except for VDB0 whose image is from the LGS V band. The HST images
were taken in either F555W or F606W. The spectroscopically determined ages in
Gyr are listed for each image, as is the camera used (“A”=ACS, “W”=WFPC2), to
aid in comparison since in general the ACS images are deeper. A comparison old
M31 globular cluster is shown, as are two candidates that turned out to be
asterisms. The scale is the same for all images; except for the two small
ones, each image is 10˝ on a side. The images are ordered by descending mass,
starting at the top left.
Barmby et al. (2007) have measured the structure of M31 clusters with
available HST imaging at the time of publication. There are 70 clusters in
their sample which we have classified as old, and 7 of our young clusters. It
is straightforward to compare the structure of the clusters they study with
the Milky Way globulars, because they use the same technique as McLaughlin &
van der Marel (2005), who have produced a careful summary of the structure of
the Milky Way globular clusters. However, it should be noted that their
fitting technique (fitting ellipses to cluster isophotes) is not well-suited
to very low-concentration clusters, and in fact one of our young clusters,
B081D, is omitted from their analysis because of its low density.
Figure 5 compares their results for old clusters from our sample with the
structure of Milky Way globulars (from the work of McLaughlin & van der Marel,
2005). It can be seen that the concentration parameter111$c$ =
log($r_{t}/r_{0}$) for King model fits, Binney and Tremaine (2008) p. 307 for
the old clusters in M31 has a similar distribution to the Milky Way globulars.
We note that although there are no old clusters in our sample with
concentration greater than 2.2, such clusters are definitely present in the
Barmby et al. (2007) sample so this absence is unlikely to be significant. The
similarity in structure is interesting, because at first sight it would seem
that the M31 clusters with high concentration would be preferentially
discovered in surveys. Perhaps the M31 globular cluster surveys (which, as we
have seen above, include a large number of non-globular clusters, as well as
the low-concentration young clusters) are now sufficiently thorough that they
are not strongly affected by this bias.
Figure 5.— Histograms of the concentration parameter from King model fits for
(top panel) old M31 clusters (young clusters shown by asterisks, the young
cluster in NGC 205 by a 5-pointed star), (middle panel) Milky Way open
clusters and (bottom panel) Milky Way globular clusters. It can be seen that
the M31 old clusters resemble the Milky Way globulars in their concentration,
while the M31 young clusters cover the range of both open and globular
clusters.
Although only seven of our young M31 clusters were analyzed by Barmby et al.
(2007), it can be seen from Figure 5 (where they are marked by asterisks and
five-pointed stars in the top panel) that their concentrations cover the whole
range of the older clusters in M31 and in the Milky Way. (The five-pointed
star represents Hubble V from NGC 205.) Our observational selection biases may
over-emphasize high-concentration clusters, but it is still interesting to see
that three of the young clusters have quite high concentration parameters. How
does their structure compare with the Milky Way open clusters? It is quite
difficult to answer this question because the available samples of Milky Way
open clusters are severely incomplete, and it is a challenging task to fit
King models to the known open clusters, because cluster membership is hard to
determine. The 2MASS database (Beichman et al., 1998; Jarrett et al., 2000)
has been used by Bonatto & Bica (2005); Santos et al. (2005); Bonatto & Bica
(2007); Bica et al. (2006); Bica & Bonatto (2008) to measure the structure of
21 open clusters. They used a CMD-fitting technique to remove contamination
from disk field stars. We have also used data from Eigenbrod et al. (2004),
who used radial velocity to decide membership. Because of the high background
in all these cases, it is possible that the “limiting radius” given by the
authors is in fact smaller than the tidal radius, in which case the cluster
concentrations would be smaller than those plotted. It can be seen in the
middle panel of Figure 5 that all these open clusters have quite low
concentrations. However, the sample of clusters with concentration
measurements is quite small, and it is quite possible that there are a few
open clusters in the Milky Way which are yet to be discovered and have high
concentrations, like the two M31 young clusters.
In summary, the young clusters in M31 show a range of structure. Most have the
low concentration typical of Milky Way open clusters, but there are a few
which have high concentrations, like most Milky Way globulars. We note that
any survey of M31 clusters will preferentially discover ones with high
concentrations. In addition, the incompleteness of Milky Way open cluster
samples and the difficulty of measuring cluster concentration in crowded
fields means that we cannot rule out the existence of such clusters in the
Milky Way.
## 5\. Ages of the Young clusters
In this section, we describe methods for determining ages from the spectra and
color magnitude diagrams for the verified clusters. Since the emphasis in this
paper is on the younger clusters, and more specifically, on their M/L ratios,
our task is first to distinguish young from old clusters, and then to obtain
accurate age measurements among the younger clusters. A more refined age
determination (for clusters older than 2 Gyr) is postponed for a later paper.
### 5.1. Ages from Spectra
Figure 6.— Spectra of a sample of young clusters, ranging in ages from 0.04 to
1 Gyr. Each object spectrum is shown with the best matching SB99 model
spectrum. The object spectra have been smoothed with a gaussian with a sigma
of 1.1Å and have been corrected for the reddening determined in this paper,
which was itself determined by matching the continuum shapes of objects and
models. Figure 7.— Mavg vs Havg for M31 cluster spectra, with different types
identified. Model indices derived from SB99 models using Padova Z=0.03
isochrones are also shown as a solid curve. Ages in Gyr are marked at selected
areas along the curve. Median error bars are shown at the right for young and
old clusters separately. The maximum errors on points in this plot are 0.45
for Mavg and 0.19 for Havg. Figure 8.— Same as Figure 7 for H$\delta/$Fe4045
and CaII. Maximum errors on points in this plot are 0.2 for both indices.
The methods for obtaining ages for young stellar populations from their
integrated spectra are similar to those used for older stellar populations,
except that instead of employing empirical stellar libraries (e.g., Worthey,
1994), modelers focussing on younger stellar populations have used synthetic
spectra, partly due to a lack of empirical spectra of young stars over a range
of metallicities. Here we have made use of the Starburst99 stellar population
modeling program (SB99, Leitherer et al., 1999).
To distinguish young from old clusters, we compared our cluster spectra with
two external sets of spectra, which served as population templates. One set
was the sample of 41 MW globular spectra obtained by Schiavon et al. (2005),
covering the abundance range of $-2<$[Fe/H]$<0$. These spectra have a
wavelength coverage from 3500 to 6300Å , a dispersion of 1Å/pixel and a
resolution of about 4.5 Å, and served as our old population templates. Our
young population templates were created from the SB99 program using the Padova
Z=0.05 interior models and Z=0.04 stellar atmospheres, since it seemed likely
the young clusters have supersolar abundances. Using a solar abundance set of
isochrones in the models has the net effect of increasing the derived ages for
clusters older than 0.1 Gyr, in a logarithmic scaling such that a 1 Gyr
supersolar model and a 2.5 Gyr solar abundance model have nearly the same
resultant spectra. A grid of 40 high resolution spectra was created for ages
from zero to 2 Gyr, with a logarithmic time step of 0.07 between models.
Our approach then was to simplify the old populations by only using a set of
MW globular cluster templates, and to simplify the young populations by
restricting ourselves to a single metal-rich chemical composition. The
evolution of integrated spectra is essentially logarithmic with time; for
example, the difference between a 5 and 12 Gyr population is relatively small
compared to the large changes which occur over the first 1 Gyr. Thus our
simplification seems warranted, particularly since in this paper we are
concerned only with identifying the younger clusters and then studying them in
detail. The major issue, as seen below, in separating old from young clusters,
is the potential degeneracy between very metal-poor old populations and young
populations, both of which are dominated by Balmer lines in their integrated
spectra.
Both sets of template spectra were rebinned to the same resolution and
dispersion as the M31 set, and then each template spectrum and M31 spectrum
had a low order fit subtracted. The significance of this step was that we did
not use the continuum shape to help determine the best matching template. A
scaling factor was then determined between each template and all the available
M31 cluster spectra, and the reduced $\chi^{2}$ calculated over the spectral
ranges of 3750-4500, 4750-5000, and 5080-5360 Å. These ranges were chosen to
exclude spectral regions where there are few lines, as well as where the MW
cluster spectra have no data due to bad columns in the CCD used for that set.
Specifically included were the Balmer lines from H$\beta$ to H$\zeta$, the Mg
b lines, Ca II H&K, and the He I lines at 4009 and 4026Å, the last being
prominent in OB stars and thus strong in the youngest clusters. The noise used
in the reduced $\chi^{2}$ calculations was that calculated from the spectra
themselves.
A few logic decisions had to be made in analyzing the resultant list of
reduced $\chi^{2}$ values. For most cases, the lowest reduced $\chi^{2}$
occurred clearly either in the SB99 set or the MW cluster set, thus allowing a
particular cluster to be identified as young or old. For the young clusters,
the age chosen was the average of ages where the reduced $\chi^{2}$ was within
10% of the lowest value. About 10% of the clusters were equally well fit by a
MW cluster, typically a metal-poor one with [Fe/H] $<-1$, and a metal-rich
SB99 model. Nearly all of these are low S/N, and thus the poor fits were not
surprising. Visual inspection of the spectrum clarified 11 of these as being
very young (with very blue continuum shapes), and the remaining 48 we grouped
as old.
Figure 6 compares the data and best-fitting models for a range of determined
ages. We estimate the errors in determined ages for the young clusters to be
about a factor of two, which leads to M/L uncertainties of 50%.
The model M/L values for each chosen age then allowed masses to be estimated
from the cluster integrated V band photometry, which is described in 5.3. To
allow a comparison of young clusters to be made, we also calculated
spectroscopic M/L values for the old M31 clusters from the models in Leonardi
& Rose (2003), by obtaining estimates of [Fe/H] for each cluster via Fe and Mg
line indices and assuming an age of 12 Gyr for each. The detailed analysis of
the old cluster spectra will be presented in future paper.
Many readers may be more familiar with extracting ages from diagrams that plot
a largely age-sensitive index versus a largely metallicity-sensitive index. To
help demonstrate the efficacy of the $\chi^{2}$ approach we show two such
diagrams. Figure 7 is a plot of a Balmer line index versus a metal line index.
We have defined Mavg=(Fe5270+Mgb)/2 and Havg=(H$\delta_{F}+$
H$\gamma_{F}+$H$\beta$)/3, all Lick indices (Worthey & Ottaviani, 1997). These
indices are equivalent widths: units are Å. For clarity in these diagrams, a
signal-to-noise ratio cutoff was made, which eliminated 20% of the clusters
from being plotted. Different symbols represent MW globular clusters (data
from Schiavon et al., 2005), and four age bins for M31 clusters, which were
determined by the $\chi^{2}$ method: very young ($<$ 0.1 Gyr), young (0.1 $<$
age $<$ 1 Gyr), intermediate (1 $<$ age $<$ 2 Gyr), and old. Clearly, there is
a sequence of M31 clusters that closely matches the sequence of MW globular
clusters, and a large spray of clusters that fall mostly in regions of higher
Balmer equivalent width. Clusters with younger ages will of course have
stronger Balmer absorption, until the very youngest ages are reached, at which
point the Balmer strength declines again.
The second diagram (figure 8) uses indices defined in Leonardi & Rose (2003),
namely the ratio of residual light in the H$\delta$ line to the nearby Fe4045
line, and the ratio of the line at 3969Å which contains both H$\epsilon$ and
CaII H, to the CaII K line. The indices are unitless. This diagram also shows
the old cluster metallicity sequence and again distinguishes them from the
young clusters, again excepting for the extremely young clusters.
The shortcoming of using such diagrams is that occasionally, due to the nature
of real data, one index will be bad, causing the cluster to look young or old
in one diagram, and the opposite in another diagram. We are thus more
confident in a fitting procedure that uses many diagnostic lines, but are
gratified that in the vast majority of cases, these two diagrams verify the
ages assigned by the $\chi^{2}$ method.
### 5.2. Ages from HST/ACS color magnitude diagrams
Williams & Hodge (2001a); Williams & Hodge (2001b) estimated ages of many
young disk clusters in M31 from WFPC2 color-magnitude diagrams (CMD) and
isochrone fitting to the main sequence or to luminous evolved stars. Four of
their clusters are bright enough to be in our spectroscopic study. Their ages
agree quite well with ours (Table Star Clusters in M31: I. A Catalog and a
Study of the Young Clusters). Additionally, as part of HST GO proposal 10407,
we obtained ACS images of several young clusters, three of which we report on
here.
The multidrizzle package (Koekemoer et al., 2002) was used to combine the 3
individual exposures taken in the F435W and F606W filters (corresponding to B
and V, respectively). Stellar photometry was obtained using the DAOPHOT
package of Stetson (1987), modeling the spatially variable PSFs for each of
the combined images separately, using only stars on those images. PSFs were
constructed using 5-10 bright stars which had no pixels above a level of
20,000 counts, the point at which an ostensible non-linearity set in and the
PSF no longer matched those of fainter stars. Aperture corrections were also
measured using these stars, to determine any photometric offset between the
psf photometry and the aperture magnitude within 0.5˝. Sirianni et al. (2005)
have provided aperture corrections from that aperture size to infinity, in all
ACS filters. Generally, two passes of photometry were run. First, a star list
was made and entered into ALLSTAR, which aside of the photometry, produces a
star-subtracted image. Stars missed in the first round were located in the
subtracted image and added to the original list. The original frame was then
measured again by ALLSTAR. The photometry was then placed on the standard
Johnson/Kron-Cousins VI system using the aperture corrections and synthetic
transformations provided in Sirianni et al. (2005). To lessen the severe
problems with crowding in these clusters, only stars that fall in an annulus
with radii of 15 and 50 pixels (0.75 and 2.5˝ ) are shown in the color-
magnitude diagrams (figure 9). The background field shown has the same area as
the cluster fields, and refers to an annulus around B049-G112 with inner
radius of 60 pixels. Isochrones with super solar abundances from the Padova
group (Cioni et al., 2006a, b) have been placed in the diagrams to allow age
determination, using a distance modulus of M31 of 24.43 and the reddenings
determined above for these two clusters (0.25 for both). These CMDs and that
of a third we have worked on (B367-G292) give ages in reasonable agreement
with those from the spectroscopic analysis (Table Star Clusters in M31: I. A
Catalog and a Study of the Young Clusters).
Figure 9.— Color magnitude diagrams obtained from HST/ACS F435W and F606W
images. Shown are two young clusters and the background field for B048-S112.
Isochrones for a range of ages at supersolar metallicity come from Cioni et
al. (2006a, b). The inset legend lists the isochrone ages shown, in log years.
A distance modulus of 24.43 and reddenings of 0.25 mag have been assumed in
positioning the isochrones. Median photometric errors at the five indicated
magnitudes are shown in the middle panel, but refer to all three panels. From
these diagrams, we have derived an age of 0.35 Gyr (log age=8.55) for
B049-G112, and 0.25 Gyr (8.4) for B458-D049.
### 5.3. Cluster integrated photometry
Multicolor photometry for most of the clusters in this project is already
collated in Barmby et al. (2000), but enough clusters are missing to warrant
remeasuring all the clusters, to allow the photometry to be used with the
spectroscopic M/L values to obtain masses. The LGS survey of M31 consisted of
10 separate but overlapping fields. Stellar photometry from these fields using
psf-fitting has been reported in Massey et al. (2006), but the aperture
photometry needed for resolved star clusters has not yet been reported. To
limit the scope of the work, we elected to measure objects in our catalogs
only in the V band. Targets from our entire catalog were located on the
images, and photometry for 12 separate apertures ranging from 0.7 to 16˝
(spaced logarithmically) was collected using DAOPHOT. Growth curves from these
apertures were constructed, and used in an automatic fashion to estimate the
aperture which enclosed the total light of the cluster. These apertures were
then inspected on the images and increased, if the clusters were in fact
larger, or decreased for cases where the apertures included substantial light
from objects clearly not part of the clusters. The local background was
measured in annuli with inner radii 1 pixel larger than the outer radius of
the aperture for the object. The apertures used are listed in Table Star
Clusters in M31: I. A Catalog and a Study of the Young Clusters. Extraneous
stars remaining in the apertures were accounted for by measuring their
magnitudes separately, and subtracting their contribution to the cluster
aperture magnitudes. The resultant instrumental magnitudes were then placed on
the standard V system using stars from the Massey et al. (2006) tables which
we were measured in the same way as the clusters. The color term in the V mag
transformation (described in Massey et al., 2006) was ignored, as it was
smaller than the errors we report. Tables Star Clusters in M31: I. A Catalog
and a Study of the Young Clusters, Star Clusters in M31: I. A Catalog and a
Study of the Young Clusters, and Star Clusters in M31: I. A Catalog and a
Study of the Young Clusters list the results from this work. The formal errors
in the standardized photometry were less than 0.03 mag, set by the
uncertainties in the transformations. However, since our goal was actually
cluster total magnitudes, our calculated uncertainties refer to the
uncertainty in setting the proper apertures and correcting for extraneous
objects within those apertures. Specifically, the uncertainties were set to be
equal to the difference in the magnitude of the aperture chosen as best
representing the limiting radius of the cluster, and that of the next larger
aperture in our logarithmic spacing of aperture sizes. In practice this means
that clusters in crowded fields have larger uncertainties than those in less
dense areas. The tables also list photometry from other sources for the
objects that are outside of the LGS images; we do not list the errors in such
cases.
Comparing our aperture magnitudes with those collected from various sources
and listed in the RBC, we find excellent agreement over the magnitude range of
14 to 18, with an rms in the differences of 0.18 mag in the set of 200 objects
in common; this in spite of the fact that no effort was made to insure that
the apertures used in the two data sets were the same. Between V=18 and 20,
our photometry tended to be fainter by 0.2 mag and the scatter increased to
0.5 mag, some of which was likely due to differences in object identification.
A further comparison was made with the 58 V magnitudes measured for M31
clusters from archival HST images in Barmby et al. (2007). Aside of one
cluster whose V magnitude appears to be a typo (B151-G205), the rms of the
differences of that set with the magnitudes presented here is 0.28, with no
apparent systematics.
## 6\. The Nature of the Young Clusters
### 6.1. Misclassified globular clusters
Morrison et al. (2004) misclassified 15 of the young clusters as old disk
globular clusters (17% of their disk globulars). This was pointed out by
Beasley et al. (2004). In some cases this was due to the low S/N of the WYFFOS
spectrum, in others the problem was misinterpretation of the spectra. Our new
study of the clusters has resolved most of this confusion and changed the
classification of a number of clusters from old and massive to younger, not
very massive. However, we still find clusters with significant masses, above
$10^{4}$M⊙, and with ages less than 150 Myr (see Table Star Clusters in M31:
I. A Catalog and a Study of the Young Clusters). Of the 10 clusters with those
physical characteristics, the HST or LGS images of three confirm them as
clusters similar in appearance to the populous clusters of the LMC (these are
B315-G038, B318-G042, and VDB0, the latter still the most massive, young
cluster known in M31, van den Bergh (1969)). Five appear more like OB
associations, and thus may not survive as bound clusters (B319-G044,
B327-G053, B442-D033, B106D and BH05). The case for the other two (B040-G102,
B043-G106) is not as clear, but their LGS images are more similar to the cases
like B315-G038 than to the OB associations.
### 6.2. Position
Figure 10.— Spitzer/MIPS 24 micron imaging (in red on top of the optical DSS
image), showing the location of clusters with ages less than 0.1 Gyr as
violet, between 0.1 and 0.32 Gyr as blue, and between 0.32 and 2 Gyr as green.
Figure 10 shows a Spitzer/MIPS 24$\mu$ mosaic of M31 (Gordon et al., 2005)
with the positions of the young clusters overlaid. Clusters younger than 0.1
Gyr, between 0.1 and 0.32 Gyr, and 0.32 and 2 Gyr are shown in different
colors. The latter two groupings divide the clusters older than 0.1 Gyr into
two equal parts. It can be seen that the spatial distribution of the young
clusters is well correlated with the star-forming regions in M31, with the
majority associated with the 10 kpc “ring of fire”. The comparison of these
young clusters and the warm dust emission is distinct from the comparison of
the latter with the location of HII regions, as we have excluded the clusters
embedded in HII regions from our sample.
Gordon et al. (2005) and Block et al. (2006) use the curious appearance of the
mid-IR ring - the split near the location of M32, creating the appearance of a
“hole” in between, and the possible offset of the ring from the nucleus - to
suggest a recent encounter of M32 and M31. Both groups suggest that the split
is caused by M32’s passage through the disk, and their models also produce
rings offset from M31’s center, albeit not as extreme as is observed (Gordon
et al., 2005, produce an offset of 1′, not 6′). An examination of our Figure
10 shows that the cluster distribution favors the outer parts of the hole, and
is generally quite symmetric about M31’s nucleus. Most models of an offset
ring assume that the inner part of the split in the observed ring is the one
which should be traced by star formation. However, this is not where most of
the clusters are found.
Figure 11.— Ages of young clusters as a function of distance from the center
of M31, in the plane of its disk.
Gordon et al. (2005) need a very recent interaction between M32 and the disk
(their model has the disk passage occurring $2\times 10^{7}$ years ago)
because the passage of M32 through the disk in their model results in a burst
of star formation that propagates outward through the disk. However, we do not
see any radial trends with cluster age, which might be expected with a
propagating ring of star formation (Figure 11).
Block et al. (2006) prefer a collision about $2\times 10^{8}$ years ago, which
triggers expanding density waves. Our young cluster ages range from 0.04 Gyr
to 1 Gyr, but most are between 108 and 109 years. If the “ring of fire” was
produced by a single event, as modeled by the above authors, we might expect
the age distribution of clusters associated with it to be more peaked. The
ages of the younger clusters presented in Williams & Hodge (2001a) range from
around $2\times 10^{7}$ to $2\times 10^{8}$ years, and there is little
evidence of a peak in star formation in this age range either. These results
seem to suggest that star formation has been fairly high in this region of M31
for 1 Gyr or more. In summary, we see no evidence of enhancement in star
formation rate or any spatial age separation, as we might expect from the M32
disk passage.
### 6.3. Kinematics
Do the kinematics of the young clusters bear out the disk origin suggested by
their close association (in projection) with star forming regions in M31’s
disk? M31’s inner disk kinematics are more complex than originally supposed,
due to M31’s bar (Beaton et al., 2007; Athanassoula & Beaton, 2006). The
velocities from our many sky fibers, taken both as a part of regular observing
and also from entire exposures devoted to offset sky in the crowded inner
regions, give us a new way of quantifying disk rotation throughout the inner
regions where the young clusters are found. (We plan to use these data in a
study of bar kinematics, Athanassoula et al. in preparation). Figure 12 shows
the disk mean velocity field.
Figure 12.— Disk mean velocity field, obtained from sky fibers. The color bar
at bottom shows the velocity scale in km/s.
We also use kinematics of HII regions that we observed as fillers in the
Hectospec fields to give us an indication of the kinematics of young disk
objects. These data will be published in Athanassoula et al. (in preparation).
Figure 13 shows the kinematics of the mean disk light, the young clusters, and
HII regions, versus major axis distance X, in kpc. We have split the sample
into objects which are close to the major axis ($<$ 1 kpc) and those projected
from $1-2$ kpc from the major axis, because the projection of a circular orbit
looks different in these two cases. Objects on the major axis in circular
orbits have all of their velocity projected on the line of sight; as we move
further away from the major axis, less of the circular velocity is projected
on the line of sight and so the tilt of the distance-velocity line becomes
smaller. This can be seen clearly in Figure 13 for all types of objects.
Figure 13.— Plot of major axis distance X vs velocity with respect to M31, for
objects within 1 kpc of the major axis (upper panel) and for objects between 1
and 2 kpc from the major axis (lower panel). Mean disk velocities obtained
from sky fibers which show absorption spectra are shown in black, from sky
fibers which show emission in green. HII region velocities are shown in cyan
and young cluster velocities (from Table Star Clusters in M31: I. A Catalog
and a Study of the Young Clusters) in red. The rotation curve from Kent (1989)
is shown as a solid line.
The curious flattening of the mean velocities from absorption line spectra for
major axis distances between 3 and 10 kpc is likely to be caused by the bar.
It can be seen that the young clusters follow the disk mean velocity curve
from absorption spectra quite well, and show an even better correlation with
the kinematics of the youngest objects: HII regions and sky fibers showing
emission spectra.
This kinematic analysis confirms our spatial association of the young clusters
with the star forming young disk in M31.
### 6.4. Masses of the clusters
The M/L values obtained from the spectroscopic age estimates can be combined
with the V band photometry to derive masses of all the observed M31 clusters,
young and old. Reddening values are of course also needed, and a large number
of E(B–V) values were derived from photometry in Barmby et al. (2000). They
and we consider only the total reddenings, foreground and internal to M31. The
methodology used in Barmby et al. (2000) meant that the reddenings would only
be valid for old clusters, and indeed few of the clusters we have identified
here as young were included in their study, thus reddenings for those objects
are needed. Therefore, we elected to rederive the reddenings for all of the
clusters in our study, young and old.
In the case of the young clusters we compared the fluxed spectra with the SB99
model spectra of the appropriate age. As described above, the ages were
obtained by matching spectral line features in the observed and model spectra,
not by comparing the continua shapes. Once the ages have been found in this
way, differences in the continua shapes may be assumed to be due to reddening,
except for a few cases where a late-type star, whether member or not, clearly
dominates the redder wavelengths as evidenced by the presence of TiO bands.
For those cases, we use the mean reddening for the young clusters of 0.28.
For the old clusters we did not use models, but rather the sample of spectra
themselves. Initial values of reddenings were obtained from Barmby et al.
(2000), and were used to deredden the spectra of those clusters with
E(B$-$V)$<0.4$, about 190 in number (there are about 350 old clusters in our
spectroscopic sample). These spectra were ordered in metallicity, which was
estimated from the spectral line indices as mentioned above, rebinned to a
coarse grid in wavelength, and normalized to have the same intensity at the
arbitrary wavelength of 5000Å. Interpolation formulae were developed from
these spectra, via a least squares method to avoid bad spectra, for intensity
as a function of both wavelength and metallicity. As a result, a cluster
spectrum of arbitrary metallicity could be created, dereddened to the accuracy
of the Barmby et al. (2000) reddenings. The individual spectra in this low
reddening sample were then compared with the appropriate interpolated spectra,
and reddenings were adjusted as needed to bring their continua shapes closer
to that of the expected template shape. The method is thus similar to methods
that use the metal abundance to predict the intrinsic broad band colors, and
then require the derived reddening to reproduce the observed colors.
The overall goal in working with the low reddening sample was to retain the
mean value of the reddenings found in Barmby et al. (2000), but to correct
those that varied significantly. After cleaning up those reddenings, the
interpolation formulae were then used to derive reddenings for the 150
clusters for which we have spectra and whose reddenings were not measured in
Barmby et al. (2000). Thus while we have not improved upon the absolute levels
of the M31 old cluster reddenings, we believe we have improved the precision
of the values in a relative sense, and as well have nearly doubled the number
of reddenings available. About 10% of the spectra were taken during nights
when the ADC was not operating properly, thus we can’t use the continuum shape
to estimate reddenings. For objects whose only spectra were taken on those
nights, we assume the average reddening of 0.28.
A comparison of our derived E(B$-$V) values (which range up to 1.4 mag) and
those in Barmby et al. (2000) results in a scatter of 0.17 mag rms, which is
good enough for our overall goal of comparing the M31 cluster system in bulk
with that of other galaxies. Interestingly, both the young cluster and old
cluster groups have clusters with E(B$-$V)$>0.5$, though the highest measured
value ( E(B$-$V)=1.4) is still found in the old cluster B037-V327, probably a
selection effect since that cluster also has the highest luminosity in all of
M31. The young cluster reddenings are listed in Table Star Clusters in M31: I.
A Catalog and a Study of the Young Clusters; those of the old clusters will be
presented in a subsequent paper. By using the position of blue-plume stars in
the color-magnitude diagram, Massey et al. (2007) estimated the average
reddening for young stars in M31 to be 0.13 mag, significantly lower than the
mean of the clusters younger than 100 Myr presented here, which may place a
constraint on the accuracy of the values presented here.
Figure 14.— Mass histograms, from top to bottom, M31 clusters (young,
intermediate and old), Milky Way open clusters, LMC massive clusters and Milky
Way globular clusters. The young M31clusters are shown in a hatched histogram,
the intermediate as solid, and the old as open.
The mass histogram for the all of the young clusters is shown in Figure 14. We
have also shown the mass distribution of Milky Way open clusters within 600 pc
of the Sun. This is based on the sample of Kharchenko et al. (2005), with mass
calculations by Lamers et al. (2005). The Kharchenko catalog is the most
homogeneous and complete catalog of open clusters in the solar neighborhood
currently available, and is based on a stellar catalog complete to V=11.5. The
cluster masses were estimated by counting the number of cluster members
brighter than the limiting magnitude, then correcting for the stars fainter
than this using a Salpeter mass function and a lower mass limit of 0.15 M⊙.
This catalog does not include the most massive clusters in the Galaxy because
of its relatively small sample size; for example, there have been recent
discoveries of more distant young clusters which may have masses as high as
$10^{5}$M⊙ (e.g. Clark et al. (2005)), and we add Westerlund 1 to the
histogram as an example. The Milky Way globular and LMC young massive cluster
histograms are shown in the bottom two panels (from McLaughlin & van der
Marel, 2005). These mass estimates are based on King model fits.
Obviously, M31 clusters with masses less than $\approx 10^{3}$M⊙ and ages
greater than a few $\times 10^{7}$ years are too faint to be part of this
study, and await a future study. Krienke & Hodge (2007) estimate over 10000
such clusters in the disk of M31; these would form the low mass tail in the
mass distribution of Figure 14.
Nonetheless, there is a trend in cluster mass, with the Milky Way open
clusters having the lowest median mass, the Milky Way and M31 globulars the
highest, and the LMC young massive clusters and the M31 young clusters in
between. This trend is consistent with a single cluster IMF plus disruption,
taking into account the small size of the volume searched for clusters in the
Milky Way.
### 6.5. Cluster survival
Would we expect these young M31 clusters to survive as they age, or to
disrupt? One of the main processes that leads to cluster disruption is 2-body
relaxation enhanced by an external tidal field (Spitzer & Harm, 1958). The
lower-mass clusters suffer more strongly from relaxation effects. Another
property of the cluster itself which will affect its survival is its density —
lower-density clusters will disrupt more quickly (Spitzer, 1958). Thus we
would expect that massive, concentrated clusters such as B018 and BH05 would
be more likely to survive.
Boutloukos & Lamers (2003) derive an empirical expression for the disruption
of clusters as a function of their mass, studying cluster populations in the
solar neighborhood, the SMC, M33 and M51. Whitmore et al. (2007) point out
that observational selection effects could mimic the decrease in the number of
clusters with age which Boutloukos et al. ascribe to cluster disruption.
However, this is almost certainly not true of the solar neighborhood open
clusters studied by Lamers et al. (2005) using a similar analysis. We show the
age-mass diagram for the young M31 clusters in Figure 15. While our sample is
clearly very incomplete below $10^{8}$ years, the diagram shows some
similarity to the LMC cluster age-mass diagram of de Grijs & Anders (2006) in
the age range we cover. Unfortunately, we do not expect our catalog to be
complete enough to permit an analysis using the techniques of Boutloukos et
al.
Figure 15.— Age-mass diagram for our young and intermediate-age clusters.
Environmental effects also control the tidal stripping of the cluster. For
stars whose orbits are mostly confined to the disk, encounters with giant
molecular clouds and spiral arms contribute to their disruption (Spitzer,
1958; Gieles et al., 2007). For clusters whose orbits are not confined to the
disk, bulge and disk shocking are more important (Ostriker et al., 1972;
Aguilar et al., 1988). The similarity of the M31 young cluster kinematics to
that of other young disk objects suggests strongly that these clusters are
confined to M31’s disk plane, so giant molecular clouds should be the relevant
external disruptor. Gieles et al. (2008) show that disruption times for
clusters in galaxies ranging in size from M51 to the SMC, scale with molecular
gas density in the expected way. M31’s molecular gas density is highest near
the ”ring of fire” where many of our clusters are found (Loinard et al.,
1999). This density is similar to the molecular gas density in the solar
neighborhood (Dame, 1993). Thus we would expect the survival due to giant
molecular cloud interactions of the M31 young clusters to be similar to that
of the solar neighborhood open clusters.
We expect that most of these young clusters will be disrupted in the next Gyr
or so (Lamers et al., 2005, derive a disruption time of 1.3 Gyr for a cluster
of mass 10${}^{4}M_{\odot}$ in the solar neighborhood). However, some of the
more massive and concentrated of the young clusters will likely survive for
longer.
## 7\. Summary
We present a new catalog of 670 M31 clusters, with accurate coordinates. In
this paper we focus on the 140 clusters (many originally classified in the
literature as globular clusters) which have ages less than 2 Gyr: most have
ages between $10^{8}$ and $10^{9}$ years. Using high-quality MMT/Hectospec
spectra, excellent ground based images, and in some cases, HST images, we
explore the nature of these clusters. With the exception of NGC 205’s young
cluster, they have spatial and kinematical properties consistent with
formation in the star-forming disk of M31. Many are located close to the 10
kpc “ring of fire” which shows active star formation. The age distribution of
our clusters, plus that of the younger clusters of Williams & Hodge (2001a),
shows no evidence for a peak in star formation there between $2\times 10^{7}$
and $10^{9}$ years ago, which we might expect if the ring was created by a
recent passage of M32 through the disk, as suggested by Gordon et al. (2005)
and Block et al. (2006).
We have estimated their masses using spectroscopic ages and M/L ratios, (in
some cases) ACS color-magnitude diagrams, and new photometry from the Local
Group Survey. The clusters have masses ranging from 250 to 150,000
$M_{\odot}$. These reach to higher values than the known Milky Way open
clusters, but it must be remembered that our sample of open clusters in the
Milky Way is far from complete. The most massive of our young clusters overlap
the mass distributions of M31’s old clusters and the Milky Way globulars.
Interestingly, although most of the young clusters show the low-concentration
structure typical of the Milky Way open clusters, a few have the high
concentrations typical of the Milky Way globulars and the old M31 clusters. We
estimate that most of these young clusters will disrupt in $1-2$ Gyr, but the
massive, concentrated clusters may well survive longer.
We would like to thank Dan Fabricant for leading the effort to design & build
the Hectospec fiber positioner and spectrograph, Perry Berlind & Mike Calkins
for help with the observations, John Roll, Maureen Conroy & Bill Joye for
their many contributions to the Hectospec software development project, and
Phil Massey, Pauline Barmby & Jay Strader for comments and data tables on M31.
HLM was supported by NSF grant AST-0607518, and would like to thank Dean
McLaughlin for helpful conversations. Work on this project has also been
supported by HST grant GO10407.
## References
* Athanassoula & Beaton (2006) Athanassoula, E., & Beaton, R. L. 2006, MNRAS, 370, 1499
* Auriere et al. (1992) Auriere, M., Coupinot, G., & Hecquet, J. 1992, A&A, 256, 95
* Aguilar et al. (1988) Aguilar, L., Hut, P., & Ostriker, J. P. 1988, ApJ, 335, 720
* Baade & Arp (1964) Baade, W., & Arp, H. 1964, ApJ, 139, 1027
* Barmby et al. (2000) Barmby, P. et al. 2000, AJ, 119, 727
* Barmby & Huchra (2001) Barmby, P., & Huchra, J. P. 2001, AJ, 122, 2458
* Barmby et al. (2007) Barmby, P., McLaughlin, D. E., Harris, W. E., Harris, G. L. H., & Forbes, D. A. 2007, AJ, 133, 2764
* Battistini et al. (1980) Battistini, P., Bonoli, F., Braccesi, A., Fusi-Pecci, F., Malagnini, M. L., & Marano, B. 1980, A&AS, 42, 357
* Battistini et al. (1987) Battistini, P., Bonoli, F., Braccesi, A., Federici, L., Fusi Pecci, F., Marano, B., & Borngen, F. 1987, A&AS, 67, 447
* Battistini et al. (1993) Battistini, P. L., Bonoli, F., Casavecchia, M., Ciotti, L., Federici, L., & Fusi-Pecci, F. 1993, A&A, 272, 77
* Beasley et al. (2004) Beasley, M. A. et al. 2004, AJ, 128, 1623
* Beaton et al. (2007) Beaton, R. L., et al. 2007, ApJ, 658, L91
* Beichman et al. (1998) Beichman, C. A., Chester, T. J., Skrutskie, M., Low, F. J., & Gillett, F. 1998, PASP, 110, 480
* Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393
* Bica et al. (2006) Bica, E., Bonatto, C., & Blumbeqrg, R. 2006, A&A, 460, 83
* Bica & Bonatto (2008) Bica, E., & Bonatto, C. 2008, MNRAS, 384, 1733
* Binney and Tremaine (2008) Binney, J. and Tremaine, S., ’Galactic Dynamics’, second edition, Princeton University Press, Princeton NJ
* Block et al. (2006) Block, D. L., et al. 2006, Nature, 443, 832
* Bonatto & Bica (2005) Bonatto, C., & Bica, E. 2005, A&A, 437, 483
* Bonatto & Bica (2007) Bonatto, C., & Bica, E. 2007, A&A, 473, 445
* Boutloukos & Lamers (2003) Boutloukos, S. G., & Lamers, H. J. G. L. M. 2003, MNRAS, 338, 717
* Burstein et al. (2004) Burstein, D., et al. 2004, ApJ, 614, 158
* Chandar et al. (1999) Chandar, R., Bianchi, L., & Ford, H. C. 1999, ApJ, 517, 668
* Cioni et al. (2006a) Cioni, M.-R. L., Girardi, L., Marigo, P., & Habing, H. J. 2006, A&A, 448, 77
* Cioni et al. (2006b) Cioni, M.-R. L., Girardi, L., Marigo, P., & Habing, H. J. 2006, A&A, 452, 195
* Clark et al. (2005) Clark, J. S., Negueruela, I., Crowther, P. A., & Goodwin, S. P. 2005, A&A, 434, 949 K., & Cameron, P. B. 2005, ApJ, 634, L45
* Cohen et al. (2005) Cohen, J. G., Matthews, K., & Cameron, P. B. 2005, ApJl, 634, L45
* Crampton et al. (1985) Crampton, D., Cowley, A. P., Schade, D., & Chayer, P. 1985, ApJ, 288, 494
* Dame (1993) Dame, T. M. 1993, Back to the Galaxy, 278, 267
* Dubath & Grillmair (1997) Dubath, P., & Grillmair, C. J. 1997, A&A, 321, 379
* Eigenbrod et al. (2004) Eigenbrod, A., Mermilliod, J.-C., Clariá, J. J., Andersen, J., & Mayor, M. 2004, A&A, 423, 189
* Elson & Walterbos (1988) Elson, R. A., & Walterbos, R. A. M. 1988, ApJ, 333, 594
* Fall & Rees (1985) Fall, S. M., & Rees, M. J. 1985, ApJ, 298, 18
* Freeman (1980) Freeman, K.C. 1980, in Star Clusters, IAU Symposium 85, edited by J. Hesser (Reidel, Dordrecht), p317
* Friel (1995) Friel, E. D. 1995, ARA&A, 33, 381
* Fusi Pecci et al. (2005) Fusi Pecci, F. et al. 2005, AJ, 130, 554
* Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 416, 917
* Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127
* Gieles et al. (2007) Gieles, M., Athanassoula, E., & Portegies Zwart, S. F. 2007, MNRAS, 376, 809
* Gieles et al. (2008) Gieles, M., Lamers, H. J. G. L. M., & Baumgardt, H. 2008, IAU Symposium, 246, 171
* Gordon et al. (2005) Gordon, K. D., et al. 2006, ApJL, 638, L87
* de Grijs & Anders (2006) de Grijs, R., & Anders, P. 2006, MNRAS, 366, 295
* Jarrett et al. (2000) Jarrett, T. H., Chester, T., Cutri, R., Schneider, S., Skrutskie, M., & Huchra, J. P. 2000, AJ, 119, 2498
* Hodge (1981) Hodge, P.W. 1981, in Astrophysical Parameters for Globular Clusters, IAU Colloquium 68, edited by A.G.D. Phillip and D.S. Hayes (Schenectady, Davis)
* Hodge (1979) Hodge, P. W. 1979, AJ, 84, 744
* Hodge et al. (1987) Hodge, P. W., Mateo, M., Lee, M. G., & Geisler, D. 1987, PASP, 99, 173
* Hubble (1932) Hubble, E. 1932, ApJ, 76, 44
* Huxor et al. (2005) Huxor, A. P., Tanvir, N. R., Irwin, M. J., Ibata, R., Collett, J. L., Ferguson, A. M. N., Bridges, T., & Lewis, G. F. 2005, MNRAS, 360, 1007
* Kharchenko et al. (2005) Kharchenko, N. et al. 2005, A&A, 438, 1163
* Kennicutt (1988) Kennicutt, R. C., Jr., & Chu, Y.-H. 1988, AJ, 95, 720
* Kent (1989) Kent, S. M. 1989, AJ, 97, 1614.
* Kinman (1963) Kinman, T. D. 1963, ApJ, 137, 213
* Kim et al. (2007) Kim, S. C., et al. 2007, AJ, 134, 706
* Koekemoer et al. (2002) Koekemoer, A. M., Fruchter, A. S., Hook, R. N., & Hack, W. 2002, The 2002 HST Calibration Workshop : Hubble after the Installation of the ACS and the NICMOS Cooling System, Proceedings of a Workshop held at the Space Telescope Science Institute, Baltimore, Maryland, October 17 and 18, 2002. Edited by Santiago Arribas, Anton Koekemoer, and Brad Whitmore. Baltimore, MD: Space Telescope Science Institute, 2002., p.337, 337
* Krienke & Hodge (2007) Krienke, O. K., & Hodge, P. W. 2007, PASP, 119, 7
* Krienke & Hodge (2008) Krienke, O. K., & Hodge, P. W. 2008, PASP, 120, 1
* Lamers et al. (2005) Lamers, H. J. G. L. M. et al. 2005, A&A, 441, 117
* Larsen (2002) Larsen, S. S. 2002, Extragalactic Star Clusters, IAUS 207, 421
* Larsen (2004) Larsen, S. S. 2004, The Formation and Evolution of Massive Young Star Clusters, 322, 19
* Lee et al. (2008) Lee, M. G., Hwang, H. S., Kim, S. C., Park, H. S., Geisler, D., Sarajedini, A., & Harris, W. E. 2008, ApJ, 674, 886
* Leonardi & Rose (2003) Leonardi, A. J., & Rose, J. A. 2003, AJ, 126, 1811
* Leitherer et al. (1999) Leitherer, C., et al. 1999, ApJS, 123, 3
* Loinard et al. (1999) Loinard, L., Dame, T. M., Heyer, M. H., Lequeux, J., & Thaddeus, P. 1999, A&A, 351, 1087
* McLaughlin & van der Marel (2005) McLaughlin, D. E., & van der Marel, R. P. 2005, ApJS, 161, 304
* Mateo (1993) Mateo, M. 1993, The Globular Cluster-Galaxy Connection, 48, 387
* Massey et al. (2006) Massey, P. et al. 2006, AJ, 131, 2478
* Massey et al. (2007) Massey, P., Olsen, K. A. G., Hodge, P. W., Jacoby, G. H., McNeill, R. T., Smith, R. C., & Strong, S. B. 2007, AJ, 133, 2393
* Mochejska et al. (1998) Mochejska, B. J., Kaluzny, J., Krockenberger, M., Sasselov, D. D., & Stanek, K. Z. 1998, Acta Astronomica, 48, 455
* Morrison et al. (2004) Morrison, H. L., Harding, P., Perrett, K., & Hurley-Keller, D. 2004, ApJ, 603, 87
* Ostriker et al. (1972) Ostriker, J. P., Spitzer, L. J., & Chevalier, R. A. 1972, ApJ, 176, L51
* Peebles & Dicke (1968) Peebles, P. J. E., & Dicke, R. H. 1968, ApJ, 154, 891
* Perrett et al. (2002) Perrett, K. M., Bridges, T. J., Hanes, D. A., Irwin, M. J., Brodie, J. P., Carter, D., Huchra, J. P., & Watson, F. G. 2002, AJ, 123, 2490
* Puzia et al. (2005) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909
* Racine (1991) Racine, R. 1991, AJ, 101, 865
* Racine & Harris (1992) Racine, R., & Harris, W. E. 1992, AJ, 104, 1068
* Santos et al. (2005) Santos, J. F. C., Jr., Bonatto, C., & Bica, E. 2005, A&A, 442, 201
* Sargent et al. (1977) Sargent, W. L. W., Kowal, C. T., Hartwick, F. D. A., & van den Bergh, S. 1977, AJ, 82, 947
* Schiavon et al. (2005) Schiavon, R. P., Rose, J. A., Courteau, S., & MacArthur, L. A. 2005, ApJS, 160, 163
* Sirianni et al. (2005) Sirianni, M., et al. 2005, PASP, 117, 1049
* Spitzer & Harm (1958) Spitzer, L. J., & Harm, R. 1958, ApJ, 127, 544
* Spitzer (1958) Spitzer, L. J. 1958, ApJ, 127, 17
* Spitzer & Shapiro (1972) Spitzer, L. J., & Shapiro, S. L. 1972, ApJ, 173, 529
* Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191
* van den Bergh (1969) van den Bergh, S. 1969, ApJS, 19, 145
* Vázquez & Leitherer (2005) Vázquez, G. A., & Leitherer, C. 2005, ApJ, 621, 695
* Vetešnik (1962) Vetešnik, M. 1962, Bulletin of the Astronomical Institutes of Czechoslovakia, 13, 180
* Whitmore et al. (2007) Whitmore, B. C., Chandar, R., & Fall, M. 2007, AJ, 133, 1067
* Williams & Hodge (2001a) Williams, B. F., & Hodge, P. W. 2001, ApJ, 559, 851
* Williams & Hodge (2001b) Williams, B. F., & Hodge, P. W. 2001, ApJ, 548, 190
* Worthey (1994) Worthey, G. 1994, ApJS, 95, 107
* Worthey & Ottaviani (1997) Worthey, G., & Ottaviani, D. L. 1997, ApJS, 111, 377
Table 1Cleaned Cluster Catalog
Object | RA | Dec | V | type | SaaSource of velocity: HS=this paper; B=Barmby et al. (2000); P=Perrett et al. (2002) | Ap | PbbSource of photometry: L=this paper; B=Barmby et al. (2000); G=Galleti et al. (2007), H=Huxor et al. (2005) | CccSource of classification as a cluster: S=spectrum from this paper indicates a cluster; L=LGS image indicates non-stellar; H=HST image indicates a cluster; objects with blank entries in this column and in the velocity source column should still be considered “candidates”
---|---|---|---|---|---|---|---|---
| 2000 | | | | ˝ | |
SH01 | 0:32:41.44 | 40:01:41.4 | 15.82 | | | | G |
G001-MII | 0:32:46.53 | 39:34:40.6 | 13.75 | old | B | | B | H
G002-MIII | 0:33:33.77 | 39:31:18.9 | 15.81 | old | B | | B |
B290 | 0:34:20.94 | 41:28:18.1 | 17.14 | old | P | | B |
BA21 | 0:34:53.83 | 39:49:40.5 | 16.64 | | | | G |
B412 | 0:34:55.28 | 41:32:26.4 | 17.36 | | | | G |
B413 | 0:35:13.00 | 41:29:07.8 | 18.12 | | | | G |
BA22 | 0:35:13.60 | 39:45:37.1 | | | | | |
B134D | 0:35:30.29 | 40:44:24.8 | 18.19 | | | | G |
B291-G009 | 0:36:04.97 | 42:02:09.3 | 16.59 | old | HS | | B | S
B292-G010 | 0:36:16.66 | 40:58:26.5 | 17.00 | old | B | | B |
B293-G011 | 0:36:20.86 | 40:53:37.2 | 16.30 | old | B | | B |
B138D | 0:36:21.66 | 41:28:33.1 | 16.87 | | | | G |
B139D | 0:36:24.67 | 39:45:07.4 | 18.59 | | | | G |
B140D | 0:36:30.78 | 41:21:52.8 | 17.67 | | | | G |
B141D | 0:36:30.99 | 41:28:38.5 | 17.43 | | | | G |
B142D | 0:36:33.83 | 41:09:07.9 | 18.63 | | | | G |
B144D | 0:36:36.64 | 41:37:03.6 | 17.72 | | | | G |
B147D | 0:36:44.40 | 41:08:27.0 | 17.96 | | | | G |
B295-G014 | 0:36:46.73 | 40:19:42.1 | 16.72 | old | HS | | B | S
B148D-SH3 | 0:36:50.17 | 41:07:10.7 | 16.31 | | | | G |
B150D | 0:36:59.91 | 41:25:30.1 | 17.67 | | | | B |
B156D | 0:37:26.32 | 41:19:02.2 | 18.16 | | | | G |
B420 | 0:37:28.43 | 41:35:44.2 | 17.85 | | | | G |
B157D | 0:37:35.27 | 40:57:52.3 | 17.83 | | | | G |
B422 | 0:37:38.45 | 41:59:59.2 | 18.11 | old | HS | | B | S
B423 | 0:37:56.66 | 40:57:35.9 | 17.87 | | | | B |
B298-G021 | 0:38:00.22 | 40:43:55.8 | 16.59 | old | B | | B | H
B165D | 0:38:04.41 | 40:55:32.1 | 17.62 | | | | G |
B426-D018 | 0:38:19.80 | 41:14:30.7 | 16.35 | | | | G |
B301-G022 | 0:38:21.59 | 40:03:37.0 | 16.93$\pm 0.26$ | old | HS | 3.8 | L | SL
B167D | 0:38:22.48 | 41:54:35.0 | 17.95 | old | HS | | B | S
B302-G023 | 0:38:33.5 | 41:20:52.2 | 16.68 | old | HS | | B | S
GC7 | 0:38:49.4 | 42:22:48.0 | 17.99$\pm 0.1$ | old | HS | | H | SH
B303-G026 | 0:38:50.55 | 40:27:31.1 | 18.00$\pm 0.15$ | young | HS | 7.7 | L | SL
B176D | 0:38:53.15 | 41:29:03.0 | 17.96 | | | | G |
DAO23 | 0:38:54.19 | 40:26:33.9 | 19.42$\pm 0.19$ | | | 3.8 | L | L
B431-G027 | 0:38:54.76 | 40:34:56.4 | 18.00$\pm 0.20$ | young | HS | 7.7 | L | SL
B304-G028 | 0:38:56.94 | 41:10:28.4 | 16.83 | old | HS | | B | S
B305-D024 | 0:38:58.85 | 40:16:32.1 | 17.49$\pm 0.16$ | old | HS | 11.6 | L | SL
B306-G029 | 0:39:08.70 | 40:34:21.2 | 16.46$\pm 0.15$ | old | HS | 7.7 | L | SL
DAO27 | 0:39:16.48 | 40:41:05.4 | 18.60$\pm 0.18$ | HII | HS | 11.6 | L | SL
B307-G030 | 0:39:18.46 | 40:32:58.2 | 17.74$\pm 0.29$ | interm | HS | 7.7 | L | SL
B309-G031 | 0:39:24.62 | 40:14:29.1 | 17.41$\pm 0.24$ | old | HS | 7.7 | L | SL
B310-G032 | 0:39:25.75 | 41:23:33.1 | 17.04 | old | HS | | B | S
B436 | 0:39:30.67 | 40:18:20.6 | 18.20$\pm 0.20$ | old | HS | 5.1 | L | SL
B181D | 0:39:30.85 | 41:28:26.4 | 17.72 | old | HS | | G | S
B311-G033 | 0:39:33.72 | 40:31:14.7 | 15.50$\pm 0.10$ | old | HS | 11.6 | L | SLH
SH07 | 0:39:37.36 | 42:09:57.1 | | | | | |
B312-G035 | 0:39:40.17 | 40:57:02.4 | 15.58 | old | HS | | B | S
B314-G037 | 0:39:44.59 | 40:14:08.1 | 17.52$\pm 0.27$ | young | HS | 7.7 | L | SL
B313-G036 | 0:39:44.60 | 40:52:55.2 | 16.45$\pm 0.18$ | old | HS | 11.6 | L | SL
B315-G038 | 0:39:48.52 | 40:31:30.6 | 16.24$\pm 0.03$ | young | HS | 15.4 | L | SLH
DAO30 | 0:39:50.78 | 40:18:14.9 | 17.81$\pm 0.04$ | young | HS | 11.6 | L | SL
B001-G039 | 0:39:51.02 | 40:58:10.6 | 16.92$\pm 0.03$ | old | HS | 15.4 | L | SL
B316-G040 | 0:39:53.58 | 40:41:39.2 | 16.90$\pm 0.03$ | interm | HS | 15.4 | L | SL
B317-G041 | 0:39:55.29 | 41:47:45.9 | 16.55 | old | HS | | B | SH
KHM31-22 | 0:39:58.71 | 40:35:23.6 | 20.10$\pm 0.09$ | young | HS | 3.8 | L | SLH
WH2 | 0:39:59.99 | 40:33:27.0 | 20.65$\pm 0.54$ | young | HS | 1.8 | L | SLH
B318-G042 | 0:40:00.85 | 40:34:08.1 | 17.13$\pm 0.17$ | young | HS | 5.1 | L | SLH
B186D | 0:40:02.25 | 39:23:12.1 | 17.84 | | | | G |
B002-G043 | 0:40:02.57 | 41:11:53.5 | 17.54 | old | HS | | B | S
B319-G044 | 0:40:03.07 | 40:33:58.6 | 17.59$\pm 0.12$ | young | HS | 7.7 | L | SLH
B003-G045 | 0:40:09.40 | 41:11:05.6 | 17.57 | old | HS | | B | S
BH02 | 0:40:10.29 | 40:36:26.2 | 19.39$\pm 0.14$ | | | 5.1 | L | L
KHM31-37 | 0:40:10.99 | 40:36:11.6 | 18.09$\pm 0.10$ | young | HS | 5.1 | L | SLH
B188D | 0:40:14.03 | 39:41:30.8 | 17.92 | | | | G |
B321-G046 | 0:40:15.37 | 40:27:46.2 | 18.00$\pm 0.17$ | young | HS | 5.1 | L | SL
B189D-G047 | 0:40:15.49 | 40:39:59.5 | 18.47$\pm 0.08$ | young | HS | 5.1 | L | SL
B322-G049 | 0:40:17.27 | 40:39:04.7 | 18.10$\pm 0.23$ | young | HS | 3.8 | L | SL
B004-G050 | 0:40:17.92 | 41:22:40.2 | 16.95 | old | HS | | B | S
B323 | 0:40:18.28 | 40:32:44.6 | 18.11$\pm 0.26$ | young | HS | 7.7 | L | SL
B442-D033 | 0:40:19.40 | 40:37:28.9 | 18.17$\pm 0.14$ | young | HS | 7.7 | L | SL
B005-G052 | 0:40:20.33 | 40:43:58.3 | 15.66$\pm 0.08$ | old | HS | 5.1 | L | SL
B324-G051 | 0:40:20.47 | 41:40:49.3 | 16.91 | young | HS | | B | SH
B443-D034 | 0:40:20.79 | 40:33:22.0 | 19.01$\pm 0.35$ | young | HS | 2.5 | L | SL
BH03 | 0:40:22.63 | 41:40:44.4 | 18.28 | na | | | G | H
B325 | 0:40:23.09 | 40:30:47.4 | 17.47$\pm 0.44$ | young | HS | 11.6 | L | SL
B327-G053 | 0:40:24.10 | 40:36:22.4 | 16.64$\pm 0.10$ | young | HS | 7.7 | L | SL
B328-G054 | 0:40:24.52 | 41:40:23.1 | 17.57 | old | HS | | B | SH
B003D | 0:40:25.02 | 41:12:23.5 | 18.50 | | | | G |
B330-G056 | 0:40:25.58 | 41:42:53.6 | 17.69 | old | HS | | B | SH
B331-G057 | 0:40:26.10 | 41:42:03.9 | 18.20 | old | HS | | B | SH
B006-G058 | 0:40:26.48 | 41:27:26.7 | 15.52 | old | HS | | B | SH
B244 | 0:40:26.49 | 41:18:35.5 | 18.26 | old | HS | | B | S
BH04 | 0:40:27.2 | 41:42:23.9 | 19.69 | old | HS | | G | SLH
VDB0-B195D | 0:40:29.43 | 40:36:14.8 | 15.19$\pm 0.30$ | young | HS | 11.6 | L | SL
B333 | 0:40:29.58 | 41:40:26.7 | 19.13 | old | HS | | B | SH
B008-G060 | 0:40:30.28 | 41:16:08.7 | 16.52 | old | HS | | B | SH
BH05 | 0:40:30.51 | 40:45:29.3 | 16.03$\pm 0.19$ | young | HS | 3.8 | L | SLH
BH06 | 0:40:30.62 | 40:44:53.9 | 18.14$\pm 0.40$ | na | | 1.8 | L | LH
B009-G061 | 0:40:30.70 | 41:36:55.6 | 16.91 | old | HS | | B | SH
B010-G062 | 0:40:31.56 | 41:14:22.5 | 16.66 | old | HS | | B | SH
B011-G063 | 0:40:31.87 | 41:39:16.9 | 16.79 | old | HS | | B | SH
B012-G064 | 0:40:32.46 | 41:21:44.2 | 15.12 | old | HS | | B | SH
B196D | 0:40:34.79 | 40:26:38.0 | 19.18$\pm 0.15$ | young | HS | 5.1 | L | SL
B448-D035 | 0:40:36.52 | 40:40:15.1 | 18.52$\pm 0.28$ | young | HS | 7.7 | L | SL
BH09 | 0:40:37.15 | 40:33:21.9 | 19.83$\pm 0.07$ | old | HS | 3.8 | L | SLH
B006D-D036 | 0:40:37.37 | 40:48:45.5 | 18.69$\pm 0.21$ | young | HS | 5.1 | L | SL
B007D | 0:40:37.57 | 40:48:11.6 | 18.37$\pm 0.09$ | | | 5.1 | L | L
B013-G065 | 0:40:38.43 | 41:25:23.7 | 17.18 | old | HS | | B | SH
B335-V013 | 0:40:41.67 | 40:38:27.9 | 17.88$\pm 0.24$ | old | HS | 11.6 | L | SL
B449-V11 | 0:40:42.3 | 40:36:04.9 | 18.64$\pm 0.31$ | old | HS | 5.1 | L | SL
BH10 | 0:40:44.86 | 40:53:07.9 | 19.61$\pm 0.10$ | young | HS | 3.8 | L | SL
B008D | 0:40:45.01 | 40:58:55.2 | 19.51$\pm 0.17$ | | | 3.8 | L | L
B015-V204 | 0:40:45.02 | 40:59:56.3 | 17.87$\pm 0.12$ | old | HS | 11.6 | L | SL
B016-G066 | 0:40:45.16 | 41:22:09.9 | 17.58 | old | HS | | B | S
PHF7-1 | 0:40:46.42 | 40:51:40.6 | 18.91$\pm 0.14$ | | | 3.8 | L | L
DAO38 | 0:40:47.01 | 40:40:57.9 | 19.06$\pm 0.27$ | old | HS | 3.8 | L | SLH
B336-G067 | 0:40:47.60 | 42:08:43.2 | 17.81 | old | HS | | B | S
V203 | 0:40:47.80 | 40:59:06.0 | 18.15$\pm 0.06$ | HII | HS | 5.1 | L | SL
V202 | 0:40:47.82 | 40:55:34.3 | 19.13$\pm 0.28$ | young | HS | 5.1 | L | SL
B452-G069 | 0:40:48.33 | 40:35:06.0 | 18.00$\pm 0.10$ | young | HS | 7.7 | L | SL
PHF7-2 | 0:40:48.38 | 40:51:58.2 | 18.43$\pm 0.20$ | young | HS | 3.8 | L | SL
B337-G068 | 0:40:48.47 | 42:12:11.0 | 16.73 | old | HS | | B | S
B017-G070 | 0:40:48.73 | 41:12:07.1 | 16.04$\pm 0.14$ | old | HS | 7.7 | L | SL
B018-G071 | 0:40:49.42 | 40:41:31.4 | 18.21$\pm 0.28$ | interm | HS | 3.8 | L | SLH
B009D | 0:40:50.01 | 41:01:39.9 | 18.75$\pm 0.09$ | | | 5.1 | L | L
BH11 | 0:40:50.83 | 40:40:38.4 | 20.05$\pm 0.15$ | old | HS | 3.8 | L | SLH
B010D | 0:40:51.1 | 41:15:04.9 | 18.92$\pm 0.08$ | young | HS | 5.1 | L | SL
B198D | 0:40:51.47 | 40:33:27.8 | 18.13$\pm 0.22$ | | | 7.7 | L | L
B011D | 0:40:51.63 | 40:44:06.1 | 18.04$\pm 0.25$ | young | HS | 2.5 | L | SLH
DAO40 | 0:40:51.96 | 40:36:02.8 | 20.35$\pm 0.07$ | HII | HS | 2.5 | L | SLH
B012D-D039 | 0:40:52.28 | 40:58:41.3 | 19.05$\pm 0.05$ | young | HS | 3.8 | L | SL
B246 | 0:40:52.29 | 40:53:55.9 | 18.63$\pm 0.14$ | old | HS | 5.1 | L | SL
B019-G072 | 0:40:52.52 | 41:18:53.8 | 14.98$\pm 0.08$ | old | HS | 7.7 | L | SLH
KHM31-74 | 0:40:52.99 | 40:35:19.8 | 19.04$\pm 0.29$ | old | HS | 7.7 | L | SLH
SK018A | 0:40:53.64 | 41:16:15.1 | 19.48$\pm 0.07$ | young | HS | 3.8 | L | SL
KHM31-77 | 0:40:53.69 | 40:36:50.8 | 19.78$\pm 0.05$ | old | HS | 5.1 | L | SLH
B020-G073 | 0:40:55.26 | 41:41:25.2 | 14.91 | old | HS | | B | S
KHM31-81 | 0:40:55.72 | 40:35:22.1 | 21.16$\pm 0.40$ | young | HS | 1.2 | L | SLH
KHM31-85 | 0:40:56.61 | 40:34:24.6 | 19.62$\pm 0.38$ | young | HS | 2.5 | L | SLH
V212 | 0:40:58.53 | 41:03:32.4 | 19.14$\pm 0.12$ | HII | HS | 3.8 | L | SLH
KHM31-97 | 0:40:58.81 | 40:34:23.9 | 19.26$\pm 0.14$ | young | HS | 5.1 | L | SLH
B338-G076 | 0:40:58.87 | 40:35:47.9 | 14.22$\pm 0.10$ | old | HS | 11.6 | L | SLH
B021-G075 | 0:40:58.99 | 41:05:39.1 | 17.78$\pm 0.14$ | old | HS | 7.7 | L | SL
B022-G074 | 0:40:59.08 | 41:24:42.0 | 17.35 | old | HS | | B | S
B339-G077 | 0:41:00.71 | 39:55:54.2 | 16.87 | old | HS | | B | S
B014D | 0:41:01.07 | 41:06:32.8 | 18.53$\pm 0.15$ | young | HS | 7.7 | L | SL
B023-G078 | 0:41:01.19 | 41:13:45.7 | 14.17$\pm 0.09$ | old | HS | 11.6 | L | SLH
V211 | 0:41:02.01 | 41:02:54.9 | 18.53$\pm 0.06$ | HII | HS | 5.1 | L | SLH
B247 | 0:41:02.27 | 41:00:32.0 | 18.26$\pm 0.40$ | old | HS | 5.1 | L | SL
KHM31-113 | 0:41:02.64 | 40:34:43.1 | 19.72$\pm 0.12$ | young | HS | 3.8 | L | SLH
B015D-D041 | 0:41:02.74 | 41:06:36.3 | 18.70$\pm 0.20$ | young | HS | 5.1 | L | SL
BH12 | 0:41:02.88 | 40:34:58.4 | 18.70$\pm 0.30$ | young | HS | 3.8 | L | SLH
B453-D042 | 0:41:03.27 | 41:00:56.9 | 18.30$\pm 0.05$ | young | HS | 5.1 | L | SL
B200D-D043 | 0:41:06.79 | 40:34:29.0 | 19.04$\pm 0.31$ | young | HS | 3.8 | L | SL
B248 | 0:41:07.94 | 40:53:01.0 | 18.06$\pm 0.05$ | old | HS | 7.7 | L | SL
B201D-D044 | 0:41:08.29 | 40:32:51.7 | 19.18$\pm 0.31$ | young | HS | 3.8 | L | SL
B341-G081ddnot a cluster in Barmby et al. (2000) | 0:41:09.15 | 40:35:52.8 | 16.27$\pm 0.03$ | old | HS | 5.1 | L | SL
B017D | 0:41:10.01 | 40:58:10.6 | 18.23$\pm 0.19$ | old | HS | 7.7 | L | SL
B024-G082 | 0:41:11.86 | 41:45:49.1 | 16.79 | old | HS | | B | S
V031 | 0:41:12.26 | 41:05:29.1 | 18.37$\pm 0.30$ | young | HS | 7.7 | L | SLH
G083-V225 | 0:41:12.45 | 41:09:49.3 | 19.06$\pm 0.16$ | young | HS | 3.8 | L | SL
B025-G084 | 0:41:12.55 | 41:00:28.3 | 16.77$\pm 0.08$ | old | HS | 5.1 | L | SL
B249 | 0:41:12.58 | 41:01:12.7 | 18.16$\pm 0.26$ | old | HS | 7.7 | L | SL
G085-V015 | 0:41:12.79 | 40:34:17.4 | 18.04$\pm 0.22$ | young | HS | 3.8 | L | SL
SK020A | 0:41:13.36 | 41:09:42.0 | 19.68$\pm 0.12$ | na | HS | 5.1 | L | SL
V014 | 0:41:13.81 | 40:33:57.9 | 17.72$\pm 0.33$ | young | HS | 5.1 | L | SL
B027-G087 | 0:41:14.54 | 40:55:50.9 | 15.61$\pm 0.06$ | old | HS | 7.7 | L | SLH
B026-G086 | 0:41:14.55 | 41:24:40.1 | 17.53 | old | HS | | B | S
V226 | 0:41:14.80 | 41:09:23.4 | 19.19$\pm 0.12$ | HII | HS | 3.8 | L | SL
B019D | 0:41:16.13 | 41:05:07.8 | 18.93$\pm 0.13$ | interm | HS | 5.1 | L | SLH
B028-G088 | 0:41:16.50 | 40:59:03.2 | 16.90$\pm 0.09$ | old | HS | 7.7 | L | SL
B020D-G089 | 0:41:17.23 | 41:08:09.1 | 17.47$\pm 0.05$ | old | HS | 7.7 | L | SLH
B029-G090 | 0:41:17.82 | 41:00:23.0 | 16.65$\pm 0.13$ | old | HS | 11.6 | L | SL
B030-G091 | 0:41:18.74 | 40:57:15.6 | 17.32$\pm 0.05$ | old | HS | 7.7 | L | SLH
B031-G092 | 0:41:20.93 | 40:59:04.2 | 17.72$\pm 0.03$ | old | HS | 5.1 | L | SL
B032-G093 | 0:41:21.51 | 41:17:30.2 | 17.61$\pm 0.03$ | old | HS | 7.7 | L | SL
B342-G094 | 0:41:24.09 | 40:36:47.0 | 18.46$\pm 0.32$ | young | HS | 5.1 | L | SLH
B033-G095 | 0:41:26.40 | 41:00:14.0 | 17.68$\pm 0.08$ | old | HS | 7.7 | L | SL
B034-G096 | 0:41:28.12 | 40:53:49.6 | 15.40$\pm 0.10$ | old | HS | 11.6 | L | SL
B457-G097 | 0:41:29.23 | 42:18:37.1 | 16.91 | old | HS | | B | S
DAO47 | 0:41:29.49 | 40:45:16.8 | 19.07$\pm 0.13$ | young | HS | 5.1 | L | SL
V034 | 0:41:30.18 | 41:05:01.9 | 19.42$\pm 0.43$ | HII | HS | 3.8 | L | SL
LGS04131.1_404612 | 0:41:31.16 | 40:46:12.6 | 19.12$\pm 0.18$ | young | HS | 3.8 | L | SL
B035 | 0:41:32.58 | 41:38:32.7 | 17.47 | old | HS | | B | S
B036 | 0:41:32.83 | 41:26:05.1 | 17.31 | old | HS | | B | S
B024D | 0:41:34.00 | 41:01:25.0 | 19.84$\pm 0.13$ | | | 2.5 | L | L
B037-V327 | 0:41:34.98 | 41:14:55.1 | 16.86$\pm 0.09$ | old | HS | 11.6 | L | SLH
B038-G098 | 0:41:35.95 | 41:19:14.8 | 16.51$\pm 0.06$ | old | HS | 5.1 | L | SL
G099-V022 | 0:41:36.86 | 40:47:25.2 | 19.35$\pm 0.80$ | young | HS | 2.5 | L | SL
B039-G101 | 0:41:37.87 | 41:20:50.1 | 16.17$\pm 0.11$ | old | HS | 7.7 | L | SL
B029D | 0:41:38.43 | 41:43:13.4 | 18.50 | | | | G |
B040-G102 | 0:41:38.84 | 40:40:54.4 | 17.60$\pm 0.15$ | young | HS | 7.7 | L | SL
PHF8-1 | 0:41:39.50 | 40:40:33.5 | 19.26$\pm 0.11$ | young | HS | 3.8 | L | SL
B206D-D048 | 0:41:40.60 | 40:50:06.8 | 18.80$\pm 0.09$ | young | HS | 3.8 | L | SL
B041-G103 | 0:41:40.81 | 41:14:45.4 | 18.57$\pm 0.11$ | old | P | 5.1 | L | LH
B042-G104 | 0:41:41.69 | 41:07:26.2 | 16.16$\pm 0.07$ | old | HS | 5.1 | L | SLH
B521 | 0:41:41.71 | 40:52:01.5 | 19.08$\pm 0.09$ | young | HS | 5.1 | L | SL
B043-G106 | 0:41:42.31 | 40:42:40.0 | 17.01$\pm 0.13$ | young | HS | 7.7 | L | SL
B044-G107 | 0:41:42.91 | 41:20:06.2 | 16.84$\pm 0.07$ | old | HS | 5.1 | L | SL
B343-G105 | 0:41:43.10 | 40:12:22.4 | 16.34 | old | HS | | B | SH
B045-G108 | 0:41:43.11 | 41:34:20.3 | 15.71$\pm 0.10$ | old | HS | 11.6 | L | SLH
B458-D049 | 0:41:44.59 | 40:51:21.9 | 18.17$\pm 0.22$ | young | HS | 7.7 | L | SLH
B046-G109 | 0:41:44.60 | 41:46:27.7 | 17.80 | old | HS | | B | S
B048-G110 | 0:41:45.53 | 41:13:30.6 | 16.58$\pm 0.08$ | old | HS | 7.7 | L | SL
B047-G111 | 0:41:45.56 | 41:42:04.1 | 17.50 | old | HS | | B | S
B049-G112 | 0:41:45.57 | 40:49:54.7 | 17.68$\pm 0.10$ | young | HS | 7.7 | L | SLH
BH13 | 0:41:45.73 | 41:33:25.4 | 20.82$\pm 0.16$ | | | 2.5 | L | L
B032D | 0:41:45.96 | 41:13:01.4 | 18.59$\pm 0.18$ | young | HS | 3.8 | L | SL
B050-G113 | 0:41:46.28 | 41:32:18.7 | 16.74$\pm 0.04$ | old | HS | 7.7 | L | SL
PHF6-2 | 0:41:46.48 | 41:18:47.8 | 19.88$\pm 0.29$ | HII | HS | 3.8 | L | SL
B051-G114 | 0:41:46.70 | 41:25:19.1 | 16.28$\pm 0.15$ | old | HS | 7.7 | L | SL
V245 | 0:41:46.74 | 41:18:46.8 | 19.53$\pm 0.20$ | HII | HS | 3.8 | L | SL
SK036A | 0:41:47.40 | 40:51:08.6 | 19.72$\pm 0.19$ | old | HS | 3.8 | L | SL
B054-G115 | 0:41:47.68 | 41:00:55.3 | 18.16$\pm 0.04$ | old | HS | 5.1 | L | SL
KHM31-330 | 0:41:48.22 | 40:52:59.3 | 20.36$\pm 0.34$ | | | 3.8 | L | L
B055-G116ddnot a cluster in Barmby et al. (2000) | 0:41:50.39 | 41:12:12.4 | 16.65$\pm 0.10$ | old | HS | 11.6 | L | SL
B035D | 0:41:50.46 | 41:20:03.2 | 18.48$\pm 0.10$ | young | HS | 5.1 | L | SL
B254 | 0:41:50.5 | 41:16:25.9 | 18.66$\pm 0.14$ | old | HS | 5.1 | L | SL
B522 | 0:41:50.95 | 40:52:48.2 | 18.66$\pm 0.14$ | old | HS | 5.1 | L | SL
B056-G117 | 0:41:51.16 | 40:57:40.2 | 17.29$\pm 0.11$ | old | HS | 11.6 | L | SLH
B057-G118 | 0:41:52.82 | 40:52:05.1 | 17.58$\pm 0.07$ | old | HS | 5.1 | L | SLH
KHM31-340 | 0:41:52.92 | 40:52:16.4 | 19.97$\pm 0.33$ | | | 3.8 | L | L
B058-G119 | 0:41:53.00 | 40:47:09.7 | 15.00$\pm 0.08$ | old | HS | 7.7 | L | SLH
KHM31-341 | 0:41:53.04 | 40:52:35.7 | 19.32$\pm 0.08$ | young | HS | 3.8 | L | SL
KHM31-345 | 0:41:53.85 | 40:50:09.8 | 20.02$\pm 0.25$ | young | HS | 3.8 | L | SL
B059-G120 | 0:41:54.11 | 41:11:00.7 | 17.18$\pm 0.06$ | old | HS | 5.1 | L | SL
KHM31-347 | 0:41:54.25 | 40:50:28.1 | 20.62$\pm 0.36$ | | | 1.8 | L |
KHM31-350 | 0:41:55.06 | 40:51:52.1 | 20.53$\pm 0.16$ | | | 2.5 | L | L
KHM31-152 | 0:41:56.93 | 40:46:31.9 | 19.56$\pm 0.17$ | young | HS | 5.1 | L | SLH
B060-G121 | 0:41:57.01 | 41:05:14.5 | 16.68$\pm 0.08$ | old | HS | 7.7 | L | SL
B255 | 0:41:59.96 | 40:48:33.7 | 18.28$\pm 0.19$ | young | HS | 5.1 | L | SL
B061-G122 | 0:42:00.14 | 41:29:35.7 | 16.59$\pm 0.10$ | old | HS | 11.6 | L | SLH
BH14 | 0:42:00.39 | 40:47:46.0 | 19.80$\pm 0.22$ | HII | HS | 5.1 | L | SLH
B038D | 0:42:00.45 | 41:12:14.2 | 19.03$\pm 0.27$ | | | 3.8 | L | L
B063-G124 | 0:42:00.88 | 41:29:09.5 | 15.73$\pm 0.05$ | old | HS | 7.7 | L | SLH
B065-G126 | 0:42:01.93 | 40:40:13.1 | 16.83$\pm 0.10$ | old | HS | 11.6 | L | SL
B064-G125 | 0:42:01.93 | 41:11:07.5 | 16.30$\pm 0.07$ | old | HS | 7.7 | L | SL
B344-G127 | 0:42:02.97 | 41:52:02.2 | 15.95 | old | HS | | B | S
B066-G128 | 0:42:03.07 | 40:44:47.1 | 17.77$\pm 0.16$ | young | HS | 5.1 | L | SL
B067-G129 | 0:42:03.19 | 41:04:23.7 | 17.20$\pm 0.08$ | old | HS | 7.7 | L | SL
B068-G130 | 0:42:03.21 | 40:58:50.2 | 16.24$\pm 0.10$ | old | HS | 11.6 | L | SLH
B257-V219 | 0:42:03.28 | 40:58:13.9 | 17.76$\pm 0.07$ | old | HS | 7.7 | L | SLH
B461-G131 | 0:42:04.24 | 42:03:26.6 | 17.52 | old | HS | | B | S
B040D | 0:42:04.3 | 41:18:07.0 | 18.55$\pm 0.06$ | young | HS | 5.1 | L | SL
B041D | 0:42:04.72 | 41:16:47.3 | 18.23$\pm 0.13$ | old | HS | 5.1 | L | SL
B069-G132 | 0:42:05.54 | 41:26:09.3 | 19.02$\pm 0.39$ | young | HS | 2.5 | L | SLH
SK044A | 0:42:06.38 | 40:53:16.8 | 19.47$\pm 0.22$ | na | HS | 3.8 | L | SLH
B070-G133 | 0:42:06.91 | 41:07:56.3 | 16.74$\pm 0.05$ | old | HS | 5.1 | L | SLH
B071 | 0:42:07.13 | 41:12:12.0 | 17.94$\pm 0.06$ | old | HS | 3.8 | L | SL
B073-G134 | 0:42:07.33 | 40:59:21.3 | 15.97$\pm 0.09$ | old | HS | 11.6 | L | SL
B072 | 0:42:07.44 | 41:22:47.6 | 17.84$\pm 0.17$ | old | HS | 2.5 | L | SL
B258 | 0:42:07.80 | 41:09:26.0 | 18.47$\pm 0.10$ | | | 3.8 | L | L
B074-G135 | 0:42:08.04 | 41:43:21.6 | 16.65 | old | HS | | B | S
B075-G136 | 0:42:08.83 | 41:20:21.3 | 17.56$\pm 0.21$ | old | HS | 5.1 | L | SL
G137 | 0:42:09.43 | 41:28:31.7 | 17.77$\pm 0.24$ | HII | HS | 5.1 | L | SLH
MITA140 | 0:42:09.51 | 41:17:45.6 | 17.00$\pm 0.04$ | old | HS | 5.1 | L | SL
B045D | 0:42:09.87 | 41:21:14.5 | 18.97$\pm 0.09$ | old | HS | 3.8 | L | SL
B076-G138 | 0:42:10.24 | 41:05:22.0 | 16.67$\pm 0.09$ | old | HS | 7.7 | L | SLH
B047D | 0:42:10.93 | 41:29:59.2 | 19.11$\pm 0.31$ | na | | 5.1 | L | LH
B077-G139 | 0:42:11.14 | 41:07:33.9 | 17.25$\pm 0.09$ | old | HS | 7.7 | L | SLH
B078-G140 | 0:42:12.17 | 41:17:58.9 | 17.69$\pm 0.11$ | old | HS | 5.1 | L | SL
B080-G141 | 0:42:12.40 | 41:19:00.6 | 17.67$\pm 0.27$ | old | HS | 3.8 | L | SL
B081-G142 | 0:42:13.59 | 40:48:39.1 | 17.48$\pm 0.17$ | young | HS | 3.8 | L | SL
B345-G143 | 0:42:14.12 | 40:17:36.5 | 16.51 | old | HS | | B | S
B462 | 0:42:14.72 | 42:01:36.7 | 18.05 | old | HS | | B | S
B082-G144 | 0:42:15.84 | 41:01:14.4 | 16.13$\pm 0.29$ | old | HS | 2.5 | L | SLH
B083-G146 | 0:42:16.44 | 41:45:20.7 | 17.09 | old | HS | | B | S
B084 | 0:42:17.45 | 41:18:55.7 | 18.00$\pm 0.09$ | old | HS | 5.1 | L | SL
B085-G147 | 0:42:18.24 | 40:39:57.2 | 16.84$\pm 0.07$ | old | HS | 7.7 | L | SL
B086-G148 | 0:42:18.65 | 41:14:02.1 | 15.09$\pm 0.08$ | old | HS | 7.7 | L | SLH
B259 | 0:42:19.0 | 41:42:13.9 | 18.36 | old | HS | | G | S
SK049A | 0:42:19.48 | 40:52:22.6 | 20.22$\pm 0.15$ | na | HS | 2.5 | L | SL
B087eenot a cluster in Crampton et al. (1985) | 0:42:19.81 | 41:38:16.2 | 18.57$\pm 0.03$ | old | HS | 5.1 | L | SL
B051D | 0:42:20.56 | 41:04:37.7 | 18.82$\pm 0.07$ | na | HS | 3.8 | L | SL
B088-G150 | 0:42:21.07 | 41:32:14.2 | 15.40$\pm 0.11$ | old | HS | 11.6 | L | SLH
B090 | 0:42:21.08 | 41:02:57.5 | 18.44$\pm 0.07$ | old | HS | 5.1 | L | SLH
SK050A | 0:42:21.57 | 41:14:19.7 | 18.61$\pm 0.04$ | na | HS | 2.5 | L | SLH
B091-G151 | 0:42:21.73 | 41:22:05.2 | 17.92$\pm 0.12$ | young | HS | 5.1 | L | SLH
B092-G152 | 0:42:22.38 | 41:08:08.7 | 16.89$\pm 0.03$ | old | HS | 7.7 | L | SL
B347-G154 | 0:42:22.89 | 41:54:27.5 | 16.49 | old | HS | | B | S
B348-G153 | 0:42:22.92 | 41:52:28.4 | 16.79 | old | HS | | B | S
B093-G155 | 0:42:23.17 | 41:21:43.5 | 16.86$\pm 0.17$ | old | HS | 5.1 | L | SLH
B349 | 0:42:24.10 | 40:37:43.9 | 18.03$\pm 0.03$ | young | HS | 7.7 | L | SL
B053D-NB20 | 0:42:24.94 | 41:12:34.7 | 19.95$\pm 0.16$ | | | 1.8 | L | L
B094-G156 | 0:42:25.06 | 40:57:17.7 | 15.61$\pm 0.09$ | old | HS | 7.7 | L | SLH
B095-G157 | 0:42:25.80 | 41:05:36.3 | 16.20$\pm 0.04$ | old | HS | 7.7 | L | SL
B096-G158 | 0:42:26.1 | 41:19:14.8 | 16.62$\pm 0.07$ | old | HS | 5.1 | L | SL
B098 | 0:42:27.40 | 40:59:36.1 | 16.25$\pm 0.04$ | old | HS | 5.1 | L | SL
B097-G159 | 0:42:27.48 | 41:25:32.1 | 16.79$\pm 0.05$ | old | HS | 7.7 | L | SL
B099-G161 | 0:42:27.59 | 41:10:02.7 | 16.83$\pm 0.08$ | old | HS | 5.1 | L | SL
B515 | 0:42:28.05 | 41:33:24.5 | 18.60$\pm 0.22$ | na | | 7.7 | L | LH
B056D | 0:42:28.36 | 41:34:27.2 | 18.54$\pm 0.08$ | na | HS | 7.7 | L | SLH
B350-G162 | 0:42:28.44 | 40:24:51.1 | 16.74 | old | HS | | B | S
B100-G163 | 0:42:28.96 | 40:49:56.0 | 17.68$\pm 0.06$ | old | HS | 7.7 | L | SL
B101-G164 | 0:42:29.04 | 41:08:15.6 | 16.91$\pm 0.03$ | old | HS | 5.1 | L | SL
NB108 | 0:42:29.28 | 41:15:15.6 | 20.17$\pm 0.37$ | | | 1.8 | L | L
B103-G165 | 0:42:29.75 | 41:17:57.5 | 15.15$\pm 0.03$ | old | HS | 7.7 | L | SLH
B104-NB5 | 0:42:29.94 | 41:17:25.7 | 17.51$\pm 0.09$ | old | HS | 5.1 | L | SLH
B105-G166 | 0:42:30.75 | 41:30:27.3 | 17.33$\pm 0.12$ | old | HS | 5.1 | L | SL
B106-G168 | 0:42:31.04 | 41:12:18.3 | 16.37$\pm 0.18$ | old | HS | 3.8 | L | SL
B108-G167 | 0:42:31.19 | 41:08:51.3 | 17.30$\pm 0.08$ | old | HS | 7.7 | L | SL
B107-G169 | 0:42:31.27 | 41:19:38.9 | 15.79$\pm 0.08$ | old | HS | 5.1 | L | SL
NB24 | 0:42:31.81 | 41:15:45.1 | 19.70$\pm 0.23$ | | | 1.8 | L | L
B109-G170 | 0:42:32.16 | 41:10:27.9 | 16.34$\pm 0.04$ | old | HS | 7.7 | L | SLH
B061D | 0:42:32.6 | 41:21:42.0 | 19.13$\pm 0.06$ | young | HS | 3.8 | L | SL
B110-G172 | 0:42:33.10 | 41:03:28.4 | 15.17$\pm 0.07$ | old | HS | 5.1 | L | SLH
NB16 | 0:42:33.12 | 41:20:16.8 | 18.47$\pm 0.13$ | old | HS | 2.5 | L | SLH
B111-G173 | 0:42:33.17 | 41:00:26.5 | 16.66$\pm 0.04$ | old | HS | 7.7 | L | SL
B260 | 0:42:33.19 | 41:31:24.8 | 18.56$\pm 0.16$ | old | HS | 7.7 | L | SL
B112-G174 | 0:42:33.26 | 41:17:42.4 | 16.58$\pm 0.16$ | old | HS | 3.8 | L | SLH
B114-G175 | 0:42:34.30 | 41:12:44.9 | 17.04$\pm 0.09$ | old | HS | 5.1 | L | SLH
B117-G176 | 0:42:34.38 | 40:57:09.3 | 16.88$\pm 0.23$ | old | HS | 2.5 | L | SLH
NB17-AU014 | 0:42:34.40 | 41:17:31.4 | 19.45$\pm 0.28$ | na | HS | 2.5 | L | SLH
B115-G177 | 0:42:34.41 | 41:14:02.0 | 15.97$\pm 0.04$ | old | HS | 5.1 | L | SLH
B116-G178 | 0:42:34.54 | 41:32:51.4 | 16.83$\pm 0.06$ | old | HS | 5.1 | L | SL
NB35-AU4 | 0:42:34.55 | 41:18:40.4 | 19.35$\pm 0.07$ | old | HS | 2.5 | L | SLH
NB29 | 0:42:35.30 | 41:17:47.2 | 19.37$\pm 0.40$ | na | | 2.5 | L | LH
B064D-NB6 | 0:42:35.54 | 41:14:34.3 | 16.41$\pm 0.07$ | old | HS | 5.1 | L | SLH
B119-NB14 | 0:42:36.11 | 41:17:35.4 | 17.47$\pm 0.09$ | old | HS | 3.8 | L | SLH
NB21-AU5 | 0:42:37.98 | 41:15:58.9 | 17.98$\pm 0.10$ | old | HS | 2.5 | L | SLH
B351-G179 | 0:42:37.98 | 42:11:30.7 | 17.55 | old | HS | | B | S
B352-G180 | 0:42:38.19 | 42:02:13.1 | 16.53 | old | HS | | B | S
B067D | 0:42:38.99 | 41:36:43.2 | 19.11$\pm 0.06$ | young | HS | 3.8 | L | SLH
B068D | 0:42:39.9 | 41:20:39.9 | 18.66$\pm 0.14$ | old | HS | 3.8 | L | SLH
B122-G181 | 0:42:40.11 | 41:33:46.8 | 17.65$\pm 0.03$ | old | HS | 7.7 | L | SL
B123-G182 | 0:42:40.66 | 41:10:33.4 | 17.38$\pm 0.08$ | old | HS | 5.1 | L | SLH
B124-NB10 | 0:42:41.44 | 41:15:23.7 | 14.73$\pm 0.07$ | old | HS | 5.1 | L | SLH
B125-G183 | 0:42:42.27 | 41:05:31.0 | 16.51$\pm 0.11$ | old | HS | 7.7 | L | SL
V270 | 0:42:42.39 | 41:31:54.6 | 18.90$\pm 0.08$ | HII | HS | 3.8 | L | SL
DAO55 | 0:42:42.5 | 40:29:27.0 | 18.68 | old | HS | | B | S
B126-G184 | 0:42:43.70 | 41:12:42.8 | 17.13$\pm 0.04$ | old | HS | 5.1 | L | SLH
NB62 | 0:42:44.21 | 41:14:23.1 | 19.53$\pm 0.13$ | | | 1.8 | L | LH
B127-G185 | 0:42:44.50 | 41:14:41.5 | 14.42$\pm 0.08$ | old | HS | 7.7 | L | SLH
NB89 | 0:42:44.78 | 41:14:44.2 | 17.51$\pm 0.20$ | na | HS | 3.8 | L | SLH
SK054A | 0:42:45.08 | 41:08:15.1 | 18.37$\pm 0.09$ | na | HS | 5.1 | L | SL
B072D | 0:42:45.79 | 41:27:27.0 | 19.06$\pm 0.08$ | old | HS | 3.8 | L | SLH
BH16 | 0:42:46.09 | 41:17:36.0 | 18.80$\pm 0.28$ | | | 1.8 | L | L
NB18 | 0:42:46.34 | 41:18:32.4 | 18.82$\pm 0.11$ | na | HS | 2.5 | L | SLH
B354-G186 | 0:42:47.64 | 42:00:24.7 | 17.81 | old | HS | | B | S
B128-G187 | 0:42:47.81 | 41:11:13.8 | 17.03$\pm 0.04$ | old | HS | 5.1 | L | SLH
NB41 | 0:42:48.18 | 41:16:00.6 | 18.09 | na | | | G | H
B129 | 0:42:48.35 | 41:25:06.6 | 17.10$\pm 0.06$ | old | HS | 7.7 | L | SLH
NB39-AU6 | 0:42:48.55 | 41:15:47.6 | 18.32$\pm 0.27$ | na | | 1.8 | L | LH
B130-G188 | 0:42:48.86 | 41:29:52.7 | 16.78$\pm 0.11$ | old | HS | 11.6 | L | SLH
AU008 | 0:42:48.97 | 41:18:11.2 | 18.15$\pm 0.30$ | na | HS | 2.5 | L | SLH
B262 | 0:42:50.05 | 41:19:28.1 | 17.74$\pm 0.11$ | old | HS | 3.8 | L | SL
BH18 | 0:42:50.73 | 41:10:33.4 | 18.10$\pm 0.09$ | old | HS | 3.8 | L | SL
B131-G189 | 0:42:50.81 | 41:17:07.3 | 15.37$\pm 0.03$ | old | HS | 5.1 | L | SLH
BH17 | 0:42:50.84 | 40:58:41.5 | 20.45$\pm 0.18$ | na | | 3.8 | L | LH
B132-NB15 | 0:42:51.44 | 41:15:40.7 | 17.78$\pm 0.12$ | old | HS | 2.5 | L | SLH
B134-G190 | 0:42:51.65 | 41:14:03.6 | 16.68$\pm 0.17$ | old | HS | 5.1 | L | SLH
B078D | 0:42:51.91 | 41:22:05.2 | 19.35$\pm 0.04$ | old | HS | 2.5 | L | SL
B135-G192 | 0:42:51.98 | 41:31:08.3 | 15.94$\pm 0.04$ | old | HS | 7.7 | L | SL
B264-NB19 | 0:42:53.19 | 41:16:14.4 | 17.70$\pm 0.10$ | old | HS | 3.8 | L | SLH
B136-G194 | 0:42:53.64 | 41:19:34.4 | 16.95$\pm 0.07$ | old | HS | 5.1 | L | SL
B137-G195 | 0:42:54.0 | 41:32:14.4 | 17.67$\pm 0.06$ | old | HS | 7.7 | L | SL
B081D | 0:42:55.22 | 41:03:07.4 | 18.27$\pm 0.05$ | young | HS | 7.7 | L | SLH
B138 | 0:42:55.62 | 41:18:35.1 | 16.94$\pm 0.09$ | old | HS | 3.8 | L | SLH
B524 | 0:42:55.89 | 41:03:13.1 | 19.31$\pm 0.11$ | old | | 3.8 | L | LH
B086D | 0:42:56.71 | 40:51:22.7 | 19.21$\pm 0.49$ | na | | 5.1 | L | LH
AU010 | 0:42:58.13 | 41:16:52.7 | 17.54$\pm 0.03$ | old | HS | 3.8 | L | SL
NB34-AU15 | 0:42:58.45 | 41:14:55.4 | 18.73$\pm 0.15$ | HII | | 3.8 | L | L
B140-G196ffnot a cluster in Racine (1991) | 0:42:58.75 | 41:08:52.7 | 17.84$\pm 0.25$ | old | HS | 3.8 | L | SLH
B087D | 0:42:58.92 | 41:09:08.8 | 17.54$\pm 0.06$ | old | HS | 5.1 | L | SL
B141-G197 | 0:42:59.29 | 41:32:47.5 | 16.83$\pm 0.08$ | old | HS | 11.6 | L | SL
B088D | 0:42:59.38 | 41:04:17.5 | 18.63$\pm 0.24$ | | | 3.8 | L | LH
B143-G198 | 0:42:59.66 | 41:19:19.3 | 15.98$\pm 0.04$ | old | HS | 7.7 | L | SLH
B144 | 0:42:59.87 | 41:16:05.7 | 16.79$\pm 0.09$ | old | HS | 3.8 | L | SLH
B090D | 0:43:01.23 | 41:16:10.4 | 17.37$\pm 0.08$ | old | HS | 3.8 | L | SL
B089D | 0:43:01.36 | 41:26:24.1 | 18.90$\pm 0.06$ | young | HS | 5.1 | L | SL
B091D-D058 | 0:43:01.44 | 41:30:17.5 | 15.44$\pm 0.07$ | old | HS | 5.1 | L | SL
B145 | 0:43:01.59 | 41:12:26.9 | 18.37$\pm 0.09$ | old | HS | 3.8 | L | SLH
B092D | 0:43:01.70 | 41:13:08.9 | 18.92$\pm 0.24$ | old | HS | 3.8 | L | SLH
B265 | 0:43:01.92 | 40:53:01.8 | 18.42$\pm 0.06$ | old | HS | 5.1 | L | SL
B146 | 0:43:02.94 | 41:15:22.6 | 16.99$\pm 0.06$ | old | HS | 3.8 | L | SLH
B147-G199ggDubath & Grillmair (1997) probably observed a different object | 0:43:03.30 | 41:21:21.5 | 15.71$\pm 0.13$ | old | HS | 11.6 | L | SLH
B266 | 0:43:03.52 | 41:40:31.2 | 18.27$\pm 0.07$ | old | HS | 7.7 | L | SLH
BH23 | 0:43:03.79 | 41:20:28.1 | 18.51$\pm 0.16$ | na | HS | 3.8 | L | SLH
B148-G200 | 0:43:03.87 | 41:18:04.8 | 15.77$\pm 0.05$ | old | HS | 5.1 | L | SLH
B220D | 0:43:04.38 | 39:50:05.3 | 16.97 | | | | B |
B149-G201 | 0:43:05.48 | 41:34:27.3 | 17.03$\pm 0.13$ | old | HS | 11.6 | L | SL
B467-G202 | 0:43:06.45 | 42:01:49.1 | 17.43 | old | HS | | B | S
B268 | 0:43:07.19 | 41:11:47.8 | 18.33$\pm 0.07$ | old | HS | 3.8 | L | SLH
B269 | 0:43:07.38 | 41:27:32.9 | 18.83$\pm 0.16$ | old | HS | 5.1 | L | SLH
B150-G203 | 0:43:07.52 | 41:20:19.6 | 16.62$\pm 0.04$ | old | HS | 7.7 | L | SL
PHF6-1 | 0:43:08.02 | 41:18:18.3 | 17.67$\pm 0.07$ | old | HS | 3.8 | L | SLH
B223D | 0:43:09.55 | 42:24:59.2 | 17.75 | | | | G |
B151-G205 | 0:43:09.56 | 41:21:32.1 | 14.77$\pm 0.04$ | old | HS | 7.7 | L | SLH
B152-G207 | 0:43:10.02 | 41:18:16.1 | 16.09$\pm 0.09$ | old | HS | 7.7 | L | SLH
B356-G206 | 0:43:10.36 | 41:50:31.3 | 16.89$\pm 0.10$ | old | HS | 11.6 | L | SL
B153 | 0:43:10.63 | 41:14:51.4 | 16.13$\pm 0.05$ | old | HS | 5.1 | L | SLH
BH25 | 0:43:11.99 | 41:02:49.0 | 18.93$\pm 0.30$ | na | | 3.8 | L | LH
B154-G208 | 0:43:12.46 | 41:16:04.9 | 16.70$\pm 0.08$ | old | HS | 5.1 | L | SLH
B357-G209 | 0:43:13.23 | 40:10:56.3 | 16.61 | old | B | | B |
B155-G210 | 0:43:13.39 | 41:03:28.3 | 17.90$\pm 0.09$ | old | HS | 5.1 | L | SLH
B225D | 0:43:13.40 | 40:01:14.9 | 18.36 | | | | B |
B156-G211 | 0:43:13.73 | 41:01:17.9 | 16.86$\pm 0.03$ | old | HS | 7.7 | L | SLH
B157-G212 | 0:43:14.00 | 41:11:19.7 | 17.59$\pm 0.12$ | old | HS | 5.1 | L | SL
B095D | 0:43:14.03 | 41:08:44.8 | 18.97$\pm 0.06$ | young | HS | 3.8 | L | SL
B158-G213 | 0:43:14.41 | 41:07:21.2 | 14.65$\pm 0.06$ | old | HS | 7.7 | L | SLH
B159 | 0:43:14.65 | 41:25:13.5 | 17.35$\pm 0.04$ | old | HS | 7.7 | L | SLH
B160-G214 | 0:43:14.93 | 41:01:35.6 | 18.03$\pm 0.09$ | old | HS | 5.1 | L | SLH
B227D | 0:43:15.29 | 40:15:42.9 | 16.91 | | | | G |
B161-G215 | 0:43:15.43 | 41:11:25.0 | 16.30$\pm 0.05$ | old | HS | 5.1 | L | SL
SK063A | 0:43:16.09 | 41:27:57.0 | 18.38$\pm 0.13$ | na | HS | 5.1 | L | SLH
B162-G216 | 0:43:16.41 | 41:24:04.5 | 17.54$\pm 0.07$ | old | HS | 5.1 | L | SLH
B097D | 0:43:16.69 | 41:06:33.4 | 18.33$\pm 0.12$ | young | HS | 7.7 | L | SL
B098D | 0:43:17.4 | 41:31:30.0 | 18.83$\pm 0.06$ | young | HS | 3.8 | L | SL
B163-G217 | 0:43:17.64 | 41:27:45.0 | 15.00$\pm 0.11$ | old | HS | 11.6 | L | SLH
B358-G219 | 0:43:17.86 | 39:49:13.1 | 15.12 | old | B | | B |
B164-V253 | 0:43:18.14 | 41:12:29.3 | 17.78$\pm 0.04$ | old | HS | 5.1 | L | SL
B165-G218 | 0:43:18.22 | 41:10:54.7 | 16.44$\pm 0.08$ | old | HS | 7.7 | L | SL
B167 | 0:43:21.13 | 41:14:08.3 | 17.34$\pm 0.07$ | old | HS | 3.8 | L | SLH
B168 | 0:43:22.52 | 41:44:05.6 | 18.41$\pm 0.19$ | old | HS | 5.1 | L | SL
B169 | 0:43:23.00 | 41:15:25.4 | 17.27$\pm 0.03$ | old | HS | 5.1 | L | SLH
B271 | 0:43:23.07 | 41:25:26.4 | 18.35$\pm 0.08$ | interm | HS | 7.7 | L | SL
B170-G221 | 0:43:23.47 | 40:50:41.8 | 17.40$\pm 0.05$ | old | HS | 5.1 | L | SL
SK066A | 0:43:24.08 | 41:38:42.2 | 19.55$\pm 0.11$ | na | HS | 5.1 | L | SLH
B272-V294 | 0:43:25.52 | 41:37:11.7 | 18.23$\pm 0.12$ | old | HS | 11.6 | L | SL
B171-G222 | 0:43:25.61 | 41:15:37.2 | 15.26$\pm 0.08$ | old | HS | 7.7 | L | SLH
B172-G223 | 0:43:26.00 | 41:21:31.6 | 16.71$\pm 0.07$ | old | HS | 5.1 | L | SL
SK068A | 0:43:28.15 | 41:00:22.0 | 19.95$\pm 0.09$ | old | HS | 3.8 | L | SL
B173-G224 | 0:43:28.76 | 41:22:37.0 | 17.45$\pm 0.06$ | old | HS | 5.1 | L | SL
B174-G226 | 0:43:30.31 | 41:38:56.2 | 15.40$\pm 0.03$ | old | HS | 15.4 | L | SLH
B176-G227 | 0:43:30.45 | 40:49:11.1 | 16.86$\pm 0.19$ | old | HS | 5.1 | L | SL
B177-G228 | 0:43:30.49 | 41:05:42.4 | 18.30$\pm 0.09$ | old | HS | 7.7 | L | SL
B178-G229 | 0:43:30.79 | 41:21:16.4 | 14.97$\pm 0.10$ | old | HS | 15.4 | L | SL
B179-G230 | 0:43:31.10 | 41:18:14.7 | 15.34$\pm 0.11$ | old | HS | 11.6 | L | SL
B180-G231 | 0:43:31.72 | 41:07:46.4 | 16.27$\pm 0.15$ | old | HS | 3.8 | L | SL
B181-G232 | 0:43:32.46 | 41:29:07.4 | 16.93$\pm 0.09$ | old | HS | 7.7 | L | SL
V254 | 0:43:34.87 | 41:09:55.3 | 17.34$\pm 0.46$ | HII | HS | 5.1 | L | SL
B182-G233 | 0:43:36.66 | 41:08:12.2 | 15.42$\pm 0.09$ | old | HS | 7.7 | L | SL
B183-G234 | 0:43:36.94 | 41:02:02.4 | 15.99$\pm 0.09$ | old | HS | 7.7 | L | SL
B185-G235 | 0:43:37.28 | 41:14:43.6 | 15.58$\pm 0.08$ | old | HS | 5.1 | L | SLH
B184-G236 | 0:43:37.52 | 41:36:34.5 | 17.46$\pm 0.14$ | old | HS | 5.1 | L | SL
B186 | 0:43:38.23 | 41:36:24.1 | 19.14$\pm 0.63$ | old | HS | 3.8 | L | SL
B187-G237 | 0:43:38.64 | 41:29:47.1 | 17.29$\pm 0.13$ | old | HS | 5.1 | L | SL
B274 | 0:43:39.36 | 41:31:18.4 | 18.77$\pm 0.22$ | old | HS | 7.7 | L | SL
B233D | 0:43:41.31 | 39:36:45.9 | 16.17 | | | | G |
B188-G239 | 0:43:41.51 | 41:24:25.6 | 17.04$\pm 0.11$ | old | HS | 7.7 | L | SL
B189-G240 | 0:43:42.42 | 41:35:23.3 | 17.06$\pm 0.22$ | old | HS | 11.6 | L | SL
B190-G241 | 0:43:43.39 | 41:34:06.0 | 16.91$\pm 0.10$ | old | HS | 7.7 | L | SL
B192-G242 | 0:43:44.52 | 41:37:26.7 | 18.54$\pm 0.09$ | young | HS | 7.7 | L | SL
M001 | 0:43:45.09 | 41:18:12.0 | 19.12$\pm 0.09$ | young | HS | 3.8 | L | SL
B194-G243 | 0:43:45.18 | 41:06:08.7 | 17.24$\pm 0.11$ | old | HS | 5.1 | L | SL
B193-G244 | 0:43:45.52 | 41:36:57.6 | 15.33$\pm 0.03$ | old | HS | 15.4 | L | SL
SK071A | 0:43:46.40 | 41:39:28.8 | 20.07$\pm 0.26$ | na | HS | 3.8 | L | SL
SK072A | 0:43:46.69 | 41:22:28.2 | 17.59$\pm 0.07$ | na | HS | 3.8 | L | SL
B103D-G245 | 0:43:47.54 | 41:27:08.0 | 17.73$\pm 0.05$ | old | HS | 7.7 | L | SL
B472-D064 | 0:43:48.42 | 41:26:53.2 | 15.21$\pm 0.06$ | old | HS | 7.7 | L | SL
SK073A | 0:43:48.52 | 41:07:48.4 | 18.07$\pm 0.08$ | na | HS | 5.1 | L | SL
B195 | 0:43:48.56 | 41:02:28.0 | 18.55$\pm 0.09$ | young | HS | 5.1 | L | SL
B196-G246 | 0:43:48.57 | 40:42:36.8 | 17.40 | old | HS | | B | S
B197-G247 | 0:43:49.72 | 41:30:10.1 | 17.77$\pm 0.09$ | old | HS | 7.7 | L | SL
B199-G248 | 0:43:49.83 | 40:58:14.8 | 17.52$\pm 0.05$ | old | HS | 7.7 | L | SLH
LGS04350.1_410223 | 0:43:50.05 | 41:02:23.2 | | na | HS | | | SL
B198-G249 | 0:43:50.11 | 41:31:52.6 | 17.91$\pm 0.16$ | old | HS | 7.7 | L | SLH
B200 | 0:43:50.44 | 41:29:22.9 | 18.29$\pm 0.16$ | old | HS | 7.7 | L | SL
B201-G250 | 0:43:52.83 | 41:09:58.1 | 16.18$\pm 0.12$ | old | HS | 7.7 | L | SL
M003 | 0:43:54.28 | 41:14:11.7 | 18.47$\pm 0.12$ | young | HS | 5.1 | L | SL
B106D | 0:43:54.45 | 41:15:14.5 | 18.28$\pm 0.14$ | young | HS | 5.1 | L | SLH
B202-G251 | 0:43:54.69 | 41:00:32.5 | 17.77$\pm 0.03$ | old | HS | 7.7 | L | SLH
B203-G252 | 0:43:55.83 | 41:32:35.1 | 16.69$\pm 0.07$ | old | HS | 11.6 | L | SLH
B204-G254 | 0:43:56.42 | 41:22:02.9 | 15.68$\pm 0.07$ | old | HS | 5.1 | L | SL
B361-G255 | 0:43:57.10 | 40:14:01.2 | 17.10 | old | HS | | B | S
B108D | 0:43:57.11 | 41:45:32.9 | 18.63$\pm 0.17$ | young | HS | 5.1 | L | SL
M005 | 0:43:58.00 | 41:21:32.9 | 19.93$\pm 0.12$ | young | HS | 2.5 | L | SL
B205-G256 | 0:43:58.17 | 41:24:38.3 | 15.44$\pm 0.09$ | old | HS | 11.6 | L | SL
B206-G257 | 0:43:58.63 | 41:30:18.1 | 15.06$\pm 0.07$ | old | HS | 7.7 | L | SLH
B110D-V296 | 0:43:59.14 | 41:36:41.3 | 18.60$\pm 0.09$ | old | HS | 3.8 | L | SLH
LGS04359.1_413843 | 0:43:59.17 | 41:38:43.8 | 18.36$\pm 0.13$ | | HS | 5.1 | L | SLH
B207-G258 | 0:43:59.44 | 41:06:10.7 | 17.27$\pm 0.05$ | old | HS | 7.7 | L | SL
B208-G259 | 0:44:00.08 | 41:23:11.6 | 18.05$\pm 0.07$ | old | HS | 7.7 | L | SL
M009 | 0:44:00.83 | 41:17:12.5 | 18.01$\pm 0.12$ | old | HS | 5.1 | L | SL
G260 | 0:44:00.85 | 42:34:48.4 | 17.01 | old | B | | B |
B209-G261 | 0:44:02.63 | 41:25:26.7 | 16.60$\pm 0.06$ | old | HS | 7.7 | L | SL
B210-M11 | 0:44:02.75 | 41:14:24.6 | 17.79$\pm 0.09$ | young | HS | 5.1 | L | SLH
M012 | 0:44:02.83 | 41:21:40.3 | 18.93$\pm 0.25$ | old | HS | 5.1 | L | SL
B211-G262 | 0:44:02.92 | 41:20:04.8 | 16.80$\pm 0.13$ | old | HS | 5.1 | L | SL
B212-G263 | 0:44:03.05 | 41:04:56.4 | 15.39$\pm 0.10$ | old | HS | 11.6 | L | SL
B213-G264 | 0:44:03.52 | 41:30:38.7 | 16.83$\pm 0.07$ | old | HS | 7.7 | L | SLH
B214-G265 | 0:44:03.96 | 41:26:18.6 | 17.65$\pm 0.07$ | old | HS | 5.1 | L | SL
B111D-D065 | 0:44:04.90 | 41:39:05.5 | 18.60$\pm 0.27$ | young | HS | 3.8 | L | SLH
B215-G266 | 0:44:06.40 | 41:31:43.7 | 17.23$\pm 0.09$ | old | HS | 7.7 | L | SLH
B240D-D066 | 0:44:06.85 | 41:40:28.1 | 18.54$\pm 0.26$ | young | HS | 2.5 | L | SL
M016 | 0:44:08.00 | 41:23:54.0 | 19.53$\pm 0.04$ | young | HS | 3.8 | L | SL
B216-G267 | 0:44:08.80 | 41:37:56.0 | 17.33$\pm 0.13$ | young | HS | 7.7 | L | SL
G268 | 0:44:10.01 | 42:46:57.8 | 16.63 | old | B | | B |
DAO67 | 0:44:10.29 | 41:58:51.5 | 18.80$\pm 0.16$ | HII | HS | 5.1 | L | SL
B217-G269 | 0:44:10.60 | 41:23:51.2 | 16.52$\pm 0.07$ | old | HS | 7.7 | L | SL
M019 | 0:44:11.71 | 41:23:54.0 | 18.35$\pm 0.09$ | old | HS | 5.1 | L | SL
SK084A | 0:44:12.35 | 41:21:11.0 | 19.42$\pm 0.17$ | na | HS | 3.8 | L | SLH
M020 | 0:44:13.94 | 41:22:18.9 | 18.53$\pm 0.03$ | young | HS | 5.1 | L | SL
B218-G272 | 0:44:14.33 | 41:19:19.4 | 14.76$\pm 0.11$ | old | HS | 11.6 | L | SL
B219-G271 | 0:44:15.04 | 40:56:47.3 | 16.39 | old | HS | | B | S
B277-M22 | 0:44:16.90 | 41:14:16.0 | 19.36$\pm 0.12$ | old | HS | 3.8 | L | SL
B363-G274 | 0:44:17.25 | 40:33:35.1 | 17.86 | old | HS | | B | S
M023 | 0:44:18.95 | 41:21:10.1 | 19.38$\pm 0.31$ | young | HS | 3.8 | L | SLH
B220-G275 | 0:44:19.44 | 41:30:35.0 | 16.51$\pm 0.13$ | old | HS | 11.6 | L | SLH
M025 | 0:44:19.60 | 41:24:09.0 | 18.88$\pm 0.14$ | young | HS | 5.1 | L | SL
M026 | 0:44:20.25 | 41:27:19.9 | 20.05$\pm 0.12$ | na | HS | 3.8 | L | SL
B112D-M27 | 0:44:21.23 | 41:19:09.8 | 18.57$\pm 0.09$ | old | HS | 5.1 | L | SLH
M028 | 0:44:22.06 | 41:33:40.1 | 19.14$\pm 0.16$ | | | 5.1 | L | L
B246D | 0:44:22.83 | 42:04:32.9 | 16.52$\pm 0.08$ | young | HS | 5.1 | L | S
B221-G276 | 0:44:23.07 | 41:33:06.4 | 16.77$\pm 0.09$ | old | HS | 11.6 | L | SL
B278-M30 | 0:44:23.33 | 41:35:04.1 | 18.79$\pm 0.08$ | young | HS | 5.1 | L | SL
M031 | 0:44:24.26 | 41:33:58.6 | 19.37$\pm 0.13$ | young | HS | 5.1 | L | SL
B222-G277 | 0:44:25.35 | 41:14:12.0 | 17.54$\pm 0.20$ | young | HS | 11.6 | L | SL
SK086A | 0:44:26.07 | 41:35:14.5 | 18.88$\pm 0.57$ | na | HS | 5.1 | L | SL
B115D-M33 | 0:44:26.52 | 41:38:57.5 | 18.53$\pm 0.25$ | na | | 5.1 | L | L
B223-G278 | 0:44:27.05 | 41:34:37.1 | 18.51$\pm 0.16$ | young | HS | 7.7 | L | SL
B224-G279 | 0:44:27.10 | 41:28:49.9 | 15.31$\pm 0.03$ | old | HS | 15.4 | L | SLH
B279-D068 | 0:44:27.99 | 41:44:10.3 | 18.60$\pm 0.18$ | old | HS | 5.1 | L | SLH
B225-G280 | 0:44:29.55 | 41:21:35.8 | 14.15 | old | HS | | B | SH
M039 | 0:44:31.34 | 41:30:04.8 | 19.18$\pm 0.12$ | young | HS | 5.1 | L | SLH
M040 | 0:44:31.51 | 41:27:55.2 | 19.71$\pm 0.09$ | old | HS | 3.8 | L | SL
B228-G281 | 0:44:33.21 | 41:41:27.8 | 16.79$\pm 0.08$ | old | HS | 11.6 | L | SLH
B229-G282 | 0:44:33.83 | 41:38:28.5 | 17.17$\pm 0.03$ | old | HS | 15.4 | L | SLH
M042 | 0:44:33.9 | 41:21:03.0 | 18.87$\pm 0.07$ | young | HS | 5.1 | L | SL
M043 | 0:44:34.36 | 41:23:11.5 | 20.16$\pm 0.14$ | old | HS | 2.5 | L | SL
M044 | 0:44:34.44 | 41:24:09.6 | 19.68$\pm 0.08$ | | | 2.5 | L | L
DAO69 | 0:44:34.80 | 41:53:27.7 | 18.32$\pm 0.48$ | young | HS | 2.5 | L | SL
B230-G283 | 0:44:35.18 | 40:57:12.2 | 16.04 | old | HS | | B | S
M045 | 0:44:36.4 | 41:35:32.9 | 19.25$\pm 0.32$ | old | HS | 5.1 | L | SLH
B365-G284 | 0:44:36.46 | 42:17:20.9 | 16.89$\pm 0.15$ | old | HS | 7.7 | L | SL
M046 | 0:44:36.67 | 41:27:14.2 | 18.93$\pm 0.19$ | na | | 5.1 | L | LH
M047 | 0:44:37.8 | 41:28:51.9 | 19.16$\pm 0.19$ | old | HS | 7.7 | L | SLH
B231-G285 | 0:44:38.59 | 41:27:47.0 | 17.26$\pm 0.03$ | old | HS | 5.1 | L | SLH
BH28 | 0:44:39.10 | 41:44:26.0 | 20.33$\pm 0.23$ | na | | 2.5 | L | LH
B118D | 0:44:39.66 | 41:24:28.2 | 19.16$\pm 0.42$ | young | HS | 2.5 | L | SL
B232-G286 | 0:44:40.23 | 41:15:00.7 | 15.62$\pm 0.08$ | old | HS | 7.7 | L | SLH
M050 | 0:44:40.67 | 41:30:06.6 | 19.47$\pm 0.27$ | young | HS | 5.1 | L | SLH
B233-G287 | 0:44:42.12 | 41:43:54.6 | 15.83$\pm 0.09$ | old | HS | 11.6 | L | SLH
B281-G288 | 0:44:42.85 | 41:20:08.6 | 17.54$\pm 0.04$ | old | HS | 15.4 | L | SL
B234-G290 | 0:44:46.38 | 41:29:17.8 | 16.81$\pm 0.08$ | old | HS | 7.7 | L | SLH
B366-G291 | 0:44:46.72 | 42:03:50.5 | 16.48$\pm 0.19$ | old | HS | 7.7 | L | SLH
B367-G292 | 0:44:47.19 | 42:05:31.9 | 18.11$\pm 0.10$ | young | HS | 7.7 | L | SLH
B368-G293 | 0:44:47.80 | 41:51:09.2 | 18.08$\pm 0.09$ | old | B | 7.7 | L | LH
B255D-D072 | 0:44:48.52 | 42:06:12.9 | 18.97$\pm 0.48$ | na | | 3.8 | L | LH
KHM31-234 | 0:44:49.37 | 41:19:35.0 | 20.74$\pm 0.23$ | young | HS | 3.8 | L | SLH
B121D | 0:44:54.42 | 41:09:28.9 | 18.94$\pm 0.04$ | | | 3.8 | L | L
KHM31-246 | 0:44:54.73 | 41:28:51.3 | 19.78$\pm 0.10$ | interm | HS | 3.8 | L | SLH
B283-G296 | 0:44:55.37 | 41:17:00.2 | 18.01$\pm 0.12$ | old | HS | 7.7 | L | SL
V298 | 0:44:55.70 | 41:29:19.3 | 18.63$\pm 0.24$ | | | 11.6 | L | LH
B475-V128 | 0:44:56.06 | 41:54:00.5 | 17.84$\pm 0.30$ | young | HS | 7.7 | L | SL
M052 | 0:44:56.20 | 41:41:36.0 | 20.83$\pm 0.34$ | | | 2.5 | L | L
M053 | 0:44:57.29 | 41:48:02.0 | 19.25$\pm 0.13$ | old | HS | 5.1 | L | SL
B235-G297 | 0:44:57.93 | 41:29:24.0 | 16.42$\pm 0.12$ | old | HS | 7.7 | L | SL
B256D | 0:44:58.7 | 41:54:36.7 | 17.35$\pm 0.17$ | young | HS | 2.5 | L | SL
M054 | 0:44:59.17 | 41:42:25.6 | 19.96$\pm 0.16$ | | | 3.8 | L | L
B257D-D073 | 0:44:59.36 | 41:54:47.3 | 18.48$\pm 0.35$ | na | | 2.5 | L | L
M055 | 0:44:59.63 | 41:33:39.4 | 19.36$\pm 0.04$ | na | | 7.7 | L | LH
V300 | 0:45:00.65 | 41:28:36.1 | 18.84$\pm 0.05$ | HII | HS | 3.8 | L | SL
M056 | 0:45:01.56 | 41:39:04.5 | 19.98$\pm 0.29$ | na | | 3.8 | L | LH
M057 | 0:45:02.75 | 41:47:02.4 | 19.76$\pm 0.24$ | old | HS | 3.8 | L | SL
M058 | 0:45:03.36 | 41:40:05.5 | 19.43$\pm 0.45$ | old | HS | 3.8 | L | SLH
M059 | 0:45:04.08 | 41:46:20.8 | 18.78$\pm 0.09$ | young | HS | 3.8 | L | SL
KHM31-264 | 0:45:05.86 | 41:35:43.3 | 20.47$\pm 0.19$ | old | HS | 2.5 | L | SLH
KHM31-267 | 0:45:07.14 | 41:37:18.3 | 19.00$\pm 0.15$ | | HS | 3.8 | L | SLH
B476-D074 | 0:45:07.18 | 41:40:31.1 | 18.59$\pm 0.06$ | old | HS | 7.7 | L | SL
M062 | 0:45:07.6 | 41:45:30.9 | 20.20$\pm 0.22$ | young | HS | 2.5 | L | SL
B477-D075 | 0:45:08.4 | 41:39:38.0 | 18.29$\pm 0.16$ | young | HS | 7.7 | L | SLH
B236-G298 | 0:45:08.90 | 40:50:28.6 | 17.38 | old | HS | | B | SH
B237-G299 | 0:45:09.22 | 41:22:34.6 | 17.31$\pm 0.25$ | old | HS | 11.6 | L | SL
M065 | 0:45:09.96 | 41:42:23.4 | 19.37$\pm 0.20$ | | | 5.1 | L | L
V133 | 0:45:10.50 | 42:00:12.0 | 18.58$\pm 0.22$ | young | HS | 5.1 | L | SL
M068 | 0:45:11.0 | 41:38:55.9 | 20.37$\pm 0.15$ | young | HS | 2.5 | L | SL
M069 | 0:45:11.3 | 41:49:20.0 | 20.26$\pm 0.28$ | young | HS | 2.5 | L | SL
M070 | 0:45:11.79 | 41:40:19.8 | 19.86$\pm 0.26$ | old | HS | 3.8 | L | SLH
M072 | 0:45:13.79 | 41:42:25.9 | 18.56$\pm 0.03$ | young | HS | 5.1 | L | SL
M071 | 0:45:13.80 | 41:42:34.0 | 20.30$\pm 0.20$ | | | 2.5 | L | L
B370-G300 | 0:45:14.39 | 41:57:40.8 | 16.28$\pm 0.09$ | old | HS | 7.7 | L | SL
B238-G301 | 0:45:14.67 | 41:19:37.1 | 16.41$\pm 0.09$ | old | HS | 11.6 | L | SL
M073 | 0:45:15.15 | 41:47:32.2 | 19.89$\pm 0.05$ | young | HS | 3.8 | L | SL
B239-M74 | 0:45:15.6 | 41:35:17.2 | 17.19$\pm 0.05$ | old | HS | 7.7 | L | SL
M075 | 0:45:15.80 | 41:44:46.1 | 19.98$\pm 0.14$ | | | 3.8 | L | L
M076 | 0:45:16.1 | 41:43:22.0 | 19.92$\pm 0.24$ | young | HS | 3.8 | L | SL
M077 | 0:45:17.39 | 41:39:33.0 | 20.25$\pm 0.18$ | HII | HS | 2.5 | L | SLH
M078 | 0:45:17.77 | 41:41:52.5 | 20.10$\pm 0.09$ | young | HS | 3.8 | L | SL
M079 | 0:45:17.80 | 41:40:58.2 | 19.70$\pm 0.19$ | young | HS | 3.8 | L | SL
M080 | 0:45:19.59 | 41:48:30.0 | 19.46$\pm 0.15$ | young | HS | 3.8 | L | SL
M081 | 0:45:22.29 | 41:47:57.0 | 20.75$\pm 0.21$ | old | HS | 2.5 | L | SL
B240-G302 | 0:45:25.04 | 41:06:22.1 | 15.23 | old | HS | | B | SH
M082 | 0:45:26.2 | 41:45:52.0 | 19.58$\pm 0.07$ | young | HS | 3.8 | L | SL
M083 | 0:45:26.93 | 41:45:43.8 | 19.90$\pm 0.26$ | | | 3.8 | L | L
B371-G303 | 0:45:27.15 | 41:43:44.9 | 18.27$\pm 0.21$ | young | HS | 7.7 | L | SL
M085 | 0:45:28.0 | 41:42:03.9 | 19.86$\pm 0.30$ | young | HS | 3.8 | L | SL
M086 | 0:45:28.49 | 41:49:29.3 | 18.98$\pm 0.25$ | young | HS | 5.1 | L | SL
B287 | 0:45:28.49 | 41:30:04.8 | 18.08$\pm 0.34$ | old | HS | 7.7 | L | SL
M087 | 0:45:32.05 | 41:49:32.0 | 18.85$\pm 0.40$ | young | HS | 3.8 | L | SL
M089 | 0:45:32.46 | 41:43:33.6 | 20.27$\pm 0.51$ | old | | 2.5 | L | L
M088 | 0:45:32.54 | 41:43:31.1 | 20.07$\pm 0.49$ | young | HS | 2.5 | L | SL
M090 | 0:45:32.98 | 41:48:23.0 | 20.22$\pm 0.10$ | | | 3.8 | L | L
M091 | 0:45:33.11 | 41:42:19.3 | 19.00$\pm 0.12$ | young | HS | 5.1 | L | SLH
B372-G304 | 0:45:33.39 | 42:00:24.4 | 16.56$\pm 0.15$ | old | HS | 7.7 | L | SL
M092 | 0:45:35.60 | 41:45:18.0 | 19.34$\pm 0.17$ | young | HS | 3.8 | L | SL
BH30 | 0:45:36.75 | 41:43:34.9 | 20.50$\pm 0.24$ | | | 2.5 | L | LH
BH29 | 0:45:37.31 | 41:36:45.3 | 19.64$\pm 0.12$ | na | | 5.1 | L | LH
M093 | 0:45:37.47 | 41:40:11.4 | 19.76$\pm 0.10$ | interm | HS | 5.1 | L | SL
M094 | 0:45:39.74 | 41:46:34.6 | 20.01$\pm 0.18$ | | | 3.8 | L | L
M095 | 0:45:39.85 | 41:44:41.6 | 19.61$\pm 0.39$ | | | 3.8 | L | L
B373-G305 | 0:45:41.85 | 41:45:33.6 | 15.70$\pm 0.09$ | old | HS | 7.7 | L | SL
B374-G306 | 0:45:44.53 | 41:41:54.9 | 18.35$\pm 0.14$ | young | HS | 5.1 | L | SLH
V129-BA4 | 0:45:44.69 | 41:51:59.4 | 17.05$\pm 0.03$ | old | HS | 7.7 | L | SL
B480-V127 | 0:45:45.55 | 41:45:52.4 | 18.24$\pm 0.09$ | young | HS | 5.1 | L | SL
B375-G307 | 0:45:45.58 | 41:39:42.4 | 17.51$\pm 0.10$ | old | HS | 11.6 | L | SL
M101 | 0:45:46.28 | 41:48:20.9 | 19.48$\pm 0.33$ | young | HS | 3.8 | L | SL
M102 | 0:45:46.8 | 41:45:23.0 | 20.26$\pm 0.06$ | HII | HS | 3.8 | L | SL
B377-G308 | 0:45:48.28 | 40:38:04.2 | 17.14 | old | HS | | B | S
B376-G309 | 0:45:48.40 | 41:42:40.1 | 18.03$\pm 0.07$ | young | HS | 7.7 | L | SL
M104 | 0:45:48.84 | 41:48:20.2 | 18.82$\pm 0.12$ | young | HS | 3.8 | L | SL
M105 | 0:45:49.70 | 41:39:26.0 | 20.18$\pm 0.43$ | old | HS | 2.5 | L | SL
B484-G310 | 0:45:53.89 | 41:47:37.0 | 18.36$\pm 0.09$ | young | HS | 5.1 | L | SL
B483-D085 | 0:45:53.92 | 42:02:18.1 | 18.43$\pm 0.16$ | young | HS | 7.7 | L | SL
B378-G311 | 0:45:57.24 | 41:53:31.1 | 17.56$\pm 0.07$ | old | HS | 7.7 | L | SL
B379-G312 | 0:45:58.83 | 40:42:31.3 | 16.13 | old | HS | | B | SH
B380-G313 | 0:46:06.20 | 42:00:53.0 | 17.53$\pm 0.27$ | young | HS | 7.7 | L | SL
B381-G315 | 0:46:06.54 | 41:20:58.8 | 15.76 | old | HS | | B | S
B486-G316 | 0:46:08.62 | 40:58:03.6 | 17.52 | old | HS | | B | S
B382-G317 | 0:46:10.32 | 41:37:40.5 | 17.36$\pm 0.06$ | old | HS | 7.7 | L | SL
B383-G318 | 0:46:11.94 | 41:19:41.4 | 15.33 | old | HS | | B | S
B384-G319 | 0:46:21.93 | 40:16:59.6 | 15.79 | old | HS | | B | SH
BH32 | 0:46:23.55 | 42:00:58.5 | 20.26$\pm 0.27$ | na | | 3.8 | L | LH
SH20 | 0:46:26.04 | 41:03:16.0 | 16.81 | | | | G |
B386-G322 | 0:46:27.00 | 42:01:52.8 | 15.69$\pm 0.09$ | old | HS | 7.7 | L | SLH
SH21 | 0:46:31.79 | 39:23:56.1 | 16.51 | | | | G |
B387-G323 | 0:46:33.51 | 40:44:13.4 | 16.98 | old | HS | | B | S
B488-G324 | 0:46:34.28 | 42:11:42.7 | 16.78$\pm 0.17$ | HII | HS | 5.1 | L | SL
B489 | 0:46:36.36 | 40:00:26.8 | 17.35 | | | | G |
DAO88 | 0:46:41.98 | 42:15:45.9 | 19.82$\pm 0.15$ | HII | HS | 3.8 | L | SL
G327-MVI | 0:46:49.49 | 42:44:46.7 | 15.94 | old | B | | B |
B297D | 0:46:55.68 | 42:19:44.9 | 17.78$\pm 0.16$ | | | 7.7 | L | L
B391-G328 | 0:46:58.10 | 41:33:56.5 | 17.28 | old | HS | | B | S
B392-G329ddnot a cluster in Barmby et al. (2000) | 0:47:00.94 | 41:54:44.5 | 16.80$\pm 0.03$ | young | HS | 15.4 | L | SL
B393-G330 | 0:47:01.20 | 41:24:06.3 | 16.93 | old | HS | | B | S
BA28 | 0:47:14.22 | 42:21:42.2 | 19.10$\pm 0.41$ | | | 7.7 | L | L
B495-G334 | 0:47:24.69 | 41:55:11.5 | 17.97$\pm 0.22$ | | | 7.7 | L | L
B396-G335 | 0:47:25.15 | 40:21:42.1 | 17.38 | old | HS | | B | S
B397-G336 | 0:47:27.23 | 41:12:10.4 | 16.53 | old | HS | | B | S
BA10 | 0:47:56.28 | 42:28:43.5 | 19.10$\pm 0.03$ | | | 7.7 | L | L
B398-G341 | 0:47:57.78 | 41:48:45.6 | 17.46 | old | HS | | B | S
B399-G342 | 0:47:59.55 | 41:35:28.3 | 17.28 | old | HS | | B | S
B400-G343 | 0:48:01.45 | 42:25:33.2 | 16.50$\pm 0.08$ | old | B | 7.7 | L | L
B401-G344 | 0:48:08.50 | 41:40:41.9 | 16.83 | old | HS | | B | S
B329D | 0:48:19.40 | 42:08:26.7 | 16.61$\pm 0.29$ | | | 11.6 | L | L
B332D | 0:48:29.33 | 41:38:10.9 | 17.21 | | | | G |
B402-G346 | 0:48:36.11 | 42:01:34.8 | 17.42$\pm 0.13$ | old | P | 7.7 | L | L
B503 | 0:48:37.41 | 39:31:03.4 | 16.30 | | | | G |
BA11 | 0:48:45.59 | 42:23:37.7 | 17.66 | old | P | | B |
DAO99 | 0:48:48.30 | 42:32:43.9 | 19.03 | | | | B |
B334D | 0:48:54.84 | 39:35:56.0 | 17.56 | | | | G |
B335D-D100 | 0:49:01.25 | 42:15:39.0 | 17.29 | | | | G |
B337D | 0:49:11.20 | 41:07:21.0 | 18.23 | old | HS | | B | S
B338D | 0:49:15.76 | 40:46:23.4 | 17.91 | | | | G |
B403-G348 | 0:49:17.62 | 41:35:08.1 | 16.22 | old | HS | | B | S
B506 | 0:49:34.90 | 40:00:28.9 | 17.04 | | | | G |
B405-G351 | 0:49:39.80 | 41:35:29.7 | 15.20 | old | HS | | B | S
B345D | 0:49:52.55 | 40:53:10.1 | 17.25 | | | | G |
B509-D108 | 0:49:52.84 | 42:09:43.3 | 17.52 | | | | G |
B406-D109 | 0:49:59.32 | 42:15:55.9 | 17.19 | | | | G |
B407-G352 | 0:50:09.95 | 41:41:00.9 | 16.09 | old | B | | B |
B347D | 0:50:13.80 | 40:56:24.2 | 16.84 | | | | G |
G353-BA13 | 0:50:18.21 | 42:35:44.1 | 17.15 | old | P | | B |
B348D | 0:50:19.21 | 40:58:02.7 | 18.14 | | | | G |
B349D | 0:50:32.00 | 41:13:20.3 | 16.68 | | | | G |
B350D | 0:50:48.85 | 42:21:43.9 | 17.77 | | | | G |
EXT8 | 0:53:14.53 | 41:33:24.4 | | | B | | |
VDB00 | 01:16:04.1 | 49:36:36.2 | 16.07 | | | | G |
Table 2Young Clusters, Ordered by Mass aafootnotetext: Not a cluster in Cohen
et al. (2005)
bbfootnotetext: Star in field
ccfootnotetext: Noted as young in van den Bergh (1969)
ddfootnotetext: Noted as young in Hodge (1979)
eefootnotetext: Noted as young in Elson & Walterbos (1988)
fffootnotetext: Noted as young in Barmby et al. (2000)
ggfootnotetext: Noted as young in Fusi Pecci et al. (2005)
hhfootnotetext: Noted as young in Burstein et al. (2004)
iifootnotetext: Noted as young in Beasley et al. (2004)
jjfootnotetext: Noted as young in Williams & Hodge (2001a)
Table 3Stars
aafootnotetext: The suffix “x” indicates that coords correspond to a real
object, but which was wrongly identified in the initial input catalog.
bbfootnotetext: All velocities from Hectospec
ccfootnotetext: Source of photometry: L=this paper; B=Barmby et al. (2000);
G=Galleti et al. (2007)
ddfootnotetext: Source of classification as a star: S=spectrum from this paper
indicates a star; L=LGS image indicates stellar FWHM; H=HST image indicates a
star; B=Barmby et al. (2000) indicated a star
eefootnotetext: NB61 is an M star, the velocity of -646 listed Galleti et al.
(2007) was likely the result of a velocity template mismatch.
fffootnotetext: Multiple stars or asterisms in M31.
ggfootnotetext: The velocity of this F supergiant star indicates is it
probably a member of the M31 giant stream. Details to be presented in a
subsequent paper.
Table 4Possible Stars
aafootnotetext: The suffix “x” indicates that coords correspond to a real
object, but which was wrongly identified in the initial input catalog.
bbfootnotetext: Source of velocity: HS=this paper; B=Barmby et al. (2000);
K=Kim et al. (2007); P=Perrett et al. (2002)
ccfootnotetext: Source of photometry: L=this paper; B=Barmby et al. (2000);
G=Galleti et al. (2007)
ddfootnotetext: Source of classification as a star: S=spectrum from this paper
exists but is inconclusive; L=LGS image indicates stellar FWHM; B=Barmby et
al. (2000) indicated a star
eefootnotetext: The colors of SK106A indicate it is an M star, the velocity of
-890 reported in Kim et al. (2007) was likely the result of a velocity
template mismatch (see also Lee et al., 2008)
Table 5Galaxies
aafootnotetext: All redshifts from Hectospec. Median velocity error is 17 km
s-1 . Entries without redshifts were classified background based on images.
bbfootnotetext: Perrett et al. (2002) had velocity indicating M31 membership
Table 6Missing objects or bad coordinates
Table 7Comparison of ages
|
arxiv-papers
| 2008-09-30T19:48:23
|
2024-09-04T02:48:58.047209
|
{
"license": "Public Domain",
"authors": "Nelson Caldwell, Paul Harding, Heather Morrison, James A. Rose,\n Ricardo Schiavon, Jeff Kriessler",
"submitter": "Nelson Caldwell",
"url": "https://arxiv.org/abs/0809.5283"
}
|
0810.0048
|
# Large structures linked to southern O-type stars
M. C. Martín 11affiliation: Instituto Argentino de Radioastronomía (CCT-La
Plata, CONICET), C.C. 5, 1894, Villa Elisa, Bs. As., Argentina C. E. Cappa
11affiliationmark: 22affiliation: Facultad de Ciencias Astronómicas y
Geofísicas, Paseo del Bosque S/N, 1990, Universidad Nacional de La Plata,
Argentina and G. A. Romero11affiliationmark: 22affiliationmark: Instituto
Argentino de Radiostronomía, Parque Pereyra Iraola Km 40,500, CC 5, Villa
Elisa, Bs. As., Argentina
###### Abstract
In our search for interstellar bubbles around massive stars we analyze the
environs of the O-type stars HD 38666, HD 124979, HD 163758, and HD 171589.
The location of the stars, which are placed far from the galactic plane,
favors the formation of large wind bubbles. We investigate the distribution of
the neutral and ionized gas based on , CO, and radio continuum data, and that
of the interstellar dust based on far infrared IRIS images. Here we report the
discovery of neutral gas cavities and slowly expanding shells associated with
the four massive stars. IR and optical counterparts were also detected for
some of the stars. We discuss the probability that the features have
originated in the action of the stellar winds on the surrounding gas.
Analizamos la distribución del material interestelar en la vecindad de las
estrellas de tipo O: HD 38666, HD 124979, HD 163758 y HD 171589, a fin de
investigar la existencia de burbujas interestelares asociadas a las estrellas.
La localización de las estrellas lejos del plano galáctico, favorece la
formación de burbujas de gran extensión. En base a datos del , continuo de
radio y CO, e imágenes en el infrarrojo lejano que revelan la emisión del
polvo interestelar, investigamos la distribución del gas neutro e ionizado en
estas regiones. Reportamos en este trabajo la detección de cavidades y
envolturas en expansión en hidrógeno neutro, asociadas a las cuatro estrellas
masivas. Se discute la posibilidad de que estas estructuras hayan sido
originadas por la acción de los vientos estelares sobre el gas circundante.
ISM: bubbles stars: individual (HD 38666, HD 124979, HD 163758, HD 171589)
## 0.1 Introduction
The strong stellar winds of massive stars interact with their environs
creating interstellar bubbles that are detected in a large range of
wavelengths. UV photons with energies h$\nu\geq$ 13.6 eV ionize the inner part
of the expanding bubbles, which are generally detected both in the optical
(Lozinskaya, 1982) and in the radio continuum ranges as shell-like thermal
sources (Goss & Lozinskaya, 1995). Should the ionizing front be trapped within
the expanding envelope, interstellar bubbles have an outer neutral region that
can be identified both in the Hi 21–cm line emission and in molecular lines
(Cappa et al., 2005)
Many interstellar bubbles are detected neither in the radio continuum nor in
optical lines (e.g. Cappa & Benaglia, 1998; Arnal et al., 1999; Cichowolski &
Arnal, 2004). A low ambient density may be responsible for the undetected
radio continuum or optical emission, as was suggested for bubbles in the LMC
by Nazé et al. (2002). As a consequence, the analysis of the Hi emission
distribution is an important tool to investigate the presence of such
structures and allows discrimination in distance based on galactic rotation
models.
Data at radio and infrared wavelengths offer an opportunity to investigate the
characteristics and distribution of the ionized and neutral gas associated
with these bubbles, as well as those of the interstellar dust.
A low-velocity massive star with constant stellar wind parameters located in
an homogeneous interstellar medium with uniform density creates a spherical
stellar wind bubble. Aspherical bubbles appear as the result of relaxing some
of these hypothesis. The combination of a powerful stellar wind and a high
space velocity makes this scenario changes drastically. Weaver et al. (1977)
pointed out that a massive star moving supersonically with respect to the
surrounding gas originates an aspherical stellar bubble, elongated in the
direction of the movement of the star. A wind bow-shock can develop ahead of
the star (Wilkin, 1996; Raga et al., 1997). As van Buren & McCray (1988) have
shown, these shocks can be identified in the far IR emission.
Here we present a large scale study of the interstellar medium around four
southern O-type stars: HD 38666, HD 124979, HD 163758, and HD 171589, the
first two of them classified as runaway stars. Our aim is to investigate the
action of the UV photon flux and the stellar winds of these stars on their
surroundings. The selected stars are located at $|$b$|>$ 3∘, allowing for the
formation of relatively large structures which are easily identified using low
angular resolution data.
In the following sections, we analyze the interstellar medium in the environs
of the stars looking for cavities and shells in the neutral gas that have
originated in the action of the massive stars on their surroundings. These
studies are important to improve theoretical models on the interplay between
massive stars and the surrounding gas.
-2cm-2cm
Table 1: Main parameters of the O-Type stars
10 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Name (l,b) Spectral $d$ $V$
$(B-V)$ $d_{spc}^{\mathrm{e}}$ $\mu_{l}\ cosb^{\mathrm{f}}$
$\mu_{b}^{\mathrm{f}}$ z Classification (∘) (kpc) (mag) (mag) (kpc) (mas yr-1)
(mas yr-1) (pc) HD 38666 237.29,–27.10 O9.5 Va 0.53a 5.17a –0.27a 0.8
+23.48$\pm$1.1 –3.52$\pm$1.1 –255 HD 124979 316.40,+9.08 O8 III ((f))b 8.52a
1.05a 3.7 –8.8$\pm$1.1 +8.2$\pm$1.1 +590 HD 163758 355.36,–6.10 O6.5 Ia (f)a
7.32a 0.03a 3.5 –2.5$\pm$1.5 –5.2$\pm$1.5 –374 HD 171589 18.65,–3.09 O7 II
(f)a 1.15c-1.5d 8.29a 0.29a 3.0 +6.0$\pm$1.5 –4.1$\pm$1.5 +160 aGOS catalogue
(Maíz-Apellániz et al. (2004)) bSee text cKozok (1985) dGarmany et al. (1982)
eSpectrophotometric distances derived in this paper. See text for details
fDerived from Tycho-2 catalogue (Hog & Murdin, 2000).
## 0.2 Data bases
The Hi 21–cm line data analyzed in this paper were extracted from the Leiden
Argentine Bonn Survey of Galactic Hi (Kalberla et al., 2005; Arnal et al.,
2000) observed using the radiotelescope of the Instituto Argentino de
Radioastronomía. The 21–cm data span the velocity range from –250 to +250 km
s-1 and were obtained with a velocity resolution of 1.3 km s-1. The angular
resolution is 30.
Radio continuum data at 4.85 GHz from the Parkes-MIT-NRAO (PMN) Southern Radio
Survey (Condon et al., 1993) are available only for HD 124979 and HD 171589,
while data at 408 MHz (Haslam et al., 1982) and at 35 MHz (Dwarakanath & Udaya
Shankar, 1990) are available for the four target stars. The angular
resolutions are 5$\arcmin$, 085, and 26$\arcmin$ x
42$\arcmin$/cos($\delta$-14∘), for the surveys at 4.85 GHz, 408 MHz, and 35
MHz, respectively.
The analysis of the molecular gas distribution for the regions of HD 163758
and HD 171589 was based on the CO survey by Dame et al. (2001), which has an
angular resolution of 88. The velocity coverage and the velocity resolution
are –160 to +160 km s-1 and 1.3 km s-1, respectively. The rms noise level is
0.2 K.
The dust distribution in the region was analyzed using the
IRIS111http://www.ias.fr/IRIS database. These images are a new generation of
IRAS images that benefit from a better zodiacal light subtraction,
calibration, and a better destriping. IRIS images have an angular resolution
of around 4 (Miville-Deschênes & Lagache, 2005).
The optical images were obtained from the Full-Sky H-Alpha survey (H-Alpha
Composite, Finkbeiner (2003))
## 0.3 Target stars
Table 1 summarizes the stellar parameters relevant to this study: the name of
the stars and their galactic coordinates are listed in the first two columns.
The spectral types, indicated in the third column, were obtained from Maíz-
Apellániz et al. (2004)(GOS catalogue). Distance estimates taken from the
literature are given in column 4. Columns 5 and 6 list the visual magnitudes
$V$ and color indices $(B-V)$, respectively. We used these values together
with absolute magnitudes from Vacca et al. (1996) and intrinsic color indices
from Wegner (1994) to derive the spectrophotometric distance listed in column
7. The components of the proper motion in the galactic coordinate system
listed in columns 8 and 9 were calculated from the Tycho-2 catalogue (Hog &
Murdin, 2000). Finally, z-distances to the galactic plane are shown in col.
10. They were derived by adopting the distances listed in col. 7.
Based on its large proper motion, HD 38666 ($\mu$ Col) was identified as a
runaway star belonging to Ori OB1 (Blaauw, 1961).
HD 124979 has been classified as an O8 ((f)) star by MacConnell & Bidelman
(1976). Following the Of typification by Walborn (1971), we have adopted a
luminosity class III. Mason et al. (1998) analyzed the binarity status, and
concluded that it maybe a spectroscopic binary, tagging it as “SB1?”. The
measured proper motion and radial velocity ($-$68 km s-1) allow to classify
this star as a runaway.
No information about the interstellar medium in the environs of the target
stars was found in the literature.
## 0.4 Hi emission towards the selected stars
Figure 1 ($\it{a}$-$\it{d}$) displays the Hi 21–cm line profiles towards the
O-type stars. They were obtained by averaging the Hi emission within an area
of about 3∘$\times$3∘ centered at the position of the massive stars. Our aim
is to analyze the main characteristics of the neutral gas emission towards the
selected stars and its relation to the galactic spiral structure.
The Hi emission profile in the region of HD 38666 (Fig. 1a) reveals a weak Hi
gas component centered at $\rm{v}\approx$ 0 km s-1 (all velocities in this
paper are referred to the LSR). Because of the high galactic latitude of the
star, this material corresponds to local gas. Bearing in mind the distances to
HD 38666 listed in Table 1, we expect that gas related to this star has low
positive velocities.
Figure 1: Hi profiles obtained by averaging the Hi emission in regions of
3∘$\times$3∘ around the O-type stars. (a) HD 38666; (b) HD 124979; (c) HD
163758; and (d) HD 171589. The intensity scale is given in brightness
temperature. Velocities are referred to the LSR.
The Hi emission profile in the region of HD 124979 (Fig. 1b) reveals three
components centered at $\rm{v}\approx$ –2, $\rm{v}\approx$ –20, and
$\rm{v}\approx$ –50 km s-1. The former gas component is associated with the
local region. The circular galactic rotation model by Brand & Blitz (1993)
predicts near and far kinematical distances $d\approx$ 1.4 and 11 kpc,
respectively, for the component at –20 km s-1, while for the component at –50
km s-1, $d\approx$ 3.7 and 8.5 kpc, respectively. The near kinematical
distances are compatible with the locations of the Sagittarius-Carina and
Scutum-Crux spiral arms (Russeil, 2003). According to the analytical fit by
Brand & Blitz, gas in the environs of this star, i.e. at 3.7 kpc should have
velocities of about –50 km s-1.
In the line of sight to HD 163758 (Fig. 1c), Hi emission was detected from –40
to +40 km-1. The profile shows a main peak of neutral gas emission centered at
$\rm{v}\approx$ +5 km s-1, probably related to the local spiral arm. The fact
that positive velocities are forbidden in the fourth galactic quadrant within
the solar circle (e.g. Brand & Blitz, 1993) indicates that most of this
material is local and associated with the Gould’s belt (Olano, 1982). Gas
linked to this star at a distance of 3.5 kpc should have velocities of about
–10 km s-1. Note however that kinematical distances have large uncertainties
in this line of sight.
The Hi emission towards HD 171589 (Fig. 1d) was detected within the velocity
interval –40 to +100 km s-1. The neutral gas emission shows major peaks at
$\rm{v}\approx$ 0, $\rm{v}\approx$ +12 km s-1, and $\rm{v}\approx$ +40 km s-1.
The lower velocity peak belongs to the local spiral arm, while peaks at
positive velocities correspond to near and far kinematical distances
$d\approx$ 1.3 and 15 kpc, and $d\approx$ 3.4 and 12.5 kpc, respectively.
These gas components are probably related to the Sagittarius-Carina and
Scutum-Crux spiral arms, respectively (Russeil 2003). A relatively low
emission component can also be identified at $\approx$ –25 km s-1. The
analytical fit by Brand & Blitz predicts that gas with this velocity is
located at $\approx$ 20 kpc, well beyond the solar circle. Material related to
the star, placed at $d\leq$ 3 kpc, should appear with velocities lower than
+33 km s-1.
## 0.5 Hi structures and their counterparts at other frequencies
In order to look for Hi structures linked to the selected stars, we analyzed
the neutral hydrogen distribution in their environs. A series of $\it{(l,b)}$
maps at maximum velocity resolution (1.3 km s-1) showing the Hi emission
distribution at different velocities were constructed. The Hi emission is
shown in Fig. 2 for HD 38666, Fig. 4 for HD 124979, Fig. 6 for HD 163758, and
Fig. 8 for HD 171589. Each individual image corresponds to a velocity interval
of 4 km s-1. Although the whole velocity range at which Hi emission is
detected was analyzed, the figures include only the images where Hi structures
probably linked to the stars are detected.
The identification of an Hi structure associated with a certain star requires
a careful inspection of the Hi images, looking for shells and voids that might
be related to the star. Several conditions are necessary to associate an Hi
structure with a certain star: (a) the star is expected to be located inside
the Hi void or close to its inner borders; (b) the kinematical distance to the
structure should coincide, whithin errors, with the spectrophotometric
distance; and (c) bearing in mind that the velocity dispersion in the
interstellar medium is about 6 km s-1, an Hi feature should remain detectable
for a larger velocity interval to be considered as a physical structure.
We have found a number of Hi structures towards the target stars that fulfill
these conditions and, consequently, can be related to the stars. In the
following sections we describe the Hi structures along with their counterparts
at other wavelengths.
The main physical parameters of each structure are summarized in Table 2,
which is described in § 6.1.
Figure 2: Hi emission distribution towards HD 38666, integrated in steps of 4
km s-1. The velocity range and the greyscale (in K) of each image are
indicated in its upper part. The contour lines are from the minimum value of
the grayscale to 12 K, in steps of 1 K. HD 38666 is indicated by the star.
Figure 3: Upper panel. Hi emission distribution within the velocity interval
–7.2 to +17.0 km s-1 showing the Hi structure related to HD 38666. The
grayscale is from 2 to 5 K, and the contour lines are 2.5, 3, 3.5, 4 and 5 K.
HD 38666 is indicated by a star. Central panel. Overlay of the IR emission at
100 $\mu$m (grayscale) and the same Hi contours of the upper panel. The
grayscale corresponds to 1.8 to 2.8 MJy ster-1). Bottom panel. Overlay of the
H-Alpha Composite image (grayscale) and the same Hi contours of the upper
panel. The grayscale is from 4.3 to 6.5 R.
### 0.5.1 The ISM around HD 38666
The Hi emission distribution towards this star within the velocity interval
from –11.8 to +21.1 km s-1 is displayed in Fig. 2. HD 38666 is indicated by a
star. The images show that the O-type star is projected onto the border of a
low emission region identified within the velocity interval from –7.7 to +17.0
km s-1, centered approximately at (l,b) = (2355,–275). Identification of Hi
structures is difficult for velocities higher than +17 km s-1 due to the lower
general Hi emission in the region.
Figure 4: Hi emission distribution towards HD 124979, as in Figure 2. For the
seven first maps, the contour lines are from 2 to 4 in steps of 0.5 K and for
5 to 10 in steps of 1 K, and for the last map, from 8 to 22 in steps of 2 K.
Figure 5: Upper panel. Probable Hi structure related to HD 124979. The image
shows the Hi column density distribution within the velocity interval –47.9,
–34.6 km s-1. The grayscale is from 3.5 to 10 K, and the contour lines are
from 3.5 K to 12 K in steps of 1 K. Central panel. Overlay of the far IR
emission distribution at 100 $\mu$m (grayscale from 21 to 35 MJy ster-1) and
the Hi column density distribution (contour lines). Bottom panel. Overlay of
the H-Alpha Composite image (grayscale) from 10 to 14 R and the Hi column
density distribution (contour lines).
The cavity and surrounding shell around this star is shown in Fig. 3 (upper
panel). The image was obtained by integrating the Hi emission within the
velocity interval from –7.7 to +14.0 km s-1. The feature is elongated in the
direction of the stellar proper motion, which is marked in the figure with a
black arrow. The central panel displays an overlay of the IRIS image at 100
$\mu$m (grayscale) and the Hi image (contour lines). The Hi shell can
partially be detected in the far infrared. The presence of a strong IR
filament at b $\approx$ –25∘is compatible with the existence of a thin Hi wall
in this region. An IR emission band is also projected onto the cavity. If
related to the structure, this emission might be linked to the approaching or
receding caps.
The bottom panel shows a superposition of the same Hi contour lines and the
H-Alpha Composite image. Optical emission is present inside the cavity and
appears bordered by the neutral envelope towards lower galactic longitudes and
higher negative galactic latitudes.
Neither catalogued Hii regions nor supernova remnants were found over the
regions under study. The image at 408 MHz (not shown here) displays a ring-
like structure centered at (l,b) = (2350,–280), with inner and outer semiaxes
of 21$\times$07 and 42$\times$19, respectively. The radio continuum feature is
clearly larger than the Hi cavity, and most of it is projected onto the
neutral envelope, contrary to what is expected for an interstellar bubble or
Hii region. Thus, the lack of coincidence between the radio continuum feature
and the Hi cavity casts doubts on its relation to the neutral shell.
The systemic velocity of the structure, defined as the velocity at which the
cavity has its larger dimensions and is better defined, is +8 km s-1.
According to the analytical fit by Brand & Blitz (1993), the low systemic
velocity derived for the Hi structure indicates a kinematical distance of
about 1.0 kpc, in agreement with the stellar distance listed in Table 1. The
location of the star with cavity, together with the agreement between the
kinematical and spectrophotometric distances strongly suggest that the Hi
structure is related to HD 38666. We have adopted a distance $d$ = 0.5$\pm$0.1
kpc for this structure.
At this distance, the stellar tangential velocity is quite large,
$\rm{v_{t}}\approx$ 57$\pm$15 km s-1, and the massive star could have created
a bow-shock structure. The presence of a bow-shock like object related to this
star was investigated by van Buren & McCray (1988), who were not succesful in
finding such structure in the IRAS images.
### 0.5.2 The ISM around HD 124979
The analysis of the Hi emission distribution maps towards HD 124979 allows
identification of a cavity and shell probably associated with the star. The Hi
minimum is detected within the velocity interval from –52.0 to –23.2 km s-1,
decreasing in angular size for velocities $\rm{v}$ $>$ –31.4 km s-1. At the
stellar position, the contour lines are distorted in the velocity range –52.0
to –39.7 km s-1.
The Hi minimum is better defined from –52.0 to –31.4 km s-1, and thus the
systemic velocity of the structure can be derived as the central value of the
mentioned interval, $\rm{v_{sys}}\approx$ –42$\pm$10 km s-1. Taking into
account a velocity dispersion of 6 km s-1, the systemic velocity corresponds
to a kinematical distance of 3.0$\pm$1.0 kpc, which is compatible with the
spectrophotometric distance. We have adopted $d$ = 3.5$\pm$1.0 kpc as the
distance to the Hi structure.
Figure 5 (upper panel) shows the Hi brightness temperature distribution
towards HD 124979, averaged in the velocity interval from –47.9 to –35.6 km
s-1, for which the Hi cavity is better defined. In order to delineate the
cavity we have considered the brightness temperature contour corresponding to
3.5 K. The centroid of the structure is (l,b) = (3172,+93). Both the cavity
and the almost complete shell are elongated in the direction of the tangential
motion of the star (marked with a black arrow).
Figure 6: Hi emission distribution towards HD 163758, as in Figure 2. For the
maps corresponding to the interval –26.3 to –5.7 km s-1, the contour lines are
from the minimum value of the grayscale to 12 K, in steps of 1 K. For the
image showing the interval –5.7 to –1.6 km s-1, the contour lines are from the
minimum value of the grayscale to 40 K, in steps of 2.5 K. HD 163658 and WR
109 are indicated by the star and the cross, respectively.
Figure 7: Upper panel: Hi emission towards HD 163758 integrated within the
velocity range –22.2 to –5.7 km s-1. The grayscale is from 6 to 12 K. The
contour lines are 4.5 and from 5 to 12 K, in steps of 1 K. Central panel:
Overlay of the H-Alpha Composite image (grayscale from 32 to 45 R) and the Hi
emission (contour lines, 4.5, 5 and from 6 to 12 K, in steps of 2 K). Bottom
panel: Overlay of the IR emission at 100 $\mu$m (grayscale from 50 to 80 MJy
ster-1) and the Hi emission contour lines of central panel.
The 100 $\mu$m IRIS image shows far IR emission bordering the region towards
the galactic plane, revealing the presence of dust associated with the bright
portions of the surrounding shell (Fig. 5, central panel). The bright IR areas
detected inside the cavity, which extend well beyond the cavity, are probably
unrelated to the Hi structure. The proper motion of the star corresponds to a
tangential velocity of $\rm{v_{t}}\approx$ 200$\pm$30 km s-1. A careful
inspection of the IRIS images does not allow to detect a bow-shock like
structure at IR wavelengths.
Neither catalogued Hii regions nor supernova remnants are linked to the
structure. The bottom panel of Fig. 5 indicates that no significant optical
emission is detected towards this region. An inspection of the image at 408
MHz do not show identifiable radio emission probably related to the Hi
structure.
No additional OB stars at a compatible distance are projected onto the Hi
structure.
Figure 8: Hi emission distribution towards HD 171589, as in Figure 2. The
grayscale is from 5 to 30 K. The contour lines are from 5 to 11 K, in steps of
2 K, and from 15 to 30 K, in steps of 5 K.
Figure 9: Upper panel: Hi column density distribution towards HD 171589,
within the velocity range +26.3 to +44.8 km s-1. The grayscale is from 7 to 20
K. Contour lines are 7, 8, and 10 K, and from 15 to 30 K, in steps of 5 K. HD
171589 is indicated by a five pointed star, and the SNR G17.4–2.3 and SNR
G17.8–2.6 are marked by crosses. Central panel: Overlay of the H-Alpha
Composite image (grayscale from 35 to 70 R) and the Hi column density
distribution shown in the upper panel (contour lines). Bottom panel: Overlay
of the IR emission distribution at 100 $\mu$m (grayscale from 70 to 130 MJy
ster-1) and the Hi column density distribution shown in the upper panel
(contour lines).
### 0.5.3 The ISM around HD 163758
Figure 6 displays Hi brightness temperature images corresponding to the
velocity interval from –26.3 to –1.6 km s-1 in the vicinity of HD 163758. An
inspection of the images shows the presence of a low emission region at b
$\leq$ –5∘.
The Hi cavity centered at (l,b) = (3560, –65) can be appreciated at velocities
in the range $\rm{v}$ from –26.3 to –5.7 km s-1. HD 163758, which is indicated
by the star symbol, appears projected near the higher density border of the
cavity, indicated by the piled-up of the contour lines.
The upper panel of Fig. 7 displays the mean Hi brightness temperature in the
range (–22.2,–5.7) km s-1, where the cavity and shell centered at (l,b) =
(3560, –65) are clearly identified in contours and grayscale. The central and
bottom panels show overlays of the Hi emission distribution and the H$\alpha$
and 100 $\mu$m IRIS emissions, respectively. The H$\alpha$ image reveals a
shell-like structure of about 15 in diameter centered at (l,b) = (3560, –65),
with the O-type star projected onto one of its borders. The IRIS image shows
the lack of IR emission in the region of the Hi cavity. Bearing in mind the
angular resolution of the Hi data, the correlation of the optical feature with
the small Hi structure is good, suggesting that the Hi feature is the neutral
gas counterpart of the optical shell. The fact that the star appears projected
onto the highest density border of the Hi structure reinforces the association
between the star and the cavity.
A search for other massive stars projected onto the region shows that the
Wolf-Rayet star WR 109 (= V617 Sgr, WN5h+?, van der Hucht, 2001) is placed
about 2∘ far away from the position of the O star. According to van der Hucht
(2001), WR 109 is located at a photometric distance of 34 kpc. The faraway
estimated distance is the result of the low optical absorption derived for the
star, along with its low apparent magnitude. This WR star, marked by a cross
symbol in Figs. 6 and 7, is also near the border of the cavity. Its position
well outside the optical shell suggests that the WR star is unconnected to the
H$\alpha$ shell.
-2cm-2cm
Table 2: Physical parameters of the Hi structures around the O stars
5 HD 38666 HD 124979 HD 163758 HD 171589 (l,b) center (2355,–275) (3172,+93)
(3560,–65) (187,–30) Velocity interval $\rm{v}_{1},\rm{v}_{2}$ (km s-1) –7,+17
–53,–25 –22,–5 +26,+45 Systemic velocity $\rm{v}_{sys}$ (km s-1) +8$\pm$1
–42$\pm$3 –10$\pm$1 +34$\pm$1 Expansion velocity $\rm{v}_{exp}$ (km s-1) 11 15
10 11 Kinematical distance (kpc) 1.0 3.0$\pm$1.0 3.3$\pm$1.0 3.0$\pm$0.5
Adopted distance (kpc) 0.5$\pm$0.1 3.5$\pm$1.0 3.5$\pm$1.0 3.0$\pm$0.6 Radius
of the Hi cavity $R_{cav}$ 16 125 06 06 Radius of the Hi structure $R_{s}$ 30
19 12 15 Radius of the Hi structure $R_{s}$ (pc) 26$\pm$5 116$\pm$33 70$\pm$20
52$\pm$9 Mass in the shell (M⊙) 600$\pm$240 58800$\pm$33600 4300$\pm$2450
6900$\pm$2300 Mass deficiency (M⊙) 80$\pm$32 6000$\pm$3400 1100$\pm$630
350$\pm$120 Swept-up mass $M_{s}$ (M⊙) 340$\pm$140 32400$\pm$18500
2700$\pm$1540 3600$\pm$1200 $n_{e}$ (cm-3) (f=1.0) 0.5 0.9 $M_{i}$ (M⊙)
(f=1.0) 24000 2900 $n_{e}^{\prime}$ (cm-3) (f=0.3) 0.9 1.8 $M_{i}^{\prime}$
(M⊙) (f=0.3) 13200 1600 Kinetic energy (1048 erg) 0.4 73-103 2.7 4.4-6.3
Dynamical age (106 yr) 1.2 4.3 3.5 3.8 Ambient density $n_{o}$ (cm-3) 0.2
0.2-0.3 0.08 0.25-0.35
The systemic velocity of the Hi structure is –10$\pm$1 km s-1. The analytical
fit to the circular galactic rotation model predicts that material at
velocities of about –10 km s-1 is placed at near and far kinematical distances
of 3.3$\pm$1.0 and 13-14 kpc, respectively. The near kinematical distance
agrees with the spectrophotometric distance to HD 163758 (see Table 1). Note
that kinematical distances have large uncertainties in this section of the
Galaxy. We adopt 3.5$\pm$1.0 kpc as the distance to the Hi feature.
According to SIMBAD, no Hii regions or SNR have been identified in this area.
No additional OB stars were found projected onto the Hi structure. The
analysis of the CO images by Dame et al. (2001) did not show detectable
molecular emission in this region.
### 0.5.4 The ISM around HD 171589
Figure 8 displays the Hi emission distribution within the velocity interval
from +20.1 to +57.2 km s-1 in a large area around the massive star. Two
cavities are clearly detected for velocities in the range +26 to +45 km s-1,
centered at (l,b) = (187,–30) and (175,–27). HD 171589 appears projected onto
the cavity at larger galactic longitudes.
The upper panel of Fig. 9 displays the Hi column density distribution within
the velocity range from +26.3 to +44.8 km s-1. Both cavities are separated in
this image. The O-type star is projected onto the Hi hole at (l,b) =
(187,–30), partially surrounded by an Hi shell.
The central and bottom panels of Fig. 9 show the superposition of the Hi
column density image (in contours) and the H$\alpha$ and IRIS images,
respectively. The central panel reveals a region without optical emission
coincident with the cavity at (l,b) = (187,–30). The bottom panel shows that
both Hi holes are partially outlined by bright IR emission at b $>$ –35. The
distribution of the Hi, optical, and IR emissions is compatible with the
presence of an interstellar bubble driven by the massive star.
The systemic velocity of the structure at (l,b) = (187,–30) is +36$\pm$2 km
s-1, corresponding to near and far kinematical distances of 3.2$\pm$0.5 and 13
kpc, respectively. The near kinematical distance agrees with the
spectrophotometric distance derived for the star. Consequently, we adopt $d$ =
3.0$\pm$0.5 kpc.
Six supernova remnants were detected in this area at galactic latitudes
$b\geq$ –2.8 (Green, 2004). Two of them (G17.4–2.3 and G17.8–2.6) appear
projected onto the cavity centered at (l,b) = (175,–27). Based on the
$\Sigma$-D relation, Guseinov et al. (2003) derived distances of about 6.3 kpc
to the SNRs. As discussed by many authors (e.g. Green 2005) distances derived
from the $\Sigma$-D relation are hardly reliable. Both SNRs are detected at 35
MHz (Dwarakanath & Udaya Shankar, 1990). An inspection of the image at 4850
MHz shows weak emission also probably related to these remnants. Two different
facts can be proposed as the origin of the Hi cavity. On one hand, the Hi
cavity and shell can be associated with the SNRs, as was found in many SNRs
(e.g. Paron et al., 2006; Reynolds et al., 2008). In this case its systemic
velocity of +36 km s-1 suggests near and far kinematical distances of 3.2 and
13 kpc, respectively. On the contrary, if the Hi hole originates in absorption
due to the radio continuum sources, the distance to the SNRs can be inferred
from the highest positive velocity at which the hole is detected (about +52 km
s-1). This suggests that the SNRs are placed at distances $\geq$ 4.5 kpc.
Further data are necessary to elucidate this question.
CO emission (Dame et al., 2001) is present mainly at b $>$ –3∘. The integrated
CO emission (not shown here) in a velocity range similar to that of the Hi
emission does not show molecular material linked to the hole at (l,b) =
(175,–27). Weak CO emission present in the interval +28.0 to +43.6 km s-1
encircles the border of the cavity related to the SNRs towards b $>$ –3∘.
## 0.6 Discussion
### 0.6.1 Physical parameters
The main physical parameters of the neutral gas structures linked to the
O-type stars are summarized in Table 2. The (l,b) centers correspond to the
approximate centroid of the features. The velocity interval indicates the
velocity range where the Hi structures can be identified, being $\rm{v_{1}}$
and $\rm{v_{2}}$ the lowest and highest velocities, respectively, at which the
features are detected.
Following Cappa et al. (2008), the expansion velocities were estimated as
$\rm{v_{exp}}=(\rm{v_{2}}-\rm{v_{1}}$)/2 + 1.3 km s-1. The derived values are
lower limits since the caps of the expanding shells are not detected in the
present cases. The extra 1.3 km s-1 takes into account the fact that the caps
may be present just outside the velocity range at which the Hi cavity is
detected, that is at velocities $\rm{v_{1}}$ \- 1.3 km s-1 and $\rm{v_{2}}$
+1.3 km s-1.
For the four selected stars the adopted distances are compatible with the
spectrophotometric and kinematical distance estimates. Uncertainties in the
adopted kinematical distances for HD 124979, HD 163758, and HD 171589 arise in
a velocity dispersion of $\pm$6 km s-1 adopted for our Galaxy.
0cm-2cm
Table 3: Energetics of the structures
5 HD 38666 HD 124979 HD 163758 HD 171589 T${}_{eff}^{\mathrm{a}}$ 34600 37100
40200 40000 $log\ L$ (L⊙) 5.0 5.6 6.03 5.7 $M$ (M⊙) 23 40 70 47 $\dot{M}$
(10-6 M⊙ yr-1) 0.00032b-0.005c 1.8 e 8.3e 2.7e 0.02d-0.35e V∞ (km s-1) 1200b
2100f 2400f 2500f Lw (1036 erg s-1) 0.0002-0.16 2.5 15.0 5.4 Dynamical age
(106 yr) 1.2 4.3 3.5 3.8 Ew (1048 erg) 0.008-6.0 350 1660 660 $\epsilon$
50-0.07 0.2-0.3 0.002 0.007-0.01 Stellar lifetimeg (106 yr) 7.0 4.5 3.0 4.0
aVacca et al. (1996) bMartins et al. (2005) cChlebowski & Garmany (1991)
dHowarth & Prinja (1989) e Estimated from Vink et al. (2000) fAdopted from
Prinja et al. (1990) gSchaller et al. (1992)
The radius of each cavity corresponds to the geometric mean of the major and
minor semiaxes, while the radii of the Hi shells ($R_{s}$) were evaluated from
the position of the maxima in the envelopes. Errors in radii come from the
distance uncertainty.
Neutral atomic masses in the shells and mass deficiencies in the cavities are
also listed in Table 2. The structures surrounding HD 124979 and HD 171589 are
easier to define over the halves further away from the galactic plane. In each
case, the other half is contaminated with diffuse emission from the galactic
plane. Thus, for these two structures, the neutral mass in the shells were
derived as twice the mass associated with the better defined half. The sewpt-
up neutral masses were obtained as mean values between the neutral mass
deficiency in the voids and the neutral mass in the envelopes. This procedure
allows to remove a first order contribution of the background emission, since
both mass determinations are probably contaminated with neutral gas unrelated
to the structures. Values listed in Table 2 include a typical interstellar He
abundance of 10%.
In the previous sections we have stated that no radio continuum emission
associated with the Hi structures was detected. Assuming that the massive
stars have ionized the surrounding gas through their strong UV photon flux,
and that this ionized material has been swept-up and is present in the inner
borders of the neutral envelopes, we can derive an upper limit for the
electron density ne and the ionized mass Mi from the rms flux density at 4.85
GHz (7.7 mJy beam-1). Adopting $R_{ou}$ = 1.0 $R_{cav}$ and $R_{in}$ = 0.9
$R_{cav}$ as the outer and inner radii of the ionized regions, we estimated
upper limits for the flux densities S4.85GHz = 3.7 and 1.0 Jy for the ionized
regions around HD 124979 and HD 171589, respectively. Estimates of the
physical parameters of the Hii regions towards these stars can be obtained
using the expressions by Mezger & Henderson (1967) for spherical ionized
regions of constant density. Adopting an electron temperature of 8$\times$103
K and a volume filling factor $f$ = 1.0, we derived the electron density
$n_{e}$ and the associated ionized mass $M_{i}$. A different electron density
and ionized mass can be estimated by considering an alternative filling
factor. For an ionized shell with outer and inner radii of $R_{ou}$ and
$R_{in}$, in which the plasma covers an area A equal to 50% of the surface of
the shell, the filling factor can be derived as $f$ = A $\times$ ($R_{ou}^{3}$
\- $R_{in}^{3}$)/$R_{ou}^{3}$ = 0.3. This value was used to derive
$n_{e}^{\prime}$ and $M_{i}^{\prime}$. The derived values for $n_{e}^{\prime}$
are consistent with the high $z$-distances of the stars. Note that electron
densities and ionized masses are upper limits. Unfortunately, the lack of
radio data at frequencies higher than 1 GHz precludes from deriving upper
limits for the regions of HD 38666 and HD 163758.
The kinetic energy $E_{\mathrm{k}}$ = $M_{\mathrm{b}}\rm{v_{exp}}^{2}/2$ was
estimated from the expansion velocities listed in Table 2 and the neutral and
ionized masses in the structures. The range in kinetic energies corresponds to
the fact that we have taken into account the neutral atomic mass only, and the
neutral atomic and ionized masses.
Dynamical ages were estimated as $t_{\mathrm{d}}$ = 0.55$\times$106
Rs/${v_{exp}}$ yr (McCray, 1983), where Rs is the radius of the bubble (pc),
$\rm{v_{exp}}$ is the expansion velocity (km s-1), and the constant represents
a mean value between the energy and the momentum conserving cases.
Finally, ambient densities $n_{o}$ were derived by uniformly distributing the
associated mass within a sphere of radius Rs. For the regions of HD 124979 and
HD 171589 two values are listed in Table 2. The first one was obtained by
distributing the swept-up neutral mass ($M_{s}$) while the second one was
derived by distributing the neutral and ionized masses ($M_{s}$ \+
$M_{i}^{\prime}$). The major source of error in radii and masses is the
distance uncertainty. The low ambient densities are consistent with the
structures being far from the galactic plane.
### 0.6.2 Energetics and origin of the structures
We will analyze here whether the massive stars can provide the energy to
create the cavities and shells found in the previous sections through their
stellar winds and UV photon fluxes. To test the former possibility, we will
estimate the mechanical energy $E_{\mathrm{w}}$ released into the ISM for the
massive stars and compare it with the kinetic energy $E_{\mathrm{k}}$ of the
structures.
Table 3 summarizes relevant parameters useful to evaluate the energetics of
the structures. The values of effective temperature $T_{eff}$, stellar
luminosity $L$, and stellar mass listed in the first three rows correspond to
the spectral classification and luminosity class of the O-type stars listed in
Table 1. These data were used to derive the mass loss rate $\dot{M}$ following
the recipe by Vink et al. (2000). Based on the mass loss rates and terminal
velocities, we estimated the mechanical wind luminosity for each star as
$L_{\mathrm{w}}=\dot{M}V_{\mathrm{w}}^{2}/2$.
The stellar wind mechanical energy $E_{\mathrm{w}}(=L_{\mathrm{w}}t)$ released
by the massive stars were derived from the dynamical ages of the bubbles. The
resulting values are included in Table 3.
The ratio $\epsilon$ between the kinetic energy $E_{\mathrm{k}}$ and the
mechanical energy $E_{\mathrm{w}}$ provided by the massive stars during the
dynamical age of the structures is also included in Table 3. Evolutionary
models of stellar wind bubbles predict that $\epsilon$ = 0.2 for the energy
conserving case and $\epsilon\leq$ 0.1 for the momentum conserving case (see
McCray, 1983). Although the uncertainty in this value is large (at least 70%
adopting a 30% error in the distance), it is clear that HD 163758 and HD
171589 are capable of blowing the observed estructures. The result for HD
124979 is still consistent with an interstellar bubble interpretation.
The obtained dynamical ages for HD 124979, HD 163758, and HD 171589 are
compatible with the ages derived from evolutionary tracks for stars with solar
abundances (see Schaller et al., 1992).
The case for HD 38666 is more complex. The large uncertainty in the estimated
mechanical energy precludes for giving a clear conclusion about the origin of
this structure. The derived dynamical age is lower than the age estimate
obtained from Schaller et al. (1992). However, Martins et al. (2005) estimate
an age $<$ (2-4)$\times$106 yr, closer to the dynamical age. On the other
hand, Hoogerwerf et al. (2001) proposed a binary-binary collision between
three stars of Trapezium cluster: AE Aur, HD 38666, and $\iota$ Ori , becoming
HD 38666 and AE Aur runaway stars. The dynamical ejection scenario took place
2.5$\times$106 yr ago. We note that a search for other catalogued massive
stars at a distance compatible with that of the structure gave negative
results. A different analysis can be done bearing in mind the observed proper
motions. HD 38666 and HD 124979 have large spatial velocities, about 57$\pm$11
km s-1 for HD 38666, and 200$\pm$30 km s-1 for HD 124979. Considering these
velocities, which include the uncertainty in the adopted distance, it took HD
38666 about (0.4-0.9)$\times$106 yr to cross the Hi structure. As regards HD
124979, the time necessary to cross the structure was about
(0.6-1.0)$\times$106 yr. Adopting 0.7$\times$106 yr for HD 38666 and
0.8$\times$106 yr for HD 124979, the mechanical energy turns out to be
(0.004-3.5)$\times$1048 erg for HD 38666 and 65$\times$1048 erg for HD 124979.
The large spatial velocity of the runaway star HD 124979 shortenes the
crossing time through the structure, which makes their origin to remain
unclear. The resulting $\epsilon$-values indicate that other energy sources
are necessary to create both structures through the stellar wind mechanism.
Additional studies are necessary to identify the agents.
No bow-shock like features were found related to HD 38666 and HD 124979. As
shown by Raga et al. (1997) and Huthoff & Kaper (2001), bow-shocks appear
associated with only about 30-40% of OB runaway stars. As indicated by the
last authors, large separation from the galactic plane, extremely high space
velocities and large distances make difficult the formation and detection of
bow-shock structures. Moreover, the physical conditions of the ambient medium
where the star is inmersed play a major role in determining the existence of
bow-shocks. Particularly, these authors find an anticorrelation between bow-
shocks and hot bubbles. Whether the observed structures originate in the mass
flow of the massive stars, the lack of bow-shocks in these cases is consistent
with their conclusions.
Taking into account the spectral classification and the luminosity class of
the associated massive stars and the ambient density where the structures are
evolving, the estimated radius of the Strömgren’s spheres is in all cases
larger than the Hi cavities. The smaller size of the cavities can be explained
taking into account that a certain amount of UV photons are used in dust
heating or drain from the patchy envelopes.
## 0.7 Summary
We have analyzed the interstellar medium in the environs of the O-type stars
HD 38666, HD 124979, HD 163758, and HD 171589. The study of the neutral
hydrogen distribution in direction to these stars allowed us to disclose Hi
structures located at kinematical distances compatible with the stellar
distances and probably related to the stars.
Assuming a stellar wind mechanism, the derived dynamical ages for the Hi
structures related to HD 124979, HD 163758, and HD 171589 are compatible with
the lifetimes of the O-type stars in the main sequence, reinforcing the
association with the stars.
We have investigated the counterparts of the structures at other frequencies.
The surrounding Hi shells around HD 38666, HD 124979, and HD 171589 have IR
counterparts, revealing the presence of dust associated with the Hi gas. The
Hi shell related to HD 163758 has IR and H$\alpha$ counterparts, with the star
projected onto one of its higher density borders.
To investigate the origin of the structures, we have compared the mechanical
energy released into the ISM for the massive stars and the kinetic energy of
the structures. Our results for the energy conversion efficiency for HD 163758
and HD 171589 indicate that the stars are capable of blowing the observed
structures. As regards HD 38666 and HD 124979, additional energy sources are
probably necesary, taking into account the derived $\epsilon$-values and the
large tangential velocities. Aditional studies are needed to clarify the
origin.
###### Acknowledgements.
We thank Dr. P. Benaglia for her help in the first stages of this paper. We
also thank Dr. J.C. Testori for his collaboration. It is a pleausure to thank
the anonymous referee for many comments and suggestions that improve this
presentation. This project was partially financed by the Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET) of Argentina under project
PIP 5886/05, Agencia PICT 14018, and UNLP under projects 11/G072.
We acknowledge the use of NASA’s SkyView facility
(http://skyview.gsfc.nasa.gov) located at NASA Goddard Space Flight Center.
The reduction and analysis of the PMN Survey data was largely the work of Mark
Griffith and Alan Wright. The FITS maps of the PMN Survey were produced by Jim
Condon (NRAO) and Niven Tasker. This research has made use of the SIMBAD
database, operated at CDS, Strasbourg, France.
## References
* Arnal et al. (2000) Arnal, E. M., Bajaja, E., Larrarte, J. J., Morras, R., & Pöppel, W. G. L. 2000, A&AS, 142, 35
* Arnal et al. (1999) Arnal, E. M., Cappa, C. E., Rizzo, J. R., & Cichowolski, S. 1999, AJ, 118, 1798
* Blaauw (1961) Blaauw, A. 1961, Bull. Astron. Inst. Netherlands, 15, 265
* Brand & Blitz (1993) Brand, J. & Blitz, L. 1993, A&A, 275, 67
* Cappa et al. (2008) Cappa, C., Niemela, V. S., Amorín, R., & Vasquez, J. 2008, A&A, 477, 173
* Cappa & Benaglia (1998) Cappa, C. E. & Benaglia, P. 1998, AJ, 116, 1906
* Cappa et al. (2005) Cappa, C. E., Rubio, M., Martín, M. C., & McClure-Griffiths, N. M. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 344, The Cool Universe: Observing Cosmic Dawn, ed. D. Lidman, C. and Alloin, 179
* Chlebowski & Garmany (1991) Chlebowski, T. & Garmany, C. D. 1991, ApJ, 368, 241
* Cichowolski & Arnal (2004) Cichowolski, S. & Arnal, E. M. 2004, A&A, 414, 203
* Condon et al. (1993) Condon, J. J., Griffith, M. R., & Wright, A. E. 1993, AJ, 106, 1095
* Dame et al. (2001) Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792
* Dwarakanath & Udaya Shankar (1990) Dwarakanath, K. S. & Udaya Shankar, N. 1990, Journal of Astrophysics and Astronomy, 11, 323
* Finkbeiner (2003) Finkbeiner, D. P. 2003, ApJS, 146, 407
* Garmany et al. (1982) Garmany, C. D., Conti, P. S., & Chiosi, C. 1982, ApJ, 263, 777
* Goss & Lozinskaya (1995) Goss, W. M. & Lozinskaya, T. A. 1995, ApJ, 439, 637
* Green (2004) Green, D. A. 2004, Bulletin of the Astronomical Society of India, 32, 335
* Guseinov et al. (2003) Guseinov, O. H., Ankay, A., & Tagieva, S. O. 2003, Serbian Astronomical Journal, 167, 93
* Haslam et al. (1982) Haslam, C. G. T., Salter, C. J., Stoffel, H., & Wilson, W. E. 1982, A&AS, 47, 1
* Hog & Murdin (2000) Hog, E. & Murdin, P. 2000, Tycho Star Catalogs: The 2.5 Million Brightest Stars (Encyclopedia of Astronomy and Astrophysics)
* Hoogerwerf et al. (2001) Hoogerwerf, R., de Bruijne, J. H. J., & de Zeeuw, P. T. 2001, A&A, 365, 49
* Howarth & Prinja (1989) Howarth, I. D. & Prinja, R. K. 1989, ApJS, 69, 527
* Huthoff & Kaper (2001) Huthoff, F. & Kaper, L. 2001, Black Holes in Binaries and Galactic Nuclei, ed. , L. KaperE. P. J. van den Heuvel & P. A. Woudt, 314
* Kalberla et al. (2005) Kalberla, P. M. W., Burton, W. B., Hartmann, D., Arnal, E. M., Bajaja, E., Morras, R., & Pöppel, W. G. L. 2005, A&A, 440, 775
* Kozok (1985) Kozok, J. R. 1985, A&AS, 62, 7
* Lozinskaya (1982) Lozinskaya, T. A. 1982, Ap&SS, 87, 313
* MacConnell & Bidelman (1976) MacConnell, D. J. & Bidelman, W. P. 1976, AJ, 81, 225
* Maíz-Apellániz et al. (2004) Maíz-Apellániz, J., Walborn, N. R., Galue, H. A., & Wei, L. H. 2004, VizieR Online Data Catalog, 5116, 0
* Martins et al. (2005) Martins, F., Schaerer, D., Hillier, D. J., Meynadier, F., Heydari-Malayeri, M., & Walborn, N. R. 2005, A&A, 441, 735
* Mason et al. (1998) Mason, B. D., Gies, D. R., Hartkopf, W. I., Bagnuolo, Jr., W. G., ten Brummelaar, T., & McAlister, H. A. 1998, AJ, 115, 821
* McCray (1983) McCray, R. 1983, Highlights of Astronomy, 6, 565
* Mezger & Henderson (1967) Mezger, P. G. & Henderson, A. P. 1967, ApJ, 147, 471
* Miville-Deschênes & Lagache (2005) Miville-Deschênes, M. A. & Lagache, G. 2005, ApJS, 157, 302
* Nazé et al. (2002) Nazé, Y., Chu, Y.-H., Guerrero, M. A., Oey, M. S., Gruendl, R. A., & Smith, R. C. 2002, AJ, 124, 3325
* Olano (1982) Olano, C. A. 1982, A&A, 112, 195
* Paron et al. (2006) Paron, S. A., Reynoso, E. M., Dubner, G. M., & Purcell, C. 2006, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 26, Revista Mexicana de Astronomia y Astrofisica Conference Series, 164
* Prinja et al. (1990) Prinja, R. K., Barlow, M. J., & Howarth, I. D. 1990, ApJ, 361, 607
* Raga et al. (1997) Raga, A. C., Noriega-Crespo, A., Cantó, J., Steffen, W., van Buren, D., Mellema, G., & Lundqvist, P. 1997, Revista Mexicana de Astronomia y Astrofisica, 33, 73
* Reynolds et al. (2008) Reynolds, S. P., Borkowski, K. J., Green, D. A., Hwang, U., Harrus, I., & Petre, R. 2008, ApJ, 680, L41
* Russeil (2003) Russeil, D. 2003, A&A, 397, 133
* Schaller et al. (1992) Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269
* Vacca et al. (1996) Vacca, W. D., Garmany, C. D., & Shull, J. M. 1996, ApJ, 460, 914
* van Buren & McCray (1988) van Buren, D. & McCray, R. 1988, ApJ, 329, L93
* van der Hucht (2001) van der Hucht, K. A. 2001, New Astronomy Review, 45, 135
* Vink et al. (2000) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362, 295
* Walborn (1971) Walborn, N. R. 1971, ApJS, 23, 257
* Weaver et al. (1977) Weaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, R. 1977, ApJ, 218, 377
* Wegner (1994) Wegner, W. 1994, MNRAS, 270, 229
* Wilkin (1996) Wilkin, F. P. 1996, ApJ, 459, L31
|
arxiv-papers
| 2008-10-01T00:28:09
|
2024-09-04T02:48:58.066906
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. C. Mart\\'in (1), C. E. Cappa (1 and 2), and G. A. Romero (1 and 2)\n ((1) Instituto Argentino de Radioastronom\\'ia (CCT-La Plata, CONICET),\n Argentina, (2) Facultad de Ciencias Astron\\'omicas y Geof\\'isicas,\n Universidad Nacional de La Plata, Argentina)",
"submitter": "Gisela Romero G.A.R",
"url": "https://arxiv.org/abs/0810.0048"
}
|
0810.0190
|
# Dynamical Dark Energy simulations: high accuracy Power Spectra at high
redshift
Luciano Casarini1,2, Andrea V. Macciò3, Silvio A. Bonometto1,2 1Department of
Physics G. Occhialini – Milano–Bicocca University, Piazza della Scienza 3,
20126 Milano, Italy 2I.N.F.N., Sezione di Milano 3Max-Planck-Institut für
Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
###### Abstract
Accurate predictions on non–linear power spectra, at various redshift $z$,
will be a basic tool to interpret cosmological data from next generation mass
probes, so obtaining key information on Dark Energy nature. This calls for
high precision simulations, covering the whole functional space of $w(z)$
state equations and taking also into account the admitted ranges of other
cosmological parameters; surely a difficult task. A procedure was however
suggested, able to match the spectra at $z=0$, up to $k\sim 3\,h$Mpc-1, in
cosmologies with an (almost) arbitrary $w(z)$, by making recourse to the
results of $N$–body simulations with $w={\rm const}$. In this paper we extend
such procedure to high redshift and test our approach through a series of
$N$–body gravitational simulations of various models, including a model
closely fitting WMAP5 and complementary data. Our approach detects $w={\rm
const.}$ models, whose spectra meet the requirement within 1$\,\%$ at $z=0$
and perform even better at higher redshift, where they are close to a permil
precision. Available Halofit expressions, extended to (constant) $w\neq-1$ are
unfortunately unsuitable to fit the spectra of the physical models considered
here. Their extension to cover the desired range should be however feasible,
and this will enable us to match spectra from any DE state equation.
###### pacs:
98.80.-k, 98.65.-r
## 1 Introduction
There seems to be little doubt left that a Dark Energy (DE) component is
required, to account for cosmological observables. Its first evidence came
from the Hubble diagram of SNIa, showing an accelerated cosmic expansion, but
a flat cosmology with $\Omega_{m}\simeq 0.25$, $\Omega_{b}\simeq 0.04$ and
$h\simeq 0.7$ is now required by CMB and LSS data and this implies that the
gap between $\Omega_{m}$ and unity is to be filled by a smooth non–particle
component ($\Omega_{m}$, $\Omega_{b}$: matter, baryon present density
parameters; $h$: present Hubble parameter in units of 100 km/s/Mpc; CMB:
cosmic microwave background; LSS: large scale structure; for SNIa data see [1,
2]; updated cosmological parameters, taking into account most available data,
are provided within the context of WMAP5 release [3].
If DE evidence seems sound, its nature is perhaps the main puzzle of
cosmology. Aside of a cosmological constant $\Lambda$, possibly related to
vacuum energy, and a scalar self–interacting field $\phi$, various pictures
have been recently discussed, ranging from a supposed back–reaction of
inhomogeneity formation to GR modifications and including even more exotic
alternatives (see, e.g., [6]). However, in most of these cases, a DE
component, with a suitable $w(a)$ state parameter ($a$: scale factor), can
still be an effective description, and a number of observational projects have
been devised, aiming first of all at constraining $w(a)$ (among them let us
quote the DUNE–EUCLIDE project [7]). Some of them are likely to be realized in
the next decade(s) and, to interpret their outcomes, we need accurate
predictions on selected observables. In particular, it has been outlined [8]
that, to fully exploit weak lensing surveys, we need predictions on non–linear
power spectra, accurate up to $\sim 1\%$.
$N$–body gravitational simulations safely predict non–linear matter evolution
up to wavenumbers $k\simeq 3\,h\,$Mpc${}^{-1}.$ When the scale of galaxy
clusters is approached, discrepancies from hydrodynamical simulations,
although small, exceed a few percents [9, 10, 11]: too much for the accuracy
required. Within the above range, safe expressions of matter fluctuation
spectra at $z=0$, for any state equation $w(a)$, can be obtained from
simulations of suitable models with $w={\rm const.}$ . This paper extends the
technique yielding such expressions, so to include higher redshift $z$, where
they will be mostly needed.
A fair approximation to the mass power spectrum for $\Lambda$CDM models is the
Halofit expression [12], based on the halo model of structure formation and
using numerical simulations to fix parameters left free by the theoretical
analysis. Halofit expressions were extended [13] to include cosmologies with a
constant state parameter ($-1.5<w<-0.5$) and for a fairly wide range of the
parameters $\Omega_{c}$, $\Omega_{b}$, $h$, $n_{s}$, $\sigma_{8}$ (here
$n_{s}:$ primeval spectral index for scalar fluctuations; $\sigma_{8}:$ linear
r.m.s. fluctuation amplitude on the scale of $8\,h^{-1}$Mpc). To this aim
suitable n–body simulations, in a box of size $L_{box}=110\,h^{-1}$Mpc and
with force resolution $\epsilon=143\,h^{-1}$kpc, were run.
We tentatively applied such generalized Halofit expressions to our cases.
Unfortunately, soon above rather low $z$’s, they fail to work. This is not
unexpected, as we had been pushing them outside the expected range of
validity. Among the models considered, however, there are cosmologies closely
fitting WMAP5 and complementary data. Accordingly, further work will be needed
to provide suitable generalized Halofit expressions.
Dynamical Dark Energy (dDE) simulations, with a variable state parameter
$w(a)$ deduced from scalar field potentials admitting a tracker solutions,
have been performed since 2003 [14] (see also [15, 16, 17]) and compared with
$w=const$ simulation outputs. Observables considered in these papers, however,
only marginally included spectra. In a recent work ([18] FLL hereafter),
however, it has been shown how spectral predictions for constant–$w$ models
can also be used to fit the spectra of cosmologies with a state parameter
$w(a)=w_{o}+(1-a)w_{a}~{},$ (1)
given by a first degree polynomial.
In fact, it had been known since several years that spectra at a given (low)
redshift $\bar{z}$ mainly depends on the comoving distance between $\bar{z}$
and the last scattering band ($d_{LSB}(\bar{z})$), at least for models where
baryons and CDM are the only matter components with a ratio not too far from
canonical values [19]. Aiming at $1\,\%$ accuracy, FLL seek the constant–$w$
model which grants the same distance from LSB of an assigned $w(a)$ function,
without varying any other parameter, and claim that spectral differences
between the two models are at the per–mil level for $k<1\,h$ Mpc-1, but still
at the percent level for $k<3\,h$ Mpc-1, at $z=0~{}.$
On this basis, at $z=0$, one could think to use the Halofit extension provided
in [13] to predict the spectra of models with variable $w$, in a large number
of cases.
FLL started from models with a constant equation of state ($w=-K$, with
$K=0.9,1,1.1$), for which the distance from the LSB is $d_{LSB}^{(K)}(z)$, and
then compared them with variable–$w$ models with the same distance from the
LSB at redshift zero: $d_{LSB}(0)=d_{LSB}^{(K)}(0)$. In the attempt to extend
the fit between spectra from $z=0$ to higher $z$, they just renormalize the
amplitudes of the constant–$w$ model spectra at higher redshift, so to meet
the low–$k$ linear behavior of the variable–$w$ models. This technique allows
them to reduce spectral discrepancies, in average still below $1\,\%$, but
attaining a maximum of a few percents at larger $k$’s.
Here we shall bypass this renormalization procedure, aiming at the per–mil
precision for any $z$. To this end we extend directly the criterion suggested
in [19] to any redshift, suitably seeking a constant–$w$ model such that
$d_{LSB}(z)/d_{LSB}^{(K)}(z)\equiv 1$, $z$ by $z$ (allowing, of course, for
any value of $K$). This also requires to follow the variations of model
parameters through their evolution dictated by the assigned $w(z)$. Two
options to do so will be outlined and we shall test the procedure of the
simpler option through a number of $N$–body simulations.
The remainder of this paper is organized as follows. Section 2 is devoted to
briefly illustrate such dDE model. In Section 4 we present our numerical
simulations and their analysis. Section 5 contains our results on the Power
Spectrum and a short discussion on them. Finally in Section 6 we present our
main conclusions.
## 2 The models considered
In this paper we run a series of simulations for two cosmological models: (i)
a model where $w(a)$ is a polynomial (1); (ii) a SUGRA model. The former model
is selected to coincide with one of the models considered by FLL, so to allow
a close comparison of the outputs.
SUGRA models are an alternative example of faster varying state equation; they
are true dDE model, where DE is a scalar field $\phi$, self–interacting
through a SUGRA potential
$V(\phi)=(\Lambda^{\alpha+4}/\phi^{\alpha})\exp(4\pi\phi^{2}/m_{p}^{2})$ (2)
admitting tracker solutions [20, 21, 22]. Here: $m_{p}=G^{-1/2}$ is the Planck
mass; $\Lambda$ and $\alpha$ are suitable parameters; however, in a spatially
flat cosmology, once the present DE density parameter is assigned, either of
them is fixed by the other one. Here we shall define our SUGRA model through
the value of $\Lambda~{}~{}(=0.1$GeV)
Figure 1: Likelihood distribution on $\log\Lambda$ for SUGRA cosmologies.
In fact, a fit of SUGRA with data, based on MCMC (MonteCarlo Markov Chains),
was recently obtained by [5]. Data include WMAP5 outputs on anisotropies and
polarization [3], SNIa [1] and 2dF data [4] on matter fluctuation spectra
(including therefore the BAO position). In Figure 1 we exhibit the likelihood
distribution on the parameter $\log(\Lambda/{\rm GeV})$, obtained from the
fit. Using $\Lambda=0.1~{}$GeV is a compromise between top likelihood and a
physically significant $\Lambda$ . The values of the other parameters in the
SUGRA simulation are then chosen quite close to the best–fit obtained once
$\Lambda=0.1~{}$GeV is fixed, and are reported in Table 1. The evolution with
the redshift of the state parameter for this model is shown in Figure 2 and
compared with a polynomial $w(a)$ ($w_{o}=-0.908$, $w_{a}=0.455$) coinciding
with SUGRA at $z=0$ and $z=10$. In general, the redshift dependence of the
SUGRA state parameter can only be approached by an expression of the form (1).
However, just looking at the recent WMAP results [3],we see that these
$w_{o},~{}w_{a}$ are at $\sim 1.5\,\sigma$’s from the best–fit polynomial
parameters.
Figure 2: Evolution of the state parameter $w$ for the SUGRA model used in the
simulation ($\Lambda=0.1$), compared with the closest polynomial $w(a)\,$.
In Table 1 we give the $z=0$ parameters of the two variable–$w$ models
considered.
${\bf Table~{}~{}1}$
$\left|~{}~{}\matrix{&\Omega_{c}&\Omega_{b}&h&\sigma_{8}&n_{s}\cr{\rm
polynom.:}&~{}~{}0.193~{}~{}&~{}~{}0.041~{}~{}&~{}~{}0.74~{}~{}&~{}~{}0.76~{}~{}&~{}~{}0.96\cr{\rm
SUGRA}&~{}~{}0.209~{}~{}&~{}~{}0.046~{}~{}&~{}~{}0.70~{}~{}&~{}~{}0.75~{}~{}&~{}~{}0.97\cr}~{}~{}\right|$
Let us remind that the former model is selected mostly for the sake of
comparison. Its polynomial coefficient read $w_{o}=-0.8$, $w_{a}=-0.732$ and
it is characterized by a rather low likelihood with respect to data (see [3])
Figure 3: Linear growth factors for different models, compared with the growth
factor of the constant–$w$ model fitting them at $z=0$. Besides of the models
treated in detail in this work (indicated in the frame), we also show the
growth factors for a further set of models treated by FLL (see text).
In Figure 3 we show the behavior of the linear growth factor $G(z)$ for both
models, normalized to the growth factor in the constant–$w$ model yielding the
same $d_{LSB}(z=0)$; i.e., for the SUGRA model, $w=-0.7634$; for the
polynomial model, as already mentioned, $w=-1$. For the sake of comparison we
report $G(z)$ for a few other models, whose $d_{LSB}(z=0)$ also coincides with
$\Lambda$CDM (in these models $w_{o}$ and $w_{a}$ hold -1.2, -1.1, -0.9 and
0.663, 0.341, -0.359, respectively). $G(z)$ is then the renormalizing factor
for the spectrum at redshift $z$ in the FLL approach. Spectra worked out with
FLL technique will be better where $G_{w(z)}/G_{w=-1}$ is smaller. We notice
that, for each model, there exist a crossover redshift $z_{co}$, such that
$G(z_{co})$ coincides with the growth factor of $\Lambda$CDM. At this
redshift, the FLL procedure does not require renormalization.
## 3 Variable vs. constant w
Figure 4: Distance between $z$ and the LSB in the SUGRA and polynomial models
of Table 1 ($\cal M$ models).
Figure 5: State parameter of the auxiliary models, as a function of redshift.
For the sake of comparison, also the intrinsic $w(z)$ dependence of each $\cal
M$ model is shown.
Figure 6: Values of $\sigma_{8}$ at $z=0$ for constant–$w$ models whose r.m.s.
fluctuation amplitude meets the $\sigma_{8}$ value of the $\cal M$ model at
$z$.
In order to find a constant–$w$ model whose spectrum approaches dDE at a
redshift $z\neq 0$, one first computes the distance $d_{LSB}(z)$, from $z$ to
the LSB, in the model $\cal M$ considered. At such redshift $z$, cosmological
parameters as $\Omega_{b}$, $\Omega_{c}$, $h$, $\sigma_{8}$ no longer keep
their $z=0$ values. However, one can easily calculate them and find a
constant–$w$ model whose parameters at that $z$ are $\Omega_{b}(z)$,
$\Omega_{c}(z)$, $h(z)$, $\sigma_{8}(z)$ and coincide with the values in $\cal
M$. In such model, then, the $z=0$ values of the parameters will be different
from their values in $\cal M$. Such model is then expected to have a
non–linear spectrum closely approaching $\cal M$ at $z$, just as FLL found at
$z=0$. The models defined in this way shall clearly be different for different
$z$ and will be said to satisfy to the strong requirement.
Among other difficulties, making use of such models to approximate high–$z$
spectra causes a technical problem which stands on the way to test this
approach through simulations. Let $L$ be the side of the box to be used. As
already outlined, for the $\cal M$ and constant–$w$ models, the $z=0$ values
of $h$ are different; therefore, if we take equal values for $Lh$ (or $L$
measured in $h^{-1}$Mpc) we shall have different $L$ values. A similar problem
occurs when we normalize, as the r.m.s. fluctuation amplitude $\sigma_{8}$
refers to the $h$ dependent scale of $8\,h^{-1}$Mpc. All that induces quite a
few complications, if we aim at comparing fairly normalized spectra of
different models, with the same seed and the same wavenumber contributions.
This difficulty can be soon overcame if we replace the above strong
requirement with the weak requirement we shall now define. Let us first notice
that, being
$H^{2}={8\pi\over 3}G\rho_{cr}={8\pi\over 3}G\rho_{m}\Omega_{m}^{-1}$ (3)
at any redshift, $\Omega_{m}h^{2}$ however scales as $a^{-3}$, independently
of DE nature. Accordingly, a constant–$w$ model whose $z=0$ values of
$\Omega_{b}$, $\Omega_{c}$ and $h^{2}$ coincide with those of $\cal M$, will
share with it the values of the reduced density parameters
$\omega_{b}=\Omega_{b}h^{2}$ and $\omega_{c}=\Omega_{c}h^{2}$ at any redshift
(attention is to be payed, all through this paper, to the different meanings
of the symbols $w$ and $\omega$, respectively state parameter and reduced
density parameters). In order to have the same $d_{LSB}(z)$, of course, it
shall have a different $h(z)$, that we however do not need to evaluate
explicitly. What we need to know are the value of the constant $w(z)$ as well
as the value $\sigma_{8}(z)$, to be assigned to the r.m.s. fluctuation
amplitude at $z=0$, to meet the $\cal M$ value of $\sigma_{8}$ at $z$.
In this paper we shall test this weak requirement (W.R.) that the dDE model
$\cal M$ and the auxiliary model $\cal W$$(z)$ have equal $d_{LSB}(z)$,
$\sigma_{8}$, $\omega_{c}$ and $\omega_{b}$, while $h$ is chosen with an
ad–hoc criterion, yielding a $z$–independent $h$ at $z=0$.
In Figures 4, 5, 6 we report the distance from the LSB $d_{LSB}(z)$, as well
as the values of $w(z)$ and $\sigma_{8}(z)$ for the auxiliary models $\cal
W$$(z)$ defined according to the W.R. .
FLL had been seeking an auxiliary model only at $z=0$, making recourse to a
different treatment at greater redshift, claimed to grant a precision $\cal
O$$(2\,$–$3\%)$. As we shall see, seeking an auxiliary model at any redshift,
according to the W.R., allows a precision $\sim 10$ times better in the
relevant $k$ range and does not lead to numerical complications. The goal
would be complete if such models were in the parameter ranges considered by
generalized Halofit expressions [13] (at $z=0$): $0.211<\Omega_{m}<0.351$,
$0.041<\Omega_{b}<0.0514$, $0.644<h<0.776$, $0.800<\sigma_{8}<0.994$,
$0.915<n_{s}<1.045$.
The selected $\Omega_{c}$, $\Omega_{b}$, $h$, $n_{s}$ are fine. Figures 5 show
that the requirement is met also by $w$. On the contrary, Figures 6 show that
the $\sigma_{8}$ value, tuned to observational data, lays outside the range
considered at any $z$, for both $\cal M$ models.
This is most unfortunate. As we shall see the generalized expression in [13]
will be scarcely useful to the present aims.
## 4 Numerical simulations
We shall compare simulations starting from realizations fixed by identical
random seeds. Transfer functions generated using a modified version of the
CAMB package [25] are used to create initial conditions with a modified
version of the PM software by Klypin & Holzmann [24], able to handle different
parameterizations of DE [14, 28]. (Possible contributions from DE clustering
on super–horizon scales are ignored in this work). Simulations were made by
using two different programs: the art code [26], courtesy of A. Klypin, and
the pkdgrav code [27], which has been modified to deal with any variable
$w(a)$ for this work. The art code has been mainly used as a benchmark to test
our modified version of pkdgrav (see appendix A); in the following we will
only present results obtained with this latter code.
An important issue, when high accuracy is sought, is a suitable handling of
cosmic (sample) variance. In order to address this point we create a series of
simulations sharing the same box size and particle number, but with different
realizations of the initial random density field.
The reference point are the simulations of the $\cal M$ models, performed in a
box with side $L_{box}=256h^{-1}$Mpc, a particle number $N=256^{3}$ and a
gravitational softening $\epsilon=25h^{-1}$ kpc. We create 4 different
realizations of them, only differing in the random seed used to sample the
phase space. These initial conditions are then evolved from $z=24$ to $z=0$
and nine outputs are saved, at redshift $z_{k}$, from $z_{0}=2.4$ to
$z_{8}=0$. For each redshift we then have the parameters, at $z=0$, of the
corresponding auxiliary $\cal W$$(z_{k})$ model, as shown in Section 3. By
using them we create suitable initial conditions for $\cal W$$(z_{k})$ models,
using the same random seed of the corresponding $\cal M$ model, and evolve the
$\cal W$$(z_{k})$ model down to $z_{k}$ (9 redshift values $z_{k}$, 9
auxiliary models $\cal W$$(z_{k})$ for each random seed). For instance, in
order to build a spectrum expected to coincide with SUGRA at $z=1.2$, we run a
simulation of a constant–$w$ model with $w=-0.7124$ and $\sigma_{8}=0.7351$ at
$z=0$.
### 4.1 Halos in simulations
Figure 7: Number of halos per $(h^{-1}$Mpc$)^{3}$ in the SUGRA model
considered and in the corresponding auxiliary $\cal W$ models. Results are
practically undiscernable and in agreement with Sheth & Tormen expressions.
The mass functions for the polynomial model behave quite similarly.
Before reporting results on spectra and their evolution, we wish to exhibit a
few results on halo formation. Since ever, mass functions have been considered
a basic test for simulations. This is also an independent test going beyond
spectral fits: spectra are related just to 2–point functions and their fitting
formulae derive from theoretical elaborations on 2–point correlations; mass
functions, instead, as the halo concentration distribution or the void
probability, depend on convolutions of n–point correlations.
We look for virialized halos using a Spherical Overdensity (SO) algorithm.
Candidate groups with a minimum of $N_{f}=200$ particles are selected using a
FoF algorithm with linking length $\phi=0.2\times d$ (the average particle
separation). We then: (i) find the point $C$ where the gravitational potential
is minimum; (ii) determine the radius $r$ of a sphere centered on $C$, where
the density contrast is $\Delta_{\rm vir}$, with respect to the critical
density of the Universe. Using all particles in the corresponding sphere we
iterate the above procedure until we converge onto a stable particle set. For
each stable particle set we obtain the virial radius, $R_{\rm vir}$, the
number of particles within the virial radius, $N_{\rm vir}$, and the virial
mass, $M_{\rm vir}$. We used a time varying virial density contrast
$\Delta_{\rm vir}$, whose value has been determined, according to linear
theory, by using the fitting formulae in [28]. We include in the halo
catalogue all the halos with more than 100 particles.
Mass functions are consistent with Sheth & Tormen predictions at all $z$,
almost always within 2$\sigma$’s (Poisson errors), while differences between
$\cal M$ and $\cal W$ models can hardly be plotted. This allows us to
formulate the conjecture that also higher order correlation functions
coincide. As future cosmic–shear surveys will enable to inspect the 3–point
function (and possibly to detect some higher order signal), we plan to compare
n–point functions in $\cal M$ and $\cal W$ models, in a forthcoming paper.
## 5 Results on the Power Spectrum
Figure 8: Spectra of SUGRA and auxiliary models at $z=0$ and 0.6. Two
realizations are shown. Differences between $\cal M$ and $\cal W$ models are
so small to be unappreciable in these plots. In the lower frames, we rather
exhibit the ratio with a Halofit expression, as well as the difference between
realization. At $z=0$, where $\cal M$ model spectra are quite close to
$\Lambda$CDM, this is of the same order of the discrepancy from Halofit. At
$z=0.6$ the discrepancy from a Halofit expression already overcomes
$5\,\%~{}$.
Figure 9: As Fig. 8, for $z=1.2$ and $z=1.8~{}.$ Notice the gradual
deterioration of Halofit, which becomes unsuitable on the whole non–linearity
range. Figure 10: As Fig. 8, for $z=2.4~{}.$ Discrepancies from Halofit exceed
20$\,\%~{}.$ Figure 11: Comparison of spectra with the fitting expression with
a correction for $w\neq-1$ [13], at $z=1.2$. Quite in general, for our models,
such expressions are no improvement in respect to Halofit (notice that the
ordinate range has doubled). This is not unexpected, as they are used out of
the allowed parameter range
Power spectra have been computed from N–body simulations by using the program
PMpowerM of the ART package. The program works out the spectrum through a FFT
(Fast Fourier Transform) of the matter density field, computed on a regular
grid $N_{G}\times N_{G}\times N_{G}$ from the particle distribution via a CIC
(Cloud in Cell) algorithm.
Figure 8, 9 and 10 present results on the Power Spectrum extracted from N–body
simulations of the SUGRA model at $z=0$, 0.6, 1.2, 1.8 and 2.4 . Their main
significance is to allow a comparison between seeds and with Halofit
expressions. The success of the weak requirement approach is so complete, that
differences between SUGRA and its $\cal W$ models cannot be appreciated, even
in the lower panel, where we report the ratio between spectra from simulations
and fitting formulae. The situation is similar for the polynomial model (not
plotted). Residual differences between $\cal M$ and $\cal W$ models will be
shown in a following plot. Here, different colors refer to two different
realizations,
The first Figure 8 shows a rather good fit with Halofit expressions at $z=0$.
Here we have another problem, which must be deepened by using simulation in a
bigger bix and with a wider dynamical range; in fact, the differences between
realizations are $<1\,\%$ only for $k>\sim 3/h{\rm Mpc}^{-1}$ (where
hydrodynamics becomes essential to describe the real world) and mostly keep
within 2$\,\%$ for $k>\sim 0.6/h{\rm Mpc}^{-1}$. At still lower $k$’s, the
seed dependence becomes more and more significant. A tentative explanation of
the effect is that the relevant long wavelengths are still not sampled well
enough, in a box of the size considered in this work, to yield a seed
independent description at $z=0$, when spectral contribution from long
wavelengths had enough time to propagate down to such high wavenumbers.
In fact, already at $z=0.6$ seed discrepancies $>1\,\%$ are circumscribed to
$k<\sim 0.6/h{\rm Mpc}^{-1}$. On the contrary, at this redshift, the
discrepancy from Halofit steadily exceeds 4–$5\,\%~{}.$
Higher redshift plots show a progressive deterioration of the Halofit
expressions, which become unsuitable in the whole non–linearity range. No
surprise about that, however, Halofit was built to meet $\Lambda$CDM spectra
and its performance in the cases considered here is better than expected.
One could presume that using the expressions [13], aimed to fit $w\neq-1$
cosmologies, could allow some improvement. As well as Halofit, they are out of
their range, but they include terms just aimed to correcting for $w\neq-1~{}.$
Unfortunately, they yield no improvement. As an example, in Figure 11, we plot
spectral ratios at $z=1.2~{}$.
Figure 12: Linear spectra ratios.
Let us now pass to the basic topic of this work and discuss discrepancies
between $\cal M$ and $\cal W$ models. To this aim, it is useful to perform a
preliminary comparison between the linear spectra. Figures 12 show spectral
ratios for the polynomial and SUGRA models at redshift values distant 0.6 up
to z=2.4 . For the polynomial model they can be confused with noise, apart of
some extremely mild signal at low $k$ ($\sim 1$–$2:100000$). Low $k$
discrepancies are greater (in the $k$ range considered they are $\sim
1:10000~{}$!) for SUGRA. Here we also appreciate a slight improvement from the
FLL to our approach: in the former case, all spectra concern the same model,
with the $w$ value deduced for $z=0$; in the latter one, we use a
$z$–dependent $w$. But, at this discrepancy level, the main point to
appreciate is that the linear spectra of $\cal W$$(z_{k})$ are quite a good
fit of the linear spectra of $\cal M$ at $z_{k}$.
Figure 13: Ratio between power spectra for the polynomial model and the
corresponding auxiliary models; dashed (solid) lines are obtained by using FLL
(our) technique. Spectra for two model realizations are plotted, but
differences are just marginally appreciable for the dashed lines. Up to
$k=3\,h$Mpc-1, the FLL approach already provides results with discrepancies
within 1$\,\%$. They are smallest at $z=2.4$, a redshift close to crossover.
By using the weak requirement (W.R.) technique described in this work, the
maximum discrepancies occur at $z=0$, where they are presumably caused by the
different non–linear history in $\cal M$ and $\cal W$ models, propagating its
effects even below $k=3\,h$Mpc-1. At $z=0.6$ there is a residual discrepancy
peaking at a value $\sim 0.3\,\%$ on a wavenumber $k\simeq 7\,h$Mpc-1.
Otherwise, discrepancies keep within a fraction of permil, at any $k$. Figure
14: Ratio between power spectra for the SUGRA model and the corresponding
auxiliary models; dashed and solid lines as in previous Figure. Again spectra
for two model realizations are plotted and some slight differences are
appreciable. At $z=0$, where FLL and W.R. approaches coincide, differences
between $\cal M$ and $\cal W$ models keep within $1\%$ just up to
$k=1.8\,h$Mpc-1; at $k=3\,h$Mpc-1, they are $\sim 1.3\,\%$. At higher
redshift, the FLL technique provides discrepancies within 1$\,\%$ just at
$z=0.6$. The W.R. technique here yields a great improvement, keeping
discrepancies within 0.6$\,\%$, 0.1$\,\%$ 0.2$\,\%$ 0.6$\,\%$ at $z=0.6$, 1.2,
1.8, 2.4, respectively.
The main results of this work are then described in the Figures 13 and 14. Let
us remind that the polynomial model is one of those already considered by FLL,
selected for the sake of comparison. As could also be expected from the linear
spectra, both techniques perform better for this model, while the SUGRA model
puts a more severe challenge.
In Figure 13 we show the ratio between power spectra for the polynomial model
and the corresponding $\cal W$$(z_{k})$ models. The spectra worked out by
using the W.R. technique are given by solid lines. We also plot those obtained
according to FLL (dashed lines). Although spectra for two model realizations
are plotted, their differences can hardly be appreciated.
Then, in Figure 14 we also show the ratio between power spectra for the SUGRA
model and the corresponding auxiliary models $\cal W$$(z_{k})$. Differences
between model realizations are more easily appreciable here, than in the
Figure 13 , but still quite negligible. What is immediately visible, instead,
is the greater difficulty that both W.R. and FLL techniques have to approach
SUGRA spectra.
Two different limitations are to be considered, when using such techniques.
The first one arises from hydrodynamics and related effects. At $z=0$
hydrodynamics pollutes N–body spectra when $k/h$Mpc-1 overcomes $\sim 3$: the
number of halos with mass $>(4\pi/3)~{}3^{3}\times 2.78\cdot
10^{11}\Delta_{vir}M_{\odot}h^{-1}$ is enough to induce a spectral distortion
$>1\,\%$. At greater redshift, the same number of halos can be found only on
smaller scales. Accordingly, N–body results are safe from hydro–pollution up
to an increasing value of $k$. It is sufficient to look at Figure 7 to
appreciate the scaling of the limiting value, which (in $h$Mpc-1 units) will
be $\sim 6$ at $z\simeq 1.2$ and $\sim 8$ at $z\simeq 2.4\,$.
The second limitation is the one we test here. Possible differences between
the spectra of the models $\cal M$ and $\cal W$$(z)$, at the redshift $z$, can
arise from the different $a(t)$ history in the two models, after the
non–linearity onset. Such differences shall be greater on greater $k$ (smaller
scales), which became non–linear earlier. At a fixed $k$, differences in the
non–linearity history will become more and more significant at later times.
These discrepancies are those we try to minimize and are shown in Figures 13,
14 . In the $k$ range considered, they are already practically absent at
$z=0.3\,$, when using the W.R. approach. This is shown by all plots for $z\neq
0$ in the above two Figures. Here, the critical $k$ for hydro distortions is
systematically below the $k$ where the non–linear history begins to matter.
Just at $z\simeq 0$ the two critical $k$’s are close, their relative setting
being somehow model dependent.
Accordingly, while the spectra of the polynomial model, also treated by FLL,
are easily met by the relative $\cal W$ model at $z=0$, SUGRA confirms to be a
harder challenge. Figures 13 and 14 however show that the W.R. technique is
successful at any $z$, mostly attaining a precision at the per mil level, and
hardly exceeding the 1$\,\%$ discrepancy even in the most difficult cases.
## 6 Conclusions
By using the “weak requirement” condition described in this paper, we showed
that even spectra of models with rapidly varying $w(a)$ are easily obtainable
from constant–$w$ spectra. We also verify that available Halofit expressions
are not a sufficient approximation, already at $z\simeq 0.6~{}$. Such
expressions were generalized to $w={\rm const.}$ models by [13], but for a
range of values of $\sigma_{8}$ which do not cover our case, tuned on recent
observational outputs.
In our opinion, this means that strong efforts are soon to be made to provide
generalized Halofit expressions, for constant–$w$ models, effective for the
whole parameter range that recent observations suggest to inspect, and working
up to reasonably high redshift.
Let us then suppose that such expressions are available and that new
observations provide direct information on density fluctuation spectra at $N$
redshifts $z_{k}$ ranging from 0 to $z_{N}$. Such spectra should then be fit
to models, directly assuming that the reduced density parameters
$\omega_{c}(z_{k})=\omega_{o,c}(1+z_{k})^{3}~{},~{}~{}~{}~{}~{}\omega_{b}(z_{k})=\omega_{o,b}(1+z_{k})^{3}~{},$
(4)
as it must however be, and seeking suitable $w_{k}$, $\sigma_{8}^{(z_{k})}$,
$h_{k}$. Using then each $w_{k},$ the $z=0$ values $\sigma_{o,8}^{(z_{k})}$
and $h_{o,k}$ should be easily reconstructed. The $z_{k}$ dependence of
$w_{k}$, $\sigma_{o,8}^{(z_{k})}$, $h_{o,k}$ should then allow to recover a
physical $w(z)$ behavior.
Let us outline, in particular, that the $w_{k}$ values directly measured are
not the physical $w(z)$. On the contrary, they can be quite far from it, as is
made clear in Figure 5: there, the measured $w_{k}$ correspond to the solid
curves, while the physical state parameter scale dependence is given by the
dotted curves.
The simultaneous use of the information on $\sigma_{o,8}^{(z_{k})}$ (fitting
curves similar to those in Figure 6, where it is simply named $\sigma_{8}$)
and $h_{o,k}$ (which can be easily determined through linear programs), can
allow a complete exploitation of forthcoming data.
Preliminary evaluations on the possible efficiency of observational techniques
probing high redshift spectra, as the planned DUNE–EUCLID experiment, should
be reviewed and possibly improved on the basis of the above conclusions.
## Acknowledgments
Thanks are due to Anatoly Klypin and Gustavo Yepes for wide discussions. We
are indebted with Joachim Stadel for granting us the use of the pkdgrav code.
We are also indebted with Giuseppe La Vacca for allowing us Figure 1, before
its publication. Most numerical simulations were performed on the PIA cluster
of the Max-Planck-Institut für Astronomie at the Rechenzentrum in Garching.
Finally, it is a pleasure to thank an anonymous referee whose suggestions
allowed us to improve and complete the presentation of our results. The
support of ASI (Italian Space Agency) through the contract I/016/07/0 “COFIS”
is acknowledged.
## Appendix A pkdgrav and Dynamical Dark Energy
The central structure in pkdgrav is a tree structure which forms the
hierarchical representation of the mass distribution. Unlike the more
traditional oct-tree pkdgrav uses a k–D tree, which is a binary tree. The
root-cell of this tree represents the entire simulation volume. Other cells
represent rectangular sub-volumes that contain the mass, center-of-mass, and
moments up to hexadecapole order of their enclosed regions. pkdgrav calculates
the gravitational accelerations using the well known tree-walking procedure of
the Barnes-Hut algorithm [31]. Periodic boundary conditions are implemented
via the Ewald summation technique [32]. pkdgrav uses the ordinary time $t$ (in
suitable units) as independent variable. The link between the expansion factor
$a$ and the ordinary time $t$, in the case ($\Lambda$)CDM models is based on
the equation:
$\dot{a}/a=H(t)=(H_{0}a^{-2})(\Omega_{r}+\Omega_{m}a+\Omega_{\kappa}a^{2}+\Omega_{\Lambda}a^{4})^{1/2}$
(5)
where all the density parameters indicate redshift zero values. Herefrom one
obtains soon that
$t=\int_{0}^{a}da/\\{aH(a)\\}=(2/3)\int_{0}^{Y(a)}dy/\\{yH[a(y)]\\}$ (6)
with
$Y(a)=a^{3/2},~{}a(y)=y^{2/3}$ (7)
and the change of variable is clearly tailored on models based on CDM. All
this procedure is modified in our algorithm. The program now creates a priori
the $t(a)$ dependence, by integrating a suitable set of differential
equations. A large number ($\sim 10000$) of $t(a)$ values are then kept in
memory and interpolated to work out $t(a)$ at any $a$ value needed during the
simulation run.
## References
## References
* [1] Astier P et al. , 2006 Astronom. Astrophys. 447 31
* [2] Riess A G et al. , 2007 Astrophys. J. 659 98
* [3] Komatsu E et al. , 2008 Preprint 0803.0547v1.
* [4] Cole S et al., Mon. Not. R. Astron. Soc. 362, 505 (2005).
* [5] La Vacca G et al. in preparation
* [6] Amendola L, Gasperini M and Ungarelli C, 2008 Phys. Rev. D 77 123526
* [7] Refregier A et al. , 2008 Preprint 0802.2522
* [8] Huterer D and Tanaka M, 2005 Astrophys. J. 23 369
* [9] White S, 2004 KITP Conf.: Galaxy-Intergalactic Medium Interactions Kavli Institute for Theoretical Physics
* [10] Jing Y P, Zhang P, Lin W P, Gao L and Springel V, 2006 Astrophys. J. 640 119
* [11] Rudd D, Zentner A and Kravtsov A, 2008 Astrophys. J. 19 672
* [12] Smith R E, Peacock J A, Jenkins A, White S D M, Frenk C S, Pearce F R, Thomas P A, Efstathiou G and Couchman H M P, 2003 Mon. Not. R. Astron. Soc. 341 1311
* [13] McDonald P, Trac H and Contaldi C, 2006 Mon. Not. R. Astron. Soc. 366 547
* [14] Klypin A, Macciò A V, Mainini R and Bonometto S A, 2003 Astrophys. J. 599 31
* [15] Macciò A V, Quercellini C, Mainini R, Amendola L, Bonometto S A, 2004 Phys. Rev. D 69 123516
* [16] Linder E V and Jenkins A, 2003 Mon. Not. R. Astron. Soc. 346 573
* [17] Solevi P, Mainini R, Bonometto S A, Macciò A V, Klypin A and Gottl ber S, 2006 Mon. Not. R. Astron. Soc. 366 1346
* [18] Francis M J, Lewis G F and Linder E V, 2007 Mon. Not. R. Astron. Soc. 380 1079
* [19] Linder E and White M, 2005 Phys. Rev. D 72, 061394
* [20] Brax P H and Martin J, 1999 Phys. Lett. B 468 40
* [21] Brax P H and Martin J, 2001 Phys. Rev. D 62 10350
* [22] Brax P, Martin J and Riazuelo A, 2000 Phys. Rev. D 61 103505
* [23] Spergel D N et al. , 2007 Astrophys. J. Suppl. 170 377
* [24] Klypin A and Holtzman J, 1997 Preprint astro-ph/9712217
* [25] Lewis A, Challinor A and Lasenby A, 2000 Astrophys. J. 538 473
* [26] Kravtsov A, Klypin A and Khokhlov A, 1997 Astrophys. J. Suppl. 111 73
* [27] Stadel J G, 2001 PhD thesis University of Washington
* [28] Mainini R, Macciò A V, Bonometto S A and Klypin A, 2003 Astrophys. J. 599 24
* [29] Crocce, M., & Scoccimarro, R. 2008, Phys. Rev. D, 77, 023533
* [30] Colombo L and Gervasi M, 2006 J. Cosmol. Astropart. Phys. 10 001
* [31] Barnes J and Hut P, 1986 Nature 324 446
* [32] Hernquist L, Bouchet F, and Suto Y, 1991 Astrophys. J. Suppl. 75 231
|
arxiv-papers
| 2008-10-01T15:12:15
|
2024-09-04T02:48:58.075953
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luciano Casarini, Andrea V. Maccio', Silvio A. Bonometto",
"submitter": "Luciano Casarini",
"url": "https://arxiv.org/abs/0810.0190"
}
|
0810.0293
|
# Meson decay in a corrected ${}^{3}P_{0}$ model
D. T. da Silva Instituto de Física, Universidade Federal do Rio Grande do Sul
Av. Bento Gonçalves, 9500, Porto Alegre, Rio Grande do Sul, CEP 91501-970,
Brazil M. L. L. da Silva Instituto de Física, Universidade Federal do Rio
Grande do Sul
Av. Bento Gonçalves, 9500, Porto Alegre, Rio Grande do Sul, CEP 91501-970,
Brazil J. N. de Quadros Instituto de Física, Universidade Federal do Rio
Grande do Sul
Av. Bento Gonçalves, 9500, Porto Alegre, Rio Grande do Sul, CEP 91501-970,
Brazil D. Hadjimichef dimiter@ufpel.edu.br; dimihadj@gmail.com Instituto de
Física e Matemática, Universidade Federal de Pelotas
Campus Universitário, Pelotas, Rio Grande do Sul, CEP 96010-900, Brazil
###### Abstract
Extensively applied to both light and heavy meson decay and standing as one of
the most successful strong decay models is the ${}^{3}P_{0}$ model, in which
$q\bar{q}$ pair production is the dominant mechanism. The pair production can
be obtained from the non-relativistic limit of a microscopic interaction
Hamiltonian involving Dirac quark fields. The evaluation of the decay
amplitude can be performed by a diagrammatic technique for drawing quark
lines. In this paper we use an alternative approach which consists in a
mapping technique, the Fock-Tani formalism, in order to obtain an effective
Hamiltonian starting from same microscopic interaction. An additional effect
is manifest in this formalism associated to the extended nature of mesons:
bound-state corrections. A corrected ${}^{3}P_{0}$ is obtained and applied, as
an example, to $b_{1}\rightarrow\omega\pi$ and $a_{1}\rightarrow\rho\pi$
decays.
###### pacs:
11.15.Tk, 12.39.Jh, 13.25.-k
††preprint: APS/123-QED
## I Introduction
A great variety of quark-based models are known that describe with reasonable
success single-hadron properties. A natural question that arises is to what
extent a model which gives a good description of hadron properties is, at the
same time, able to describe the complex hadron-hadron interaction or by the
same principles hadron decay. In particular, the theoretical aspects of strong
decay have been challenged by QCD exotica (glueballs and hybrids) where a
consistent understanding of the mixing schemes for these states is still an
open question close -klempt2 . The nature of the family of “new mesons”
$X,Y,Z$ barnes-xyz is another unsolved puzzle: are they actually new
$q\bar{q}$ mesons, hadronic molecules or something else? In the direction of
clarifying these questions is the successful decay model, the ${}^{3}P_{0}$
model, which considers only OZI-allowed strong-interaction decays. This model
was introduced over thirty years ago by Micu micu and applied to meson decays
in the 1970 by LeYaouanc et al. leyaouanc . This description is a natural
consequence of the constituent quark model scenario of hadronic states.
T. Barnes et al. barnes1 -barnes4 have made an extensive survey of meson
states in the light of the ${}^{3}P_{0}$ model. Two basic parameters of their
formulation are $\gamma$ (the interaction strength) and $\beta$ (the wave
function’s extension parameter). Although they found the optimum values near
$\gamma=0.5$ and $\beta=0.4$ GeV, for light 1S and 1P decays, these values
lead to overestimates of the widths of higher-L states. In this perspective a
modified $q\bar{q}$ pair-creation interaction, with $\gamma=0.4$ was
preferred.
In the present work, we employ a mapping technique in order to obtain an
effective interaction for meson decay. A particular mapping technique long
used in atomic physics girar1 , the Fock-Tani formalism (FTf), has been
adapted, in previous publications annals -mario , in order to describe hadron-
hadron scattering interactions with constituent interchange. Now this
technique has been extended in order to include meson decay. We start from the
microscopic $q\bar{q}$ pair-creation interaction, as will be shown, in lower
order, the ${}^{3}P_{0}$ results are reproduced. An additional and interesting
feature appears in higher orders of the formalism: corrections due to the
bound-state nature of the mesons and a natural modification in the $q\bar{q}$
interaction strength.
In the Fock-Tani formalism one starts with the Fock representation of the
system using field operators of elementary constituents which satisfy
canonical (anti) commutation relations. Composite-particle field operators are
linear combinations of the elementary-particle operators and do not generally
satisfy canonical (anti) commutation relations. “Ideal” field operators acting
on an enlarged Fock space are then introduced in close correspondence with the
composite ones. Next, a given unitary transformation, which transforms the
single composite states into single ideal states, is introduced. Application
of the unitary operator on the microscopic Hamiltonian, or on other hermitian
operators expressed in terms of the elementary constituent field operators,
gives equivalent operators which contain the ideal field operators. The
effective Hamiltonian in the new representation has a clear physical
interpretation in terms of the processes it describes. Since all field
operators in the new representation satisfy canonical (anti)commutation
relations, the standard methods of quantum field theory can then be readily
applied.
In this paper we shall extend the FTf to meson decay processes. In the next
section we review the basic aspects of the formalism. Section III is dedicated
to obtain an effective decay Hamiltonian. In section IV, two light mesons
decays examples are calculated $b_{1}\rightarrow\omega\pi$ and
$a_{1}\rightarrow\rho\pi$. The summary and conclusions are followed by
appendixes which detail the method employed throughout this work.
## II Mapping of mesons
This section reviews the formal aspects of the mapping procedure and how it is
implemented to quark-antiquark meson states annals . The starting point of the
Fock-Tani formalism is the definition of single composite bound states. We
write a single-meson state in terms of a meson creation operator
$M_{\alpha}^{\dagger}$ as
$\displaystyle|\alpha\rangle=M_{\alpha}^{\dagger}|0\rangle,$ (1)
where $|0\rangle$ is the vacuum state. The meson creation operator
$M_{\alpha}^{\dagger}$ is written in terms of constituent quark and antiquark
creation operators $q^{\dagger}$ and $\bar{q}^{\dagger}$,
$\displaystyle
M^{\dagger}_{\alpha}=\Phi_{\alpha}^{\mu\nu}q_{\mu}^{\dagger}{\bar{q}}_{\nu}^{\dagger},$
(2)
$\Phi_{\alpha}^{\mu\nu}$ is the meson wave function and
$q_{\mu}|0\rangle=\bar{q}_{\nu}|0\rangle=0$. The index $\alpha$ identifies the
meson quantum numbers of space, spin and isospin. The indices $\mu$ and $\nu$
denote the spatial, spin, flavor, and color quantum numbers of the constituent
quarks. A sum over repeated indices is implied. It is convenient to work with
orthonormalized amplitudes,
$\displaystyle\langle\alpha|\beta\rangle=\Phi_{\alpha}^{*\mu\nu}\Phi_{\beta}^{\mu\nu}=\delta_{\alpha\beta}.$
(3)
The quark and antiquark operators satisfy canonical anticommutation relations,
$\displaystyle\\{q_{\mu},q^{\dagger}_{\nu}\\}=\\{\bar{q}_{\mu},\bar{q}^{\dagger}_{\nu}\\}=\delta_{\mu\nu},$
$\displaystyle\\{q_{\mu},q_{\nu}\\}=\\{\bar{q}_{\mu},\bar{q}_{\nu}\\}=\\{q_{\mu},\bar{q}_{\nu}\\}=\\{q_{\mu},\bar{q}_{\nu}^{\dagger}\\}=0.$
(4)
Using these quark anticommutation relations, and the normalization condition
of Eq. (3), it is easily shown that the meson operators satisfy the following
non-canonical commutation relations
$[M_{\alpha},M^{\dagger}_{\beta}]=\delta_{\alpha\beta}-\Delta_{\alpha\beta},\hskip
42.67912pt[M_{\alpha},M_{\beta}]=0,$ (5)
where
$\displaystyle\Delta_{\alpha\beta}=\Phi_{\alpha}^{*{\mu\nu}}\Phi_{\beta}^{\mu\sigma}\bar{q}^{\dagger}_{\sigma}\bar{q}_{\nu}+\Phi_{\alpha}^{*{\mu\nu}}\Phi_{\beta}^{\rho\nu}q^{\dagger}_{\rho}q_{\mu}.$
(6)
In addition,
$\displaystyle[q_{\mu},M_{\alpha}^{\dagger}]=\Phi^{\mu\nu}_{\alpha}{\bar{q}}_{\nu}^{\dagger}\hskip
14.22636pt,\hskip
14.22636pt[{\bar{q}}_{\nu},M_{\alpha}^{\dagger}]=-\Phi^{\mu\nu}_{\alpha}q_{\mu}^{\dagger},\hskip
28.45274pt$
$\displaystyle[q_{\mu},M_{\alpha}]=[{\bar{q}}_{\nu},M_{\alpha}]=0.$ (7)
The presence of the operator $\Delta_{\alpha\beta}$ in Eq. (5) is due to the
composite nature of the mesons. This term enormously complicates the
mathematical description of processes that involve the hadron and quark
degrees of freedom. The usual field theoretic techniques used in many-body
problems, such as the Green’s functions method, Wick’s theorem, etc, apply to
creation and annihilation operators that satisfy canonical relations.
Similarly, the non-vanishing of the commutators
$[q_{\mu},M_{\alpha}^{\dagger}]$ and $[{\bar{q}}_{\nu},M_{\alpha}^{\dagger}]$
is a manifestation of the lack of kinematic independence of the meson operator
from the quark and antiquark operators. Therefore, the meson operators
$M_{\alpha}$ and $M_{\alpha}^{\dagger}$ are not convenient dynamical variables
to be used.
A transformation is defined such that a single-meson state $|\alpha\rangle$ is
redescribed by an (“ideal”) elementary-meson state by
$\displaystyle|\alpha\rangle\longrightarrow
U^{-1}|\alpha\rangle=m^{\dagger}_{\alpha}|0\rangle,$ (8)
where $m^{\dagger}_{\alpha}$ an ideal meson creation operator. The ideal meson
operators $m^{\dagger}_{\alpha}$ and $m_{\alpha}$ satisfy, by definition,
canonical commutation relations
$[m_{\alpha},m^{\dagger}_{\beta}]=\delta_{\alpha\beta},\hskip
42.67912pt[m_{\alpha},m_{\beta}]=0.$ (9)
The state $|0\rangle$ is the vacuum of both $q$ and $m$ degrees of freedom in
the new representation. In addition, in the new representation the quark and
antiquark operators $q^{\dagger}$, $q$, $\bar{q}^{\dagger}$ and $\bar{q}$ are
kinematically independent of the $m^{\dagger}_{\alpha}$ and $m_{\alpha}$
$\displaystyle[q_{\mu},m_{\alpha}]=[q_{\mu},m^{\dagger}_{\alpha}]=[\bar{q}_{\mu},m_{\alpha}]=[\bar{q}_{\mu},m^{\dagger}_{\alpha}]=0.$
(10)
The unitary operator $U$ of the transformation is
$\displaystyle U(t)=\exp\left[t\,F\right],$ (11)
where $F$ is the generator of the transformation and $t$ a parameter which is
set to $-\pi/2$ to implement the mapping. The next step is to obtain the
transformed operators in the new representation. The basic operators of the
model are expressed in terms of the quark operators. Therefore, in order to
obtain the operators in the new representation, one writes
$\displaystyle q(t)=U^{-1}\,q\,U,\hskip
28.45274pt{\bar{q}}(t)=U^{-1}\,{\bar{q}}\,U.$ (12)
The generator $F$ of the transformation is
$\displaystyle
F=m^{{\dagger}}_{\alpha}\,\tilde{M}_{\alpha}-\tilde{M}^{{\dagger}}_{\alpha}m_{\alpha}$
(13)
where
$\displaystyle\tilde{M}_{\alpha}=\sum_{i=0}^{n}\tilde{M}^{(i)}_{\alpha},$ (14)
with
$\displaystyle[\tilde{M}_{\alpha},\tilde{M}^{\dagger}_{\beta}]=\delta_{\alpha\beta}\hskip
14.22636pt+\hskip 14.22636pt{\cal O}(\Phi^{n+1}),$
$\displaystyle[\tilde{M}_{\alpha},\tilde{M}_{\beta}]=[\tilde{M}^{\dagger}_{\alpha},\tilde{M}^{\dagger}_{\beta}]=0.$
(15)
It is easy to see from (13) that $F^{{\dagger}}=-F$ which ensures that $U$ is
unitary. The index $i$ in (14) represents the order of the expansion in powers
of the wave function $\Phi$. The $\tilde{M}_{\alpha}$ operator is determined
up to a specific order $n$ consistent with (15). The examples studied in
annals required the determination of $\tilde{M}^{(i)}_{\alpha}$ up to order 3
as shown below
$\displaystyle\tilde{M}_{\alpha}^{(0)}$ $\displaystyle=$ $\displaystyle
M_{\alpha}\hskip 14.22636pt;\hskip 14.22636pt\tilde{M}_{\alpha}^{(1)}=0$
$\displaystyle\tilde{M}_{\alpha}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta_{\alpha\beta}M_{\beta}\hskip 2.84544pt;\hskip
2.84544pt\tilde{M}_{\alpha}^{(3)}=\frac{1}{2}M^{\dagger}_{\beta}\,\,T_{\alpha\beta\gamma}\,\,M_{\gamma},$
(16)
with$T_{\alpha\beta\gamma}=-\left[M_{\alpha},\,\Delta_{\beta\gamma}\right]$.
In the “zero-order” approximation, overlap among mesons is neglected and terms
of the same power in the bound-state wave function $\Phi_{\alpha}$
($\Phi^{*}_{\alpha}$) are collected. In order to have a consistent power
counting scheme, the implicit $\Phi_{\alpha}$ ($\Phi^{*}_{\alpha}$) entering
via Eq. (2) are not counted. The consequence of this is that the equations for
$m_{\alpha}$ and $\tilde{M}_{\alpha}$ are manifestly symmetric,
$\displaystyle{dm_{\alpha}(t)\over dt}$ $\displaystyle=$
$\displaystyle[m_{\alpha}(t),F]=\tilde{M}_{\alpha}(t),$
$\displaystyle{d\tilde{M}_{\alpha}(t)\over dt}$ $\displaystyle=$
$\displaystyle[\tilde{M}_{\alpha}(t),F]=-m_{\alpha}(t),$ (17)
and their solutions involve only trigonometric functions of $t$,
$\displaystyle m_{\alpha}(t)$ $\displaystyle=$
$\displaystyle\tilde{M}_{\alpha}\sin t+m_{\alpha}\cos t,$
$\displaystyle\tilde{M}_{\alpha}(t)$ $\displaystyle=$
$\displaystyle\tilde{M}_{\alpha}\cos t-m_{\alpha}\sin t.$ (18)
The equations of motion for the quark operators $q$ and $\bar{q}$ can be
obtained by making use of Eq. (7) in a similar way,
$\displaystyle{dq_{\mu}(t)\over dt}$ $\displaystyle=$
$\displaystyle\left[q_{\mu}(t),F\right]\hskip 14.22636pt;\hskip
14.22636pt{d\bar{q}_{\mu}(t)\over dt}=\left[\bar{q}_{\mu}(t),F\right].$ (19)
In the zero-order approximation, the effects of the meson structure are
neglected resulting
$\displaystyle q^{(0)}_{\mu}(t)=q_{\mu},\hskip
14.22636pt\bar{q}^{(0)}_{\nu}(t)=\bar{q}_{\nu},$ $\displaystyle
m^{(0)}_{\alpha}(t)=m_{\alpha}\cos t+M_{\alpha}\sin t,$ $\displaystyle
M^{(0)}_{\alpha}(t)=M_{\alpha}\cos t-m_{\alpha}\sin t.$ (20)
In first order one has
$\displaystyle
q^{(1)}_{\mu}(t)=-\,\Phi^{\mu\nu_{1}}_{\alpha}\bar{q}^{\dagger}_{\nu_{1}}\left[m_{\alpha}\sin
t+M_{\alpha}\left(1-\cos t\right)\right],$
$\displaystyle\bar{q}^{(1)}_{\nu}(t)=\Phi^{\mu_{1}\nu}_{\alpha}q^{\dagger}_{\mu_{1}}\left[m_{\alpha}\sin
t+M_{\alpha}\left(1-\cos t\right)\right],$ $\displaystyle
m^{(1)}_{\alpha}(t)=0,\hskip 28.45274ptM^{(1)}_{\alpha}(t)=0.$ (21)
The second and third order solutions to (19) were calculated in reference
annals and appear again, for completeness, in appendix A, together with the
higher order operators required in our calculation.
Once a microscopic interaction Hamiltonian $H$ is defined, at the quark level,
a new transformed Hamiltonian can be obtained. This effective interaction we
shall call the Fock-Tani Hamiltonian and is evaluated by the application of
the unitary operator $U$ on the microscopic Hamiltonian ${\cal H}_{\rm
FT}=U^{-1}HU$. The transformed Hamiltonian ${\cal H}_{\rm FT}$ describes all
possible processes involving mesons and quarks. The general structure of
${\cal H}_{\rm FT}$ is of the following form
$\displaystyle{\cal H}_{\rm FT}={\cal H}_{\rm\bf q}+{\cal H}_{\rm\bf m}+{\cal
H}_{\rm\bf mq},$ (22)
where the first term involves only quark operators, the second one involves
only ideal meson operators, and ${\cal H}_{\rm\bf mq}$ involves quark and
meson operators.
In ${\cal H}_{\rm FT}$ there are higher order terms that provide bound-state
corrections (also called orthogonality corrections) to the lower order ones.
The basic quantity for these corrections is the bound-state kernel
$\Delta(\rho\tau;\lambda\nu)$ defined as
$\displaystyle\Delta(\rho\tau;\lambda\nu)=\Phi^{\rho\tau}_{\alpha}\Phi^{\ast\lambda\nu}_{\alpha}.$
(23)
To discuss the physical meaning of the bound-state corrections and how they
modify the fundamental quark interaction we shall present an example, in a toy
model similar to the model studied in annals , where the basic arguments are
outlined. In this example, the starting point is a two-body microscopic quark-
antiquark Hamiltonian of the form
$\displaystyle H_{2q}$ $\displaystyle=$ $\displaystyle
T\left(\mu\right)q^{\dagger}_{\mu}q_{\mu}+T\left(\nu\right)\bar{q}_{\nu}^{\dagger}\bar{q}_{\nu}+V_{q\bar{q}}(\mu\nu;\sigma\rho)q^{\dagger}_{\mu}\bar{q}^{\dagger}_{\nu}\bar{q}_{\rho}q_{\sigma}$
$\displaystyle\\!\\!\\!\\!+\frac{1}{2}V_{qq}(\mu\nu;\sigma\rho)q^{\dagger}_{\mu}q^{\dagger}_{\nu}q_{\rho}q_{\sigma}+\frac{1}{2}V_{\bar{q}\bar{q}}(\mu\nu;\sigma\rho)\bar{q}^{\dagger}_{\mu}\bar{q}^{\dagger}_{\nu}\bar{q}_{\rho}\bar{q}_{\sigma}.$
The transformation ${\cal H}_{\rm FT}=U^{-1}\,H_{2q}\,U$ is implemented again
by transforming each quark and antiquark operator in Eq. (LABEL:qHamilt),
where a similar structure to Eq. (22) is obtained. In free space, the wave
function $\Phi$ of Eq. (2) satisfy the following equation
$H(\mu\nu;\sigma\rho)\Phi_{\alpha}^{\sigma\rho}=\epsilon_{[\alpha]}\Phi_{[\alpha]}^{\mu\nu},$
(25)
where $H(\mu\nu;\sigma\rho)$ is the Hamiltonian matrix
$\displaystyle H(\mu\nu;\sigma\rho)$ $\displaystyle=$
$\displaystyle\delta_{\mu[\sigma]}\delta_{\nu[\rho]}\left[T([\sigma])+T([\rho])\right]$
(26) $\displaystyle+V_{q\bar{q}}(\mu\nu;\sigma\rho),$
$\epsilon_{[\alpha]}$ is the total energy of the meson. There is no sum over
repeated indices inside square brackets.
The effective quark Hamiltonian ${\cal H}_{\rm\bf q}$ has an identical
structure to the microscopic quark Hamiltonian, Eq. (LABEL:qHamilt), except
that the term corresponding to the quark-antiquark interaction is modified as
follows
$\displaystyle\\!\\!\\!\\!{\cal V}_{q\bar{q}}$ $\displaystyle=$
$\displaystyle\left[\,V_{q\bar{q}}-H\,\Delta-\Delta\,H+\Delta\,H\,\Delta\,\right]\,,$
(27)
where $V_{q\bar{q}}\equiv V_{q\bar{q}}(\mu\nu;\sigma\rho)$ and the contraction
$H\,\Delta\equiv H(\mu\nu;\tau\xi)\,\Delta(\tau\xi;\sigma\rho)$. An important
property of the bound-state kernel is
$\Delta(\mu\nu;\sigma\rho)\Phi^{\sigma\rho}_{\alpha}=\Phi^{\mu\nu}_{\alpha},$
(28)
which follows from the wave function’s orthonormalization, Eq. (3). In the
case that $\Phi$ is a solution of Eq. (25), the new quark-antiquark
interaction term becomes
$\displaystyle{\cal
V}_{q\bar{q}}(\mu\nu;\sigma\rho)=V_{q\bar{q}}(\mu\nu;\sigma\rho)-\sum_{\alpha}\epsilon_{\alpha}\Phi^{\ast\mu\nu}_{\alpha}\Phi^{\sigma\rho}_{\alpha}.$
(29)
The spectrum of the modified quark Hamiltonian, ${\cal H}_{\rm\bf q}$, is
positive semi-definite and hence has no bound-states girar1 . This result is
exactly the same as in Weinberg’s quasiparticle method quasi , where the
bound-states are redescribed by ideal particles. The new ${\cal V}_{q\bar{q}}$
is a weaker potential, modified in such a way that no new bound-states are
formed.
In the quark-meson sector of Eq. (22) in ${\cal H}_{\rm\bf mq}$ appears a term
related to spontaneous meson break-up
$\displaystyle H_{\rm m\to
q\bar{q}}=V(\mu\nu;\alpha)\,q^{\dagger}_{\mu}\bar{q}^{\dagger}_{\nu}m_{\alpha}$
(30)
with
$\displaystyle V(\mu\nu;\alpha)$ $\displaystyle=$ $\displaystyle
H(\mu\nu;\sigma\rho)\Phi^{\sigma\rho}_{\alpha}-\Delta(\mu\nu;\sigma\rho)H(\sigma\rho;\tau\lambda)\Phi^{\tau\lambda}_{\alpha}.$
Again, in the case that $\Phi$ is a solution of Eq. (25), a straightforward
calculation demonstrates that $H_{\rm m\to q\bar{q}}=0$. When there is no
external interaction, this result is a direct consequence of the bound-state’s
stability against spontaneous break-up. This term can be interesting in
studies related to dense hadronic mediums. For these systems the wave function
is, in general, not a solution of Eq. (25) and the strength of the potential
$V(\mu\nu;\alpha)$ is now only decreased sergio .
In the ideal meson sector ${\cal H}_{\rm\bf m}$ many similar approaches to FTf
annals have obtained the meson-meson scattering interaction in the Born
approximation: Resonating Group Method (RGM) oka , Quark Born Diagram
Formalism (QBDF) qbd ,
$H_{mm}=T_{mm}+V_{mm},$ (32)
where $T_{mm}$ is the kinetic term and $V_{mm}$ is the meson-meson interaction
potential with constituent interchange. This potential is given by
$\displaystyle V_{mm}=V_{mm}^{dir}+V_{mm}^{exc}+V_{mm}^{int}\,,$ (33)
where $V_{mm}^{dir}$ is the direct potential (no quark interchange),
$V_{mm}^{exc}$ the quark exchange term and $V_{mm}^{int}$ the intra-exchange
term. As shown in Ref. annals and sergio , if one extends the FT calculation
to higher orders a new meson-meson Hamiltonian is obtained
$\displaystyle\bar{H}_{mm}=H_{mm}+\delta H_{mm}$ (34)
where $\delta H_{mm}$ is the bound-state correction Hamiltonian. If the wave
function $\Phi$ is chosen to be an eigenstate of the microscopic quark
Hamiltonian, then the intra-exchange term $V_{mm}^{int}$ is cancelled
$\displaystyle V_{mm}^{int}+\delta H_{mm}=0.$ (35)
In summary, these examples reveal an important and common feature of bound-
state corrections: they weaken the quark-antiquark potential. In the next
section we shall follow the same procedure for a quark pair creation
interaction, which is fundamental for the description of meson decay. Similar
to the toy model, the resulting interaction that describes meson decay, will
contain a Born order contribution and a bound-state correction.
## III The ${}^{3}P_{0}$ Decay Model in the Fock-Tani Formalism
In the paper of E. S. Ackleh, T. Barnes and E. S. Swanson barnes1 a
formulation of the ${}^{3}P_{0}$ model is presented. It regards the decay of
an initial state meson in the presence of a $q\bar{q}$ pair created from the
vacuum. The pair production is obtained from the non-relativistic limit of the
interaction Hamiltonian $H_{I}$ involving Dirac quark fields
$\displaystyle H_{I}=2\,m_{q}\,\gamma\,\int
d\vec{x}\,\bar{\psi}(\vec{x})\,\psi(\vec{x})\,,$ (36)
where $\gamma$ is the pair production strength. For a $q\bar{q}$ meson $A$ to
decay to mesons $B+C$ we must have
$(q\bar{q})_{A}\to(q\bar{q})_{B}+(q\bar{q})_{C}$. To determine the decay rate
a matrix element of (36) is evaluated
$\displaystyle\langle
BC|H_{I}|A\rangle=\delta(\vec{P}_{A}-\vec{P}_{B}-\vec{P}_{C})\,h_{fi}.$ (37)
The evaluation of $h_{fi}$ is performed by diagrammatic technique for drawing
quark lines. The $h_{fi}$ decay amplitude is combined with relativistic phase
space, resulting in the differential decay rate
$\displaystyle\frac{d\Gamma_{A\to
BC}}{d\Omega}=2\pi\,P\,\frac{E_{B}\,E_{C}}{M_{A}}|h_{fi}|^{2}$ (38)
which after integration in the solid angle $\Omega$ a usual choice for the
meson momenta is made: $\vec{P}_{A}=0$ ($P=|\vec{P}_{B}|=|\vec{P}_{C}|$).
In our approach, the starting point for the Fock-Tani $h_{fi}$ is also the
microscopic Hamiltonian $H_{I}$ in (36). The momentum expansion of the quark
fields, color and flavor are not represented explicitly, is
$\displaystyle\psi(\vec{x})=\sum_{s}\int\frac{d^{3}k}{(2\pi)^{3/2}}[u_{s}(\vec{k})q_{s}(\vec{k})$
$\displaystyle+v_{s}(-\vec{k})\bar{q}_{s}{{}^{\dagger}}(-\vec{k})]e^{i\vec{k}\cdot\vec{x}}\,.$
(39)
In the product $\bar{\psi}(x)\psi(x)$ we shall retain only the
$q^{{\dagger}}\bar{q}^{{\dagger}}$ term, which yields from Eq. (36) a
Hamiltonian in a compact form,
$\displaystyle H_{I}=V_{\mu\nu}\,q^{{\dagger}}_{\mu}\bar{q}^{{\dagger}}_{\nu}$
(40)
where sum (integration) is again implied over repeated indexes. In the compact
notation, the quark and antiquark momentum, spin, flavor and color are written
as $\mu=(\vec{p}_{\mu},s_{\mu},f_{\mu},c_{\mu})$;
$\nu=(\vec{p}_{\nu},s_{\nu},f_{\nu},c_{\nu})$, while the pair creation
potential $V_{\mu\nu}$ is given by
$\displaystyle V_{\mu\nu}\equiv
2\,m_{q}\,\gamma\,\,\delta(\vec{p}_{\mu}+\vec{p}_{\nu})\,\bar{u}_{s_{\mu}f_{\mu}c_{\mu}}(\vec{p}_{\mu})\,v_{s_{\nu}f_{\nu}c_{\nu}}(\vec{p}_{\nu}).$
(41)
It should be noted that since Eq. (36) is meant to be taken in the
nonrelativistic limit, Eq. (41) should be as well. In the meson decay
calculations, of the next section, this limit is considered.
Applying the Fock-Tani transformation to $H_{I}$ one obtains the effective
Hamiltonian
$\displaystyle{\cal H}_{FT}=U^{-1}\,\,H_{I}\,\,U.$ (42)
The physical quantities in the FTf appear in a second quantization notation.
The effective decay amplitude will be a product of the ideal meson operators
with the following structure in the ideal meson sector:
$m^{{\dagger}}m^{{\dagger}}m$. To obtain this product corresponds to expand,
in powers of the wave function, up to third order. A Hamiltonian that
describes this decay process, which we shall call $H_{m}$, can be extracted
from the mapping (42) by the following products
$\displaystyle
H_{m}=V_{\mu\nu}\,q^{{\dagger}(3)}_{\mu}\bar{q}^{{\dagger}(0)}_{\nu}+V_{\mu\nu}\,q^{{\dagger}(1)}_{\mu}\bar{q}^{{\dagger}(2)}_{\nu}.$
(43)
After the substitution of Eqs. (20), (21), (59) and (61) into (43) results in
the effective meson decay Hamiltonian
$\displaystyle
H_{m}=f^{\mu\nu}(\alpha,\beta,\gamma)\,V_{\mu\nu}\,m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}m_{\gamma}$
(44)
where
$\displaystyle
f^{\mu\nu}(\alpha,\beta,\gamma)=-\Phi^{\ast\mu\tau}_{\alpha}\Phi^{\ast\rho\nu}_{\beta}\Phi^{\rho\tau}_{\gamma}.$
(45)
In the ideal meson space the new initial and final states involve only ideal
meson operators $|A\rangle=m^{{\dagger}}_{\gamma}|0\rangle$ and
$|BC\rangle=m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}|0\rangle$. The
${}^{3}P_{0}$ amplitude is obtained in the FTf by an expression equivalent to
Eq. (37),
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!!\\!\\!\langle
BC|H_{I}|A\rangle$ $\displaystyle=$ $\displaystyle\langle
0|m_{\alpha}m_{\beta}\,H_{m}\,m^{{\dagger}}_{\gamma}|0\rangle$ (46)
$\displaystyle=$ $\displaystyle
f^{\mu\nu}(\alpha,\beta,\gamma)V_{\mu\nu}+f^{\mu\nu}(\beta,\alpha,\gamma)V_{\mu\nu}.$
The term $f^{\mu\nu}(\beta,\alpha,\gamma)$ of (46) is shown in Fig. (1a), the
term $f^{\mu\nu}(\alpha,\beta,\gamma)$ corresponds to the same diagram with
$\alpha\leftrightarrow\beta$.
Figure 1: Diagrams representing the C${}^{3}P_{0}$ model. Diagram (a)
corresponds to the ${}^{3}P_{0}$ amplitude. Diagrams (b), (c) and (d) are the
bound state corrections. The complete $h_{fi}$ amplitude includes the diagrams
above plus diagrams with $\alpha\leftrightarrow\beta$.
In the FTf perspective a new aspect is introduced to meson decay: bound-state
corrections. The lowest order correction is one that involves only one bound-
state kernel $\Delta(\mu\nu;\sigma\rho)$. This implies that the Hamiltonian
representing this correction must be of fifth order in the power expansion of
the wave function.
We shall call this new Hamiltonian, with the same basic operatorial structure
$m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}m_{\gamma}$, of $\delta H_{m}$.
The only combinations $q^{{\dagger}\,(i)}\bar{q}^{{\dagger}\,(j)}$ that
results in a fifth order Hamiltonian are
$\displaystyle\delta H_{m}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left[q^{{\dagger}(3)}_{\mu}\bar{q}^{{\dagger}(2)}_{\nu}+q^{{\dagger}(1)}_{\mu}\bar{q}^{{\dagger}(4)}_{\nu}+q^{{\dagger}(5)}_{\mu}\bar{q}^{{\dagger}(0)}_{\nu}\right]V_{\mu\nu}.$
(47)
Details of this calculation is found in the appendix B. The bound-state
corrected ${}^{3}P_{0}$ Hamiltonian, which shall be called the $C^{\,3}P_{0}$
Hamiltonian, is
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H^{\rm C3P0}$
$\displaystyle=$ $\displaystyle H_{m}+\delta H_{m}$ (48) $\displaystyle=$
$\displaystyle-\Phi^{\ast\rho\xi}_{\alpha}\Phi^{\ast\lambda\tau}_{\beta}\Phi^{\omega\sigma}_{\gamma}\,V^{\rm
C3P0}\,m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}m_{\gamma}$
where $V^{\rm C3P0}$ is a condensed notation for
$\displaystyle V^{\rm C3P0}$ $\displaystyle=$
$\displaystyle\left[\delta_{\mu\lambda}\delta_{\xi\nu}\delta_{\omega\rho}\delta_{\sigma\tau}-\frac{1}{2}\delta_{\sigma\xi}\,\delta_{\lambda\omega}\,\,\Delta(\rho\tau;\mu\nu)\right.$
(49)
$\displaystyle+\frac{1}{4}\delta_{\sigma\xi}\,\delta_{\mu\lambda}\,\,\Delta(\rho\tau;\omega\nu)$
$\displaystyle+\left.\frac{1}{4}\delta_{\xi\nu}\,\delta_{\lambda\omega}\,\,\Delta(\rho\tau;\mu\sigma)\right]V_{\mu\nu}.$
## IV Light Meson Decay Examples
The light meson sector is an interesting test ground where the effects of the
bound-state correction can be compared to the usual ${}^{3}P_{0}$ model. In
particular, as examples, two specific decay processes will be studied:
$b_{1}\rightarrow\omega\pi$ and $a_{1}\rightarrow\rho\pi$. The wave function
and details of the matrix elements are found in the appendix C. The general
decay amplitude can be written as
$\displaystyle h_{fi}^{\rm C3P0}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\,{\cal M}_{fi}^{\rm
C3P0}.$ (50)
For the first decay process, $b_{1}\rightarrow\omega\pi$, results in a decay
amplitude given by
$\displaystyle{\cal M}_{fi}^{\rm C3P0}$ $\displaystyle=$
$\displaystyle{\mathcal{C}}_{01}\,Y_{00}\left(\Omega_{x}\right)+{\mathcal{C}}_{21}\,Y_{20}\left(\Omega_{x}\right)\,,$
(51)
with
$\displaystyle{\mathcal{C}}_{01}$ $\displaystyle\equiv$
$\displaystyle-\frac{2^{4}}{3^{5/2}}\left[1-\frac{2}{9}x^{2}\right]e_{1}(x)+\frac{2^{5}}{7^{5/2}3}\left[1-\frac{8}{21}x^{2}\right]e_{2}(x)$
$\displaystyle{\mathcal{C}}_{21}$ $\displaystyle\equiv$
$\displaystyle-\,x^{2}\left[\frac{2^{11/2}}{3^{9/2}}\,\,e_{1}(x)-\frac{2^{17/2}}{7^{7/2}9}\,\,e_{2}(x)\right]$
(52)
where $x=P/\beta$ and
$\displaystyle
e_{1}(x)=\exp\left(-\frac{x^{2}}{12}\right)\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,e_{2}(x)=\exp\left(-\frac{9x^{2}}{28}\right)\,.$
(53)
The decay rate has a straightforward evaluation, by substituting (51), (52) in
(50) and then in (38) obtaining
$\Gamma_{b_{1}\rightarrow\omega\pi}=2\sqrt{\pi}\,x\,\frac{E_{\omega}E_{\pi}}{M_{b_{1}}}\,{\gamma}^{2}\,\left({\mathcal{C}}_{01}^{2}+{\mathcal{C}}_{21}^{2}\right).$
(54)
The second decay process, $a_{1}\rightarrow\rho\pi$, is similar to the former
one and results in the following amplitude
$\displaystyle{\cal M}_{fi}^{\rm C3P0}$ $\displaystyle=$
$\displaystyle{\mathcal{C}}_{01}\,Y_{00}\left(\Omega_{x}\right)+{\mathcal{C}}_{21}\,Y_{20}\left(\Omega_{x}\right)\,,$
(55)
with
$\displaystyle{\mathcal{C}}_{01}$ $\displaystyle\equiv$
$\displaystyle\frac{2^{9/2}}{3^{5/2}}\left[1-\frac{2}{9}x^{2}\right]\,e_{1}(x)-\frac{2^{11/2}}{7^{5/2}\,3}\left[1-\frac{8}{21}x^{2}\right]\,e_{2}(x)$
$\displaystyle{\mathcal{C}}_{21}$ $\displaystyle\equiv$
$\displaystyle-\,x^{2}\left[\frac{2^{5}}{3^{9/2}}\,e_{1}(x)-\frac{2^{7}\,5}{3^{2}\,7^{7/2}}\,e_{2}(x)\right]$
(56)
and by a similar procedure one obtains
$\Gamma_{a_{1}\rightarrow\rho\pi}=2\sqrt{\pi}\,x\,\frac{E_{\rho}E_{\pi}}{M_{a_{1}}}\,{\gamma}^{2}\,\left({\mathcal{C}}_{01}^{2}+{\mathcal{C}}_{21}^{2}\right).$
(57)
In the former equations, $\,e_{2}(x)=0$, recovers the original ${}^{3}P_{0}$
results.
In addition to the decay widths $\Gamma$, $b_{1}$ and $a_{1}$ mesons have
$D/S$ ratios, which give a sensitive test for decay models. By definition,
these quantities are obtained from the ratios of ${\cal C}_{21}$ and ${\cal
C}_{01}$ coefficients, in equations (52) and (56).
$\displaystyle{D\over S}\bigg{|}_{a_{1}\to\rho\pi}$ $\displaystyle=$
$\displaystyle\frac{-\,x^{2}\left[\frac{2^{1/2}}{3^{2}}e_{1}(x)-\frac{3^{1/2}2^{5/2}\,5}{7^{7/2}}e_{2}(x)\right]}{\left[1-\frac{2}{9}x^{2}\right]e_{1}(x)-\frac{3^{3/2}\,2}{7^{5/2}}\left[1-\frac{8}{21}x^{2}\right]e_{2}(x)}$
$\displaystyle{D\over S}\bigg{|}_{b_{1}\to\omega\pi}$ $\displaystyle=$
$\displaystyle\frac{x^{2}\left[\frac{2^{3/2}}{3^{2}}e_{1}(x)-\frac{2^{9/2}3^{1/2}}{7^{7/2}}e_{2}(x)\right]}{\left[1-\frac{2}{9}x^{2}\right]e_{1}(x)-\frac{3^{3/2}\,2}{7^{5/2}}\left[1-\frac{8}{21}x^{2}\right]e_{2}(x)}\,.$
The meson masses assumed in the numerical calculation were $M_{\pi}=138$ MeV;
$M_{\rho}=775$ MeV; $M_{a_{1}}=1230$ MeV; $M_{b_{1}}=1229$ MeV;
$M_{\omega}=782$ MeV PDG .
The choice of SHO wave functions allow exact evaluations of the decay
amplitudes even in the corrected model. A first new aspect that appears is the
presence of a new dependence in the exponential of the corrected term. This
implies in a different range for the bound-state correction due to the fact
that $e_{2}(x)/e_{1}(x)\rightarrow 0$ as $x\rightarrow\infty$.
The correction introduces the bound-state kernel, Eq. (23), to the calculation
of the decay processes. A general sum over the meson index $\alpha$ is present
and as stated before, this index represents the quantum numbers of space, spin
and isospin. The OZI-allowed decays represent, flavor conserved continuous
(anti)quark lines. A direct consequence of this fact is the possibility to sum
over a larger set of mesons in the $\alpha$ index. In our calculation the sum
was restricted only to the final state mesons. In the
$b_{1}^{+}\rightarrow\omega\pi^{+}$ decay, there are two bound-state kernel
contributions one associated to $\omega$ meson and the other to $\pi^{+}$.
Similarly, the $a_{1}^{+}\rightarrow\rho^{+}\pi^{0}$ decay has two bound-state
kernel contributions one associated to $\rho^{+}$ meson and the other to the
$\pi^{0}$.
In this example, the parameters were chosen in order to give a closer fit to
the experimental data. In the $b_{1}$ decay, width and partial waves are known
with accuracy. The ${}^{3}P_{0}$ model’s optimum fit for the $b_{1}$ data
($\Gamma$ and $D/S$ ratio) is achieved with $\gamma=0.506$ and $\beta=0.397$
GeV. In the C${}^{3}P_{0}$ model a similar fit is obtained with $\gamma=0.539$
and $\beta=0.396$ GeV. These parameters are used in the two models to describe
the $a_{1}$ decay. The results for $\Gamma$ as a function of $\beta$ appear in
figure 2 and specific values are presented in table 1. In figure 3, the $D/S$
ratios for the two models are plotted.
Table 1: Decay rates ${}^{3}P_{0}$ ($\gamma=0.506$ e $\beta=0.397$ GeV ) and $C^{3}P_{0}$ ($\gamma=0.539$ e $\beta=0.396$ GeV ) | | | $\Gamma$ (MeV) | | | | | D/S | |
---|---|---|---|---|---|---|---|---|---|---
Decay | | Exp PDG | ${}^{3}P_{0}$ | C${}^{3}P_{0}$ | | Exp PDG | | ${}^{3}P_{0}$ | | C${}^{3}P_{0}$
$b_{1}\rightarrow\omega\pi$ | | 142 | 143 | 142 | | $0.277(27)$ | | $0.288$ | | $0.273$
$a_{1}\rightarrow\rho\pi$ | | 250 to 600 | 543 | 543 | | $-0.108(16)$ | | $-0.149$ | | $-0.113$
Figure 2: Decay rates for $b_{1}\rightarrow\omega\pi$ and
$a_{1}\rightarrow\rho\pi$ decays, for ${}^{3}P_{0}$ ($\gamma=0.506$) and
C${}^{3}P_{0}$ ($\gamma=0.539$) models . Figure 3: $D/S$ ratios in
$b_{1}\rightarrow\omega\pi$ and $a_{1}\rightarrow\rho\pi$ decays, for
${}^{3}P_{0}$ and C${}^{3}P_{0}$ models.
## V Summary and Conclusions
In this paper we have presented an alternative approach for meson decay which
consists in a mapping technique, known as the Fock-Tani formalism, long used
in atomic physics. This formalism has been applied to hadron-hadron scattering
interactions with constituent interchange. The challenge, resided in extending
the approach to include meson decay. After demonstrating that in lower order
the result obtained was equivalent to the ${}^{3}P_{0}$ model, an additional
feature pointed out was the appearance of bound-state corrections in the
effective decay Hamiltonian. These corrections present a natural modification
in the $q\bar{q}$ interaction strength. As an example, we studied two decay
processes $b_{1}\rightarrow\omega\pi$ and $a_{1}\rightarrow\rho\pi$. The $D/S$
ratios, in Fig. (3), show that a common range of $\beta$ values for mesons is
obtained. In a new calculation with the inclusion of other decay processes it
might require different $\beta$ values faessler . The corrected model presents
an interesting feature that for these two mesons the decay width differs
slightly when compared with the ${}^{3}P_{0}$, but $D/S$ ratios are improved.
The examples studied here are encouraging but a more extensive survey of the
light meson sector would be a necessary next step. The inclusion of the full
meson octet, in the evaluation of the bound-state correction, may provide a
fine tuning for the model.
###### Acknowledgements.
The authors would like to thank H. Stöcker, J. Aichelin and W. Greiner for
important and enlightening discussions. This research was supported by
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq),
Universidade Federal do Rio Grande do Sul (UFRGS) and Universidade Federal de
Pelotas (UFPel).
## Appendix A Second and third order operators
The second order operators
$\displaystyle q^{(2)}_{\mu}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi^{\ast\mu_{2}\nu_{1}}_{\alpha}\Phi^{\mu\nu_{1}}_{\beta}\,M_{\alpha\beta}\,q_{\mu_{2}}$
$\displaystyle\bar{q}^{(2)}_{\nu}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi^{\ast\mu_{1}\nu_{2}}_{\alpha}\Phi^{\mu_{1}\nu}_{\beta}\,M_{\alpha\beta}\,\bar{q}_{\nu_{2}}\;,$
(59)
where
$\displaystyle M_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
m^{{\dagger}}_{\alpha}M_{\beta}\sin t\cos
t-m^{{\dagger}}_{\alpha}m_{\beta}\sin^{2}t$ (60)
$\displaystyle-M^{{\dagger}}_{\alpha}M_{\beta}(2-2\cos t-\sin^{2}t)$
$\displaystyle-M^{{\dagger}}_{\alpha}m_{\beta}\left(2\sin t-\sin t\cos
t\right).$
The third order operators are
$\displaystyle q^{(3)}_{\mu}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\mu\sigma}_{\beta}\Phi^{\rho\sigma_{1}}_{\gamma}\bar{q}^{{\dagger}}_{\sigma_{1}}\,M_{\alpha\beta\gamma}\,$
$\displaystyle-\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\mu\nu_{1}}_{\alpha}\Phi^{\rho\sigma_{1}}_{\beta}\bar{q}^{{\dagger}}_{\nu_{1}}\bar{q}^{{\dagger}}_{\sigma_{1}}\bar{q}_{\sigma}\,\overline{M}_{\beta}$
$\displaystyle-\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\mu\nu_{1}}_{\alpha}\Phi^{\rho_{1}\sigma}_{\beta}\bar{q}^{{\dagger}}_{\nu_{1}}q^{{\dagger}}_{\rho_{1}}q_{\rho}\overline{M}_{\beta}$
$\displaystyle\bar{q}^{(3)}_{\nu}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\rho\nu}_{\beta}\Phi^{\rho_{1}\sigma}_{\gamma}q^{{\dagger}}_{\rho_{1}}M_{\alpha\beta\gamma}\,$
(61)
$\displaystyle+\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\mu_{1}\nu}_{\alpha}\Phi^{\rho\sigma_{1}}_{\beta}q^{{\dagger}}_{\mu_{1}}\bar{q}^{{\dagger}}_{\sigma_{1}}\bar{q}_{\sigma}\,\overline{M}_{\beta}$
$\displaystyle+\frac{1}{2}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\mu_{1}\nu}_{\alpha}\Phi^{\rho_{1}\sigma}_{\beta}q^{{\dagger}}_{\mu_{1}}q^{{\dagger}}_{\rho_{1}}q_{\rho}\,\overline{M}_{\beta}\,,$
where
$\displaystyle M_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle
m^{{\dagger}}_{\alpha}m_{\beta}m_{\gamma}\sin^{3}t+M^{{\dagger}}_{\alpha}M_{\beta}m_{\gamma}\left(\sin
t-\sin^{3}t\right)$
$\displaystyle+M^{{\dagger}}_{\alpha}m_{\beta}M_{\gamma}\left(2\sin t-\sin
t\cos t-\sin^{3}t\right)$
$\displaystyle+\left(M^{{\dagger}}_{\alpha}m_{\beta}m_{\gamma}+m^{{\dagger}}_{\alpha}M_{\beta}m_{\gamma}\right)\left(-\cos
t+\cos^{3}t\right)$
$\displaystyle+m^{{\dagger}}_{\alpha}m_{\beta}M_{\gamma}\left(-\cos
t+\cos^{3}t+\sin^{2}t\right)$
$\displaystyle+M^{{\dagger}}_{\alpha}M_{\beta}M_{\gamma}\left(2-\cos
t-\cos^{3}t-\sin^{2}t\right)$
$\displaystyle+m^{{\dagger}}_{\alpha}M_{\beta}M_{\gamma}\left(\sin t-\sin
t\cos t-\sin^{3}t\right)$ $\displaystyle\overline{M}_{\beta}$ $\displaystyle=$
$\displaystyle 2M_{\beta}\left(1-\cos t\right)+m_{\beta}\sin t\,.$ (62)
## Appendix B The $\delta H_{m}$ Hamiltonian
The $\delta H_{m}$ Hamiltonian is evaluated from Eq. (47). The
$q^{{\dagger}(3)}_{\mu}\bar{q}^{{\dagger}(2)}_{\nu}$ combination can be
obtained from (59) and (61)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta H_{1}$
$\displaystyle=$ $\displaystyle
q^{{\dagger}(3)}_{\mu}\bar{q}^{{\dagger}(2)}_{\nu}\,V_{\mu\nu}=\delta
f^{\mu\nu}_{1}(\alpha,\beta,\gamma)\,V_{\mu\nu}\,m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}m_{\gamma}$
(63)
with
$\displaystyle\delta
f^{\mu\nu}_{1}(\alpha,\beta,\gamma)=\frac{1}{4}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\ast\mu\tau}_{\beta}\Delta(\rho\tau;\lambda\nu)\Phi^{\lambda\sigma}_{\gamma}.$
(64)
The $q^{{\dagger}(1)}_{\mu}\bar{q}^{{\dagger}(4)}_{\nu}$ combination has an
important feature: a contribution from a higher order operator. A new
generator $\tilde{M}_{\alpha}$ has to be evaluated, with the inclusion of the
following fourth order term
$\displaystyle\tilde{M}_{\alpha}^{\left(4\right)}$ $\displaystyle=$
$\displaystyle\frac{3}{8}\Delta_{\alpha\beta}\Delta_{\beta\gamma}M_{\gamma}-\frac{1}{8}M_{\beta}^{\dagger}\left[\Delta_{\alpha\gamma},\,\Delta_{\beta\delta}\right]M_{\delta}M_{\gamma}$
(65)
$\displaystyle-\frac{1}{4}M_{\beta}^{\dagger}\left[M_{\alpha},T^{{\dagger}}_{\delta\gamma\beta}\right]M_{\gamma}M_{\delta}.$
The only relevant term in the $\bar{q}_{\nu}^{\dagger(4)}$ for meson decay is
$\displaystyle\bar{q}_{\nu}^{(4)}(t)$ $\displaystyle\approx$
$\displaystyle-\frac{1}{8}\Phi_{\alpha}^{\ast\sigma\eta}\Delta\left(\sigma\nu;\rho\tau\right)\Phi_{\beta}^{\rho\eta}M_{\alpha}^{\dagger\left(0\right)}(t)\bar{q}_{\tau}M_{\beta}^{\left(0\right)}(t).$
The resulting contribution is then
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta H_{2}$
$\displaystyle=$ $\displaystyle
q^{{\dagger}(1)}_{\mu}\bar{q}^{{\dagger}(4)}_{\nu}\,V_{\mu\nu}=\delta
f^{\mu\nu}_{2}(\alpha,\beta,\gamma)V_{\mu\nu}\,m^{{\dagger}}_{\alpha}m^{{\dagger}}_{\beta}m_{\gamma}$
(67)
where
$\displaystyle\delta
f^{\mu\nu}_{2}(\alpha,\beta,\gamma)=-\frac{1}{8}\Phi^{\ast\rho\sigma}_{\alpha}\Phi^{\ast\mu\tau}_{\beta}\Delta(\rho\tau;\lambda\nu)\Phi^{\lambda\sigma}_{\gamma}.$
(68)
The $q^{{\dagger}(5)}_{\mu}\bar{q}^{{\dagger}(0)}_{\nu}$ combination implies
in a fifth order generator to obtain the complete set of equations of motion
(17) and (19)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\tilde{M}_{\alpha}^{\dagger\left(5\right)}$
$\displaystyle=$ $\displaystyle-
M_{\beta}^{\dagger}Z_{\alpha\gamma\beta}^{\dagger}M_{\gamma}+\frac{1}{8}M_{\omega}^{\dagger}M_{\beta}^{\dagger}W_{\alpha\omega\beta\gamma\delta}^{\dagger}M_{\delta}M_{\gamma}$
(69)
where
$\displaystyle Z_{\alpha\gamma\beta}^{\dagger}$ $\displaystyle=$
$\displaystyle-\frac{3}{8}T^{{\dagger}}_{\alpha\delta\beta}\Delta_{\delta\gamma}-\frac{5}{8}\Delta_{\beta\delta}T^{{\dagger}}_{\alpha\gamma\delta}-\frac{1}{4}T^{{\dagger}}_{\delta\gamma\beta}\Delta_{\delta\alpha}$
$\displaystyle W_{\alpha\omega\beta\gamma\delta}^{\dagger}$ $\displaystyle=$
$\displaystyle\left[M_{\alpha}^{\dagger},\,Q_{\omega\beta\gamma\delta}\right]-\left[\Delta_{\omega\gamma},\,T^{{\dagger}}_{\alpha\delta\beta}\right]$
$\displaystyle Q_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\Delta_{\alpha\gamma},\,\Delta_{\beta\delta}\right]-\left[M_{\alpha},T^{{\dagger}}_{\delta\gamma\beta}\right].$
(70)
The only relevant terms in the $q_{\mu}^{\dagger(5)}$ for meson decay are
$\displaystyle q_{\mu}^{\dagger(5)}(t)$ $\displaystyle\approx$
$\displaystyle\left[\frac{1}{2}\Delta\left(\rho\tau;\mu\omega\right)\Phi_{\alpha}^{\ast\rho\sigma}\Phi_{\beta}^{\ast\lambda\tau}\Phi_{\gamma}^{\lambda\sigma}\right.$
(71)
$\displaystyle-\frac{1}{4}\Delta\left(\rho\tau;\mu\sigma\right)\Phi_{\alpha}^{\ast\rho\omega}\Phi_{\beta}^{\ast\lambda\tau}\Phi_{\gamma}^{\lambda\sigma}$
$\displaystyle\left.-\frac{3}{8}\Delta\left(\rho\tau;\lambda\omega\right)\Phi_{\alpha}^{\ast\mu\tau}\Phi_{\beta}^{\ast\rho\sigma}\Phi_{\gamma}^{\lambda\sigma}\right]$
$\displaystyle\times
M_{\alpha}^{\dagger\left(0\right)}(t)M_{\beta}^{\dagger\left(0\right)}(t)\bar{q}_{\omega}M_{\gamma}^{\left(0\right)}(t).$
The resulting contribution is
$\displaystyle\delta H_{3}$ $\displaystyle=$ $\displaystyle\delta
f^{\mu\nu}_{3}(\alpha,\beta,\gamma)V_{\mu\nu}\,\,m_{\alpha}^{\dagger}m_{\beta}^{\dagger}m_{\gamma}$
(72)
where
$\displaystyle\delta f^{\mu\nu}_{3}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Phi_{\alpha}^{\ast\rho\sigma}\Phi_{\beta}^{\ast\lambda\tau}\Delta(\rho\tau;\mu\nu)\Phi_{\gamma}^{\lambda\sigma}$
(73)
$\displaystyle-\frac{1}{4}\Phi_{\alpha}^{\ast\sigma\tau}\Phi_{\beta}^{\ast\rho\nu}\Delta(\rho\tau;\mu\lambda)\Phi_{\gamma}^{\sigma\lambda}$
$\displaystyle-\frac{3}{8}\Phi_{\alpha}^{\ast\rho\sigma}\Phi_{\beta}^{\ast\mu\tau}\Delta(\rho\tau;\lambda\nu)\Phi_{\gamma}^{\lambda\sigma}.$
The complete $\delta H_{m}$ Hamiltonian is
$\displaystyle\delta H_{m}$ $\displaystyle=$ $\displaystyle\delta H_{1}+\delta
H_{2}+\delta H_{3}$ (74) $\displaystyle=$ $\displaystyle\delta
f^{\mu\nu}(\alpha,\beta,\gamma)V_{\mu\nu}\,\,m_{\alpha}^{\dagger}m_{\beta}^{\dagger}m_{\gamma}$
with
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta
f^{\mu\nu}(\alpha,\beta,\gamma)$ $\displaystyle=$ $\displaystyle\delta
f^{\mu\nu}_{1}+\delta f^{\mu\nu}_{2}+\delta f^{\mu\nu}_{3}$ (75)
$\displaystyle=$
$\displaystyle\,\,\,\,\,\frac{1}{2}\Phi_{\alpha}^{\ast\rho\sigma}\Phi_{\beta}^{\ast\lambda\tau}\Delta(\rho\tau;\mu\nu)\Phi_{\gamma}^{\lambda\sigma}$
$\displaystyle-\frac{1}{4}\Phi_{\alpha}^{\ast\rho\sigma}\Phi_{\beta}^{\ast\mu\tau}\Delta(\rho\tau;\lambda\nu)\Phi_{\gamma}^{\lambda\sigma}.$
$\displaystyle-\frac{1}{4}\Phi_{\alpha}^{\ast\sigma\tau}\Phi_{\beta}^{\ast\rho\nu}\Delta(\rho\tau;\mu\lambda)\Phi_{\gamma}^{\sigma\lambda}.$
## Appendix C Decay, Wave function and matrix elements
We will use the decay $b_{1}^{+}(+\hat{z})\rightarrow\omega(+\hat{z})\pi^{+}$
to illustrate the nature of our formalism and, simply quote the other case in
the text.
### C.1 The wave function
The general meson wave function can be written as
$\displaystyle\Phi_{\alpha}^{\mu\nu}=\chi_{S_{\alpha}}^{s_{1}s_{2}}f_{f_{\alpha}}^{f_{1}f_{2}}C^{c_{1}c_{2}}\Phi_{nl}^{\vec{P}_{\alpha}-\vec{p}_{1}-\vec{p}_{2}}\;,$
(76)
a direct product of the spin $\chi_{S_{\alpha}}^{s_{1}s_{2}}$ [the indexes
$s_{1}$ and $s_{2}$ are the quark (antiquark) spin projections with
$(s=1\Rightarrow\uparrow$ and $s=2\Rightarrow\downarrow)$; the index
$S_{\alpha}$ denotes the meson spin]; flavor $f_{f_{\alpha}}^{f_{1}f_{2}}$;
color $C^{c_{1}c_{2}}$ and space
$\Phi_{nl}^{\vec{P}_{\alpha}-\vec{p}_{1}-\vec{p}_{2}}$ components.
In all our calculations the color component will be given by
$\displaystyle C^{c_{1}c_{2}}=\frac{1}{\sqrt{3}}\,\,\delta^{c_{1}c_{2}}.$ (77)
The spatial part is defined as harmonic oscillator wave functions
$\displaystyle\Phi_{nl}^{\vec{P}_{\alpha}-\vec{p}_{1}-\vec{p}_{2}}=\delta(\vec{P}_{\alpha}-\vec{p}_{1}-\vec{p}_{2})\,\,\Phi_{nl}(\vec{p}_{1},\vec{p}_{2})$
(78)
where $\Phi_{nl}(\vec{p}_{i},\vec{p}_{j})$ is given by
$\displaystyle\Phi_{nl}(\vec{p}_{i},\vec{p}_{j})$ $\displaystyle=$
$\displaystyle(\frac{1}{2\beta})^{l}\,N_{nl}\,|\vec{p}_{i}-\vec{p}_{j}|^{l}\,\exp\left[-\frac{(\vec{p}_{i}-\vec{p}_{j})^{2}}{8\beta^{2}}\right]\,$
(79) $\displaystyle\times{\cal
L}_{n}^{l+\frac{1}{2}}\left[\frac{(\vec{p}_{i}-\vec{p}_{j})^{2}}{4\beta^{2}}\right]Y_{lm}(\Omega_{\vec{p}_{i}-\vec{p}_{j}})$
with $p_{i(j)}$ the internal momentum, the spherical harmonic $Y_{lm}$,
$\beta$ a scale parameter, $N_{nl}$ the normalization constant dependent on
the radial and orbital quantum numbers
$\displaystyle
N_{nl}=\left[\frac{2(n!)}{\beta^{3}\,\Gamma(n+l+3/2)}\right]^{\frac{1}{2}}.$
(80)
The Laguerre polynomials ${\cal L}_{n}^{l+\frac{1}{2}}(p)$ are defined as
$\displaystyle{\cal
L}_{n}^{l+\frac{1}{2}}(p)=\sum_{k=0}^{n}\frac{(-)^{k}\,\Gamma(n+l+3/2)^{(n-k)!}}{k!\,\Gamma(k+l+3/2)}\,\,p^{k}\,.$
(81)
In this paper two kinds of light non-strange mesons will be studied:
1. 1.
$L_{q\bar{q}}=0$
$\displaystyle\varphi(\vec{p})\equiv\Phi_{00}(\vec{p})=\frac{1}{\pi^{3/4}\beta^{3/2}}\exp\left[-\frac{p^{2}}{8\beta^{2}}\right]\,$
(82)
2. 2.
$L_{q\bar{q}}=1$
$\displaystyle\Phi_{1m}(\vec{p})=\phi(\vec{p})\,\,Y_{1m}(\Omega_{\vec{p}})$
(83)
with
$\displaystyle\phi(\vec{p})=\left[\frac{2}{3\sqrt{\pi}\beta^{5}}\right]^{\frac{1}{2}}\,\,p\,\,\exp\left[-\frac{p^{2}}{8\beta^{2}}\right].$
(84)
Returning to our example the pion, has $J=0$ and $b_{1}$ $J=1$. We choose the
$\left(+\hat{z}\right)$ direction for this calculation, so the spin wave
functions become
$\displaystyle\left|b_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\uparrow\bar{\downarrow}\right\rangle-\left|\downarrow\bar{\uparrow}\right\rangle\right)$
$\displaystyle\left|\omega\right\rangle$ $\displaystyle=$
$\displaystyle\left|\uparrow\bar{\uparrow}\,\right\rangle$
$\displaystyle\left|\pi\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\uparrow\bar{\downarrow}\right\rangle-\left|\downarrow\bar{\uparrow}\right\rangle\right)$
(85)
or in the $\chi$ notation
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\chi_{\omega}^{11}=1,\quad\,\chi_{\omega}^{12}=\chi_{\omega}^{21}=\chi_{\omega}^{22}=0$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\chi_{\pi,b_{1}}^{11}=\chi_{\pi,b_{1}}^{22}=0\,\,\,\,\,\,;\,\,\,\,\,\,\chi_{\pi,b_{1}}^{12}=-\chi_{\pi,b_{1}}^{21}=\frac{1}{\sqrt{2}}.$
(86)
The flavor component $f_{f_{\alpha}}^{f_{\mu}f_{\nu}}$ follows the same logic
as the spin part
$\displaystyle\left|b_{1}^{\,+}\,\right\rangle$ $\displaystyle=$
$\displaystyle\left|\pi^{+}\right\rangle=-\left|u\bar{d}\right\rangle$
$\displaystyle\left|\omega\,\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|u\bar{u}\right\rangle+\left|d\bar{d}\right\rangle\right)$
(87)
The $b_{1}^{+}$ and the $\pi^{+}$ mesons have the same flavor contribution
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!f_{b_{1}^{+},\pi^{+}}^{12}=-1\,;f_{b_{1}^{+},\pi^{+}}^{11}$
$\displaystyle=f_{b_{1}^{+},\pi^{+}}^{21}=f_{b_{1}^{+},\pi^{+}}^{22}=$
$\displaystyle 0\,.$ (88)
For $\omega$, one has
$\displaystyle f_{\omega}^{11}$ $\displaystyle=f_{\omega}^{22}=$
$\displaystyle\frac{1}{\sqrt{2}};\qquad f_{\omega}^{12}=f_{\omega}^{21}=0.$
(89)
### C.2 The spin matrix elements
In the evaluation of a decay amplitude, the following spin matrix element is
necessary
$\chi_{s^{\prime}}^{\ast}\left(\vec{\sigma}\cdot\vec{{P}}\right)\chi_{s}^{c}$
(90)
with
$\chi_{1}=\left(\begin{array}[]{c}1\\\
0\end{array}\right);\;\chi_{2}=\left(\begin{array}[]{c}0\\\
1\end{array}\right);\;\chi_{1}^{c}=\left(\begin{array}[]{c}0\\\
1\end{array}\right);\;\chi_{2}^{c}=\left(\begin{array}[]{c}-1\\\
0\end{array}\right).$ (91)
By direct calculation one can show
$\displaystyle\chi_{1}^{\ast}\left(\vec{\sigma}\cdot\vec{{P}}\right)\chi_{1}^{c}$
$\displaystyle=$ $\displaystyle{P}_{x}-i{P}_{y}$
$\displaystyle\chi_{1}^{\ast}\left(\vec{\sigma}\cdot\vec{{P}}\right)\chi_{2}^{c}$
$\displaystyle=$ $\displaystyle-{P}_{z}$
$\displaystyle\chi_{2}^{\ast}\left(\vec{\sigma}\cdot\vec{{P}}\right)\chi_{1}^{c}$
$\displaystyle=$ $\displaystyle-{P}_{z}$
$\displaystyle\chi_{2}^{\ast}\left(\vec{\sigma}\cdot\vec{{P}}\right)\chi_{2}^{c}$
$\displaystyle=$ $\displaystyle-({P}_{x}+i{P}_{y})\,.$ (92)
### C.3 Matrix elements: $b_{1}^{+}\rightarrow\omega\pi^{+}$ Decay
The transition considered is of the form $m_{\gamma}\rightarrow
m_{\alpha}+m_{\beta}$, where the initial state is
$\left|A\right\rangle=m_{\gamma}^{\dagger}\left|0\right\rangle$ and the final
state is given by
$\left|BC\right\rangle=m_{\alpha}^{\dagger}m_{\beta}^{\dagger}\left|0\right\rangle$.
The matrix element of the uncorrected part results in
$\displaystyle\left\langle BC\left|H_{m}\right|A\right\rangle=-d_{1}-d_{2},$
(93)
$d_{1}$ and $d_{2}$ are defined as
$\displaystyle d_{1}$ $\displaystyle\equiv$
$\displaystyle\Phi_{\alpha}^{\star\rho\nu}\Phi_{\beta}^{\star\mu\lambda}\Phi_{\gamma}^{\rho\lambda}V_{\mu\nu}$
$\displaystyle d_{2}$ $\displaystyle\equiv$
$\displaystyle\Phi_{\alpha}^{\star\mu\lambda}\Phi_{\beta}^{\star\rho\nu}\Phi_{\gamma}^{\rho\lambda}V_{\mu\nu}.$
(94)
Equations (94) can be decomposed according to the sector of the wave function
they correspond: flavor, color, spin-space:
$\displaystyle d_{1}$ $\displaystyle=$ $\displaystyle
d_{1}^{f}\,d_{1}^{c}\,d_{1}^{s-e}$ $\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle d_{2}^{f}\,d_{2}^{c}\,d_{2}^{s-e}.$ (95)
The matrix elements of the bound-state correction refer to diagrams (b), (c)
and (d) of figure (1). The bound-state kernel’s definition as
$\Phi^{\rho\tau}_{\alpha}\Phi^{\ast\lambda\nu}_{\alpha}$ implies in an
additional element, due to the contraction in the $\alpha$ index, a sum over
species requirement girar1 . A question that naturally arises is, which states
to include in this sum? We shall adopt in our calculation a restrictive
choice: include in the sum only the particles that are present in the final
state. For the $b_{1}^{+}$ decay, $\Delta\left(\rho\tau;\lambda\nu\right)$
will have two contributions: $\omega$ and $\pi^{+}$. Similarly, the
$a_{1}^{+}$ decay shall be corrected by the final state mesons $\rho^{+}$ and
$\pi^{0}$.
Due to the parity assignment of the spatial part, the integration of diagram
(1b) is zero. Spatial symmetry also implies that the matrix elements of
diagrams (1c) and (1d) are equal. This simplifies our calculation, reducing
the problem to the evaluation of diagram (1c) only. The bound-state correction
(bsc) matrix element reduces to evaluate the following expression
$\displaystyle\left\langle BC\left|\delta
H_{m}\right|A\right\rangle=-d_{1}^{\rm bsc}-d_{2}^{\rm bsc}$ (96)
where
$\displaystyle d_{1}^{\rm bsc}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(d_{1\omega}+d_{1\pi}\right)$ $\displaystyle
d_{2}^{\rm bsc}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(d_{2\omega}+d_{2\pi}\right)$ (97)
with
$\displaystyle d_{1j}$ $\displaystyle=$
$\displaystyle\Phi_{\alpha}^{\star\rho\sigma}\Phi_{\beta}^{\star\mu\tau}\Delta_{j}(\rho\tau;\lambda\nu)\Phi_{\gamma}^{\lambda\sigma}V_{\mu\nu}\;\equiv\;d_{1j}^{f}d_{1j}^{c}d_{1j}^{s-e}$
$\displaystyle d_{2j}$ $\displaystyle=$
$\displaystyle\Phi_{\alpha}^{\star\mu\tau}\Phi_{\beta}^{\star\rho\sigma}\Delta_{j}(\rho\tau;\lambda\nu)\Phi_{\gamma}^{\lambda\sigma}V_{\mu\nu}\;\equiv\;d_{2j}^{f}d_{2j}^{c}d_{2j}^{s-e}.$
In (LABEL:4.3-5d) $j$ refers to mesons $\omega$ and $\pi^{+}$.
### C.4 $b_{1}^{+}\rightarrow\omega\pi^{+}$ Decay (uncorrected)
$\bullet$ Flavor:
$\displaystyle d_{1}^{\,f}$ $\displaystyle=$ $\displaystyle
d_{2}^{\,f}=f_{\pi}^{\,f_{\rho}f_{\nu}}f_{\omega}^{\\!f_{\mu}f_{\lambda}}f_{b_{1}}^{\\!f_{\rho}f_{\lambda}}\delta_{f_{\mu}f_{\nu}}=\frac{1}{\sqrt{2}}.$
(99)
$\bullet$ Color:
$\displaystyle\\!\\!\\!\\!\\!\\!d_{1}^{c}=d_{2}^{c}=\frac{1}{3\sqrt{3}}\delta^{\,c_{\rho}c_{\nu}}\delta^{c_{\mu}c_{\lambda}}\delta^{c_{\rho}c_{\lambda}}\delta^{c_{\mu}c_{\nu}}=\frac{1}{\sqrt{3}}.$
(100)
$\bullet$ Spin-space:
The spin matrix element is
$\displaystyle d_{1}^{\,s}$ $\displaystyle=$ $\displaystyle
d_{2}^{\,s}=\chi_{\pi}^{\,s_{\rho}s_{\nu}}\chi_{\omega}^{\\!s_{\mu}s_{\lambda}}\chi_{b_{1}}^{\\!s_{\rho}s_{\lambda}}V_{s_{\mu}s_{\nu}}^{s-e}=\frac{1}{2}\,V_{11}^{s-e}\left(\vec{p_{\mu}},\vec{p_{\nu}}\right)$
where
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!V_{11}^{s-e}\left(\vec{p_{\mu}},\vec{p_{\nu}}\right)=-\gamma\,\delta(\vec{p_{\mu}}+\vec{p_{\nu}})\chi^{\ast}_{1}\,[\vec{\sigma}\cdot(\vec{p_{\mu}}-\vec{p_{\nu}})]\,\chi^{C}_{1}.$
(102)
Using (92) to evaluate (102) and after integrating momentum conservation
deltas one arrives in
$\displaystyle d_{1}^{s-e}$ $\displaystyle=$ $\displaystyle-\gamma\,\,\int
d^{3}{K}\>\left({K}_{x}-i{K}_{y}\right)\varphi\left({\vec{P}}-2\vec{{K}}\right)$
$\displaystyle\times$
$\displaystyle\phi\left(2{\vec{P}}-2\vec{{K}}\right)Y_{11}\left(\Omega_{2{\vec{P}}\hat{-}2\vec{{K}}}\right)\,\varphi\left({\vec{P}}-2\vec{{K}}\right)\,.$
Introducing the spatial wave function and integrating
$\displaystyle d_{1}^{\,s-e}$ $\displaystyle=$
$\displaystyle\left(\frac{2^{7/2}}{3^{5/2}}\right)\,\left(\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\right)\left\\{\left[1-\frac{2}{9}x^{2}\right]Y_{00}\left(\Omega_{x}\right)\right.$
(104)
$\displaystyle\left.+\frac{2}{3^{2}\sqrt{5}}x^{2}\,Y_{20}\left(\Omega_{x}\right)\right\\}e_{1}(x).$
$d_{2}^{\,s-e}$ is obtained from $d_{1}^{\,s-e}$ by $\vec{P}\to-\vec{P}$. The
decay amplitude results
$\displaystyle h_{fi}$ $\displaystyle=$
$\displaystyle-\left(\frac{2^{4}}{3^{5/2}}\right)\,\left(\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\right)\left\\{\left[1-\frac{2}{9}x^{2}\right]Y_{00}\left(\Omega_{x}\right)\right.$
(105)
$\displaystyle\left.+\frac{2}{3^{2}\sqrt{5}}x^{2}\,Y_{20}\left(\Omega_{x}\right)\right\\}e_{1}(x).$
### C.5 $b_{1}^{+}\rightarrow\omega\pi^{+}$ Decay (bound-state corrected)
The quantities between $[\ldots]$ in the following expressions are related to
the bound-state kernel.
$\bullet$ Flavor:
$\displaystyle d_{1\omega}^{\,f}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{\,f_{\rho}f_{\sigma}}f_{\omega}^{\\!f_{\mu}f_{\tau}}\left[f_{\omega}^{\,f_{\rho}f_{\tau}}f_{\omega}^{\\!f_{\lambda}f_{\mu}}\right]f_{b_{1}}^{\\!f_{\lambda}f_{\sigma}}=\frac{1}{2\sqrt{2}}$
$\displaystyle d_{1\pi}^{\,f}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{\,f_{\rho}f_{\sigma}}f_{\omega}^{\\!f_{\mu}f_{\tau}}\left[f_{\pi}^{\,f_{\rho}f_{\tau}}f_{\pi}^{\\!f_{\lambda}f_{\mu}}\right]f_{b_{1}}^{\\!f_{\lambda}f_{\sigma}}=\frac{1}{\sqrt{2}}$
$\displaystyle d_{2\omega}^{\,f}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{\\!f_{\mu}f_{\tau}}f_{\omega}^{\,f_{\rho}f_{\sigma}}\left[f_{\omega}^{\,f_{\rho}f_{\tau}}f_{\omega}^{\\!f_{\lambda}f_{\mu}}\right]f_{b_{1}}^{\\!f_{\lambda}f_{\sigma}}=\frac{1}{2\sqrt{2}}$
$\displaystyle d_{2\pi}^{\,f}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{\\!f_{\mu}f_{\tau}}f_{\omega}^{\,f_{\rho}f_{\sigma}}\left[f_{\pi}^{\,f_{\rho}f_{\tau}}f_{\pi}^{\\!f_{\lambda}f_{\mu}}\right]f_{b_{1}}^{\\!f_{\lambda}f_{\sigma}}=0.$
(106)
$\bullet$ Color:
$\displaystyle d_{1\omega}^{\,c}$ $\displaystyle=$ $\displaystyle
d_{1\pi}^{\,c}=d_{2\omega}^{\,c}=d_{2\pi}^{\,c}$ (107) $\displaystyle=$
$\displaystyle\frac{1}{9\sqrt{3}}\delta^{\,c_{\rho}c_{\sigma}}\delta^{c_{\mu}c_{\tau}}[\delta^{c_{\rho}c_{\tau}}\delta^{c_{\lambda}c_{\nu}}]\delta^{c_{\lambda}c_{\sigma}}\delta^{c_{\mu}c_{\nu}}$
$\displaystyle=$ $\displaystyle\frac{1}{3\sqrt{3}}.$
$\bullet$ Spin-space:
The spin matrix element is
$\displaystyle d_{1\omega}^{\,s}$ $\displaystyle=$
$\displaystyle\chi_{\pi}^{\,s_{\rho}s_{\sigma}}\chi_{\omega}^{\\!s_{\mu}s_{\tau}}\left[\chi_{\omega}^{\,s_{\rho}s_{\tau}}\chi_{\omega}^{\\!s_{\lambda}s_{\nu}}\right]\chi_{b_{1}}^{\\!s_{\lambda}s_{\sigma}}V_{s_{\mu}s_{\nu}}^{s-e}$
$\displaystyle=$
$\displaystyle\frac{1}{2}V_{11}^{s-e}\left(\vec{p_{\mu}},\vec{p_{\nu}}\right)$
$\displaystyle d_{1\pi}^{\,s}$ $\displaystyle=$
$\displaystyle\chi_{\pi}^{\,s_{\rho}s_{\sigma}}\chi_{\omega}^{\\!s_{\mu}s_{\tau}}\left[\chi_{\pi}^{\,s_{\rho}s_{\tau}}\chi_{\pi}^{\\!s_{\lambda}s_{\nu}}\right]\chi_{b_{1}}^{\\!s_{\lambda}s_{\sigma}}V_{s_{\mu}s_{\nu}}^{s-e}$
$\displaystyle=$
$\displaystyle\frac{1}{4}V_{11}^{s-e}\left(\vec{p_{\mu}},\vec{p_{\nu}}\right)$
$\displaystyle d_{2\omega}^{\,s}$ $\displaystyle=$
$\displaystyle\chi_{\pi}^{\\!s_{\mu}s_{\tau}}\chi_{\omega}^{\,s_{\rho}s_{\sigma}}\left[\chi_{\omega}^{\,s_{\rho}s_{\tau}}\chi_{\omega}^{\\!s_{\lambda}s_{\nu}}\right]\chi_{b_{1}}^{\\!s_{\lambda}s_{\sigma}}V_{s_{\mu}s_{\nu}}^{s-e}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle d_{2\pi}^{\,s}$
$\displaystyle=$
$\displaystyle\chi_{\pi}^{\\!s_{\mu}s_{\tau}}\chi_{\omega}^{\,s_{\rho}s_{\sigma}}\left[\chi_{\pi}^{\,s_{\rho}s_{\tau}}\chi_{\pi}^{\\!s_{\lambda}s_{\nu}}\right]\chi_{b_{1}}^{\\!s_{\lambda}s_{\sigma}}V_{s_{\mu}s_{\nu}}^{s-e}$
(108) $\displaystyle=$
$\displaystyle\frac{1}{4}V_{11}^{s-e}\left(\vec{p_{\mu}},\vec{p_{\nu}}\right).$
Due to symmetries in the spatial part the following relations are true
$\displaystyle d_{1\pi}^{\,s-e}$ $\displaystyle=$ $\displaystyle
d_{2\pi}^{\,s-e}=\frac{1}{2}\,d_{1\omega}^{\,s-e};\qquad
d_{2\omega}^{\,s-e}=0\,,$ (109)
where $d_{1\pi}^{\,s-e}$ is given by
$\displaystyle d_{1\pi}^{s-e}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}\int d^{3}K\,d^{3}q\,(K_{x}-i\,K_{y})$
$\displaystyle\\!\\!\\!\times\varphi\left(2\vec{q}-\vec{{P}}\right)\varphi\left(2\vec{K}-\vec{{P}}\right)$
$\displaystyle\\!\\!\\!\times\left[\varphi\left(\vec{q}+\vec{K}-2\vec{{P}}\right)\varphi\left(\vec{q}+\vec{K}\right)\right]\phi\left(2\vec{q}\right)Y_{11}\left(\Omega_{2\vec{q}}\right).$
After integration one finds
$\displaystyle d_{1\pi}^{\,s-e}$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\left(\frac{2^{11/2}}{7^{5/2}}\right)\,\left(\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\right)\left\\{\left[1-\frac{8}{21}x^{2}\right]Y_{00}\left(\Omega_{x}\right)\right.$
(111)
$\displaystyle\left.+\frac{2^{7/2}}{21}\sqrt{\frac{1}{10}}\,x^{2}Y_{20}\left(\Omega_{x}\right)\right\\}e_{2}(x).$
The decay amplitude for the bound-state correction
$\displaystyle h_{fi}^{\rm bsc}$ $\displaystyle=$
$\displaystyle\left(\frac{2^{4}}{7^{5/2}3}\right)\,\left(\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\right)\,\left\\{\left[1-\frac{8}{21}\,x^{2}\right]Y_{00}\left(\Omega_{x}\right)\right.$
(112)
$\displaystyle\left.+\frac{2^{7/2}}{21}\,x^{2}Y_{20}\left(\Omega_{x}\right)\right\\}\,e_{2}(x).$
The total amplitude will be
$\displaystyle h_{fi}^{\rm C3P0}=h_{fi}+2\,h_{fi}^{\rm
bsc}=\frac{\gamma}{\pi^{1/4}\,\beta^{1/2}}\,{\cal M}_{fi}^{\rm C3P0},$ (113)
which are expressions (50) and (51).
## References
* (1) F. E. Close and A. Kirk, Eur. Phys. J. C 21, 531 (2001).
* (2) U. Thoma, Eur. Phys. J. A 18, 135 (2003).
* (3) E. Klempt, A. Zaitsev, Phys. Rept. 454, 1 (2007).
* (4) T. Barnes, Int. J. Mod. Phys. A 21, 5583 (2006).
* (5) L. Micu, Nucl. Phys. B 10, 512 (1969).
* (6) A. Leyaouanc, L. Oliver, O. Pène and J. Raynal, Phys. Rev. D 8, 2223 (1973); ibid. D 9, 1415 (1974); D 11, 680 (1975); D 11, 1272 (1975); Phys. Rev. Lett. B 71, 397 (1977).
* (7) E. S. Ackleh, T. Barnes and E. S. Swanson, Phys. Rev. D 54, 6811 (1996).
* (8) T. Barnes, F. E. Close, P. R. Page and E. S. Swanson, Phys. Rev. D 55, 4157 (1997).
* (9) T. Barnes, N. Black and P.R. Page, Phys. Rev. D 68, 054014 (2003).
* (10) T. Barnes, S. Godfrey and E. S. Swanson, Phys. Rev. D 72, 054026 (2005).
* (11) M.D. Girardeau, Phys. Rev. Lett. 27, 1416 (1971), ibid. J. Math. Phys. 16, 1901 (1975); Phys Rev. A 26, 217 (1982).
* (12) D. Hadjimichef, G. Krein, S. Szpigel and J. S. da Veiga, Ann. of Phys. 268, 105 (1998); ibid. Phys. Lett. B 367, 317 (1996).
* (13) S. Szpigel, Interação Méson-Méson no Formalismo Fock-Tani. PhD thesis, Instituto de Física, Universidade de São Paulo, São Paulo, 1995.
* (14) D. Hadjimichef, J. Haidenbauer, G. Krein, Phys. Rev. C 63, 035204 (2001); ibid. C 66, 055214 (2002).
* (15) D.T. da Silva, D. Hadjimichef, J. Phys. G 30, 191 (2004).
* (16) M. L. L. Silva, D. Hadjimichef, C. A. Z. Vasconcellos, B. E. J. Bodmann, J. Phys. G 32, 475 (2006).
* (17) S. Weinberg, Phys. Rev. 130, 776 (1963); ibid. 131, 440 (1963); M. Scadron and S. Weinberg, Phys. Rev. 133, B1589 (1964); M. Scadron, S. Weinberg and J. Wright, Phys. Rev. 135 B202 (1964).
* (18) M. Oka and K. Yazaki, Prog. Theor. Phys. 66 556 (1981); ibid. 572 (1981).
* (19) E. S. Swanson, Ann. of Phys. 220, 73 (1992); T. Barnes and E. S. Swanson, Phys. Rev. D 46, 131 (1992); T. Barnes, S. Capstick, M. D. Kovarik and E. S. Swanson, Phys. Rev. C 48, 539 (1993).
* (20) Particle Data Group, W.-M. Yao et al., J. Phys. G 33, 1 (2006).
* (21) M. Strohmeier-Prešiček, T. Gutsche, R. Vinh Mau, and A. Faessler, Phys. Rev. D 60, 054010 (1999).
|
arxiv-papers
| 2008-10-02T14:03:01
|
2024-09-04T02:48:58.084082
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. T. da Silva, M. L. L. da Silva, J. N. de Quadros and D. Hadjimichef",
"submitter": "Dimiter Hadjimichef",
"url": "https://arxiv.org/abs/0810.0293"
}
|
0810.0300
|
# DESY 08-136
LTH 805
Wilson twisted mass fermions in the epsilon regime
K. Jansen, A. Nube,
DESY Zeuthen, Platanenallee 6
D-15738 Zeuthen, Germany
E-mail karl.jansen@desy.de annube@ifh.de
Theoretical Physics Division,
Dept. of Mathematical Sciences,
University of Liverpool
Liverpool L69 7ZL, UK
E-mail
###### Abstract:
In this proceeding contribution we report on the ongoing effort to understand
and simulate Wilson twisted mass fermions in the so called $\epsilon$ regime.
## 1 Introduction
Simulations in the $\epsilon$ regime are complementary to standard large
volume simulations. They allow to extract low energy constants of the chiral
Lagrangian, in some cases with less contaminations from chiral logs coming
from higher order corrections. For a long time it has been thought that
simulations in the $\epsilon$ regime are restricted to chirally invariant
lattice formulations. In ref. [1] we have argued that actually this is not the
case, and that simulations in the $\epsilon$ regime can be performed also with
non chirally invariant lattice actions such as Wilson like fermions.
In particular in [1] we suggested that a combination of algorithmic and
theoretical understanding of Wilson twisted mass makes it possible to actually
perform simulations in the $\epsilon$ regime with Wilson twisted mass
fermions.
Recently it has been shown that with suitable and related algorithmic ideas
[2] it possible to reach or get close to the $\epsilon$ regime also with
improved Wilson fermions. At this lattice conference further results in this
directions have been presented [3].
In this proceeding we consider a second lattice spacing and we extend to NLO
the analysis performed in [1]. Our setup is a $L^{3}\times T$ euclidean
lattice with spacing $a$. The lattice action
$S[\chi,\overline{\chi},U]=S_{G}[U]+S_{F}[\chi,\overline{\chi},U],$ (1)
is the sum of the so called tree-level improved Symanzik gauge action [4]
$S_{G}[U]=\frac{\beta}{3}\sum_{x}\left\\{b_{0}\sum_{\mu<\nu}\mathbb{R}{\rm
e}~{}{\rm{Tr}}\left[\mathbbm{1}-P^{(1\times
1)}(x;\mu,\nu)\right]+b_{1}\sum_{\mu\neq\nu}\mathbb{R}{\rm
e}~{}{\rm{Tr}}\left[\mathbbm{1}-P^{(2\times 1)}(x;\mu,\nu)\right]\right\\},$
(2)
where
$b_{0}=1-8b_{1},\quad b_{1}=-{1\over 12},$ (3)
with Wilson twisted mass [5]
$S_{\rm
F}[\chi,\overline{\chi},U]=a^{4}\sum_{x}\overline{\chi}(x)\Big{[}D_{\rm
W}+i\mu_{\rm q}\gamma_{5}\tau^{3}\Big{]}\chi(x),$ (4)
where
$D_{\rm
W}=\frac{1}{2}\\{\gamma_{\mu}(\nabla_{\mu}+\nabla^{*}_{\mu})-a\nabla^{*}_{\mu}\nabla_{\mu}\\}+m_{0}.$
(5)
The basic idea of [1] is that by sampling all topological sectors in the
ensemble generation, it is not necessary to have an unambiguous definition of
topology at finite lattice spacing. To achieve this goal it was suggested [1]
to use a PHMC algorithm [6] treating the low modes exactly and reweighting the
observables. This could allow to perform simulations at very low quark masses
without encountering instabilities or metastabilities.
## 2 $\epsilon$ expansion
Lowering the quark mass at finite lattice spacing with Wilson-like fermions
requires a detailed understanding of the interplay between the genuine chiral
behaviour induced by the ’pion’ dynamics and the one generated by cutoff
effects. A review on the phase diagram and cutoff effects with Wilson twisted
mass (Wtm) can be found in ref. [7]. In the $\epsilon$ regime this is
equivalent to saying that it is necessary to understand the coupling of the
zero modes with the relevant operators describing the effect of the lattice
artifacts. The actual values of the lattice spacing, the physical volume and
the quark mass determine the appropriate power counting, which ought to be
used to perform computations using chiral effective theories. In the continuum
the exact integration over the constant zero modes can be achieved in the
chiral effective theory modifying the $p$ regime power counting, in the so
called $\epsilon$ expansion where the would-be pion mass is small compared to
the linear size of the box
$\frac{1}{T}={\rm O}(\epsilon),\quad\frac{1}{L}={\rm O}(\epsilon),\quad
M_{\pi}={\rm O}(\epsilon^{2}).$ (6)
As a result of the exact integration the order parameter, or the equivalent
ratio $R=\frac{\langle\bar{q}q\rangle}{B_{0}F^{2}}$, vanishes in the chiral
limit at fixed finite volume [8], obtaining restoration of chiral symmetry.
One possible way to include the effects of the lattice artifacts in this
analysis is to include with an appropriate power counting the lattice spacing.
Here we modify the standard power counting in the following way [9]
$M=O(\epsilon^{4}),\quad\frac{1}{L}=O(\epsilon),\quad\frac{1}{T}=O(\epsilon)\quad{a^{2}=O(\epsilon^{4})},$
(7)
where $M$ indicates generically a quark mass. The partition function at
leading order reads
${\mathcal{Z}}=\int{\mathcal{D}}[U_{0}]{\rm
e}^{\frac{c_{1}V}{2}{\rm{Tr}}\left[U_{0}+U_{0}^{\dagger}\right]-\frac{c_{2}V}{4}{\rm{Tr}}\left[U_{0}+U_{0}^{\dagger}\right]^{2}+\frac{c_{3}V}{2}{\rm{Tr}}\left[i\tau^{3}\left(U_{0}^{\dagger}-U_{0}\right)\right]},$
(8)
where the scaling variables are
$z_{1}=c_{1}V=B_{0}F^{2}m^{\prime}V,\qquad
z_{2}=c_{2}V=-\frac{F^{2}w^{\prime}Va^{2}}{4},\qquad
z_{3}=c_{3}V=B_{0}F^{2}\mu_{\rm R}V.$ (9)
To argue that this is a proper power counting for actual numerical simulations
we list here some values
$M\simeq 5{\rm MeV},\quad a\simeq 0.1{\rm fm},\quad L\simeq 1.5{\rm fm}$ (10)
$\quad F\simeq 90{\rm MeV},\quad B_{0}\simeq 5.5{\rm
GeV},\quad|w^{\prime}|\simeq(570{\rm MeV})^{4}.$ (11)
Using these values to estimate the relevant scaling variables in this regime
one obtains
$MF^{2}B_{0}V\simeq 0.75,\quad a^{2}F^{2}|w^{\prime}|V\simeq
0.75,\quad\frac{MB_{0}}{a^{2}|w^{\prime}|}\simeq 1,$ (12)
which indicates that this is an appropriate power counting.111If the lattice
spacing is much smaller a different power counting ought to be used where the
lattice artifacts only appear at NNLO.The chiral condensate can be computed in
the standard way
$R=\frac{1}{N_{\rm f}}\frac{\partial}{\partial z_{3}}\log{\mathcal{Z}},\qquad
z_{1}=0,$ (13)
Figure 1: Quark mass (left plot) and lattice spacing (right plot) dependence
for the single flavour chiral condensate normalized with its LO value in the
continuum and infinite volume
and fig. 1 shows the quark mass (left plot) and lattice spacing (right plot)
dependence of the chiral condensate. We can certainly conclude that the
dependence on the quark mass is, as expected, smooth, and the cutoff effects
are under control. Extension of this computation to NLO including standard
2-point functions is currently in progress [9]. The power counting introduced
is general and valid also for plain Wilson fermions ($z_{3}=0$). The same
power counting could be used to develop an expansion with staggered fermions
and to check the chiral properties of the spectrum in the presence of roots of
the staggered determinant.
## 3 Numerical results
Details of the algorithm used to generate the gauge ensemble can be found in
ref. [1]. In this proceeding we complement the results obtained in [1] with a
second lattice spacing with a NLO analysis. The inversions for the quark
propagator have been performed with a stochastic $Z_{2}\times Z_{2}$ source
located randomly along the euclidean time. Table 1 summarizes the simulation
setup.
$\beta$ | $\kappa$ | $L/a$ | $T/a$ | $a\mu_{\rm q}$
---|---|---|---|---
$4.05$ | $0.157010$ | $20$ | $40$ | $0.00039$
$N_{\rm traj}$ | $N_{\rm ana}$ | $\tau_{\rm int}(P)$ | $\tau_{\rm int}(m_{\rm PCAC})$
---|---|---|---
$2500$ | $421$ | $\sim 0.5$ | $\sim 0.5$
$r_{0}/a$ | $a[{\rm fm}]$ | $L[{\rm fm}]$ | $am_{\rm PCAC}$
---|---|---|---
$6.61(3)$ | $0.0656(11)$ | $1.31$ | $0.00045(12)$
Table 1: Summary of the simulation setup and of the basic ensemble parameters.
In the left plot of fig. 2, we show in the first strip the plaquette MC
history and its distribution. In the second strip we show the MC history and
distribution of the lowest eigenvalue, compared with the value of the infrared
cutoff (horizontal red line) provided by the twisted mass. In the third strip
we show the MC history of the reweighting factor and its distribution.
Figure 2: Left plot: MC histories and distributions of the plaquette (first
strip) smallest eigenvalue (second strip) and reweighting factor (third
strip). The smallest eigenvalue is compared with the infrared cutoff provided
by the twisted mass (horizontal line). Right plot: MC history and distribution
at $x_{0}=T/4$ (first strip) together with the euclidean time dependence of
the PCAC mass (second strip).
One crucial parameter for stability issues and for controlling discretization
errors is the PCAC mass. In the right plot of fig. 2, we show the MC history
and the distribution of the PCAC mass at $x_{0}=T/4$, together with the
euclidean time dependence of the PCAC mass. It is remarkable that there is
almost no sign of boundary O($a$) cutoff effects. The analysis gives with the
corresponding Z factors [10]
$am_{\rm PCAC}=0.00045(12)\qquad\Rightarrow\qquad aM_{\rm
R}^{{\rm\overline{MS\kern-0.35002pt}\kern 0.35002pt}}(2{\rm GeV})=0.0012(2),$
(14)
where
$M_{\rm R}^{{\rm\overline{MS\kern-0.35002pt}\kern 0.35002pt}}(2{\rm
GeV})=\frac{1}{Z_{\rm P}}M\qquad M=\sqrt{\left(Z_{\rm A}m_{\rm
PCAC}\right)^{2}+\mu_{\rm q}^{2}}.$ (15)
We are clearly not at full twist. It is important to remark that this is not
so relevant in the regime where chiral symmetry is restored. Automatic O($a$)
improvement [11] actually holds in a finite volume and with suitable boundary
conditions also for massless Wilson fermions [12]. This is somehow related to
the fact that in the region where chiral symmetry is restored only O($am_{\rm
PCAC}$) cutoff effects are expected, i.e. very small effects. On the other
hand if the mass is of O($a^{2}$) in general the latter could be the cutoff
effects to become visible.
### 3.1 Low energy constants
The values of the low energy constants (LEC) can be extracted comparing the
results of the numerical simulations for the euclidean time dependence of
basic two-point functions with the prediction of $\chi$PT [13, 14]. In this
proceeding we consider the correlation function
$C_{\rm P}(x_{0})=\frac{a^{3}}{L^{3}}\sum_{\bf x}C_{\rm P}({\bf
x},x_{0})\qquad\delta^{ab}C_{\rm P}({\bf x},x_{0})=\langle P^{a}({\bf
x},x_{0})P^{b}({\bf 0},0)\rangle$ (16)
between charged pseudoscalar currents
$P^{a}(x)=\overline{\chi}(x)i\gamma_{5}\frac{\tau^{a}}{2}\chi(x).$ (17)
The euclidean time dependence of the correlation function in $\chi$PT is given
by
$C_{\rm P}(x_{0})=a_{\rm P}+\frac{T}{L^{3}}b_{\rm
P}\left[\frac{y^{2}}{2}-\frac{1}{24}\right]+\ldots\qquad
y=\frac{x_{0}}{T}-\frac{1}{2},$ (18)
where we have defined the following variables
$a_{\rm P}=\frac{B_{0}^{2}F^{4}\rho^{2}}{8}G_{1}(u),\qquad b_{\rm
P}=F^{2}B_{0}^{2}\left[1-\frac{1}{8}G_{1}(u)\right].$ (19)
Details on the definitions of $\rho$, $u$ and $g_{1}$ can be found in [13,
14]. We can thus fit the results from the numerical simulations with the
following fit formulæ
$C_{\rm P}(x_{0})=A_{0}+A_{2}y^{2}\qquad\Rightarrow\qquad a_{\rm
P}=A_{0}+\frac{A_{2}}{12}\qquad b_{\rm P}=A_{2}\frac{2L^{3}}{T}$ (20)
In the left plot of fig. 3, we show the euclidean time dependence of $C_{\rm
P}(x_{0})$ together with the result of the fit (red curve). The line is solid
along the fit range and it becomes dashed where the points are not included in
the fit.
Figure 3: Left plot: Euclidean time dependence of $C_{\rm P}(x_{0})$ together
with the result of the fit. The solid line indicates the fit range, while the
dashed line indicates the same curve outside the fit range. Right plot: fit
results for chiral condensate (first strip) and decay constant (second strip)
as a function of the number of points included in the fit around the middle
point $T/2$. The circles indicate the actual values quoted in the text and the
dashed lines indicate the range of values used to determine the systematic
error.
The results of the analysis are
$a^{3}L^{3}A_{0}=(5.94(36))\cdot 10^{-3},\qquad
a^{3}L^{3}A_{2}=(4.81(30))\cdot 10^{-2}$ (21)
where the errors have been computed with a nested Jackknife/bootstrap
procedure. It is important to check the stability of the fit results with
respect to the number of data $N_{\rm data}$ included in the fit. This is
especially important if we want to make sure that the parabolic time
dependence is a real feature coming from simulating in the $\epsilon$ regime
and not just accidental, i.e. coming from the standard $cosh$ dependence which
can reproduce a fake parabolic behaviour around the $T/2$.
In the right plot of fig. 3, we show the stability of the effective chiral
condensate and decay constant as a function of the number of data points (i.e
time slices) around the middle point included in the fits. While the chiral
condensate shows a remarkably stable result including more points in the fit,
the decay constant shows a somehow not completely flat dependence on the
number of data included in the fit. Although this is not worrisome, it might
be an indication of a physical volume not sufficiently large to suppress
higher order corrections. A perfectly well defined way to proceed would be to
include in the systematic error for $F$ the spread of its value in the region
between the 2 dashed lines. The preliminary result of this analysis is
$r_{0}\Sigma^{1/3}=0.620(8),\qquad r_{0}F=0.220(8)(10)$ (22)
which agrees rather well with results obtained in the $\epsilon$ regime using
improved Wilson fermions [15].
## 4 Conclusions and outlook
We are establishing the basic knowledge to simulate with Wilson-like fermions
in the $\epsilon$ regime. To do this we have introduced a power counting to
study the $\epsilon$ expansion with Wilson-like fermions. The LO computation
for the chiral condensate confirms the absence of any phase transition, and a
NLO extension for the condensate and other observables is currently ongoing.
Numerical simulations in the $\epsilon$ regime with Wtm are feasible using a
PHMC with exact reweighting. This particular choice allows to lower
significantly the quark mass. The extraction of LEC such as $\Sigma$ and $F$
becomes then feasible. Moreover the NLO $\epsilon$ expansion is not
contaminated by chiral logs, which could be a benefit in reducing the
systematic errors.
Computations in this extreme region with Wilson-like fermions require a
detailed understanding of the usual systematics: discretization errors, quark
mass and volume dependence.
We remark that it might be advantageous to combine $p$ and $\epsilon$ regime
simulations both as a tool to attack $2+1(+1)$ simulations, and to further
constrain the values of the LEC.
###### Acknowledgments.
We thank the organizers of “Lattice 2008” for the very interesting conference
realized in Williamsburg. We thank J. Gonzalez Lopez, C. Michael and G.C.
Rossi for a careful reading of this proceeding, and all the members of ETMC
for valuable discussions.
## References
* [1] K. Jansen, A. Nube, A. Shindler, C. Urbach and U. Wenger, Exploring the epsilon regime with twisted mass fermions, PoS LAT2007 (2007) 084 [0711.1871].
* [2] A. Hasenfratz, R. Hoffmann and S. Schaefer, Reweighting towards the chiral limit, Phys. Rev. D78 (2008) 014515 [0805.2369].
* [3] A. Hasenfratz and F. Palombi. Talks presented at this lattice conference.
* [4] P. Weisz, Continuum Limit Improved Lattice Action for Pure Yang- Mills Theory. 1, Nucl. Phys. B212 (1983) 1.
* [5] Alpha Collaboration, R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058 [hep-lat/0101001].
* [6] R. Frezzotti and K. Jansen, A polynomial hybrid Monte Carlo algorithm, Phys. Lett. B402 (1997) 328–334 [hep-lat/9702016].
* [7] A. Shindler, Twisted mass lattice QCD, Phys. Rept. 461 (2008) 37–110 [0707.4093].
* [8] J. Gasser and H. Leutwyler, Thermodynamics of Chiral Symmetry, Phys. Lett. B188 (1987) 477.
* [9] A. Shindler, In preparation, .
* [10] P. Dimopoulos, R. Frezzotti, V. Lubicz and G. Rossi, ETMC internal notes, .
* [11] R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014].
* [12] S. Sint, The Schroedinger functional with chirally rotated boundary conditions, PoS LAT2005 (2006) 235 [hep-lat/0511034].
* [13] P. Hasenfratz and H. Leutwyler, Goldstone boson related finite size effects in field theory and critical phenomena with O(N) symmetry, Nucl. Phys. B343 (1990) 241–284.
* [14] F. C. Hansen, Finite size effects in spontaneously broken SU(N) x SU(N) theories, Nucl. Phys. B345 (1990) 685–708.
* [15] A. Hasenfratz, R. Hoffmann and S. Schaefer, Low energy chiral constants from epsilon-regime simulations with improved Wilson fermions, 0806.4586.
|
arxiv-papers
| 2008-10-01T21:51:07
|
2024-09-04T02:48:58.090331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Jansen, A. Nube, A. Shindler",
"submitter": "Andrea Shindler",
"url": "https://arxiv.org/abs/0810.0300"
}
|
0810.0463
|
# Solutions for real dispersionless Veselov-Novikov hierarchy
Jen-Hsu Chang and Yu-Tung Chen
Department of Computer Science,
National Defense University, Taoyuan, Taiwan
E-mail: jhchang@ndu.edu.tw
###### Abstract
We investigate the dispersionless Veselov-Novikov (dVN) equation based on the
framework of dispersionless two-component BKP hierarchy. Symmetry constraints
for real dVN system are considered. It is shown that under symmetry
reductions, the conserved densities are therefore related to the associated
Faber polynomials and can be solved recursively. Moreover, the method of
hodograph transformation as well as the expressions of Faber polynomials are
used to find exact real solutions of the dVN hierarchy.
## 1 Introduction
The Veselov-Novikov equation
$u_{\tau}=(uV)_{z}+(u\bar{V})_{\bar{z}}+u_{zzz}+u_{\bar{z}\bar{z}\bar{z}},\quad
V_{\bar{z}}=-3u_{z}$ (1)
was invented in [1] as a certain two-dimensional integrable extension of the
KdV equation. Here $z=x+iy$ and the subscripts $z,\bar{z},\tau$ denote partial
derivatives. The important subclass of this equation is the so-called
dispersionless Veselov-Novikov (dVN) equation that has been considered [2, 3,
4] by taking the quasi-classical limit of (1), in which the dispersion effect
had been dropped. Namely,
$u_{\tau}=(uV)_{z}+(u\bar{V})_{\bar{z}},\quad V_{\bar{z}}=-3u_{z}.$ (2)
Recently, it was demonstrated that in [3, 5, 6, 7] the dVN hierarchy is
amenable to the semiclassical $\bar{\partial}$-dressing method. Also, the dVN
equation and dVN hierarchy have appeared in aspects of symmetries and relevant
in the description of geometrical optics phenomena [4, 5, 7, 6]. In [4], some
symmetry constraints for dVN equations were proposed to be efficient ways of
construction of reductions (see also [8, 9, 10] for symmetry constraints of
dispersionless integrable equations). It was also shown that the dVN equation
can be reduced into (1+1)-dimensional hydrodynamic type systems under the
symmetry constraint.
The dispersionless Hirota equations for the two-component BKP system was first
derived by Takasaki [11] as the dispersionless limit of the differential Fay
identity. Later, the Hirota equations was rederived [12] from the method of
kernel formulas provided by Carroll and Kodama [13]. As observed in [5, 6,
11], the Hamilton-Jacobi equation arising from extra equation of these Hirota
equations can be related to the dVN equation as well as the Eikonal equation
in the geometrical optics limit of Maxwell equations. Inspire by these
observations, we are interested in connecting dVN hierarchy to the
dispersionless two-component BKP (2-dBKP) hierarchy. In this paper, we study
the dVN equation based on the framework of the 2-dBKP hierarchy [11] (see also
[12] for the extended dBKP hierarchy). The Hirota equations of 2-dBKP provide
an effective way of constructing Faber polynomials [14, 15] and the associated
Hamilton-Jacobi equations. Real symmetry constraints will be imposed to find
the real solutions of the dVN hierarchy.
This paper is organized as follows. In section 2, we recall the 2-dBKP
hierarchy and give an identification of hierarchy flows for the 2-dBKP system
and the dVN hierarchy. In section 3, we derive Faber polynomials of the dVN
hierarchy by means of its Hirota equations. In particular, we obtain the
recursion formulas of the Faber polynomials. In section 4, under symmetry
constraint for the dVN hierarchy, we show that the corresponding Faber
polynomials characterize the second derivatives of free energy and the
conserved densities. In section 5, we present the hodograph solutions for the
dVN hierarchy by choosing some suitable initial data. The solutions for $S$
function are also given as examples. In section 6, we discuss $2N$-component
symmetry constraint for the dVN hierarchy and derive the corresponding
conserved densities. Section 7 is devoted to the concluding remarks.
## 2 The dVN hierarchy
The 2-dBKP system can be characterized by the following Hirota equations [11,
12]
$\displaystyle\frac{p(\lambda)-p(\mu)}{p(\lambda)+p(\mu)}$ $\displaystyle=$
$\displaystyle\exp(-D(\lambda)S(\mu)),$ (3)
$\displaystyle\frac{\tilde{p}(\lambda)-\tilde{p}(\mu)}{\tilde{p}(\lambda)+\tilde{p}(\mu)}$
$\displaystyle=$ $\displaystyle\exp(-\tilde{D}(\lambda)\tilde{S}(\mu)),$ (4)
$\displaystyle\frac{p(\lambda)-\tilde{q}(\mu)}{p(\lambda)+\tilde{q}(\mu)}$
$\displaystyle=$
$\displaystyle\exp(-D(\lambda)\tilde{S}(\mu))=\exp(D(\lambda)\tilde{D}(\mu)\mathcal{F}),$
(5) $\displaystyle\frac{\tilde{p}(\mu)-q(\lambda)}{\tilde{p}(\mu)+q(\lambda)}$
$\displaystyle=$
$\displaystyle\exp(-\tilde{D}(\mu)S(\lambda))=\exp(\tilde{D}(\mu)D(\lambda)\mathcal{F}),$
(6)
where the generating functions $S(\lambda),\tilde{S}(\lambda)$ are defined by
$S(\lambda)=\sum_{n=0}^{\infty}t_{2n+1}\lambda^{2n+1}-D(\lambda)\mathcal{F},\quad\tilde{S}(\lambda)=\sum_{n=0}^{\infty}\tilde{t}_{2n+1}\lambda^{2n+1}-\tilde{D}(\lambda)\mathcal{F},$
and
$D(\lambda)=\sum_{n=0}^{\infty}\frac{2\lambda^{-2n-1}}{2n+1}\partial_{t_{2n+1}}$,
$\tilde{D}(\lambda)=\sum_{n=0}^{\infty}\frac{2\lambda^{-2n-1}}{2n+1}\partial_{\tilde{t}_{2n+1}}$
denote the vertex operators [9]; morever,
$p(\lambda),q(\lambda),\tilde{p}(\lambda),\tilde{q}(\lambda)$ are defined by
$\displaystyle p(\lambda)=\frac{\partial S(\lambda)}{\partial
t_{1}}=\lambda-D(\lambda)\partial_{t_{1}}\mathcal{F},\quad
q(\lambda)=\frac{\partial
S(\lambda)}{\partial\tilde{t}_{1}}=-D(\lambda)\partial_{\tilde{t}_{1}}\mathcal{F},$
(7)
$\displaystyle\tilde{p}(\lambda)=\frac{\partial\tilde{S}(\lambda)}{\partial\tilde{t}_{1}}=\lambda-\tilde{D}(\lambda)\partial_{\tilde{t}_{1}}\mathcal{F},\quad\tilde{q}(\lambda)=\frac{\partial\tilde{S}(\lambda)}{\partial
t_{1}}=-\tilde{D}(\lambda)\partial_{t_{1}}\mathcal{F}.$ (8)
By equating Eqs. (5) and (6), one has the equation [11]:
$p(\lambda)q(\lambda)=\tilde{p}(\mu)\tilde{q}(\mu)$, from which, after letting
$\lambda,\mu\rightarrow\infty$ one obtains
$-2\mathcal{F}_{t_{1},\tilde{t}_{1}}=-2\mathcal{F}_{\tilde{t}_{1},t_{1}}\equiv
u,$ (9)
where $u=u(t_{1},t_{2},\ldots;\tilde{t}_{1},\tilde{t}_{2},\ldots)$ is a scalar
function. Morever, for arbitrary $\lambda$, one has
$p(\lambda)q(\lambda)=\tilde{p}(\lambda)\tilde{q}(\lambda)=u.$ (10)
Denoting $H_{2n+1}=2\partial_{t_{2n+1}}\partial_{t_{1}}\mathcal{F}$,
$\hat{H}_{2n+1}=2\partial_{t_{2n+1}}\partial_{\tilde{t}_{1}}\mathcal{F}$,
$\tilde{H}_{2n+1}=2\partial_{\tilde{t}_{2n+1}}\partial_{\tilde{t}_{1}}\mathcal{F}$
and
$\tilde{\hat{H}}_{2n+1}=2\partial_{\tilde{t}_{2n+1}}\partial_{t_{1}}\mathcal{F}$,
the Eq. (9) can be treated as the evolution of $u$ with respect to $t_{2n+1}$
and $\tilde{t}_{2n+1}$, respectively, namely
$\displaystyle\frac{\partial u}{\partial t_{2n+1}}$ $\displaystyle=$
$\displaystyle-(H_{2n+1})_{\tilde{t}_{1}}=-(\hat{H}_{2n+1})_{t_{1}},$ (11)
$\displaystyle\frac{\partial u}{\partial\tilde{t}_{2n+1}}$ $\displaystyle=$
$\displaystyle-(\tilde{H}_{2n+1})_{t_{1}}=-(\tilde{\hat{H}}_{2n+1})_{\tilde{t}_{1}}.$
(12)
Now we define the $\tau_{2n+1}$-flow by setting
$\partial_{\tau_{2n+1}}=\partial_{t_{2n+1}}+\partial_{\tilde{t}_{2n+1}}$ and
identify $\tilde{t}_{2n+1}$ as the complex conjugate of $t_{2n+1}$, in
particular, $t_{1}:=z$ and $\tilde{t}_{1}:=\bar{z}$, where $z=x+iy$. From now
on, the functions $\tilde{H}_{2n+1}$, $\tilde{\hat{H}}_{2n+1}$ can also be
taken as the complex conjugate of $H,\hat{H}$:
$\tilde{H}_{2n+1}=\bar{H}_{2n+1},\quad\tilde{\hat{H}}_{2n+1}=\bar{\hat{H}}_{2n+1}.$
Thus, incorporating (11) and (12) for $n\geq 1$ together with $n=0$ in (11)
(or (12)) we obtain $\tau_{2n+1}$-flow of $u$
$u_{\tau_{2n+1}}=-(H_{2n+1})_{\bar{z}}-(\bar{H}_{2n+1})_{z}=-(\hat{H}_{2n+1})_{z}-(\bar{\hat{H}}_{2n+1})_{\bar{z}},\quad
u_{z}=-(H_{1})_{\bar{z}},$ (13)
which is what we call the dVN hierarchy. As in what follows, we shall show
that for $n=1$, the corresponding equation reduces to the dVN equation (2).
## 3 Faber polynomials of 2-dBKP
According to Takasaki’s observations [11], the left hand side of (3) (or (5))
can be expanded as the form
$\log\frac{p(\lambda)-w}{p(\lambda)+w}=-\sum_{n=0}^{\infty}\frac{2\Phi_{2n+1}(w)}{2n+1}\lambda^{-2n-1},$
(14)
where $w=p(\mu)$ in (3) (or $w=\tilde{q}(\mu)$ in (5)) and $\Phi_{n}(w)$ is
the $n$-th Faber polynomial of $p(\lambda)$ defined by
$\log\frac{p(\lambda)-w}{\lambda}=-\sum_{n=1}^{\infty}\frac{\Phi_{n}(w)}{n}\lambda^{-n}.$
(15)
Eq. (15) is analytic for large $\lambda$ for fixed $w\in\mathbb{C}$ and, (14)
is obtained from (15) by taking into account the symmetry conditions
$p(-\lambda)=-p(\lambda),\qquad\Phi_{n}(-w)=(-1)^{n}\Phi_{n}(w).$
If we replace $w$ in (14) by $p(\mu)$, then Eq. (3) can be reduced to the
following system of Hamilton-Jacobi equation
$\frac{\partial S(\mu)}{\partial t_{2n+1}}=\Phi_{2n+1}(p(\mu)).$ (16)
Likewise, replacing $w$ by $\tilde{q}(\mu)$, Eq. (5) reduces to the following
Hamilton-Jacobi equation
$\frac{\partial\tilde{S}(\mu)}{\partial
t_{2n+1}}=\Phi_{2n+1}(\tilde{q}(\mu)).$ (17)
After differentiating Eqs. (16), (17) with respect to $z,\bar{z}$, we have
time evolutions of $p(\mu),q(\mu),\tilde{p}(\mu)$ and $\tilde{q}(\mu)$ in
$t_{2n+1}$-flow in the following form
$\displaystyle\frac{\partial p(\mu)}{\partial
t_{2n+1}}=\partial_{z}\Phi_{2n+1}(p(\mu)),\quad\frac{\partial q(\mu)}{\partial
t_{2n+1}}=\partial_{\bar{z}}\Phi_{2n+1}(p(\mu)),$ (18)
$\displaystyle\frac{\partial\tilde{p}(\mu)}{\partial
t_{2n+1}}=\partial_{\bar{z}}\Phi_{2n+1}(\tilde{q}(\mu)),\quad\frac{\partial\tilde{q}(\mu)}{\partial
t_{2n+1}}=\partial_{z}\Phi_{2n+1}(\tilde{q}(\mu)).$ (19)
In the same way, for the Hirota equations (4) and (6), one can derive the
corresponding Hamilton-Jacobi equations via the expression of Faber
polynomials as
$\log\frac{\tilde{p}(\lambda)-w}{\tilde{p}(\lambda)+w}=-\sum_{n=0}^{\infty}\frac{2\tilde{\Phi}_{2n+1}(w)}{2n+1}\lambda^{-2n-1}.$
(20)
From which, substitutions of $w=\tilde{p}(\mu)$ and $w=q(\mu)$ yielding the
following systems of Hamilton-Jacobi equations
$\frac{\partial\tilde{S}(\mu)}{\partial\tilde{t}_{2n+1}}=\tilde{\Phi}_{2n+1}(\tilde{p}(\mu)),\quad\frac{\partial
S(\mu)}{\partial\tilde{t}_{2n+1}}=\tilde{\Phi}_{2n+1}(q(\mu)).$
Therefore, we have the following time evolutions of
$p(\mu),q(\mu),\tilde{p}(\mu)$ and $\tilde{q}(\mu)$ with respect to
$\tilde{t}_{2n+1}$-flow
$\displaystyle\frac{\partial\tilde{p}(\mu)}{\partial\tilde{t}_{2n+1}}=\partial_{\bar{z}}\tilde{\Phi}_{2n+1}(\tilde{p}(\mu)),\quad\frac{\partial\tilde{q}(\mu)}{\partial\tilde{t}_{2n+1}}=\partial_{z}\tilde{\Phi}_{2n+1}(\tilde{p}(\mu)),$
(21) $\displaystyle\frac{\partial
p(\mu)}{\partial\tilde{t}_{2n+1}}=\partial_{z}\tilde{\Phi}_{2n+1}(q(\mu)),\quad\frac{\partial
q(\mu)}{\partial\tilde{t}_{2n+1}}=\partial_{\bar{z}}\tilde{\Phi}_{2n+1}(q(\mu)).$
(22)
To see how the Faber polynomials will generate functions
$H_{2n+1},\hat{H}_{2n+1},\tilde{H}_{2n+1}$ and $\tilde{\hat{H}}_{2n+1}$ shown
in Eqs. (11), (12), similar derivations in Teo’s paper [15] (see also [14]),
we differentiate (14) to the both sides with respect to $\lambda$ and obtain
$\frac{wp^{\prime}(\lambda)}{p^{2}(\lambda)-w^{2}}=\sum_{n=0}^{\infty}\Phi_{2n+1}(w)\lambda^{-2n-2}.$
Putting
$p(\lambda)=\lambda-\sum_{n=0}^{\infty}\frac{H_{2n+1}}{2n+1}\lambda^{-2n-1}$
into this expression, then we have
$w+w\sum_{n=0}^{\infty}H_{2n+1}\lambda^{-2n-2}=\left[\left(\lambda-\sum_{n=0}^{\infty}\frac{H_{2n+1}}{2n+1}\lambda^{-2n-1}\right)^{2}-w^{2}\right]\left(\sum_{n=0}^{\infty}\Phi_{2n+1}(w)\lambda^{-2n-2}\right).$
Comparing coefficients of all powers of $\lambda$ on both sides, we have
$\displaystyle\Phi_{1}(w)$ $\displaystyle=$ $\displaystyle w,$
$\displaystyle\Phi_{3}(w)$ $\displaystyle=$ $\displaystyle w^{3}+3H_{1}w$ (23)
and the recursion formula
$\displaystyle\Phi_{2n+5}(w)$ $\displaystyle=$ $\displaystyle
w^{2}\Phi_{2n+3}(w)-\sum_{m=0}^{n}\sum_{k=0}^{n-m}\frac{H_{2n-2m-2k+1}H_{2k+1}}{(2n-2m-2k+1)(2k+1)}\Phi_{2m+1}(w)$
$\displaystyle+2\sum_{m=0}^{n+1}\frac{H_{2n-2m+3}}{2n-2m+3}\Phi_{2m+1}+wH_{2n+3},\qquad
n=0,1,2,\ldots,$
which can be used to solve for $\Phi_{n}$. The first few of $\Phi_{n}(w)$ are
given by
$\displaystyle\Phi_{5}(w)$ $\displaystyle=$ $\displaystyle
w^{5}+5H_{1}w^{3}+5(H_{1}^{2}+H_{3}/3)w,$ $\displaystyle\Phi_{7}(w)$
$\displaystyle=$ $\displaystyle
w^{7}+7H_{1}w^{5}+7(2H_{1}^{2}+H_{3}/3)w^{3}+7(H_{1}^{3}+(2/3)H_{1}H_{3}+H_{5}/5)w,$
$\displaystyle\Phi_{9}(w)$ $\displaystyle=$ $\displaystyle
w^{9}+9H_{1}w^{7}+9(3H_{1}^{2}+H_{3}/3)w^{5}+3(10H_{1}^{3}+4H_{1}H_{3}+(3/5)H_{5})w^{3}$
(24)
$\displaystyle+9(H_{1}^{4}+H_{1}^{2}H_{3}+H_{3}^{2}/9+(2/5)H_{1}H_{5}+H_{7}/7)w.$
Similarly, differentiating (20) with respect to $\lambda$, we have
$\frac{w\tilde{p}^{\prime}(\lambda)}{\tilde{p}^{2}(\lambda)-w^{2}}=\sum_{n=0}^{\infty}\tilde{\Phi}_{2n+1}(w)\lambda^{-2n-2}.$
Now putting
$\tilde{p}(\lambda)=\lambda-\sum_{n=0}^{\infty}\frac{\bar{H}_{2n+1}}{2n+1}\lambda^{-2n-1}$
into this expression and comparing coefficients of powers of $\lambda$, we
derive the first few expressions of Faber polynomials
$\displaystyle\tilde{\Phi}_{1}(w)$ $\displaystyle=$ $\displaystyle w,$
$\displaystyle\tilde{\Phi}_{3}(w)$ $\displaystyle=$ $\displaystyle
w^{3}+3\bar{H}_{1}w,$ $\displaystyle\tilde{\Phi}_{5}(w)$ $\displaystyle=$
$\displaystyle w^{5}+5\bar{H}_{1}w^{3}+5(\bar{H}_{1}^{2}+\bar{H}_{3}/3)w,$
$\displaystyle\tilde{\Phi}_{7}(w)$ $\displaystyle=$ $\displaystyle
w^{7}+7\bar{H}_{1}w^{5}+7(2\bar{H}_{1}^{2}+\bar{H}_{3}/3)w^{3}+7(\bar{H}_{1}^{3}+(2/3)\bar{H}_{1}\bar{H}_{3}+\bar{H}_{5}/5)w,$
$\displaystyle\tilde{\Phi}_{9}(w)$ $\displaystyle=$ $\displaystyle
w^{9}+9\bar{H}_{1}w^{7}+9(3\bar{H}_{1}^{2}+\bar{H}_{3}/3)w^{5}+3(10\bar{H}_{1}^{3}+4\bar{H}_{1}\bar{H}_{3}+(3/5)\bar{H}_{5})w^{3}$
(25)
$\displaystyle+9(\bar{H}_{1}^{4}+\bar{H}_{1}^{2}\bar{H}_{3}+\bar{H}_{3}^{2}/9+(2/5)\bar{H}_{1}\bar{H}_{5}+\bar{H}_{7}/7)w,$
$\displaystyle\vdots$
in which $\tilde{\Phi}_{2n+1}$ obey the recurrence relations
$\displaystyle\tilde{\Phi}_{2n+5}(w)$ $\displaystyle=$ $\displaystyle
w^{2}\tilde{\Phi}_{2n+3}(w)-\sum_{m=0}^{n}\sum_{k=0}^{n-m}\frac{\bar{H}_{2n-2m-2k+1}\bar{H}_{2k+1}}{(2n-2m-2k+1)(2k+1)}\tilde{\Phi}_{2m+1}(w)$
$\displaystyle+2\sum_{m=0}^{n+1}\frac{\bar{H}_{2n-2m+3}}{2n-2m+3}\tilde{\Phi}_{2m+1}+w\bar{H}_{2n+3},\qquad
n=0,1,2,\ldots.$
## 4 Symmetry constraint of dVN hierarchy and conserved densities
One way of determine the conserved densities of the dVN hierarchy is to impose
the desired symmetry constraints [4], so that the explicit formulas of these
densities can be connected to the corresponding Faber polynomials and be
solved recursively. The main symmetry constraint we considere is of the form
$u_{x}=(S^{i})_{z\bar{z}},$ (26)
where $S^{i}=S(\mu_{i})$ is evaluated at some point $\mu_{i}$ and we assume
$S^{i}$ is real number. In this section, we would like to show that all of the
conserved densities can be derived by means of the associated Faber
polynomials under this symmetry reduction. We discuss these relations along
the following two ways.
(I) Let us take the derivatives of $S(\lambda)$ with respect to $z,\bar{z},x$
and, noticing that $-2\mathcal{F}_{z\bar{z}}=u$, we have
$\frac{\partial^{3}S(\lambda)}{\partial z\partial\bar{z}\partial
x}=-D(\lambda)\mathcal{F}_{z\bar{z}x}=\frac{1}{2}D(\lambda)u_{x}=\frac{1}{2}D(\lambda)S^{i}_{z\bar{z}},$
(27)
in which $\partial S^{i}/\partial z=p(\mu_{i})=p^{i}$, $\partial
S^{i}/\partial\bar{z}=q(\mu_{i})=q^{i}=\bar{(}\partial S^{i}/\partial
z)=\bar{p}^{i}$ obey the algebraic relation $u=p^{i}q^{i}=p^{i}\bar{p}^{i}$
and then $u$ is positive real number. We remark here that in the context of
nonlinear geometry optics, the quantity $\sqrt{u}$ is proportion to the
refractive index and $u=S^{i}_{z}S^{i}_{\bar{z}}$ is nothing but the standard
Eikonal equation arises from the high-frequency limit of Maxwell equations [4,
5, 7, 6].
Integrating (27) with respect to $z,\bar{z}$ respectively and considering (3),
it follows that
$\displaystyle(p(\lambda))_{x}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}D(\lambda)S^{i}\right)_{z}=-\frac{1}{2}\partial_{z}\left(\log\frac{p(\lambda)-p^{i}}{p(\lambda)+p^{i}}\right),$
(28) $\displaystyle(q(\lambda))_{x}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}D(\lambda)S^{i}\right)_{\bar{z}}=-\frac{1}{2}\partial_{\bar{z}}\left(\log\frac{p(\lambda)-p^{i}}{p(\lambda)+p^{i}}\right).$
(29)
Using (14) with $w$ replaced by $p^{i}$ and the expansions of $p(\lambda)$ and
$q(\lambda)$, Eqs. (28) and (29) can be rewritten respectively by
$\displaystyle\partial_{x}H_{2n+1}$ $\displaystyle=$
$\displaystyle-\partial_{z}\Phi_{2n+1}(p^{i}),$ (30)
$\displaystyle\partial_{x}\hat{H}_{2n+1}$ $\displaystyle=$
$\displaystyle-\partial_{\bar{z}}\Phi_{2n+1}(p^{i}),$ (31)
where $H_{2n+1}\equiv 2\partial_{z}\partial_{t_{2n+1}}\mathcal{F}$ and
$\hat{H}_{2n+1}\equiv 2\partial_{\bar{z}}\partial_{t_{2n+1}}\mathcal{F}$.
Hence, Eqs. (30), (31) provide the Hamilton-Jacobi equations (18), which can
now be read as
$\frac{\partial p^{i}}{\partial t_{2n+1}}=-\frac{\partial H_{2n+1}}{\partial
x},\quad\frac{\partial\bar{p}^{i}}{\partial
t_{2n+1}}=-\frac{\partial\hat{H}_{2n+1}}{\partial x}.$ (32)
As the result, the functions $H_{2n+1}$ and $\hat{H}_{2n+1}$ appear to be the
_conserved densities_ that characterized by the associated Hamilton-Jabobi
equations. Furthermore, from (30), (31) we see that $H_{2n+1}$ and
$\hat{H}_{2n+1}$ are related by the compatibility relations
$\partial_{\bar{z}}H_{2n+1}=\partial_{z}\hat{H}_{2n+1},$
and can be obtained by solving Eqs. (30) and (31).
(II) In the similar way, the differentiation of $\tilde{S}(\lambda)$ with
respect to $z,\bar{z},x$ shows that
$\frac{\partial^{3}\tilde{S}(\lambda)}{\partial z\partial\bar{z}\partial
x}=-\tilde{D}(\lambda)\mathcal{F}_{z\bar{z}x}=\frac{1}{2}\tilde{D}(\lambda)u_{x}=\frac{1}{2}\tilde{D}(\lambda)S^{i}_{z\bar{z}},$
Then we get
$\displaystyle(\tilde{p}(\lambda))_{x}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\tilde{D}(\lambda)S^{i}\right)_{\bar{z}}=-\frac{1}{2}\partial_{\bar{z}}\left(\log\frac{\tilde{p}(\lambda)-\bar{p}^{i}}{\tilde{p}(\lambda)+\bar{p}^{i}}\right),$
(33) $\displaystyle(\tilde{q}(\lambda))_{x}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\tilde{D}(\lambda)S^{i}\right)_{z}=-\frac{1}{2}\partial_{z}\left(\log\frac{\tilde{p}(\lambda)-\bar{p}^{i}}{\tilde{p}(\lambda)+\bar{p}^{i}}\right).$
(34)
Using (20) with $w$ replaced by $\bar{p}^{i}$ and the expansion of
$\tilde{p}(\lambda)$, we rewrite (33) and (34) as
$\displaystyle\partial_{x}\bar{H}_{2n+1}$ $\displaystyle=$
$\displaystyle-\partial_{\bar{z}}\tilde{\Phi}_{2n+1}(\bar{p}^{i}),$ (35)
$\displaystyle\partial_{x}\bar{\hat{H}}_{2n+1}$ $\displaystyle=$
$\displaystyle-\partial_{z}\tilde{\Phi}_{2n+1}(\bar{p}^{i}).$ (36)
where $\bar{H}_{2n+1}\equiv
2\partial_{\bar{z}}\partial_{\bar{t}_{2n+1}}\mathcal{F}$ and
$\bar{\hat{H}}_{2n+1}\equiv
2\partial_{z}\partial_{\bar{t}_{2n+1}}\mathcal{F}$. Therefore, the Hamilton-
Jacobi equations (22) can now be read by the conservation laws:
$\frac{\partial\bar{p}^{i}}{\partial\bar{t}_{2n+1}}=-\frac{\partial\bar{H}_{2n+1}}{\partial
x},\quad\frac{\partial
p^{i}}{\partial\bar{t}_{2n+1}}=-\frac{\partial\bar{\hat{H}}_{2n+1}}{\partial
x}.$ (37)
Also, the conserved densities $\bar{H}_{2n+1}$ and $\bar{\hat{H}}_{2n+1}$ in
(35) and (36) satisfy the compatibilities
$\partial_{z}\bar{H}_{2n+1}=\partial_{\bar{z}}\bar{\hat{H}}_{2n+1}$
and can be solved according to (35), (36). Notice that the Faber polynomials
$\tilde{\Phi}_{2n+1}(\bar{p}^{i})$ have become the complex conjugate of
$\Phi_{2n+1}(p^{i})$ i.e.,
$\tilde{\Phi}_{2n+1}(\bar{p}^{i})=\overline{\Phi_{2n+1}(p^{i})}$. We remark
here that since $u=p^{i}\bar{p}^{i}$ in those $H_{n}$’s, $\Phi_{2n+1}$ is
understood as functions of $p^{i},\bar{p}^{i}$.
In the following, under the symmetry constraint (26) we shall give some
examples to demonstrate how to solve conserved densities $H_{2n+1}$,
$\hat{H}_{2n+1}$. Then, $\bar{H}_{2n+1}$ and $\bar{\hat{H}}_{2n+1}$ are given
automatically by taking the complex conjugate of $H_{2n+1}$ and
$\hat{H}_{2n+1}$, respectively. For simplifying calculations, we shall use
Faber polynomials (23), (24) and the useful identities
$\displaystyle p^{i}_{z}$ $\displaystyle=$ $\displaystyle p^{i}_{x}-u_{x},$
(38) $\displaystyle u_{z}p^{i}$ $\displaystyle=$ $\displaystyle
up^{i}_{x}-uu_{x}+u_{x}(p^{i})^{2}.$ (39)
We determine the relationship of $\hat{H}_{2n+1}$ and $H_{2n+1}$ from the
equation (10). Noting that $p(\lambda)$ and $q(\lambda)$ are defined by
$p(\lambda)=\lambda-\sum_{n=0}^{\infty}\frac{H_{2n+1}}{2n+1}\lambda^{-2n-1},\quad
q(\lambda)=-\sum_{n=0}^{\infty}\frac{\hat{H}_{2n+1}}{2n+1}\lambda^{-2n-1},$
and putting $p(\lambda)$ and $q(\lambda)$ into (10), we have the expression:
$u=-\sum_{n=0}^{\infty}\frac{\hat{H}_{2n+1}}{2n+1}\lambda^{-2n}+\sum_{n,m=0}^{\infty}\frac{H_{2n+1}\hat{H}_{2m+1}}{(2n+1)(2m+1)}\lambda^{-2n-2m-2}.$
(40)
Identifying the coefficients of all powers of $\lambda$ at the both sides, we
obtain
$\hat{H}_{1}=-u$
and the recursion relation of $\hat{H}_{2n+1}$ and $H_{2n+1}$ by
$\hat{H}_{2n+3}=(2n+3)\sum_{k=0}^{n}\frac{H_{2n-2k+1}\hat{H}_{2k+1}}{(2n-2k+1)(2k+1)},\qquad
n=0,1,2,\ldots.$ (41)
Some of them are given by
$\displaystyle\hat{H}_{3}$ $\displaystyle=$ $\displaystyle-3uH_{1},$
$\displaystyle\hat{H}_{5}$ $\displaystyle=$
$\displaystyle-\frac{5}{3}u(3H_{1}^{2}+H_{3}),$ $\displaystyle\hat{H}_{7}$
$\displaystyle=$
$\displaystyle-\frac{7}{3}u(3H_{1}^{3}+2H_{1}H_{3}+\frac{3}{5}H_{5}).$
Examples of constructing conserved densities
Example 1. By (30), for $n=0$
$H_{1x}=-\Phi_{1z}=-p^{i}_{z}=(u-p^{i})_{x},$
where we have used (38). Integrating both sides with respect to $x$ yields
$H_{1}=u-p^{i}.$ (42)
Example 2. By (30), for $n=1$,
$\displaystyle H_{3x}$ $\displaystyle=$
$\displaystyle-\Phi_{3z}=-\left((p^{i})^{3}+3(u-p^{i})p^{i}\right)_{z},$
$\displaystyle=$
$\displaystyle\left(3(u-p^{i})^{2}-(p^{i})^{3}\right)_{x}=\left(3H_{1}^{2}-(p^{i})^{3}\right)_{x},$
where we have used Eqs. (23), (42) in the first line and (38), (39) to obtain
the second line. After integrating both sides with respect to $x$, we get
$H_{3}=3H_{1}^{2}-(p^{i})^{3}.$ (43)
Example 3. For $n=2$ in (30), using (24), (42), (43) and the identities (38),
(39) we have
$\displaystyle H_{5x}$ $\displaystyle=$
$\displaystyle-\Phi_{5z}=-\left((p^{i})^{5}+5(u-p^{i})(p^{i})^{3}+5\left(2(u-p^{i})^{2}-\frac{1}{3}(p^{i})^{3}\right)p^{i}\right)_{z},$
$\displaystyle=$
$\displaystyle\left(10(u-p^{i})^{3}-\frac{20}{3}(u-p^{i})(p^{i})^{3}-(p^{i})^{5}\right)_{x}=\left(10H_{1}^{3}-\frac{20}{3}H_{1}(p^{i})^{3}-(p^{i})^{5}\right)_{x},$
$\displaystyle=$
$\displaystyle\left(-10H_{1}^{3}+\frac{20}{3}H_{1}H_{3}-(p^{i})^{5}\right)_{x},$
where we have used the substitution for $(p^{i})^{3}$ by (43) to obtain the
last equality. Integrating both sides with respect to $x$, we have
$H_{5}=-10H_{1}^{3}+\frac{20}{3}H_{1}H_{3}-(p^{i})^{5}.$ (44)
Example 4. For $n=3$, similarly, we have
$\displaystyle H_{7x}$ $\displaystyle=$ $\displaystyle-\Phi_{7z},$
$\displaystyle=$
$\displaystyle-\left\\{(p^{i})^{7}+7(u-p^{i})(p^{i})^{5}+7\left(3(u-p^{i})^{2}-\frac{1}{3}(p^{i})^{3}\right)(p^{i})^{3}\right.$
$\displaystyle+7\left[(u-p^{i})^{3}+2(u-p^{i})\left((u-p^{i})^{2}-\frac{1}{3}(p^{i})^{3}\right)\right.$
$\displaystyle\left.\left.-2(u-p^{i})^{3}+4(u-p^{i})\left((u-p^{i})^{2}-\frac{1}{3}(p^{i})^{3}\right)-\frac{1}{5}(p^{i})^{5}\right]p^{i}\right\\}_{z},$
$\displaystyle=$ $\displaystyle
7\left(5(u-p^{i})^{4}-5(u-p^{i})^{2}(p^{i})^{3}-\frac{6}{5}(u-p^{i})(p^{i})^{5}+\frac{1}{3}(p^{i})^{6}-\frac{1}{7}(p^{i})^{7}\right)_{x}.$
Again, with substitutions for $(p^{i})^{3}$ and $(p^{i})^{5}$ obtained by (43)
and (44) respectively and integrating over $x$, we solve
$H_{7}=7\left(5H_{1}^{4}-5H_{1}^{2}H_{3}+\frac{6}{5}H_{1}H_{5}+\frac{1}{3}H_{3}^{2}-(p^{i})^{7}/7\right).$
(45)
## 5 Hodograph solutions of dVN hierarchy
Having set up the Faber polynomials in terms of $p^{i},\bar{p}^{i}$ for the
dVN hierarchy that underline the imposed symmetry constraint (26), now we
would like to use the hodograph method to find the hodograph solutions of
$p^{i}(z,\bar{z},\tau_{2n+1})$ and $\bar{p}^{i}(z,\bar{z},\tau_{2n+1})$. Hence
we shall obtain solutions of the dVN equation.
From (30)-(32) and (35)-(37), the $t_{2n+1}$\- and $\bar{t}_{2n+1}$-flows of
$p^{i},\bar{p}^{i}$ can be written respectively by
$\displaystyle\begin{pmatrix}p^{i}\\\ \bar{p}^{i}\end{pmatrix}_{t_{2n+1}}$
$\displaystyle=$
$\displaystyle\partial_{p^{i}}\Phi_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{z}+\partial_{\bar{p}^{i}}\Phi_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\bar{z}},$ $\displaystyle\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\bar{t}_{2n+1}}$ $\displaystyle=$
$\displaystyle\partial_{p^{i}}\overline{\Phi_{2n+1}}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{z}+\partial_{\bar{p}^{i}}\overline{\Phi_{2n+1}}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\bar{z}},$
where we have used the fact that $p^{i}_{\bar{z}}=\bar{p}^{i}_{z}$. Therefore,
the $\tau_{2n+1}$-flow of the dVN hierarchy is governed by
$\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\tau_{2n+1}}=\partial_{p^{i}}M_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{z}+\partial_{\bar{p}^{i}}M_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\bar{z}},\quad n\geq 1$ (46)
where $M_{2n+1}\equiv\Phi_{2n+1}+\overline{\Phi_{2n+1}}$. Note that the first
equation $n=0$ of the hierarchy says that $p^{i}$ and $\bar{p}^{i}$ depend on
$\tau_{1}$ and $x$ only through the linear combination $\tau_{1}+x$. It is
easy to see that the above equation has the following implicit form of
hodograph equations
$\displaystyle\begin{split}&z+\sum_{n=1}^{\infty}f_{2n+1}(p^{i},\bar{p}^{i})\tau_{2n+1}=F(p^{i},\bar{p}^{i}),\\\
&\bar{z}+\sum_{n=1}^{\infty}g_{2n+1}(p^{i},\bar{p}^{i})\tau_{2n+1}=G(p^{i},\bar{p}^{i}),\end{split}$
(47)
where $F$ and $G$ are the initial data at $\tau_{2n+1}=0$, and
$f_{2n+1}=\partial_{p^{i}}M_{2n+1},\;g_{2n+1}=\partial_{\bar{p}^{i}}M_{2n+1}$.
One can show that, because of commutativity of the $\tau_{2n+1}$-flows of
$p^{i},\bar{p}^{i}$, $G$ and $F$ obey the following constraints
$\displaystyle F_{\bar{p}^{i}}$ $\displaystyle=$ $\displaystyle G_{p^{i}},$
(48) $\displaystyle p^{i}F_{p^{i}}$ $\displaystyle=$
$\displaystyle-\bar{p}^{i}G_{\bar{p}^{i}}-(1-p^{i}-\bar{p}^{i})G_{p^{i}}.$
(49)
It turns out that in (48) there exists a function $\varphi(p^{i},\bar{p}^{i})$
such that $F=\partial_{p^{i}}\varphi$, $G=\partial_{\bar{p}^{i}}\varphi$.
Substituting into (49) we have the defining equation for $\varphi$
$p^{i}\varphi_{p^{i}p^{i}}+\bar{p}^{i}\varphi_{\bar{p}^{i}\bar{p}^{i}}+(1-p^{i}-\bar{p}^{i})\varphi_{p^{i}\bar{p}^{i}}=0.$
(50)
Let $p^{i}=(\rho_{1}-i\rho_{2})/2$, $\bar{p}^{i}=(\rho_{1}+i\rho_{2})/2$, we
have $\partial/\partial
p^{i}=\partial/\partial\rho_{1}-i\partial/\partial\rho_{2}$ and
$\partial/\partial\bar{p}^{i}=\partial/\partial\rho_{1}+i\partial/\partial\rho_{2}$.
Then the defining equation (50) becomes
$\varphi_{\rho_{1}\rho_{1}}+2\rho_{2}\varphi_{\rho_{1}\rho_{2}}+(1-2\rho_{1})\varphi_{\rho_{2}\rho_{2}}=0.$
(51)
In fact, due to the existence of $\varphi$, the functions $F,G$ can be chosen
as a natural setting in the linear combination of $f$ and $g$ defined by (47).
Namely, $F=\sum_{n\geq 0}\mu_{n}f_{2n+1}$ and $G=\sum_{n\geq
0}\xi_{n}g_{2n+1}$ with constraint $\mu_{n}=\xi_{n}$. We deduce that $\varphi$
has the polynomial type expansion in $\rho_{1},\rho_{2}$:
$\varphi=\sum_{n=0}^{\infty}\mu_{n}M_{2n+1}(\rho_{1},\rho_{2})$ (52)
satisfies (51). For instance, some cases are established as follows.
(i) $\varphi=M_{1}=\Phi_{1}+\overline{\Phi_{1}}=\rho_{1}$. It is obvious.
(ii)
$\varphi=M_{3}=\Phi_{3}+\overline{\Phi_{3}}=\rho_{1}^{3}-3\rho_{1}^{2}+\frac{3}{2}(\rho_{1}^{2}+\rho_{2}^{2})$.
(iii)
$\varphi=M_{5}=\Phi_{5}+\overline{\Phi_{5}}=\rho_{1}^{5}-\frac{20}{3}\rho_{1}^{4}+10\rho_{1}^{3}+\frac{5}{3}\rho_{1}^{2}(\rho_{1}^{2}+\rho_{2}^{2})-\frac{15}{2}\rho_{1}(\rho_{1}^{2}+\rho_{2}^{2})+\frac{5}{3}(\rho_{1}^{2}+\rho_{2}^{2})^{2}$.
(iv) $\varphi=M_{7}=\Phi_{7}+\overline{\Phi_{7}}$ has expression as
$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\rho_{1}^{7}-{\frac{259}{30}}\,\rho_{1}^{6}+{\frac{35}{2}}\,\rho_{1}^{5}+{\frac{21}{10}}\,\rho_{1}^{4}\rho_{2}^{2}+\frac{7}{5}\,(\rho_{1}^{2}+\rho_{2}^{2})^{2}\rho_{1}^{2}-{\frac{35}{2}}\,\rho_{1}^{3}\rho_{2}^{2}$
$\displaystyle+35\,\rho_{1}^{2}\rho_{2}^{2}-{\frac{35}{4}}\,(\rho_{1}^{2}+\rho_{2}^{2})^{2}\rho_{1}-{\frac{35}{8}}\,(\rho_{1}^{2}+\rho_{2}^{2})^{2}+{\frac{28}{15}}\,(\rho_{1}^{2}+\rho_{2}^{2})^{3}.$
Remark. One can also find several simple solutions of the certain PDEs in Eq.
(51) in the following ways: (a) $\varphi_{\rho_{1}\rho_{1}}=0$,
$2\rho_{2}\varphi_{\rho_{1}\rho_{2}}+(1-2\rho_{1})\varphi_{\rho_{2}\rho_{2}}=0$,
(b) $\varphi_{\rho_{1}\rho_{2}}=0$,
$\varphi_{\rho_{1}\rho_{1}}+(1-2\rho_{1})\varphi_{\rho_{2}\rho_{2}}=0$, and
(c) $\varphi_{\rho_{2}\rho_{2}}=0$,
$\varphi_{\rho_{1}\rho_{1}}+2\rho_{2}\varphi_{\rho_{1}\rho_{2}}=0$. It can be
shown that cases (a) and (b) have solutions of polynomial type involved in
(52), while (c) is not the case. For example, case (c) has solutions of the
form:
$\varphi=c_{0}+c_{1}\rho_{1}+c_{2}\rho_{2}+c_{3}\rho_{2}\exp(-2\rho_{1})$.
### Example
To find the (2+1)-dimensional solutions involving $(z,\bar{z},\tau)$ that
satisfy (46), using $\Phi_{3}=(p^{i})^{3}+3H_{1}p^{i}$ where
$H_{1}=p^{i}\bar{p}^{i}-p^{i}$, we expand the hodograph equation (47) up to
$\tau_{3}=\tau$:
$\displaystyle\begin{split}F(p^{i},\bar{p}^{i})&=z+f_{3}\,\tau=z+\bigg{(}3(p^{i}+\bar{p}^{i})^{2}-6p^{i}\bigg{)}\tau,\\\
G(p^{i},\bar{p}^{i})&=\bar{z}+g_{3}\,\tau=\bar{z}+\bigg{(}3(p^{i}+\bar{p}^{i})^{2}-6\bar{p}^{i}\bigg{)}\tau.\end{split}$
Choosing $F=1,G=1$, the above equations can be easily solved by
$\displaystyle p^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12\tau}\left(3\tau+(z-\bar{z})\pm\sqrt{9\tau^{2}-6\tau(z+\bar{z}-2)}\right),$
$\displaystyle\bar{p}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12\tau}\left(3\tau-(z-\bar{z})\pm\sqrt{9\tau^{2}-6\tau(z+\bar{z}-2)}\right).$
Then $u$ is read as
$u=p^{i}\bar{p}^{i}=\frac{1}{144\tau^{2}}\bigg{(}18\tau^{2}-6\tau(z+\bar{z}-2)-(z-\bar{z})^{2}\pm
6\tau\sqrt{9\tau^{2}-6\tau(z+\bar{z}-2)}\bigg{)}.$ (53)
One can verify that (53) satisfies the dVN equation (2) with
$\tau=\tau_{3},V=3H_{1}$. Furthermore, if we choose $F=f_{3},G=g_{3}$, we get
$\displaystyle p^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12(\tau-1)}\left(3(\tau-1)+(z-\bar{z})\pm\sqrt{9(\tau-1)^{2}-6(\tau-1)(z+\bar{z})}\right),$
$\displaystyle\bar{p}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{12(\tau-1)}\left(3(\tau-1)-(z-\bar{z})\pm\sqrt{9(\tau-1)^{2}-6(\tau-1)(z+\bar{z})}\right).$
Therefore,
$u=\frac{18(\tau-1)^{2}-6(\tau-1)(z+\bar{z})-(z-\bar{z})^{2}\pm
6(\tau-1)\sqrt{9(\tau-1)^{2}-6(\tau-1)(z+\bar{z})}}{144(\tau-1)^{2}}.$
More new solutions can be given in this manner, but the main difficulty we
have to confront with is to solve higher order algebraic equations. Finally,
we want to solve $S^{i}$ function of the above example via the partial
differentiations $\partial S^{i}/\partial z=p^{i}$ and $\partial
S^{i}/\partial\bar{z}=\bar{p}^{i}$. It is easy to obtain that the expression
of $S^{i}$ is given by
$S^{i}(z,\bar{z},\tau)=\frac{3(\tau-1)(z-\bar{z})^{2}+18(\tau-1)^{2}(z+\bar{z}+4C)-2\sqrt{3}\left(3(\tau-1)^{2}-2(\tau-1)(z+\bar{z})\right)^{3/2}}{72(\tau-1)^{2}},$
where $C$ is an arbitrary constant.
## 6 $2N$-component case
In this section, we give an $2N$-component reduction of the dVN hierarchy
under a more general symmetry constraint, and construct the corresponding
hodograph equation. Let us consider the symmetry constraint of the form [4]
$u_{x}=\sum_{i=1}^{N}\epsilon_{i}S^{i}_{z\bar{z}}.$ (54)
Particularly, we impose two assumptions: $u=p^{i}\bar{p}^{i}$,
$\forall\,i=1,\ldots,N$ and $\sum_{i=1}^{N}\epsilon_{i}=1$. Similar
calculations in Sec.4, we have the following relations between conserved
densities and the associated Faber polynomials:
$\displaystyle(H_{2n+1})_{x}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{N}\epsilon_{i}\partial_{z}\Phi_{2n+1}(p^{i}),$
$\displaystyle(\hat{H}_{2n+1})_{x}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{N}\epsilon_{i}\partial_{\bar{z}}\Phi_{2n+1}(p^{i}),$
where the Faber polynomials $\Phi_{2n+1}(p^{i})$ are defined as before, in
which the conserved densities have different forms and can also be determined
recursively. Some of $H_{2n+1}$ for the $2N$-reduction system are given by
$\displaystyle H_{1}=u-\sum_{i=1}^{N}\epsilon_{i}p^{i},\quad
H_{3}=3H_{1}^{2}-\sum_{i=1}^{N}\epsilon_{i}(p^{i})^{3},\quad
H_{5}=-10H_{1}^{3}+\frac{20}{3}H_{1}H_{3}-\sum_{i=1}^{N}\epsilon_{i}(p^{i})^{5},$
$\displaystyle
H_{7}=35H_{1}^{4}-35H_{1}^{2}H_{3}+\frac{42}{5}H_{1}H_{5}+\frac{7}{3}H_{3}^{2}-\sum_{i=1}^{N}\epsilon_{i}(p^{i})^{7}.$
In terms of these $H_{n}$’s, the expressions of $\hat{H}_{2n+1}$ follow the
same as the presented form in (41). Under the symmetry constraint, the
Hamilton-Jacobi equations can now be written in the following way:
$\displaystyle\frac{\partial p^{k}}{\partial
t_{2n+1}}=\partial_{z}\Phi_{2n+1}(p^{k};p^{1},\ldots,p^{N},\bar{p}^{1},\ldots,\bar{p}^{N}),\quad\frac{\partial\bar{p}^{k}}{\partial
t_{2n+1}}=\partial_{\bar{z}}\Phi_{2n+1}(p^{k};p^{1},\ldots,p^{N},\bar{p}^{1},\ldots,\bar{p}^{N}),$
$\displaystyle\frac{\partial
p^{k}}{\partial\bar{t}_{2n+1}}=\partial_{z}\bar{\Phi}_{2n+1}(\bar{p}^{k};p^{1},\ldots,p^{N},\bar{p}^{1},\ldots,\bar{p}^{N}),\quad\frac{\partial\bar{p}^{k}}{\partial\bar{t}_{2n+1}}=\partial_{\bar{z}}\bar{\Phi}_{2n+1}(\bar{p}^{k};p^{1},\ldots,p^{N},\bar{p}^{1},\ldots,\bar{p}^{N}),$
where $k=1,\ldots,N$. After incorporating the above evolution equations to the
$\tau_{2n+1}$-flow of dVN hierarchy and noting that
$p^{i}_{\bar{z}}=\bar{p}^{i}_{z}$ for $i=1,\ldots,N$, we arrive the hodograph
equation of $2N$-component system
$\begin{pmatrix}p^{k}\\\
\bar{p}^{k}\end{pmatrix}_{\tau_{2n+1}}=\sum_{i=1}^{N}f^{i}_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{z}+\sum_{i=1}^{N}g^{i}_{2n+1}\begin{pmatrix}p^{i}\\\
\bar{p}^{i}\end{pmatrix}_{\bar{z}},\qquad k=1,\ldots,N,\quad n\geq 1,$ (55)
where
$f^{i}_{2n+1}(p^{k},\bar{p}^{k})=\partial_{p^{i}}(\Phi_{2n+1}(p^{k})+\bar{\Phi}_{2n+1}(\bar{p}^{k}))$
and
$g^{i}_{2n+1}(p^{k},\bar{p}^{k})=\partial_{\bar{p}^{i}}(\Phi_{2n+1}(p^{k})+\bar{\Phi}_{2n+1}(\bar{p}^{k}))$.
For example, in the case of $N=2$ we have
$\begin{pmatrix}p^{1}\\\ \bar{p}^{1}\\\ p^{2}\\\
\bar{p}^{2}\end{pmatrix}_{\tau_{2n+1}}=\begin{pmatrix}\mbox{~{}}&\mbox{~{}}\\\\[-2.84544pt]
f^{1}(p^{1},\bar{p}^{1})I_{2}+g^{1}(p^{1},\bar{p}^{1})\mathbb{A}(p^{1})&f^{2}(p^{1},\bar{p}^{1})I_{2}+g^{2}(p^{1},\bar{p}^{1})\mathbb{A}(p^{2})\\\\[11.38092pt]
f^{1}(p^{2},\bar{p}^{2})I_{2}+g^{1}(p^{2},\bar{p}^{2})\mathbb{A}(p^{1})&f^{2}(p^{2},\bar{p}^{2})I_{2}+g^{2}(p^{2},\bar{p}^{2})\mathbb{A}(p^{2})\\\\[-5.69046pt]
\mbox{~{}}&\mbox{~{}}\end{pmatrix}\begin{pmatrix}p^{1}\\\ \bar{p}^{1}\\\
p^{2}\\\ \bar{p}^{2}\end{pmatrix}_{z},$
where $I_{2}$ is the $2\times 2$ identity matrix and
$\mathbb{A}(p^{i})=\begin{pmatrix}0&1\\\
-\frac{\bar{p}^{i}}{p^{i}}&\frac{1-p^{i}-\bar{p}^{i}}{p^{i}}\end{pmatrix},\qquad
i=1,2.$
## 7 Concluding remarks
In this paper we have studied dVN hierarchy from the framework of the 2-dBKP
system. One demonstrates how to derive the associated Faber polynomials and
their recursion relation via the Hirota equations of 2-dBKP hierarchy. Under
the symmetry constraint (26), we solve conserved densities by the derived
Faber polynomials. Also, we provide a set of hodograph equation of the dVN
hierarchy, expanded by the derivatives of its associated Faber polynomials.
Explicitly, we obtain the hodograph solutions to the dVN equation as an
example.
For the more general symmetry constraint, we construct the $2N$-component
reduction system by the generalized Faber polynomials and wrote down the
corresponding hodograph equation. However, the main difficulty is to find the
explicit solutions of the $2N$-reduction system (55). We hope to address this
problem elsewhere.
### Acknowledgments
The author J.H. Chang will thank Prof. Konopelchenko and Dr. Moro for their
stimulating discussions. This work is supported in part by the National
Science Council of Taiwan (Grant Nos. NSC 96-2115-M-606-001-MY2, J.-H. C., and
NSC 96-2811-M-606-001, Y.-T. C.).
## References
* [1] A. P. Veselov and S. P. Novikov, Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations, Soviet Math. Dokl. 30 (1984) 588–591.
* [2] I. M. Krichever, Method of averaging for two-dimensioal ”integrable” equations, Functional Anal. Appl. 22 (1988) 200–213.
* [3] B. G. Konopelchenko and L. Martínez Alonso, Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations, Stud. Appl. Math. 109 (2002) 313–336.
* [4] L. V. Bogdanov, B. G. Konopelchenko, and A. Moro, Symmetry constraints for real dispersionless Veselov-Novikov equation, J. Math. Science 136 (2006) 4411–4418.
* [5] B. G. Konopelchenko and A. Moro, Geometrical optics in nonlinear media and integrable equations, J. Phys. A: Math. Gen. 37 (2004) L105–L111.
* [6] B. G. Konopelchenko and A. Moro, Integrable equations in nonlinear geometrical optics, Stud. Appl. Math. 113 (2004) 325–352.
* [7] B. G. Konopelchenko and A. Moro, Light propagation in a Cole-Cole nonlinear medium via the Burgers-Hopf equation, Theoret. Math. Phys. 144 (2005) 968–974.
* [8] L. V. Bogdanov and B. G. Konopelchenko, Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type, Phys. Lett. A 330 (2004) 448–459.
* [9] L. V. Bogdanov and B. G. Konopelchenko, On dispersionless BKP hierarchy and its reductions, J. Nonlinear Math. Phys. 12 suppl. 1 (2005) 64–73.
* [10] J. H. Chang, On the waterbag model of the dispersionless KP hierarchy (II), J. Phys. A: Math. Theor. 40 (2007) 12973–12985.
* [11] K. Takasaki, Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2, Paper 057 (2006).
* [12] Y. T. Chen and M. H. Tu, On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy, J. Math. Phys. 47 (2006) 102702.
* [13] R. Carroll and Y. Kodama, Solution of the dispersionless Hirota equations, J. Phys. A 28 (1995) 6373–6387.
* [14] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV.
* [15] L. P. Teo, Analytic functions and integrable hierarchies–characterization of tau functions, Lett. Math. Phys. 64 (2003) 75–92.
|
arxiv-papers
| 2008-10-02T15:50:15
|
2024-09-04T02:48:58.096519
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jen-Hsu Chang and Yu-Tung Chen",
"submitter": "Jen-Hsu Chang",
"url": "https://arxiv.org/abs/0810.0463"
}
|
0810.0509
|
# Continuous Paranematic-to-Nematic Ordering Transitions of
Liquid Crystals in Tubular Silica Nanochannels
Andriy V. Kityk1 andriy.kityk@univie.ac.at Matthias Wolff2 Klaus Knorr2 Denis
Morineau3 Ronan Lefort3 Patrick Huber2 p.huber@physik.uni-saarland.de
1Institute for Computer Science, Czestochowa University of Technology, Al.
Armii Krajowej 17, P-42200 Czestochowa, Poland
2Faculty of Physics and Mechatronics Engineering, Saarland University, D-66041
Saarbrücken, Germany
3Institut de Physique de Rennes, CNRS-UMR 6251, Université de Rennes 1,
F-35042 Rennes, France
###### Abstract
The optical birefringence of rod-like nematogens (7CB, 8CB), imbibed in
parallel silica channels with 10 nm diameter and 300 micrometer length, is
measured and compared to the thermotropic bulk behavior. The orientational
order of the confined liquid crystals, quantified by the uniaxial nematic
ordering parameter, evolves continuously between paranematic and nematic
states, in contrast to the discontinuous isotropic-to-nematic bulk phase
transitions. A Landau-de Gennes model reveals that the strength of the
orientational ordering fields, imposed by the silica walls, is beyond a
critical threshold, that separates discontinuous from continuous paranematic-
to-nematic behavior. Quenched disorder effects, attributable to wall
irregularities, leave the transition temperatures affected only marginally,
despite the strong ordering fields in the channels.
###### pacs:
61.30.Gd, 42.25.Lc, 64.70.Nd
Spatial confinement on the micro- and nano-scale can affect the physics of
liquid crystals (LCs) markedly. Modified phase transition behavior has been
found in experiments on LCs imbibed into a variety of porous media I1 ; Kralj
, in aerogels I2 , in semi-confined thin film geometries Garcia2008 , and at
the free surface of bulk LCs Ocko1986 .
For example, the heat capacity anomaly typical of the second-order nematic-to-
smectic-A (N-SmA) transition in rod-like LCs immersed in aerogels is absent or
greatly broadened. This allowed a detailed study of the influence of quenched
disorder introduced by random spatial confinement on this archetypical phase
transition I2 .
It has also been demonstrated experimentally I1 ; Kralj , in agreement with
expectations from theory Sheng1976 ; Steuer2005 , that there is no ”true”
isotropic-nematic (I-N) transition for LCs confined in geometries spatially
restricted in at least one direction to a few nanometers. The anchoring at the
confining walls, quantified by a surface field, imposes a partial
orientational, that is a partially nematic ordering of the confined LCs, even
at temperatures $T$ far above the bulk I-N transition temperature $T^{\rm
b}_{IN}$. The symmetry breaking doesn’t occur spontaneously, as characteristic
of a genuine phase transition, but is enforced over relevant distances by the
interaction with the walls Stark2002 . Thus confinement here plays a similar
role as an external magnetic field for a spin system: The strong first order
I-N transition is replaced by a weak first order or continuous paranematic-to-
nematic (P-N) transition, depending on the strength of the surface
orientational field.
Whereas a qualitative understanding of this behavior has been achieved for a
variety of spatially mesoconfined LCs I1 ; Kralj , more detailed comparisons
with theoretical predictions have been extremely challenging in the past,
mainly due to the complex, tortuous, multiply connected pore networks or
unknown surface/interface-LC interactions in the aforementioned studies. The
advent of arrays of straight nanochannels of silica and silicon with simpler
channel geometries may allow to gain deeper, quantitative insights into this
phenomenology.
In this Letter, we present a high-resolution optical birefringence study of
rod-like nematogens (7CB and 8CB) confined to an array of parallel, non-
tortuous channels of 10 nm mean diameter and 300 micrometer length in a
monolithic silica membrane. We demonstrate that the optical transparency, the
straight channel geometry of the host along with the sensitivity of the
modulated beam technique employed allows us to precisely characterize the
orientational order of the nematogens with respect to the long axis of the
nanochannels. The simple type of restricted geometry will allow us to compare
our measurements quantitatively with a Landau-de Gennes model for the P-N
transition and, consequently, to determine both the strength of the
orientational ordering field, imposed by the silica walls, and the influence
of quenched disorder, attributable to channel irregularities.
Figure 1: (color online). Birefringence of 7CB and 8CB measured in the bulk
state, panel (a), (b), and in the silica nanochannels, panel (c) and (d),
resp., as a function of temperature in comparison to fits (solid lines) based
on the KKLZ-model discussed in the text. The final birefringence
characteristic of the paranematic phases are shaded down to the $P-N$
”transition” temperatures, $T_{\rm PN}$. The dashed lines mark the bulk I-N
and N-SmA transition temperatures. As insets in (a) and (c), the bulk
isotropic ($I^{\rm b}$) as well as the bulk nematic ($N^{\rm b}$) phases upon
homeotropic alignment, and the confined paranematic (P) and nematic (N) phases
are illustrated, respectively.
For rod-like molecules the degree of orientational molecular order can be
quantified by the uniaxial order parameter $Q=\frac{1}{2}\left\langle
3\cos^{2}\phi-1\right\rangle$, where $\phi$ is the angle between the long axis
of a single molecule and a direction of preferred orientation of that axis,
the director. The brackets denote an averaging over all molecules under
consideration. The orientation of the director may vary locally. However, it
can be dictated by external fields or by surface anchoring conditions over
macroscopic distances. Planar silica surfaces enforce planar anchoring of 7CB
and 8CB without a preferred lateral orientation Kumar2001 . Additionally, the
director is expected to be oriented parallel to the long axis in a cylindrical
silica channel GrohDietrich1999 . A statement which shall be explored in the
following by birefringence measurements.
The propagation speed of light and thus the refractive index $n$ in a LC
sensitively depends on the orientation of the polarization with respect to the
molecular orientation of the anisotropic nematogens. Conversely, the state of
molecular order in a LC can be inferred from optical polarization
measurements. To a good approximation, $Q$ is proportional to the optical
birefringence $\Delta n=n_{\rm e}-n_{\rm o}$, where $n_{\rm o}$ and $n_{\rm
e}$ refer to polarizations perpendicular and parallel to the local optical
axis, the so-called ordinary and extraordinary refractive indices Haller ,
respectively. In a nematic LC the local optical axis agrees with the director.
Thus, in principle it is sufficient to determine the experimentally accessible
$\Delta n$ in order to determine the molecular arrangement in an LC. However,
there are weak, but final $T$-dependencies of the bare refractive indices,
which do not originate from changes of the averaged collective molecular
orientations but from changes in the anisotropic molecular polarizabilities of
the single molecules as a function of $T$. In order to separate out these
effects, we resort to the quantity $\Delta n^{*}=n_{\rm e}^{2}(T)-n_{\rm
o}^{2}(T)\propto\Delta n(T)\cdot(n_{\rm e}(T)+n_{\rm o}(T))/2$ which can be
shown to be solely proportional to $Q(T)$ Lau . For simplicity, we will refer
to $\Delta n^{*}$ as ”effective birefringence” in the following.
For our measurements, a monolithic silica membrane permeated by an array of
parallel aligned, non-interconnected channels of 300 $\mu$m length was
prepared by thermal oxidation of a free-standing silicon membrane Gruener2008
at 800 ∘C for 12 hours. The mean channel diameter $D$ and porosity $P$ was
determined by recording a volumetric nitrogen sorption isotherm at
$T~{}=~{}77~{}K$ to $D$=10.0$\pm 0.5$ nm and $P$ = 50$\pm$2%, respectively.
Electron micrographs of the channels Gruener2008 indicate sizeable 1.0$\pm
0.5$ nm mean square deviations of their surfaces from an ideal cylindrical
form. The membrane was completely filled with the LCs by spontaneous
imbibition Huber2007 . For the bulk measurement, we used a 50 $\mu$m thick
glass cell containing homeotropically aligned LCs, see Fig. 1(a).
We used a high-resolution optical modulated beam method for the accurate
determination of the phase retardation $R$ between two perpendicularly
polarized components of light transmitted through the samples. The setup, see
Fig. 2, employs an optical photoelastic modulator and a dual lock-in detection
scheme in order to minimize the influence of uncontrolled light-intensity
fluctuations Skarabot ; Kumar2001 . After passing the sample the laser light
intensity ($\lambda=632.8~{}nm$) was detected by a photodiode and two lock-in
amplifiers, which simultaneously determined the amplitudes of the first
($U_{\rm\Omega}$) and second ($U_{\rm 2\Omega}$) harmonics, respectively. The
phase retardation by the sample $R=\arctan[(U_{\rm\Omega}J_{\rm 2}(A_{\rm
0}))/(U_{\rm 2\Omega}J_{\rm 1}(A_{\rm 0}))]$ (here $J_{\rm 1}(A_{\rm 0})$ and
$J_{\rm 2}(A_{\rm 0})$ are the Bessel functions corresponding to the PEM
retardation amplitude $A_{\rm 0}=0.383\lambda$) was measured for an incident
angle $\theta=43.5$ deg between laser beam and sample surface and thus between
beam and long axes of the silica channels, see inset of Fig. 2. For such a
tilted sample geometry the conversion of the retardation $R$ to $\Delta n$ and
$\Delta n^{*}$ was performed by numerically solving Berek’s compensator
formula Berek .
In Fig. 1 $\Delta n^{*}$ is plotted for bulk 7CB and 8CB upon slow cooling
($\sim$0.01 K/min) to the solidification temperature. There is a jump in
$\Delta n^{*}(T)$ of bulk 7CB typical of the first-order I-N phase transition
at $T^{\rm b}_{\rm IN}\approx$ 42 ∘CLau . $\Delta n^{*}(T)$ of 8CB exhibits
the signatures characteristic of the first-order I-N transition at $T^{\rm
b}_{\rm IN}\approx$ 41 ∘C and an almost continuous N-SmA transition at $T^{\rm
b}_{\rm NSmA}\approx$34 ∘CI1 ; Kas . Any pretransitional effects are clearly
absent in the bulk isotropic phase of both LCs investigated, $\Delta
n^{*}(T)=0$ for $T>T^{\rm b}_{\rm IN}$.
The nanoconfined LCs reveal a considerably different behavior, see Fig. 1
lower panels. In agreement with neutron diffraction experiments on 8CB in
nanochannels, there is no indication of a sharp N-SmA phase transition
Guegon2006 . More interestingly, there exists a residual $\Delta n^{*}$
characteristic of a paranematic LC state at $T$s far above $T^{\rm b}_{\rm
IN}$. Upon further cooling $\Delta n^{*}$ increases continuously and at the
lowest $T$s investigated, the absolute magnitude of $\Delta n^{*}$ is
compatible with an 80% (75%) alignment of the 7CBs’ (8CBs’) long axes parallel
to the channel axes. Hence, the silica nanochannel confinement dictates indeed
a substantial molecular alignment, as proposed in the introduction and
illustrated in the inset of Fig. 1. More importantly, it renders the
transition continuous.
Figure 2: (color online). Schematic experimental setup of a high-resolution
birefringence measurement consisting of a He-Ne laser, an optical polarizer
(P), a temperature controlled (TC) optical cell (OC), an optical analyzer (A),
a photoelastic modulator (PEM-90), a photodiode as a detector (PD) and a
”lock-in” detection and analyzing unit. The inset depicts the orientation of
the silicon nanochannel membrane with respect to the incident laser beam.
In the following we are going to analyze the peculiar $\Delta n^{*}$ behavior
within a Landau-de Gennes model for the $I-N$ transition in confinement
suggested by Kutnjak, Kralj, Lahajnar, and Zumer (KKLZ-model) Kralj ;
Sheng1976 . The dimensionless free energy density of a nematic phase spatially
confined in a cylindrical geometry with planar anchoring conditions reads in
the KKLZ-model as:
$f=tq^{2}-2q^{3}+q^{4}-q\sigma+\kappa q^{2}$
where $q=Q/Q(T^{\rm b}_{\rm IN})$ is the scaled nematic order parameter, $t$
is a reduced temperature, and $\sigma$ is the effective surface field. The
last term in Eq. 1 describes quenched disordering effects due to surface-
induced deformations (wall irregularities) I1 . Minimalization of $f$ yields
the equilibrium order parameter $q_{\rm e}$, which is shown in Fig. 3 for
selected values of $\sigma$ and $\kappa$ as a function of $t$. In the KKLZ-
model, the I-N transition is of first order for $\sigma<\sigma_{\rm c}=0.5$.
The jump of $q_{\rm e}$ approaches zero while $\sigma\rightarrow 0.5$, see
inset in Fig. 3. Thus, $\sigma_{\rm c}$ marks a critical threshold separating
first-order, discontinous from continuous I-N behavior.
In the following we apply the KKLZ-model to our measured $\Delta n^{*}(T)$.
The solid lines in Figs. 1(a),(b) are the best fits of the dependencies
$\Delta n^{*}(T)$ as obtained by rescaling $q_{\rm e}$ and $t$ while assuming
an absence of any surface ordering and quenched disorder fields in the bulk
state ($\sigma(bulk)=\sigma(D=\infty)$=0, $\kappa=0$). Thereby, we achieve an
encouraging agreement between the measured $\Delta n^{*}(T)$ curves in the
proximity of $T^{\rm b}_{\rm IN}$ and deep into the nematic phase for both
bulk LCs. The sizeable deviations for 7CB below 230 ∘C originate in the
neglect of higher order terms in the KKLZ-model, necessary to produce the
saturation behavior of $q$ at low $T$.
More importantly, we achieve also an excellent agreement between measured and
KKLZ-modelled $\Delta n^{*}(T)$ over the entire P-N transition regime for the
nanoconfined LCs, provided we assume final surface ordering fields of
magnitude $\sigma(D=10$nm$)$ = 0.81, 1.15 and strengths of the quenched
disorder parameter $\kappa$ = 0.83, 1.4 for 7CB and 8CB, resp., see panel (c)
and (d) in Fig. 1, respectively. As expected from the observed continuous
behavior of $\Delta n^{*}(T)$ in both cases $\sigma(D)$ is well above
$\sigma_{\rm c}=0.5$. Moreover, the upward shift in $T_{\rm PN}$ due to the
anchoring surface field, predicted by the KKLZ-model, is slightly overbalanced
by a downward shift due to quenched disorder effects for this set of KKLZ-
parameters - see also Fig. 3.
The bare geometrical mechanism which favors an alignment of rod-like molecules
in cylindrical confining geometries is expected to increase in strength with
the length of the molecules GrohDietrich1999 . Also the alignment effect of
the anchoring field of the confining walls increases with this length.
Therefore the $\sigma$-increase of $30\%$ appears not too surprising, despite
the relatively small 10% length-increase between 8CB and 7CB.
According to the KKLZ-model $\sigma(D)$ scales with $1/D$, which also allows
us to estimate $\sigma$ as a function of $D$ for other confining silica
geometries. In particular, we arrive at a $D_{\rm c}$ of 23 nm and 16 nm for
the ”critical” silica pore diameter of 8CB and 7CB, resp., ($\sigma(D_{\rm
c})=\sigma_{\rm c}=0.5$) separating continuous from discontinuous behavior.
This is in quantitative agreement with the still weakly discontinuous behavior
reported for 8CB confined in 24 nm xerogel mesopores Kralj and the continuous
behavior found for 7CB in 7 nm mean pore diameter Vycor glass I1 .
Figure 3: (color online). Order parameter $q_{\rm e}$ in the KKLZ-model as a
function of reduced temperature $t$ for selected effective surface fields
$\sigma$ in the absence of quenched disorder, $\kappa=0$ (solid lines) and for
the $\sigma$-, $\kappa$-values which yield the best fits of $\Delta n^{*}(T)$
measured for the confined LCs (dashed lines). Inset: Order parameter jump at
the I-N (P-N) transition as a function of the surface field $\sigma$
($\kappa=0$).
In conclusion, our measurements indicate that in $10$ nm straight silica
channels the surface anchoring fields render the bulk discontinuous I-N
transition to a continuous P-N transition. The transition temperature is
changed only marginally, due to a balance of its surface ordering induced
upward and its quenched disorder induced downward shift, similarly as has been
observed for LCs imbibed in tortuous Vycor glass pores and a variety of other
mesoporous matrices I1 . The purely phenomenological findings presented here
would profit from more microscopic information, for example gained by Monte
Carlo or Molecular Dynamics simulations Steuer2005 or from x-ray and neutron
scattering experiments.
Finally we would like to state, that the rheology of bulk LC changes abruptly
at the I-N transition Miesowicz1946 ; Jadzyn2001 . By contrast, the gradual
P-N transition reported here should lead to a continuous $T$-evolution of the
fluidity of LC in silica nanochannels. Moreover, velocity slippage at the
walls, crucial in the emerging field of nanofluidics MNFluidics ; Huber2007 ,
is expected to be associated with the molecular alignments in the channels
Heidenreich2007 . Thus, we hope the peculiarities reported here will stimulate
LC flow experiments with silica nanochannels, which would be important
extensions of previous experiments on the fluidity of LCs in thin film
geometries RheologyThinFilms .
###### Acknowledgements.
We thank the DFG for support within the priority program 1164, Nano- &
Microfluidics (Hu 850/2) and acknowledge support by the DAAD and the French
Ministry of Foreign Affairs within the French-German PROCOPE program.
## References
* (1) G. S. Iannacchione et. al., Phys. Rev. Lett. 71, 2595 (1993); R. J. Ondris-Crawford et. al., Phys. Rev. E 48, 1998 (1993); G. S. Iannacchione and D. Finotello et. al., ibid. 50, 4780 (1994); S. Qian, G. S. Iannacchione, and D. Finotello, ibid. 57, 4305 (1998); K.A. Crandall, C. Rosenblatt, and F.M. Aliev, ibid. 53, 636 (1996); M.D. Dadmun and M. Muthukumar, J. Chem. Phys. 98, 4850 (1993); M.C. Choi et. al., Proc. Nat. Acad. Sc. 101, 17340 (2004); S. Cloutier et. al., Phys. Rev. E 73, 051703 (2006).
* (2) Z. Kutnjak et. al., ibid. 68, 021705 (2003); Z. Kutnjak et. al., ibid. 70, 051703 (2004).
* (3) T. Bellini et. al., Science 294, 1074 (2001).
* (4) V. Designolle et. al., Langmuir 22, 363 (2005); R. Garcia, E. Subashi, and M. Fukuto, Phys. Rev. Lett. 100, 197801 (2008).
* (5) B.M. Ocko et. al., Phys. Rev. Lett. 57, 94 (1986).
* (6) P. Sheng, Phys. Rev. Lett. 37, 1059 (1976); A. Poniewierski and T.S. Sluckin, Liq. Crys. 2, 281 (1987).
* (7) H. Steuer, S. Hess, and M. Schoen, Phys. Rev. E 69, 031708 (2004); D. Cheung and F. Schmid, Chem. Phys. Lett. 418, 392 (2005).
* (8) H. Stark, Phys. Rev. E 66, 032701 (2002).
* (9) I. Drevensek Olenik et. al., Eur. Phys. J. E 11, 169 (2003); S. Kumar (ed.), Liquid Crystals, Cambridge UP, Cambridge (2001); P. S. Pershan, Structure of Liquid Crystal Phases, World Sc., Singapore (1988); W.H. de Jeu, Physical Properties of Liquid Crystals, Acad. Press, NY (1980).
* (10) B. Groh and S. Dietrich, Phys. Rev. E 59, 4216 (1999).
* (11) I. Haller, Prog. Solid State Chem. 10 103 (1975).
* (12) S. Gruener and P. Huber, Phys. Rev. Lett. 100, 064502 (2008); P. Kumar et. al., J. Appl. Phys. 103, 024303 (2008).
* (13) P. Huber et. al., E. Phys. J. Spec. Top 141 101 (2007).
* (14) M. Skarabot et. al., Phys. Rev. E 58 575 (1998).
* (15) M. Born and E. Wolf, Principles of Optics, p. 694, Pergamon Press, 6th edition, Oxford (1975).
* (16) G. B. Kasting, C. W. Garland, and K. J. Lushington, J. Phys. (Paris) 41, 879 (1980); J. Thoen, H. Marynissen, and W. Van Dael, Phys. Rev. A 26,2886 (1982); I. Hatta and T. Nakayama, Mol. Cryst. Liq. Cryst. 66, 97 (1980).
* (17) R. Guégan et. al., Phys. Rev. E 73, 011707 (2006); R. Guégan et. al., Eur. Phys. J. E 26, 261 (2008).
* (18) Y. G. J. Lau et. al., Liq. Crys. 34, 421 (2007); H. Mada and S. Kobayashi, Appl. Phys. Lett. 35, 4 (1979).
* (19) M. Miesowicz, Nature 158, 261 (1946).
* (20) J. Jadzyn and G. Czechowski, J. Phys.: Condens. Matter 13, L261 (2001).
* (21) T.M. Squires and S.R. Quake, Rev. Mod. Phys. 77, 977 (2005); J.C.T. Eijkel and A.v.d. Berg, Micro. Nanofl. 1, 249 (2005).
* (22) S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E 75, 066302 (2007).
* (23) M. Ruths, S. Steinberg, and J.N. Israelachvili, Langmuir 12, 6637 (1996); M. Ruths and S. Granick, Langmuir 16, 8368 (2000); J. Janik, R. Tadmor, and J. Klein, Langmuir 17, 5476 (2001); Y. Golan et. al., Phys. Rev. Lett. 86, 1263 (2001); K.L. Yung et. al., Polymer 47, 4454 (2006).
|
arxiv-papers
| 2008-10-02T19:59:14
|
2024-09-04T02:48:58.103102
|
{
"license": "Public Domain",
"authors": "Andriy V. Kityk, Matthias Wolff, Klaus Knorr, Denis Morineau, Ronan\n Lefort, and Patrick Huber",
"submitter": "Patrick Huber",
"url": "https://arxiv.org/abs/0810.0509"
}
|
0810.0556
|
# Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction
L.V. Bogdanov, Jen-Hsu Chang‡ and Yu-Tung Chen† L.D. Landau ITP, Kosygin str.
2, Moscow 119334, Russia Department of Computer Science, National Defense
University, Taoyuan, Taiwan, E-mail: jhchang@ndu.edu.tw, ${\ddagger}$
corresponding author.
###### Abstract
We study Manakov-Santini equation, starting from Lax-Sato form of associated
hierarchy. The waterbag reduction for Manakov-Santini hierarchy is introduced.
Equations of reduced hierarchy are derived. We construct new coordinates
transforming non-hydrodynamic evolution of waterbag reduction to non-
homogeneous Riemann invariants form of hydrodynamic type.
Keywords: Manakov-Santini hierarchy, Lax representation, Waterbag reduction,
Non-homogeneous systems of hydrodynamic type, Riemann invariants
PACS: 02.30.Ik
## 1 Introduction
In this paper we study an integrable system introduced recently by Manakov and
Santini [1] (see also [2, 3]). This system is connected with commutation of
general 2-dimensional vector fields (containing derivative on spectral
variable). Reduction to Hamiltonian vector fields leads to the well-known
dispersionless KP (or Khokhlov-Zabolotskaya) equation. Alternatively, a
natural reduction to 1-dimensional vector fields reduces Manakov-Santini
system to the equation introduced by Pavlov [4] (see also [5, 6, 7]). Using
general construction of the works [8, 11], we introduce the hierarchy for
Manakov-Santini system in Lax-Sato form and generating equation for it (the
hierarchy in terms of recursion operator was introduced in [2]). We introduce
waterbag ansatz for Manakov-Santini hierarchy and derive equations of the
reduced hierarchy. Using rational form of the G function (see below), one can
introduce new coordinates such that the non-hydrodynamic evolution of waterbag
reduction transforms to non-homogeneous Riemann invariants form of hydro-
dynamic type.
This paper is organized as follows. In section 2, GdKP hierarchy is described,
connection to Manakov-Santini system is demonstrated. In section 3, waterbag
reduction for Manakov-Santini hierarchy is introduced, equations of reduced
hierarchy are derived (in non-hydrodynamic form). In section 4, we introduce
new coordinates transforming the evolution of waterbag reduction to non-
homogeneous Riemann invariants form of hydro-dynamic type. The examples are
given. Section 5 is devoted to the concluding remarks.
## 2 Generalized dKP hierarchy
To introduce generalized dKP (Manakov-Santini) hierarchy, we use general
construction of the works [8, 11]. The hierarchy is described by the Lax-Sato
equations
$\displaystyle\frac{\partial\psi}{\partial
t_{n}}=A_{n}\frac{\partial\psi}{\partial x}-B_{n}\frac{\partial\psi}{\partial
p},\qquad\psi=\left(\begin{array}[]{c}\mathcal{L}\\\
\mathcal{M}\end{array}\right),$ (3)
or, equivalently, by the generating equation
$(J_{0}^{-1}\mathrm{d}\mathcal{L}\wedge\mathrm{d}\mathcal{M})_{-}=0,$ (4)
where $A_{n}\equiv(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial p)_{+}$,
$B_{n}\equiv(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial x)_{+}$ with the Lax
and Orlov operators $\mathcal{L}(p),\mathcal{M}(p)$ being the Laurent series
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\sum_{n=1}^{\infty}u_{n}(x)p^{-n},$ (5) $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\sum_{n=1}^{\infty}v_{n}(x)\mathcal{L}^{-n}.$
(6)
Here $(\cdots)_{+}$ ($(\cdots)_{-}$) denote respectively the projection on the
polynomial part (negative powers), and $J_{0}$ is defined by
$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\frac{\partial\mathcal{M}}{\partial x}-\frac{\partial\mathcal{L}}{\partial
x}\frac{\partial\mathcal{M}}{\partial p}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\left(\left.\frac{\partial\mathcal{M}}{\partial\mathcal{L}}\right|_{t_{n},v_{n}\
\rm{fixed}}\frac{\partial\mathcal{L}}{\partial
x}+\left.\frac{\partial\mathcal{M}}{\partial x}\right|_{\mathcal{L}\
\rm{fixed}}\right)-\frac{\partial\mathcal{L}}{\partial
x}\left.\frac{\partial\mathcal{M}}{\partial\mathcal{L}}\right|_{t_{n},v_{n}\
\rm{fixed}}\frac{\partial\mathcal{L}}{\partial p}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\left.\frac{\partial\mathcal{M}}{\partial x}\right|_{\mathcal{L}\
\rm{fixed}}=1+v_{1x}p^{-1}+(v_{2x}-u_{1})p^{-2}+\cdots.$
We list some of $A_{n}$ and $B_{n}$ as follows
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
A_{2}$ $\displaystyle=$ $\displaystyle 2p-2v_{1x},$ $\displaystyle A_{3}$
$\displaystyle=$ $\displaystyle 3p^{2}-3v_{1x}p+6u_{1}+3(v_{1x})^{2}-3v_{2x},$
$\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle
4p^{3}-4v_{1x}p^{2}+(12u_{1}+4(v_{1x})^{2}-4v_{2x})p$ (7)
$\displaystyle+12u_{2}-4v_{3x}+8v_{1x}v_{2x}-4(v_{1x})^{3}-8u_{1}v_{1x},$
and
$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B_{2}$ $\displaystyle=$ $\displaystyle 2u_{1x},$ $\displaystyle B_{3}$
$\displaystyle=$ $\displaystyle 3u_{1x}p-3u_{1x}v_{1x}+3u_{2x},$
$\displaystyle B_{4}$ $\displaystyle=$ $\displaystyle
4u_{1x}p^{2}+(4u_{2x}-4u_{1x}v_{1x})p$ (8)
$\displaystyle+4u_{1x}(4u_{1}+(v_{1x})^{2}-v_{2x})-4u_{2x}v_{1x}+4u_{3x}.$
The $t_{1}$ flow of the generalized dKP hierarchy (3) says that the dependence
on $t_{1}$ and $x$ appear in the linear combination $t_{1}+x$.
###### Proposition 2.1
The compatibility of the commuting flow
$[\partial_{t_{m}},\partial_{t_{n}}]\psi=0$ requires $A_{n},B_{n}$ to satisfy
$\displaystyle\partial_{t_{m}}A_{n}-\partial_{t_{n}}A_{m}$ $\displaystyle=$
$\displaystyle\langle A_{m},A_{n}\rangle_{x}+B_{n}A_{mp}-B_{m}A_{np},$
$\displaystyle\partial_{t_{m}}B_{n}-\partial_{t_{n}}B_{m}$ $\displaystyle=$
$\displaystyle\langle B_{n},B_{m}\rangle_{p}+A_{m}B_{nx}-A_{n}B_{mx},$ (9)
where $\langle U,V\rangle_{i}:=U(\partial_{i}V)-(\partial_{i}U)V$.
Proof. Substituting (3) into
$\partial_{t_{m}}\partial_{t_{n}}\psi=\partial_{t_{n}}\partial_{t_{m}}\psi$,
and comparing the coefficients of independent variables $\psi_{x}$ and
$\psi_{p}$ respectively to the both sides, we obtain (9). $\square$
The evolution of $\mathcal{L},\mathcal{M}$ with respect to $t_{2}=y$ in (3)
are given by
$\displaystyle\frac{1}{2}\frac{\partial\mathcal{L}}{\partial y}$
$\displaystyle=$ $\displaystyle(p-v_{1x})\frac{\partial\mathcal{L}}{\partial
x}-u_{1x}\frac{\partial\mathcal{L}}{\partial p},$ (10)
$\displaystyle\frac{1}{2}\frac{\partial\mathcal{M}}{\partial y}$
$\displaystyle=$ $\displaystyle(p-v_{1x})\frac{\partial\mathcal{M}}{\partial
x}-u_{1x}\frac{\partial\mathcal{M}}{\partial p}.$ (11)
Using the convention $(\sum_{n}a_{n}p^{n})_{[s]}=a_{s}$ for a formal Laurent
series, then from Eq.(10) we have
$\displaystyle\frac{1}{2}u_{1y}$ $\displaystyle=$
$\displaystyle\Big{(}(p-v_{1x})\mathcal{L}_{x}-u_{1x}\mathcal{L}_{p}\Big{)}_{[-1]}=u_{2x}-v_{1x}u_{1x},$
(12) $\displaystyle\frac{1}{2}u_{2y}$ $\displaystyle=$
$\displaystyle\Big{(}(p-v_{1x})\mathcal{L}_{x}-u_{1x}\mathcal{L}_{p}\Big{)}_{[-2]}=u_{3x}-v_{1x}u_{2x}+u_{1}u_{1x},$
(13)
On the other hand, the expression of Eq.(11) together with (10) gives
$\mathcal{L}+\frac{1}{2}\sum_{n=1}^{\infty}v_{ny}\mathcal{L}^{-n}=(p-v_{1x})\left(1+\sum_{n=1}^{\infty}v_{nx}\mathcal{L}^{-n}\right).$
Comparing the coefficients of powers $p^{-1}$ and $p^{-2}$ to the above, we
have
$\displaystyle v_{2x}$ $\displaystyle=$ $\displaystyle
u_{1}+v_{1x}^{2}+\frac{1}{2}v_{1y},$ (14) $\displaystyle v_{3x}$
$\displaystyle=$ $\displaystyle
u_{2}+\frac{1}{2}v_{2y}+u_{1}v_{1x}+v_{1x}v_{2x}.$ (15)
Similarly, the evolution of $\mathcal{L},\mathcal{M}$ w.r.t. $t_{3}=t$ are
given by
$\displaystyle\frac{1}{3}\frac{\partial\mathcal{L}}{\partial t}$
$\displaystyle=$
$\displaystyle\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\frac{\partial\mathcal{L}}{\partial
x}-\left(u_{1x}p+\frac{1}{2}u_{1y}\right)\frac{\partial\mathcal{L}}{\partial
p},$ (16) $\displaystyle\frac{1}{3}\frac{\partial\mathcal{M}}{\partial t}$
$\displaystyle=$
$\displaystyle\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\frac{\partial\mathcal{M}}{\partial
x}-\left(u_{1x}p+\frac{1}{2}u_{1y}\right)\frac{\partial\mathcal{M}}{\partial
p},$ (17)
Then the $t$-flow of $u_{1}$ can be read by Eq.(16) by taking the coefficient
of $p^{-1}$:
$\displaystyle\frac{1}{3}u_{1t}$ $\displaystyle=$ $\displaystyle
u_{3x}-v_{1x}u_{2x}+(u_{1}-\frac{1}{2}v_{1y})u_{1x}+u_{1}u_{1x},$ (18)
$\displaystyle=$
$\displaystyle\frac{1}{2}u_{2y}-\frac{1}{2}u_{1x}v_{1y}+u_{1}u_{1x},$
where we have used (13) to reach the second line. Also, the expression of
Eq.(17) together with (16) gives
$\mathcal{L}^{2}+\frac{1}{3}\sum_{n=1}^{\infty}v_{nt}\mathcal{L}^{-n}=\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\left(1+\sum_{n=1}^{\infty}v_{nx}\mathcal{L}^{-n}\right),$
in which the coefficient of $p^{-1}$ gives
$\frac{1}{3}v_{1t}=-u_{2}+\frac{1}{2}v_{2y}+u_{1}v_{1x}-\frac{1}{2}v_{1x}v_{1y},$
(19)
where we have used Eq.(15). Now differentiating Eqs.(18), (19) respectively
with respect to $x$ and eliminating $u_{2x}$ and $v_{2x}$ by Eqs.(12) and
(14), we obtain the following two coupled equations for $u_{1}:=u$ and
$v_{1}=v$:
$\displaystyle\frac{1}{3}u_{xt}$ $\displaystyle=$
$\displaystyle\frac{1}{4}u_{yy}+(uu_{x})_{x}+\frac{1}{2}v_{x}u_{xy}-\frac{1}{2}u_{xx}v_{y},$
$\displaystyle\frac{1}{3}v_{xt}$ $\displaystyle=$
$\displaystyle\frac{1}{4}v_{yy}+uv_{xx}+\frac{1}{2}v_{x}v_{xy}-\frac{1}{2}v_{xx}v_{y}.$
(20)
Eq.(20) is the so called _Manakov-Santini equation_ [1, 2, 3]. The Lax pair
for this equation is defined by linear equations (10,11) and (16,17). Notice
that for $v=0$ reduction, the system reduces to the dKP equation
$\frac{1}{3}u_{xt}=\frac{1}{4}u_{yy}+(uu_{x})_{x}.$ (21)
Respectively, $u=0$ reduction gives an equation [4] (see also [5, 6, 7])
$\frac{1}{3}v_{xt}=\frac{1}{4}v_{yy}+\frac{1}{2}v_{x}v_{xy}-\frac{1}{2}v_{xx}v_{y}.$
(22)
###### Proposition 2.2
Equation (3) can be written in Hamilton-Jacobi type equation
$\left.\frac{\partial p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}\
\rm{fixed}}=\left.A_{n}(p(\mathcal{L}))\frac{\partial p(\mathcal{L})}{\partial
x}\right|_{\mathcal{L}\ \rm{fixed}}+B_{n}(p(\mathcal{L})),$ (23)
where $A_{n}(p)=(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial p)_{+}$ and
$B_{n}(p)=(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial x)_{+}$.
Proof. By taking into account the partial derivatives with respect to $t_{n}$
for fixed $p$ or $\mathcal{L}$, it is easy to show that
$\frac{\partial p}{\partial t_{n}}=0=\left.\frac{\partial
p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}}+\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
t_{n}},$
or
$\left.\frac{\partial p(\mathcal{L})}{\partial
t_{n}}\right|_{\mathcal{L}}=-\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
t_{n}}.$ (24)
Using (3), and (24) with $n=1$, we have
$\left.\frac{\partial p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}\
\rm{fixed}}=-\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\left(A_{n}(p)\frac{\partial\mathcal{L}}{\partial
x}-B_{n}(p)\frac{\partial\mathcal{L}}{\partial
p}\right)=\left.A_{n}(p(\mathcal{L}))\frac{\partial p(\mathcal{L})}{\partial
x}\right|_{\mathcal{L}\ \rm{fixed}}+B_{n}(p(\mathcal{L})).\,\square$
###### Proposition 2.3
The function
$J_{0}=\partial_{p}\mathcal{L}\partial_{x}\mathcal{M}-\partial_{x}\mathcal{L}\partial_{p}\mathcal{M}$
and its inverse $G=J_{0}^{-1}$ satisfy
$\displaystyle\partial_{t_{n}}J_{0}$ $\displaystyle=$
$\displaystyle\left(A_{n}J_{0}\right)_{x}-\left(B_{n}J_{0}\right)_{p},$ (25)
$\displaystyle\partial_{t_{n}}G$ $\displaystyle=$ $\displaystyle\langle
A_{n},G\rangle_{x}-\langle B_{n},G\rangle_{p},$ (26)
where $\langle U,V\rangle_{i}:=U(\partial_{i}V)-(\partial_{i}U)V$.
Proof. Using the $t_{n}$-flows of $\mathcal{L},\mathcal{M}$ in (3) and the
definition of $J_{0}$, we have
$\displaystyle\partial_{t_{n}}J_{0}$ $\displaystyle=$
$\displaystyle(\mathcal{L}_{p})_{t_{n}}\mathcal{M}_{x}+\mathcal{L}_{p}(\mathcal{M}_{x})_{t_{n}}-(\mathcal{L}_{x})_{t_{n}}\mathcal{M}_{p}-\mathcal{L}_{x}(\mathcal{M}_{p})_{t_{n}},$
$\displaystyle=$ $\displaystyle-
B_{np}J_{0}+A_{nx}J_{0}+A_{n}J_{0x}-B_{n}J_{0p},$ $\displaystyle=$
$\displaystyle(A_{n}J_{0})_{x}-(B_{n}J_{0})_{p}.$
Moreover, substituting $J_{0}=G^{-1}$ into the above we obtain (26). $\square$
As we will see, Proposition 2.3 can provide a crucial way to determine the
hierarchy flows.
## 3 Waterbag-type reduction
Consider the waterbag-type reduction of the generalized dKP hierarchy
represented by [8]
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\sum_{i=1}^{N}\epsilon_{i}\log(p-U_{i}),$ (27) $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\sum_{i=1}^{M}\delta_{i}\log(p-V_{i}),$
(28)
where $\epsilon_{i}$ and $\delta_{i}$ are assumed to satisfy
$\sum_{i=1}^{N}\epsilon_{i}=\sum_{i=1}^{M}\delta_{i}=0.$ (29)
The ansatz (27,28) is consistent with the dynamics defined by Manakov-Santini
hierarchy (3), i.e., the form of ansatz is preserved by the dynamics.
Condition (29) guarantees that expansion of $\mathcal{L}$, $\mathcal{M}$ at
infinity is of the form (5,6). Reduced hierarchy is represented as infinite
set of (1+1)-dimensional systems of equations for the functions $U_{i}$,
$V_{i}$, which are obtained by the substitution of ansatz (27,28) to equations
of Manakov-Santini hierarchy (3).
Let us consider first flows of reduced hierarchy. For expansion of
$\mathcal{L}$, $\mathcal{M}$ at infinity from (27,28) we get
$\displaystyle\mathcal{L}=p-\sum_{n=1}^{\infty}\left(\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{n}}{n}\right)p^{-n},$
(30)
$\displaystyle\mathcal{M}=\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}-\sum_{n=1}^{\infty}\left(\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{n}}{n}\right)p^{-n}.$
(31)
Comparing these expansions with formulae (5,6), we come to the conclusion that
$u_{n}=-\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{n}}{n}$. To calculate $v_{n}$,
we should invert the series (30) to find $p(\mathcal{L})$ that can be done
recursively, and substitute $p(\mathcal{L})$ to (31) . For the first
coefficients $u_{n}$, $v_{n}$ we get
$\displaystyle u_{1}=-\sum_{i=1}^{N}\epsilon_{i}U_{i},\quad
u_{2}=-\frac{1}{2}\sum_{i=1}^{N}\epsilon_{i}U_{i}^{2},$ $\displaystyle
v_{1}=-\sum_{i=1}^{M}\delta_{i}V_{i},\quad
v_{2}=-\frac{1}{2}\sum_{i=1}^{M}\delta_{i}V_{i}^{2}.$
Substituting these expressions to relations (7), (8) and using equations (3),
we obtain equations of reduced hierarchy. Equations of the flow corresponding
to $y=t_{2}$ read
$\displaystyle\partial_{y}U_{k}=\Big{(}2U_{k}+\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\Big{)}\partial_{x}U_{k}-2\partial_{x}\Big{(}\sum_{i=1}^{N}\epsilon_{i}U_{i}\Big{)},$
$\displaystyle\partial_{y}V_{k}=\Big{(}2V_{k}+\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\Big{)}\partial_{x}V_{k}-2\partial_{x}\Big{(}\sum_{i=1}^{N}\epsilon_{i}U_{i}\Big{)}.$
(32)
For the flow corresponding to $t=t_{3}$ we get
$\displaystyle\partial_{t}U_{k}=\left(3U_{k}^{2}+3U_{k}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}-6\sum_{i=1}^{N}\epsilon_{i}U_{i}+3\bigl{(}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\bigr{)}^{2}+3\partial_{x}\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{2}}{2}\right)\partial_{x}U_{k}-$
$\displaystyle\qquad\left(3U_{k}\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i}+3(\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i})(\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i})+3\partial_{x}\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{2}}{2}\right),$
$\displaystyle\partial_{t}V_{k}=\left(3V_{k}^{2}+3V_{k}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}-6\sum_{i=1}^{N}\epsilon_{i}U_{i}+3\bigl{(}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\bigr{)}^{2}+3\partial_{x}\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{2}}{2}\right)\partial_{x}V_{k}-$
$\displaystyle\qquad\left(3V_{k}\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i}+3(\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i})(\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i})+3\partial_{x}\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{2}}{2}\right).$
(33)
A common solution to the systems (32), (33) gives a solution to Manakov-
Santini equation (20) defined as $u=-\sum_{i=1}^{N}\epsilon_{i}U_{i}$,
$v=-\sum_{i=1}^{M}\delta_{i}V_{i}$.
## 4 Diagonal form of reduced hierarchy
For the waterbag reduction (27, 28) one can show that the $G$ function can be
expressed in the following form
$G=J_{0}^{-1}=\frac{\prod_{i=1}^{N}(p-U_{i})\prod_{j=1}^{M}(p-V_{j})}{F(U_{n},U_{nx},V_{m},V_{mx};p)},\qquad
n=1,\ldots,N;\,m=1,\ldots,M,$ (34)
where the function $F$ in denominator is a polynomial of $p$ with degree
$N+M$. In general, $F$ can also be factorized into
$\prod_{k=1}^{N+M}(p-W_{k})$, for which
$W_{k}=W_{k}(U_{n},U_{nx},V_{m},V_{mx})$ are roots of $J_{0}$. We like to
mention here that the derivatives $U_{nx},V_{mx}$ can be inversely expressed
as function of the form
$U_{nx}=f_{n}(U_{i},V_{j},W_{k}),\quad V_{mx}=g_{m}(U_{i},V_{j},W_{k}).$ (35)
Therefore, we have
$J_{0}=\frac{\prod_{k=1}^{N+M}(p-W_{k})}{\prod_{i=1}^{N}(p-U_{i})\prod_{j=1}^{M}(p-V_{j})},\qquad
n=1,\ldots,N;\,m=1,\ldots,M.$ (36)
As the result, the evaluation of $G$ at $U_{i}$ or $V_{i}$, i.e.,
$G(p=U_{i})=0$ or $G(p=V_{i})=0$ shows that Eq.(26) can be written into the
following evolution equations of $U_{i}$, $V_{i}$:
$\displaystyle\frac{\partial U_{i}}{\partial t_{n}}$ $\displaystyle=$
$\displaystyle A_{n}(p=U_{i})\frac{\partial U_{i}}{\partial
x}+B_{n}(p=U_{i}),$ (37) $\displaystyle\frac{\partial V_{i}}{\partial t_{n}}$
$\displaystyle=$ $\displaystyle A_{n}(p=V_{i})\frac{\partial V_{i}}{\partial
x}+B_{n}(p=V_{i}).$ (38)
Similarly, Eq.(25) with $J_{0}(p=W_{i})=0$ gives rise
$\frac{\partial W_{i}}{\partial t_{n}}=A_{n}(p=W_{i})\frac{\partial
W_{i}}{\partial x}+B_{n}(p=W_{i}).$ (39)
In summary, combining (37), (38), (39) and replacing those $U_{nx}$’s and
$V_{mx}$’s in $A_{n},B_{n}$ with the transformations (35), we obtain the _non-
homogeneous Riemann invariant form_ as
$\partial_{t_{n}}R_{i}=A_{n}(p=R_{i})R_{ix}+B_{n}(p=R_{i}),\quad
i=1,\ldots,2N+2M,$ (40)
for which
$(R_{1},\ldots,R_{2N+2M})=(U_{1},\ldots,U_{N};V_{1},\ldots,V_{M};W_{1},\ldots,W_{N+M})$.
Some linearly degenerate non-homogeneous Riemann invariants forms, associated
with commuting quadratic Hamiltonians and the Killing vector fields of the
given metric, were investigated in [9, 10]. However, in our case equation (40)
is obviously not linearly degenerate.
Remark. For the type of non-homogeneous Riemann invariant form
$\partial_{t_{n}}R^{i}=\Lambda_{n}^{i}(\mathbf{R})R^{i}_{x}+Q_{n}^{i}(\mathbf{R}),$
(41)
the requirements of the commutativity are equivalent to the following
restrictions on their characteristic speeds and non-homogeneous terms (see
appendix A)
$\frac{\partial_{j}\Lambda_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}\Lambda_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}},\quad\frac{\partial_{j}Q_{n}^{i}}{Q_{n}^{j}}=\frac{\partial_{j}Q_{m}^{i}}{Q_{m}^{j}},\quad\frac{Q_{n}^{j}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{Q_{m}^{j}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}},\quad
i\neq j,\quad n\neq m.$
where $\partial_{i}\equiv\partial/\partial R^{i}$.
Example 1. $(N,M)=(1,1)$ reduction. In this case,
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle p+\log(1-U/p),$
$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\log(1-V/p).$
Comparing to the expansion of (5,6) we have $u_{n}=-U^{n}/n$ for $n\geq 1$ and
$v_{1}=-V,v_{2}=-V^{2}/2,v_{3}=UV-V^{3}/3$, etc. These transformations allow
us to get $A_{n},B_{n}$ (by Eqs.(7), (8)) which correspond to the reduced
system. The $G$ function is given by
$G=\frac{p(p-U)(p-V)}{\prod_{i=1}^{3}(p-W_{i})},$ (42)
where $W_{i}$ satisfy
$\sum_{i=1}^{3}W_{i}=U+V+V_{x},\quad\sum_{\scriptstyle
i,j=1\atop\scriptstyle(i>j)}^{3}W_{i}W_{j}=U+UV+UV_{x},\quad\prod_{i=1}^{3}W_{i}=UV+UV_{x}-U_{x}V.$
(43)
Notice that (42) is not coincident with that in (34), there is one more root
of $p=0$ to be considered. By (26), it turns out that the evaluation of $p=0$
gives an additional condition, namely
$UVB_{n}(p=0)=0,\quad\forall n\geq 1.$ (44)
There are two simple cases: (i) $V=0,U\neq 0$, (ii) $V\neq 0,U=0$. One can
easily deduce considering $t_{2}$-flow of (40) that case (i) is a trivial
reduction. For the case (ii), we have the fact that $B_{n}(U=0)=0$ for $n\geq
1$, and Eq.(43) will reveal us the only one relation: $V_{x}=W-V$. To this
end, system (40) reduces to the type of homogeneous one in (41) with
$Q_{n}^{i}=0$, namely
$\partial_{t_{n}}R^{i}=\Lambda_{n}^{i}(\mathbf{R})\partial_{x}R^{i},$ (45)
where $\mathbf{R}=(R^{1},R^{2})=(V,W)$ and the characteristic speeds
$\Lambda_{n}^{i}=A_{n}(p=R^{i},U=0)$. For instance, for $t_{2}=y$ flow, we
have $A_{2}(U=0)=2p+2V_{x}=2p+2(W-V)$, then Eq.(45) becomes
$\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{y}=\left(\begin{array}[]{cc}2W&0\\\
0&4W-2V\end{array}\right)\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{x}.$ (46)
For $t_{3}=t$ flow, we derive $A_{3}(U=0)=3p^{2}+3(W-V)p+3(W-V)^{2}+3V(W-V)$,
thus
$\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{t}=\left(\begin{array}[]{cc}3W^{2}&0\\\
0&9W^{2}-6VW\end{array}\right)\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{x}.$ (47)
From the two nontrivial flows (46), (47), we readily obtain the following set
of hodograph equation
$\displaystyle x+2Wy+3W^{2}t$ $\displaystyle=$ $\displaystyle\hat{F}(V,W),$
$\displaystyle x+(4W-2V)y+(9W^{2}-6VW)t$ $\displaystyle=$
$\displaystyle\hat{G}(V,W),$ (48)
where $\hat{F}$ and $\hat{G}$ satisfy the linear equations
$\displaystyle(W-V)\hat{G}_{V}$ $\displaystyle=$
$\displaystyle\hat{F}-\hat{G},$ $\displaystyle(W-V)\hat{F}_{W}$
$\displaystyle=$ $\displaystyle\hat{G}-\hat{F}.$
Dividing these two equations for $V\neq W$ we get $\hat{G}_{V}=-\hat{F}_{W}$.
It follows that there exists a function $\phi$ such that
$\hat{F}=\phi_{V},\hat{G}=-\phi_{W}$, whence $\phi$ satisfies the defining
equation
$(V-W)\phi_{VW}=\phi_{V}+\phi_{W}.$ (49)
Eq. (49) has a general solution of the form
$\phi=(V-W)\left(f(W)+\int\frac{g(V)}{(V-W)^{2}}\,dV\right),$
where $f(W)$ and $g(V)$ are arbitrary functions of $W$ and $V$, respectively.
Choosing, for example $f(W)=W^{3}$, $g(V)=\mbox{Const.}$, then we have
$\hat{F}=W^{3}$ and $\hat{G}=-3VW^{2}+4W^{3}$. Substituting back into the
hodograph equation (48) we solve
$\displaystyle V=W$ $\displaystyle=$
$\displaystyle\frac{1}{6}h^{-1/3}(24y+36t^{2}+6th^{1/3}+h^{2/3}),$
$\displaystyle h$ $\displaystyle=$ $\displaystyle
216yt+108x+216t^{3}+12\sqrt{-96y^{3}-108y^{2}t^{2}+324ytx+81x^{2}+324xt^{3}},$
which satisfies the $t_{2}$\- and $t_{3}$-flows (46), (47). However, $V=W$
contradicts to the relation $W=V+V_{x}$ and $V$ does not satisfy equation
(22). Actually, from equation (49) we can see that when $V=W$ one can get
$\hat{F}=\hat{G}$. Then we obtain all the solutions will satisfy $V=W$.
Consequently, there is no (1,1)- reduction. Similar considerations can show
that there are no (1,2)- and (2,1)- reductions, either.
Example 2. $(N,M)=(2,2)$ reduction. In this case,
$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\epsilon_{1}\log\frac{p-U_{1}}{p-U_{2}},$ $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\delta_{1}\log\frac{p-V_{1}}{p-V_{2}}.$
For simplicity, we set $\epsilon_{1}=\delta_{1}=1$. Comparing to the expansion
of (5,6), we have $u_{n}=(U_{2}^{n}-U_{1}^{n})/n$ for $n\geq 1$ and
$v_{1}=V_{2}-V_{1},v_{2}=(V_{2}^{2}-V_{1}^{2})/2,v_{3}=u_{1}v_{1}+(V_{2}^{3}-V_{1}^{3})/3,\ldots$.
Now we expand the hierarchy flow of $U_{i}$, $V_{i}$ and $W_{i}$ up to
$t_{2}=y,t_{3}=t$. From (34) with $(N,M)=(2,2)$ we have
$G=\frac{\prod_{i=1}^{2}(p-U_{i})\prod_{j=1}^{2}(p-V_{j})}{\prod_{k=1}^{4}(p-W_{k})}.$
where $W_{i}$ satisfy
$\displaystyle\sum_{i=1}^{4}W_{i}$ $\displaystyle=$ $\displaystyle
U_{1}+U_{2}+V_{1}+V_{2}+V_{1x}-V_{2x},$ (50) $\displaystyle\sum_{\scriptstyle
i,j=1\atop\scriptstyle(i>j)}^{4}W_{i}W_{j}$ $\displaystyle=$ $\displaystyle
U_{1}-U_{2}+U_{1}U_{2}+V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x}$ (51)
$\displaystyle+(U_{1}+U_{2})(V_{1}+V_{2}+V_{1x}-V_{2x}),$
$\displaystyle\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}$ $\displaystyle=$
$\displaystyle(U_{1}+U_{2})(V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x})+(U_{2x}-U_{1x})(V_{1}-V_{2})$
(52) $\displaystyle+(U_{1}-U_{2}+U_{1}U_{2})(V_{1}+V_{2}+V_{1x}-V_{2x}),$
$\displaystyle\prod_{i=1}^{4}W_{i}$ $\displaystyle=$
$\displaystyle(U_{1}-U_{2}+U_{1}U_{2})(V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x})$
(53) $\displaystyle-(V_{1}-V_{2})(U_{1x}U_{2}-U_{1}U_{2x}),$
from which, one can substitute into $A_{n},B_{n}$ to eliminate
$U_{ix},V_{ix}$, etc. For $n=2$, using (50), (52) we have
$\displaystyle A_{2}(p)$ $\displaystyle=$ $\displaystyle
2p+2(V_{1x}-V_{2x})=2p+2\Big{(}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)},$
$\displaystyle=$ $\displaystyle
2(p-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8}),$
and the non-homogeneous term
$\displaystyle B_{2}(p)=2(U_{2x}-U_{1x}),$
$\displaystyle=\frac{2}{V_{1}-V_{2}}\Bigg{[}\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}+(U_{1}+U_{2})\bigg{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}^{4}W_{i}W_{j}\bigg{)}$
$\displaystyle\quad+\Bigg{.}\bigg{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\bigg{)}\bigg{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\bigg{)}\Bigg{]},$
$\displaystyle=\frac{2}{R_{3}-R_{4}}\Big{[}(R_{1}+R_{2}-R_{5}-R_{6}-R_{7}-R_{8})(R_{1}-R_{2}-R_{1}R_{2}-R_{1}^{2}-R_{2}^{2})\Big{.}$
$\displaystyle\quad+(R_{1}+R_{2})(R_{1}-R_{2}+R_{1}R_{2}-R_{5}R_{6}-R_{5}R_{7}-R_{5}R_{8}-R_{6}R_{7}-R_{6}R_{8}-R_{7}R_{8})$
$\displaystyle\quad+R_{5}R_{6}R_{7}+R_{6}R_{7}R_{8}+R_{7}R_{8}R_{5}+R_{8}R_{5}R_{6}\Big{]}.$
Then the $t_{2}=y$ flow in (40) is now read
$\partial_{y}R_{i}=2(R_{i}-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8})R_{ix}+B_{2}.$
(54)
For $n=3$, Eq.(40) becomes
$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle A_{3}(p=R_{i})R_{ix}+B_{3}(p=R_{i}),$ $\displaystyle=$
$\displaystyle\left[3p^{2}+3(V_{1x}-V_{2x})p+6(U_{2}-U_{1})+3(V_{1x}-V_{2x})^{2}+\frac{3}{2}(V_{1}^{2}-V_{2}^{2})_{x}\right]_{p=R_{i}}R_{ix}$
$\displaystyle+\left[3(U_{2x}-U_{1x})p-3(U_{2x}-U_{1x})(V_{2x}-V_{1x})+\frac{3}{2}(U_{2}^{2}-U_{1}^{2})_{x}\right]_{p=R_{i}}$
$\displaystyle=$
$\displaystyle\left[3R_{i}^{2}+3R_{i}(V_{1x}-V_{2x})+6(U_{2}-U_{1})+3(V_{1x}-V_{2x})^{2}+\frac{3}{2}(V_{1}^{2}-V_{2}^{2})_{x}\right]R_{ix}$
$\displaystyle+3R_{i}(U_{2x}-U_{1x})-3(U_{2x}-U_{1x})(V_{2x}-V_{1x})+3(U_{2}U_{2x}-U_{1}U_{1x}).$
Using Eqs.(50)–(53) we arrive
$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle
3R_{ix}\Bigg{[}U_{2}-U_{1}+R_{i}\Big{(}R_{i}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)}$
$\displaystyle+U_{1}U_{2}-V_{1}V_{2}-V_{1}^{2}-V_{2}^{2}-\sum_{i>j}^{4}W_{i}W_{j}+\Big{(}U_{1}+U_{2}+V_{1}+V_{2}-\sum_{i=1}^{4}W_{i}\Big{)}^{2}$
$\displaystyle\qquad+(U_{1}+U_{2}+V_{1}+V_{2})\Big{(}-U_{1}-U_{2}+\sum_{i=1}^{4}W_{i}\Big{)}\Bigg{]}$
$\displaystyle+\frac{3R_{i}}{V_{1}-V_{2}}\Bigg{[}(U_{1}+U_{2})\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}^{4}W_{i}W_{j}\Big{)}+\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}$
$\displaystyle\quad+\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}\Bigg{]}+$
$\displaystyle+3\Big{(}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)}\times$
$\displaystyle\times\frac{1}{V_{1}-V_{2}}\Bigg{(}\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}$
$\displaystyle\qquad+(U_{1}+U_{2})\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}W_{i}W_{j}\Big{)}+\sum_{i-1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}\Bigg{)}$
$\displaystyle+\frac{3}{V_{1}-V_{2}}\Bigg{(}(U_{1}+U_{2})\sum_{i-1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}+(U_{1}+U_{2})^{2}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}W_{i}W_{j}\Big{)}$
$\displaystyle-\prod_{i=1}^{4}W_{i}+(U_{1}+U_{2})\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}$
$\displaystyle+(U_{1}-U_{2}+U_{1}U_{2})\Big{(}\sum_{i>j}^{4}W_{i}W_{j}-(U_{1}-U_{2})-U_{1}U_{2}+(U_{1}+U_{2})^{2}-(U_{1}+U_{2})\sum_{i=1}^{4}W_{i}\Big{)}\Bigg{)}.$
Expressing in terms of $R_{i},i=1,\ldots,8$, we get
$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle
3R_{ix}\Bigg{[}R_{2}-R_{1}+R_{i}(R_{i}-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8})$
$\displaystyle\quad+R_{1}R_{2}+R_{3}R_{4}+R_{5}R_{6}+R_{5}R_{7}+R_{5}R_{8}+R_{6}R_{7}+R_{6}R_{8}+R_{7}R_{8}$
$\displaystyle\quad+R_{1}R_{3}+R_{1}R_{4}-R_{1}R_{5}-R_{1}R_{6}-R_{1}R_{7}-R_{1}R_{8}+R_{2}R_{3}+R_{2}R_{4}$
$\displaystyle\quad-
R_{2}R_{5}-R_{2}R_{6}-R_{2}R_{7}-R_{2}R_{8}-R_{3}R_{5}-R_{3}R_{6}-R_{3}R_{7}-R_{3}R_{8}$
$\displaystyle\quad-
R_{4}R_{5}-R_{4}R_{6}-R_{4}R_{7}-R_{4}R_{8}+R_{5}^{2}+R_{6}^{2}+R_{7}^{2}+R_{8}^{2}\Bigg{]}$
$\displaystyle+\frac{3R_{i}}{R_{3}-R_{4}}\Big{(}2R_{1}^{2}-2R_{2}^{2}-R_{1}R_{5}-R_{1}R_{6}-R_{1}R_{7}-R_{1}R_{8}+R_{2}R_{5}+R_{2}R_{6}$
$\displaystyle\quad+R_{2}R_{7}+R_{2}R_{8}-R_{1}^{3}-R_{2}^{3}-R_{1}^{2}R_{2}+R_{1}^{2}R_{5}+R_{1}^{2}R_{6}+R_{1}^{2}R_{7}+R_{1}^{2}R_{8}$
$\displaystyle\quad-
R_{1}R_{2}^{2}+R_{2}^{2}R_{5}+R_{2}^{2}R_{6}+R_{2}^{2}R_{7}+R_{2}^{2}R_{8}+R_{1}R_{2}R_{5}+R_{1}R_{2}R_{6}$
$\displaystyle\quad+R_{1}R_{2}R_{7}+R_{1}R_{2}R_{8}-R_{1}R_{5}R_{6}-R_{1}R_{5}R_{7}-R_{1}R_{5}R_{8}-R_{1}R_{6}R_{7}$
$\displaystyle\quad-
R_{1}R_{6}R_{8}-R_{1}R_{7}R_{8}-R_{2}R_{5}R_{6}-R_{2}R_{5}R_{7}-R_{2}R_{5}R_{8}-R_{2}R_{6}R_{7}$
$\displaystyle\quad-
R_{2}R_{6}R_{8}-R_{2}R_{7}R_{8}+R_{5}R_{6}R_{7}+R_{6}R_{7}R_{8}+R_{7}R_{8}R_{5}+R_{8}R_{5}R_{6}\Big{)}+$
$\displaystyle+\frac{3}{R_{3}-R_{4}}\Big{(}-R_{1}^{2}-R_{2}^{2}+2R_{1}R_{2}+R_{1}^{3}-R_{2}^{3}-R_{1}^{2}R_{2}+R_{1}R_{2}^{2}-2R_{1}^{2}R_{3}+2R_{2}^{2}R_{3}$
$\displaystyle\quad-2R_{1}^{2}R_{4}+2R_{2}^{2}R_{4}+R_{1}^{2}R_{5}-R_{2}^{2}R_{5}+R_{1}^{2}R_{6}-R_{2}^{2}R_{6}+R_{1}^{2}R_{7}-R_{2}^{2}R_{7}$
$\displaystyle\quad+R_{1}^{2}R_{8}-R_{2}^{2}R_{8}-R_{1}R_{5}^{2}+R_{2}R_{5}^{2}+R_{1}R_{6}^{2}-R_{2}R_{6}^{2}+R_{1}R_{7}^{2}-R_{2}R_{7}^{2}$
$\displaystyle\quad+R_{1}R_{8}^{2}-R_{2}R_{8}^{2}+R_{1}^{3}R_{2}+R_{1}^{3}R_{3}+R_{1}^{3}R_{4}-R_{1}^{3}R_{5}-R_{1}^{3}R_{6}-R_{1}^{3}R_{7}-R_{1}^{3}R_{8}$
$\displaystyle\quad+R_{1}R_{2}^{3}+R_{2}^{3}R_{3}+R_{2}^{3}R_{4}-R_{2}^{3}R_{5}-R_{2}^{3}R_{6}-R_{2}^{3}R_{7}-R_{2}^{3}R_{8}$
$\displaystyle\quad+R_{1}^{2}R_{2}^{2}+R_{1}^{2}R_{5}^{2}+R_{2}^{2}R_{5}^{2}-R_{1}^{2}R_{6}^{2}-R_{2}^{2}R_{6}^{2}-R_{1}^{2}R_{7}^{2}-R_{2}^{2}R_{7}^{2}-R_{1}^{2}R_{8}^{2}-R_{2}^{2}R_{8}^{2}$
$\displaystyle\quad+R_{1}R_{5}R_{6}+R_{1}R_{5}R_{7}+R_{1}R_{5}R_{8}+3R_{1}R_{6}R_{7}+3R_{1}R_{6}R_{8}+3R_{1}R_{7}R_{8}$
$\displaystyle\quad-
R_{2}R_{5}R_{6}-R_{2}R_{5}R_{7}-R_{2}R_{5}R_{8}-3R_{2}R_{6}R_{7}-3R_{2}R_{6}R_{8}-3R_{2}R_{7}R_{8}$
$\displaystyle\quad+R_{3}R_{5}R_{1}-R_{3}R_{5}R_{2}-R_{3}R_{6}R_{1}+R_{3}R_{6}R_{2}-R_{3}R_{7}R_{1}+R_{3}R_{7}R_{2}$
$\displaystyle\quad-
R_{3}R_{8}R_{1}+R_{3}R_{8}R_{2}+R_{4}R_{5}R_{1}-R_{4}R_{5}R_{2}-R_{4}R_{6}R_{1}+R_{4}R_{6}R_{2}$
$\displaystyle\quad-
R_{4}R_{7}R_{1}+R_{4}R_{7}R_{2}-R_{4}R_{8}R_{1}+R_{4}R_{8}R_{2}-2R_{5}R_{1}^{2}R_{2}-2R_{6}R_{1}^{2}R_{2}$
$\displaystyle\quad-2R_{7}R_{1}^{2}R_{2}-2R_{1}R_{6}R_{2}^{2}-2R_{1}R_{7}R_{2}^{2}-2R_{1}R_{8}R_{2}^{2}-2R_{8}R_{1}^{2}R_{2}-2R_{1}^{2}R_{6}R_{7}$
$\displaystyle\quad-2R_{1}^{2}R_{6}R_{8}-2R_{1}^{2}R_{7}R_{8}-2R_{2}^{2}R_{6}R_{7}-2R_{2}^{2}R_{6}R_{8}-2R_{2}^{2}R_{7}R_{8}-2R_{1}R_{5}R_{2}^{2}$
$\displaystyle\quad+R_{5}^{2}R_{6}R_{7}+R_{7}R_{8}R_{5}^{2}+R_{8}R_{5}^{2}R_{6}+R_{5}^{2}R_{1}R_{2}-R_{1}R_{5}^{2}R_{6}-R_{1}R_{5}^{2}R_{7}-R_{1}R_{5}^{2}R_{8}$
$\displaystyle\quad-
R_{2}R_{5}^{2}R_{6}-R_{2}R_{5}^{2}R_{7}-R_{2}R_{5}^{2}R_{8}+R_{5}R_{6}^{2}R_{7}+R_{6}^{2}R_{7}R_{8}+R_{8}R_{5}R_{6}^{2}-R_{6}^{2}R_{1}R_{2}$
$\displaystyle\quad-
R_{1}R_{5}R_{6}^{2}-R_{1}R_{6}^{2}R_{7}-R_{1}R_{6}^{2}R_{8}-R_{2}R_{5}R_{6}^{2}-R_{2}R_{6}^{2}R_{7}-R_{2}R_{6}^{2}R_{8}+R_{5}R_{6}R_{7}^{2}$
$\displaystyle\quad+R_{6}R_{7}^{2}R_{8}+R_{7}^{2}R_{8}R_{5}-R_{7}^{2}R_{1}R_{2}-R_{1}R_{5}R_{7}^{2}-R_{1}R_{6}R_{7}^{2}-R_{1}R_{7}^{2}R_{8}-R_{2}R_{5}R_{7}^{2}$
$\displaystyle\quad-
R_{2}R_{6}R_{7}^{2}-R_{2}R_{7}^{2}R_{8}+R_{6}R_{7}R_{8}^{2}+R_{7}R_{8}^{2}R_{5}+R_{8}^{2}R_{5}R_{6}-R_{8}^{2}R_{1}R_{2}-R_{1}R_{5}R_{8}^{2}$
$\displaystyle\quad-
R_{1}R_{6}R_{8}^{2}-R_{1}R_{7}R_{8}^{2}-R_{2}R_{5}R_{8}^{2}-R_{2}R_{6}R_{8}^{2}-R_{2}R_{7}R_{8}^{2}+R_{3}R_{1}^{2}R_{2}+R_{3}R_{1}R_{2}^{2}$
$\displaystyle\quad-
R_{3}R_{5}R_{1}^{2}-R_{3}R_{5}R_{2}^{2}+R_{3}R_{6}R_{1}^{2}+R_{3}R_{6}R_{2}^{2}+R_{3}R_{7}R_{1}^{2}+R_{3}R_{7}R_{2}^{2}+R_{3}R_{8}R_{1}^{2}$
$\displaystyle\quad+R_{3}R_{8}R_{2}^{2}+R_{4}R_{1}^{2}R_{2}+R_{4}R_{1}R_{2}^{2}-R_{4}R_{5}R_{1}^{2}-R_{4}R_{5}R_{2}^{2}+R_{4}R_{6}R_{1}^{2}+R_{4}R_{6}R_{2}^{2}$
$\displaystyle\quad+R_{4}R_{7}R_{1}^{2}+R_{4}R_{7}R_{2}^{2}+R_{4}R_{8}R_{1}^{2}+R_{4}R_{8}R_{2}^{2}-3R_{1}R_{5}R_{6}R_{7}-3R_{1}R_{6}R_{7}R_{8}$
$\displaystyle\quad-3R_{1}R_{5}R_{7}R_{8}-3R_{1}R_{5}R_{6}R_{8}-3R_{2}R_{5}R_{6}R_{7}-3R_{2}R_{6}R_{7}R_{8}+3R_{5}R_{6}R_{7}R_{8}$
$\displaystyle\quad-3R_{2}R_{7}R_{8}R_{5}-3R_{2}R_{8}R_{5}R_{6}+R_{1}R_{2}R_{5}R_{6}+R_{1}R_{2}R_{5}R_{7}+R_{1}R_{2}R_{5}R_{8}$
$\displaystyle\quad-
R_{1}R_{2}R_{6}R_{7}-R_{1}R_{2}R_{6}R_{8}-R_{1}R_{2}R_{7}R_{8}-R_{3}R_{5}R_{6}R_{7}-R_{3}R_{6}R_{7}R_{8}$
$\displaystyle\quad-
R_{3}R_{7}R_{8}R_{5}-R_{3}R_{8}R_{5}R_{6}-R_{3}R_{5}R_{1}R_{2}+R_{3}R_{6}R_{1}R_{2}+R_{3}R_{7}R_{1}R_{2}$
$\displaystyle\quad+R_{3}R_{8}R_{1}R_{2}+R_{3}R_{1}R_{5}R_{6}+R_{3}R_{1}R_{5}R_{7}+R_{3}R_{1}R_{5}R_{8}+R_{3}R_{1}R_{6}R_{7}$
$\displaystyle\quad+R_{3}R_{1}R_{6}R_{8}+R_{3}R_{1}R_{7}R_{8}+R_{3}R_{2}R_{5}R_{6}+R_{3}R_{2}R_{5}R_{7}+R_{3}R_{2}R_{5}R_{8}$
$\displaystyle\quad+R_{3}R_{2}R_{6}R_{7}+R_{3}R_{2}R_{6}R_{8}+R_{3}R_{2}R_{7}R_{8}-R_{4}R_{5}R_{6}R_{7}-R_{4}R_{6}R_{7}R_{8}$
$\displaystyle\quad-
R_{4}R_{7}R_{8}R_{5}-R_{4}R_{8}R_{5}R_{6}-R_{4}R_{5}R_{1}R_{2}+R_{4}R_{6}R_{1}R_{2}+R_{4}R_{7}R_{1}R_{2}$
$\displaystyle\quad+R_{4}R_{8}R_{1}R_{2}+R_{4}R_{1}R_{5}R_{6}+R_{4}R_{1}R_{5}R_{7}+R_{4}R_{1}R_{5}R_{8}+R_{4}R_{1}R_{6}R_{7}$
$\displaystyle\quad+R_{4}R_{1}R_{6}R_{8}+R_{4}R_{1}R_{7}R_{8}+R_{4}R_{2}R_{5}R_{6}+R_{4}R_{2}R_{5}R_{7}+R_{4}R_{2}R_{5}R_{8}$
$\displaystyle\quad+R_{4}R_{2}R_{6}R_{7}+R_{4}R_{2}R_{6}R_{8}+R_{4}R_{2}R_{7}R_{8}\Big{)}.$
## 5 Concluding Remarks
In this article, we investigate the Manakov-Santini equation starting from
Lax-Sato formulation of associated hierarchy and obtain equations (25), (26),
which generalize the results of [6]. From these, one can introduce new
coordinates (34 ) such that the non-hydrodynamic evolution (32), (33) of
waterbag reduction transforms to non-homogeneous Riemann invariants form of
hydrodynamic type (40). The equation (40) is not linearly degenerate. Hence
the generalization of [9, 10] from linearly degenerate case to the general one
could be interesting. Also, the solution structures of (40) having infinite
symmetries should be investigated. These issues will be published elsewhere.
## Appendix
## Appendix A Commutability properties of the non-homogeneous diagonal system
We start from the commutability of (41) by
$\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$:
$\displaystyle\partial_{m}\partial_{n}R^{i}$
$\displaystyle=\partial_{m}(\Lambda_{n}^{i}R^{i}_{x})+\partial_{m}Q_{n}^{i},$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\partial_{m}R^{j})R^{i}_{x}+\Lambda_{n}^{i}\partial_{x}(\partial_{m}R^{i})+\sum_{j}(\partial_{j}Q_{n}^{i})(\partial_{m}R^{j}),$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j})R^{i}_{x}+\Lambda_{n}^{i}\partial_{x}(\Lambda_{m}^{i}R^{i}_{x}+Q_{m}^{i})+\sum_{j}(\partial_{j}Q_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j}),$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j})R^{i}_{x}+\Lambda_{n}^{i}\Big{(}\sum_{j}(\partial_{j}\Lambda_{m}^{i})R^{j}_{x}R^{i}_{x}+\Lambda_{m}^{i}R^{i}_{xx}+\sum_{j}(\partial_{j}Q_{m}^{i})R^{j}_{x}\Big{)}$
$\displaystyle\quad+\sum_{j}(\partial_{j}Q_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j}),$
$\displaystyle=\sum_{j}\Big{[}(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{j}+(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{i}\Big{]}R^{j}_{x}R^{i}_{x}+\sum_{j}(\partial_{j}\Lambda_{n}^{i})Q_{m}^{j}R^{i}_{x}+\Lambda_{n}^{i}\Lambda_{m}^{i}R^{i}_{xx}$
$\displaystyle\quad+\sum_{j}\Big{[}(\partial_{j}Q_{m}^{i})\Lambda_{n}^{i}+(\partial_{j}Q_{n}^{i})\Lambda_{m}^{j}\Big{]}R^{j}_{x}+\sum_{j}(\partial_{j}Q_{n}^{i})Q_{m}^{j}.$
Similarly,
$\displaystyle\partial_{n}\partial_{m}R^{i}$ $\displaystyle=$
$\displaystyle\partial_{n}(\Lambda_{m}^{i}R^{i}_{x})+\partial_{n}Q_{m}^{i},$
$\displaystyle=$
$\displaystyle\sum_{j}\Big{[}(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{j}+(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{i}\Big{]}R^{j}_{x}R^{i}_{x}+\sum_{j}(\partial_{j}\Lambda_{m}^{i})Q_{n}^{j}R^{i}_{x}+\Lambda_{m}^{i}\Lambda_{n}^{i}R^{i}_{xx}$
$\displaystyle\quad+\sum_{j}\Big{[}(\partial_{j}Q_{n}^{i})\Lambda_{m}^{i}+(\partial_{j}Q_{m}^{i})\Lambda_{n}^{j}\Big{]}R^{j}_{x}+\sum_{j}(\partial_{j}Q_{m}^{i})Q_{n}^{j}.$
Then, $\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$ provide
the following compatibility conditions:
* (i)
Taking the coefficients of $R^{j}_{x}R^{i}_{x}$ we have
$(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{j}+(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{i}=(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{j}+(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{i},$
which implies
$\frac{\partial_{j}\Lambda_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}\Lambda_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.1)
* (ii)
Taking the coefficients of $R^{i}_{x}$ we have
$(\partial_{j}\Lambda_{n}^{i})Q_{m}^{j}=(\partial_{j}\Lambda_{m}^{i})Q_{n}^{j}.$
Combining (A.1), the above equation can be written as
$\frac{Q_{n}^{j}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{Q_{m}^{j}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.2)
* (iii)
Taking the coefficients of $R^{j}_{x}$ we get
$\frac{\partial_{j}Q_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}Q_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.3)
* (iv)
The zero-th term of
$\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$ give us
$\frac{\partial_{j}Q_{n}^{i}}{Q_{n}^{j}}=\frac{\partial_{j}Q_{m}^{i}}{Q_{m}^{j}}.$
(A.4)
Notice that according to (A.2), equation (A.4) is equivalent to (A.3). To
summarize, we have three compatibility conditions (A.1), (A.2) and (A.4).
## Acknowledgments
The first author (LVB) is grateful to National Defense University (Taoyuan,
Taiwan), where a part of this work was done, for hospitality. This research
was particularly supported by the Russian-Taiwanese grant 95WFE0300007 (RFBR
grant 06-01-89507) and NSC-95-2923-M-606-001-MY3. LVB was also supported in
part by RFBR grant 07-01-00446.
## References
* [1] S. V. Manakov and P. M. Santini, The Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation, JETP Lett. 83 (2006) 462-466.
* [2] S. V. Manakov and P. M. Santini, A hierarchy of integrable PDEs in 2+1 dimensions associated with 2-dimensional vector fields, Theor. Math. Phys. 152 (2007) 1004–1011.
* [3] S. V. Manakov and P. M. Santini, On the solutions of the dKP equation: the nonlinear, Riemann-Hilbert problem, longtime behaviour, implicit solutions and wave breaking, J Phys. A: Math. Theor. 41 (2008) 055204.
* [4] M. V. Pavlov, Integrable hydrodynamic chains, J. Math. Phys. 44(9) (2003) 4134–4156.
* [5] M. Dunajski, A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type, J. Geom. Phys. 51 (2004) 126–137.
* [6] L. Martínez Alonso and A. B. Shabat, Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A 300 (2002) 58–64.
* [7] L. Martínez Alonso and A. B. Shabat, Hydrodynamic reductions and solutions of a universal hierarchy, Theor. Math. Phys. 140 (2004) 1073–1085.
* [8] L. V. Bogdanov, V. S. Dryuma and S. V. Manakov, On the dressing method for Dunajski anti-self-duality equation, nlin/0612046 (2006).
* [9] A.P. Fordy, Systems of hydrodynamic type from Poisson commuting Hamiltonians, ”Integrability: The Seiberg-Witten and Whitham Equations”, Edited by H.W.Braden and I.M. Krichever, Gordon and Breach Science Publishers, 2000
* [10] E.V.Ferapontov and A.P. Fordy, Non-Homogeneous systems of hydrodynamic type, related to quadratic Hamiltonians with electromagnetic terms, Physica D, 108, p.350-364(1997)
* [11] L. V. Bogdanov, V. S. Dryuma and S. V. Manakov, Dunajski generalization of the second heavenly equation: dressing method and the hierarchy, J Phys. A: Math. Theor. 40 (2007) 14383–14393.
|
arxiv-papers
| 2008-10-03T00:07:12
|
2024-09-04T02:48:58.107881
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L.V. Bogdanov, Jen-Hsu Chang and Yu-Tung Chen",
"submitter": "Jen-Hsu Chang",
"url": "https://arxiv.org/abs/0810.0556"
}
|
0810.0569
|
# Ultracool Subdwarfs: The Halo Population Down to the Substellar Limit
Adam J. Burgasser Sebastién Lépine Nicolas Lodieu Ralf-Dieter Scholz
Phillippe Delorme Wei-Chun Jao Brandon J. Swift Michael C. Cushing
###### Abstract
Ultracool subdwarfs are low luminosity, late-type M and L dwarfs that exhibit
spectroscopic indications of subsolar metallicity and halo kinematics. Their
recent discovery and ongoing investigation have led to new insights into the
role of metallicity in the opacity structure, chemistry (e.g. dust formation)
and evolution of low-temperature atmospheres; the long-term evolution of
magnetic activity and angular momentum amongst the lowest-mass stars; the form
of the halo luminosity and mass functions down to the hydrogen-burning mass
limit; and even fundamental issues such as spectral classification and
absolute brightness scales. This Splinter Session was devoted to bringing
advances in observational and theoretical ultracool subdwarf research to the
attention of the low-mass stellar and brown dwarf communities, as well as to
share results among ultracool subdwarf enthusiasts.
###### Keywords:
stars:subdwarfs – stars: abundances – techniques: spectroscopic
###### :
97.10.Ex,97.10.Ri,
97.10.Tk,97.10.Vm,97.10.Wn,97.20.Tr,97.20.Vs,97.80.Di,98.35.Gi
## 1 Introduction
The late-type M and L dwarf classes of ultracool dwarf stars and brown dwarfs
are now well-sampled, due in large part to wide-field red and infrared imaging
surveys such as 2MASS, DENIS and SDSS. We are now beginning to uncover the
metal-poor, halo counterparts to these disk dwarfs, the so-called ultracool
subdwarfs 2005ESASP.560..237B . These sources have been found primarily in
wide-field imaging surveys and the first generations of red/near-infrared
proper motion surveys. Discoveries include the first L-type—and possibly
T-type—subdwarfs, extending our knowledge of the Galactic halo down to the
substellar (brown dwarf) regime. As the low-mass ultracool subdwarfs, with
their extremely long nuclear burning lifetimes, were presumably formed early
in the Galaxy’s history, they are important tracers of Galactic structure and
chemical enrichment history. In addition, detailed studies of their complex
spectral energy distributions are facilitating new insights on the role of
metallicity in the opacity structure, chemistry and evolution of cool
atmospheres; the long-term evolution of magnetic activity and angular
momentum; and fundamental issues of spectral classification and
temperature/luminosity scales.
This Splinter was devoted to highlighting advances in ultracool subdwarf
research, both observational and theoretical. It was organized into three
subtopics: Discoveries, Classification and Fundamental Parameters. Individual
presentations will be made available at http://www.browndwarfs.org/cs15.
## 2 Discoveries
### 2.1 An Extremely Wide and Very Low-Mass Common Proper Motion Pair -
Representatives of a Nearby Halo Stream?
(R.-D. Scholz et al.)
A pair of faint stars sharing exactly the same very large proper motion of
about 860 mas/yr and separated by about six degrees has been discovered in a
high proper motion survey of the southern sky using multi-epoch positions and
photometry from the SuperCOSMOS Sky Surveys. The two stars, SSSPM J2003$-$4433
and SSSPM J1930$-$4311, have been classified as a late-type (M7) dwarf and an
ultracool subdwarf (sdM7) (Figure 1) with individually estimated spectroscopic
distances of 38 pc and 72 pc, respectively. In view of the accurate agreement
in their large proper motions a common distance of about 50 pc and a projected
physical separation of about 5 pc has been assumed, ruling out a physical
binary. The mean heliocentric space velocity of the pair ($U,V,W$) =
($-232,-170,+74$) km s-1 is typical of the Galactic halo population. These
values rely on a preliminary radial velocity measurement and on the assumption
of a common distance and velocity vector. The large separation and the
different metallicities of dwarf and subdwarf make a common formation scenario
as a wide binary (later disrupted) improbable. It seems more likely that this
wide pair is part of an old halo stream 2008A&A…487..595S .
Figure 1: Low-resolution classification spectra of SSSPM J2003$-$4433 (M7) and
SSSPM J1930$-$4311 (sdM7) and some comparison sources (from 2008A&A…487..595S
).
### 2.2 A Metal-poor Mid-T dwarf from the CFBDS Survey
(P. Delorme et al.)
We report the discovery of CFBDS J150000-182407, a T subdwarf candidate which
we identified during the Canada France Brown Dwarf Survey (CFBDS). CFDBS
2008A&A…484..469D is an $i^{\prime}-z^{\prime}$ wide-field search for
ultracool brown dwarfs which uses the MegaCam wide-field camera on the Canada-
France-Hawaii Telescope (CFHT).
CFBDS1500 is a peculiar T4.5 dwarf, with strong spectroscopic evidence for a
subsolar metallicity. Comparison of the overall shape of its spectrum with
synthetic spectra from 2006ApJ…640.1063B suggests [M/H]$\sim$-0.3, while the
complete absence of the $\sim$1.25$\mu$m K I doublet (Figure 2) rather
suggests that [M/H]$\leq$-0.5 when comparing with the same models. The
kinematics of CFBDS1500 imply an 80% probability that it belongs to the thick
disk, but leaves $\sim$10% probability that it instead is an older member of
the thin disk or a member of the halo. [M/H]$\sim$-0.5 is consistent with
either thick disc membership or an older thin disk population, while
[M/H]$\sim$-0.3 would lean slightly towards the older thin disk but remains
easily compatible with thick disk membership.
Figure 2: Spectra of T subdwarf candidate CFBDS1500 (black line) and 2M2254
(red line, from 2008ApJ…678.1372C ), the latter a solar-metallicity T dwarf
template that best matches the spectrum of CFBDS1500. Left panel shows the
whole NIR spectrum while the right panel zooms in on $J$-band region. The
vertical lines mark the locations of the K I doublet.
Spectroscopy and kinematics together make CFBDS1500 a strong thick disk
candidate, and as such the first incontrovertibly substellar object which
would not belong to the thin disk. As one of the few low-metallicity brown
dwarfs known, it is a major benchmark for synthetic spectra. The inconsistency
between the metallicity determined from the strength of the K I doublet and
from the general shape of the spectrum will need to be explained. We tend to
trust the metallicity constrain from the K I doublet more, since it depends
(besides, admittedly, the structure of the atmosphere) on a single line pair
with very well understood atomic physics, rather than on millions of
incompletely characterized molecular lines.
If the [M/H]$\sim$-0.5 metallicity holds, the observed spectrum would suggest
that the models overestimate the sensitivity of the general shape of the
spectrum to metallicity, especially in the $J$-band. That would hardly be a
surprise, given the many poorly constrained physical inputs that need to be
taken into account. Since the discovery of brown dwarfs belonging to old
Galactic populations is now possible, it becomes crucial to interpret their
spectra with state-of-the-art evolutionary models. The only one for low
metallicity brown dwarfs (to our knoweldge) was calculated without taking into
account clouds in the atmosphere 2001RvMP…73..719B . New theoretical work on
this topic (e.g., 2008arXiv0808.2611S ) is now a critical point of T sub-
dwarfs characterization.
## 3 Classification
### 3.1 A Spectral Sequence of K- and M-type Subdwarfs
(W.-C. Jao et al.)
Using new spectra of 88 K and M-type subdwarfs, we have considered novel
methods for assigning spectral types and take steps toward developing a
comprehensive spectral sequence for subdwarf types K3.0 to M6.0. The types are
assigned based on the overall morphology of spectra covering 6000Å to 9000Å
through the understanding of GAIA model grids. The types and sequence
presented link the spectral types of cool subdwarfs to their main sequence
counterparts, with emphasis on the relatively opacity-free region from
8200–9000Å. When available, supporting abundance, kinematic, and trigonometric
parallax information is used to provide more complete portraits of the
observed subdwarfs. We find that the CaHn (n= 1–3) and TiO5 indices often used
for subdwarf spectral typing are affected in complicated ways by combinations
of subdwarfs’ temperatures, metallicities, and gravities, and we use model
grids to evaluate the trends in all three parameters. Because of the complex
interplay of these three characteristics, it is not possible to identify a
star as an “extreme” subdwarf simply based on very low metallicity, and we
suggest that the modifiers “extreme” or “ultra” only outline locations on
spectroscopic indices plots, and should not be used to imply low or very low
metallicity stars. In addition, we propose that “VI” be used to identify a
star as a subdwarf, rather than the confusing “sd” prefix, which is also used
for hot O and B subdwarfs that are unrelated to the cool subdwarfs. These
results have been published in 2008AJ….136..840J .
### 3.2 Spectroscopic Sequences of Cool and Ultra-cool Subdwarfs
(sdM/esdM/usdM) from the Sloan Digital Sky Survey
(S. Lépine et al.)
We present new spectra of cool and ultra-cool subdwarfs, obtained from the
Sloan Digital Sky Survey (SDSS). Thousands of these sources have now been
observed in SDSS (see 2008ApJ…681L..33L and poster by Lépine, these
proceedings). The stars were categorized in a new, recently expanded
classification system 2007ApJ…669.1235L as subdwarfs (sdM), extreme subdwarfs
(esdM), and ultrasubdwarfs (usdM), based on the ratio of the TiO to CaH
molecular bands. It is argued that the M/sdM/esdM/usdM subclasses represent a
metallicity sequence (cf. 1997PASP..109.1233G ; 2006PASP..118..218W ), with
the usdM being the most metal-poor of the subdwarfs.
While it has been suggested by others that the morphology of the subdwarfs
depends on both their metallicity and gravity, observational evidence supports
the idea that metallicity is the dominant parameter, and that the current
classification system effectively ranks the stars according to mass (subtype)
and metallicity (subclass). As demonstrated in Figures 2 and 3 of
2007ApJ…669.1235L , stars kinematically selected from the Galactic disk
display only small variations in their TiO/CaH ratio, while stars
kinematically selected from the Galactic halo display a very broad range of
TiO/CaH values. The disk population has a broader range of stellar ages, and
should also have a broader range of surface gravities than the halo stars,
which are uniformly old. The fact that the disk stars have a much smaller
range of TiO/CaH than the halo stars indicates than metallicity effects must
dominate in the subdwarfs. Furthermore, atmospheric models indicate that
gravity variations not only change the TiO/CaH ratio, but they also notably
affect the strength of atomic spectral lines such as KI and NaI. However SDSS
spectra show no clear variation in the equivalent widths of the KI, NaI, and
CaII lines for stars of a given subclass and subtype, which again suggest that
stars of similar subtypes have similar surface gravities. Gravity, in the
nearby halo subdwarfs, is not a free parameter but is essentially set by the
initial mass and metallicity of the subdwarf.
Ultra-cool subdwarfs are particularly useful for refining the current
classification scheme, because their molecular bands and atomic lines are more
prominent. Spectra of many cool and ultra-cool subdwarfs are now available in
the SDSS database, and new ones should become available in the future as halo
subdwarfs are now specifically targeted for follow-up spectroscopy. The new
SDSS spectra also have a broad spectral coverage (4000Å-9000Å), and spectra
from many late-type subdwarfs can be combined to obtain high signal-to-noise
spectral templates. A fit to these templates yield a more reliable
classification than the narrowly defined TiO5, CaH2, and CaH3 spectral
indices. The use of the spectral indices in the classification is useful, but
should be phased out in favor of the new SDSS classification templates
(Lépine, in preparation).
### 3.3 L Subdwarfs: Classification, Distance Scale and Low-Metallicity
Condensate Formation
(A. Burgasser et al.)
L subdwarfs are the metal-poor counterparts to the L dwarf class of low mass
stars and brown dwarfs, effectively extending our sampling of halo stars to
the (metallicity-dependent) hydrogen-burning mass limit. With strong metal
hydride bands, weak metal oxides and red optical spectral energy
distributions, the L subdwarfs share many of the same spectral characteristics
as L dwarfs, but differ in their considerably bluer near-infrared colors
($J-K_{s}$ $\approx$ 0 vs 1.5–2.5) and much stronger metal hydride, metal
oxide and alkali line absorption in the red optical (Figure 3). The first L
subdwarfs were identified in 2003; today there are at least three reported in
the literature 2003ApJ…592.1186B ; 2004ApJ…614L..73B ; siv04 , not including
the unusual source LSR 1610-0040 2003ApJ…591L..49L discussed in detail below.
One of the recent advances in L subdwarf research is a formalized spectral
classification scheme for these objects. While the classification of M
subdwarfs is actively debated due to the large samples now available (see
contributions from S. Lépine and W.-C. Jao), there are too few L subdwarfs to
accurately define the class. 2007ApJ…657..494B have proposed that L subdwarfs
be classified according to the closest match to the L dwarf standards of
1999ApJ…519..802K in the 7300–9000 Å range (see Figure 3), a region in which
peculiarities are minimized. This had provided preliminary types for 2MASS
0532+82 (sdL7 2003ApJ…592.1186B ), 2MASS 1626+39 (sdL4 2004ApJ…614L..73B ) and
SDSS 1256-02 (sdL3.5 siv04 ).
Figure 3: Left: Comparison of the red optical spectra of the sdL4 2MASS
1626+39 to the L4 spectral standard 2MASS 1155+23, illustrating their
similarities in the 7300–9000 Å range (from 2007ApJ…657..494B ). Right:
$M_{K_{s}}$/spectral type relation for late-type M and L dwarfs (black points,
polynomial fit indicated by the solid line with dashed lines indicating
$\pm$1$\sigma$) compared to measurements for the sdM7 LHS 377, sdM8 LSR
1425+71 and sdL7 2MASS 0532+82 (red points; 1992AJ….103..638M ;
2008arXiv0806.2336D ; 2008ApJ…672.1159B ). The predicted $M_{K_{s}}$ for the
sdL3.5 SDSS 1256-02 is indicated by the hatched region (from bur1256 ).
As the first L subdwarf identified, 2MASS 0532+82 has been the best studied.
It is the first L subdwarf to have its astrometric parallax measured ($\pi$ =
37.5$\pm$1.7 mas 2008ApJ…672.1159B ), and combined with photometry from 2MASS
and Spitzer 2006ApJ…651..502P we have measured a luminosity of
$\log_{10}{L_{bol}/L_{\odot}}$ = $-4.24{\pm}0.06$ and estimate Teff =
1730$\pm$90 K for this source. These parameters are comparable to those of
mid-type L field dwarfs, suggesting that L subdwarf classifications are
“later” than corresponding L dwarf types. Interesting, these values make the
hydrogen burning status of 2MASS 0532+82 somewhat ambiguous, since depending
on its metallicity (which is largely unconstrained) this source may be a star,
a brown dwarf, or a brown dwarf which will ultimately reach the main sequence
and become a star! Examining absolute magnitude relations, it appears that the
$M_{K}$/spectral type relation of L subdwarfs is roughly coincident with those
of L dwarfs (Figure 3), although more astrometric work is needed (see also
2008arXiv0806.2336D ). 2MASS 0532+82 is also the only L subdwarf with high
resolution spectral observations 2006AJ….131.1806R , which indicate rapid
rotation ($v\sin{i}=65{\pm}15$ km s-1) suggesting the absence of angular
momentum evolution for very low mass stars. The $UVW$ velocities of this
source unambiguously confirm it as a halo object, with a retrograde Galactic
orbit relative to the Galactic disk.
We are currently examining theoretical spectral models for the sdL3.5 SDSS
1256-02 (bur1256 ; also see next contribution by B. Swift). Using the Phoenix-
Drift models 2008ApJ…675L.105H , we have attempted to reproduce the observed
optical and near-infrared colors and spectra of this source. Our results so
far indicate a Teff $\approx$ 2100–2500 K and [M/H] $\approx$ -1.5 – -1.0 dex,
although there remain strong discrepancies particularly in the red optical.
Interestingly, we find that even with the most advanced treatment of
(metallicity-dependent) cloud formation, clouds in the models are too thick
and condensation too efficient to match the observations of SDSS 1256-02,
consistent with the “cloud suppression” in ultracool subdwarf atmospheres
suggested in prior studies 2003ApJ…592.1186B ; 2006AJ….131.1806R ;
2006AJ….132.2372G ; 2007ApJ…657..494B
## 4 Fundamental Parameters
### 4.1 Theoretical Modeling of L Subdwarf Spectra
(B. Swift et al.)
We present fits of atmospheric models of varying metallicity to the published
optical and near-infrared spectrum and IRAC photometry of the sdL7 2MASS
J05325346+8246465 2003ApJ…592.1186B . This source is the best-observed member
of a growing population of what are suspected to be metal-poor ultracool
dwarfs, and the only one with a parallax distance. The model selection was
made using a goodness-of-fit statistic in conjunction with Monte Carlo
simulated data to account for uncertainties in the absolute flux; this
statistic was then examined in various bands to study systematic issues in the
models and to produce a concordance fit for the properties of this object in
Teff, $\log{g}$ and [M/H] space.
### 4.2 Astrometric Observations of LSR 1610-0040
(M. Cushing et al.)
Since its discovery 2003ApJ…591L..49L , LSR 1610-0040 has defied explanation.
Its red optical spectrum and high proper motion suggest that LSR 1610-0040 is
an early-type L subdwarf, yet its near-infrared spectrum indicates a mid-type
dM or sdM, albeit with numerous peculiar spectral features 2006AJ….131.1797C ;
2006AJ….131.1806R . Based on a comparison of its near-infrared spectrum to
that of field M dwarfs, 2006AJ….131.1797C assign LSR 1610-0040 a spectral
type of sd/dM6.
We present new astrometric observations of LSR 1610-0040 2008arXiv0806.2336D .
At a measured distance of $\sim$32 pc, its position in color/ absolute
magnitude diagrams ($M_{V}$ vs $V-I$, $M_{K_{s}}$ vs $I-K_{s}$, $M_{K_{s}}$ vs
$J-K_{s}$) is consistent with mid-type M dwarfs. However, its $B-V$ color is
1.2 mags redder than both M subdwarfs and dwarfs of similar spectral types. We
speculate that the $B$-band magnitude of LSR 1610-0040 is suppressed due to
enhanced AlH absorption since Al appears over-abundant relative to solar based
on the strength of the 1.313 $\mu$m Al I doublet.
Perhaps most interesting is that the astrometric observations indicate that
LSR 1610-0040 is an unresolved binary with a period of 1.66 yr. The
photocentric orbit has a moderate eccentricity of 0.44$\pm$0.02, a semi-major
axis of 0.28$\pm$0.01 AU, and an inclination of 83$\pm$1o. Under the
assumption that the secondary contributes little to no light to the system,
and using a near-infrared mass luminosity relation, we estimate the masses of
the components to be MA = 0.095 M⊙ and MB = 0.059 to 0.082 M⊙. We speculate
that LSR 1610-0040A was originally a 0.05 Msun star with [Fe/H] $\sim$ $-2$
which later accreted $<$0.05 M⊙ of material from a massive AGB star that has
undergone hot bottom burning. Pollution by such material, enhanced in Al and
Na and depleted in O, would then explain the peculiar spectrum of LSR
1610-0040A. LSR 1610-0040B has too little mass to be the remnant white dwarf
of such a hypothetical AGB star so the AGB star in question must have been a
more distant companion that has since been lost from the system.
We thank the conference organizers for providing a forum to discuss this
topic.
## References
* (1) A. J. Burgasser, J. D. Kirkpatrick, and S. Lépine, “Ultracool subdwarfs: metal-poor stars and brown dwarfs extending into the late-type M, L and T dwarf regimes,” in _13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun_ , edited by F. Favata, G. A. J. Hussain, and B. Battrick, 2005, vol. 560 of _ESA Special Publication_ , pp. 237.
* (2) R.-D. Scholz, N. V. Kharchenko, N. Lodieu, and M. J. McCaughrean, _A &A_ 487, 595–599 (2008).
* (3) P. Delorme, C. J. Willott, T. Forveille, X. Delfosse, C. Reylé, E. Bertin, L. Albert, E. Artigau, A. C. Robin, F. Allard, R. Doyon, and G. J. Hill, _A &A_ 484, 469–478 (2008).
* (4) A. Burrows, D. Sudarsky, and I. Hubeny, _ApJ_ 640, 1063–1077 (2006).
* (5) M. C. Cushing, M. S. Marley, D. Saumon, B. C. Kelly, W. D. Vacca, J. T. Rayner, R. S. Freedman, K. Lodders, and T. L. Roellig, _ApJ_ 678, 1372–1395 (2008).
* (6) A. Burrows, W. B. Hubbard, J. I. Lunine, and J. Liebert, _Reviews of Modern Physics_ 73, 719–765 (2001).
* (7) D. Saumon, and M. S. Marley, _ArXiv e-prints_ 808 (2008).
* (8) W.-C. Jao, T. J. Henry, T. D. Beaulieu, and J. P. Subasavage, _AJ_ 136, 840–880 (2008).
* (9) S. Lépine, and R.-D. Scholz, _ApJL_ 681, L33–L36 (2008).
* (10) S. Lépine, R. M. Rich, and M. M. Shara, _ApJ_ 669, 1235–1247 (2007).
* (11) J. Gizis, and I. Reid, _PASP_ 109, 1232-1236 (1997).
* (12) V. M. Woolf, and G. Wallerstein, _PASP_ 118, 218–226 (2006).
* (13) A. J. Burgasser, J. D. Kirkpatrick, A. Burrows, J. Liebert, I. N. Reid, J. E. Gizis, M. R. McGovern, L. Prato, and I. S. McLean, _ApJ_ 592, 1186–1192 (2003).
* (14) A. J. Burgasser, _ApJL_ 614, L73–L76 (2004).
* (15) T. Sivarani, A. K. Kembhavi, and J. Gupchup, _ArXiv e-prints_ (2004).
* (16) S. Lépine, R. M. Rich, and M. M. Shara, _ApJL_ 591, L49–L52 (2003).
* (17) A. J. Burgasser, K. L. Cruz, and J. D. Kirkpatrick, _ApJ_ 657, 494–510 (2007).
* (18) J. D. Kirkpatrick, I. N. Reid, J. Liebert, R. M. Cutri, B. Nelson, C. A. Beichman, C. C. Dahn, D. G. Monet, J. E. Gizis, and M. F. Skrutskie, _ApJ_ 519, 802–833 (1999).
* (19) D. G. Monet, C. C. Dahn, F. J. Vrba, H. C. Harris, J. R. Pier, C. B. Luginbuhl, and H. D. Ables, _AJ_ 103, 638–665 (1992).
* (20) C. C. Dahn, H. C. Harris, S. E. Levine, T. Tilleman, A. K. B. Monet, R. C. Stone, H. H. Guetter, B. Canzian, J. R. Pier, W. I. Hartkopf, J. Liebert, and M. Cushing, _ArXiv e-prints_ 806 (2008).
* (21) A. J. Burgasser, F. J. Vrba, S. Lépine, J. A. Munn, C. B. Luginbuhl, A. A. Henden, H. H. Guetter, and B. C. Canzian, _ApJ_ 672, 1159–1166 (2008).
* (22) A. J. Burgasser, S. Witte, C. Helling, and P. H. Hauschildt, _ApJ_ (in preparation).
* (23) B. M. Patten, J. R. Stauffer, A. Burrows, M. Marengo, J. L. Hora, K. L. Luhman, S. M. Sonnett, T. J. Henry, D. Raghavan, S. T. Megeath, J. Liebert, and G. G. Fazio, _ApJ_ 651, 502–516 (2006).
* (24) A. Reiners, and G. Basri, _AJ_ 131, 1806–1815 (2006).
* (25) C. Helling, M. Dehn, P. Woitke, and P. H. Hauschildt, _ApJL_ 675, L105–L108 (2008).
* (26) J. E. Gizis, and J. Harvin, _AJ_ 132, 2372–2375 (2006).
* (27) M. C. Cushing, and W. D. Vacca, _AJ_ 131, 1797–1805 (2006).
|
arxiv-papers
| 2008-10-03T05:00:52
|
2024-09-04T02:48:58.113340
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Adam J. Burgasser (MIT), Sebastien Lepine (AMNH), Nicolas Lodieu\n (IAC), Ralf-Dieter Scholz (Potsdam), Phillippe Delorme (Grenoble), Wei-Chun\n Jao (Georgia State), Brandon J. Swift (Steward Observatory) and Michael C.\n Cushing (U. Hawaii IfA)",
"submitter": "Adam J. Burgasser",
"url": "https://arxiv.org/abs/0810.0569"
}
|
0810.0570
|
# Clouds, Brightening and Multiplicity Across the L Dwarf/T Dwarf Transition
Adam J. Burgasser
###### Abstract
The transition between the two lowest-luminosity spectral classes of brown
dwarfs—the L dwarfs and T dwarfs—is traversed by nearly all brown dwarfs as
they cool over time. Yet distinct features of this transition, such as the
”J-band bump” and an unusually high rate of multiplicity, remain outstanding
problems, although evidence points to condensate cloud evolution as a critical
component. Using a Monte Carlo population simulation that incorporates the
empirical spectral properties of unresolved brown dwarfs in magnitude-limited
samples, I demonstrate that the J-band bump and enhanced multiplicity
naturally emerge from a short timescale of photospheric cloud dissipation.
This timescale may help constrain future evolutionary models exploring the
cloud dissipation process.
###### Keywords:
Stars: binaries: spectroscopic—Stars: fundamental parameters—Stars: low-mass,
brown dwarfs—Stars: luminosity function, mass function
###### :
97.10.Ri, 97.10.Xq, 97.20.Vs, 97.80.Fk
## 1 Context
The transition between the two coldest classes of brown dwarfs, the L dwarfs
and T dwarfs, marks a dramatic change in the chemical abundances, condensate
cloud properties and spectral energy distributions of late-type brown dwarfs.
This transition is traversed by nearly all brown dwarfs during their cooling
lifetimes. Yet it remains one of the most poorly understood phases of brown
dwarf evolution, both in terms of empirical characterization (relatively few
examples are known, most are too faint to be well characterized) and
theoretical reproduction of observational data 2002ApJ…571L.151B ;
2004AJ….127.3553K ; 2008ApJ…678.1372C . The L dwarf/T dwarf transition is also
distinguished by two notable peculiar features. The “J-band bump” is an
apparent brightening in the 1.0-1.3 $\mu$m region from late-type L to mid-type
T 2002AJ….124.1170D which is observed as a “flux reversal” between the
components of L/T binary systems 2006ApJS..166..585B ; 2006ApJ…647.1393L ;
2008arXiv0803.0544L . There is also an excess of binary systems amongst L/T
transition objects, up to twice as frequent as warmer L dwarf and cooler T
dwarf systems 2006ApJS..166..585B . The disappearance of condensate clouds
across this transition is likely to be a key factor in the unusual properties
of the L/T transition. However, given the complexity 2001ApJ…556..872A ;
2006A&A…455..325H and ubiquity of condensate formation in the dynamic, low-
temperature atmospheres of both late-type brown dwarfs and exoplanets, the
peculiar properties of L/T transition objects can provide important empirical
constraints for cloud formation theories.
## 2 Simulations
To understand the origin of the binary excess amongst L/T transition objects,
I simulated a volume-complete population of L and T dwarfs, including
unresolved binary systems, building off of prior mass function/luminosity
function Monte Carlo simulations (2004ApJS..155..191B ; see Figure 1). Power
law forms of the mass function (dN/dM $\propto$ M-α, where $\alpha$ =
0,0.5,1.0,1.5) as well as a lognormal form (parameters from 2002ApJ…567..304C
) were examined. A constant star formation rate was assumed. Evolutionary
models from both 2001RvMP…73..719B and 2003A&A…402..701B were used to
convert masses and ages to luminosities (solar metallicity was assumed).
Luminosities were then converted to spectral types using an empirical relation
based on single sources 2004AJ….127.3516G and individual components of
resolved binaries 2004A&A…413.1029M ; 2005ApJ…634..616L ; 2006ApJS..166..585B
. Simulated binary populations were constructed assuming intrinsic binary
fractions ranging from 5–70%, and both exponential and constant mass ratio
distributions were considered 2006ApJS..166..585B . Binary spectra were
produced by flux-calibrating low-resolution template spectra (obtained with
the SpeX spectrograph 2003PASP..115..362R ) according to empirical MK/spectral
type relations 2006ApJ…647.1393L ; 2007ApJ…659..655B . The binary spectra,
assumed to be unresolved, were classified using calibrated spectral indices.
Space density and binary fraction distributions as functions of spectral type
for both volume-limited and magnitude-limited samples (the latter taking into
account the overluminosity of unresolved binary systems) were calculated for
the full range of parameters examined.
Figure 1: Steps in the L/T binary simulation, from left to right: luminosity
function of brown dwarfs based on Monte Carlo mass function simulations,
empirical luminosity/spectral type relation, empirical MK/spectral type
relation, exponential fit to observed mass ratio distribution of very low mass
stars and brown dwarfs, and synthesized combined light (unresolved) binary
spectra (from 2004ApJS..155..191B ; 2006ApJS..166..585B ; 2007ApJ…659..655B ).
## 3 Results
The shallow slope of the luminosity/spectral type relation from L7 to T4
(Figure 1) amplifies the dip in the luminosity function seen in prior mass
function simulations, predicting $\sim$10$\times$ fewer early-type T dwarfs
than other L or T types in a given volume (Figure 2). The rarity of individual
early-type T dwarfs allows them to be outnumbered by hybrid L dwarf + T dwarf
pairs, particularly in magnitude-limited samples.
The frequency of binaries as a function of spectral type for both volume-
limited and magnitude-limited samples (Figure 2) shows a clear peak at the L/T
transition. For an inherent binary fraction of 11${}^{+4}_{-2}$%, we reproduce
the observed (magnitude-limited) resolved binary fraction distribution in
detail. The higher binary fraction of L/T transition objects is attributable
to both the paucity of single early-type T dwarfs and the fact that systems
comprised of (more common) late-type L dwarf plus T dwarf components resemble
early-type T dwarfs. If the binary fraction across the L/T transition is
closer to 66%, as suggested by 2006ApJ…647.1393L , than the intrinsic binary
fraction of brown dwarfs may be as high as 40%, nearly twice current estimates
2006AJ….132..663B ; 2006ApJS..166..585B .
The shallow luminosity/spectral type relation inferred from the binary
fraction peak implies that brown dwarfs evolve between types L and T—and lose
their photospheric condensate clouds in the process—over a relatively short
period. A 0.03 M⊙ brown dwarf makes the jump from L8 to T3 in a mere 100 Myr.
Current equilibrium cloud models predict a much more gradual settling of
clouds (e.g., 2002ApJ…568..335M ; 2006ApJ…640.1063B ). Global non-equilibrium
effects, reflected in either cloud fragmentation 2002ApJ…571L.151B or
enhanced condensate rain-out 2004AJ….127.3553K ; 2008ApJ…678.1372C must be an
inherent feature of the L/T transition.
Figure 2: Simulation results. (Left) Volume-limited spectral type distribution
of singles (solid black lines), secondaries (dash-dot blue lines) and single +
binary systems (dashed red lines). A deep minimum is found across the L/T
transition. (Right) Observed resolved binary fraction distributions (black
points with error bars) compared to predictions for inherent binary fractions
of (top to bottom) 5%, 10%, 15%, 20% and 30% (from 2007ApJ…659..655B ).
## 4 New Work
As binary systems are the best probes of empirical trends across the L/T
transition, several groups are attempting to uncover new L/T binaries through
resolved imaging studies. We have recently developed a spectral template
matching technique that identifies and characterizes unresolved binaries from
combined-light, low resolution, near-infrared spectroscopy 2007AJ….134.1330B ;
2008ApJ…681..579B (Figure 3). One of the systems discovered, the M8.5 + T5
dwarf pair 2MASS 0320-0446, has been independently identified as a radial
velocity variable 2008ApJ…678L.125B . By finding more systems like these, a
more robust measure of the intrinsic brown dwarf binary fraction unaffected by
separation limitations is possible, and precise constraints on luminosity and
brightness trends across the L/T transition may be made.
Figure 3: Binaries found by the spectral template matching technique. (Left)
the L4.5 + T5 binary SDSS 0805+4812 2007AJ….134.1330B . (Right) the M8.5 + T5
binary 2MASS 0320-0446 2008ApJ…681..579B , independently identified as a
radial velocity variable 2008ApJ…678L.125B .
## References
* (1) A. J. Burgasser, M. S. Marley, A. S. Ackerman, et al., _ApJL_ 571, L151–L154 (2002).
* (2) G. R. Knapp, S. K. Leggett, X. Fan, et al., _AJ_ 127, 3553–3578 (2004).
* (3) M. C. Cushing, M. S. Marley, D. Saumon, et al., _ApJ_ 678, 1372–1395 (2008).
* (4) C. C. Dahn, H. C. Harris, F. J. Vrba, et al., _AJ_ 124, 1170–1189 (2002).
* (5) A. J. Burgasser, J. D. Kirkpatrick, K. L. Cruz, et al., _ApJS_ 166, 585–612 (2006).
* (6) M. C. Liu, S. K. Leggett, D. A. Golimowski, et al., _ApJ_ 647, 1393–1404 (2006).
* (7) D. L. Looper, C. R. Gelino, A. J. Burgasser, and J. D. Kirkpatrick, _ArXiv e-prints_ 803 (2008).
* (8) A. S. Ackerman, and M. S. Marley, _ApJ_ 556, 872–884 (2001).
* (9) C. Helling, and P. Woitke, _A &A_ 455, 325–338 (2006).
* (10) A. J. Burgasser, _ApJS_ 155, 191–207 (2004).
* (11) G. Chabrier, _ApJ_ 567, 304–313 (2002).
* (12) A. Burrows, W. B. Hubbard, J. I. Lunine, and J. Liebert, _Reviews of Modern Physics_ 73, 719–765 (2001).
* (13) I. Baraffe, G. Chabrier, T. S. Barman, F. Allard, and P. H. Hauschildt, _A &A_ 402, 701–712 (2003).
* (14) D. A. Golimowski, S. K. Leggett, M. S. Marley, et al., _AJ_ 127, 3516–3536 (2004).
* (15) M. J. McCaughrean, L. M. Close, R.-D. Scholz, et al., _A &A_ 413, 1029–1036 (2004).
* (16) M. C. Liu, and S. K. Leggett, _ApJ_ 634, 616–624 (2005).
* (17) J. T. Rayner, D. W. Toomey, P. M. Onaka, et al., _PASP_ 115, 362–382 (2003).
* (18) A. J. Burgasser, _ApJ_ 659, 655–674 (2007).
* (19) G. Basri, and A. Reiners, _AJ_ 132, 663–675 (2006).
* (20) M. S. Marley, S. Seager, D. Saumon, K. Lodders, A. S. Ackerman, R. S. Freedman, and X. Fan, _ApJ_ 568, 335–342 (2002).
* (21) A. Burrows, D. Sudarsky, and I. Hubeny, _ApJ_ 640, 1063–1077 (2006).
* (22) A. J. Burgasser, _AJ_ 134, 1330–1336 (2007).
* (23) A. J. Burgasser, M. C. Liu, M. J. Ireland, K. L. Cruz, and T. J. Dupuy, _ApJ_ 681, 579–593 (2008).
* (24) C. H. Blake, D. Charbonneau, R. J. White, G. Torres, M. S. Marley, and D. Saumon, _ApJL_ 678, L125–L128 (2008).
|
arxiv-papers
| 2008-10-03T05:06:19
|
2024-09-04T02:48:58.117298
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Adam J. Burgasser (MIT)",
"submitter": "Adam J. Burgasser",
"url": "https://arxiv.org/abs/0810.0570"
}
|
0810.0750
|
# The Calculation of $f_{\pi}$ and $m_{\pi}$ at Finite Chemical Potential
Yu Jiang1, Yuan-mei Shi1, Hua Li1, Wei-min Sun1,2 and Hong-shi Zong1,2 1
Department of Physics, Nanjing University, Nanjing 210093, China 2 Joint
Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
###### Abstract
Based on the previous work in [Y. Jiang, Y.M. Shi, H.T. Feng, W.M. Sun and
H.S. Zong, Phys. Rev. C 78, 025214 (2008)] on the quark-meson vertex and pion
properties at finite quark chemical potential, we provide an analytical
analysis of the weak decay constant of the pion ($f_{\pi}[\mu]$) and the pion
mass ($m_{\pi}[\mu]$) at finite quark chemical potential using the model quark
propagator proposed in [R. Alkofer, W. Detmold, C.S. Fischer and P. Maris,
Phys. Rev. D 70, 014014 (2004)]. It is found that when $\mu$ is below a
threshold value $\mu_{0}$ (which equals $0.350~{}\mathrm{GeV}$,
$0.377~{}\mathrm{GeV}$ and $0.341~{}\mathrm{GeV}$, for the $\mathrm{2CC}$,
$\mathrm{1R1CC}$ and $\mathrm{3R}$ parametrizations of the model quark
propagator, respectively.), $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ are kept
unchanged from their vacuum values. The value of $\mu_{0}$ is intimately
connected with the pole distribution of the model quark propagator and is
found to coincide with the threshold value below which the quark-number
density vanishes identically. Numerical calculations show that when $\mu$
becomes larger than $\mu_{0}$, $f_{\pi}[\mu]$ exhibits a sharp decrease
whereas $m_{\pi}[\mu]$ exhibits a sharp increase. A comparison is given
between the results obtained in this paper and those obtained in previous
literatures.
Key-words: weak decay constant of pion, pion mass, finite quark chemical
potential
E-mail: zonghs@chenwang.nju.edu.cn.
PACS Numbers: 11.10.Wx, 11.10.St, 11.15.Tk, 14.40.Aq
The in-medium modification of the properties of the pion is of fundamental
interest in hadron physics. The pion is identified as a Goldstone boson
arising from the spontaneous breakdown of chiral symmetry which is essential
for describing low-energy hadronic phenomena. Since chiral symmetry is
expected to be restored at high enough density, the change of pion properties
in medium will provide crucial information on the restoration of chiral
symmetry. Among these, the weak decay constant of the pion $f_{\pi}$ and the
pion mass $m_{\pi}$ are the two most important quantities, since they are
closely related to the spontaneous breakdown of chiral symmetry of Quantum
Chromodynamics (QCD). Unfortunately, so far it has not been possibile to
obtain detailed information about modification of pion properties in medium
directly from QCD. In this situation, different models have been used to study
this sort of problems Delorme ; Kirchbach ; Kaiser ; Meissner ; Kim ; Mallik ;
Nam ; Maris ; Bender ; Bender1 . Just as was pointed out in Ref. Maris1 , the
pion has a dual role: it can be identified as a quark-antiquark bound state as
well as a Goldstone boson arising from the spontaneous breakdown of chiral
symmetry. From the point of view that the pion can be regarded as a quark-
antiquark bound state, the full dynamical information of the pion is contained
in the corresponding Bethe-Salpeter Amplitude (BSA): $\Gamma_{\pi}(k,p)$ ($k$
is the relative and $p$ the total momentum of the quark-antiquark pair), which
is the one-particle-irreducible, fully-amputated quark-meson vertex. The
Dyson-Schwinger equations (DSEs) of QCD provide a nonperturbative, continuum
framework for analyzing such quark-meson vertices directly Maris1 ; DSE1 ;
DSE2 ; DSE3 ; DSE4 . The aim of this paper is to study the change of $f_{\pi}$
and $m_{\pi}$ with quark chemical potential $\mu$ in the framework of this
nonperturbative QCD model.
The DSEs of QCD have been used extensively at zero temperature and zero quark
chemical potential to extract hadronic observables DSE1 ; DSE2 ; DSE3 ; DSE4 .
However, this is very difficult at finite quark chemical potential due to the
fact that the number of independent Lorentz structures of the quark-meson
vertex at finite $\mu$ is much larger than that of the corresponding one at
$\mu=0$. In Ref. fpi1 , using the method of studying the dressed quark
propagator at finite $\mu$ given in Ref. Zong05 , the authors have given a new
approach for tackling this problem. Based on the rainbow-ladder approximation
of the DSEs and the assumption of analyticity of the quark-meson vertex in the
neighborhood of $\mu=0$ and neglecting the $\mu$-dependence of the dressed
gluon propagator, the authors show that the general quark-meson vertex at
finite $\mu$ can be obtained from the corresponding one at $\mu=0$ by a shift
of variable: $\Gamma[\mu](k,p)=\Gamma(\tilde{k},p)$, where
$\tilde{k}=(\vec{k},k_{4}+i\mu)$. From this result the authors of Ref. fpi1
numerically calculated $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ for
$\mu<300~{}\mathrm{MeV}$. It is found that $f_{\pi}[\mu]$ increases slowly
(with an increase of less than about $0.01\%$) and $m_{\pi}[\mu]$ falls slowly
(with a decrease of less than about $0.06\%$) with increasing $\mu$.
Numerically the change of $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ is so small that
one can think $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ does not change with $\mu$ for
$\mu<300~{}\mathrm{MeV}$ within numerical errors. One of our motivations for
this work is to explore the mathematical reason behind this. Based on the work
in fpi1 , in this paper we provide an analytic analysis of $f_{\pi}[\mu]$ and
$m_{\pi}[\mu]$. It is found that when $\mu$ is below a critical value
$\mu_{0}$, $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ are kept unchanged from their
vacuum values. Moreover, numerical calculations show that when $\mu$ becomes
larger than $\mu_{0}$, $f_{\pi}[\mu]$ exhibits a sharp decrease whereas
$m_{\pi}[\mu]$ exhibits a sharp increase.
According to Ref. fpi1 , the pion decay constant at finite $\mu$ can be
expressed as the following
$\delta^{ij}f_{\pi}[\mu]p_{\nu}=\int_{q}\,\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}\gamma_{\nu}S(\tilde{q}_{+})\Gamma_{\pi}^{j}(\tilde{q};p)S(\tilde{q}_{-})\right],$
(1)
where $S(q)$ is the full dressed quark propagator,
$\tilde{q}_{\pm}=\tilde{q}\pm p/2$, $\tilde{q}=(\vec{q},q_{4}+i\mu)$,
$\frac{\tau^{i}}{2}$ are the flavor $SU(2)$ generators and
$\int_{q}\equiv\int\,d^{4}q/(2\pi)^{4}$. In the present paper we will not
write the renormalisation constants explicitly because one would find that in
the final result the renormalisation constants cancel each other. In fact, Eq.
(1) is the expression of $f_{\pi}$ which is independent of the renormalisation
point and the regularisation mass-scale Maris1 .
The integral of the right-hand-side of Eq. (1) can be rewritten as:
$\int_{q}\equiv\int\,\frac{d^{4}q}{(2\pi)^{4}}\equiv\int\,\frac{d^{4}\tilde{q}}{(2\pi)^{4}}.$
(2)
Contracting both sides of Eq. (1) with $p_{\nu}$ and using Eq. (2), we obtain
the following:
$\displaystyle\delta^{ij}f_{\pi}[\mu]$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\int\limits_{-\infty+i\mu}^{+\infty+i\mu}\,\frac{dq_{4}}{(2\pi)}\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}S(q_{+})\Gamma_{\pi}^{j}(q;p)S(q_{-})\right]$
(3) $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\int_{C_{1}}\frac{dq_{4}}{(2\pi)}\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}S(q_{+})\Gamma_{\pi}^{j}(q;p)S(q_{-})\right]$
where the integration path $C_{1}$ is depicted in Fig. 1.
FIG.1. The integration path in the complex $q_{4}$ plane.
Let us use $z_{n}=\chi_{n}+i\omega_{n}~{}(\omega_{n}>0),n=1,2\cdots$ to denote
the poles of the function
$F^{ij}(q_{4})\equiv\frac{1}{p^{2}}\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}S(q_{+})\Gamma_{\pi}^{j}(q;p)S(q_{-})\right]$
(4)
located in the upper half complex $q_{4}$ plane. According to Cauchy’s theorem
we obtain the following from Eq. (3):
$\displaystyle\delta^{ij}f_{\pi}[\mu]$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\int_{C_{1}}\frac{dq_{4}}{(2\pi)}\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}S(q_{+})\Gamma_{\pi}^{j}(q;p)S(q_{-})\right]$
(5) $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\int_{C_{0}}\frac{dq_{4}}{(2\pi)}\mbox{tr}\left[\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}S(q_{+})\Gamma_{\pi}^{j}(q;p)S(q_{-})\right]$
$\displaystyle-i\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\sum_{n}\theta(\mu-\omega_{n})\mbox{Res}\\{F^{ij}(z);z_{n}\\}$
$\displaystyle=$
$\displaystyle\delta^{ij}f_{\pi}-i\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\sum_{n}\theta(\mu-\omega_{n})\mbox{Res}\\{F^{ij}(z);z_{n}\\}.$
From Eq. (5) it is easily seen that when $\mu<min\\{\omega_{n}\\}$, the
function $F^{ij}(q_{4})$ has no pole in the region $\Omega$ (the region
enclosed by $C_{1}$ and $C_{0}$, see Fig. 1) and therefore
$f_{\pi}[\mu]=f_{\pi}$, which means that for small enough $\mu$ the pion decay
constant should be independent of $\mu$. Of course, when
$\mu>min\\{\omega_{n}\\}$ the pion decay constant can have an explicit
$\mu$-dependence.
In the chiral limit, expanding the trace term of the right-hand-side of Eq.
(4) to $\mathcal{O}(p^{2})$ near $p=0$ DSE1 , we have the following:
$\displaystyle F^{ij}(q_{4})$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\mbox{tr}\left\\{\frac{\tau^{i}}{2}\gamma_{5}{\not\\!p}\bigg{[}S+\frac{1}{2}p\cdot\partial
S\bigg{]}\bigg{[}\Gamma_{\pi}^{j}(q,0)+\mathcal{O}(p)\gamma_{5}\bigg{]}\bigg{[}S-\frac{1}{2}p\cdot\partial
S\bigg{]}\right\\},$ (6)
where we have adopted the approximation DSE1
$\Gamma_{\pi}^{j}(q,p)=\Gamma_{\pi}^{j}(q,0)+\mathcal{O}(p)\gamma_{5}.$ (7)
With this approximation $\Gamma_{\pi}^{j}(q,0)$ can be expressed as DSE1 ;
Frank96
$\Gamma_{\pi}^{j}(q,0)=\tau^{j}\gamma_{5}\cdot\frac{iB(q^{2})}{f_{\pi}},$ (8)
where $B(q^{2})$ is the scalar part of $S^{-1}(q)$. Noticing that
$\mbox{tr}\big{[}\gamma_{5}{\not\\!p}S\gamma_{5}S\big{]}=0$, we obtain the
following:
$\displaystyle F^{ij}(q_{4})$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\mbox{tr}\left\\{\frac{\tau^{i}}{4}\gamma_{5}{\not\\!p}\bigg{[}p\cdot\partial
S\Gamma_{\pi}^{j}(q,0)S-S\Gamma_{\pi}^{j}(q,0)p\cdot\partial
S\bigg{]}\right\\}+\mathcal{O}(p).$ (9)
Substituting Eq. (8) into Eq. (9) and using
$\mbox{tr}(\tau^{i}\tau^{j})=2\delta^{ij}$, we obtain
$\displaystyle F^{ij}(q_{4})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2p^{2}}\delta^{ij}\frac{iB(q^{2})}{f_{\pi}}\mbox{tr}\left\\{\gamma_{5}{\not\\!p}\bigg{[}p\cdot\partial
S\gamma_{5}S-S\gamma_{5}p\cdot\partial S\bigg{]}\right\\}.$ (10)
Adopting the following expression of $S(q)$
$S(q)=\frac{1}{i{\not\\!q}A(q^{2})+B(q^{2})}=-i{\not\\!q}\sigma_{v}(q^{2})+\sigma_{s}(q^{2}),$
(11)
we obtain
$\displaystyle F^{ij}(q_{4})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\delta^{ij}\frac{1}{f_{\pi}}\frac{8\sigma_{s}}{\sigma_{v}^{2}q^{2}+\sigma_{s}^{2}}\left[\sigma_{s}\sigma_{v}+\frac{2(p\cdot
q)^{2}}{p^{2}}(\sigma_{s}\sigma_{v}^{\prime}-\sigma_{s}^{\prime}\sigma_{v})\right]$
(12) $\displaystyle=$ $\displaystyle\delta^{ij}\frac{1}{f_{\pi}}F(q_{4}),$
(13)
where ′ means $d/dq^{2}$ and
$\displaystyle F(q_{4})$ $\displaystyle\equiv$
$\displaystyle\frac{4\sigma_{s}}{\sigma_{v}^{2}q^{2}+\sigma_{s}^{2}}\left[\sigma_{s}\sigma_{v}+2\frac{(p\cdot
q)^{2}}{p^{2}}(\sigma_{s}\sigma_{v}^{\prime}-\sigma_{s}^{\prime}\sigma_{v})\right].$
(14)
Then Eq. (5) can be written as
$f_{\pi}[\mu]\simeq
f_{\pi}-\frac{i}{f_{\pi}}\int\limits_{-\infty}^{+\infty}\,\frac{d^{3}\vec{q}}{(2\pi)^{3}}\sum_{n}\theta(\mu-\omega_{n})\mbox{Res}\\{F(z);z_{n}\\}.$
(15)
To determine the pole distribution of function $F(q_{4})$, we should first
specify the form of the dressed quark propagator. Here, as in Refs. fpi1 ;
con1 we adopt the following propagator proposed in Ref. Alkofer04 :
$S(q)=\sum_{j=1}^{n_{P}}\left(\frac{r_{j}}{i{\not\\!q}+a_{j}+ib_{j}}+\frac{r_{j}}{i{\not\\!q}+a_{j}-ib_{j}}\right).$
(16)
The propagator of this form has $n_{P}$ pairs of complex conjugate poles
located at $a_{j}\pm ib_{j}$. When some $b_{j}$ is set to zero, the pair of
complex conjugate poles degenerates to a real pole. The restrictions of the
parameters $r_{j}$, $a_{j}$ and $b_{j}$ in the chiral limit are Alkofer04
$\displaystyle\sum_{j=1}^{n_{P}}\,r_{j}=\frac{1}{2},$ (17)
$\displaystyle\sum_{j=1}^{n_{P}}\,r_{j}a_{j}=0.$ (18)
If we are not in the chiral limit, the right hand side of Eq. (18) should be
replaced by the current quark mass. The value of these parameters are shown in
Table I, where 2CC, 1R1CC and 3R stand for three meromorphic forms of the
quark propagator, respectively: two pairs of complex conjugate poles, one real
pole and one pair of complex conjugate poles, three real poles.
Table I. The parameters used in the calculation of $F(q_{4})$ and $f_{\pi}$.
These parameters are taken directly from Ref. Alkofer04 .
Parameterization | $r_{1}$ | $a_{1}$ (GeV) | $b_{1}$ (GeV) | $r_{2}$ | $a_{2}$ (GeV) | $b_{2}$ (GeV) | $r_{3}$ | $a_{3}$ (GeV)
---|---|---|---|---|---|---|---|---
2CC | 0.360 | 0.351 | 0.08 | 0.140 | -0.899 | 0.463 | - | -
1R1CC | 0.354 | 0.377 | - | 0.146 | -0.91 | 0.45 | - | -
3R | 0.365 | 0.341 | - | 1.2 | -1.31 | - | -1.06 | -1.40
Without losing generality we assume $p_{\nu}=(\vec{0},p)$ (i.e. the pion is at
rest) and write
$\frac{(p\cdot q)^{2}}{p^{2}}=\frac{q_{4}^{2}p^{2}}{p^{2}}=q_{4}^{2}.$ (19)
Now let us calculate $F(q_{4})$. With the quark propagator given in Eq. (16)
we can obtain
$\displaystyle F(q_{4})$ $\displaystyle=$
$\displaystyle\frac{\Xi(q_{4}^{2})}{\prod\limits_{j}[q^{2}+(a_{j}+ib_{j})^{2}]^{2}[q^{2}+(a_{j}-ib_{j})^{2}]^{2}\prod\limits_{k}(q^{2}+\eta_{k}^{2})},$
(20)
where $\Xi$ is a polynomial of $q_{4}^{2}$ (for the detailed calculation of
$F(q_{4})$, $\Xi$ and $\eta_{k}$, see the Appendix). The values of $\eta_{k}$
are shown in Table II ($\eta_{k}$ are ordered from small to large according to
their real part).
Table II. The calculated values of $\eta_{k}$.
Parameterization | $\eta_{1}$ (GeV) | $\eta_{2}$ (GeV) | $\eta_{3}$ (GeV) | $\eta_{4}$ (GeV) | $\eta_{5}$ (GeV)
---|---|---|---|---|---
2CC | 0.350 | 0.723-0.351i | 0.723+0.351i | - | -
1R1CC | 0.377 | 0.723-0.328i | 0.723+0.328i | - | -
3R | 0.341 | 0.617 | 1.31 | 1.40 | 1.849
Here it should be noticed that when some $b_{j}=0$ (the quark propagator has a
real pole), some $\eta_{k}$ must exactly equal the corresponding $|a_{j}|$
(see the Appendix). For 1R1CC case, $b_{1}=0$ and $\eta_{1}=|a_{1}|$. For 3R
case, all $b_{j}=0$ and $\eta_{1}=|a_{1}|$, $\eta_{3}=|a_{2}|$,
$\eta_{4}=|a_{3}|$. For 2CC case, because $b_{1}=0.08~{}\mbox{GeV}$ is very
close to zero, the value of $\eta_{1}$ is very close to $a_{1}$.
Because $q^{2}=q_{4}^{2}+\vec{q}^{2}$, according to Eq. (20) the poles of
$F(q_{4})$: $z_{n}=\chi_{n}+i\omega_{n}$ are decided by the following equation
$\displaystyle(\chi_{n}+i\omega_{n})^{2}+\vec{q}^{2}+(\xi_{nR}+i\xi_{nI})^{2}=0,$
(21)
where $\xi_{nR}$ and $\xi_{nI}$ are the real and imaginary part of $\eta_{k}$
or $a_{j}\pm ib_{j}$. One can easily find
$\displaystyle\omega_{n}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(\vec{q}^{2}+\xi_{nR}^{2}-\xi_{nI}^{2})+\sqrt{(\vec{q}^{2}+\xi_{nR}^{2}-\xi_{nI}^{2})^{2}+4\xi_{nR}^{2}\xi_{nI}^{2}}}{2}}$
(22) $\displaystyle\chi_{n}$ $\displaystyle=$
$\displaystyle-\frac{\xi_{nR}\xi_{nI}}{\omega_{n}}.$ (23)
From Eq. (22) we find that for $\mu<|\xi_{nR}|$ the corresponding $\omega_{n}$
is always larger than $\mu$, irrespective of $\vec{q}$. For $\mu>|\xi_{nR}|$,
$\omega_{n}<\mu$ when
$\vec{q}^{2}<\mu^{2}-(\xi_{nR}^{2}\xi_{nI}^{2}/\mu^{2})-\xi_{nR}^{2}+\xi_{nI}^{2}$,
and $\omega_{n}>\mu$ when
$\vec{q}^{2}>\mu^{2}-(\xi_{nR}^{2}\xi_{nI}^{2}/\mu^{2})-\xi_{nR}^{2}+\xi_{nI}^{2}$.
Therefore Eq. (15) can be written as
$f_{\pi}[\mu]=f_{\pi}-\frac{i}{2\pi^{2}f_{\pi}}\sum_{n}\theta(\mu-|\xi_{nR}|)\int\limits_{0}^{\Lambda_{n}(\mu)}\,d|\vec{q}|\,\vec{q}^{2}\mbox{Res}\\{F(z);z_{n}\\},$
(24)
where
$\displaystyle\Lambda_{n}(\mu)$ $\displaystyle=$
$\displaystyle\sqrt{\mu^{2}-(\xi_{nR}^{2}\xi_{nI}^{2}/\mu^{2})-\xi_{nR}^{2}+\xi_{nI}^{2}}\,.$
(25)
From Eq. (24) and the values of $a_{j}$, $b_{j}$ and $\eta_{k}$ in Table I and
II we find that when $\mu$ is below some threshold value $\mu_{0}$, the pion
decay constant at finite chemical potential $f_{\pi}[\mu]$ is kept unchanged
from its vacuum value. The threshold value $\mu_{0}$, which equals the minimum
of the real part of $a_{j}\pm ib_{j}$ and $\eta_{k}$, is shown in Table III.
Table III. The calculated values of $\mu_{0}$.
Parameterization | $\mu_{0}$ (GeV)
---|---
2CC | 0.350
1R1CC | 0.377
3R | 0.341
Here we note that in Ref. Zong08 it is found that when $\mu$ is below the
same threshold value $\mu_{0}$, the quark-number density vanishes identically.
Namely, $\mu=\mu_{0}$ is a singularity which separates two regions with
different quark-number densities. In fact, in Ref. Halasz98 , based on a
universal argument, it is pointed out that the existence of some singularity
at the point $\mu=\mu_{0}$ and $T=0$ is a robust and model-independent
prediction. Below $\mu=\mu_{0}$, the QCD system at finite $\mu$ remains in the
vacuum (ground state) of QCD at $\mu=0$, so the properties of the Goldstone
boson excited from this vacuum does not change with $\mu$. Thus the result
that $f_{\pi}[\mu]$ is kept unchanged from its vacuum value is just to be
expected. Here it should also be noticed that in our method the value of
$\mu_{0}$ is intimately connected with the pole distribution of the quark
propagator.
In Ref. con1 , with the same quark propagator the authors find that the quark
condensate at finite chemical potential is kept unchanged from its vacuum
value when $\mu<\mu_{0}$. From the Gell-Mann-Oakes-Renner relation
$f_{\pi}^{2}[\mu]m_{\pi}^{2}[\mu]=2m\langle\bar{q}q\rangle_{0}[\mu]+\mathcal{O}(m^{2})$
Maris1 ; fpi1 (where $m_{\pi}[\mu]$ is the pion mass at finite $\mu$, $m$ is
the current quark mass and $\langle\bar{q}q\rangle_{0}[\mu]$ is the quark
condensate in the chiral limit at finite $\mu$) one would also conclude that
$m_{\pi}[\mu]$ is kept unchanged from its value at $\mu=0$ when $\mu<\mu_{0}$.
In Ref. fpi1 , the authors did not made an analytical analysis of
$f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ by the method of pole analysis, but instead
made a direct numerical calculation. There exist numerical errors in this
calculation. Within numerical errors $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ do not
change with $\mu$ for $\mu<300~{}\mathrm{MeV}$. The analytical analysis made
in this paper explains the numerical results obtained in fpi1 .
For $\mu>\mu_{0}$, one can calculate $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$
numerically based on Eq. (24) and the Gell-Mann-Oakes-Renner relation. The
behaviors of $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ for $\mu>\mu_{0}$ are shown in
Fig. 2 and Fig. 3. One sees that $f_{\pi}[\mu]$ exhibits a sharp decrease
whereas $m_{\pi}[\mu]$ exhibits a sharp increase near $\mu_{0}$ for all three
cases. This result is quite different from the result in previous literatures.
For example, in a recent work Nam , those authors also investigated $f_{\pi}$
and $m_{\pi}$ at finite density within the framework of the nonlocal quark
model from the instanton vacuum. Their results show that in the range
$0\leq\mu\leq 320~{}\mathrm{MeV}$, $f_{\pi}$ falls slowly whereas $m_{\pi}$
increases slowly. This behavior of $f_{\pi}$ and $m_{\pi}$ is qualitatively
different from that found in this paper.
Finally, we should emphasize that in obtaining our results about
$f_{\pi}[\mu]$, $m_{\pi}[\mu]$ and $\langle\bar{q}q\rangle_{0}[\mu]$ in this
paper, we have made these approximations and assumptions: (1) we adopt the
rainbow-ladder approximation of the DSEs; (2) we assume the quark propagator
and quark-meson vertex are analytic in the neighborhood of $\mu=0$; (3) we
have neglected the $\mu$-dependence of the dressed gluon propagator. (for a
discussion about these approximations and assumptions, see Ref. fpi1 ). For
further study one should consider improvements on these approximations.
FIG.2. The $\mu$ dependence of $f_{\pi}$ near $\mu_{0}$.
FIG.3. The $\mu$ dependence of $m_{\pi}$ near $\mu_{0}$.
To summarize, based on the previous work in Ref. fpi1 on the quark-meson
vertex and pion properties at finite quark chemical potential, we provide an
analytical analysis of the weak decay constant of the pion ($f_{\pi}[\mu]$)
and the pion mass ($m_{\pi}[\mu]$) at finite quark chemical potential using
the model quark propagator proposed in Ref. Alkofer04 . It is found that when
$\mu$ is below a threshold value $\mu_{0}$ (which equals
$0.350~{}\mathrm{GeV}$, $0.377~{}\mathrm{GeV}$ and $0.341~{}\mathrm{GeV}$, for
the $\mathrm{2CC}$, $\mathrm{1R1CC}$ and $\mathrm{3R}$ parametrizations of the
model quark propagator, respectively.), $f_{\pi}[\mu]$ and $m_{\pi}[\mu]$ are
kept unchanged from their vacuum values. The value of $\mu_{0}$ is intimately
connected with the pole distribution of the model quark propagator and is
found to coincide with the threshold value below which the quark-number
density vanishes identically. Numerical calculations show that when $\mu$
becomes larger than $\mu_{0}$, $f_{\pi}[\mu]$ exhibits a sharp decrease
whereas $m_{\pi}[\mu]$ exhibits a sharp increase. These results are quite
different from those obtained in previous literatures. For example, our
results are qualitatively different from those reported in Ref. Nam , which
uses the nonlocal chiral quark model from the instanton vacuum to investigate
$f_{\pi}$ and $m_{\pi}$ at finite density.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of
China (under Grant No 10575050) and the Research Fund for the Doctoral Program
of Higher Education (under Grant No 20060284020).
## Appendix A The Analysis of the Poles
### A.1 General Analysis
With the quark propagator given by Eq. (16) one can find the following
$\displaystyle\sigma_{v}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[\frac{r_{j}}{q^{2}+(a_{j}+ib_{j})^{2}}+\frac{r_{j}}{q^{2}+(a_{j}-ib_{j})^{2}}\right]=\frac{f_{v}}{f_{0}}$
(26) $\displaystyle\sigma_{s}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[\frac{r_{j}(a_{j}+ib_{j})}{q^{2}+(a_{j}+ib_{j})^{2}}+\frac{r_{j}(a_{j}-ib_{j})}{q^{2}+(a_{j}-ib_{j})^{2}}\right]=\frac{f_{s}}{f_{0}}$
(27)
with
$\displaystyle f_{v}$ $\displaystyle=$
$\displaystyle\sum_{j}2r_{j}\left(q^{2}+a_{j}^{2}-b_{j}^{2}\right)\prod_{k\neq
j}[q^{2}+(a_{k}+ib_{k})^{2}][q^{2}+(a_{k}-ib_{k})^{2}]$ (28) $\displaystyle
f_{s}$ $\displaystyle=$
$\displaystyle\sum_{j}2r_{j}a_{j}\left(q^{2}+a_{j}^{2}+b_{j}^{2}\right)\prod_{k\neq
j}[q^{2}+(a_{k}+ib_{k})^{2}][q^{2}+(a_{k}-ib_{k})^{2}]$ (29) $\displaystyle
f_{0}$ $\displaystyle=$
$\displaystyle\prod_{j}[q^{2}+(a_{j}+ib_{j})^{2}][q^{2}+(a_{j}-ib_{j})^{2}].$
(30)
Then one has
$\displaystyle F(q_{4})$ $\displaystyle=$
$\displaystyle\frac{4\sigma_{s}}{\sigma_{v}^{2}q^{2}+\sigma_{s}^{2}}\left[\sigma_{s}\sigma_{v}+2q_{4}^{2}(\sigma_{s}\sigma_{v}^{\prime}-\sigma_{s}^{\prime}\sigma_{v})\right]=\frac{1}{f_{0}}\frac{\Xi}{f_{v}^{2}q^{2}+f_{s}^{2}},$
(31)
where
$\Xi=4f_{s}\left[f_{s}f_{v}+2(q^{2}-\vec{q}^{2})(f_{s}f_{v}^{\prime}-f_{s}^{\prime}f_{v})\right].$
(32)
For convenience let us use $x^{2}=q^{2}$ with $x$ a complex number. Then the
denominator of the right-hand-side of Eq. (31) can be decomposed as
$\displaystyle f_{v}^{2}x^{2}+f_{s}^{2}$ $\displaystyle=$
$\displaystyle(f_{v}x+if_{s})(f_{v}x-if_{s}).$ (33)
$f_{v}$ and $f_{s}$ can be expressed as
$\displaystyle f_{v}$ $\displaystyle=$
$\displaystyle\sum_{j}r_{j}\left[\frac{f_{0}}{x^{2}+(a_{j}+ib_{j})^{2}}+\frac{f_{0}}{x^{2}+(a_{j}-ib_{j})^{2}}\right]$
(34) $\displaystyle f_{s}$ $\displaystyle=$
$\displaystyle\sum_{j}r_{j}\left[\frac{f_{0}(a_{j}+ib_{j})}{x^{2}+(a_{j}+ib_{j})^{2}}+\frac{f_{0}(a_{j}-ib_{j})}{x^{2}+(a_{j}-ib_{j})^{2}}\right],$
(35)
so one has the following
$\displaystyle(f_{v}x+if_{s})(f_{v}x-if_{s})$ (36) $\displaystyle=$
$\displaystyle\left\\{\sum_{j}r_{j}f_{0}\left[\frac{x+i(a_{j}+ib_{j})}{x^{2}+(a_{j}+ib_{j})^{2}}+\frac{x+i(a_{j}-ib_{j})}{x^{2}+(a_{j}-ib_{j})^{2}}\right]\right\\}$
$\displaystyle\times\left\\{\sum_{j}r_{j}f_{0}\left[\frac{x-i(a_{j}+ib_{j})}{x^{2}+(a_{j}+ib_{j})^{2}}+\frac{x-i(a_{j}-ib_{j})}{x^{2}+(a_{j}-ib_{j})^{2}}\right]\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\sum_{j}r_{j}f_{0}\left[\frac{1}{x-i(a_{j}+ib_{j})}+\frac{1}{x-i(a_{j}-ib_{j})}\right]\right\\}$
$\displaystyle\times\left\\{\sum_{j}r_{j}f_{0}\left[\frac{1}{x+i(a_{j}+ib_{j})}+\frac{1}{x+i(a_{j}-ib_{j})}\right]\right\\}.$
$f_{0}$ can be expressed as
$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle\prod_{k_{1}}[x+i(a_{k_{1}}+ib_{k_{1}})][x+i(a_{k_{1}}-ib_{k_{1}})]$
(37)
$\displaystyle\times\prod_{k_{2}}[x-i(a_{k_{1}}+ib_{k_{1}})][x-i(a_{k_{1}}-ib_{k_{1}})].$
Therefore one obtains
$\displaystyle\sum_{j}r_{j}f_{0}\left[\frac{1}{x-i(a_{j}+ib_{j})}+\frac{1}{x-i(a_{j}-ib_{j})}\right]$
(38) $\displaystyle=$
$\displaystyle\sum_{j}\bigg{\\{}r_{j}[x-i(a_{j}-ib_{j})+x-i(a_{j}+ib_{j})]\prod_{k_{1}}[x+i(a_{k_{1}}+ib_{k_{1}})][x+i(a_{k_{1}}-ib_{k_{1}})]$
$\displaystyle\times\prod_{k_{2}\neq
j}[x-i(a_{k_{2}}+ib_{k_{2}})][x-i(a_{k_{2}}-ib_{k_{2}})]\bigg{\\}}$
$\displaystyle=$
$\displaystyle\prod_{k_{1}}[x+i(a_{k_{1}}+ib_{k_{1}})][x+i(a_{k_{1}}-ib_{k_{1}})]$
$\displaystyle\times\sum_{j}2r_{j}(x-ia_{j})\prod_{k\neq
j}[x-i(a_{k}+ib_{k})][x-i(a_{k}-ib_{k})]$
and
$\displaystyle\sum_{j}r_{j}f_{0}\left[\frac{1}{x+i(a_{j}+ib_{j})}+\frac{1}{x+i(a_{j}-ib_{j})}\right]$
(39) $\displaystyle=$
$\displaystyle\sum_{j}\bigg{\\{}r_{j}[x+i(a_{j}-ib_{j})+x+i(a_{j}+ib_{j})]\prod_{k_{1}\neq
j}[x+i(a_{k_{1}}+ib_{k_{1}})][x+i(a_{k_{1}}-ib_{k_{1}})]$
$\displaystyle\times\prod_{k_{2}}[x-i(a_{k_{2}}+ib_{k_{2}})][x-i(a_{k_{2}}-ib_{k_{2}})]\bigg{\\}}$
$\displaystyle=$
$\displaystyle\prod_{k_{2}}[x-i(a_{k_{2}}+ib_{k_{2}})][x-i(a_{k_{2}}-ib_{k_{2}})]$
$\displaystyle\times\sum_{j}2r_{j}(x+ia_{j})\prod_{k\neq
j}[x+i(a_{k}+ib_{k})][x+i(a_{k}-ib_{k})].$
With Eq. (37) one can find the following
$\displaystyle f_{v}^{2}x^{2}+f_{s}^{2}$ $\displaystyle=$
$\displaystyle(f_{v}x+if_{s})(f_{v}x-if_{s})$ (40) $\displaystyle=$
$\displaystyle f_{0}\left\\{\sum_{j}2r_{j}(x-ia_{j})\prod_{k\neq
j}[x-i(a_{k}+ib_{k})][x-i(a_{k}-ib_{k})]\right\\}$
$\displaystyle\times\left\\{\sum_{j}2r_{j}(x+ia_{j})\prod_{k\neq
j}[x+i(a_{k}+ib_{k})][x+i(a_{k}-ib_{k})]\right\\}.$
Hence, in order to determine the poles of $F(q_{4})$, one should solve the
following three equations:
$\displaystyle
f_{0}=\prod_{j}[x^{2}+(a_{j}+ib_{j})^{2}][x^{2}+(a_{j}-ib_{j})^{2}]$
$\displaystyle=$ $\displaystyle 0,$ (41)
$\displaystyle\sum_{j}2r_{j}(x-ia_{j})\prod_{k\neq
j}[x-i(a_{k}+ib_{k})][x-i(a_{k}-ib_{k})]$ $\displaystyle=$ $\displaystyle 0,$
(42) $\displaystyle\sum_{j}2r_{j}(x+ia_{j})\prod_{k\neq
j}[x+i(a_{k}+ib_{k})][x+i(a_{k}-ib_{k})]$ $\displaystyle=$ $\displaystyle 0.$
(43)
Here it should be noted that if some $b_{j}=0$ then $x=ia_{j}$
($\mathrm{or}~{}x=-ia_{j}$) must be the solution of Eq. (42) (or Eq. (43)).
One should also be aware that after finding the roots of the above equations
one should substitute them into $\Xi$ to ensure that $\Xi(x)\neq 0$ (we will
see it in the discussion of 1R1CC and 3R case below). For general $n_{P}$ Eq.
(42) (or Eq. (43)) is an equation of degree $2n_{P}-1$ in $x$ and it is almost
impossible to give the analytic form of the solution for general $r_{j}$,
$a_{j}$ and $b_{j}$ when $n_{P}\geq 2$.
### A.2 Detailed Calculation of the Poles
For 2CC case one can find the following
$\displaystyle f_{v}$ $\displaystyle=$ $\displaystyle
q^{6}+d_{v1}q^{4}+d_{v2}q^{2}+d_{v3}$ (44) $\displaystyle f_{s}$
$\displaystyle=$ $\displaystyle d_{s1}q^{4}+d_{s2}q^{2}+d_{s3}$ (45)
$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle[q^{4}+2(a_{1}^{2}-b_{1}^{2})q^{2}+(a_{1}^{2}+b_{1}^{2})^{2}][q^{4}+2(a_{2}^{2}-b_{2}^{2})q^{2}+(a_{2}^{2}+b_{2}^{2})^{2}],$
(46)
where $d_{v1},d_{v2},d_{v3},d_{s1},d_{s2},d_{s3}$ are coefficients decided by
$r_{j},a_{j},b_{j}$. With parameters shown in Table I the solutions of Eq.
(42) and Eq. (43) are found to be
$\eta_{1}=0.350~{}\mathrm{GeV},\eta_{2,3}=(0.723\pm 0.351i)~{}\mathrm{GeV}$.
Of course, one can directly verify
$\displaystyle f_{v}^{2}q^{2}+f_{s}^{2}$ $\displaystyle=$ $\displaystyle
f_{0}(q^{2}+\eta_{1}^{2})(q^{2}+\eta_{2}^{2})(q^{2}+\eta_{3}^{2}).$ (47)
So the poles of $F(q_{4})$ for 2CC parameters (in the upper half complex
$q_{4}$ plane) are
$\displaystyle z_{1}$ $\displaystyle=$ $\displaystyle
i\sqrt{\vec{q}^{2}+\eta_{1}^{2}}\,\,\,\,(\mbox{simple pole})$ (48)
$\displaystyle z_{2}$ $\displaystyle=$
$\displaystyle\chi_{2}+i\omega_{2}\,\,\,\,(\mbox{simple pole})$ (49)
$\displaystyle z_{3}$ $\displaystyle=$
$\displaystyle\chi_{3}+i\omega_{3}\,\,\,\,(\mbox{simple pole})$ (50)
$\displaystyle z_{4}$ $\displaystyle=$
$\displaystyle\chi_{4}+i\omega_{4}\,\,\,\,(\mbox{double pole})$ (51)
$\displaystyle z_{5}$ $\displaystyle=$
$\displaystyle\chi_{5}+i\omega_{5}\,\,\,\,(\mbox{double pole})$ (52)
$\displaystyle z_{6}$ $\displaystyle=$
$\displaystyle\chi_{6}+i\omega_{6}\,\,\,\,(\mbox{double pole})$ (53)
$\displaystyle z_{7}$ $\displaystyle=$
$\displaystyle\chi_{7}+i\omega_{7}\,\,\,\,(\mbox{double pole})$ (54)
with
$\displaystyle\omega_{2}$ $\displaystyle=$ $\displaystyle\omega_{3}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\vec{q}^{2}+(\mbox{Re}\eta_{2})^{2}-(\mbox{Im}\eta_{2})^{2}+\sqrt{[\vec{q}^{2}+(\mbox{Re}\eta_{2})^{2}-(\mbox{Im}\eta_{2})^{2}]^{2}+4(\mbox{Re}\eta_{2})^{2}(\mbox{Im}\eta_{2})^{2}}}{2}}$
$\displaystyle\chi_{2}$ $\displaystyle=$
$\displaystyle-\chi_{3}=-\frac{(\mbox{Re}\eta_{2})(\mbox{Im}\eta_{2})}{\omega_{2}}$
(56) $\displaystyle\omega_{4}$ $\displaystyle=$
$\displaystyle\omega_{5}=\sqrt{\frac{\vec{q}^{2}+a_{1}^{2}-b_{1}^{2}+\sqrt{(\vec{q}^{2}+a_{1}^{2}-b_{1}^{2})^{2}+4a_{1}^{2}b_{1}^{2}}}{2}}$
(57) $\displaystyle\chi_{4}$ $\displaystyle=$
$\displaystyle-\chi_{5}=-\frac{a_{1}b_{1}}{\omega_{4}}$ (58)
$\displaystyle\omega_{6}$ $\displaystyle=$
$\displaystyle\omega_{7}=\sqrt{\frac{\vec{q}^{2}+a_{2}^{2}-b_{2}^{2}+\sqrt{(\vec{q}^{2}+a_{2}^{2}-b_{2}^{2})^{2}+4a_{2}^{2}b_{2}^{2}}}{2}}$
(59) $\displaystyle\chi_{6}$ $\displaystyle=$
$\displaystyle-\chi_{7}=-\frac{a_{2}b_{2}}{\omega_{6}}.$ (60)
For 1R1CC (and 3R) case, the analysis is similar except a little modification
for correctly analyzing the degree of the poles. Because $b_{1}=0$ for 1R1CC
case (for 3R case, all $b_{j}$ equal zero) the function $f_{v}$ and $f_{s}$
has a factor of $q^{2}+a_{1}^{2}$ (see Eq. (28) and Eq. (29)) which would be
canceled by the same factor in $f_{0}$. Therefore for 1R1CC case one should
adopt the following modified expressions
$\displaystyle f_{v1}$ $\displaystyle=$
$\displaystyle\frac{f_{v}}{q^{2}+a_{1}^{2}}$ (61) $\displaystyle=$
$\displaystyle
2r_{1}[q^{2}+(a_{2}+ib_{2})^{2}][q^{2}+(a_{2}-ib_{2})^{2}]+r_{2}(q^{2}+a_{1}^{2})[q^{2}+(a_{2}-ib_{2})^{2}]$
$\displaystyle+r_{2}(q^{2}+a_{1}^{2})[q^{2}+(a_{2}+ib_{2})^{2}]$
$\displaystyle f_{s1}$ $\displaystyle=$
$\displaystyle\frac{f_{s}}{q^{2}+a_{1}^{2}}$ (62) $\displaystyle=$
$\displaystyle
2r_{1}a_{1}[q^{2}+(a_{2}+ib_{2})^{2}][q^{2}+(a_{2}-ib_{2})^{2}]+r_{2}(a_{2}+ib_{2})(q^{2}+a_{1}^{2})[q^{2}+(a_{2}-ib_{2})^{2}]$
$\displaystyle+r_{2}(a_{2}-ib_{2})(q^{2}+a_{1}^{2})[q^{2}+(a_{2}+ib_{2})^{2}]$
$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{f_{0}}{q^{2}+a_{1}^{2}}$ (63) $\displaystyle=$
$\displaystyle(q^{2}+a_{1}^{2})[q^{4}+2(a_{2}^{2}-b_{2}^{2})q^{2}+(a_{2}^{2}+b_{2}^{2})^{2}].$
According to the decomposition in Eq. (40) one has
$\displaystyle f_{v}^{2}q^{2}+f_{s}^{2}$ $\displaystyle=$ $\displaystyle
f_{0}(q^{2}+\eta_{1}^{2})(q^{2}+\eta_{2}^{2})(q^{2}+\eta_{3}^{2})$ (64)
with $\eta_{1,2,3}$ being obtained by solving Eqs. (42)-(43). Then
$\displaystyle F(q_{4})$ $\displaystyle=$
$\displaystyle\frac{4f_{s}[f_{s}f_{v}+2(q^{2}-\vec{q}^{2})(f_{s}f_{v}^{\prime}-f_{v}f_{s}^{\prime})]}{f_{0}^{2}(q^{2}+\eta_{1}^{2})(q^{2}+\eta_{2}^{2})(q^{2}+\eta_{3}^{2})}$
(65) $\displaystyle=$
$\displaystyle\frac{4f_{s1}[f_{s1}f_{v1}+2(q^{2}-\vec{q}^{2})(f_{s1}f_{v1}^{\prime}-f_{v1}f_{s1}^{\prime})]}{f_{1}^{2}(q^{2}+\eta_{1}^{2})(q^{2}+\eta_{2}^{2})(q^{2}+\eta_{3}^{2})}(q^{2}+a_{1}^{2}).$
(66)
Because $\eta_{1}$ equal $a_{1}$ exactly, one would find that
$i\sqrt{\vec{q}^{2}+a_{1}^{2}}$ is a double pole. The analysis for 3R case is
similar.
## References
* (1) J. Delorme, G. Chanfray, and M. Ericson, Nucl. Phys. A 603, 239 (1996).
* (2) M. Kirchbach and A. Wirzba, Nucl. Phys. A 616, 648 (1997).
* (3) N. Kaiser and W. Weise, Phys. Lett. B 512, 283 (2001).
* (4) U.G. Meissner, J.A. Oller and A. Wirzba, Annals. Phys. 297, 27 (2002).
* (5) H.C. Kim and M. Oka, Nucl. Phys. A 720, 386 (2003).
* (6) S. Mallik and S. Sarkar, Phys. Rev. C 69, 015204 (2004).
* (7) S.I. Nam and H.C. Kim, Phys. Lett. B 666, 324 (2008).
* (8) P. Maris, C.D. Roberts, and S. Schmidt, Phys. Rev. C 57, R2821 (1998).
* (9) A. Bender et al., Phys. Lett. B 431, 263 (1998).
* (10) A. Bender, W. Detmold, and A.W. Thomas, Phys. Lett. B 516, 54 (2001).
* (11) P. Maris, C.D. Roberts, and P.C. Tandy, P]hys. Lett. B 420, 267 (1998).
* (12) C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994), and references therein.
* (13) C.D. Roberts and S.M. Schmidt, Prog. Part. Nucl. Phys. 45S1, 1 (2000), and references therein.
* (14) P. Maris and C.D. Roberts, Int. J. Mod Phys. E 12, 297 (2003).
* (15) R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001); C.S. Fischer and R. Alkofer, Phys. Rev. D 67, 094020 (2003), and references therein.
* (16) Y. Jiang, Y.M. Shi, H.T. Feng, W.M. Sun and H.S. Zong, Phys. Rev. C 78, 025214 (2008).
* (17) H.S. Zong, L. Chang, F.Y.Hou, W.M. Sun and Y.X. Liu, Phys. Rev. C 71, 015205 (2005).
* (18) M.R. Frank and C.D. Roberts, Phys. Rev. C 53, 390 (1996).
* (19) Y. Jiang, Y.B. Zhang, W.M. Sun and H.S. Zong, Phys. Rev. D 78, 014005 (2008).
* (20) R. Alkofer, W. Detmold, C.S. Fischer and P. Maris, Phys. Rev. D 70, 014014 (2004).
* (21) H.S. Zong and W.M. Sun, Phys. Rev. D 78, 054001 (2008).
* (22) M.A. Halasz, A.D. Jackson, R.E. Shrock, M.A. Stephanov and J.J.M. Verbaarschot, Phys. Rev. D 58, 096007 (1998).
|
arxiv-papers
| 2008-10-04T08:51:48
|
2024-09-04T02:48:58.123420
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Jiang, Yuan-mei Shi, Hua Li, Wei-min Sun and Hong-shi Zong",
"submitter": "Wei-Min Sun",
"url": "https://arxiv.org/abs/0810.0750"
}
|
0810.0824
|
# Coherent transport on Apollonian networks and continuous-time quantum walks
Xin-Ping Xu1,3 Wei Li1,2 Feng Liu1 1Institute of Particle Physics, HuaZhong
Normal University, Wuhan 430079, China
2Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig,
Germany
3Institute of High Energy Physics, Chinese Academy of Science, Beijing 100049,
China
###### Abstract
We study the coherent exciton transport on Apollonian networks generated by
simple iterative rules. The coherent exciton dynamics is modeled by
continuous-time quantum walks and we calculate the transition probabilities
between two nodes of the networks. We find that the transport depends on the
initial nodes of the excitation. For networks less than the second generation
the coherent transport shows perfect revivals when the initial excitation
starts at the central node. For networks of higher generation, the transport
only shows partial revivals. Moreover, we find that the excitation is most
likely to be found at the initial nodes while the coherent transport to other
nodes has a very low probability. In the long time limit, the transition
probabilities show characteristic patterns with identical values of limiting
probabilities. Finally, the dynamics of quantum transport are compared with
the classical transport modeled by continuous-time random walks.
###### pacs:
05.60.Gg, 05.60.Cd, 71.35.-y, 89.75.Hc, 89.75.-k
The problem of coherent exciton transport modeled by quantum walks is widely
studied and relevant to many distinct fields, such as polymer physics, solid
state physics, biological physics and quantum computation rn1 ; rn2 ; rn3 .
Such studies have been done in the framework of continuous-time quantum walks
(CTQWs) and on various discrete systems rn4 ; rn5 . It has been shown that the
dynamics of coherent transport are strongly influenced by the structure of the
underlying discrete systems rn6 ; rn7 ; rn8 . Most of previous studies focus
CTQWs on graphs with simple structures rn9 ; rn10 ; rn11 , coherent dynamics
on general graphs have not received much attention in the scientific
community. To this end, it is natural to consider quantum transport on graphs
with general structure embedded in nature.
An important and universal feature of networked systems (or graphs) in nature
is that they have the small-world and scale-free property rn12 ; rn13 . The
Apollonian networks (ANs) rn14 ; rn15 are a very useful toy model that
captures all these features simultaneously, thus provide a good facility to
study the dynamical processes taking place on networked systems, including
percolation, electrical conduction, etc rn14 ; rn16 ; rn17 .
In this paper, we consider coherent exciton transport on 2D Apollonian
networks (ANs). The network can be generated as follows rn14 : At the initial
generation $g=0$, the network is composed of three fully connected nodes
marked as $1$, $2$, and $3$. At the subsequent generation, a new node is added
inside each (newly established) triangle and linked to the three vertices of
the triangle. Using this simple rule, we can obtain a deterministic 2D ANs of
size $N=3+(3^{G}-1)/2$ ($G$ is the number of generation) rn14 . Many
topological properties of this network model have been well-studied in the
literature rn15 ; rn18 . Fig. 1 shows the structure of an AN in four
generations ($G=4$).
Figure 1: Apollonian network generated by simple iterative rules in four
generations ($G=4$). The nodes at each generation $g$ are marked as
consecutive numbers.
Figure 2: (Color online) Time evolution of transition probabilities
$\pi_{k,4}(t)$ for different values of $k$ (marked as different types of
curves) on ANs of $G=3$ (a) and $G=4$ (b). The excitation starts at central
node $4$. The insets are enlarged linear-scale plots of return probability
$\pi_{4,4}(t)$.
The coherent exciton transport on a connected network is modeled by the
continuous-time quantum walks (CTQWs), which is obtained by replacing the
Hamiltonian of the system by the classical transfer matrix, i.e., $H=-T$ rn19
; rn20 . The transfer matrix $T$ relates to the Laplace matrix by $T=-\gamma
A$, where for simplicity we assume the transmission rates $\gamma$ of all
bonds to be equal and set $\gamma\equiv 1$ in the following rn19 ; rn20 . The
Laplace matrix $A$ has nondiagonal elements $A_{ij}$ equal to $-1$ if nodes
$i$ and $j$ are connected and $0$ otherwise. The diagonal elements $A_{ii}$
equal to degree of node $i$, i.e., $A_{ii}=k_{i}$. The states $|j\rangle$
endowed with the node $j$ of the network form a complete, ortho-normalised
basis set, which span the whole accessible Hilbert space. The time evolution
of a state $|j\rangle$ starting at time $t_{0}$ is given by
$|j,t\rangle=U(t,t_{0})|j\rangle$, where $U(t,t_{0})=exp[-iH(t-t_{0})]$ is the
quantum mechanical time evolution operator. The transition amplitude
$\alpha_{k,j}(t)$ from state $|j\rangle$ at time $0$ to state $|k\rangle$ at
time $t$ reads $\alpha_{k,j}(t)=\langle k|U(t,0)|j\rangle$ and obeys
Schrödinger s equation rn21 . Then the classical and quantum transition
probabilities to go from the state $|j\rangle$ at time $0$ to the state
$|k\rangle$ at time $t$ are given by $p_{k,j}(t)=\langle k|e^{-tA}|j\rangle$
and $\pi_{k,j}(t)=|\alpha_{k,j}(t)|^{2}=|\langle k|e^{-itH}|j\rangle|^{2}$
rn19 , respectively. Using $E_{n}$ and $|q_{n}\rangle$ to represent the $n$th
eigenvalue and eigenvector of $H$, the classical and quantum transition
probabilities between two nodes can be written as rn19 ; rn20 ; rn21
$p_{k,j}(t)=\sum_{n}e^{-tE_{n}}\langle k|q_{n}\rangle\langle q_{n}|j\rangle,$
(1) $\pi_{k,j}(t)=|\alpha_{k,j}(t)|^{2}=|\sum_{n}e^{-itE_{n}}\langle
k|q_{n}\rangle\langle q_{n}|j\rangle|^{2}.$ (2)
Generally, to get $p_{k,j}(t)$ and $\pi_{k,j}(t)$, all the eigenvalues $E_{n}$
and eigenvectors $|q_{n}\rangle$ are required. In the following we will
consider $p_{k,j}(t)$ and $\pi_{k,j}(t)$ obtained from diagonalizing the
Hamiltonian $H$ by using the standard software package Mathematica 5.0.
Figure 3: (Color online) Time evolution of the classical probabilities
$p_{k,4}(t)$ for different values of $k$ (marked as different types of curves)
on ANs of $G=3$ (a) and $G=4$ (b). The excitation starts at central node $4$.
The classical $p_{k,4}(t)$ approach the equip-partitioned probability $1/N$ at
long time scale.
We start our analysis by considering transport dynamics on ANs of $G=3$
($N=16$) and $G=4$ ($N=43$) when the excitation starts at the central node
$4$. The nodes are numbered according to Fig. 1 and network of $G=3$ (nodes
labeled as $1\sim 16$) is a subgraph of $G=4$ (nodes labeled as $1\sim 43$).
Due to rotational symmetry, the transition probabilities from node $4$ to
certain groups of nodes are equal. Thus, we choose several different
transition probabilities, namely, $\pi_{4,4}(t)$, $\pi_{1,4}(t)$,
$\pi_{5,4}(t)$, $\pi_{8,4}(t)$ and $\pi_{14,4}(t)$ for further study.
Fig. 2 shows these quantum transition probabilities for ANs of $G=3$ and
$G=4$. We find that there is a high probability to find the excitation at the
initial node ($\pi_{4,4}(t)$ marked as solid curves in the Fig. 2). To see the
behavior of $\pi_{4,4}(t)$ clearly, we display $\pi_{4,4}(t)$ in an enlarged
linear scale (See inserted plots in Fig. 2). For AN of $G=3$, $\pi_{4,4}(t)$
shows regular oscillations, as generation increases, $\pi_{4,4}(t)$ becomes
irregular and its average value increases (Compare the inserted plots in Fig.
2 (a) and (b)). Transition probabilities between the initial node and other
nodes are considerably low compared to the return probability $\pi_{4,4}(t)$
(See the dashed curves in (a) and (b)). The corresponding transition
probabilities $\pi_{k,4}(t)$ ($k\neq 4$) of a $G=3$ AN is higher than those of
a $G=4$ AN. This may be attributed to the fact that the return probability
$\pi_{4,4}(t)$ on a $G=4$ AN is larger than that on a $G=3$ AN.
Figure 4: (Color online) (a) Transition probabilities $\pi_{k,j}(t)$ for AN of
$G=1$. (b) Transition probabilities $\pi_{k,4}(t)$ for AN of $G=2$. Both
results are numerically obtained by diagonalizing the Hamiltonian $H$ and
consistent with the analytical results in Eqs. (3) and (4).
Fig. 3 shows the classical transition probabilities $p_{k,4}(t)$ for different
values of $k$. It is found that the classical transition probabilities
approach the equipartition $1/N$ very quickly and $p_{14,4}(t)$ reaches $1/N$
much slower than other transition probabilities. This can be explained by the
shortest path length from the initial excitation node $4$. The number of bonds
between node $4$ and $14$ is larger than the distance between other pairs of
nodes. In addition, because the shortest path lengths between $4$ and $1$, $4$
and $5$, $4$ and $8$ are equal, the classical $p_{1,4}(t)$, $p_{5,4}(t)$ and
$p_{8,4}(t)$ are comparable (Compare the curves in Fig. 3). Noting that the
long time averaged $\pi_{4,4}(t)$ is much higher than equip-partitioned
probability $1/N$ and other (long time averaged) quantum transition
probabilities is less than $1/N$, we conclude that the classical transport to
other nodes (non-initial node) of the network is more efficient than quantum
transport.
Interestingly, for $G=1$ ($N=4$) and $G=2$ ($N=7$) ANs, the quantum transition
probabilities are fully periodic when the coherent excitation starts from the
central node $4$. In this case we obtain, based on the analytically determined
eigenvalues and eigenvectors rn22 , that for $G=1$, $\pi_{k,j}(t)$ have the
following, periodic form:
$\pi_{k,j}(t)=\left\\{\begin{array}[]{ll}(5+3\cos 4t)/8,&k=j,\\\ (1-\cos
4t)/8,&k\neq j.\end{array}\right.$ (3)
And for $G=2$ we have,
$\pi_{k,4}(t)=\left\\{\begin{array}[]{ll}(37+12\cos 7t)/49,&k=4,\\\ (2-2\cos
7t)/49,&k\neq 4.\end{array}\right.$ (4)
Fig. 4 shows the behavior of $\pi_{k,j}(t)$ obtained by numerically
diagonalizing the Hamiltonian $H$ for $G=1$ and $G=2$ ANs. This agrees the
analytical results in Eqs. (3) and (4). We find that there is a perfect
revival of the initial state for each $t=2n\pi/N$ ($n\in$ Integers), where $N$
is the number of nodes of the considered network. This revival of the initial
probability distribution resembles the results obtained for continuous and
discrete quantum carpets rn23 ; rn7 , in which the revival is only perfect for
small size of cycles rn24 ; rn7 . The case for ANs is analogous: The revivals
are perfect for small ANs of $G\leqslant 2$, when the network size becomes
larger ($G\geqslant 3$), there are only partial revivals of the initial state
(Compare Fig. 2 and Fig. 4).
Figure 5: (Color online) Time evolution of transition probabilities
$\pi_{k,1}(t)$ for different values of $k$ (marked as different types of
curves) on ANs of $G=3$ (a) and $G=4$ (b). The excitation starts at noncentral
node $1$. The return probability $\pi_{1,1}(t)$ is nearly periodic for both
the networks.
Now we turn to the case when the initial excitation starts at other positions
of the network. Fig. 5 shows the transition probabilities when the initial
excitation is placed at node $1$. For both the $G=3$ and $G=4$ ANs, the return
probabilities $\pi_{1,1}(t)$ display regular oscillations. The return
probability $\pi_{1,1}(t)$ is much larger than other transition probabilities
$\pi_{k,1}(t)$ ($k\neq 1$) at most time intervals. It is interesting to note
that except for the high return probability $\pi_{1,1}(t)$, there is also
considerable transport to nodes $2$ and $3$ (Note $\pi_{2,1}(t)=\pi_{3,1}(t)$
because of axis-symmetry). Nevertheless, transport to other nodes (such as
$4$, $5$ etc) is particularly low. This suggests that the excitation is
preferably located on the nodes of the same generation of the initial node.
If the initial excitation starts from other noncentral nodes, the results are
similar but some details change. The oscillation amplitude and period are
different and there is also a relative high probability to find the excitation
at the initial node.
Figure 6: (Color online) Long time limiting probabilities $\chi_{k,j}$ for
different node $j$ of initial excitation on the $G=3$ (a) and $G=4$ (b) ANs.
The squares, dots, triangles and rhombus denote initial excitation at node
$4$, $1$, $5$ and $8$ respectively.
In order to discuss what happens at long times, we consider the long time
averages of the transition probabilities $p_{j,k}(t)$ and $\pi_{j,k}(t)$. On
finite ANs, the transition probability converges to a certain value, this
value is determined by the long time average. Classically, the long time
averaged transition probabilities equal to the equal-partitioned probability
$1/N$. However, the quantum transport does not lead to equipartition. The long
time average of $\pi_{j,k}(t)$ is defined as
$\begin{array}[]{ll}\chi_{k,j}&=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\pi_{k,j}(t)dt\\\
&=\sum_{n,l}\delta(E_{n}-E_{l})\langle k|q_{n}\rangle\langle
q_{n}|j\rangle\langle j|q_{l}\rangle\langle q_{l}|k\rangle,\end{array}$ (5)
where $\delta(E_{n}-E_{l})=1$ for $E_{n}=E_{l}$ and $\delta(E_{n}-E_{l})=0$
else. Some eigenvalues of $H$ may be degenerate, so the sum in the equation
contain terms belonging to different eigenstates. Here, we consider the
limiting transition probabilities $\chi_{k,j}$ according to this equation.
Fig. 6 (a) and (b) show the limiting probability distributions for $G=3$ and
$G=4$ ANs. In the figure, we find that $\chi_{j,j}$ is larger than other
transition probabilities $\chi_{k,j}$ ($k\neq j$). This indicates the
excitation is most likely to be found at the initial node, which is accord
with the observation in Figs. 2 and 5.
An interesting feature related to the limiting probabilities is that different
nodes, $k$ and $l$, may have the same transition probabilities, i.e.,
$\chi_{k,j}=\chi_{l,j}$. Concretely, for an excitation starting from the
central node $j=4$, transport to nodes of certain cluster has identical
limiting transition probabilities (See the black squares in Fig. 6). For
instance, $\chi_{1,4}$, $\chi_{2,4}$ and $\chi_{3,4}$ are equal to each other;
$\chi_{5,4}=\chi_{6,4}=\chi_{7,4}$; $\chi_{k,4}$ are equal for $8\leqslant
k\leqslant 13$; … . The nodes of clusters having the same transition
probabilities in such case are easy to be identified due to rotation-symmetry
of the central node. Furthermore, nodes of the same generation or having the
same connectivity may have different limiting probabilities (compare the
values of $\chi_{k,4}$ for cluster $8\leqslant k\leqslant 13$ and cluster
$14\leqslant k\leqslant 16$ in the Fig. 6).
For an excitation starting from noncentral node, the situation is quite
different. When the excitation starts at node $1$ (See the dots in Fig 6),
$\chi_{2,1}$ equals to $\chi_{3,1}$. Such identical values of transition
probabilities are also easy to be distinguished and can be understood as a
result of the axis-symmetry. The case for excitation starting at node $5$ is
analogous (See the triangles in the plots). Particularly, if the excitation
starts at node $8$, nodes $10$ and $12$ have the same limiting probability,
i.e., $\chi_{10,8}=\chi_{12,8}$ (rhombus indicated in Fig. 6). Such kind of
identical probability is not straightforward to be realized but also can be
ascribed to the rotation symmetry of the structure of ANs. If triangle $\Delta
145$ is rotated $\pi/3$ and $2\pi/3$, the initial node $8$ changes to the
positions $12$ and $10$ respectively. Except for the equal value of
$\chi_{10,8}$ and $\chi_{12,8}$ on both the $G=3$ and $G=4$ ANs, we find that
there are more identical probabilities on the $G=4$ AN. For instance, we find
the following equal transition probabilities: $\chi_{31,8}=\chi_{33,8}$,
$\chi_{19,17}=\chi_{21,17}$, $\chi_{26,23}=\chi_{28,23}$,
$\chi_{10,29}=\chi_{12,29}$, $\chi_{31,29}=\chi_{33,29}$,
$\chi_{37,35}=\chi_{39,35}$, etc. For ANs of higher generation, the previous
identical limiting probabilities are preserved and additional identical values
of transition probabilities are observed due to structural symmetry of the
network.
In summary, we have studied coherent exciton transport modeled by continuous-
time quantum walks on ANs. The quantum transport exhibits a very distinct
behavior compared to the classical random walks. For networks up to the second
generation the coherent transport shows perfect recurrences when the initial
excitation starts at the central node rn7 . For networks of higher generation,
the recurrence ceases to be perfect, which resembles results for discrete
quantum carpets rn7 . The excitation depends on the initial nodes and is most
likely to be found at the original nodes while the coherent transport to other
nodes is particularly low. In the long time limit, the transition
probabilities show identical values between different nodes, which reflects
the symmetry of the network structure.
We would like to point out that although CTQWs on ANs show oscillation and
revivals like the results of the 1D case, there are some difference in the
quantum dynamics between the two structures. For ANs, we find that the return
probabilities at the central nodes are nearly periodic, in contrast to the 1D
case where the (maximums of) return probability shows a power law decay as
$\pi(t)\sim t^{-1}$ rn21 ; rn25 . In Ref. rn25 , the authors find that for a
1D chain, quantum revivals do not repeat indefinitely but become less and less
accurate as time progresses rn25 . They also find that the quantum walks
displays Anderson localizations or decoherence in the presence static or
dynamic disorder rn25 . For ANs, there are also considerable localizations on
the initial nodes (See Fig. 6). Such localizations may relate to the network
structures and requires a further study.
This work is supported by National Natural Science Foundation of China under
Project Nos 10575042, 10775058 and MOE of China under contract number IRT0624
(CCNU).
## References
* (1) J. Kempe, Contemp. Phys. 44, 307 (2002).
* (2) G. H. Weiss, _Aspect and Applications of the Random Walk_ (North-Holland, Amsterdam, 1994).
* (3) J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990).
* (4) D. Solenov and L. Fedichkin, Phys. Rev. A 73, 012313(2003).
* (5) H. Krovi and T. A. Brun, Phys. Rev. A 73, 032341 (2006).
* (6) E. M. Bollt and D. ben-Avraham, New J. Phys. 7 26 (2005).
* (7) O. Mülken, V. Bierbaum and A. Blumen, J. Chem. Phys 124, 124905 (2006).
* (8) A. Blumen, V. Bierbaum and O. Mülken, Physica A 371, 10 (2006).
* (9) N. Ashwin and V. Ashvin, quant-ph/0010117.
* (10) O. Mülken and A. Blumen, Phys. Rev. E 71, 016101 (2005).
* (11) O. Mülken, V. Pernice and A. Blumen, Phys. Rev. E 76, 051125 (2007).
* (12) R. Albert and A.-L. Barabäsi, Rev. Mod. Phys 74, 47 (2002).
* (13) S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Phys. Rep. 424, 175 (2006).
* (14) J.S. Andrade, H.J. Herrmann, R.F.S. Andrade, and L.R.da Silva, Phys. Rev. Lett. 94, 018702 (2005).
* (15) J. P. K. Doye and C. P. Massen, Phys. Rev. E 71, 016128 (2005).
* (16) A. M. Souza and H. Herrmann, Phys. Rev. B 75, 054412 (2007).
* (17) A. P. Vieira, J. S. Andrade, H. J. Herrmann and R. F. Andrade, Phys. Rev. E 76, 026111 (2007).
* (18) P. G. Lind, J. A. C. Gallas and H. J. Herrmann, Phys. Rev. E 70, 056207 (2004).
* (19) O. Mülken and A. Blumen, Phys. Rev. E 71, 016101 (2005).
* (20) O. Mülken and A. Blumen, Phys. Rev. E 71, 036128 (2005).
* (21) X. P. Xu, Phys. Rev. E 77, 061127 (2008).
* (22) R.F.S. Andrade and J.G.V. Miranda, Physica A 356, 1 (2005).
* (23) R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, Phys. Rev. Lett. 95, 053902 (2005).
* (24) F. Grossmann, J.-M. Rost, and W. P. Schleich, J. Phys. A 30, L277 (1997).
* (25) Y. Yin, D. E. Katsanos, and S. N. Evangelou, Phys. Rev. A 77, 022302 (2008).
|
arxiv-papers
| 2008-10-06T02:00:11
|
2024-09-04T02:48:58.128887
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xinping Xu, Wei Li, Feng Liu",
"submitter": "Xinping Xu",
"url": "https://arxiv.org/abs/0810.0824"
}
|
0810.0905
|
# Construction of empirical formulas for prediction of experimental data
Marijan Ribarič and Luka Šušteršič
Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
e-mail address: luka.sustersic@ijs.si
###### Abstract
We consider construction of empirical formulas for predicting new averaged
experimental data from a finite number of the old ones so as to point out the
relevant technical problems. The main problem is that there is always only a
finite number of inaccurate data at our disposal. As an example, we construct
empirical formulas for describing Planck’s law in terms of averaged
experimental data.
PASC numbers: 01.65.+q, 07.05.Kf, 02-60.-x, 32.30.-v
UDC: 53(091), 53.08, 519.657, 535.33
Keywords: empirical formulas, Planck’s law, experimental data, Hooke’s law,
Ohm’s law
## 1 Introduction
Physics attempts to predict results of new measurements on the basis of
already obtained experimental results. So to say, it is concerned with
interpolation and extrapolation of experimental data by mathematical formulas,
starting with empirical ones. Historically, many empirical relationships have
been stepping stones to theories providing physical laws that generalize and
extend them. Let us give just four examples:
* 1)
Ptolemaic system espoused in the 2nd century AD, evolved over Copernican
system to Newton’s law of universal gravitation, cf. e.g.[1].
* 2)
Hooke’s law stated in 1676 has been generalized to a tensor expression to
study deformations of various materials, cf. e.g.[2].
* 3)
Ohm’s law published in 1827 remains an extremely useful formula in
electrical/electronic engineering, cf. e.g.[3]
* 4)
Balmer formula proposed in 1855 to predict the spectral line emissions of the
hydrogen atom, and its generalization, Rydberg’s formula proposed in 1888 to
predict spectral lines of hydrogen-like atoms of chemical elements, were
incorporated into the Bohr’s model in 1913; a stepping stone to quantum
mechanics, cf. e.g.[4, 5]
How empirical observations of physical behaviour and empirical laws determine
the form and content of physics and its theoretical structurs has been
considered in straightforward untechnical style by Cook [6]. Dealing with
basic aspects of the general framework of physics, he provides a lucid
examination of issues of fundamental importance in the empirical and
metaphysical foundations of physics. As a supplement, we will enumerate only
the activities necessary when constructing empirical formulas based on
experimental data. We briefly describe the kind and amount of work that is
necessary to this end. There are two crucial facts: there is always only a
finite number of experimental data, and they are not accurate. To avoid
uniluminating complications, we will consider only the averaged experimental
data.
To point out the problems we encounter using experimental data to infer
empirical laws, let us simulate an empirical derivation of Planck’s law. This
law describes the spectral radiance of electromagnetic radiation of a black
body as a function of frequency $\nu$ and its temperature $T$:
$I(\nu,T)=2hc^{-2}\nu^{3}\Big{/}(e^{h\nu/kT}-1)\,.$ (1)
Take a set of $J$ distinct blackbody states, with $T_{j}$ denoting the
temperature of state $j=1$, …, $J$. Let us observe the spectral radiance of
each state at $I$ different frequencies $\nu_{i}$, $i=1$, …, $I$, repeating
each measurement $N$ times, cf.[6], Sec.1.2. We denote the actually measured
spectral radiance by $i_{j}(\nu_{i};n)$, $n=1$, …, $N$, and their averages by
$i_{j}(\nu_{i})=N^{-1}\sum_{n=1}^{N}i_{j}(\nu_{i};n)\,.$ (2)
Having obtained a finite number of averaged experimental data about the
spectral radiance of the $j$-th blackbody state, say
$D_{j}=\Bigl{\\{}i_{j}(\nu_{i}),i=1,\ldots,I\Bigr{\\}}\,,$ (3)
we are going to consider two problems:
* (A)
How to construct empirical formulas by using only a few of the averaged
experimental data $i_{j}(v_{i})$ to predict the rest of them.
* (B)
How to construct empirical equations that represent the physical law that
underlies $D_{j}$; that is, how to obtain some quantitative information about
Planck’s law from experimental data.
## 2 Empirical formulas
An empirical formula is a mathematical equation whose parameters have been
calculated from a few experimental data so as to predict the rest of them.
Such empirical formulas should be constructed parsimoniously [8], trying to
achieve the required accuracy111This accuracy should be chosen with regard to
the accuracy of measured spectral radiances $i_{j}(\nu_{i};n)$. of predictions
using as few parameters as possible.
Experience suggests we start with a linear ansatz
$E_{1}(\nu)=c_{1}+c_{2}\nu\,.$ (4)
There are many ways to specify parameters $c_{1}$ and $c_{2}$ in terms of
$i_{j}(\nu_{i})$. Fitting two of them, say $i_{j}(\nu_{1})$ and
$i_{j}(\nu_{2})$, we get
$E_{1}(\nu)=L(\nu;i_{j}(\nu),\nu_{1},\nu_{2})\,,$ (5)
where
$L(\nu;i_{j}(\nu),\nu_{1},\nu_{2})=i_{j}(\nu_{1})+[i_{j}(\nu_{2})-i_{j}(\nu_{1})](\nu-\nu_{1})\big{/}(\nu_{2}-\nu_{1})\,.$
(6)
This empirical formula reproduces $i_{j}(\nu)$ well if $\nu$ is sufficiently
close to $\nu_{1}$ or $\nu_{2}$.222For simplicity, and to avoid being tedious,
we will not formally specify what is meant by these words. When a linear
ansatz does not reproduce the averaged experimental data well enough, we might
try a higher-order polynomial ansatz with more parameters. However, if too
many parameters are required, different empirical formulas might be more
appropriate.
By plotting $\ln\nu_{i}^{-3}i_{j}(\nu_{i})$ versus $\nu_{i}$, we could
conclude that
$E_{\infty}(\nu)=\nu^{3}\exp\bigl{[}L(\nu;\ln\nu^{-3}i_{j}(\nu),\nu_{1},\nu_{2})\bigr{]}$
(7)
is a suitable333There is a computationally efficient method [9, 10] that
generalizes such graphical procedures. Using it, we can also discover more
intricate functional dependencies than those that can be discovered using
graph paper. empirical formula for predicting the asymptotic values of
$i_{j}(\nu)$ for large frequencies $\nu>\to\infty$ if we choose $\nu_{1}$ and
$\nu_{2}$ large enough.2
Similarly, on plotting $\ln i_{j}(\nu_{i})$ versus $\ln\nu_{i}$, we could
conclude that the empirical formula
$E_{0}(\nu)=i_{j}(\nu_{0})(\nu/\nu_{0})^{2}$ (8)
is suitable for predicting the values of $i_{j}(\nu_{i})$ for small
frequencies $\nu_{i}\to 0$ if we choose $\nu_{0}$ small enough.2
One may combine empirical formulas to obtain a more general and parsimonious
empirical formula for predicting the averaged experimental data.
## 3 Local empirical approximations of Planck’s law for radiance of a
blackbody state
As the preceding three empirical formulas turn out useful for predicting the
averaged experimental data $D_{j}$ about the radiance of the $j$-th state, we
construct related local approximations to the underlying physical law. No such
local approximation can be proved or falsified by the results of a finite
number of available experimental measurement. They can only show how
compatible it is with them.
We start with two basic assumptions about the measurement uncertainties of
data $D_{j}$:
* A1
The averaged experimental data $i_{j}(\nu_{i})$ tend toward the values of
spectral radiance as the number of repeated measurements $N$ becomes very
large, i.e., for any $i,j$
$\lim_{N\to\infty}i_{j}(\nu_{i})=I(\nu_{i},T_{j})\in(0,\infty)\,.$ (9)
* A2
The maximal relative observational error of data $D_{j}$ tends to zero as the
number of repeated observations $N$ becomes large, i.e.,
$\max_{i,j}|i_{j}(\nu_{i})-I(\nu_{i},T_{j})|\big{/}I(\nu_{i},T_{j})\to
0\qquad\hbox{as}\quad N\to\infty\,.$ (10)
In addition, we make the following three assumptions about the limiting
behaviour of the spectral radiance $I(\nu,T)$:
* A3
The limit
$c_{1j}=\lim_{\nu_{2}\to\nu_{1}}[I(\nu_{2},T_{j})-I(\nu_{1},T_{j})]\big{/}(\nu_{2}-\nu_{1})$
(11)
exists for all $\nu_{1}>0$.
* A4
There is the limit
$c_{2j}=\lim_{\nu\to 0}I(\nu,T_{j})\big{/}\nu^{2}\,.$ (12)
* A5
There is the limit
$c_{\infty j}=-\lim_{\nu\to\infty}\nu^{-1}\ln[I(\nu,T_{j})\big{/}\nu^{3}]\,.$
(13)
* A6
There is the limit
$c_{3j}=\lim_{\nu\to\infty}\nu^{-3}e^{c_{\infty j}\nu}i_{j}(\nu)\,.$ (14)
These limits define four constants that describe tha intensive physical
properties of the $j$-th blackbody state. Evaluation of these limits requires
an infinite number of averaged experimental data, but the number of available
experimental data is always finite. Therefore, such limits are theoretical
constructs that are evaluated by how useful they are in making empirical
approximations to the underlying physical law. If, in some application, we
need the actual value of such a limit, we have to make do with an estimate
based on the available experimental data $D_{j}$, and denote it with the same
letter written with a hat: $\hat{c}_{1j}$, $\hat{c}_{2j}$, $\hat{c}_{3j}$,
$\hat{c}_{\infty j}$. Metrological methods provide various estimates to this
end.
Inspecting the values of $\hat{c}_{3j}$, and of the product
$\hat{c}_{2j}\hat{c}_{\infty j}$ for the states considered, we might conclude
that the values $c_{3j}$ and $c_{2j}c_{\infty j}$ are state-independent and
equal the same value, say $c_{3}$, i.e. that
$c_{2j}c_{\infty j}=c_{3j}=c_{3}\qquad\hbox{for all}\quad j=1,2,\ldots,J\,.$
(15)
### 3.1 Linear approximation
The empirical formula $E_{1}(\nu)$ suggests the following linear approximation
to Planck’s law:
$L_{1}(\nu,T_{j})=I(\nu_{1},T_{j})+c_{1j}(\nu-\nu_{1})+O((\nu-\nu_{1})^{2})\qquad\hbox{as}\quad\nu\to\nu_{1}\,.$
(16)
Testing it on the available data $D_{j}$, we get an estimate $\hat{c}_{1j}$ of
the value of constant $c_{1j}$, and an estimate of for how large values of
$|\nu-\nu_{1}|$ this linear approximation is still useful. Hooke’s law and
Ohm’s law are two examples of such linear approximations.
### 3.2 Asymptotic behaviour for low frequencies
The usefullness of the empirical formula $E_{0}(\nu)$ indicates that spectral
radiance $I(\nu,T_{j})$ has the following asymptotic behaviour for low
frequencies:
$L_{0}(\nu,T_{j})=c_{2j}\nu^{2}+O(\nu^{3})\qquad\hbox{as}\quad\nu\to 0\,.$
(17)
The first term, the Rayleigh-Jeans formula, is totally useless for predicting
the spectral radiance for high frequencies.
### 3.3 Asymptotic behaviour for high frequencies
The usefullness of the empirical formula $E_{\infty}(\nu)$ for predicting the
values of $i_{j}(\nu_{i})$ for high frequencies suggests that
$L_{\infty}(\nu,T_{j})=c_{3}\nu^{3}e^{-c_{\infty
j}\nu}+O(\nu^{3}e^{-2c_{\infty j}\nu})\qquad\hbox{as}\quad\nu\to\infty$ (18)
describes the asymptotic behaviour of $I(\nu,T_{j})$ for high frequencies. The
first term, Wien’s law, predicts also the correct limiting value of spectral
radiance for $\nu=0$, but does not imply the correct asymptotic behaviour for
low frequencies.
## 4 Global approximations to Planck’s law for the radiance of a blackbody
state
As the local approximations $L_{1}$, $L_{0}$ and $L_{\infty}$ turn out to
predict the averaged experimental data $D_{j}$ well enough locally, we will
use them to construct a global approximation to Planck’s law for the $j$-th
state. They suggest we assume that
* A7
Spectral radiance is a non-negative, analytic function in a vicinity of any
positive frequency.
After some juggling, we construct a mathematical formula that unifies local,
asymptotic approximations $L_{0}$ and $L_{\infty}$ by an interpolation and
agrees with assumption A7:
$U(\nu;c_{3},c_{\infty j},c_{j})=c_{3}\nu^{2}(c_{j}+\nu)\Big{/}(e^{c_{\infty
j}\nu}-1+c_{j}c_{\infty j})\,,$ (19)
where $c_{j}$ is a non-negative empirical constant. On testing this formula
with $c_{j}=0$, we find that it represents data $D_{j}$ well enough for any
state if $N$ is large enough. So we could presume that $U(\nu;c_{3},c_{\infty
j},0)$ represents Planck’s law for the radiance of the $j$-th blackbody state;
it is determined by two physical properties $c_{3}$ and $c_{\infty j}$ of this
state.
## 5 Empirical form of Planck’s law
On measuring the temperatures $T_{j}$ of various states and plotting
$T^{-1}_{j}$ versus $\hat{c}_{\infty j}$, we might conclude that there is a
positive constant $c_{\infty}$ such that
$c_{\infty j}=c_{\infty}T_{j}^{-1}\,,j=1,2,\ldots,J\,.$ (20)
And so we could put forward the formula
$U(\nu;c_{3},c_{\infty}/T,0)$ (21)
as an empirical form of Planck’s law that is valid for all temperatures $T$
and depends only on two empirical constants, $c_{3}$ and $c_{\infty}$.
## 6 Comments
### 6.1 Identifying the states
The blackbody states are identified by the index $j$, giving the order in
which they were researched. In principle, we could identify the blackbody
states by their physical properties. So, were they known to us, we could use
the values of $c_{\infty j}$ that are inversly proportional to temperature, or
the values of spectral radiance for some frequency $\nu_{m}$. However, we
could make do also with estimates $\hat{c}_{\infty j}$ and
$\hat{I}(\nu_{m},T_{j})$, were they so accurate that they did not overlap for
the states considered.
### 6.2 Qualitative properties of Planck’s law
Planck’s law is such that
$I(-\nu,-T)=-I(\nu,T)\,,\qquad I(T\nu,T)=T^{3}I(\nu,1)\,,$ (22)
and $I(\nu,T)$ is an analytic function of the complex variable $\nu$
everywhere but at $h\nu=2n\pi kT$, $n=1$, 2, …, where it has a first-order
pole, and at $|\nu|=\infty$, where it has an essential singularity. However,
these properties cannot be directly inferred from experimental data.
Note that only the constant function is analytic in the whole complex plane.
So, an analytic funtion is, up to a constant, uniquely determined by its
singularities. An estimate of a singular point of $I(\nu,T)$ close to a point
on the positive real axis can be obtained from the convergence of the Taylor
expansion at this point.
Were some physical law an even function of $\nu$, this property would not be
directly evident from experimental data, though equations that presume this
property would be more efficient, especially so in the vicinity of $\nu=0$.
The asymptotic behaviour $L_{0}(\nu,T_{j})$ is compatible with the hypothesis
that Planck’s law is an even function of frequency. It is only the asymptotic
behaviour $L_{\infty}(\nu,T_{j})$ that falsifies this hypothesis.
### 6.3 The time we have for experiments is limited
The maximal relative observational error of the available data $D_{j}$ is in
general positive, though we expect it to get smaller as we increase the number
$N$ of repetitions. To obtain $J$ sets $D_{j}$, each containing $I$ data, we
must perform $NIJ$ mesurements. And each one takes a certain amount of time,
say at least $t_{m}$. However, there is obviously an upper limit, say $T_{M}$,
on the amount of time available for observing physical phenomena. So the total
number of possible measurements is at most $T_{M}/t_{m}$, and we must have
$IJN<T_{M}/t_{m}\,.$ (23)
Let us point out some consequences of this limit on the possible number of
available experimental data.
A) When $T_{m}<t_{m}$, the phenomenon is practically unobservable; an example
is provided by the blackbody radiation of visual frequencies at room
temperatures. So parts of Planck’s law are practically untestable, cf.
e.g.[7].
B) For each physical law $L(x)$, there is an infinite number of alternatives
$L_{a}(x)=L(x)(1+\epsilon\phi(x))$, with $\phi(x)$ bounded: so the relative
difference
$|(L(x)-L_{a}(x))/L(x)|\leq|\epsilon|\sup_{x}|\phi(x)|$ (24)
is arbitrarily small for sufficiently small $|\epsilon|$. And when relative
differences between predictions of two physical laws are sufficiently small,
we will never be able to tell them apart experimentally. Such a problem occurs
with the empirical form of Planck’s law (21), since the relative difference
between $U(\nu;c_{3},c_{\infty}/T,c_{j})$ and $U(\nu;c_{3},c_{\infty}/T,0)$
tends to zero uniformly as $c_{j}\to 0$. Thus we will never be able to falsify
the assumption that $c_{j}$ is a very small positive constant. It is expedient
to choose $c_{j}=0$,444Such is the case with the mass of the photon. For some
theoretical calculations it is convenient to limit the photon mass to zero
only in their final stage, though many theoretical considerations take as
their basic presumption that photons have no mass, cf. Veltman[11]. though
$U(\nu;c_{3},c_{\infty}/T,c_{j}\neq 0)$ might turn out to be a theoreticaly
significant modification of Planck’s law.
## References
* [1] Geocentric model, En.wikipedia.
* [2] Hooke’s law, En.wikipedia.
* [3] Ohm’s law, En.wikipedia.
* [4] Balmer series, En.wikipedia.
* [5] Rydberg formula, En.wikipedia.
* [6] A. H. Cook, The observational foundations of physics, Cambridge University Press, Cambridge (1994).
* [7] Planck’s law, En.wikipedia.
* [8] M. Ribarič et al, Computational methods for Parsimonious data fitting. Compstat Lectures 2, Physica Verlag, Vienna 1984.
* [9] M. Ribarič, D. Stojanovski and B. Žekš, Fizika (Zagreb) 11 (1979) 17.
* [10] M. Ribarič and B. Žekš, Chem. Phys. 41 (1979) 221.
* [11] M. Veltman, Diagrammatica, Cambridge University Press, Cambridge (1994).
|
arxiv-papers
| 2008-10-06T08:20:01
|
2024-09-04T02:48:58.134127
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Marijan Ribaric and Luka Sustersic",
"submitter": "Marjan Ribari\\v{c} Prof.",
"url": "https://arxiv.org/abs/0810.0905"
}
|
0810.0907
|
# Top Quark Mass from CDF
Petteri Mehtälä1
(on behalf of the CDF collaboration) University of Helsinki - Dept of Physics
and Helsinki Institute of Physics
Gustaf Hällströmin katu 2a, FI-00014 University of Helsinki - Finland
###### Abstract
This letter presents results on the precision measurement of the top quark
mass and a combination of the best CDF top mass measurements. A combination by
the TevEWWG (Tevatron electroweak working group) of the best top mass results
from CDF and D0 in Run 1 and Run 2 of the Tevatron is also presented. This
result is the current world average, and offers an uncertainty less than 1%.
The new mass value has been included in traditional LEP EWWG fits to precision
electroweak data, and implications for the Standard Model Higgs have been
derived.
## I INTRODUCTION
The top quark, discovered in 1995 by both CDF and D0 cdftopdisc ; d0topdisc ,
is the heaviest known elementary particle. The mass of the top quark is of
particular interest, as it can be used together with accurate measurement of
the W boson mass to limit mass range of the Standard Model (SM) Higgs boson
searches. The precision measurements of the top quark and W boson mass serve
as a consistency check of the SM if the Higgs boson is discovered, and they
help to answer whether a discovered scalar particle is indeed the SM Higgs
boson. This letter describes measurement of the top quark mass using data up
to 3.0 fb-1 collected by the CDF collaboration during Run II at the Tevatron.
## II PRODUCTION AND DECAY
Top quarks are predominantly produced in pairs at the Tevatron with a cross-
section of about 7 $pb$. According to SM, they decay into W boson and a b
quark with probability of about one. The decay of the W boson defines the
topology of the $t\bar{t}$ event. The dilepton channel, where both W bosons
decay leptonically to an electron or muon and their corresponding neutrino,
has a clean signal, but suffers from small branching fraction (about 5%) and
underconstrained kinematics. When one W boson decays leptonically and the
other one hadronically, into two quarks, the channel is called lepton+jets
channel. This channel has overconstrained kinematics, event though it has one
an undetected neutrino and has branching fraction of about 30%. The all-
hadronic channel, where both W bosons decay hadronically, does not have
undetected neutrino and has large branching fraction of about 44%, but suffers
from large QCD multijet background and jet ambiguities.
## III DILEPTON ANALYSES
Due to the unconstrained kinematics, top quark mass measurements in the
dilepton channel must integrate over some unknown quantities.Neutrino
Weighting Method scans over the azimuthal angles of the two neutrinos and
reconstructs the top mass estimator using kinematic fit. The solution
corresponding to the minimum of the goodness-of-the fit, $\chi^{2}$ over
azimuthal angles is selected. Each possible neutrino longitudinal and W boson
b-jet pairing solution is weighted by the goodness-of-the fit. A template is
constructed using the weighted top masses, where the weight is given by
$e^{-\chi^{2}/2}$. The measurement on the data corresponding to integrated
luminosity of $2.8~{}fb^{-1}$ yields ${\rm M_{top}}=165.1^{+3.4}_{-3.3}({\rm
stat})\pm 3.1({\rm syst})~{}{\rm GeV}/c^{2}$ cdfdilnw . A second measurement
presented here, uses matrix element method, where the theoretical production
and decay matrix elements are used to construct most probable top quark mass
for each event. A matrix element method that is applied on the data of
$1.9~{}fb^{-1}$ uses a evolutionary neural network, where the weights and the
topology of the network are optimized for the analysis sensitivity by
selecting the strong performers of a population of the networks into the
successive generations. The evolutionary neural network is applied in the
selection stage to improve the a priori statistical uncertainty on the top
quark mass by 20%. The analysis measures ${\rm M_{top}}=171.2\pm 2.7({\rm
stat})\pm 2.9({\rm syst})~{}{\rm GeV}/c^{2}$ cdfdilmat .
## IV LEPTON+JETS ANALYSES
Using a matrix element multivariate analysis technique, weighting jet-parton
assignments using tagging probability and using neural network based event-by-
event discriminant for background rejection a measurement on data
corresponding to integrated luminosity of $2.7~{}fb^{-1}$ yields ${\rm
M_{top}}=172.7\pm 1.8({\rm stat+JES})\pm 1.2({\rm syst})~{}{\rm GeV}/c^{2}$.
The measurement is on July 2008 the most precise single measurement for top
mass in the world cdfljmtm .
The top quark mass can be measured using the decay length of b-jets and lepton
transverse energy in the lepton+jets channel. Both these quantities are
roughly linearly proportional to the top mass and the measurement has minimal
dependence on the jet energy scale, $JES$. The method applied on the data
corresponding to integrated luminosity of $1.9~{}fb^{-1}$ yields ${\rm
M_{top}}=175.3\pm 6.2({\rm stat+JES})\pm 3.0({\rm syst})~{}{\rm GeV}/c^{2}$
cdfljlxy . This analysis is statistically limited, but if this analysis is
done at the LHC statistics will no longer be an issue. Further, since some of
the dominant systematics are statistically limited, the results of these
techniques could well become competitive with conventional top mass analyses,
and due to their reduced correlation with conventional top measurements they
should help reduce the uncertainty on the world average top mass in a
combination.
## V ALL-HADRONIC ANALYSES
The all-hadronic channel is challenging due to large QCD multijet background.
A dedicated event selection is required to increase the signal-to-background
ratio to an acceptable level for the mass measurement. A neural network based
events selection is used in CDF cdfnn . Two recent measurements in all-
hadronic channel, both using integrated luminosity of $1.9~{}fb^{-1}$ from CDF
have been performed. Both analyses use in situ jet energy scale calibration,
data-driven background model to estimate the shape and amount of the
background events with neural network discrimination. The first analysis uses
events with six to eight jets, requires one or two b-jets. It uses top mass
and dijet mass templates for top mass determination. The measurement yields
${\rm M_{top}}=176.9\pm 3.8({\rm stat+JES})\pm 1.7({\rm syst})~{}{\rm
GeV}/c^{2}$ cdfahtmt .
The second measurement uses Ideogram technique, that uses simplified matrix
element for the signal and templates in top mass and dijet mass for the
background and for signal events that have misreconstruction of the jets. This
analysis requires exactly six jets and at least two b-jets in the event making
the signal-to-background fraction much higher, about 2/3. It also measures the
signal fraction from the data simultaneously. The measurement yields ${\rm
M_{top}}=165.2\pm 4.4({\rm stat+JES})\pm 1.7({\rm syst})~{}{\rm GeV}/c^{2}$
cdfahideo . The individual systematic uncertainties for this analysis are
shown in Figure 1.
Figure 1: The systematic uncertainties for the Ideogram measurement in the
all-hardonic decay channel.
## VI TEVATRON COMBINATION
The combination of D0 and CDF results is performed using BLUE technique
bluemethod . In most of the analyses, the jet energy scale is still dominant
systematic uncertainty, but it is expected to decrease as it is measured in
situ when more data will be available and therefore many analyses still
benefit from increased integrated luminosity. The world average of the best
independent measurements of the top quark mass as of March 2008 from D0 and
CDF yields ${\rm M_{top}}=172.4\pm 0.7({\rm stat})\pm 1.0({\rm syst})~{}{\rm
GeV}/c^{2}$ and is of 0.7% accuracy tevcomb .
---
Figure 2: The top quark mass world average as on July 2008 (left). The limits
on the Standard Model Higgs boson mass as on July 2008 (right).
This result can be combined with the current W boson mass measurement to
obtain a constraint for the SM Higgs boson. With 95% CL SM Higgs boson mass is
less than 154$~{}{\rm GeV}/c^{2}$. And the most probable Higgs boson mass
value of $m_{H}=84^{+34}_{-26}GeV/c^{2}$. Fig. 2 compares the world average
with measurements from both experiments and shows the current limits on the
Standard Model Higgs boson mass in top mass - W boson mass plane.
## References
* (1) F. Abe et al., The CDF Collaboration, Phys. Rev. Lett., 74, 2626 (1995).
* (2) S. Abachi et al., The D0 Collaboration, Phys. Rev. Lett., 74, 2632 (1995).
* (3) The CDF Collaboration, CDF Conference Note 9456 (2008).
* (4) The CDF Collaboration, CDF Conference Note 9130 (2007).
* (5) The CDF Collaboration, CDF Conference Note 9427 (2008).
* (6) The CDF Collaboration, CDF Conference Note 9414 (2008).
* (7) The CDF Collaboration Phys. Rev. D76 072009 (2007).
* (8) The CDF Collaboration, CDF Conference Note 9165 (2008).
* (9) The CDF Collaboration, CDF Conference Note 9265 (2007).
* (10) L. Lyons, D. Gibaut and P. Clifford, Nucl. Inst. Meth., A270, 110 (1988).
* (11) The CDF and D0 Collaborations, CDF Conf Note 9449, D0 Conf Note 5751 (2008), hep-ex/0808.1089.
|
arxiv-papers
| 2008-10-06T08:38:01
|
2024-09-04T02:48:58.137813
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Mehtala (The CDF Collaboration)",
"submitter": "Petteri Meht\\\"al\\\"a",
"url": "https://arxiv.org/abs/0810.0907"
}
|
0810.0962
|
# Closed 1-forms in topology and geometric group theory
Michael Farber, Ross Geoghegan and Dirk Schütz Department of Mathematics,
University of Durham, Durham DH1 3LE, UK michael.farber@durham.ac.uk
Department of Mathematics, SUNY Binghamton, NY 13902-6000, USA
ross@math.binghamton.edu Department of Mathematics, University of Durham,
Durham DH1 3LE, UK dirk.schuetz@durham.ac.uk
###### Abstract.
In this article we describe relations of the topology of closed 1-forms to the
group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a
survey, we extend these Sigma invariants to finite CW-complexes and show that
many properties of the group theoretic version have analogous statements. In
particular we show the relation between Sigma invariants and finiteness
properties of certain infinite covering spaces. We also discuss applications
of these invariants to the Lusternik-Schnirelmann category of a closed 1-form
and to the existence of a non-singular closed 1-form in a given cohomology
class on a high-dimensional closed manifold.
To S.P. Novikov on the occasion of his 70-th birthday
## Introduction
The last three decades have seen a growing interest in the topology of closed
1-forms ever since S.P. Novikov [23, 24] introduced Morse theoretic techniques
to study classical problems in mathematical physics. In analogy to the Morse-
Smale complex of an ordinary Morse function on a closed manifold, he
constructed a chain complex, now called the Novikov complex, associated to a
closed 1-form whose singularities are non-degenerate in the sense of Morse.
While Novikov’s interest was to study such problems as Kirchhoff type
equations [24, 25, 26], other applications in different areas of mathematics
would soon become apparent. Gradient vector fields of closed 1-forms, for
example, give rise to fascinating results in dynamical systems, and some of
the topics arising this way have been covered in a recent monograph [13] and
survey article [18].
Another quite surprising link has been made to geometric group theory, when it
became clear that the work of Sikorav [29], who applied Novikov theory to
symplectic topology, was closely related to the work of Bieri, Neumann,
Strebel and Renz [6, 7]. More specifically, the geometric invariants
$\Sigma^{k}(G)$ of a group $G$, which contain important information on the
finiteness properties of certain subgroups of $G$ and whose definition is
recalled in Section 1, can be described in terms of vanishing results of a
generalized Novikov homology.
The feature that combines these areas is that closed 1-forms represent
cohomology classes $\xi\in H^{1}(X;{\mathbb{R}})$. If $X$ is a smooth manifold
this is a special case of de Rham theory, and the first-named author has
developed a theory of closed 1-forms on topological spaces for which this
result holds more generally, see [11]. Now if $X$ is connected,
$H^{1}(X;{\mathbb{R}})$ can be identified with the set of homomorphisms ${\rm
Hom}(\pi_{1}(X),{\mathbb{R}})$, where ${\mathbb{R}}$ is considered as a group
with the usual addition. Indeed, the invariants $\Sigma^{k}(G)$ can be viewed
as subsets of the unit sphere in ${\rm Hom}(G,{\mathbb{R}})$.
One purpose of this article is to describe these relations in a general
setting, and to develop the theory of Bieri-Neumann-Strebel-Renz using the
language of the topology of closed 1-forms. In particular, we extend the
notion of Sigma invariants to finite CW-complexes. These invariants
$\Sigma^{k}(X)$ for $k\geq 1$ are defined as generalizations of the group
theoretic versions, and they have similar properties. This generalization is
motivated by the fact that the group theoretic invariants regularly occur in
the topology of closed 1-forms. For example, the condition
$\xi\in\Sigma^{2}(\pi_{1}(M))$ appears implicitly in the work of Latour [21]
as a necessary condition for the existence of a non-singular closed 1-form in
$\xi\in H^{1}(M;{\mathbb{R}})$, where $M$ is a high-dimensional closed
manifold ($\dim M\geq 6$). Another necessary condition of [21], which in fact
implies $\xi\in\Sigma^{2}(\pi_{1}(M))$, is the contractibility of a certain
function space. It turns out that contractibility of this space is equivalent
to $\xi\in\Sigma^{k}(M)$ for all $k\geq 1$. We remark that for compact
3-manifolds the condition $\xi\in\Sigma^{1}(\pi_{1}(M))$ is sufficient for the
existence of a non-singular closed 1-form representing $\xi$, see [6].
Another property of these new Sigma invariants is that they reflect finiteness
properties of infinite abelian covering spaces $q:\overline{X}\to X$. By an
abelian covering we mean a regular covering with
$\pi_{1}(X)/\pi_{1}(\overline{X})$ an abelian group. Denote
$\displaystyle S(X,\overline{X})$ $\displaystyle=$
$\displaystyle\left\\{0\not=\xi:\pi_{1}(X)\to{\mathbb{R}}\,\left|\,\xi|_{\pi_{1}(\overline{X})}=0\right.\right\\}.$
###### Theorem.
Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. For $k\geq 1$,
the following properties are equivalent.
1. (1)
$\overline{X}$ is homotopy equivalent to a CW-complex with finite
$k$-skeleton.
2. (2)
$S(X,\overline{X})\subset\Sigma^{k}(X)$.
Furthermore, if $S(X,\overline{X})\subset\Sigma^{\dim X}(X)$, then
$\overline{X}$ is finitely dominated.
The proof of this theorem is given in Section 7.
A common feature in the definitions and techniques is the notion of movability
of subsets of a space $X$. Here movability is meant with respect to a closed
1-form $\omega$ representing a cohomology class $\xi\in
H^{1}(X;{\mathbb{R}})$. Roughly, movability of a set $A\subset X$ means that
there is a homotopy $H$ of $A$ into $X$ starting with the inclusion and such
that for every point $a\in A$ the integral of $\omega$ along the path
$t\mapsto H_{t}(a)$ is large. While homological versions, using chain
homotopies, of this appeared already in Bieri and Renz [7, Thm.C], a
topological version was formulated in a quite different context in developing
a Lusternik-Schnirelmann theory for closed 1-forms, see [11, 12, 13, 14, 18].
Due to the similar nature one expects a closer relation, which we derive in
Section 4. A movability notion for homology classes is developed in Section 8,
which has applications to cup-length estimates for the Lusternik-Schnirelmann
theory of a closed 1-form.
This article is written as a companion to the recent survey article [18] which
was focussing on applications of closed 1-forms in dynamical systems. The
present paper contains a significant amount of new material, although parts of
it are also meant as a survey.
## 1\. Bieri-Neumann-Strebel-Renz Invariants
Let $G$ be a finitely generated group. We want to recall the definition of the
Bieri-Neumann-Strebel-Renz invariants $\Sigma^{k}(G;{\mathbb{Z}})$, introduced
in [6, 7]. We denote
$\displaystyle S(G)$ $\displaystyle=$ $\displaystyle({\rm
Hom}(G,{\mathbb{R}})-\\{0\\})/{\mathbb{R}}_{+},$
that is, we identify nonzero homomorphisms, if one is a positive multiple of
the other. This is a sphere of dimension $r-1$, where $r$ denotes the rank of
the abelianization of $G$. We will identify $S(G)$ with the unit sphere in
${\rm Hom}(G,{\mathbb{R}})$ (after choosing an inner product on the latter)
and simply write $\xi\in S(G)$.
Given $\xi\in S(G)$, we denote
$\displaystyle{\mathbb{Z}}G_{\xi}$ $\displaystyle=$
$\displaystyle\left\\{\left.\sum_{g\in
G}n_{g}g\in{\mathbb{Z}}G\,\right|\,n_{g}=0\mbox{ for }\xi(g)<0\right\\},$
a subring of ${\mathbb{Z}}G$.
We say that the trivial ${\mathbb{Z}}G$-module ${\mathbb{Z}}$ is of type
$FP_{k}$ over ${\mathbb{Z}}G_{\xi}$, if there exists a resolution
(1) $\ldots\longrightarrow F_{i}\longrightarrow
F_{i-1}\longrightarrow\ldots\longrightarrow
F_{0}\longrightarrow\mathbb{Z}\longrightarrow 0$
of ${\mathbb{Z}}$ by free ${\mathbb{Z}}G_{\xi}$-modules with each $F_{i}$
finitely generated for $i\leq k$.
###### Definition 1.
The Bieri-Neumann-Strebel-Renz invariants are now defined as
$\displaystyle\Sigma^{k}(G;{\mathbb{Z}})$ $\displaystyle=$
$\displaystyle\\{\xi\in S(G)\,|\,{\mathbb{Z}}\mbox{ is of type }FP_{k}\mbox{
over }{\mathbb{Z}}G_{\xi}\\}.$
The power of these invariants lies in the fact that they are closely related
to the finiteness properties of subgroups of $G$. Let us recall the relevant
finiteness properties.
###### Definition 2.
For $k\geq 1$ a group $G$ is of type $FP_{k}$, if there is a resolution (1) of
${\mathbb{Z}}$ by free ${\mathbb{Z}}G$-modules with each $F_{i}$ finitely
generated for $i\leq k$. Also, we say that $G$ is of type $F_{k}$, if there is
an Eilenberg-MacLane space for $G$ with finite $n$-skeleton.
We get that $G$ of type $F_{k}$ implies type $FP_{k}$ by looking at the
cellular chain complex of the universal cover of the Eilenberg-MacLane space,
and type $FP_{1}$ is equivalent to type $F_{1}$ which simply means finitely
generated. But type $FP_{2}$ does not imply finitely presented, as the
examples of Bestvina and Brady show [2]. For more information on these
finiteness properties see Brown [9] and Geoghegan [19].
Notice that ${\mathbb{Z}}G$, when viewed as a ${\mathbb{Z}}G_{\xi}$-module for
any $\xi$, is a direct limit of free ${\mathbb{Z}}G_{\xi}$-modules. It is
therefore a flat ${\mathbb{Z}}G_{\xi}$-module. Furthermore, for every
${\mathbb{Z}}G$-module $A$ we have
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}A\cong A$. Thus if
$\Sigma^{k}(G;{\mathbb{Z}})\not=\emptyset$ for some $k\geq 1$, we can apply
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}-$ to the resolution (1) for some
$\xi\in\Sigma^{k}(G;{\mathbb{Z}})$, to get that $G$ is itself of type
$FP_{k}$.
The following theorem, which we generalize in Section 7, relates finiteness
properties of certain subgroups to the invariants.
###### Theorem 1 (Bieri-Renz, [7]).
Let $G$ be a group of type $FP_{k}$, $N$ a subgroup of $G$ containing the
commutator subgroup of $G$. Then $N$ is of type $FP_{k}$ if and only if
$\Sigma^{k}(G;{\mathbb{Z}})$ contains the subsphere $S(G,N)=\\{\xi\in
S(G)\,|\,N\leq{\rm Ker}\,\xi\\}$.
There also exists a version of Theorem 1 which gives a criterion for $N$ to be
of type $F_{k}$, involving a homotopical invariant $\Sigma^{k}(G)$. We will
see more about this invariant below. Another result, proven in [7] is that all
$\Sigma^{k}(G;{\mathbb{Z}})$ are open subsets of $S(G)$.
In [6] it was shown that even the particular case $k=1$ has very important
applications to group theory.
###### Theorem 2 (Bieri-Neumann-Strebel, [6]).
Let $G$ be a finitely presented group without non-abelian free subgroups. Then
$\displaystyle\Sigma^{1}(G;{\mathbb{Z}})\cup-\Sigma^{1}(G;{\mathbb{Z}})$
$\displaystyle=$ $\displaystyle S(G).$
Here $-\Sigma^{1}(G)$ denotes the image of $\Sigma^{1}(G)$ under the antipodal
map.
We now want to give a more geometrical interpretation of these invariants, for
the moment we will confine ourselves to the case $k=1$ and for simplicity we
will assume that $G$ is finitely presented. Let $X$ be a finite CW-complex
with $\pi_{1}(X)\cong G$ and let $q:\overline{X}\to X$ be the universal
abelian covering. Given a non-zero homomorphism $\xi:G\to{\mathbb{R}}$ we can
build a map $h:\overline{X}\to{\mathbb{R}}$ with $h(gx)=\xi(g)+h(x)$ for all
$g\in G$ and $x\in\overline{X}$ by induction over the skeleta of
$\overline{X}$. Note that $G$ acts on $\overline{X}$ by covering
transformations, with the commutator subgroup acting trivially. Write
$N=h^{-1}([0,\infty))$, then $N$ need not be connected, but has a unique
component on which $h$ is unbounded, see [6, Lm.5.2]. In the result below we
assume that the basepoint of $N$ is chosen in this component.
###### Theorem 3 (Bieri-Neumann-Strebel, [6]).
We have $\xi\in\Sigma^{1}(G;{\mathbb{Z}})$ if and only if
$i_{\\#}:\pi_{1}(N)\to\pi_{1}(\overline{X})$ is an epimorphism, where
$i:N\to\overline{X}$ is the inclusion.
This geometric criterion is closely related to a concept of movability of a
subset of $X$ with respect to a given $\xi$ and which has recently been
studied in connection with a Lusternik-Schnirelmann theory of such $\xi$, see
[11, 13, 18]. Let us recall the definition of a closed 1-form on a topological
space $X$.
###### Definition 3.
A continuous closed 1-form $\omega$ on a topological space $X$ is defined as a
collection $\\{f_{U}\\}_{U\in\mathcal{U}}$ of continuous real-valued functions
$f_{U}:U\to{\mathbb{R}}$ where $\mathcal{U}=\\{U\\}$ is an open cover of $X$
such that for any pair $U,V\in\mathcal{U}$ the difference
$f_{U}|_{U\cap V}-f_{V}|_{U\cap V}:U\cap V\to{\mathbb{R}}$
is a locally constant function. Another such collection
$\\{g_{V}\\}_{V\in\mathcal{V}}$ (where $\mathcal{V}$ is another open over of
$X$) defines an equivalent closed 1-form if the union collection
$\\{f_{U},g_{V}\\}_{U\in\mathcal{U},V\in\mathcal{V}}$ is a closed 1-form,
i.e., if for any $U\in\mathcal{U}$ and $V\in\mathcal{V}$ the function
$f_{U}-g_{V}$ is locally constant on $U\cap V$.
These closed 1-forms behave in the same way as smooth closed 1-forms on
manifolds; they can be integrated along paths $\gamma:[a,b]\to X$, and
integration along loops defines a homomorphism
$\xi_{\omega}:\pi_{1}(X)\to{\mathbb{R}}$. Furthermore, every such homomorphism
can be realized by a closed 1-form. See [18, §3] for details.
###### Example 1.
A continuous function $f:X\to S^{1}$ determines a closed 1-form in the
following way. Think of $S^{1}$ as ${\mathbb{R}}/{\mathbb{Z}}$ and let
$p:{\mathbb{R}}\to{\mathbb{R}}/{\mathbb{Z}}$ be the projection. If
$I=(a,b)\subset{\mathbb{R}}$ is an open interval with $b-a\leq 1$, then $I$ is
homeomorphic to the open subset $p(I)\subset{\mathbb{R}}/{\mathbb{Z}}$ via
$p$. The collection
$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\left\\{(p|_{I})^{-1}\circ f|_{f^{-1}(p(I))}:f^{-1}(p(I))\to
I\right\\}$
then defines a closed 1-form. Furthermore,
$\xi_{\omega}:\pi_{1}(X)\to{\mathbb{R}}$ can be identified with
$f_{\\#}:\pi_{1}(X)\to\pi_{1}(S^{1})$, if the standard generator of
$\pi_{1}(S^{1})$ is identified with $1\in{\mathbb{R}}$.
###### Definition 4.
Let $X$ be a finite connected CW-complex, $G=\pi_{1}(X)$ and $\omega$ a closed
1-form on $X$. A subset $A\subset X$ is called $n$-movable with respect to
$\omega$ and control $C\geq 0$ (where $n$ is an integer), if there is a
homotopy $H:A\times[0,1]\to X$ such that $H(a,0)=a$ or all $a\in A$, and
$\displaystyle\int_{a}^{H_{1}(a)}\omega$ $\displaystyle\geq$ $\displaystyle n$
and
$\displaystyle\int_{a}^{H_{t}(a)}\omega$ $\displaystyle\leq$ $\displaystyle-C$
for all $a\in A$ and $t\in[0,1]$. Here the integral is taken over the path
$s\mapsto H(a,s)$ for $s\in[0,t]$.
This notion of movability has its roots in the Lusternik-Schnirelmann theory
of a closed 1-form, compare [11, 18]. The next Proposition shows that it also
gives a criterion for $\Sigma^{1}(G;{\mathbb{Z}})$.
###### Proposition 1.
Let $X$ be a finite connected CW-complex, $G=\pi_{1}(X)$ and
$\xi:G\to{\mathbb{R}}$ a homomorphism which is represented by a closed 1-form
$\omega$. Then the following are equivalent.
1. (1)
$\xi\in\Sigma^{1}(G;{\mathbb{Z}})$.
2. (2)
There is a $C\geq 0$ such that the 1-skeleton $X^{(1)}\subset X$ is
$n$-movable with respect to $\omega$ and control $C$ for every $n>0$.
###### Proof.
(1) $\Longrightarrow$ (2): Let $\overline{X}$ be the universal abelian cover
of $X$ and $h:\overline{X}\to{\mathbb{R}}$ be obtained from the pullback of
$\omega$ to $\overline{X}$. It is easy to see that we have $h(gx)=\xi(g)+h(x)$
for all $x\in\overline{X}$ and $g\in G$. We choose $N=h^{-1}([0,\infty))$, and
let $N^{\prime}\subset N$ be the component such that $h$ is unbounded on
$N^{\prime}$ by [6, Lemma 5.2]. For every cell $\sigma$ of $X$ pick a lift
$\bar{\sigma}\subset h^{-1}((-\infty,-1])$. If $\sigma$ is a 0-cell, we can
find a cellular map $H_{\sigma}:[0,1]\to\overline{X}$ such that
$H_{\sigma}(0)=\bar{\sigma}$ and $H_{\sigma}(1)\in N^{\prime}$. Here $[0,1]$
has the standard cell structure with two 0-cells and one 1-cell. Using
equivariance, this gives an equivariant cellular homotopy
$H^{0}:\overline{X}^{(0)}\times[0,1]\to\overline{X}$. Note that $G$ acts on
$\overline{X}\times[0,1]$ by $g(x,t)=(gx,t)$, and $H^{0}$ induces a homotopy
on $X^{(0)}$ which gives $1$-movability of the 0-skeleton. As the image of
$H^{0}_{1}$ is in the 0-skeleton, we can iterate this homotopy to obtain
$n$-movability for any $n>0$.
Now pick a cell $\bar{\sigma}\subset h^{-1}((-\infty,-1])$ for every 1-cell
$\sigma$ of $X$ and let $u,v$ be the boundary points of $\bar{\sigma}$. By
possibly iterating $H^{0}$, we can assume that $H^{0}(u,1),H^{0}(v,)\in
N^{\prime}$ and since $N^{\prime}$ is connected, we can find a cellular map
$H_{u,v}:[0,1]\to N^{\prime}$ connecting these points.
Note that $H_{u,v}([0,1])$, $H^{0}(\\{u,v\\}\times[0,1])$ and $\bar{\sigma}$
combine to a closed loop in $\tilde{X}$ when suitably oriented. But by Theorem
3 this loop is representable by a loop in $N^{\prime}$. In other words, by
changing the path $H_{u,v}$ suitably in $N^{\prime}$, we can assume that this
loop bounds. Therefore we can extend $H^{0}$ to
$H_{\sigma}:(\overline{X}^{(0)}\cup\bar{\sigma})\times[0,1]\to\overline{X}$
cellularly such that $H_{\sigma}(\bar{\sigma},1)=H_{u,v}([0,1])$. Doing this
for every 1-cell of $X$ and extending equivariantly gives a cellular and
equivariant homotopy $H^{1}:\overline{X}^{(1)}\times[0,1]\to\overline{X}$ such
that $H^{1}_{0}$ is inclusion and $h(H^{1}(x,1))-h(x)\geq 1$.
Since $H^{1}$ is equivariant we get a homotopy $H:X^{(1)}\times[0,1]\to X$
which shows that $X^{(1)}$ is 1-movable with respect to $\omega$. By
compactness there is a $C>0$ such that $X^{(1)}$ is 1-movable with control
$C$. Again the image of $H_{1}$ is in the 1-skeleton of $X$, so we can iterate
the homotopy. Note that iterating the homotopy does not increase the control,
so we get that $X^{(1)}$ is $n$-movable with respect to $\omega$ with control
$C$ for every $n>0$.
(2) $\Longrightarrow$ (1): Let $h:\overline{X}\to{\mathbb{R}}$ and
$N^{\prime}\subset N=h^{-1}([0,\infty))$ be as above. Pick a basepoint
$x_{0}\in N^{\prime}$ with $f(x_{0})\geq C+1$. Let
$\gamma:(S^{1},1)\to(\tilde{X},x_{0})$ be a loop which can be assumed
cellular. By compactness of $S^{1}$ there is a $K\leq 0$ such that
$\gamma(S^{1})\subset f^{-1}([K,\infty))$.
By assumption there is a homotopy
$\bar{H}:\overline{X}^{(1)}\times[0,1]\to\overline{X}$ with $\bar{H}_{0}$ is
inclusion and
$\displaystyle h(\bar{H}_{1}(x))-h(x)$ $\displaystyle\geq$ $\displaystyle C-K$
$\displaystyle h(\bar{H}_{t}(x))-h(x)$ $\displaystyle\geq$ $\displaystyle-C$
for all $x\in\overline{X}$. Let $\mu:\tilde{X}\to[0,1]$ be a map with
$\mu|f^{-1}([C+1,\infty))\equiv 0$ and $\mu|f^{-1}((-\infty,C])\equiv 1$. Now
define $A:S^{1}\times[0,1]\to\tilde{X}$ by
$A(x,t)=\bar{H}(\gamma(x),\mu(\gamma(x))\cdot t)$. Then $A(x,0)=\gamma(x)$,
$A(x,1)=\bar{H}(\gamma(x),\mu(\gamma(x)))\in N^{\prime}$ and
$A(x_{0},t)=x_{0}$ for all $t\in[0,1]$. Therefore
$\xi\in\Sigma^{1}(G;{\mathbb{Z}})$ by Theorem 3. ∎
A criterion analogous to condition (2) of Proposition 1 leads to the
homotopical version of the Bieri-Neumann-Strebel-Renz invariants
$\Sigma^{k}(G)$, introduced in [7]. For this we assume that $X$ is an
Eilenberg-MacLane space with finite $n$-skeleton for some $n\geq 1$.
###### Definition 5.
Let $X$ be as above, $\xi\in S(G)$, $\omega$ a closed 1-form on $X$
representing $\xi$ and $k\geq 0$. We say that $\xi\in\Sigma^{k}(G)$, if there
is $\varepsilon>0$ and a cellular homotopy $H:X^{(k)}\times[0,1]\to X$ such
that $H(x,0)=x$ for all $x\in X^{(k)}$ and
$\displaystyle\int_{\gamma_{x}}\omega$ $\displaystyle\geq$
$\displaystyle\varepsilon$
for all $x\in X^{(k)}$, where $\gamma_{x}:[0,1]\to X$ is given by
$\gamma_{x}(t)=H(x,t)$. Here $X^{(k)}$ denotes the $k$-skeleton of $X$.
The condition that $H$ is cellular ensures that $H_{1}$ has image in
$X^{(k)}$, so that the homotopy can be iterated. As a result we see that
$\varepsilon$ can be arbitrarily large which shows that the definition does
not depend on the particular $\omega$. These iterations all have the same
control $C\geq 0$. Using cellular approximations it is easy to see that
condition (2) of Proposition 1 is equivalent to $\xi\in\Sigma^{1}(G)$.
The above definition is not the usual definition of $\Sigma^{k}(G)$, but it
follows from Proposition 2 below that it agrees with the definition given in
Bieri and Renz [7, §6].
Even though it is generally quite difficult to describe $\Sigma^{k}(G)$ and
$\Sigma^{k}(G;{\mathbb{Z}})$, there are some important classes of groups for
which the Sigma invariants can be determined, for example right-angled Artin
groups, see [22], and Thompson’s group $F$, see [5]. For more information and
applications of these invariants see, for example, [3, 4, 6, 7, 20].
## 2\. Sigma invariants for CW-complexes
There is no particular reason for $X$ to be aspherical in Definition 5, so we
can extend this definition to more general $X$. For simplicity we will assume
that $X$ is a finite connected CW-complex, but it is possible to consider the
case where $X$ has finite $n$-skeleton for some $n\geq 1$, in which case we
can define $\Sigma^{k}(X)$ for $k\leq n$. Let us first define
$\displaystyle S(X)$ $\displaystyle=$ $\displaystyle({\rm
Hom}(H_{1}(X),{\mathbb{R}})-\\{0\\})/{\mathbb{R}}_{+}$
where we again identify a homomorphism with its positive multiples. Clearly
$S(X)=S(G)$ with $G=\pi_{1}(X)$.
The Sigma invariants for CW-complexes are now defined in analogy to Definition
5.
###### Definition 6.
Let $X$ be a finite connected CW-complex, $\xi\in S(X)$, $\omega$ a closed
1-form on $X$ representing $\xi$ and $k\geq 0$. We say that
$\xi\in\Sigma^{k}(X)$, if there is $\varepsilon>0$ and a cellular homotopy
$H:X^{(k)}\times[0,1]\to X$ such that $H(x,0)=x$ for all $x\in X^{(k)}$ and
$\displaystyle\int_{\gamma_{x}}\omega$ $\displaystyle\geq$
$\displaystyle\varepsilon$
for all $x\in X^{(k)}$, where $\gamma_{x}:[0,1]\to X$ is given by
$\gamma_{x}(t)=H(x,t)$.
Let $p:\tilde{X}\to X$ be the universal covering space. Just as in the smooth
manifold case, a closed 1-form $\omega$ pulls back to an exact form on
$\tilde{X}$, $p^{\ast}\omega=dh$ for some $h:\tilde{X}\to{\mathbb{R}}$ with
(2) $\displaystyle h(gx)$ $\displaystyle=$ $\displaystyle h(x)+\xi(g)$
for all $g\in\pi_{1}(X)$ and $x\in\tilde{X}$. A function
$h:\tilde{X}\to{\mathbb{R}}$ with property (2) is called a height function.
Such a height function defines a closed 1-form representing $\xi$. A subset
$N\subset\tilde{X}$ is called a neighborhood of $\infty$ with respect to
$\xi$, if there exists a height function $h_{\xi}:\tilde{X}\to{\mathbb{R}}$
and $a\in{\mathbb{R}}$ such that
$\displaystyle h_{\xi}^{-1}([a,\infty))$ $\displaystyle\subset$ $\displaystyle
N.$
It is easy to check that if $N$ is a neighborhood of $\infty$ with respect to
$\xi$ for some height function, it is also a neighborhood of $\infty$ with
respect to $\xi$ for every other height function.
If a particular height function $h_{\xi}$ is given, we write
$N_{i}=h_{\xi}^{-1}([i,\infty))$ for every $i\in{\mathbb{R}}$.
We can describe $\Sigma^{k}(X)$ in terms of height functions alone. For this
we need one more definition.
###### Definition 7.
Let $X$ be a finite connected CW-complex, $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a
height function and $\xi\in H^{1}(X;{\mathbb{R}})$ be nonzero. A path to
$\infty$ with respect to $\xi$ is a map $\gamma:[0,\infty)\to\tilde{X}$ such
that for every neighborhood $N$ of $\infty$ there is a $T\geq 0$ such that
$\gamma(t)\in N$ for all $t\geq T$.
Given a path $\gamma$ to $\infty$ we can pick points $\gamma(T_{N})\in N$ for
every neighborhood $N$ of $\infty$ and get an inverse system
$\\{\pi_{\ast}(\tilde{X},N,\gamma(T_{N}))\\}$ where the basepoint change is
done via $\gamma|[T_{N},T_{N^{\prime}}]$. We will often suppress the
basepoints but we want to point out that there is always a path to $\infty$ in
the background. The next proposition shows that the basepath is not important
for our purposes.
###### Proposition 2.
Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a height function. The
following are equivalent.
1. (1)
$\xi\in\Sigma^{k}(X)$.
2. (2)
There is a $\lambda\geq 0$ such that
$j_{\\#}:\pi_{l}(\tilde{X},N_{i})\to\pi_{l}(\tilde{X},N_{i-\lambda})$ is
trivial for all $l\leq k$ and every $i\in{\mathbb{R}}$.
3. (3)
For every neighborhood $N$ of $\infty$ with respect to $\xi$, there is another
neighborhood $N^{\prime}\subset N$ such that
$j_{\\#}:\pi_{l}(\tilde{X},N^{\prime})\to\pi_{l}(\tilde{X},N)$ is trivial for
all $l\leq k$.
4. (4)
There is an $\varepsilon>0$ and an equivariant cellular homotopy
$\tilde{H}:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$, such that $\tilde{H}_{0}$
is inclusion and
$h_{\xi}(\tilde{H}_{1}(\tilde{x}))-h_{\xi}(\tilde{x})\geq\varepsilon$ for all
$\tilde{x}\in\tilde{X}^{(k)}$.
###### Proof.
(1) $\Longleftrightarrow$ (4) as we can lift $H$ to $\tilde{H}$ and
$\tilde{H}$ determines $H$. (2) $\Longrightarrow$ (3) is obvious. (3)
$\Longrightarrow$ (4): We define $H$ by induction on the skeleta of
$\tilde{X}$. The homotopy can always be defined on $\tilde{X}^{(0)}$ as
$\tilde{X}$ is connected and $\xi$ nonzero. Assume that
$H:\tilde{X}^{(k-1)}\times[0,1]\to\tilde{X}$ satisfies the conclusion of (4).
Let $N=N_{0}=h_{\xi}^{-1}([0,\infty))$ be a neighborhood of $\infty$ with
respect to $\xi$ and $N^{\prime}\subset N$ as in (3), and choose a
$\varepsilon>0$. Choose a lift
$\tilde{\sigma}\subset\tilde{X}-N_{-\varepsilon}$ for every $k$-cell $\sigma$
of $X$. By iterating $H$ if necessary, we can assume that
$H_{1}(\partial\tilde{\sigma})\subset N^{\prime}$. Given a characteristic map
$\chi_{\sigma}:(D^{k},S^{k-1})\to(\tilde{X}^{(k)},\tilde{X}^{(k-1)})$, we can
compose $\chi_{\sigma}|S^{k-1}$ with $H$ to get an element of
$\pi_{k}(\tilde{X},N^{\prime})$ which restricts to the trivial element of
$\pi_{k}(\tilde{X},N)$. This gives a homotopy
$\chi:D^{k}\times[0,1]\to\tilde{X}$ with $\chi_{0}=\chi_{\sigma}$ and
$\chi_{1}(D^{k})\subset N$. We can use this homotopy to extend $H$
equivariantly to the $k$-skeleton such that (4) is satisfied.
To see that (4) $\Longrightarrow$ (2) observe that $H$ can be used to homotop
every map $\varphi:(D^{k},S^{k-1})\to(\tilde{X},N_{i})$ to a map with image in
any neighborhood of $\infty$. As the base point should not be moved during the
homotopy, we have to modify $H$ on a ‘buffer zone’ $N_{i-\lambda}-N_{i}$ so
that points mapped to $N_{i}$ will not be changed. Nevertheless we can find
$N_{i-\lambda}$ such that every $\varphi$ can be homotoped to a map
$(D^{k},S^{k-1})\to(N_{i-\lambda})$ which gives (1). ∎
###### Remark 1.
Notice that in (4) we can choose $\varepsilon>0$ arbitrary large: as the
homotopy is cellular, we can simply iterate it to increase the $\varepsilon$.
Our next result shows that the Sigma invariants are in fact open subsets of
$S(X)$, for the group theoretic version of this statement see [6, 7]. For the
proof we need a version of an Abel-Jacobi map. Let $q:\overline{X}\to X$ be
the universal abelian cover of $X$ and let $r=b_{1}(X)$, the first Betti
number of $X$. Then $H_{1}(X)$ acts on $\overline{X}$ by covering translations
and on ${\mathbb{R}}^{r}=H_{1}(X)\otimes{\mathbb{R}}$ by translation.
There exists an equivariant map $A:\overline{X}\to{\mathbb{R}}^{r}$, canonical
up to homotopy, called an Abel-Jacobi map, see [15, Prop.1]. For a different
construction, note that we have a canonical epimorphism
$\pi_{1}(X)\to{\mathbb{Z}}^{r}$ factoring through $H_{1}(X)$. Then $A$ is a
lift of the resulting classifying map $X\to(S^{1})^{r}$.
###### Theorem 4.
Let $X$ be a finite connected CW-complex. For every $k\geq 0$ the set
$\Sigma^{k}(X)$ is open and $\Sigma^{n}(X)=\Sigma^{\dim X}(X)$ for $n\geq\dim
X$.
###### Proof.
Let $h:\tilde{X}\to{\mathbb{R}}^{r}$ by the composition of the covering map
$\bar{p}:\tilde{X}\to\overline{X}$ with the Abel-Jaobi map
$A:\overline{X}\to{\mathbb{R}}^{r}$. Then, given
$\xi:\pi_{1}(X)\to{\mathbb{R}}$ we get a height function by
$h_{\xi}=l_{\xi}\circ h$ where
$l_{\xi}:\pi_{1}(X)/[\pi_{1}(X),\pi_{1}(X)]\otimes{\mathbb{R}}\to{\mathbb{R}}$
is defined by $l_{\xi}([g]\otimes t)=\xi(g)\cdot t$. Now let $\tilde{H}$ be a
homotopy as in Proposition 2 (4) for a $\xi$. Define
$\tilde{K}:S(X)\times\tilde{X}^{(k)}\to{\mathbb{R}}$ by
$\displaystyle\tilde{K}(\xi^{\prime},\tilde{x})$ $\displaystyle=$
$\displaystyle
h_{\xi^{\prime}}(H_{1}(\tilde{x}))-h_{\xi^{\prime}}(\tilde{x}).$
By the choice of $H$ we get $\tilde{K}(\xi,\tilde{x})\geq\varepsilon$ for all
$\tilde{x}\in\tilde{X}$ and some $\varepsilon>0$. It also induces a map
$K:S(X)\times X^{(k)}\to{\mathbb{R}}$. From the compactness of $X^{(k)}$, we
get
$\displaystyle\tilde{K}(\xi^{\prime},\tilde{x})$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{2}$
for all $\tilde{x}\in\tilde{X}^{(k)}$and all $\xi^{\prime}$ in a neighborhood
of $\xi$ in $S(X)$. Therefore Proposition 2 (4) is satisfied for all such
$\xi^{\prime}$. ∎
We get the following relation between $\Sigma^{k}(X)$ and the group-theoretic
version $\Sigma^{k}(\pi_{1}(X))$.
###### Proposition 3.
Let $X$ be a finite connected CW-complex and $k\geq 0$. If $\tilde{X}$ is
$k$-connected, then $\Sigma^{k}(X)=\Sigma^{k}(\pi_{1}(X))$ and
$\Sigma^{k+1}(X)\subset\Sigma^{k+1}(\pi_{1}(X))$.
The inclusion can be proper, as the example
$X={\mathbf{S}}^{1}\vee{\mathbf{S}}^{k}$ with $k\geq 2$ shows.
###### Proof.
Note that we can build a $K(\pi_{1}(X),1)$ out of $X$ by attaching $n$-cells
for $n\geq k+2$. Denote this Eilenberg-MacLane space by $Y$. If
$\xi\in\Sigma^{k+1}(X)$, the cellular homotopy
$\tilde{H}:\tilde{X}^{(k+1)}\times[0,1]\to\tilde{X}$ from Proposition 2 (4)
induces a homotopy
$\tilde{H}^{\prime}:\tilde{Y}^{(k+1)}\times[0,1]\to\tilde{Y}$, since
$\tilde{Y}^{(k+1)}=\tilde{X}^{(k+1)}$ and $\tilde{X}\subset\tilde{Y}$.
Therefore $\xi\in\Sigma^{k+1}(\pi_{1}(X))$.
If $\xi\in\Sigma^{k}(\pi_{1}(X))$, we get a cellular homotopy
$\tilde{H}:\tilde{Y}^{(k)}\times[0,1]\to\tilde{Y}^{(k+1)}$ as in Proposition 2
(4), and since $\tilde{Y}^{(k+1)}=\tilde{X}^{(k+1)}$, this gives
$\xi\in\Sigma^{k}(X)$. ∎
In analogy to the homological invariants $\Sigma^{k}(G,{\mathbb{Z}})$ of Bieri
and Renz [7] we now want to define homological invariants
$\Sigma^{k}(X,{\mathbb{Z}})$. We will in fact introduce a more general
definition for chain complexes which will also generalize the invariants of
[7]. We assume that all chain complexes satisfy $C_{i}=0$ for $i<0$.
###### Definition 8.
Let $R$ be a ring, $n$ a non-negative integer and $C$ a chain complex over
$R$. Then $C$ is of finite $n$-type, if there is a finitely generated
projective chain complex $C^{\prime}$ and a chain map $f:C^{\prime}\to C$ with
$f_{i}:H_{i}(C^{\prime})\to H_{i}(C)$ an isomorphism for $i<n$ and an
epimorphism for $i=n$. In this situation we call $f$ an $n$-equivalence.
It is clear that this is equivalent to the existence of a free $R$-chain
complex $D$ and a chain map $f:D\to C$ inducing an isomorphism on homology,
and such that $D_{i}$ is finitely generated for $i\leq n$.
###### Definition 9.
Let $C$ be a chain complex over ${\mathbb{Z}}G$ and $k\geq 0$. Then
$\displaystyle\Sigma^{k}(C)$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(G)\,|\,C\mbox{ is of finite }k\mbox{-type over }{\mathbb{Z}}G_{\xi}\\}.$
###### Definition 10.
If $X$ is a finite connected CW-complex, we set
$\displaystyle\Sigma^{k}(X;\mathbb{Z})$ $\displaystyle=$
$\displaystyle\Sigma^{k}(C_{\ast}(\tilde{X})).$
The invariants $\Sigma^{n}(G;A)$ of Bieri and Renz [7] are given by
$\Sigma^{k}(G;A)=\Sigma^{k}(P)$, where $P$ is a projective ${\mathbb{Z}}G$
resolution of the $\mathbb{Z}G$-module $A$.
###### Remark 2.
Notice that ${\mathbb{Z}}G$ is a flat ${\mathbb{Z}}G_{\xi}$-module, as
${\mathbb{Z}}G$ is a direct limit of free ${\mathbb{Z}}G_{\xi}$-modules. As
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}A\cong A$ for every
${\mathbb{Z}}G$-module $A$, we see that $\xi\in\Sigma^{n}(C)$ implies that $C$
is of finite $n$-type over ${\mathbb{Z}}G$. If the chain complex $C$ is free,
we then get that $C$ is chain-homotopy equivalent to a free chain complex $D$
such that $D_{i}$ is finitely generated for $i\leq n$.
To get an analogue of Proposition 2 we will define a chain complex version of
a height function.
###### Definition 11.
Let $C$ be a finitely generated free chain complex over ${\mathbb{Z}}G$ and
$\xi:G\to{\mathbb{R}}$ a non-zero homomorphism. A valuation on $C$ extending
$\xi$ is a sequence of maps $v:C_{k}\to{\mathbb{R}}_{\infty}$ satisfying the
following
1. (1)
$v(a+b)\geq\min\\{v(a),v(b)\\}$ for all $a,b\in C_{k}$.
2. (2)
$v(ga)=\xi(g)+v(a)$ for all $g\in G$, $a\in C_{k}$.
3. (3)
$v(-a)=v(a)$ for all $a\in C_{k}$.
4. (4)
$v(\partial a)\geq v(a)$ for all $a\in C_{k}$.
5. (5)
$v^{-1}(\\{\infty\\})=\\{0\\}$.
Here ${\mathbb{R}}_{\infty}$ denotes the reals together with an element
$\infty$ with the obvious extension of addition and $\geq$.
To define a valuation on a free ${\mathbb{Z}}G$ complex $C$ which is finitely
generated in every degree, let $X_{i}$ be a basis for $C_{i}$. We begin with
setting $v(x)=0$ for all $x\in X_{0}$. The valuation can then be extended in
the obvious way to $C_{0}$. Inductively we now define for $x\in C_{i}$ the
valuation by $v(x)=0$ if $\partial x=0$, or $v(x)=v(\partial x)$, if $\partial
x\not=0$. Again we can extend $v$ to $C_{i}$ which gives the existence of
valuations on $C$.
###### Proposition 4.
Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $v:C_{\ast}(\tilde{X})\to{\mathbb{R}}_{\infty}$ a valuation
extending $\xi$. The following are equivalent.
1. (1)
$\xi\in\Sigma^{k}(X,{\mathbb{Z}})$.
2. (2)
There is a $\lambda\geq 0$ such that $j_{\ast}:H_{l}(\tilde{X},N_{i})\to
H_{l}(\tilde{X},N_{i-\lambda})$ is trivial for all $l\leq k$ and every
$i\in{\mathbb{R}}$.
3. (3)
For every neighborhood $N$ of $\infty$ with respect to $\xi$, there is another
neighborhood $N^{\prime}\subset N$ such that
$j_{\ast}:H_{l}(\tilde{X},N^{\prime})\to H_{l}(\tilde{X},N)$ is trivial for
all $l\leq k$.
4. (4)
Given $\varepsilon>0$ there exists a ${\mathbb{Z}}G$-chain map
$A:C_{\ast}(\tilde{X})\to C_{\ast}(\tilde{X})$ chain homotopic to the identity
with $v(A(x))\geq v(x)+\varepsilon$ for all non-zero $x\in C_{i}(\tilde{X})$
with $i\leq k$.
The equivalences of (2),(3) and (4) are similar to the proof of Proposition 2
and will be omitted. For the equivalence to (1) we refer to Appendix A which
treats a more general version.
###### Corollary 5.
Let $X$ be a finite connected CW-complex. For every $k\geq 0$ the set
$\Sigma^{k}(X,{\mathbb{Z}})$ is open and
$\Sigma^{n}(X,{\mathbb{Z}})=\Sigma^{\dim X}(X,{\mathbb{Z}})$ for $n\geq\dim
X$.∎
###### Corollary 6.
Let $X$ be a finite connected CW-complex. Then
1. (1)
$\Sigma^{1}(X)=\Sigma^{1}(X,{\mathbb{Z}})$.
2. (2)
$\Sigma^{k}(X)\subset\Sigma^{k}(X,{\mathbb{Z}})$ for $k\geq 2$.
###### Proof.
It is easy to see that Condition (3) of Proposition 2 implies Condition (3) of
Proposition 4 so that $\Sigma^{k}(X)\subset\Sigma^{k}(X,{\mathbb{Z}})$ for
$k\geq 1$. Also, for $k=1$ the chain homotopy of Proposition 4 (3) can easily
be used to realize a homotopy as in Proposition 2 (3). ∎
It follows from the examples of Bestvina and Brady [2] that in general
$\Sigma^{2}(X)\not=\Sigma^{2}(X,{\mathbb{Z}})$, see [22].
## 3\. Novikov rings and homology
Let $G$ be a finitely generated group and $\xi\in S(G)$. We then let
$\displaystyle\widehat{{\mathbb{Z}}G}_{\xi}$ $\displaystyle=$
$\displaystyle\left\\{\left.\sum_{g\in G}n_{g}g\,\right|\,\mbox{for all
}t\in{\mathbb{R}}\,\,\,\\#\\{g\,|\,n_{g}\not=0\mbox{ and
}\xi(g)>t\\}<\infty\right\\},$
a ring containing the group ring ${\mathbb{Z}}G$. If $\xi:G\to{\mathbb{R}}$ is
injective, this is the Novikov ring first defined in [24]. For general $\xi$
we call it the Novikov-Sikorav ring, first introduced in [29].
The relation to the Bieri-Neumann-Strebel-Renz invariants was immediately
apparent once the work of Sikorav in [29] became known. This relation also
extends to our situation and is explained in Proposition 7 below.
If $X$ is a finite connected CW-complex, $p:\overline{X}\to X$ a regular
covering space with $\pi_{1}(\overline{X})\subset{\rm Ker}\,\xi$, we get an
induced homomorphism, also denoted by $\xi:G\to{\mathbb{R}}$, where
$G=\pi_{1}(X)/\pi_{1}(\overline{X})$. We then obtain a finitely generated free
$\widehat{{\mathbb{Z}}G}_{\xi}$-chain complex by setting
$\displaystyle C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi})$ $\displaystyle=$
$\displaystyle\widehat{{\mathbb{Z}}G}_{\xi}\otimes_{{\mathbb{Z}}G}C_{\ast}(\overline{X})$
and we denote by
$\displaystyle H_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi})$ $\displaystyle=$
$\displaystyle H_{\ast}(C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi}))$
the resulting homology, called the Novikov-Sikorav homology (Novikov homology
if $\xi$ is injective).
The case of the universal cover is the one directly related to
$\Sigma^{k}(X)$, but the case of abelian covers plays an important role for
the cup-length estimates of the Lusternik-Schnirelmann categories.
###### Proposition 7.
Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ non-
zero, $\tilde{X}$ the universal cover of $X$ and $G=\pi_{1}(X)$. Then the
following are equivalent.
1. (1)
$\xi\in\Sigma^{k}(X;{\mathbb{Z}})$.
2. (2)
$H_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})=0$ for $i\leq k$.
Note that for testing $\xi\in\Sigma^{k}(X;{\mathbb{Z}})$ we have to use the
Novikov-Sikorav ring with respect to $-\xi$. The reason for this is that we
allow infinitely many non-zero coefficients in elements of
$\widehat{{\mathbb{Z}}G}_{\xi}$ in the “negative direction”, which is in line
with the original definition of the Novikov ring [23]. But to adhere to the
convention of [7] one has to complete in the “positive direction”. In order to
stick to the convention used in [18], we have to introduce the minus-sign
above.
###### Proof of Proposition 7.
(1) $\Longrightarrow$ (2): Let $A:C_{\ast}(\tilde{X})\to C_{\ast}(\tilde{X})$
be the chain map chain homotopic to the identity, given by Proposition 4 (4).
Then ${\rm id}-A:C_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})\to
C_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ is an isomorphism with inverse ${\rm
id}+A+A^{2}+\ldots$, which converges over $\widehat{{\mathbb{Z}}G}_{-\xi}$ by
the valuation property in Proposition 4(4). So the map on homology is both an
isomorphism and zero, which means that the homology vanishes.
(2) $\Longrightarrow$ (1): As $C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ is
free and bounded below, the vanishing of its homology groups up to degree $k$
guarantees the existence of a chain homotopy
$\delta:C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{-\xi})\to
C_{\ast+1}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ with
$\partial_{i+1}\delta_{i}+\delta_{i-1}\partial_{i}={\rm id}$ for $i\leq k$.
“Cutting off” gives a chain homotopy $\bar{\delta}:C_{\ast}(\tilde{X})\to
C_{\ast}(\tilde{X})$ with
$\partial_{i+1}\bar{\delta}_{i}+\bar{\delta}_{i-1}\partial_{i}={\rm id}-A$.
Then $A$ is a chain map homotopic to the identity, and by approximating
$\delta$ sufficiently well with $\bar{\delta}$, condition (4) of Proposition 4
is satisfied. ∎
## 4\. Relations to the Lusternik-Schnirelmann category of closed 1-forms
We now want to describe a connection to the Lusternik-Schnirelmann theory of
closed 1-forms, which has been introduced in a series of papers [11, 12, 14]
and for which more information, in particular on applications and
calculations, can be found in [13, 18]. Indeed, there exist various different
notions, and the one related closest to $\Sigma^{k}(X)$ is denoted by
${\rm{Cat}}(X,\xi)$. Here again $X$ is a finite CW-complex and $\xi\in
H^{1}(X;{\mathbb{R}})$. Let us recall the definition from [18].
###### Definition 12.
Let $X$ be a finite CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$. Fix a
closed 1-form $\omega$ representing $\xi$. Then ${\rm{Cat}}(X,\xi)$ is defined
as the minimal integer $k$ such that there exists an open subset $U\subset X$
satisfying
1. (1)
${\rm{cat}}_{X}(X-U)\leq k$.
2. (2)
for some homotopy $h:U\times[0,\infty)\to X$ one has
$h(x,0)=x\quad\mbox{ and
}\quad\lim_{t\to\infty}\int_{x}^{h_{t}(x)}\omega=-\infty$
for any point $x\in U$.
3. (3)
the limit in (2) is uniform in $x\in U$.
The integral is taken along the path $\gamma:[0,t]\to X$ given by
$\gamma(\tau)=h(x,\tau)$.
Here ${\rm{cat}}_{X}(A)$ for $A\subset X$ is the minimal number $i$ such that
there exist open sets $U_{1},\ldots,U_{i}\subset X$ covering $A$, each of
which is null-homotopic in $X$. The invariant ${\rm{Cat}}(X,\xi)$ originally
appeared in [12]. It is easy to see that it does not depend on positive
multiples of $\xi$, therefore it is well defined for $\xi\in S(X)$. Note that
it is also defined for $\xi=0$, in which case we recover the original
Lusternik-Schnirelmann category of $X$, see [18] for details.
The connection to $\Sigma^{k}(X)$ can now be described as follows. Because of
our sign conventions, we obtain another minus-sign in front of a $\xi$.
###### Theorem 5.
Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero. Write $n=\dim X$. If $-\xi\in\Sigma^{k}(X)$ for some $k\leq n$. Then
${\rm{Cat}}(X,\xi)\leq n-k$.
###### Proof.
Let $U$ be a small open neighborhood of $X^{(k)}$ in $X$ which deformation
retracts to $X^{(k)}$. Let $H:X^{(k)}\times[0,1]\to X$ be a homotopy starting
with inclusion which lifts to a homotopy as in Proposition 2 (3). Now define
$h:X^{(k)}\times[0,\infty)\to X$ by $h(x,t)=H(H^{m}(x,1),t-m)$, where $m$ is
an integer with $t\in[m,m+1]$. Notice that $H$ is cellular so $H^{m}(x,1)\in
X^{(k)}$. It is easy to see that $h$ combined with the deformation retraction
of $U$ to $X^{(k)}$ gives a homotopy as required in Definition 12. It is well
known that ${\rm{cat}}_{X}(X-X^{(k)})\leq\dim(X)-k$ so the result follows. ∎
There are other definitions for a Lusternik-Schnirelmann category of $\xi$,
denoted by ${\rm{cat}}(X,\xi)$ and ${\rm{cat}}^{1}(X,\xi)$, see [11, 13, 14,
18], satisfying
${\rm{cat}}(X,\xi)\leq{\rm{cat}}^{1}(X,\xi)\leq{\rm{Cat}}(X,\xi).$
Therefore Theorem 5 provides an upper bound for these as well.
## 5\. A Hurewicz type result
We can get a converse of Corollary 6 if we assume that
$\xi\in\Sigma^{2}(\pi_{1}(X))$. This can be described in the following way.
###### Definition 13.
Let $G$ be a finitely presented group and $\xi:G\to{\mathbb{R}}$ a nonzero
homomorphism. Let $X$ be a finite connected CW-complex with $\pi_{1}(X)=G$. We
say $\xi\in\Sigma^{2}(G)$, if there is a $\lambda\geq 0$ such that
$j_{\\#}:\pi_{l}(N_{i})\to\pi_{l}(N_{i-\lambda})$ is trivial for $l\leq 1$ and
every $i\in{\mathbb{R}}$.
Note that we can think of $\xi\in H^{1}(X;{\mathbb{R}})$ so neighborhoods of
$\infty$ are defined as above.
###### Theorem 6.
Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. Let $k\geq 2$. If
$\xi\in\Sigma^{2}(\pi_{1}(X))\cap\Sigma^{k}(X,{\mathbb{Z}})$, then
$\xi\in\Sigma^{k}(X)$.
The theorem will follow from two lemmas.
###### Lemma 8.
Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. If $\xi\in\Sigma^{2}(\pi_{1}(X))$, then $\\{\pi_{2}(\tilde{X},N)\\}$
and $\\{H_{2}(\tilde{X},N)\\}$ are pro-isomorphic.
###### Proof.
Notice that
$\displaystyle{\rm
Im}(\pi_{2}(\tilde{X},N_{i})\to\pi_{2}(\tilde{X},N_{i-\lambda}))$
$\displaystyle=$ $\displaystyle{\rm
Im}(\pi_{2}(\tilde{X})\to\pi_{2}(\tilde{X},N_{i-\lambda}))$
if every 1-sphere in $N_{i}$ bounds in $N_{i-\lambda}$, in particular the
images become abelian. As $\xi\in\Sigma^{2}(\pi_{1}(X))$, we can define a
homomorphism $\pi_{2}(\tilde{X},N_{i})\to H_{2}(\tilde{X},N_{i-\lambda})$ as
in a typical proof of the classical Hurewicz Theorem (after possibly
increasing $\lambda$), see, for example, Spanier [30]. The details will be
left to the reader. ∎
###### Lemma 9.
Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. Let $k\geq 3$ and $\xi\in\Sigma^{k-1}(X)$. Then
$\\{\pi_{k}(\tilde{X},N)\\}$ and $\\{H_{k}(\tilde{X},N)\\}$ are pro-
isomorphic.
###### Proof.
This is similar to the proof of Lemma 8, but easier, as we can define the
homomorphism directly. ∎
These two Lemmas combine to a proof of Theorem 6.
###### Remark 3.
Theorem 6 also follows from Latour’s Theorem 5.10 and 5.3 [21].
## 6\. Functoriality properties
###### Proposition 10.
Let $X$ and $Y$ be finite CW-complexes, $f:X\to Y$ and $g:Y\to X$ maps with
$fg\simeq{\rm id}_{Y}$. Then for all $k\geq 0$ we have
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X))$ $\displaystyle\subset$
$\displaystyle\Sigma^{k}(Y)$
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X,{\mathbb{Z}}))$
$\displaystyle\subset$ $\displaystyle\Sigma^{k}(Y,{\mathbb{Z}}).$
###### Proof.
We give a proof for the homotopy invariant $\Sigma^{k}(Y)$. Choose liftings
$\tilde{f}:\tilde{X}\to\tilde{Y}$ and $\tilde{g}:\tilde{Y}\to\tilde{X}$ with
$\tilde{f}\tilde{g}\simeq{\rm id}_{\tilde{Y}}$ equivariantly. Let $\xi\in
S(Y)$ satisfy $f^{\ast}(\xi)\in\Sigma^{k}(X)$. Let
$h_{\xi}:\tilde{Y}\to{\mathbb{R}}$ be a height function. Then
$h_{\xi}\circ\tilde{f}:\tilde{X}\to{\mathbb{R}}$ is a height function for
$f^{\ast}\xi$.
By cocompactness, there exists a $C\geq 0$ such that
$\displaystyle|h_{\xi}(\tilde{f}\tilde{g}(\tilde{y}))-h_{\xi}(\tilde{y})|$
$\displaystyle\leq$ $\displaystyle C$
for every $\tilde{y}\in\tilde{Y}$.
By assumption, Proposition 2 and Remark 1, there is an equivariant homotopy
$\tilde{H}:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$ starting with inclusion,
such that
$\displaystyle h_{\xi}\tilde{f}(H(\tilde{x},1))-h_{\xi}\tilde{f}(\tilde{x})$
$\displaystyle\geq$ $\displaystyle C+1$
for all $\tilde{x}\in\tilde{X}$. Therefore
$\displaystyle h_{\xi}\tilde{f}(H(\tilde{g}(\tilde{y}),1))-h_{\xi}(\tilde{y})$
$\displaystyle\geq$ $\displaystyle 1.$
Now $\tilde{f}H(\tilde{g},\cdot)$ can be combined with the homotopy
$\tilde{f}\tilde{g}\simeq{\rm id}_{\tilde{Y}}$ to show that
$\xi\in\Sigma^{k}(Y)$. ∎
###### Corollary 11.
Let $X$ and $Y$ be finite connected CW-complexes and $h:X\to Y$ a homotopy
equivalence. Then $h^{\ast}(\Sigma^{k}(Y))=\Sigma^{k}(X)$ and
$h^{\ast}(\Sigma^{k}(Y,{\mathbb{Z}}))=\Sigma^{k}(X,{\mathbb{Z}})$ for all
$k\geq 0$.
###### Proposition 12.
Let $X$ and $Y$ be finite CW-complexes and $f:X\to Y$ $m$-connected with
$m\geq 1$. Then
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X))$ $\displaystyle\subset$
$\displaystyle\Sigma^{k}(Y)$
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X,{\mathbb{Z}}))$
$\displaystyle\subset$ $\displaystyle\Sigma^{k}(Y,{\mathbb{Z}})$
for all $k\leq m$.
###### Proof.
Add cells of dimension $\geq m+1$ to $X$ to get a (possibly infinite) CW-
complex $X^{\prime}$ containing $X$ such that $f$ extends to a homotopy
equivalence $f^{\prime}:X^{\prime}\to Y$. Let $g:Y\to X^{\prime}$ be a
cellular homotopy inverse. Then $g(Y^{(m)})\subset X$. If
$f^{\ast}(\xi)\in\Sigma^{k}(X)$, the homotopy
$H:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$ can now be used as in the proof of
Proposition 10 to show that $\xi\in\Sigma^{k}(Y)$ for $k\leq m$. ∎
###### Example 2.
Let $X$ be a finite connected CW-complex and $f:X\to X$ a map. The mapping
torus $M_{f}$ is the quotient space $M_{f}=X\times[0,1]/\,\sim$, where
$(x,0)\sim(f(x),1)$. There is a natural map $g:M_{f}\to{\mathbf{S}}^{1}$ given
by $g([x,t])=\exp 2\pi it$. Let
$\xi=[g]\in[M_{f},{\mathbf{S}}^{1}]=H^{1}(M_{f};{\mathbb{Z}})\subset
H^{1}(M_{f};{\mathbb{R}}).$
The homotopy $h:M_{f}\times[0,1]\to M_{f}$ given by
$\displaystyle H([x,t],s)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rc}[x,t-s]&t\geq s\\\
\,[f(x),1+t-s]&t\leq s\end{array}\right.$
shows that $-\xi\in\Sigma^{k}(M_{f})$ for all $k\geq 0$.
###### Proposition 13.
Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{Z}})$ be
nonzero. Let $q:\bar{X}\to X$ be the infinite cyclic covering space
corresponding to ${\rm Ker}\,\xi$. Assume that $\bar{X}$ is homotopy
equivalent to a CW-complex $Y$ with finite $k$-skeleton. Then
$\pm\xi\in\Sigma^{k}(X)$.
###### Proof.
Let $h:\bar{X}\to{\mathbb{R}}$ be induced by a height function
$h_{\xi}:\tilde{X}\to{\mathbb{R}}$ and $\zeta:\bar{X}\to\bar{X}$ the
generating covering transformation with $h\zeta(x)>h(x)$ for all
$x\in\bar{X}$. Let $a:Y\to\bar{X}$ and $b:\bar{X}\to Y$ be mutually inverse
homotopy equivalences. The there is a homotopy equivalence $g:M_{b\zeta a}\to
X$ given by
$M_{b\zeta a}\simeq M_{\zeta ab}\simeq M_{\zeta}\simeq X$
where the last homotopy equivalence is given by $[\bar{x},t]\to q(\bar{x})$
for $\bar{x}\in\bar{X}$.
We can assume that $b\zeta a:Y\to Y$ sends the $k$-skeleton to the
$k$-skeleton. Let $\varphi:Y^{(k)}\to Y^{(k)}$ be the restriction of $b\zeta
a$ to $Y^{(k)}$. The induced map $M_{\varphi}\to M_{b\zeta a}$ is
$k$-connected and so there is a $k$-connected map $f:M_{\varphi}\to X$. By
Example 2 $-f^{\ast}(\xi)\in\Sigma^{k}(M_{\varphi})$. It follows from
Proposition 12 that $-\xi\in\Sigma^{k}(X)$. To get $\xi\in\Sigma^{k}(X)$ as
well, replace $\zeta$ by $\zeta^{-1}$. ∎
In the next section we will show that the converse of Proposition 13 also
holds.
## 7\. Domination results for covering spaces
Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. Then we define
$\displaystyle S(X,\overline{X})$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(X)\,|\,q^{\ast}\xi=0\\}.$
In particular, if $\overline{X}$ is the universal abelian covering of $X$,
then $S(X,\overline{X})=S(X)$. More generally, $S(X,\overline{X})$ is a sphere
of dimension $d-1$, where $d$ is the rank of the finitely generated abelian
group $\pi_{1}(X)/\pi_{1}(\overline{X})$.
###### Theorem 7.
Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. Let $k\geq 1$.
Then the following are equivalent.
1. (1)
$\overline{X}$ is homotopy equivalent to a CW-complex with finite
$k$-skeleton.
2. (2)
$S(X,\overline{X})\subset\Sigma^{k}(X)$.
Furthermore, if $S(X,\overline{X})\subset\Sigma^{\dim X}(X)$, then
$\overline{X}$ is finitely dominated.
We derive Theorem 7 from a more general version for chain complexes which is a
generalization of [7, Thm.B]. To get an alternative proof for (1)
$\Longrightarrow$ (2) one can use the techniques of [28, Thm.3.2].
If $N$ is a normal subgroup of $G$ with $G/N$ abelian, we write
$\displaystyle S(G,N)$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(G)\,|\,N\leq{\rm Ker}\,\xi\\}$
###### Theorem 8.
Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is finitely generated in
every degree $i\leq n$, and $N$ a normal sugroup of $G$ such that $G/N$ is
abelian. Then $C$ is of finite $n$-type over ${\mathbb{Z}}N$ if and only if
$S(G,N)\subset\Sigma^{n}(C)$.
Theorem 7 follows from Theorem 8 by the work of Wall [32, 33].
###### Proof.
Assume that $C$ is of finite $n$-type over ${\mathbb{Z}}N$. Let $\xi\in
S(G,N)$ and denote $Q=G/N$. Then $\xi$ induces a homomorphism, also denoted
$\xi$, $\xi:Q\to{\mathbb{R}}$. As ${\mathbb{Z}}G$ is free over
${\mathbb{Z}}N$, there is a chain homotopy equivalence $f:P\to C$ over
${\mathbb{Z}}N$ with $P_{j}$ finitely generated free for all $j\leq n$. Then
$f:{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}P\to{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$
shows that ${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$ is of finite $n$-type
over ${\mathbb{Z}}G_{\xi}$. Also
${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C\cong{\mathbb{Z}}Q_{\xi}\otimes C$
where the chain complex on the right has ${\mathbb{Z}}G_{\xi}$ acting
diagonally. The isomorphism is given by $g\otimes c\mapsto\pi(g)\otimes gc$,
where $\pi:G\to Q$ is projection. By [7, Lm.5.2] there is a free resolution
$E_{\ast}\to{\mathbb{Z}}$ over ${\mathbb{Z}}Q_{\xi}$ which is finitely
generated in every degree. Therefore each $E_{p}\otimes C$ is of finite
$n$-type over ${\mathbb{Z}}G_{\xi}$. Let $f:P_{p\,q}\to E_{p}\otimes C_{q}$ be
the corresponding chain map, notice that $P_{p\,q}$ is just a positive power
of ${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}P_{q}$ depending on the rank of
$E_{p}$. As $C$ is free over ${\mathbb{Z}}N$, we get that
${\mathbb{Z}}Q_{\xi}\otimes C\cong{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$
is free over ${\mathbb{Z}}G_{\xi}$, and we can assume that $f$ is a chain
homotopy equivalence with inverse $g_{q}:E_{p}\otimes C_{q}\to P_{p\,q}$.
Denote by $L:E_{p}\otimes C_{q}\to E_{p}\otimes C_{q+1}$ the chain homotopy
$L:fg\simeq 1$.
For $k\geq 0$ define
$F^{k}:P_{p\,q}\to E_{p-k}\otimes C_{q+k}\mbox{ by }F^{k}=(Ld)^{k}f,$
where $d:E_{p}\to E_{p-1}$ is the boundary of the resolution. Also define
$K^{i}:P_{p\,q}\to P_{p-i\,q-1+i}\mbox{ by }K^{i}=gdF^{i-1}$
for $i\geq 1$. We also set $K^{0}=\partial:P_{p\,q}\to P_{p-1\,q}$.
###### Lemma 14.
Let $\partial$ denote the boundaries $\partial:P_{p\,q}\to P_{p\,q-1}$ and
$\partial:E_{p}\otimes C_{q}\to E_{p}\otimes C_{q-1}$. Then $\partial
F^{0}=F^{0}\partial$ and for $m\geq 1$ we have
$\displaystyle\partial F^{m}+(-1)^{m+1}F^{m}\partial$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m-1}(-1)^{k}F^{k}K^{m-k}-dF^{m-1}.$
###### Proof.
The proof by induction is straightforward. ∎
###### Lemma 15.
For $m\geq 0$ we have
$\displaystyle\sum_{s=0}^{m}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle 0.$
###### Proof.
For $m=0$ this means $\partial\partial=0$, so assume the statement holds for
$m\geq 0$. Then
$\displaystyle\sum_{s=0}^{m+1}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle gdF^{m}\partial+(-1)^{m+1}\partial
gdF^{m}+\sum_{s=1}^{m}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle(-1)^{m+1}\left(gd\sum_{k=0}^{m-1}(-1)^{k}F^{k}K^{m-k}-gddF^{m-1}\right)+$
$\displaystyle+\sum_{s=1}^{m}(-1)^{s}gdF^{m-s-1}K^{s}$ $\displaystyle=$
$\displaystyle 0$
by Lemma 14 and since $dd=0$. ∎
Now define a chain complex $TP$ by $TP_{k}=\bigoplus\limits_{p+q=k}P_{p\,q}$
and $\delta:TP_{k}\to TP_{k-1}$ by
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}(-1)^{p}K^{s}$
where $(-1)^{p}$ refers to $P_{p\,k-p}$. By Lemma 15 we get that
$\delta\delta=0$. Also define $TE_{k}=\bigoplus\limits_{p+q=k}E_{p}\otimes
C_{q}$ with $\delta=(-1)^{p}(\partial+d)$. We get a chain map $F:TP\to TE$ by
setting
$\displaystyle F$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}(-1)^{k}F^{k}$
That $F$ is indeed a chain map follows from Lemmata 14 and 15.
Using the filtrations $(TP^{(m)})_{k}=\bigoplus\limits_{p+q=k,p\leq
m}P_{p\,q}$ and
$(TE^{(m)})_{k}=\bigoplus\limits_{p+q=k,p\leq m}E_{p}\otimes C_{q}$ we see
that $F$ induces a chain homotopy equivalence $\bar{F}:TP^{(m)}/TP^{(m-1)}\to
TE^{(m)}/TE^{(m-1)}$ and by a spectral sequence argument $F$ is a chain
homotopy equivalence. Another spectral sequence argument gives a homology
isomorphism from $TE$ to $C$. As $TP_{k}$ is finitely generated free over
${\mathbb{Z}}G_{\xi}$, we now get that $C$ is of finite $n$-type over
${\mathbb{Z}}G_{\xi}$.
Now assume that $S(G,N)\subset\Sigma^{n}(C)$.
Let $X_{i}$ be a ${\mathbb{Z}}G$-basis of $C_{i}$ for $i\leq n$, a finite set
by assumption. Given $c\in C_{i}$, we can therefore write
$\displaystyle c$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{i}}n^{i}_{j}x^{i}_{j}$
with $n^{i}_{j}\in{\mathbb{Z}}G$ and
$\displaystyle\partial c$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{i-1}}n^{i-1}_{j}x_{j}^{i-1}$
here the $x^{i}_{j}$ denote the elements of $X_{i}$. For $y\in{\mathbb{Z}}G$
we denote ${\rm supp}\,y$ as the elements of $G$ with nonzero coefficient and
for $c\in C_{i}$ as above we let
$\displaystyle{\rm supp}\,c$ $\displaystyle=$
$\displaystyle\bigcup_{j=1}^{m_{i}}{\rm
supp}\,n_{j}^{i}\cup\bigcup_{j=1}^{m_{i-1}}{\rm supp}\,n_{j}^{i-1}$
In particular, we get ${\rm supp}\,\partial c\subset{\rm supp}\,c$. The
support depends on the chosen basis, but we fix the basis once and for all.
We also denote $\pi:G\to G/N=Q$. By choosing an inner product on
$Q_{\mathbb{R}}=\mathbb{R}\otimes Q$ we get a norm $\|\cdot\|$ on
$Q_{\mathbb{R}}$ and we can think of $S(G,N)\subset Q_{\mathbb{R}}$ as the
unit sphere in this normed vector space. We extend the norm to $C$ by setting
$\displaystyle\|c\|$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\max\\{\|\pi(g)\|\,|\,g\in{\rm
supp}\,c\\}&c\not=0\\\ 0&c=0\end{array}\right.$
Notice that we set $\|0\|=0$ despite the fact that $\|0\|=-\infty$ in [7]. For
$\xi\in S(G,N)\subset Q_{\mathbb{R}}$ we also obtain a valuation $v_{\xi}$ by
setting
$\displaystyle v_{\xi}(c)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\min\\{\langle\pi(g),\xi\rangle\,|\,g\in{\rm
supp}\,c\\}&c\not=0\\\ \infty&c=0\end{array}\right.$
Let us also set for $a,b\in C$
$\displaystyle{\rm diam}(a,b)$ $\displaystyle=$
$\displaystyle\max\\{\|\pi(g)-\pi(h)\|\,|\,g\in{\rm supp}\,a,h\in{\rm
supp}\,b\\}$
with the convention that ${\rm diam}(a,b)=0$ if $a$ or $b$ is zero. Finally,
for $r>0$ and $c\in C$ we let
$\displaystyle B_{r}(c)$ $\displaystyle=$ $\displaystyle\\{d\in C\,|\,{\rm
diam}(c,d)\leq r\\}.$
Given $r>0$, as $S(G,N)\subset\Sigma^{n}(C)$ we can find for every $\xi\in
S(G,N)$ a chain map $\varphi^{\xi}:C\to C$ and a chain homotopy
$H^{\xi}:1\simeq\varphi^{\xi}$ such that
$v_{\xi}(\varphi^{\xi}c)-v_{\xi}(c)\geq 2r$ for all $c\in C$. Furthermore,
there is an open neighborhood $U_{\xi}$ of $\xi$ in $S(G,N)$ such that
$\displaystyle v_{\eta}(\varphi^{\xi}c)-v_{\eta}(c)$ $\displaystyle\geq$
$\displaystyle r$
for all $\eta\in U_{\xi}$ and $c\in C_{l}$, $l\leq n$.
As $S(G,N)$ is compact, finitely many of the $U_{\xi}$ suffice to cover
$S(G,N)$, so let $(U_{i},\varphi^{i},H^{i})$ for $i=1,\ldots,k$ be triples
where $U_{i}$ cover $S(G,N)$, $\varphi^{i}:C\to C$ chain map with
$\displaystyle v_{\eta}(\varphi^{i}c)-v_{\eta}(c)$ $\displaystyle\geq$
$\displaystyle r$
for all $\eta\in U_{i}$ and all $c\in C_{l}$, $l\leq n$, and
$H^{i}:1\simeq\varphi^{i}$ a chain homotopy.
As we deal only with finitely many chain homotopies, there is a $M\geq 0$ such
that
$\displaystyle v_{\eta}(c)-v_{\eta}(H^{i}c)$ $\displaystyle\leq$
$\displaystyle M$
for all $\eta\in U_{i}$ and all $c\in C_{l}$, $l\leq n$. Furthermore, by
replacing the chain homotopy $H^{i}$ by
$H^{i}-\varphi^{i}H^{i}:1\simeq(\varphi^{i})^{2}$ we can increase $r>0$
without increasing $M>0$. Therefore we can assume that
(6) $\displaystyle r$ $\displaystyle>$ $\displaystyle 3Mn$
Finally, there exists an $L>0$ with
$\displaystyle{\rm diam}(x,x)$ $\displaystyle\leq$ $\displaystyle L$
$\displaystyle{\rm diam}(x,\varphi^{i}x)$ $\displaystyle\leq$ $\displaystyle
L$ $\displaystyle{\rm diam}(x,H^{i}x)$ $\displaystyle\leq$ $\displaystyle L$
for all $i=1,\ldots,k$ and all $x\in X_{l}$, $l\leq n$, as there are only
finitely many such conditions.
Notice that $C$ is a free ${\mathbb{Z}}N$ chain complex, and a basis is given
by $TX_{i}=\\{tx\,|\,t\in T,x\in X_{i}\\}$, where $T\subset G$ is a subset
such that $\pi|T$ induces a bijection from $T$ to $Q$.
###### Lemma 16.
If $S(G,N)\subset\Sigma^{n}(C)$, there exist constants $r>0$, $M>0$, $A>0$, a
${\mathbb{Z}}N$-chain map $\psi:C\to C$ and a ${\mathbb{Z}}N$-chain homotopy
$K:1\simeq\psi$ such that for $m\leq n$ we have $\psi_{m}(z)=z$ if $\|z\|\leq
A$, and for $tx\in TX_{m}$ with $\|tx\|>A$ we get
$\displaystyle\|\psi_{m}(tx)\|$ $\displaystyle\leq$ $\displaystyle\|tx\|-r.$
Furthermore $\|K_{m}(z)\|\leq\|z\|+M$ for all $z\in C_{m}$.
###### Proof.
In order to define $\psi$ and $K$, we define them on $TX_{m}$. Let $r>0$ and
$M>0$ be as above Lemma 16.
###### Lemma 17.
Let $tx\in TX_{m}$ with
$\|tx\|\geq\max\\{\frac{3}{4}r+\frac{L^{2}}{r},\frac{L^{2}}{M}\\}$. Let $i$ be
such that $\xi_{t}/\|\xi_{t}\|\in U_{i}$, where
$\xi_{t}(g)=\langle\pi(g),\pi(t)\rangle$ for $g\in G$. Then
$\displaystyle\|\varphi^{i}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{1}{2}r$ $\displaystyle\|H^{i}(tx)\|$
$\displaystyle\leq$ $\displaystyle\|tx\|+\frac{3}{2}M.$
###### Proof.
Let $g\in{\rm supp}\,tx$ satisfy $\|\pi(g)\|=\|tx\|$. As ${\rm
diam}(tx,\varphi^{i}tx)\leq L$, we get for $h\in{\rm supp}\,\varphi^{i}(tx)$
$\displaystyle\|\pi(h)\|^{2}$ $\displaystyle\leq$
$\displaystyle(\|\pi(g)\|-r)^{2}+L^{2}$ $\displaystyle=$
$\displaystyle\|\pi(g)\|^{2}-2r\|\pi(g)\|+r^{2}+L^{2}$ $\displaystyle\leq$
$\displaystyle\|\pi(g)\|^{2}-r\|\pi(g)\|-\frac{3}{4}r^{2}-L^{2}+r^{2}+L^{2}$
$\displaystyle=$ $\displaystyle(\|\pi(g)\|-\frac{r}{2})^{2}$
So $\|\varphi^{i}(tx)\|\leq\|tx\|-\frac{r}{2}$. Similarly, using
$\|\pi(g)\|\geq\frac{L^{2}}{M}$, we get $H^{i}(tx)\|\leq\|tx\|+\frac{3}{2}M$.
∎
If $\|tx\|>L$, then $\pi({\rm supp}\,tx)\subset Q_{\mathbb{R}}-\\{0\\}$.
Denote $p:Q_{\mathbb{R}}-\\{0\\}\to S(G,N)$ the standard retraction. Using the
Lebesgue number of the covering $U_{1}\cup\ldots\cup U_{k}$, we can find a
constant $A^{\prime}>0$ with $\|tx\|>A^{\prime}$ implying
(7) $\displaystyle p(\pi({\rm supp}\,B_{L^{\prime}}(tx)))$
$\displaystyle\subset$ $\displaystyle U_{i}$
for some $i$, where
$L^{\prime}=\max\\{\frac{3}{4}r+\frac{L^{2}}{r},\frac{L^{2}}{M}\\}+(2n+1)L$.
Let $A=A^{\prime}+L^{\prime}$.
Define $K_{0}:C_{0}\to C_{1}$ by $K_{0}(tx)=0$ if $\|tx\|\leq A$, and
$K_{0}(tx)=H^{i}_{0}(tx)$ if $\|tx\|>A$, where $i$ is the smallest number with
(7) satisfied. Also define $\psi_{0}:C_{0}\to C_{0}$ by
$\displaystyle\psi_{0}(tx)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ccc}tx&\mbox{if}&\|tx\|\leq A\\\
\varphi^{i}(tx)&\mbox{if}&\|tx\|>A\end{array}\right.$
where $i$ is again the smallest number such that (7) holds. It follows that
$\partial K_{0}=1-\psi_{0}$. By Lemma 17 we get
$\displaystyle\|\psi_{0}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{r}{2}$ $\displaystyle\|K_{0}(tx)\|$
$\displaystyle\leq$ $\displaystyle\|tx\|+\frac{3}{2}M$
for $\|tx\|>A$. Also
$\displaystyle{\rm diam}(\psi_{0}(tx),tx)$ $\displaystyle\leq$ $\displaystyle
L$ $\displaystyle{\rm diam}(K_{0}(tx),tx)$ $\displaystyle\leq$ $\displaystyle
L$
for all basis elements $tx$.
Assume now by induction that we have $K_{j}:C_{j}\to C_{j+1}$,
$\psi_{j}:C_{j}\to C_{j}$ with
$\displaystyle\partial K_{j}+K_{j-1}\partial=1-\psi_{j}$
and
(9) $\displaystyle\|\psi_{j}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{r}{2}+j\frac{3}{2}M$ (10)
$\displaystyle\|K_{j}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|+\frac{3}{2}M(j+1)$
for $\|tx\|>A$ and
(11) $\displaystyle{\rm diam}(\psi_{j}(tx),tx)$ $\displaystyle\leq$
$\displaystyle(2j+1)L$ (12) $\displaystyle{\rm diam}(K_{j}(tx),tx)$
$\displaystyle\leq$ $\displaystyle(2j+1)L$
for all basis elements $tx$, for all $j\leq m-1$ with $m\leq n$.
Then define $K_{m}:C_{m}\to C_{m+1}$ by $K_{m}(tx)=0$ if $\|tx\|\leq A$ and
$\displaystyle K_{m}(tx)$ $\displaystyle=$ $\displaystyle
H^{i}_{m}(tx)-H^{i}_{m}K_{m-1}\partial(tx)$
for $\|tx\|>A$, where $i$ satisfies (7). It is straightforward to check that
$\displaystyle(\partial K_{m}+K_{m-1}\partial)(tx)$ $\displaystyle=$
$\displaystyle tx-\varphi^{i}_{m}(tx)+\varphi^{i}_{m}(K_{m-1}\partial
tx)-H^{i}_{m-1}\psi_{m-1}\partial tx$
for $\|tx\|>A$. So define $\psi_{m}:C_{m}\to C_{m}$ by $\psi_{m}(tx)=tx$ for
$\|tx\|\leq A$ and
$\displaystyle\psi_{m}(tx)$ $\displaystyle=$
$\displaystyle\varphi^{i}_{m}(tx)-\varphi_{m}^{i}(K_{m-1}\partial
tx)+H^{i}_{m-1}\psi_{m-1}\partial tx$
if $\|tx\|>A$. It follows that $\partial K_{m}+K_{m-1}\partial=1-\psi_{m}$. As
$\|\varphi^{i}_{m}(tx)-\varphi_{m}^{i}(K_{m-1}\partial
tx)+H^{i}_{m-1}\psi_{m-1}\partial tx\|\,\,\,\leq\hskip 56.9055pt$
$\max\\{\|\varphi^{i}_{m}(tx)\|,\|\varphi_{m}^{i}(K_{m-1}\partial
tx)\|,\|H^{i}_{m-1}\psi_{m-1}\partial tx\|\\}$
it is easy to see that (9) also holds for $j=m$. Here we use that
${\rm diam}(K_{m-1}\partial tx,tx)\leq(2m-1)L+L$, so that Lemma 17 still
applies to all basis elements occuring in $K_{m-1}\partial tx$ by the choice
of $A$.
Similarly, (10), (11) and (12) hold for $j=m$. For $m>n$ we set $K_{m}=0$ and
define $\psi_{m}$ such that
$\displaystyle\partial K_{m}+K_{m-1}\partial=1-\psi_{m}.$
As the identity is a chain map, we get that $\psi$ is a chain map. Replacing
$r$ by $\frac{r}{2}-\frac{3}{2}nM$ and $M$ by $\frac{3}{2}M(n+1)$ we get Lemma
16. Note that the new $r>0$ by (6). ∎
###### Lemma 18.
If $S(G,N)\subset\Sigma^{n}(C)$, then there exists a free ${\mathbb{Z}}N$
chain complex $D$ with $D_{i}$ finitely generated for $i\leq n$, and
${\mathbb{Z}}N$-chain maps $a:D\to C$, $b:C\to D$ with $ab$ chain homotopic to
the identity on $C$. Also, $a:D_{i}\to C_{i}$ can be assumed to be inclusion
for $i\leq n$, and $D_{i}=C_{i}$ for $i>n$.
###### Proof.
Let $r,M,A>0$, $\psi$ and $K$ as provided by Lemma 16. If $l$ is a positive
integer, $B\geq 0$ and $z\in C_{m}$ with $m\leq n$, we have
(13) $\|\psi^{l}(z)\|\leq A+B\hskip 22.76228pt\mbox{if}\hskip
22.76228pt\|z\|\leq A+l\cdot r+B$
We define a ${\mathbb{Z}}N$-chain homotopy $\Phi:C\to C_{+1}$ as follows. For
$s\leq n$, set
$\displaystyle\Phi_{s}(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}K_{s}\psi^{j}tx-K_{s}\psi^{j}\Phi_{s-1}\partial
tx$
where $l$ is an integer such that $\|tx\|\in(A+l\cdot r,A+(l+1)r]$, and for
$s>n$ we simply set $\Phi_{s}=0$. We get a ${\mathbb{Z}}N$-chain map
$\zeta:C\to C$ by setting $\zeta=1-\partial\Phi-\Phi\partial$, in particular,
$\zeta$ is chain homotopy equivalent to the identity.
Using induction, we see that
$\displaystyle\|\Phi_{s}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|+(s+1)M$
for $s\leq n-1$. It follows that
$\displaystyle\|\Phi_{s}(z)\|$ $\displaystyle\leq$ $\displaystyle\|z\|+(s+1)M$
for all $z\in C_{s}$.
We claim that for $s\leq n$ we get
(14) $\displaystyle\|\zeta(z)\|$ $\displaystyle\leq$ $\displaystyle A+s\cdot
M$
for all $z\in C_{s}$.
This holds for $s=0$ by (13), as $\zeta(tx)=\psi^{l+1}(tx)$ if
$\|tx\|\in(A+l\cdot r,A+(l+1)r]$.
Now notice that
$\displaystyle(\partial\Phi_{s}+\Phi_{s-1}\partial)(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}(K_{s}\psi^{j}tx-K_{s}\psi^{j}\Phi_{s-1}\partial
tx)+\Phi_{s-1}\partial(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}((\partial
K_{s}\psi^{j}+K_{s-1}\partial\psi^{j})(tx)$
$\displaystyle-\sum_{j=0}^{l}(\partial
K_{s}\psi^{j}\Phi_{s-1}+K_{s-1}\partial\psi^{j}\Phi_{s-1})(\partial tx)$
$\displaystyle+\sum_{j=0}^{l}(K_{s-1}\psi^{j}\partial\Phi_{s-1}\partial tx-
K_{s-1}\partial\psi^{j}tx)+\Phi_{s-1}\partial tx$ $\displaystyle=$
$\displaystyle tx-\psi^{l+1}tx-\Phi_{s-1}\partial
tx+\psi^{l+1}\Phi_{s-1}\partial tx$
$\displaystyle-\sum_{j=0}^{l}K_{s-1}\partial\psi^{j}tx+\Phi_{s-1}\partial tx$
$\displaystyle+\sum_{j=0}^{l}(K_{s-1}\psi^{j}\partial tx-
K_{s-1}\psi^{j}\zeta\partial tx)$ $\displaystyle=$ $\displaystyle
tx-\psi^{l+1}tx+\psi^{l+1}\Phi_{s-1}\partial
tx-\sum_{j=0}^{l}K_{s-1}\psi^{j}\zeta\partial tx$
It follows that
$\displaystyle\zeta_{s}(tx)$ $\displaystyle=$
$\displaystyle\psi^{l+1}tx-\psi^{l+1}\Phi_{s-1}\partial
tx+\sum_{j=0}^{l}K_{s-1}\psi^{j}\zeta_{s-1}\partial tx$
Using induction, we see again by (13), that (14) holds.
Define a chain complex $D$ by
$\displaystyle D_{s}$ $\displaystyle=$ $\displaystyle\\{z\in
C_{s}\,|\,\|z\|\leq A+nM\\}$
for $s\leq n$. For $s>n$ we let $D_{s}=C_{s}$. The boundary map
$\partial_{D}:D_{n+1}\to D_{n}$ is given by
$\partial_{D}=\zeta\circ\partial_{C}$. We can define a chain map $b:C\to D$ by
using $\zeta$ for $s\leq n$ and the identity for $s>n$ and a chain map $a:D\to
C$ by using inclusion for $s\leq n$ and $\zeta$ for $s>n$. Then $ab=\zeta:C\to
C$ is chain homotopic to the identity. As only finitely many $q\in Q$ satisfy
$\|q\|\leq A+nM$, we get that $D_{s}$ is finitely generated free over
${\mathbb{Z}}N$, compare [7, 5.4]. This finishes the proof of Lemma 18. ∎
To finish the proof of Theorem 8, note that by Lemma 18 the chain complex $C$
is dominated over ${\mathbb{Z}}N$ by a chain complex $D$ with $D_{i}$ finitely
generated free for $i\leq n$. By a standard construction, compare Ranicki [27,
§3] or Wall [33], there exists a ${\mathbb{Z}}N$ chain complex $E$ chain
homotopy equivalent to $C$ with $E_{i}$ finitely generated free for $i\leq n$.
∎
###### Corollary 19.
Let $C$ be a finitely generated free ${\mathbb{Z}}G$-chain complex with
$C_{i}=0$ for $i>n$. Then $C$ is ${\mathbb{Z}}N$-chain homotopy equivalent to
a finitely generated projective ${\mathbb{Z}}N$-chain complex $D$ with
$D_{i}=0$ for $i>n$ if and only if $S(G,N)\subset\Sigma^{n}(C)$.
###### Proof.
If $C$ is ${\mathbb{Z}}N$-chain homotopy equivalent to a finitely generated
projective ${\mathbb{Z}}N$-complex, it is, by definition, of finite $n$-type
over ${\mathbb{Z}}N$, hence $S(G,N)\subset\Sigma^{n}(C)$ by Theorem 8.
If $S(G,N)\subset\Sigma^{n}(C)$, then by Lemma 18 there is a chain complex
$D^{\prime}$, finitely generated free over ${\mathbb{Z}}N$ with
$D_{i}^{\prime}=0$ for $i>n$ which dominates $C$. By [27, Prop.3.1], the
required chain complex $D$ exists. ∎
## 8\. Movability of homology classes
Recall from Proposition 7 that the vanishing and non-vanishing of Novikov-
Sikorav homology groups in the universal cover case determine
$\Sigma^{k}(X,{\mathbb{Z}})$. We now want to take a closer look at the
homology of other coverings, in particular coverings with abelian covering
transformation group.
In this section, $R$ is a ring, although we have mainly the cases
$R={\mathbb{Z}}$ and $R={\mathbf{k}}$ a field in mind.
Let $p:\overline{X}\to X$ be a regular cover, where $X$ is again a finite
connected CW-complex. Denote $G=\pi_{1}(X)/\pi_{1}(\overline{X})$. Recall that
$S(X,\overline{X})$ consists of those $\xi\in S(X)$ with $p^{\ast}\xi=0$. For
such $\xi$ we can define a height function $h:\overline{X}\to{\mathbb{R}}$ and
neighborhoods $N\subset\overline{X}$ of infinity with respect to $\xi$ as in
the case of the universal covering.
###### Definition 14.
A homology class $z\in H_{q}(\overline{X};R)$ is said to be movable to
infinity in $\overline{X}$ with respect to $\xi\in S(X,\overline{X})$, if $z$
can be realized by a singular cycle in any neighborhood $N$ of infinity with
respect to $\xi$.
In other words, $z\in H_{q}(\overline{X};R)$ is movable to infinity with
respect to $\xi$, if $z$ is an element of
$\bigcap_{N}{\rm Im}(H_{q}(N;R)\to H_{q}(\overline{X};R))$
where the intersection is taken over all neighborhoods of infinity with
respect to $\xi$.
Note that we have an inverse system of $R$-modules given by
$\\{H_{q}(\overline{X},N;R)\leftarrow H_{q}(\overline{X},N^{\prime};R)\\}$
which runs over neighborhoods $N^{\prime}\subset N$ of infinity with respect
to $\xi$. So $z$ is movable to infinity with respect to $\xi$ if and only if
$z\in{\rm
Ker}(H_{q}(\overline{X};R)\to\lim\limits_{\leftarrow}H_{q}(\overline{X},N;R)).$
Just as with integer coefficients, we can define a Novikov-Sikorav ring
$\widehat{RG}_{\xi}$ with coefficients in an arbitrary ring $R$. As this ring
can be expressed as an inverse limit, standard methods give the following
exact sequence, see [15] or [19] for details.
(15)
$0\rightarrow{\lim\limits_{\leftarrow}}^{1}H_{q+1}(\overline{X},N;R)\rightarrow
H_{q}(X;\widehat{RG}_{-\xi})\rightarrow\lim\limits_{\leftarrow}H_{q}(\overline{X},N;R)\rightarrow
0$
Let us give a simple criterion for $z\in H_{i}(\overline{X};R)$ to be movable
to infinity.
###### Definition 15.
An element $\Delta\in RG$ is said to have $\xi$-lowest coefficient 1, if
$\Delta=1-y$, with $y=\sum a_{j}g_{j}\in RG$ and such that each $g_{j}\in G$
satisfies $\xi(g_{j})>0$.
Such a $\Delta$ is invertible over $\widehat{RG}_{-\xi}$. So if $\Delta\cdot
z=0\in H_{q}(\overline{X};R)$, we get that the image of $z$ in
$H_{q}(X;\widehat{RG}_{-\xi})$ is zero, and by (15), $z$ is movable to
infinity.
Under certain conditions, this is in fact necessary for movability to
infinity. The following theorem is taken from [15, 17].
###### Theorem 9.
Let $X$ be a finite connected CW-complex and $p:\overline{X}\to X$ a regular
covering with covering transformation group $G\cong{\mathbb{Z}}^{r}$. Let
$\xi\in S(X,\overline{X})$ induce an injective homomorphism
$G\to{\mathbb{R}}$. Let $R$ be either ${\mathbb{Z}}$ or a field. For $z\in
H_{q}(\overline{X};R)$, the following are equivalent:
1. (1)
$z$ is movable to infinity with respect to $\xi$.
2. (2)
$i_{\ast}(z)=0\in H_{q}(X;\widehat{RG}_{-\xi})$, where
$i_{\ast}:H_{q}(\overline{X};R)\cong H_{q}(X;RG)\to
H_{q}(X;\widehat{RG}_{-\xi})$ is change of coefficients.
3. (3)
There is $\Delta\in RG$ with $\xi$-lowest coefficient 1 such that $\Delta\cdot
z=0$.
In the case that $R$ is a field, condition (3) is equivalent to the existence
of a non-zero $\Delta\in RG$ with $\Delta\cdot z=0\in H_{q}(\overline{X};R)$,
as we can find $r\in R$ and $g\in G$ such that $\Delta\cdot rg$ has
$\xi$-lowest coefficient 1.
Notice that $\Delta\in RG$ having $\xi$-lowest coefficient 1 is an open
condition in $\xi\in S(X,\overline{X})$; in particular, if $z$ is movable to
infinity with respect to $\xi$, it is also movable to infinity with respect to
nearby $\xi^{\prime}$, under the conditions of Theorem 9.
The equivalence of (1) and (2) is obtained by showing that the
${\lim\limits_{\leftarrow}}^{1}$-term in (15) vanishes. This is done in [15,
17] under the conditions of Theorem 9. Using Usher [31, Thm.1.3] one can see
that the ${\lim\limits_{\leftarrow}}^{1}$ term in (15) vanishes in fact for
any abelian covering and any Noetherian ring $R$.
Theorem 9 is an important ingredient in obtaining lower bounds for
${\rm{cat}}(X,\xi)$ and ${\rm{cat}}^{1}(X,\xi)$ via cup-lengths. We refer the
reader to [16, 17] for details.
## 9\. Function spaces of paths to infinity
If $\gamma:[0,\infty)\to X$ is a map, we can lift it to a map
$\tilde{\gamma}:[0,\infty)\to\tilde{X}$ into the universal cover. We want to
look at those maps, which lift to paths to $\infty$ with respect to a given
$\xi$, compare Definition 7. This does not depend on the particular lift of
$\gamma$.
We set
$\mathcal{C}_{\xi}(X)\,=\,\\{\gamma:[0,\infty)\to X\,|\,\tilde{\gamma}\mbox{
is a path to infinity with respect to }\xi\\}.$
It is equipped with the following topology: For $a,b\in[0,\infty)$ and $U$
open in $X$ let
$\displaystyle W(a,b;U)$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\gamma([a,b])\subset U\\}$
and for $a,A\in[0,\infty)$ let
$\displaystyle W(a,A)$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\forall t\geq
a\,\,\,h_{\xi}\tilde{\gamma}(t)-h_{\xi}\tilde{\gamma}(0)>A\\}.$
These sets form a subbasis for the topology of $\mathcal{C}_{\xi}(X)$. Notice
that the sets $W(a,b;U)$ provide the compact-open topology on
$\mathcal{C}_{\xi}(X)$ while the sets $W(a,A)$ give a ”control at infinity”.
The evaluation $e:\mathcal{C}_{\xi}(X)\to X$ given by $e(\gamma)=\gamma(0)$ is
a fibration and for $x_{0}\in X$ we have the fiber
$\displaystyle\mathcal{M}_{\xi}$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\gamma(0)=x_{0}\\},$
compare [21].
###### Remark 4.
If we consider $\mathcal{M}_{\xi}$ with the compact-open topology, we get that
$\mathcal{M}_{\xi}$ is contractible. To see this, choose a
$\gamma_{\infty}\in\mathcal{M}_{\xi}$. For any $\gamma\in\mathcal{M}_{\xi}$
and $t\in[0,\infty)$, let $\gamma_{t}\in\mathcal{M}_{\xi}$ be given by
$\displaystyle\gamma_{t}(s)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\gamma_{\infty}(s)&0\leq s\leq t\\\
\gamma_{\infty}(2t-s)&t\leq s\leq 2t\\\ \gamma(s-2t)&2t\leq
s\end{array}\right.$
It is easy to see that
$H:\mathcal{M}_{\xi}\times[0,\infty]\to\mathcal{M}_{\xi}$ given by
$H(\gamma,t)=\gamma_{t}$ is continuous in the compact-open topology, and hence
defines a contraction. Of course $H$ is no longer continuous with the $W(a,A)$
open in $\mathcal{M}_{\xi}$.
###### Remark 5.
Let us give another interpretation of the topology on $\mathcal{M}_{\xi}$. Let
$\tilde{X}_{\infty}=\tilde{X}\cup\\{\infty\\}$, that is, we add a point
$\infty$ to $\tilde{X}$; the topology on $\tilde{X}_{\infty}$ is generated by
the open sets of $\tilde{X}$ and sets $N\cup\\{\infty\\}$, where $N$ is an
open neighborhood of $\infty$ with respect to $\xi$ (in the sense of Section
2).
Write
$\displaystyle\mathcal{P}(\tilde{X}_{\infty})$ $\displaystyle=$
$\displaystyle\\{\gamma:[0,\infty]\to\tilde{X}_{\infty}\,|\,\gamma(\infty)=\infty\\},$
which we topologize with the compact open topology. This is a usual path space
with $\infty$ as the basepoint. The space $\mathcal{C}_{\xi}$ of paths to
infinity with respect to $\xi$ can be identified with the subspace of
$\mathcal{P}(\tilde{X}_{\infty})$ consisting of those $\gamma$ with
$\gamma([0,\infty))\subset\tilde{X}$. Similarly, $\mathcal{M}_{\xi}$ can be
identified with a subspace of
$\Omega(\tilde{X}_{\infty})=e^{-1}(\\{\tilde{x_{0}}\\})$, where
$\tilde{x_{0}}\in p^{-1}(\\{x_{0}\\})\subset\tilde{X}$ and
$e:\mathcal{P}(\tilde{X}_{\infty})\to\tilde{X}_{\infty}$ the usual fibration.
Note that $\mathcal{C}_{\xi}$ is a covering space of $\mathcal{C}_{\xi}(X)$
with covering group $\pi_{1}(X)$.
Given $\gamma_{0}\in\mathcal{M}_{\xi}$, we want to examine the homotopy groups
$\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$ for $k\geq 0$. For this let
$g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ be a map. It gives rise to a
map $\phi_{g}:S^{k}\times[0,\infty)\to X$ by $\phi_{g}(x,t)=g(x)(t)$ and since
$\phi_{g}(x,0)=x_{0}$, a map $\Phi_{g}:{\mathbb{R}}^{k+1}\to X$ such that
$\Phi_{g}(x\cdot t)=\phi(x,t)$ for $x\in S^{k}\subset{\mathbb{R}}^{k+1}$. If
we assume that $\ast\in S^{k}$ corresponds to
$(1,0,\ldots,0)\in{\mathbb{R}}^{k+1}$, we have
$\Phi_{g}(t,0,\ldots,0)=\gamma_{0}(t)$. Furthermore, if we lift $\Phi_{g}$ to
a map $\tilde{\Phi}_{g}:{\mathbb{R}}^{k+1}\to\tilde{X}$, we get that
$h_{\xi}\circ\tilde{\Phi}(x)\to\infty$, as $|x|\to\infty$. A homotopy between
two maps $g_{0},g_{1}:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ relative
to the basepoint corresponds to a homotopy
$\Phi:{\mathbb{R}}^{k+1}\times[0,1]\to X$ between $\Phi_{g_{0}}$ and
$\Phi_{g_{1}}$ relative to $[0,\infty)\times\\{0\\}$ and such that
$h_{\xi}\circ\tilde{\Phi}(x,s)\to\infty$ as $|x|\to\infty$ uniformly in
$s\in[0,1]$, for a lifting $\tilde{\phi}$.
Now assume we have a sequence $(N_{i})_{i\geq 0}$ of neighborhoods of $\infty$
with respect to $\xi$ such that $N_{i}\subset N_{i-1}$ for all $i$ and
$\bigcap_{i\geq 0}N_{i}=\emptyset$. Let $\tilde{x}_{0}\in\tilde{X}$ be a
lifting of $x_{0}\in X$. Also, let $\tilde{\gamma}_{0}:[0,\infty)\to\tilde{X}$
be the lifting of $\gamma_{0}$ with $\tilde{\gamma}_{0}(0)=\tilde{x}_{0}$.
Pick a sequence $t_{i}>0$ such that $t_{i+1}>t_{i}$ for all $i\geq 0$ such
that $\tilde{\gamma}_{0}(t)\in N_{i}$ for all $t\geq t_{i}$. This sequence
exists by the definition of $\gamma_{0}\in\mathcal{M}_{\xi}$. We let
$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}_{0}(t_{i})\in N_{i}$
be the sequence of basepoints of $N_{i}$ for all $i\geq 0$. We get a natural
homomorphism
$\chi_{i}:\pi_{k}(\tilde{X},N_{i+1},y_{i+1})\to\pi_{k}(\tilde{X},N_{i},y_{i})$
induced by inclusion where the change of basepoint is done using the path
$\tilde{\gamma}_{0}|[t_{i},t_{i+1}]$. This gives rise to an inverse system
$(\chi_{i}:\pi_{k}(\tilde{X},N_{i+1},y_{i+1})\to\pi_{k}(\tilde{X},N_{i},y_{i}))$
for every $k\geq 0$. Note that this is an inverse system of pointed sets for
$k\leq 1$.
For every $i\geq 0$ and $k\geq 0$ define
$\varphi_{i}:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\to\pi_{k+1}(\tilde{X},N_{i},y_{i})$
in the following way. Given $g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$,
we define $\varphi_{g}:(D^{k+1},S^{k},\ast)\to(\tilde{X},N_{i},y_{i})$ by
$\varphi_{g}(x\cdot t)=\tilde{\Phi}_{g}(\lambda(x)\cdot x\cdot t)$ for $x\in
S^{k}$ and $t\in[0,1]$, where $\lambda:S^{k}\to(0,\infty)$ is a map such that
$\lambda(\ast)=t_{i}$, and for every $x\in S^{k}$ we have $g(x)(t)\in N_{i}$
for all $t\geq\lambda(x)$. It is clear that the homotopy class of
$\varphi_{g}$ does not depend on the particular choice of $\lambda$. Basically
we use the map $\tilde{\Phi}_{g}$ and restrict it to a large enough ball in
${\mathbb{R}}^{k+1}$.
This induces the map
$\varphi_{i}:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\to\pi_{k+1}(\tilde{X},N_{i},y_{i})$
and we clearly have $\chi_{i}\varphi_{i+1}=\varphi_{i}$. Thus we get an
induced map
$\displaystyle\varphi:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$
$\displaystyle\longrightarrow$
$\displaystyle\lim\limits_{\leftarrow}\,\pi_{k+1}(\tilde{X},N_{i},y_{i})$
which is a group homomorphism for $k\geq 1$ and a map of pointed sets for
$k=0$.
Next we define a map $\psi^{\prime}:\prod_{i\geq
0}\pi_{k+2}(\tilde{X},N_{i},y_{i})\to\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$,
so let $a_{i}\in\pi_{k+2}(\tilde{X},N_{i},y_{i})$. Represent $a_{i}$ by a map
$g_{i}:(D^{k+2},S^{k+1},\ast)\to(\tilde{X},N_{i},y_{i})$. Furthermore, let
$G^{\prime}:{\mathbb{R}}^{k+1}\to\tilde{X}$ be given by $G^{\prime}(x\cdot
t)=\tilde{\gamma}_{0}(t)$ for $x\in S^{k}$ and $t\in[0,\infty)$. Let
$B(t_{i})$ be a small disc with center at $(-t_{i},0)\in{\mathbb{R}}^{k+1}$
such that the $B(t_{i})$ are pairwise disjoint. We use $(-t_{i},0)$ as the
center since we want to change $G^{\prime}$ on the $B(t_{i})$ without changing
it on $[0,\infty)\times\\{0\\}\subset{\mathbb{R}}^{k+1}$. So homotop
$G^{\prime}$ to a map $G$ relative to ${\mathbb{R}}^{k+1}-\bigcup
B^{\prime}(t_{i})$ such that $G$ is constant to $\tilde{\gamma}_{0}(t_{i})$ on
$B(t_{i})$ for all $i\geq 0$. Here $B^{\prime}(t_{i})$ is a slightly bigger
disc such that they are still pairwise disjoint.
If we restrict the map $g_{i}$ to $S^{k+1}$, we can think of this map as a map
$\tilde{g}_{i}:(D^{k+1},S^{k})\to(N_{i},y_{i})$. Now we can replace $G$ by a
map $\tilde{G}:{\mathbb{R}}^{k+1}\to\tilde{X}$ such that
$\tilde{G}|B(t_{i})\equiv\tilde{g}_{i}$ and $\tilde{G}$ agrees with $G$
everywhere else. Clearly $\tilde{G}$ induces a map
$g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ and this defines a map
$\displaystyle\psi^{\prime}:\prod_{i\geq 0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$
$\displaystyle\longrightarrow$
$\displaystyle\pi_{k}(\mathcal{M}_{\xi},\gamma_{0}).$
Note here that if
$g^{\prime}_{i}:(D^{k+2},S^{k+1},\ast)\to(\tilde{X},N_{i},y_{i})$ also
represents $a_{i}$ for every $i\geq 0$, we get that $g_{i}|S^{k+1}$ is
homotopic to $g^{\prime}_{i}|S^{k+1}$ within $N_{i}$, so the resulting maps
$g$ and $g^{\prime}$ represent the same element in
$\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$. It is worth pointing out that
$\tilde{G}$ is homotopic to $G$ relative to $[0,\infty)\times\\{0\\}$, but the
resulting function $H:S^{k}\times[0,1]\to\mathcal{M}_{\xi}$ need not be
continuous, since the maps $g_{i}$ are only null homotopic in $\tilde{X}$, but
not necessarily with control.
Let us recall the definition of the derived limit for our inverse system. Two
sequences $(a_{i}),(b_{i})\in\prod_{i\geq 0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$
are called equivalent, if there exists a sequence $(c_{i})\in\prod_{i\geq
0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$ such that $b_{i}=c_{i}\cdot
a_{i}\cdot\chi_{i}(c_{i+1})$ for all $i\geq 0$. Then
${\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})$ is the set
of equivalence classes. For $k\geq 1$ this has the structure of an abelian
group, but for $k=0$ we only get a pointed set.
It is easy to see that $\psi^{\prime}$ induces a map
$\displaystyle\psi:{\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})$
$\displaystyle\longrightarrow$
$\displaystyle\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$
which is a homomorphism for $k\geq 1$ and a map of pointed sets for $k=0$.
###### Proposition 20.
With the above notation there is a short exact sequence
$1\to{\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\lim\limits_{\leftarrow}\,\pi_{k+1}(\tilde{X},N_{i},y_{i})\to
1$
which is a short exact sequence of groups in the case $k\geq 1$ and of pointed
sets in the case $k=0$. If $k=0$, $\psi$ is also injective.
The proof is standard and will be omitted, see also [19]. Notice the
similarity between this sequence and (15).
We can use this to give another equivalent definition for
$\xi\in\Sigma^{k}(X)$.
###### Proposition 21.
Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $k\geq 1$. Then $\xi\in\Sigma^{k}(X)$ if and only if
$\mathcal{M}_{\xi}$ is $(k-1)$-connected.
###### Proof.
If $\mathcal{M}_{\xi}$ is $(k-1)$-connected, then by Proposition 20 we get
that the inverse system $\\{\pi_{l}(\tilde{X},N)\\}$ is pro-trivial for $l\leq
k$ which gives $\xi\in\Sigma^{k}(X)$ by Proposition 2.
To get the other direction, we have to worry about
${\lim\limits_{\leftarrow}}^{1}\,\pi_{k+1}(\tilde{X},N_{i})$. But by the next
lemma $\\{\pi_{k+1}(\tilde{X},N)\\}$ is semi-stable, so the
${\lim\limits_{\leftarrow}}^{1}$-term vanishes. It follows from Proposition 20
that $\mathcal{M}_{\xi}$ is $(k-1)$-connected. ∎
###### Lemma 22.
Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a height function. If
$\xi\in\Sigma^{k}(X)$, then $\\{\pi_{k+1}(\tilde{X},N)\\}$ is semi-stable.
###### Proof.
Use the homotopy from Proposition 2(4) to push any $k$-sphere in $N$
arbitrarily far away, this gives semi-stability as the pushing may be done in
$N^{\prime}$ slightly bigger than $N$ (depends on $H$ only). ∎
## Appendix A Sigma invariants of chain complexes
In this appendix we show how $\Sigma^{k}(C)$ is related to criteria involving
chain homotopies on $C$.
If the chain complex $C$ consists of flat $R$-modules, we have the following
criterion.
###### Proposition 23.
Let $C$ be a chain complex of flat $R$-modules and $n$ a non-negative integer.
Then the following are equivalent.
1. (1)
$C$ is of finite $n$-type.
2. (2)
For every index set $J$, the natural map
$H_{k}(C,\prod_{J}R)\to\prod_{J}H_{k}(C)$ is an isomorphism for $k<n$ and an
epimorphism for $k=n$.
This is basically [8, Thm.2], but we allow $C$ to be flat and not necessarily
projective so we only have a homology criterion. The proof goes through for
flat modules.
By a filtration of $C$ we mean a family
$\\{C^{\alpha}\\}_{\alpha\in\mathcal{A}}$ of sub-chain complexes where
$\mathcal{A}$ is a directed set, $C^{\alpha}\subset C^{\beta}$ for
$\alpha\leq\beta$ and $C=\bigcup C^{\alpha}$.
Given a filtration, we define $D^{\alpha}=C/C^{\alpha}$.
The analogue of [10, Thm.2.2] is
###### Theorem 10.
Let $C$ be an $R$-chain complex with a filtration
$\\{C^{\alpha}\\}_{\alpha\in\mathcal{A}}$ of finite $n$-type complexes
$C^{\alpha}$ of flat $R$-modules. Then $C$ is of finite $n$-type if and only
if the direct system $\\{H_{i}(D^{\alpha})\\}$ is essentially trivial for
$i\leq n$.
###### Proof.
The proof is similar to the proof of [10, Thm.2.2], but one has to use
Proposition 23. We omit the details. ∎
Let $G$ be a finitely generated group and $\xi:G\to{\mathbb{R}}$ a non-zero
homomorphism. Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is
finitely generated in every degree. Given a valuation on $C$ extending $\xi$,
we can define a subcomplex
$\displaystyle C^{\xi}$ $\displaystyle=$ $\displaystyle\\{x\in C\,|\,v(x)\geq
0\\}$
Valuations are determined by their value on basis elements, so it is easy to
see that $C^{\xi}$ is a finitely generated free chain complex over
${\mathbb{Z}}G_{\xi}$, compare [7, Lemma 3.1]. Given $g\in G$, we can also
look at the subcomplex $gC^{\xi}\subset C$ which is isomorphic to $C^{\xi}$.
Denote $D^{g}=C/gC^{\xi}$.
We use Theorem 10 to get
###### Proposition 24.
Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is finitely generated in
every degree, $v$ a valuation extending $\xi$ and $n$ a non-negative integer.
Then the following are equivalent.
1. (1)
$\xi\in\Sigma^{n}(C)$.
2. (2)
The direct system $\\{H_{i}(D^{g})\\}$ is essentially trivial for $i\leq n$.
3. (3)
There exists a chain map $\varphi:C\to C$ chain homotopic to the identity such
that $v(\varphi(x))>v(x)$ for all non-zero $x\in C_{i}$ with $i\leq n$.∎
## References
* [1]
* [2] M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445-470.
* [3] R. Bieri, Finiteness length and connectivity length for groups, in: Geometric group theory down under (Canberra, 1996), 9-22, de Gruyter, Berlin, 1999.
* [4] R. Bieri, R. Geoghegan, Connectivity properties of group actions on non-positively curved spaces, Mem. Amer. Math. Soc. 161 (2003), no. 765.
* [5] R. Bieri, R. Geoghegan, D. Kochloukova, The Sigma invariants of Thompson’s group $F$, preprint.
* [6] R. Bieri, W. Neumann, R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), 451-477.
* [7] R. Bieri, B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63 (1988), 464-497.
* [8] K. Brown, Homological criteria for finiteness, Comment. Math. Helv. 50 (1975), 129-135.
* [9] K. Brown, Cohomology of groups, Springer-Verlag, New York-Berlin, 1982.
* [10] K. Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), 45-75.
* [11] M. Farber, Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory, Topol. Methods Nonlinear Anal. 19 (2002), 123-152.
* [12] M. Farber, Lusternik-Schnirelman theory and dynamics, ”Lusternik-Schnirelmann Category and Related Topics”, Contemporary Mathematics, 316(2002), 95 - 111.
* [13] M. Farber, Topology of closed one-forms, Mathematical Surveys and Monographs, AMS, 2004.
* [14] M. Farber, T. Kappeler, Lusternik-Schnirelman theory and dynamics, II, Proc. Steklov Inst. Math. 2004, no. 4 (247), 232–245.
* [15] M. Farber, D. Schütz, Moving homology classes to infinity, Forum Math. 19 (2007), 281-296.
* [16] M. Farber, D. Schütz, Cohomological estimates for $cat(X,\xi)$, Geom. Top. 11 (2007), 1255-1289.
* [17] M. Farber, D. Schütz, Homological category weights and estimates for ${\rm{cat}}^{1}(X,\xi)$, J. Eur. Math. Soc. 10 (2008), 243-266.
* [18] M. Farber, D. Schütz, Closed 1-forms in topology and dynamics, preprint.
* [19] R. Geoghegan, Topological methods in group theory, Springer-Verlag, New York, 2008.
* [20] J. Harlander, D. Kochloukova, The $\Sigma^{3}$-conjecture for metabelian groups, J. London Math. Soc. (2) 67 (2003), 609-625.
* [21] F. Latour, Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rham, Publ. IHES 80 (1994), 135-194.
* [22] J. Meier, H. Meinert, L. van Wyk, Higher generation subgroup sets and the $\Sigma$-invariants of graph groups, Comment. Math. Helv. 73 (1998), 22-44.
* [23] S. P. Novikov, Multi-valued functions and functionals. An analogue of Morse theory, Soviet Math. Doklady, 24(1981), pp. 222–226.
* [24] S.P. Novikov, The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russian Math. Surveys 37 (1982), 1-56.
* [25] S. P. Novikov, Variational methods and periodic solutions of equations of Kirchhoff type, II, Functional Analysis and Its Applications, 15(1981), pp. 263 - 274.
* [26] S. P. Novikov, I. Smel’tser, Periodic solutions of the Kirhhoff equations for the free motions of a rigid body in a liquid, and extended Lusternik-Schnirelman-Morse theory, I Functional Analysis and Its Applications, 15(1981), pp. 197 - 207.
* [27] A. Ranicki, The algebraic theory of finiteness obstruction, Math. Scand. 57 (1985), 105-126.
* [28] D. Schütz, Finite domination, Novikov homology and nonsingular closed 1-forms, Math. Z. 252 (2006), 623-654.
* [29] J.-Cl. Sikorav, Thèse, Université Paris-Sud, 1987.
* [30] E.H. Spanier, Algebraic topology. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966.
* [31] M. Usher, Spectral numbers in Floer theories, preprint, available as `arXiv:0709.1127`
* [32] C.T.C. Wall, Finiteness conditions for ${\rm CW}$-complexes, Ann. of Math. (2) 81 (1965), 56-69.
* [33] C.T.C. Wall, Finiteness conditions for CW-complexes II, Proc. Roy. Soc. Ser. A 295 (1966), 129-139.
|
arxiv-papers
| 2008-10-06T15:34:13
|
2024-09-04T02:48:58.143093
|
{
"license": "Public Domain",
"authors": "Michael Farber, Ross Geoghegan and Dirk Schuetz",
"submitter": "Michael Farber",
"url": "https://arxiv.org/abs/0810.0962"
}
|
0810.1017
|
# On linear resolution of powers of an ideal
Keivan Borna Faculty of Mathematical Sciences and Computer, Tarbiat Moallem
University, Tehran, Iran borna@ipm.ir
###### Abstract.
In this paper we give a generalization of a result of Herzog, Hibi, and Zheng
providing an upper bound for regularity of powers of an ideal. As the main
result of the paper, we give a simple criterion in terms of Rees algebra of a
given ideal to show that high enough powers of this ideal have linear
resolution. We apply the criterion to two important ideals $J,J_{1}$ for which
we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if
$k\neq 2.$ The procedures we include in this work is encoded in computer
algebra package CoCoA [3].
###### Key words and phrases:
Castelnuovo-Mumford regularity, powers of ideals, Rees algebra
###### 2000 Mathematics Subject Classification:
Primary 13D02; Secondary 13P10
The author is grateful to Dipartimento di Matematica, Universitá di Genova,
Italia
## 1\. Introduction
Let $S=K[x_{1},\cdots,x_{r}]$ and let
$\displaystyle\mathbb{F}:\cdots\rightarrow F_{i}\rightarrow
F_{i-1}\rightarrow\cdots$
be a graded complex of free $S$-modules, with $F_{i}=\sum_{j}S(-a_{i,j})$. The
Castelnuovo-Mumford regularity, or simply regularity, of $\mathbb{F}$ is the
supremum of the numbers $a_{i,j}-i.$ The regularity of a finitely generated
graded S-module M is the regularity of a minimal graded free resolution of M.
We will write $\rm{reg}(M)$ for this number. The regularity of an ideal is an
important measure of how complicated the ideal is. The above definition of
regularity shows how the regularity of a module governs the degrees appearing
in a minimal resolution. As Eisenbud mentions in [8] Mumford defined the
regularity of a coherent sheaf on projective space in order to generalize a
classic argument of Castelnuovo. Mumford’s definition [12] is given in terms
of sheaf cohomology. The definition for modules, which extends that for
sheaves, and the equivalence with the condition on the resolution used above
definition, come from Eisenbud and Goto [9]. Alternate formulations in terms
of $\rm{Tor}$, $\rm{Ext}$ and local cohomology are given in the following. Let
$I$ be a graded ideal, $\mathfrak{m}=(x_{1},\cdots,x_{r})$ the maximal ideal
of $S$, and $n=\dim(S/I).$ Let
$a_{i}(S/I)=\max\\{t;\mbox{H}^{i}_{\mathfrak{m}}(S/I)_{t}\neq 0\\},0\leq i\leq
n,$
where $\mbox{H}^{i}_{\mathfrak{m}}(S/I)$ is the $i$th local cohomology module
with the support in $\mathfrak{m}$ (with the convention
$\max\,\emptyset=-\infty$). Then the regularity is the number
$\rm{reg}(S/I)=\max\\{a_{i}(S/I)+i;0\leq i\leq n\\}.$
Note that $\rm{reg}(I)=\rm{reg}(S/I)+1.$ We may also compute $\rm{reg}(I)$ in
terms of $\rm{Tor}$ by the formula
$\rm{reg}(I)=\max_{k}\\{t_{k}(I)-k\\},$
where $t_{p}(I):=\max\\{\mbox{degree of the minimal}\,p-\mbox{th syzygies
of}\,I\\}.$ Simply this definition may be rewritten as
$\displaystyle\begin{split}\rm{reg}(I)=&\max_{i,j}\\{j-i:\rm{Tor}_{i}(I,k)_{j}\neq
0\\},\\\ =&\max_{i,j}\\{j-i;\beta_{i,j}(I)\neq 0\\}.\end{split}$
Anyway, from local duality one see that the two ways of expressing the
regularity are also connected termwise by the inequality $t_{k}(I)-k\geq
a_{i}(S/I)+n-k.$ Regularity is a kind of universal bound for important
invariants of graded algebras, such as the maximum degree of the syzygies and
the maximum non-vanishing degree of the local cohomology modules. One has
often tried to find upper bounds for the Castelnuovo-Mumford regularity in
terms of simpler invariants which reflect the complexity of a graded algebra
like dimension and multiplicity. Clearly $t_{0}(I^{k})\leq k\,t_{0}(I)$ and
one may expect to have the same inequality for regularity, that is,
$\rm{reg}(I^{k})\leq k\,\rm{reg}(I)$. Unfortunately this is not true in
general. However, in [6] Cutkosky, Herzog, and Trung and in [11] Kodiyalam
studied the asymptotic behavior of the Castelnuovo-Mumford regularity and
independently showed that the regularity of $I^{k}$ is a linear function for
large $k$, i.e.,
(1.1) $\displaystyle\rm{reg}(I^{k})=a(I)k+b(I),\quad\forall k\geq c(I).$
Now assume that $I$ is an equigenerated ideal, that is, generated by forms of
the same degree $d$. Then one has $a(I)=d$ and hence,
$\rm{reg}(I^{k+1})-\rm{reg}(I^{k})=d$ for all $k\geq c(I)$. Hence we have
(1.2) $\displaystyle\rm{reg}(I^{k})=(k-c(I))d+\rm{reg}(I^{c(I)}),\quad\forall
k\geq c(I).$
One says that the regularity of the powers of $I$ jumps at place $k$ if
$\rm{reg}(I^{k})-\rm{reg}(I^{k-1})>d.$ In [4] the author gives several
examples of ideals generated in degree $d$ ($d=2,3$), with linear resolution
(i.e., $\rm{reg}(I)=d$), and such that the regularity of the powers of $I$
jumps at place $2$, i.e., such that $\rm{reg}(I^{2})>2d.$ As it is indicated
in [4], the first example of such an ideal was given by Terai. Throughout this
paper we use $J$ for this ideal. Geometrically speaking, this is an example of
Reisner which corresponds to the (simplicial complex of a) triangulation of
the real projective plane $\mathbb{P}^{2}$; see Fig. 1 and [2] for more
details. Let $R:=K[x_{1},\cdots,x_{6}]$ one has
(1.3)
$\displaystyle\begin{split}J=(&x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{3}x_{5},x_{1}x_{4}x_{6},x_{1}x_{5}x_{6},x_{2}x_{3}x_{6},x_{2}x_{4}x_{5},x_{2}x_{5}x_{6},x_{3}x_{4}x_{5},\\\
&x_{3}x_{4}x_{6}).\end{split}$
It is known that $J$ is a square-free monomial ideal whose Betti numbers,
regularity and projective dimension depend on the characteristic of the base
field. Indeed whenever $\rm{char}(K)\neq 2,$ $R/J$ is Cohen-Macaulay (and
otherwise not), moreover one has $\rm{reg}(J)=3$ and
$\rm{reg}(J^{2})=7\,(\text{which is of course}>2\times 3).$ If
$\rm{char}(K)=2,$ then $J$ itself has no linear resolution. So the following
natural question arises:
### Question A
How it goes on for the regularity of powers of $J$?
Figure 1. The ideal of triangulation of the real projective plane
$\mathbb{P}^{2}$.
By the help of (1.1) we are able to write $\rm{reg}(J^{k})=3k+b(J),\,\forall
k\geq c(J)$. But what are $b(J)$ and $c(J)$? In this paper we give an answer
to this question and prove that $J^{k}$ has linear resolution (in
$\rm{char}(K)=0$) $\forall k\neq 2$, that is, $b(J)=0$ and $c(J)=3$. That is
$\displaystyle\rm{reg}(J^{k})=3k,\quad\forall k\neq 2.$
To answer Question A we develop a general strategy and to this end we need to
follow the literature a little bit. In [13] Römer proved that
(1.4) $\displaystyle\rm{reg}(I^{n})\leq nd+\rm{reg}_{x}(R(I)),$
where $R(I)$ is the Rees ring of $I,$ which is naturally bigraded, and
$\rm{reg}_{x}$ refers to the x-regularity of $R(I)$, that is,
$\rm{reg}_{x}(R(I))=\max\\{b-i:\rm{Tor}_{i}(R(I),K)_{(b,d)}=0\\},$
as defined by Aramova, Crona and De Negri [1]. In Section 2 we study Rees
rings and their bigraded structure in more details. It follows from (1.4) that
if $\rm{reg}_{x}(R(I))=0$, then each power of $I$ admits a linear resolution.
Based on Römer’s formula, in [10, Theorem 1.1 and Corollary 1.2] Herzog, Hibi
and Zheng showed the following:
###### Theorem 1.1.
Let $I\subseteq K[x_{1},\cdots,x_{n}]:=S$ be an equigenerated graded ideal.
Let $m$ be the number of generators of $I$ and let $T:=S[t_{1},\cdots,t_{m}],$
and let $R(I)=T/P$ be the Rees algebra associated to $I$. If for some term
order $<$ on $T,$ $P$ has a Gröbner basis $G$ whose elements are at most
linear in the variables $x_{1},\cdots,x_{n},$ that is $\deg_{x}(f)\leq 1$ for
all $f\in G,$ then each power of $I$ has a linear resolution.
Throughout this paper we simply write $S=K[\mathrm{\underline{x}}]$ and
$T=S[\mathrm{\underline{t}}].$ One can easily see that for $J,$ (1.3), one has
at least 3 elements in $\rm{in}(P)$ with $deg_{x}>1$, no matter if we take
initial ideal w.r.t. term ordering
$\mathrm{\underline{x}}>\mathrm{\underline{t}}$ or
$\mathrm{\underline{t}}>\mathrm{\underline{x}}$ in either Lex or DegRevLex
order as it is reported in Table 1. Note that for example if one starts in
DegRevLex order and $\mathrm{\underline{x}}>\mathrm{\underline{t}}$ then there
is 4 elements in $\rm{in}(P)$ which have x-degree $>1$ ($=2\,$ actually) and
among them 2 term has t-degree 1 and 2 term is in t-degree 2.
| $\mathrm{\underline{x}}>\mathrm{\underline{t}}$ | $\mathrm{\underline{t}}>\mathrm{\underline{x}}$
---|---|---
DegRevLex | (1,2):2,(2,2):2 | (1,2):2,(2,2):1
Lex | (1,2):2,(2,2):1 | (1,2):2,(2,2):1
Table 1. Count of elements of $\rm{in}(P)$ with $deg_{x}>1$ for the ideal of
(1.3).
The main motivation for our work is to generalize Herzog, Hibi and Zheng’s
techniques in order to apply them to a wider class. Furthermore, we will
indicate the least exponent $k_{0}$ for which $I^{k}$ has linear resolution
for all $k\geq k_{0}$. Indeed our generalization works for all ideals which
admit the following condition:
###### Theorem 1.2.
Let $Q\subseteq S=K[x_{1},\cdots,x_{r}]$ be a graded ideal which is generated
by m polynomials all of the same degree $d$, and let $I=\rm{in}(g(P))$ for
some linear bi-transformation $g\in\rm{GL}_{r}(K)\times\rm{GL}_{m}(K)$. Write
$I=G+B$ where $G$ is generated by elements of $\deg_{x}\leq 1$ and $B$ is
generated by elements of $\deg_{x}>1$. If $I_{(k,j)}=G_{(k,j)}$ for all $k\geq
k_{0}$ and for all $j\in\mathbb{Z}$, then $Q^{k}$ has linear resolution for
all $k\geq k_{0}$. In other words, $\rm{reg}(Q^{k})=kd$ for all $k\geq k_{0}$.
Another motivation for our paper is an example that Conca considered in [4].
###### Example 1.3.
Let $J_{1}$ be the ideal of 3-minors of a $4\times 4$ symmetric matrix of
linear forms in 6 variables, that is, 3-minors of
$\left[\begin{array}[]{cccc}0&x_{1}&x_{2}&x_{3}\\\ x_{1}&0&x_{4}&x_{5}\\\
x_{2}&x_{4}&0&x_{6}\\\ x_{3}&x_{5}&x_{6}&0\\\ \end{array}\right].$
As an ideal of $S=\mathbb{Q}[x_{1},\cdots,x_{6}]$ one has:
(1.5)
$\displaystyle\begin{split}J_{1}:&=(2x_{1}x_{2}x_{4},2x_{1}x_{3}x_{5},2x_{2}x_{3}x_{6},2x_{4}x_{5}x_{6},x_{1}x_{3}x_{4}+x_{1}x_{2}x_{5}-x_{1}^{2}x_{6},x_{3}x_{4}x_{6}+\\\
&x_{2}x_{5}x_{6}-x_{1}x_{6}^{2},-x_{2}x_{3}x_{4}+x_{2}^{2}x_{5}-x_{1}x_{2}x_{6},-x_{3}^{2}x_{4}+x_{2}x_{3}x_{5}+x_{1}x_{3}x_{6},-x_{3}x_{4}^{2}+\\\
&x_{2}x_{4}x_{5}+x_{1}x_{4}x_{6},-x_{3}x_{4}x_{5}+x_{2}x_{5}^{2}-x_{1}x_{5}x_{6}).\end{split}$
As Conca mentioned in his paper [4, Remark 3.6] and as we will show in this
paper, the ideals $J,J_{1}$ are very closely related. For instance, we prove
that
$\displaystyle\rm{reg}(J_{1}^{k})=3k,\quad\forall k\neq 2.$
Similar to the ideal of (1.3), one can easily check that $\rm{in}(P_{1})$,
where $P_{1}$ is the associated ideal to Rees ring of $J_{1}$, has at least 9
elements with $deg_{x}>1$, no matter if we take initial ideal w.r.t. term
ordering $\mathrm{\underline{x}}>\mathrm{\underline{t}}$ or
$\mathrm{\underline{t}}>\mathrm{\underline{x}}$ in Lex or DegRevLex order; see
Table 2 for more details.
| $\mathrm{\underline{x}}>\mathrm{\underline{t}}$ | $\mathrm{\underline{t}}>\mathrm{\underline{x}}$
---|---|---
DegRevLex | (1,2):6,(2,2):5,(1,3):1,(4,2):1 | (1,2):6,(2,2):3,(1,3):1
Lex | (1,2):6,(2,2):3 | (1,2):6,(2,2):5
Table 2. Count of elements of $\rm{in}(P_{1})$ with $deg_{x}>1$ for
$J_{1}$,(1.5).
We also show that $J$ and $J_{1}$ and their powers have the same Hilbert
series ($\rm{HS}$ for short) correspondingly:
$\rm{HS}(S/J^{k})=\rm{HS}(S/J_{1}^{k}),\quad\forall k.$
Indeed we have computed the multigraded Hilbert series of the corresponding
ideals to the Rees algebra of $J$ and $J_{1}$ and observed that they are the
same. As a result we conclude that all of the powers of $J$ and $J_{1}$ have
the same graded Betti numbers as well:
$\beta_{i,j}(J^{k})=\beta_{i,j}(J_{1}^{k}),\quad\forall i,j,\forall k.$
## 2\. Main results
Let $K$ be a field, $I=(f_{1},\cdots,f_{m})$ be a graded ideal of
$S=K[x_{1},\cdots,x_{r}]$ generated in a single degree, say $d$. The Rees
algebra of $I$ is known to be
$R(I)=\bigoplus_{j\geq 0}I^{j}t^{j}=S[f_{1}t,\cdots,f_{m}t]\subseteq S[t].$
Let $T=S[t_{1},\cdots,t_{m}]$. Then there is a natural surjective homomorphism
of bigraded $K$-algebras $\varphi:T\longrightarrow R(I)$ with
$\varphi(x_{i})=x_{i}$ for $i=1,\cdots,r$ and $\varphi(y_{j})=f_{j}t$ for
$j=1,\cdots,m.$ So one can write $R(I)=T/P$. In this paper we consider $T$,
and so $R(I)$, as a standard bigraded polynomial ring with $\deg(x_{i})=(0,1)$
and $\deg(t_{j})=(1,0)$. Indeed if we start with the natural bigraded
structure $\deg(x_{i})=(0,1)$ and $\deg(f_{j}t)=(d,1)$ then
$R(I)_{(k,vd)}=(I^{k})_{vd}$, but the standard bidegree normalizes the
bigrading in the following sense:
(2.1) $\displaystyle R(I)_{(k,j)}=(I^{k})_{kd+j}$
For each $k\in\mathbb{Z}$ we define a functor $F_{k}$ from the category of
bigraded $T$-modules to the category of graded $S$-modules with bigraded maps
of degree zero. Let $M$ be a bigraded $T$-module, define
$F_{k}(M)=\bigoplus_{j\in\mathbb{Z}}M_{(k,j)},$
obviously $F_{k}$ is an exact functor and associates to each free
$K[{\mathrm{\underline{x}}},{\mathrm{\underline{t}}}]$-module a free
$K[{\mathrm{\underline{x}}}]$-module. Sometimes we simply write
$M_{(k,\star)}$ instead of $F_{k}(M)$. Using (2.1) we get
(2.2)
$\displaystyle[T/P]_{(k,\star)}=R(I)_{(k,\star)}=\bigoplus_{j\in\mathbb{Z}}R(I)_{(k,j)}=\bigoplus_{j\in\mathbb{Z}}(I^{k})_{kd+j}=I^{k}(kd),$
which provides the link between $I$ and its Rees ring $R(I)$. In the sequel we
need to know what is $F_{k}(T(-a,-b))$. For the convenience of reader we
provide a proof.
###### Remark 2.1.
For each integer $k$ we have
(2.5) $\displaystyle T(-a,-b)_{(k,\star)}=\left\\{\begin{array}[]{c
l}0&\text{if $k<a$},\\\ S(-b)^{N}&\text{otherwise}.\end{array}\right.$
Where
$N:=\\#\\{{\mathrm{\underline{t}}}^{\alpha}:|\alpha|=k-a\\}=\binom{m-1+k-a}{m-1}.$
###### Proof.
(2.6)
$\displaystyle\begin{split}T(-a,-b)_{(k,\star)}&=\bigoplus_{j\in\mathbb{Z}}T(-a,-b)_{(k,j)}=\bigoplus_{j\in\mathbb{Z}}T_{(k-a,j-b)}\\\
&=\bigoplus_{j\in\mathbb{Z}}<\mathbf{\underline{t}}^{\alpha}\mathbf{\underline{x}}^{\beta}:\,|\alpha|=k-a,|\beta|=j-b>,\end{split}$
where the last equality is as vector spaces. From (2.6) the proof is immediate
when $k<a.$ Considering as an $S=K[{\mathrm{\underline{x}}}]$-module the last
module in (2.6) is free. Since $|\beta|=j-b$ could be any integer where $j$
changes over $\mathbb{Z}$, a shift by $-b$ is required for the representation
of the graded free module $T(-a,-b)_{(k,\star)}$ and finally the proposed $N$
will take care of the required copies. ∎
Note that in the spacial case $a=b=0,$ we have
(2.7) $\displaystyle T_{(k,\star)}=S^{\binom{m-1+k}{m-1}}.$
As we mentioned in Introduction, Theorem 1.1 is subject to condition that
$\rm{in}(P)=(u_{1},\cdots,u_{m})$ and $\deg_{x}(u_{i})\leq 1$. So the natural
way to generalize it is to change the upper bound for x-degree of $u_{i}$ with
some number $t.$ As one may expect, we end up with $\rm{reg}(I^{n})\leq
nd+(t-1)\,\rm{pd}(T/\rm{in}(P))$. The proof is mainly as that of Theorem 1.1
but for the convenience of reader we bring it here.
###### Proposition 2.2.
Let $I\subseteq S$ be an equigenerated graded ideal and let $R(I)=T/P$. If
$\rm{in}(P)=(u_{1},\cdots,u_{m})$ and $\deg_{x}(u_{i})\leq t$, then
$\rm{reg}(I^{n})\leq nd+(t-1)\,\rm{pd}(T/\rm{in}(P))$.
###### Proof.
Let $C_{\bullet}$ be the Taylor resolution of $\rm{in}(P)$. The module $C_{i}$
has the basis $e_{\sigma}$ with
$\sigma={j_{1}<j_{2}<\cdots<j_{i}}\subseteq[m]$. Each basis element
$e_{\sigma}$ has the multidegree $(a_{\sigma},b_{\sigma})$ where
$x^{a_{\sigma}}.y^{b_{\sigma}}=\rm{lcm}\\{u_{j_{1}},\cdots,u_{j_{m}}\\}$. It
follows that $\deg_{x}(e_{\sigma})\leq ti$ for all $e_{\sigma}\in C_{i}$.
Since the shifts of $C_{\bullet}$ bound the shifts of a minimal multigraded
resolution of in(P ), we conclude that
$\displaystyle\rm{reg}_{x}(T/P)\leq\rm{reg}_{x}(T/\rm{in}(P))$
$\displaystyle=\max_{i,j}\\{a_{ij}-i\\}$ $\displaystyle\leq ti-i=(t-1)i$
$\displaystyle\leq(t-1)\,\rm{pd}(T/\rm{in}(P)).$
Now (1.4) completes the proof. ∎
One can see that now Theorem 1.1 is the special case of Proposition 2.2 with
$t=1.$ However, this approach seems to be less effective. Our approach to
generalize Theorem 1.1 is to change $P$ with an isomorphic image $g(P)$ so
that $\rm{in}(g(P))_{(k,\star)}$ only consists of terms with x-degree$\leq$ 1,
for some $k$. To this end, we need a simple fact.
Let $<$ be any term order on $S=K[\mathrm{\underline{x}}]$ and let $V\subseteq
S$ be a $K$-vector space. Then with respect to the monomial order on $S$
obtained by restricting $<$, by definition $V$ is homogenous if for any
element $f$ of $V$, $f=\displaystyle\sum_{i=0}^{n}f_{i}$, where $f_{i}$ is an
element of $S$ of degree $i$, we have $f_{i}\in V,\,\forall i=0,\cdots,n.$
That is to say
$V=\displaystyle\bigoplus_{i=0}^{\infty}V_{i}\,,\,V_{i}=V\bigcap S_{i}.$ It
yields that $\rm{in}(V)=\displaystyle\bigoplus_{i=0}^{\infty}\rm{in}(V_{i})$
and so, $\rm{in}(V)_{i}=\rm{in}(V_{i}).$ Generalizing this idea to bigraded
(or multigraded) situation is also well understood. Let $F$ be a free
$S$-module with a fixed basis and $M$ a bigraded subvector space of it. Then
$\rm{in}(M)_{(i,j)}=\rm{in}(M_{(i,j)}),$
and so
(2.8)
$\displaystyle\rm{in}(M)_{(k,\star)}:=\displaystyle\bigoplus_{j\in\mathbb{Z}}\rm{in}(M)_{(k,j)}=\displaystyle\bigoplus_{j\in\mathbb{Z}}\rm{in}(M_{(k,j)})=\rm{in}(M_{(k,\star)}).$
See [7] chapter 15.2 for more details. Furthermore since
$\beta_{ij}^{S}(F/M)\leq\beta_{ij}^{S}(F/\rm{in}(M)),$ it is easy to conclude
with
(2.9) $\displaystyle\rm{reg}(F/M)\leq\rm{reg}(F/\rm{in}(M)).$
###### Lemma 2.3.
Let $P$ be the associated ideal of Rees ring $R(I)$ and let $T=R/P$. Then
$\rm{reg}([T/P]_{(k,\star)})\leq\rm{reg}([T/\rm{in}(P)]_{(k,\star)}).$
###### Proof.
Since $P$ is a naturally bigraded ideal of T, and since easily $T_{(k,\star)}$
is a free $S$-module (see (2.7)), (2.8) implies that
$\rm{in}(P)_{(k,\star)}=\rm{in}(P_{(k,\star)}).$ Applying (2.9) for
$F:=T_{(k,\star)}$ and $M:=P$ we obtain
$\rm{reg}(T_{(k,\star)}/P_{(k,\star)})\leq\rm{reg}(T_{(k,\star)}/\rm{in}(P_{(k,\star)})).$
Finally putting all together we get the required inequality.
$\displaystyle\begin{split}\rm{reg}([T/P]_{(k,\star)})=\rm{reg}(T_{(k,\star)}/P_{(k,\star)})&\leq\rm{reg}(T_{(k,\star)}/\rm{in}(P_{(k,\star)}))\\\
&=\rm{reg}(T_{(k,\star)}/\rm{in}(P)_{(k,\star)})\\\
&=\rm{reg}([T/\rm{in}(P)]_{(k,\star)}).\end{split}$
∎
In the following the proof of Theorem 1.2 is given.
###### Proof.
First of all notice that, since
$g:K[\mathrm{\underline{x}},\mathrm{\underline{t}}]\longrightarrow
K[\mathrm{\underline{x}},\mathrm{\underline{t}}]$ is an invertible bi-
homogenous transformation, we have the following bi-homogenous isomorphism:
$\frac{K[\mathrm{\underline{x}},\mathrm{\underline{t}}]}{P}\simeq\frac{K[\mathrm{\underline{x}},\mathrm{\underline{t}}]}{g(P)},$
and so we can simply take $g=id$ in the rest of proof. Write down the so-
called Taylor resolution of $T/G$:
(2.15)
$\displaystyle\begin{array}[]{ccccccccccc}\\!&\\!&F_{2,0}&\\!&\\!&\\!&\\!&\\!&\\!&\\!&\\!\\\
\\!&\\!&\bigoplus&\\!&F_{1,0}&\\!&\\!&\\!&\\!&\\!&\\!\\\
\cdots&\longrightarrow&F_{2,1}&\longrightarrow&\bigoplus&\longrightarrow&T&\longrightarrow&T/G&\longrightarrow&0,\\\
\\!&\\!&\bigoplus&\\!&F_{1,1}&\\!&\\!&\\!&\\!&\\!&\\!\\\
\\!&\\!&F_{2,2}&\\!&\\!&\\!&\\!&\\!&\\!&\\!&\\!\\\ \end{array}$
where $F_{i,j}=\bigoplus_{a\in\mathbb{Z}}T(-a,-j)^{\beta_{i,(a,j)}(T/G)}.$
Note that $\beta_{i,(a,j)}(T/G),$ is an integer number which depends on $i$,
$a$, and $j$. Since $(k,\star)$ is an exact functor, the following complex of
$K[\mathrm{\underline{x}}]$-modules is exact:
(2.21)
$\displaystyle\begin{array}[]{ccccccccccc}\\!&\\!&(F_{2,0})_{(k,\star)}&\\!&\\!&\\!&\\!&\\!&\\!&\\!&\\!\\\
\\!&\\!&\bigoplus&\\!&(F_{1,0})_{(k,\star)}&\\!&\\!&\\!&\\!&\\!&\\!\\\
\cdots&\longrightarrow&(F_{2,1})_{(k,\star)}&\longrightarrow&\bigoplus&\longrightarrow&T_{(k,\star)}&\longrightarrow&[T/G]_{(k,\star)}&\longrightarrow&0.\\\
\\!&\\!&\bigoplus&\\!&(F_{1,1})_{(k,\star)}&\\!&\\!&\\!&\\!&\\!&\\!\\\
\\!&\\!&(F_{2,2})_{(k,\star)}&\\!&\\!&\\!&\\!&\\!&\\!&\\!&\\!\\\ \end{array}$
Using formula (2.5) we obtain $T(-a,-b)_{(k,\star)}=S(-b)^{N_{a,k}}$, so for
$F_{i,j}$ we get
(2.22)
$\displaystyle(F_{i,j})_{(k,\star)}=\bigoplus_{a\in\mathbb{Z}}S(-j)^{N_{a,k}\,\beta_{i,(a,j)}(T/G)}.$
It follows that (2.21) is a (possibly non-minimal) graded free
$K[\mathrm{\underline{x}}]$\- resolution of $[T/G]_{(k,\star)}$. Since
$\deg_{x}(G)\leq 1$, from (2.21) and (2.22) we conclude that
(2.23) $\displaystyle\rm{reg}([T/G]_{(k,\star)})=0\quad\text{for all $k$}.$
Now we have
(2.24)
$\displaystyle\begin{split}dk\leq\rm{reg}(Q^{k})\leq\rm{reg}([T/P]_{(k,\star)})+dk&\leq\rm{reg}([T/\rm{in}(P)]_{(k,\star)})+dk\\\
&=\rm{reg}([T/G]_{(k,\star)})+dk\quad\text{for all $k\geq k_{0}$}\\\
&=0+dk=dk,\end{split}$
where the second (in)equality in (2.24) follows from (2.2), the third
inequality is due to Lemma 2.3, and the forth comes from the easy argument
$[T/\rm{in}(P)]_{(k,\star)}=T_{(k,\star)}/\rm{in}(P)_{(k,\star)}=T_{(k,\star)}/G_{(k,\star)}=[T/G]_{(k,\star)}.$
Finally (2.24) implies that $\rm{reg}(Q^{k})=kd$ for all $k\geq k_{0}$ as
desired. ∎
## 3\. Examples and applications
In this section we provide some applications of Theorem 1.2. But before that
we examine our condition on the decomposition of $\rm{in}(P)$ in a closer
view. In the following a reformulation of our results is provided.
With the assumptions and notation introduced in Theorem 1.2 assume that
$B=(m_{1},\cdots,m_{p})$ and $\rm{bideg}(m_{i})=(t_{i},\geq 2).$ By
$(t_{i},\geq 2)$ we mean that the $\deg_{x}(m_{i})\geq 2$. It is harmless to
assume that $t_{1}\leq\cdots\leq t_{p}.$ If for all $i=1,\cdots,p$ and all
$\alpha\in\mathbb{N}^{m}$ with $\mid\alpha\mid=t_{p}+1-t_{i}$ we have
$\mathrm{\underline{t}}^{\alpha}m_{i}\subseteq G$ then
$I_{(k,\star)}=G_{(k,\star)}$ for all $k>t_{p}+1.$
Using this strategy and as an application for our main result we give an
answer to the Question A proposed in the Introduction.
###### Example 3.1.
Let $S=\mathbb{Q}[x_{1},\cdots,x_{6}]$ and let $J$ be the ideal of (1.3). Let
$T=\mathbb{Q}[x_{1},\cdots,x_{6},t_{1},\cdots,t_{10}]$ with order
$\mathrm{\underline{x}}>\mathrm{\underline{t}}$ (and DegRevLex). We also use
$J$ for the ideal of $T$ generated by the same generators as of $J$ in $S$.
Let $P$ be the defining ideal of the Rees ring of $J$, so $R(J)=T/P$. One can
check that $P$ has 15 elements of bidegree (1,1), 10 elements of bidegree
(3,0), and 15 elements of bidegree (4,0). Take $G$ and $B$ as in Theorem 1.2.
We have checked that
$|G|=60,\,B=\mbox{Ideal}(t_{6}x_{4}x_{5},t_{4}x_{3}x_{5},t_{4}t_{6}x_{5}^{2}),$
and so $\max\\{\deg_{t}(h)\mid h\in B\\}=2.$ But
$(\mathrm{\underline{t}})^{2}(t_{6}x_{4}x_{5})\nsubseteq
G,(\mathrm{\underline{t}})^{2}(t_{4}x_{3}x_{5})\nsubseteq
G,(\mathrm{\underline{t}})(t_{4}t_{6}x_{5})\nsubseteq G.$ So in DegRevLex
(also Lex) order and $\mathrm{\underline{x}}>\mathrm{\underline{t}},$ we were
unable to admit the conditions of Theorem 1.2. We have observed that the same
story happens for ordering $\mathrm{\underline{t}}>\mathrm{\underline{x}}$
either DegRevLex or Lex. One could try to take $g$ ”generic”, as in (3.1).
(3.1) $\displaystyle\begin{split}&g:=g_{1}\times g_{2},\\\
&g_{1}:=x_{i}\longmapsto\mbox{Random}(\rm{Sum}(x_{1},\cdots,x_{6})),\\\
&g_{2}:=t_{j}\longmapsto\mbox{Random}(\rm{Sum}(t_{1},\cdots,t_{10})),\end{split}$
for all $i=1,\cdots,6$ and all $j=1,\cdots,10$, where by
$\mbox{Random}(\rm{Sum}(x_{1},\cdots,x_{6}))$ we mean a linear combination of
$x_{1},\cdots,x_{6}$ with random coefficients and the same interpretation for
$t_{1},\cdots,t_{10}.$ But we realized that a properly chosen sparse random
upper triangular $g$ does the job as well. We continue in DegRevLex order and
$\mathrm{\underline{t}}>\mathrm{\underline{x}}$.
We have implemented some functions (in CoCoA) to look for a desired upper
triangular bi-change of coordinates. For example, the following $g$ works fine
for $J$, indeed there exists many of such $g$:
$\displaystyle g:=g_{1}\times
g_{2}\in\rm{GL}_{6}(\mathbb{Q})\times\rm{GL}_{10}(\mathbb{Q}),$
where
$g_{1}:\mathbb{Q}[\mathrm{\underline{x}}]\longrightarrow\mathbb{Q}[\mathrm{\underline{x}}]$
is given by
$\displaystyle x_{4}\longmapsto x_{1}+x_{4},$ $\displaystyle x_{6}\longmapsto
x_{3}+x_{6},$
and sends $x_{i}$ for $i\neq 4,6$ to itself and let $g_{2}$ to be the identity
map over $\mathbb{Q}[\mathrm{\underline{t}}]$. One can compute that $\mid
G\mid=98,\,B=(t_{7}x_{3}^{2},t_{4}t_{6}x_{5}^{2}).$ It is easy to verify that
(3.4) $\displaystyle
I_{(k,\star)}=G_{(k,\star)},\,\text{for\;}k>2\Longleftrightarrow\left\\{\begin{array}[]{cc}(t_{7}x_{3}^{2})(t_{1},\cdots,t_{10})^{2}\subseteq
G,\\\ (t_{4}t_{6}x_{5}^{2})(t_{1},\cdots,t_{10})\subseteq
G,\end{array}\right.$
and since in the right side of (3.4) both containments are valid we conclude
with $\rm{reg}(J^{k})=3k$ for all $k>2.$
Taking several ideas from Example 3.1 now we are able to quickly find an
answer to Question A for $J_{1}.$ In the following we show that
$\rm{reg}(J_{1}^{k})=3k,$ for all $k>2.$
###### Example 3.2.
Let $S=\mathbb{Q}[x_{1},\cdots,x_{6}]$ and let $J_{1}$ be the ideal of (1.5).
Let $T=\mathbb{Q}[t_{1},\cdots,t_{10},x_{1},\cdots,x_{6}]$ in DegRevLex order,
and let $P_{1}$ be the defining ideal of the Rees ring of $J_{1}$, so
$R(J_{1})=T/P_{1}$. One can observe that $P$ has 15 elements of bidegree
(1,1), 10 elements of bidegree (3,0), and 12 elements of bidegree (4,0). Take
$g$ to be the following simple upper triangular bi-transformation:
$\displaystyle g:=g_{1}\times
g_{2}\in\rm{GL}_{6}(\mathbb{Q})\times\rm{GL}_{10}(\mathbb{Q}),$
where
$g_{1}:\mathbb{Q}[\mathrm{\underline{x}}]\longrightarrow\mathbb{Q}[\mathrm{\underline{x}}]$
shall be given by
$\displaystyle x_{4}\longmapsto x_{2}+x_{4},$ $\displaystyle x_{6}\longmapsto
x_{1}+x_{6},$
and sending the rest to themselves and take
$g_{2}:\mathbb{Q}[\mathrm{\underline{t}}]\longrightarrow\mathbb{Q}[\mathrm{\underline{t}}]$
to be
$\displaystyle t_{8}\longmapsto t_{7}+t_{8},$
and for $i\neq 8,\,t_{i}\longmapsto t_{i}$. Computations by CoCoA shows that
$|G|=144,\,B=(t_{10}x_{2}x_{3},t_{2}t_{4}x_{5}^{2}).$ Since
$I:=\rm{in}(g(P))=G+B$, we have
(3.7) $\displaystyle
I_{(k,\star)}=G_{(k,\star)},\,\text{for\;}k>2\Longleftrightarrow\left\\{\begin{array}[]{cc}(t_{10}x_{2}x_{3})(t_{1},\cdots,t_{10})^{2}\subseteq
G,\\\ (t_{2}t_{4}x_{5}^{2})(t_{1},\cdots,t_{10})\subseteq
G,\end{array}\right.$
and since it is easy to check that the right side of (3.7) is holding, we
obtain that $\rm{reg}(J_{1}^{k})=3k$ for all $k>2.$
We conclude with the following two corollaries which indicate that ideals $J,$
(1.3), and $J_{1}$, (1.5), are very tightly related.
###### Corollary 3.3.
When the characteristic of the base field is zero, all the powers of $J$, and
$J_{1}$, but the second power have linear resolution.
Since the least exponent $k_{0}$ for $J^{k}$, and also for $J_{1}^{k}$ in
order to have linear resolution for all $k>k_{0}$ is 2, the following question
seems to be interesting to discover:
### Question B
Does there exist an ideal $Q$ with generators of the same degree $d$ over some
polynomial ring $S=K[x_{1},\cdots,x_{r}]$, for which
$\rm{reg}(Q^{k})=kd,\,\forall k\neq 3$ or $\forall k\neq 2,3$?
As we mentioned in Introduction, it is easy to check that $T/P$ and $T/P_{1}$
have the same multigraded Hilbert series, where $P$, and $P_{1}$ are the
defining ideals of Rees rings of $J$ and $J_{1}$ correspondingly. The
immediate result is as follows:
###### Corollary 3.4.
$\rm{HS}(S/J^{k})=\rm{HS}(S/J_{1}^{k})\,\forall k,$ and so
$\beta_{i,j}(J^{k})=\beta_{i,j}(J_{1}^{k})\,\forall i,j,\forall k.$
## Acknowledgment
The results of this paper were obtained during the visit of the author to
Dipartimento di Matematica, Universitá di Genova, Italia. The author would
like to express his deep gratitude to Professor Aldo Conca for the kind
invitation and for his warm hospitality whose guidance and support were
crucial for the successful completion of this project. During the stay in
Genova, the author was supported by a grant within the frame of the Italian
network on “Commutative, Combinatorial, Computational Algebra” PRIN 2006-07,
directed by Professor Valla. It is a pleasure for the author to warmly thank
him and also the kind staff in DIMA as well. Finally, the role of the software
package CoCoA in computations of concrete examples as we worked on this
project is acknowledged.
## References
* [1] A. Aramova, K. Crona, and E. de Negri, _Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions,_ J. Pure Appl. Algebra 150 (2000), 215- 235. MR1769356 (2001f:13046)
* [2] W. Bruns, J. Herzog, _Cohen-Macaulay Rings,_ Revised Edition, Cambridge University Press, Cambridge, 1996.
* [3] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
* [4] A. Conca, _Regularity jumps for powers of ideals,_ Commutative algebra, Lect. Notes Pure Appl. Math., 244, Chapman & Hall/CRC, Boca Raton, FL, (2006) 21 -32. MR2184787 (2007c:13017)
* [5] A. Conca, J. Herzog, _Castelnuovo-Mumford regularity of products of ideals,_ Collect. Math. 54 (2003), no. 2, 137 -152. MR95c:13026. (Zbl 809.13010)
* [6] S. D. Cutkosky, J. Herzog, and N. V. Trung, _Asymptotic behaviour of the Castelnuovo-Mumford regularity,_ Compositio Math. 118 (1999), 243 -261. MR1711319 (2000f:13037)
* [7] D. Eisenbud, _Commutative Algebra With A View Toward Algebraic Geometry,_ Graduate Texts in Mathematics 150, Springer-Verlag, 1995.
* [8] by same author, _The Geometry of Syzygies, A second course in Commutative Algebra and Algebraic Geometry,_ University of California, Berkeley, 2002.
* [9] D. Eisenbud, S. Goto, _Linear free resolutions and minimal multiplicity,_ J. Algebra 88 (1984), no. 1, 89–133. MR741934 (85f:13023)
* [10] J. Herzog, T. Hibi, X. Zheng, _Monomial ideals whose powers have a linear resolution,_ Math. Scand. 95 (2004), no. 1, 23–32. MR2091479 (2005f:13012)
* [11] V. Kodiyalam, _Asymptotic behaviour of Castelnuovo-Mumford regularity,_ Proc. Amer. Math. Soc. 128 (2000), 407–411. MR1621961 (2000c:13027)
* [12] D. Mumford, _Lectures on curves on an algebraic surface,_ Princeton Univ. Press, Princeton (1966), MR0209285 (35:187)
* [13] T. Römer, _On minimal graded free resolutions,_ Illinois J. Math. 45 (2001), no. 2, 1361- 1376. MR1895463 (2003d:13015)
|
arxiv-papers
| 2008-10-06T17:21:27
|
2024-09-04T02:48:58.152222
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Keivan Borna",
"submitter": "Keivan Borna Lorestani",
"url": "https://arxiv.org/abs/0810.1017"
}
|
0810.1072
|
# Nonextensive Entropy, Prior PDFs and Spontaneous Symmetry Breaking
Fariel Shafee
Department of Physics
Princeton University
Princeton, NJ 08540
###### Abstract
We show that using nonextensive entropy can lead to spontaneous symmetry
breaking when a parameter changes its value from that applicable for a
symmetric domain, as in field theory. We give the physical reasons and also
show that even for symmetric Dirichlet priors, such a definition of the
entropy and the parameter value can lead to asymmetry when entropy is
maximized.
## 1 Introduction
Nonextensive entropies, such as that defined by Tsallis[1], or more recently
by us[2], among others[3, 4] differ from the conventional Boltzmann-Shannon
form, which is extensive in the sense of being additive when two subsystems in
equilibrium are joined together. In nonextensive forms the combined value of
entropy may be, in general, higher or lower than the sum of the entropies for
the subunits joined. The deviation is , therefore, ascribable to interactions
of a nonrandom nature among the microsystems comprising each subunit.
The maximum value of extensive entropy occurs when the probabilities are
equally distributed among all the possible states of the system. In other
words the conventional entropy is maximal for the most symmetric distribution
of the microsystems. In the present paper we show that for nonextensive
entropies defined on terms of phase cell deformations, the maximal entropy may
not correspond to an equidistribution of probability among the states.
## 2 Nonextensive Entropy
The classical entropy
$S=-\sum_{i}p_{i}\log p_{i}$ (1)
may be modified in several ways. The well-known Tsallis form generalizes the
logarithm:
$\log p\rightarrow{\bf Log}_{q}p\equiv\frac{1-p^{q}}{1-q}$ (2)
For our entropy we make the measure a fractal:
$S=-\sum_{i}p_{i}^{q}\log p_{i}$ (3)
with $q=1$ giving the classical Shannon entropy, as in the Tsallis case. In
ref. [2] we have given detailed account of the different physical
considerations that lead to our expression and also comparison of the
statistical mechanical properties of the three entropies.
The justification of choosing Shannon or any other more generalized entropy,
such as that of Tsallis or Renyi, or the one we have presented elsewhere[2,
6], lies eventually in the relevance or “good fit” such an entropy would
produce in the data corresponding to a situation where the presence or lack of
interactions among the members or other considerations suggest the need for a
proper choice. However, data are always finite, and probability distribution
is the limit of relative frequencies with an infinite sample. One, therefore
faces the problem of estimating the best PDF from a finite sample [7]. This
PDF may be subject to the constraint of a known entropy, in whatever way
defined, as a functional of the PDF.
Mathematically, the problem of determining the best posterior PDF, given a
rough prior PDF and data points, is expressed formally by Bayes Theorem.
However, the constraint of the constant entropy makes the functional integral
impossible to handle even for a fairly simple prior as found by Wolpert and
Wolf [8] and by Nemenman, Shafee and Bialek[5]. The integrals involved were
first considered in a general context by [8], and the question of priors was
addressed in [7, 5]. It was discovered that, though the integral for the
posterior was intractable, the moments of the entropy could be calculated with
relative ease.
In [5] it has also been shown that for Dirichlet type priors [9]
$P(p_{i})=\prod p_{i}^{\beta}$ (4)
in particular (which give nice analytic moments with exact integrals, and
hence, are hard to ignore) the Shannon entropy is fixed by the exponent
$\beta$ of the probabilities chosen for small data samples, and hence, not
much information is obtained for unusual distributions,such as that of Zipf,
i.e. a prior has to be wisely guessed for any meaningful outcome. As a
discrete set of bins has no metric, or even useful topology that can be made
use of in Occam razor type of smoothing, in this paper other tricks were
suggested to overcome the insensitiveness of the entropy.
We have noted already that the PDF associated with our proposed entropy
differs from that of the Shannon entropy by only a power of $p_{i}$, but this
changes the symmetry of the integrations for the moments for the different
terms for different bins. We,therefore, shall examine in this chapter if the
nature of the moments are sufficiently changed by our entropy to indicate
cases where data can pick this entropy in preference to Shannon or other
entropies.
## 3 Priors and Moments of Entropy
For completeness, we mention here the formalism developed by Wolpert and Wolf
[8]. The uniform PDF is given by
$\displaystyle{\mathcal{P}}_{\rm unif}(\\{p_{i}\\})={1\over Z_{\rm
unif}}\,\delta\left(1-\sum_{i=1}^{K}p_{i}\right)$ $\displaystyle Z_{\rm
unif}=\int_{\mathcal{A}}dp_{1}dp_{2}\cdots
dq_{K}\,\delta\left(1-\sum_{i=1}^{K}p_{i}\right)$ (5)
where the $\delta$ function is for normalization of probabilities, $Z_{\rm
unif}$ is the total volume occupied by all models. The integration domain
${\mathcal{V}}$ is bounded by each $p_{i}$ in the range $[0,1]$. Because of
the normalization constraint, any specific set of $\\{p_{i}\\}$ chosen from
this distribution is not uniformly distributed and “uniformity” means simply
that all distributions that obey the normalization constraint are equally
likely a priori.
We can find the probability of the model $\\{p_{i}\\}$ with Bayes rule as
$\displaystyle
P(\\{p_{i}\\}|\\{n_{i}\\})=\frac{P(\\{n_{i}\\}|\\{p_{i}\\}){\mathcal{P}}_{\rm
unif}(\\{p_{i}\\})}{P_{\rm unif}(\\{n_{i}\\})}$ $\displaystyle
P(\\{n_{i}\\}|\\{p_{i}\\})=\prod_{i=1}^{K}(p_{i})^{n_{i}}.$ (6)
Generalizing these ideas, we have considered priors with a power-law
dependence on the probabilities calculated as
${\mathcal{P}}_{\beta}(\\{p_{i}\\})={1\over
Z(\beta)}\delta\left(1-\sum_{i=1}^{K}p_{i}\right)\prod_{i=1}^{K}p_{i}^{\beta-1}\,,$
(7)
It has been shown [5] that if $p_{i}$’s are generated in sequence [
$i=1\rightarrow K$] from the Beta–distribution
$\displaystyle
P(p_{i})=B\left(\frac{q_{i}}{1-\sum_{j<i}p_{j}};\beta,(K-i)\beta\right)$
$\displaystyle B\left(x;a,b\right)=\frac{x^{a-1}(1-x)^{b-1}}{B(a,b)}$ (8)
gives the probability of the whole sequence $\\{p_{i}\\}$ as
${\mathcal{P}}_{\beta}(\\{p_{i}\\})$.
Random simulation of PDF’s with different shapes (a few bins occupied, versus
more spread out ones) show that the entropy depends largely on the parameter
$\beta$ of the prior and hence, sparse data has virtually no role in getting
the output distribution shape. This would seem unsatisfactory, and some
adjustments appear to be needed to get any useful information out.
We shall not here repeat the methods and results of [5], which considers only
Shannon entropy.
## 4 Comparison of Shannon and Our Entropy
In our case with the entropy function given by Eqn. 3, we note that it does
not involve a simple replacement of the exponents $n_{i}$ of $p_{i}$ by
$n_{i}+q-1$ in the case of the Dirichlet prior (Eqn. 4) in the product
involved in the moment determination integrals given in [8], but a complete
re-calculation of the moment, using the same techniques given in [8].
Apparently, the maximal value of entropy should correspond to the most flat
distribution, i.e.
$S_{max}=K^{(1-q)}\log(K)$ (9)
In the limit of extremely sparse, nearly zero data ($n_{i}=0$), we get for the
first moment, i.e. the expected entropy,
$\langle S_{1}\rangle/\langle
S_{0}\rangle=K\frac{\Gamma(\beta+q)}{\Gamma(\beta)}\frac{\Gamma(\beta
K)}{\Gamma(\beta K+q)}\Delta\Phi^{0}(\beta K+q,\beta+q)$ (10)
where we have for conciseness used the notation of ref. [8]
$\Delta\Phi^{p}(a,b)=\Psi^{(p-1)}(a)-\Psi^{(p-1)}(b)$ (11)
$\Psi^{n}(x)$ being the polygamma function of order $n$ of the argument $x$.
It can be checked easily that this expression reduces to that in ref. [5] when
$q=1$, i.e. when we use Shannon entropy.
## 5 Results for Mean Entropy
So, we now have, unlike Shannon, a parameter $q$ that may produce the
difference from the Shannon case, where $q$ is fixed at unity. In Figs. 1 \- 3
we show the variation of the ratio of $\langle S_{1}\rangle/S_{max}$ with
variable bin number $K$. In ref. [5] we have commented how insensitive the
Dirichlet prior [9] is when Shannon entropy is considered in the
straightforward manner given in ref. [8]. In our generalized form of the
entropy, we note that by changing the parameter, specific to our form of the
entropy, for $q>1$, we get a peak for small $\beta$ and large $K$ values.
Figure 1: Ratio of expected value (first moment) of new entropy plotted
against bin number $K$ and prior exponent $\beta$ for entropy parameter
$q=0.5$. Figure 2: Same as Fig. 1, but for $q=1.0$, i.e. Shannon entropy.
Figure 3: As previous two figures, but for $q=1.5$.
This peak allows us to choose uniform Dirichlet priors with appropriate $q$
value, that would nevertheless lead to asymmetry not possible with Shannon
entropy. In other words, instead of the priors, we can feed the information
about expected asymmetry of the PDF to the entropy with no need to choose
particular bins. The nonextensivity of our entropy, coming possibly from
interaction among the units, gives rise to situations where the entropy maxima
do not increase with the number of bins like $\log(K)$, but being
$K^{(1-q)}\log(K)$, may be extended or squeezed, according to the value of $q$
being less than or greater than unity.
## 6 Spontaneous Symmetry Breaking
The interesting thing to note is that for $q>1$ and large $K$, at small
prioric parameter $\beta$, the entropy peak exceeds the normally expected
expression in Eqn. 9, with full $K$, so, the expected value of entropy is seen
to exceed the formal maximum. The clustering or repulsive effects, change the
measure of disorder from the Shannon type entropy. So, the highest expected
value of entropy may correspond not to a uniformly distributed population, but
to that corresponding to one with a smaller subset that is populated. This
means that for our entropy the most uniform distribution is not the least
informative, the $p^{q}$ weighting distorts it to an uneven distribution for
the expected maximal entropy value. This result is in some ways similar to
spontaneous symmetry breaking in field theory, where the variation of a
parameter leads to broken-symmetry energy minima.
A neater view of these results can be seen in Figs. 4\- 6 with $K$ values
fixed.
Figure 4: Clearer view in 2-dimensional plot, with $K=10$. Red, green and
blue lines are for $q=0.5,1.0$ and $1.5$ respectively Figure 5: As Fig. 4, but
for bin number $K=100$. Figure 6: As previous two figures, but for $K=1000$.
We have not obtained the second moment, i.e. the standard deviation , or the
spread, of the entropy distribution, because, with our entropy and an
arbitrary $q$, the expressions cannot be obtained in the simple form of ref.
[5]. We, can however, expect that the variation of the higher moments from the
Shannon case will be less than the first moment, because higher derivatives of
the $\Gamma$ functions are smoother. We shall assume the spreads are narrow
enough to concentrate on the first moments only.
Apart from the PDF estimates above, this picture of broken symmetry for the
maximal entropy when the parameter $q>1$ is also manifest directly in an
explicit calculation of the entropy using our prescription with a simple
three-state system. The symmetric expected maximal entropy in this case should
be
$S_{max}=-3p^{q}\log p$ (12)
with $p=1/3$.
With two of the probabilities $p_{1}$ and $p_{2}$ running free from $0$ to $1$
with the constraint $p_{1}+p_{2}+p_{3}=1$, the plot for the entropy
$S=-p_{1}^{q}\log p_{1}-p_{2}^{q}\log
p_{2}-(1-p_{1}-p_{2})\log(1-p_{1}-p_{2})$ (13)
we plot $S/S_{max}$ in Figs. 7, 8. For $q=2.44$ we obtain the most interesting
behavior, with a local maximum at the point of symmetry
$p_{1}=p_{2}=p_{3}=1/3$, which is not the global maximum. For $q\leq 1$ the
symmetry point gives the global maximum.
Figure 7: Our entropy for a three-state system, with parameter $q=2.44$, as
two independent probabilities $p_{1}$ and $p_{2}$ are varied wit the
constraint $p_{1}+p_{2}+p_{3}=1$. The expected maximum at the symmetry point
$p_{1}=p_{2}=p_{3}$ turns out to be a local maximum. The global maxima are not
at the end points with one of the probabilities going up to unity and the
others vanishing, which gives zero entropy as expected, but occurs near such
end points, as shown clearly in the next figure. Figure 8: Two-dimensional
version of the previous Fig. 7, with $p_{1}=1/3$ fixed, so that only $p_{2}$
varies. This shows a clearer picture of local maximum at the symmetry point
and global maxima near the end points.
A physical explanation of the SSB may be the distortion introduced by
nonrandom interactions in the volumes of the ’phase space’ of the states. In
ref. [2] we have shown how the new entropy is related to such volumes, in
terms of Shannon coding theorem. Apparently this distortion introduces a
mixing of states that reduces the weights of clearly defined states and hence
introduces a new measure of uncertainty not present in the case of Shannon
entropy. As a result it is entropically preferable to leave some states
underpopulated to increase the total entropy by overpopulating others. In
other words we have a reduction of the problem from $N$ states to less, but
with a measure factor with less dimunition that overcompensates the decrease
in the logarithmic factor. In a field-theoretic model with the Lagrangain
$\displaystyle L=|\partial\phi|^{2}-V_{I}$ $\displaystyle
V_{I}=-\mu^{2}|\phi|^{2}+\lambda|\phi|^{4}$ (14)
for $\mu^{2}$ and $\lambda$ with opposite signs, the lowest energy state is
the symmetric vacuum (no state occupied), and for same sign the vacuum becomes
a local maximum, with a ring of minima at $|\phi|=|\mu/\surd{(2\lambda)}|$,
which forces us to choose a unique vacuum with a particular complex $\phi$
having this magnitude. In the case of entropy, for $q\leq 1$ we have the
highest entropy for all states equally populated, and for $q>1$ the
configuration with a symmetric flat PDF is no longer the one with the highest
entropy.
## 7 Conclusions
We have seen that average entropies corresponding to uniformly symmetric
Dirichlet type priors can be obtained exactly even for nonextensive entropies
of a type we have described earlier. Remarkably this entropy shows maxima for
asymmetric probability distributions, which can be considerably higher than
the symmetric distribution, unlike Shannon entropy. We think this asymmetry is
a consequence of the distortion of the ‘phase cells’ associated with the
states, which may in turn be due to nonrandom interactions.
The author thanks Prof. Phil Broadbridge of the Australian Mathematical
Sciences Institute for encouragement.
## References
* [1] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat.Phys. 52, 479–487 (1988).
* [2] F. Shafee, Lambert function and a new non-extensive form of entropy, IMA Journal of Applied Mathematics 72, 785–800 (2007)
* [3] G. Kaniadakis, Nonlinear kinetics underlying generalized statistics. Physica A 296, 405–425 (2001).
* [4] G. Kaniadakis, Statistical mechanics in the context of special relativity. Phys. Rev. E 66, 056125 (2002).
* [5] I. Nemenman, F. Shafee and W. Bialek, Entropy and Inference, Revisited, in Adv. Neur. Info. Processing 14, eds. T.G. Dietterich, S. Becker and Z. Ghahramani (MIT Press, Cambridge, 2002) pp. 471–478.
* [6] F. Shafee, Generalized Entropy with Clustering and Quantum Entangled States. cond-mat/0410554 (accepted by Chaos, Solitons and Fractals)(2008)
* [7] W. Bialek, C.G. Callan and S.P. Strong, Field theories for learning probability distributions.Phys. Rev. Lett. 77, 4693–4697 (1996)
* [8] D. Wolpert and D. Wolf, Estimating functions of probability distributions from a finite set of samples, Phys. Rev. E, 52, 6841–6854 (1995).
* [9] E.T. Jaynes, Monkeys, Kangaroos, and N , University of Cambridge Physics Dept. Report 1189 (1984).
|
arxiv-papers
| 2008-10-06T22:26:09
|
2024-09-04T02:48:58.157766
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fariel Shafee",
"submitter": "Fariel Shafee",
"url": "https://arxiv.org/abs/0810.1072"
}
|
0810.1118
|
# Interstellar Plasma Turbulence Spectrum Toward the Pulsars PSR B0809$+$74
and B0950$+$08
T. V. Smirnova V. I. Shishov Pushchino Radio Astronomy Observatory, Astro
Space Center of the Lebedev Institute of Physics, Russian Academy of Sciences,
Pushchino, Moscow oblast’, 142290 Russia
###### Abstract
Interstellar scintillations of pulsars PSR B0809$+$74 and B0950$+$08 have been
studied using observations at low frequencies (41, 62, 89, and 112 MHz).
Characteristic temporal and frequency scales of diffractive scintillations at
these frequencies have been determined. The comprehensive analysis of the
frequency and temporal structure functions reduced to the same frequency has
shown that the spectrum of interstellar plasma inhomogeneities toward both
pulsars is described by a power law. The exponent of the spectrum of
fluctuations of interstellar plasma inhomogeneities toward PSR B0950$+$08
($n=3.00\pm 0.05$) appreciably differs from the Kolmogorov exponent. Toward
PSR B0809$+$74 the spectrum is a power law with an exponent $n=3.7\pm 0.1$. A
strong angular refraction has been detected toward PSR B0950$+$08\. The
distribution of inhomogeneities along the line of sight has been analyzed; it
has been shown that the scintillations of PSR B0950$+$08 take place on a
turbulent layer with enhanced electron density, which is localized at
approximately 10 pc from the observer. For PSR B0809$+$74 the distribution of
inhomogeneities is quasi-uniform. Mean-square fluctuations of electron density
on inhomogeneities with a characteristic scale $\rho_{0}=10^{7}$ m toward four
pulsars have been estimated. On this scale the local turbulence level in the
10-pc layer is 20 times higher than in an extended region responsible for the
scintillations of PSR B0809$+$74.
## 1 INTRODUCTION
Pulsars represent a good tool for the study of interstellar plasma (ISP),
because they possess very small angular sizes and intense emission. The
investigation of their intensity fluctuations in the frequency–time domain
allows us to study the ISP spectrum in various directions of our Galaxy. As
was shown in [1, 2], the observational data on pulsar scintillations are
statistically well described by a power-law spectrum of inhomogeneities with
an exponent $n=3.67$ (i.e., by the Kolmogorov spectrum) in a very broad range
of spatial scales $\rho$. However, a scatter in the values of the spectrum
exponent $n$ is observed for the same value of $\rho$ for different sources;
this testifies that in local directions of the Galaxy the spectrum can differ
from the Kolmogorov spectrum. The application of a new method for the study of
ISP using a comprehensive analysis of a structure function obtained from
multifrequency observations of pulsar scintillations has revealed a difference
of the spectrum from Kolmogorov’s for PSR B0329$+$54, B0437$-$47, and
B1642$-$03 [3–5]. The purpose of this paper was a study of the ISP spectrum
ISP toward nearby and powerful at meter wavelengths pulsars PSR B0809$+$74 and
B0950$+$08\. The distances to these pulsars and their velocities are known
from parallax measurements [6]: $R=433$ pc, $V=102$ km/s and $R=262$ pc,
$V=36.6$ km/s for PSR B0809$+$74 and B0950$+$08 respectively. Since they are
the nearest pulsars, the analysis of their scintillations allows us to study
the region of interstellar plasma nearest to the Sun.
The obtained earlier observational data demonstrate the presence of three
components of interstellar turbulent plasma in the solar neighborhood at
distances of the order of 1 kpc and less. The first component is turbulent
plasma with a statistically quasi-uniform distribution in the space between
spiral arms (component A in the classification of [7, 8]). The second
component represents a cavern with a depleted electron density ($n_{e}\cong
0.005$ cm-3) in the solar neighborhood with a scale of about 200–300 pc
perpendicularly to the Galactic plane and 50–100 pc in the Galactic plane.
This cavern has been detected in X-ray observations [9]. The presence of a
cavern with a lowered turbulence level on a scale of the order of 100 pc has
been revealed also by the analysis of scintillations of pulsars with
dispersion measures from 3 to 35 pc cm-3 [10]. The third component corresponds
to a layer with an enhanced turbulence level at a distance of about 10 pc from
the Sun. This layer has been detected in observations of interstellar
scintillations of quasars in the centimeter waveband [11, 12]. In [5] this
layer was classified as component C. In [5] it was also shown that the pulsar
PSR J0437$-$4715 scintillates on inhomogeneities of layer C.
Table 1: Parameters of the observations Parameter | $f$, MHz
---|---
| 41 | 62.43 | 88.57 | 111.87
PSR B0809$+$74
$\Delta t$, ms | 5.12; 2.56 | 5.15 | 5.38 | 2.56
$N_{\textrm{ch}}$ | 128; 640 | 96 | 96 | 128
$\Delta f$, kHz | 1.25; 200. | 20 | 20 | 20
$T,$ s | 51.68 | 51.68 | 38.76 | 19.38
PSR B0950$\leavevmode\nobreak\ +\leavevmode\nobreak\ $08
$\Delta t$, ms | 5.12; 2.56 | 2.56 | 2.56 | 2.56
$N_{\textrm{ch}}$ | 6400. | 96 | 128 | 128
$\Delta f$, kHz | 1.25; 200. | 20 | 20 | 20
$T$, s | 20.24 | 15.18 | 20.24 | 5.06
## 2 OBSERVATIONS AND DATA PREPROCESSING
The observations of PSR B0809$+$74 and B0950$+$08 were carried out on the BSA
and DKR-1000 radio telescopes of the Pushchino Radio Astronomy Observatory
(Astro Space Center, Lebedev Institute of Physics, Russian Academy of
Sciences) at frequencies 41, 62.43, 88.57 MHz (DKR-1000), and 111.87 MHz (BSA)
in December 2001 – January 2004. As the interference situation at low
frequencies was complex, we have used for the analysis only those records
where interference was small. We received linearly polarized emission. Two
multichannel receivers were used: a 128-channel receiver with a channel
bandwidth $\Delta f=20$ kHz at frequencies 88.57, 62.43, and 111.87 MHz as
well as a 128-channel receiver with a channel bandwidth $\Delta f=1.25$ kHz at
a frequency of 41 MHz. The time of observations on BSA (frequency 111.87 MHz)
in each session was 12 and 3.3 min for PSR B0809$+$74 and B0950$+$08
respectively. At low frequencies it was 35.5 min for PSR B0809$+$74 and 15.63
min for PSR B0950$+$08\. Table 1 lists the time and frequency resolution
($\Delta t$ and $\Delta f$) and the used number of channels
($N_{\textrm{ch}}$) at each frequency. Individual pulses of the pulsars were
recorded on the computer disk in all channels with a period synchronized with
the precomputed topocentric pulse arrival time. Then the signal in all
channels was shifted in accordance with the dispersion shift at the given
frequency (reduction to the highest frequency channel); in channels affected
by interference (if any) the signal was replaced by an average value found
from the adjacent channels. The amplification in all channels was reduced to
the same value by normalization, so that the noise dispersion in all channels
were equal to its average value found from channels not affected by
interference. At frequencies 41–89 MHz the record was performed in a window
1.8 $P_{1}$ ($P_{1}$ is the pulsar period) and at 112 MHz in a window 0.9
$P_{1}$. To improve the signal-to-noise ratio ($S/N$), we averaged individual
pulses: from 15 pulses at 112 MHz to 40 pulses at low frequencies. The
averaging time $T$ for both pulsars at all frequencies is listed in Table 1.
## 3 CORRELATION ANALYSIS OF THE DATA
To analyze the intensity variations of the pulse radiation as a function of
frequency (channel number) within the bandwidth of the multichannel receiver,
we formed spectra of individual pulses $I(f_{k})$ by averaging the signal in a
selected range of longitudes of the pulse and in a region outside the pulse,
on the noise segment of the record (for the same number of longitudes)
$I_{N}(f_{k})$ for each channel. Here $f_{k}=f+(k-1)\Delta f$ is the frequency
of the $k$th channel, and $f$ is the frequency of observation. Then we
subtracted the noise component $I_{N}(f_{k})$ from $I(f_{k})$, and the spectra
of individual pulses thus obtained were analyzed with the purpose to get
information in the frequency as well as in the time domains.
Figure 1: (a) Mean pulse profile of PSR B0809$+$74 at a frequency of 111.87
MHz as observed on April 6, 2001. Vertical axis: intensity in relative
(computer) units; horizontal axis: time inside the pulse in milliseconds.
Longitude ranges (interval 1 and interval 2) in which the pulse intensity was
averaged are marked. (b) Mean normalized CCF between the spectra of adjacent
pulses whose intensities were averaged in the above-mentioned longitude
ranges: 1 (lower panel) and 2 (upper panel).
Characteristic frequency ($f_{\textrm{dif}}$) and temporal
($t_{\textrm{dif}}$) scales of scintillations were determined using the
correlation analysis. We computed the mean cross-correlation function (CCF)
between the spectra of adjacent, noiseless pulses averaged in the given range
of longitudes of the pulse emission; CCF was normalized to the product of rms
deviations $\sigma_{1}\sigma_{2}$. We considered a pulse as noiseless if its
amplitude exceeded a level of 4$\sigma_{N}$. The mean profile of the pulsar
was derived by addition of all individual pulses in the given observational
session. Since the pulse averaging time $T$ (Table 1) was considerably shorter
than the scintillation timescale, the decorrelation of the spectra of adjacent
(averaged) pulses was insignificant; in return, we completely eliminated the
uncorrelated component of noise. The frequency scale $f_{\textrm{dif}}$ was
determined as a frequency shift at which CCF decreased by a factor of two.
Since we receive linearly polarized emission, we must take into consideration
the influence of polarization on its frequency–time structure. At low
frequencies the degree of polarization for pulsars studied by us is high:
about 60% for PSR B0809$+$74 [13] at the longitudes of the leading part of the
mean profile and much lower in its trailing part; (70–80)% for PSR B0950$+$08
[14] throughout the profile. The rotation measure is $\textrm{RM}={-}11.7$
rad/m2 for PSR B0809$+$74\. The frequency of the Faraday rotation is
$\displaystyle P_{F}\textrm{\leavevmode\nobreak\
[MHz]}=17.475f^{3}/{\textrm{RM}},$ (1)
where $f$ is the observational frequency in hundreds of megahertz.
Accordingly, $P_{F}=2140$ kHz at 112 MHz and $P_{F}=59$ kHz at 41 MHz. The
influence of polarization on the correlation analysis for PSR B0809$+$74 is
shown in Fig. 1. Longitude ranges in which we averaged the intensity to obtain
the spectra used in the calculation of the average CCF are shown. When
averaging the leading part of the pulse (interval 1), a superposition of two
structures—narrow-band (produced by propagation of the radiation in the
interstellar plasma) and broadband (due to a much slower intensity variation
with frequency as a result of the rotation of the polarization plane across
the receiver bandwidth)—is visible. When averaging the trailing part (interval
2), the slow component is virtually absent; this confirms the small degree of
polarization at these longitudes. In the frequency–time data analysis of PSR
B0809$+$74 we have used the average over the longitudes of the trailing part
of the mean profile.
Figure 2: (a) Mean profile of PSR B0950$+$08 at a frequency of 41 MHz as
observed on January 16, 2004. Vertical axis: intensity in relative (computer)
units; horizontal axis: time inside the pulse in milliseconds. The time
resolution is 2.56 ms, frequency resolution is 20 kHz. Arrows show longitudes
at which the pulse intensity was selected in all channels for the calculation
of the correlation functions. (b) Mean CCF between the pulse spectra taken at
longitudes 1–1, 1–2, and 1–3. The curve in the lowermost graph is the fitted
sine wave with a frequency of 600 kHz. Figure 3: Mean normalized CCF for the
spectra of adjacent pulses at 88.57 MHz (upper graphs) and 62.43 MHz (lower
graphs) for PSR B0809$+$74 (left) and PSR B0950$+$08 (right). Figure 4: Mean
profiles of PSR B0950$+$08 and PSR B0809$+$74 at 41 MHz as observed on
December 25, 2003 (left) and mean normalized CCF between the spectra of
adjacent pulses on this frequency (right). The time resolution is 5.12 ms, and
the frequency resolution 1.25 kHz. In the left graphs, the vertical axis plots
the intensity in relative (computer) units, and the horizontal axis the time
inside the pulse in milliseconds. Figure 5: Mean normalized CCF for the
spectra of pulses taken at appropriate time intervals (horizontal axis); left:
for PSR B0809$+$74 at 111.87 MHz, right: for PSR B0950$+$08 at 88.57 MHz.
For PSR B0950$+$08 the rotation measure is considerably smaller than for PSR
B0809$+$74; however, in the literature its values appreciably differ: in the
catalog of pulsars [15] $\textrm{RM}=(1.35\pm 0.15)$ rad/m2, in [14]
$\textrm{RM}=(4\textrm{--}6)$ rad/m2, and in [16] $\textrm{RM}=({-}0.66\pm
0.04)$ rad/m2. Probably, authors [14] underestimated the contribution of the
ionosphere to the obtained rotation measure. For $\textrm{RM}=1$ rad/m2 the
Faraday rotation frequency is $P_{F}=20.4$ MHz at 112 MHz and $P_{F}=1200$ kHz
at 41 MHz. The influence of polarization on our observations at 41 MHz is
shown in Fig. 2. The profile averaged over the session (Fig. 2a) is double-
peaked with a separation between the components of 13 ms. Arrows shown
longitudes at which we selected pulse intensities over all channels to form
the spectrum at the given longitude. Then we calculated mean normalized CCF
between the spectra at longitudes 1–1 (autocorrelation function, ACF), 1–2,
and 1–3 for all noiseless pulses. Figure 2b shows mean CCF between the spectra
at the corresponding longitudes. In these observations we used the receiver
with a channel bandwidth of 20 kHz. The narrow unresolved feature
corresponding to the frequency scale of diffractive scintillations at 41 MHz
has a maximum at the zero frequency shift, while the slow component is shifted
to the left with an increase in the longitude offset of the spectra; this is
due to the change in the position angle across the pulsar profile. Fitting a
sine wave to the slow CCF component 1–3 yields the modulation frequency
$P_{F}=(600\pm 60)$ kHz, which corresponds to $\textrm{RM}=(2\pm 0.2)$ rad/m2.
The observations were carried out in the nighttime, and we suppose that the
contribution of the ionosphere to this value was not more than 1 rad/m2.
Consequently, RM toward PSR B0950$+$08 is about 1 rad/m2, and the effect of
polarization on the frequency–time structure of the emission for this pulsar
is considerably smaller than for PSR B0809$+$74\. Therefore, for obtaining the
spectra we used the average over all longitudes determined by the level of
0.25 of the mean profile maximum of the pulsar.
Table 2: Characteristic scales of diffractive scintillations Parameter | $f$, MHz
---|---
| 41 | 62.43 | 88.57 | 111.87
PSR B0809$+$74
$f_{\textrm{dif}},$ kHz | 0.$2\pm 0.6$ | $7\pm 3$ | $20\pm 10$ | $45\pm 5$0
$t_{\textrm{dif}},$ s | – | – | $450\pm 70$0 | $600\pm 100$
PSR B0950$\leavevmode\nobreak\ +\leavevmode\nobreak\ $08
$f_{\textrm{dif}},$ kHz | $1.5\pm 0.4$ | $25\pm 10$ | $100\pm 40$0 | $220\pm 60$0
$t_{\textrm{dif}},$ s | $350\pm 100$ | $400\pm 100$ | $450\pm 120$ | ${>}200$
Figure 3 shows for both pulsars mean CCF between the spectra of adjacent
pulses at 88.57 MHz (upper graphs) and 62.43 MHz (lower graphs) obtained in
individual observational sessions. Table 2 lists, together with their errors,
the average values of characteristic scales of scintillations
$f_{\textrm{dif}}$ found from all sessions. The errors correspond to standard
deviations from the mean. The slow CCF component is due to the effect of
polarization. It is visible that with decreasing frequency the relative
amplitude of the slow component increases; this testifies to an increase in
the degree of polarization with frequency. Figure 4 presents the mean profiles
of PSR B0809$+$74 and B0950$+$08 for one of the observational sessions at the
low frequency 41 MHz (left-hand graphs) and mean CCF between the spectra of
adjacent pulses at this frequency (right-hand graphs). The frequency
resolution here was 1.25 kHz for both pulsars. To improve the signal-to-noise
ratio for PSR B0950$+$08 observed at $f=41$ MHz with the narrow-band receiver,
we used a time constant of 10 ms; therefore, the components of the mean
profile merge. For PSR B0809$+$74 a well-resolved two-peak profile with a
separation between components of 96 ms is observed.
We determined the characteristic timescale of scintillations as the time shift
at which the coefficient of correlation between the spectra separated by an
appropriate time interval decreases by a factor of two. Figure 5 presents the
corresponding functions for both pulsars obtained from observations at one of
the frequencies: 111.87 MHz for PSR B0809$+$74 (left) and 88.57 MHz for PSR
B0950$+$08 (right). The average values of the characteristic scintillation
scales $t_{\textrm{dif}}$ found from all sessions are given together with
their errors in Table 2. The errors correspond to standard deviations from the
mean. For PSR B0809$+$74 we did not manage to determine $t_{\textrm{dif}}$ at
41 and 62 MHz because of a poor signal-to-noise ratio.
## 4 STRUCTURAL ANALYSIS OF THE DATA
We used the obtained frequency–time correlation functions for the structural
analysis of the data and, accordingly, for the study of the spectrum of ISP
inhomogeneities. The power-law spectrum of electron density fluctuations is
defined as
$\displaystyle\Phi_{Ne}(q)=C_{Ne}^{2}q^{-n},$ (2)
where $C_{Ne}^{2}$ characterizes the plasma turbulence level, $q=2\pi/\rho$
and $\rho$ are, respectively, the spatial frequency and spatial scale of an
inhomogeneity in the plane perpendicular to the line of sight (generally it is
a three-dimensional vector). The structure function of phase fluctuations
$D(\rho)$ and spectrum $\Phi_{Ne}$ are related by the Fourier transform. In
the case of a statistically uniform distribution of inhomogeneities in the
medium $D(\rho)$ is described by relationship [17]
$\displaystyle D_{s}(\rho)=(k\Theta_{0}\rho)^{n-2},$ (3)
$\displaystyle(k\Theta_{0})^{n-2}=A(n)(\lambda r_{e})^{2}C_{Ne}^{2}L/(n-1),$
$\displaystyle A(n)=\frac{2^{4-n}\pi^{3}}{[\Gamma^{2}(n/2)\sin(\pi n/2)]}.$
Here $\Theta_{0}$ is the characteristic scattering angle, $\lambda$ is the
wavelength, $k=2\pi/\lambda$ is the wavenumber, $r_{e}$ is the electron
classical radius, $L$ is the effective distance to the turbulent layer or
distance to the pulsar in the case of a uniform distribution of turbulence
along the line of sight. The spatial scale is related to the timescale by a
simple relationship $\rho=V\Delta t$, where $V$ is the velocity of motion of
the line of sight across the picture plane. If the pulsar velocity
considerably exceeds the velocity of the observer and of the motion of the
turbulent medium, then it determines the displacement of the line of sight. As
was shown in [3], for small time shifts $\Delta t$ the phase structure
function $D_{s}(\Delta t)$ can be obtained from the correlation function of
intensity fluctuations $B_{I}(\Delta t)$:
$\displaystyle D_{s}(\Delta t)={{B_{I}(0)-B_{I}(\Delta t)}\over{\langle
I\rangle^{2}}}\quad{\textrm{at}}\quad\Delta t\leq t_{\textrm{dif}},$ (4)
where $\langle I\rangle$ is the mean intensity for an observational session.
In the frequency domain we have used the relationship
$\displaystyle D_{s}(\Delta f)={{B_{I}(0)-B_{I}(\Delta f)}\over{\langle
I\rangle^{2}}}\quad{\textrm{for}}\quad\Delta f\leq f_{\textrm{dif}},$ (5)
where $\Delta f$ is the frequency shift. To reduce the data from different
frequencies to a single frequency $f_{0}$, it is necessary to scale temporal
and frequency structure functions in accordance with the law:
$\displaystyle D_{s}(f_{0},\Delta t(f_{0}),\Delta f(f_{0}))=D_{s}(f,\Delta
t,\Delta f)(f/f_{0})^{2}.$ (6)
As noted in [3], $\Delta t(f_{0})=\Delta t(f)$; however, $\Delta
f(f)\neq\Delta f(f_{0})$. In the case of purely diffractive scintillations we
have
$\displaystyle\Delta f_{d}(f_{0})=(f_{0}/f)^{2}\Delta f(f).$ (7)
In the presence of a strong angular refraction, i.e., when the refraction
angle considerably exceeds the scattering angle, the frequency dependence
$\Delta f$ is quite different:
$\displaystyle\Delta f_{r}(f_{0})=(f_{0}/f)^{3}\Delta f(f).$ (8)
We have used these formulas in the analysis of our data considering two
models, diffractive and refractive.
Figure 6: Frequency structure functions of phase fluctuations for PSR
B0809$+$74 in two models: refractive (a) and diffractive (b). Open circles:
111.87-MHz data; asterisks: 88.57 MHz; square: 62.43 MHz; triangle: 41 MHz;
filled circle: 408 MHz [18]. The data for all frequencies have been reduced to
the same frequency $f_{0}=1000$ MHz. The straight line least-square-fitted to
the first points of structure functions for the diffractive model has a slope
$\beta=1.3\pm 0.2$. Figure 7: Same as in Fig. 6, but for PSR B0950$+$08\. The
fitted straight line has a slope $\beta=0.96\pm 0.05$. Figure 8: Temporal
structure function of phase fluctuations for PSR B0809$+$74\. Open circles:
111.87 MHz; asterisks: 88.57 MHz; small open squares: 933 MHz [19]. The data
for all frequencies have been reduced to the same frequency $f_{0}=1000$ MHz.
The straight line has been least-square-fitted to the initial points of the
structure functions. Top: axis of the corresponding spatial scales. Figure 9:
Temporal structure function of phase fluctuations for PSR B0950$+$08\.
Asterisks: 88.57 MHz; small filled squares: 62.43 MHz. The data for all
frequencies have been reduced to the same frequency $f_{0}=1000$ MHz. The
straight line has been least-square-fitted. Top: axis of the corresponding
spatial scales.
Figures 6 and 7 show for both models frequency structure functions of phase
fluctuations in the double logarithmic scale for PSR B0809$+$74 and PSR
B0950$+$08 respectively. All the data were reduced to the same frequency
$f_{0}=1000$ MHz. As in our previous publication on the study of ISP spectra
toward a number of pulsars [3–5], we have used this value of $f_{0}$ for
convenience of a comprehensive analysis of all the data. In the figures the
data at different frequencies are shown by different symbols. In Fig. 6 a
filled circle marks a point obtained from observations at 408 MHz [18].
Statistical errors of the structure functions were estimated using equation
(B12) from [19]. From Fig. 6 it is visible that for PSR B0809$+$74 the
diffractive model (Fig. 6b) describes the experimental data much better than
the refractive model (Fig. 6a). A straight line least-square-fitted to the
first points of the structure functions has a slope $\beta=1.3\pm 0.2$. For
fitting we must take the values at shifts smaller than or of the order of the
characteristic frequency scale of scintillations. For PSR B0950$+$08 (Fig. 7)
the data at different frequencies in both models differ not so strongly as for
PSR B0809$+$74; however, the scatter of the points for the refractive model
(Fig. 7a) is smaller. The largest difference takes place for a point obtained
from observations at 41 MHz (filled triangle). The slope of the fitted
straight line is $\beta=0.96\pm 0.05$.
Figures 8 and 9 show temporal structure functions of phase fluctuations for
PSR B0809$+$74 and PSR B0950$+$08 respectively. In addition to our data, we
have used in Fig. 8 also a structure function obtained in [19] from
observations at 933 MHz (open squares). It is visible that the data at
different frequencies for PSR B0809$+$74 presented in the double logarithmic
scale are well described by a unified power-law spectrum. A straight line
fitted to the experimental points has a slope $\alpha=1.7\pm 0.04$. As it is
visible from Fig. 9, for PSR B0950$+$08 temporal structure functions at
different frequencies reduced to 1000 MHz satisfy a power law with an exponent
$\alpha=1.0\pm 0.05$. The scale of the upper horizontal axis in Fig. 8 and 9
corresponds to spatial scales of interstellar plasma inhomogeneities
$\rho=V\Delta t$. Here we have used the pulsar velocities measured in [6]:
$V=102$ and 36.6 km/s for PSR B0809$+$74 and PSR B0950$+$08 respectively.
Figure 10: Characteristic frequency scale of scintillations as a function of
the frequency of observation for PSR B0950$+$08 (left) and PSR B0809$+$74
(right). The filled squares show the scales obtained in this work, the
asterisks the data of [20], the crust the data of [21], the open circle the
data of [22], the filled circle the data of [10], the triangle the data of
[18], and the open squares: data from [7]. The lines are the result of least-
squares fits. The slope for PSR B0950$+$08 using our data is $\gamma=4.9\pm
0.6$, while the slope for the data at frequencies ${\geq}112$ MHz is
$\gamma=2\pm 0.6$. The fitting for PSR B0809$+$74 fitting using all the points
yielded the slope of $\gamma=3.4\pm 0.15$. The dashed line with a slope of 4.3
is the expected $f_{\textrm{dif}}(f)$ dependence. Figure 11: Characteristic
temporal scale of scintillations as a function of the frequency of
observation; top: PSR B0809$+$74, bottom: PSR B0950$+$08\. Filled squares:
scales obtained in this work, triangle: the data from [23], filled asterisks:
[20], the open asterisk: [24], the diamond the data of [25], open circle:
[19]. A straight line has been least-square-fitted to the data for PSR
B0809$+$74; its slope is 1.1. Asymptotic straight lines with slopes of
$\alpha=2$ and $\alpha={-}0.5$ corresponding to equations (10) and (14) are
shown for PSR B0950$+$08.
## 5 DISCUSSION
As shown above, the temporal structure functions for PSR B0809$+$74 and PSR
B0950$+$08 are described by a power law, and, accordingly, the spectrum of
interstellar plasma inhomogeneities toward these pulsars is a power-law one
with an exponent $n=\alpha+2$; this corresponds to $n=3.0\pm 0.05$ for PSR
B0950$+$08 and $n=3.7\pm 0.04$ for PSR B0809$+$74\. Whereas for PSR B0809$+$74
the measured turbulence spectrum matches to the Kolmogorov form ($n=3.67$),
for PSR B0950$+$08 it is much flatter. In addition, we can say that toward PSR
B0950$+$08 an appreciable angular refraction takes place. As shown in [3], for
the diffractive model the slope of the frequency structure function is related
to that of the temporal structure function as $\beta=\alpha/2$, and for the
refractive model $\beta=\alpha$. For PSR B0950$+$08 we have
$\beta\approx\alpha$. For PSR B0809$+$74 the accuracy of the determination of
$\beta$ is low, and within the $3\sigma$ error limits the relationship
$\beta=\alpha/2$ is fulfilled, though a more convincing evidence of the
absence of angular refraction in this direction is a strong deviation of the
data on the frequency structural function from this model (Fig. 6).
Figures 10 and 11 present the characteristic frequency and temporal scales of
scintillations as functions of the frequency of observation for both pulsars.
Filled squares show our measurements, other symbols denote the data taken from
[7, 10, 18–25]. For the reduction of the scales $f_{d}$ obtained by a
different method in [21] to our measurements, we have used the relationship
$f_{\textrm{dif}}=0.3f_{d}$. We have least-square-fitted straight lines to the
observation data. For PSR B0950$+$08 (Fig. 10) fitting was done separately to
the low-frequency data of this work (slope $\gamma=4.9\pm 0.6$), and to the
data at frequencies $f>112$ MHz (slope $\gamma=2\pm 0.6$). For PSR B0809$+$74
fitting for all points (Fig. 10) yields $\gamma=3.4\pm 0.15$. As shown above,
the frequency CCF of the flux fluctuations of the pulsar PSR B0809$+$74 is
described by the model of diffractive scintillations on inhomogeneities of the
power-law spectrum of interstellar plasma turbulence with an exponent $n=3.7$.
For the diffractive model the characteristic frequency scale of scintillations
is [17]
$\displaystyle f_{\textrm{dif}}=c/(\pi R\Theta_{0})^{2}\propto f^{2n/(n-2)},$
(9)
where $c$ is the velocity of light. For $n=3.7$ the expected for PSR
B0809$+$74 value $\gamma=4.3$ is shown in Fig. 10 with a dotted line. This
line is also consistent with the experimental points, except for the high-
frequency ones. It is possible that the values of $f_{\textrm{dif}}$ measured
at high frequencies are underestimated because of the limited total frequency
band of the observation. The characteristic timescale of the scintillations
for the diffractive model is
$\displaystyle t_{\textrm{dif}}\approx 1/k\Theta_{0}V\propto f^{2/(n-2)},$
(10)
where $k$ is the wavenumber and $V$ is the velocity of motion of the line of
sight with respect to the turbulent medium. For $n=3.7$ the expected for PSR
B0809$+$74 value $2/(n-2)=1.18$; this agrees well with the experimentally
obtained figure 1.1 (Fig.11). In [26] it was shown that the scintillation
parameters for the pulsar PSR B0809$+$74 and quasar B0917$+$624, which is
located close to the pulsar on the celestial sphere, are determined by the
same turbulent medium and this medium is distributed almost uniformly on a
scale of the order of 500 pc.
For the pulsar PSR B0950$+$08 the temporal and frequency correlation functions
of the flux fluctuations are described by a more sophisticated model.
Firstly, we should take into account that the observational data correspond to
two modes of scintillations. At high frequencies ($f>f_{\textrm{cr}}$)
scintillations are weak, and at low frequencies ($f<f_{\textrm{cr}}$) the
observation data correspond to the diffractive component of strong
scintillations. Near the critical frequency $f_{\textrm{cr}}$ scintillations
correspond to the focusing mode, which has been poorly studied, either
theoretically or experimentally. According to our data, the scintillations
index is about unity at all observational frequencies; thus, scintillations
are strong at frequencies $f\leq 112$ MHz. Consequently, the critical
frequency $f_{\textrm{cr}}>112$ MHz. According to the data [25] obtained at
$f=4.8$ GHz, scintillations of the pulsar PSR B0950$+$08 are weak. In the same
paper an estimate of the critical frequency $f_{\textrm{cr}}<700$ MHz is
given. In [27] on the basis of the compilation of the observational data of
1970s an estimate of the critical frequency $f_{\textrm{cr}}\approx 300$ MHz
is given.
Secondly, it is necessary to take into account the presence of a strong
angular refraction. In this case, the characteristic frequency scale of
diffractive (strong) scintillations is described by the relationship [3]
$\displaystyle f_{\textrm{dif}}\approx c/\pi
r_{\textrm{ef}}\Theta_{0}\Theta_{\textrm{ref}}\propto f^{2+n/(n-2)},$ (11)
$\displaystyle f<f_{\textrm{cr}},$
where $r_{\textrm{ef}}$ is the effective distance to the turbulent layer,
$\Theta_{\textrm{ref}}$ is the characteristic angle of refraction,
$\Theta_{0}$ is the characteristic scattering angle determined by (3). For
$n=3$ equation (11) yields $f_{\textrm{dif}}\propto f^{5}$. The observational
data indeed correspond to a power law with a turnover. At low frequencies
($f\leq 112$ MHz) the exponent $\gamma=4.9\pm 0.6$; this is consistent with
the expected value for the mode of strong scintillations in the presence of
refraction. Therefore, we can assert that the critical frequency
$f_{\textrm{cr}}>112$ MHz. In the case of weak scintillations we should
replace the scattering angle $\Theta_{0}$ with the Fresnel angle
$\displaystyle\Theta_{\textrm{Fr}}=(1/kr_{\textrm{ef}})^{1/2}.$ (12)
As a result we obtain
$\displaystyle f_{\textrm{dif}}\approx c/\pi
r_{\textrm{ef}}\Theta_{\textrm{Fr}}\Theta_{\textrm{ref}}\propto f^{2.5}.$ (13)
At high frequencies $f\geq 112$ MHz the exponent $\gamma=2\pm 0.6$; this also
agrees with the expected value for the mode of weak scintillations.
The characteristic timescale of scintillations for PSR B0950$+$08 is also
described by a power law with a turnover. At low frequencies scintillations
are strong, and the characteristic timescale of scintillations is determined
by formula (10). At high frequencies scintillations are weak, and the
characteristic timescale of scintillations is
$\displaystyle
t_{\textrm{dif}}\approx(r_{\textrm{ef}}/k)^{1/2}/V_{\textrm{ef}}\propto
f^{-1/2},$ (14)
where $V_{\textrm{ef}}$ is the velocity of motion of the line of sight with
respect to the turbulent layer. The asymptotic relationships (10) and (14) are
shown in Fig. 11 with straight lines. We see that the theoretical dependences
agree well enough with the experimental points. From the timescale of weak
scintillations we can determine the distance to the layer responsible for the
radiowave modulation. If $L$ is the distance from the observer to the
turbulent layer and $R$ is the distance from the observer to the pulsar, the
effective distance is
$\displaystyle 1/r_{\textrm{ef}}=1/(R-L)+1/L.$ (15)
The effective velocity is
$\displaystyle V_{\textrm{ef}}=V_{\textrm{o}}(R-L)/R+V_{p}L/R,$ (16)
where $V_{\textrm{o}}$ and $V_{\textrm{p}}$ are the components of the
velocities of the observer and pulsar in the picture plane respectively. The
pulsar velocity is $V_{p}=36.6$ km/s [6]; therefore, $V_{p}\cong
V_{\textrm{o}}$, and $V_{\textrm{ef}}\cong V_{p}\cong 30$ km/s. Using this
value of $V_{\textrm{ef}}$ in (14) and supposing $t_{0}=15$ min at the
frequency $f=4.8$ GHz [25], we obtain
$\displaystyle L(R-L)/R\cong 9.7\textrm{\leavevmode\nobreak\ pc}.$ (17)
For this equation we find the solutions $L_{1}=10$ pc and $L_{2}=252$ pc.
Therefore, the turbulent layer is either near the observer or near the pulsar.
It should be noted that parameters of the scintillations of the pulsar PSR
B0950$+$08 are rather close to those of the pulsar B0437$-$47 [5], which is
located at approximately the same galactic longitude. For the latter pulsar it
was shown in [5] that its scintillations as well as those of the quasar PKS
0405$-$385, at a small angular distance, are determined by a layer of the
medium with enhanced turbulence at 10 pc from the observer. Therefore, we can
accept that the pulsar PSR B0950$+$08 scintillates on the same layer of the
turbulent medium and that the right solution is $L_{1}=10$ pc.
Table 3: Local turbulence levels PSR | $D_{s}(\rho_{0})$, | $L$, | $D_{s}(\rho_{0})/L$, | $\Delta N_{e}$,
---|---|---|---|---
| rad2 | pc | rad2/pc | cm-3
B0329$+$54 | 0.034 | 1000 | $3\times 10^{-5}$ | $7.7\times 10^{-4}$
B0437$-$47 | 0.0033 | 10 | $3\times 10^{-4}$ | $2.4\times 10^{-3}$
B0809$+$74 | 0.0003 | 433 | $1\times 10^{-6}$ | $1.4\times 10^{-4}$
B0950$+$08 | 0.002 | 10 | $2\times 10^{-4}$ | $2\times 10^{-3}$
Let us estimate the turbulence level toward PSR B0809$+$74 and PSR
B0950$+$08\. Note that we cannot use for the comparative analysis the values
of $C_{Ne}^{2}$, because they change their dimension with changing spectrum
exponent $n$. We will use the values of the structure function on a selected
scale $\rho_{0}=10^{7}$ m at a frequency $f=1$ GHz. The local level of
turbulence is related to the gradient $D_{s}(\rho_{0})$ as follows:
$(d/L)D_{s}(\rho_{0})\approx D_{s}(\rho_{0})/L$, where $L$ is the
characteristic thickness of the layer of the medium. The value of
$D_{s}(\rho_{0})/L$ is proportional to the mean square of electron density
fluctuations on inhomogeneities with a characteristic scale $\rho_{0}$; for
$\rho_{0}=10^{7}$ m we have:
$\displaystyle D_{s}(\rho_{0})/L\textrm{\leavevmode\nobreak\
[rad${}^{2}$/pc]}$ (18) $\displaystyle{}\cong 50\langle[\Delta
N_{e}(\rho_{0})]^{2}\rangle\textrm{\leavevmode\nobreak\ [\leavevmode\nobreak\
cm${}^{-6}$]}.$
Table 3 lists the values of $D_{s}(\rho_{0})$ at a frequency $f=1$ GHz
together with the values of $D_{s}(\rho_{0})/L$ and $\Delta N_{e}$ for the
pulsars studied by us in [3, 5] and in this paper. The distribution of the
estimated local turbulence levels corresponds to three components of the
interstellar medium mentioned in the Introduction. The greatest values of
$\Delta N_{e}$ correspond to the layer with an enhanced turbulence level
(layer C): the pulsars PSR B0437$-$47 and PSR B0950$+$08 scintillate on its
inhomogeneities. The minimum value of $\Delta N_{e}$ obtained toward the
pulsar PSR B0809$+$74 corresponds to a cavern. The intermediate value of
$\Delta N_{e}$ is obtained toward the pulsar PSR B0329$+$54; it corresponds to
turbulent plasma in the space between spiral arms (component A).
## 6 CONCLUSION
As a result of the analysis of the phase structure functions obtained by us
from the study of interstellar scintillations toward PSR B0809$+$74 and PSR
B0950$+$08 at low frequencies 41–112 MHz with invoking the previously
published higher frequency data, we have shown that the spectrum of
interstellar plasma inhomogeneities toward both pulsars is described by a
power law. The spectrum exponent toward PSR B0950$+$08 differs appreciably
from the Kolmogorov exponent; it is $n=3.00\pm 0.05$. Toward PSR B0809$+$74
the spectrum is a power law with an exponent $n=3.7\pm 0.1$; within the error
limits this correspond to the Kolmogorov spectrum. The analysis of the
frequency dependence of the diffraction parameters has shown that toward PSR
B0950$+$08 the transition to the weak scintillation mode takes place
approximately at a frequency of 200–300 MHz; here is a turnover in
relationships $t_{\textrm{dif}}(f)$ and $f_{\textrm{dif}}(f)$. The
experimental data match well enough the dependence expected by the theory.
As a result of the analysis of the temporal and frequency phase structural
functions for a purely diffractive model and model with a strong angular
refraction, we have shown that toward PSR B0950$+$08 there exists a strong
angular refraction. The conducted analysis of scintillations of this pulsar at
different frequencies has shown that the distribution of inhomogeneities along
the line of sight is not uniform and that the scintillations of PSR B0950$+$08
take place on a turbulent layer with enhanced electron density (layer C),
which is located at a distance of about 10 pc from the observer. The same
layer is responsible for the scintillations of PSR B0437$-$47 and and of the
quasar PKS 0405$-$385\. Here the local turbulence level on a scale
$\rho_{0}=10^{7}$ m is 20 times higher than in the extended region with a size
of the order of 430 pc responsible for the scintillations of PSR B0809$+$74.
## ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research (project
codes 06-02-16810 and 06-02-16888) and by the Program of Basic Research of the
Presidium of the Russian Academy of Sciences ‘‘Origin and evolution of stars
and galaxies’’.
## References
* (1) J. W. Armstrong, B. J. Rickett, and S. R. Spangler, Astrophys. J. 443, 209 (1995).
* (2) V. I. Shishov and T. V. Smirnova, Astron. Zh. 79, 810 (2002) [Astron Rep. 46, 731 (2002)].
* (3) V. I. Shishov, T. V. Smirnova, W. Sieber, et al., Astron. and Astrophys. 404, 557 (2003).
* (4) T. V. Smirnova, V. I. Shishov, W. Sieber, et al., Astron. and Astrophys. 455, 195 (2006).
* (5) T. V. Smirnova, C. R. Gwinn, and V. I. Shishov, Astron. and Astrophys. 453, 453 (2006).
* (6) W. E. Brisken, J. M. Benson, W. M. Goss, and S. E. Thorsett, Astrophys. J. (Letters) 573, 111 (2002).
* (7) J. M. Cordes, J. M. Weisberg, and V. Boriakoff, Astrophys. J. 288, 221 (1985).
* (8) A. V. Pynzar’ and V. I. Shishov, Astron. Zh. 74, 663 (1997) [Astron Rep. 41, 586 (1997)].
* (9) S. L. Snowden, D. P. Cox, D. McCammon, and W. T. Sanders, Astrophys. J. 354, 211 (1990).
* (10) N. D. Bhat, Y. Gupta, and A. Pramesh Rao, Astrophys. J. 500, 262 (1998).
* (11) J. Dennett-Thorpe and A. G. de Bruyn, Astrophys. and Space Sci. 278, 101 (2001).
* (12) B. J. Rickett, L. Kedziora-Chudczer, and D. L. Jauncy, Astrophys. J. 581, 103 (2002).
* (13) J. M. Rankin, R. Ramachandran, and S. A. Suleymanova, Astron. and Astrophys. 429, 999 (2005).
* (14) T. V. Shabanova and Yu. P. Shitov, Astron. and Astrophys. 418, 203 (2004).
* (15) J. H. Taylor, R. N. Manchester, and A. G. Lyne, Astrophys. J. 88, 529 (1993).
* (16) S. Johnston, G. Hobbs, S. Vigeland, et al., Monthly Not. Roy. Astron. Soc. 364, 1397 (2005).
* (17) T. V. Smirnova, V. I. Shishov and D. R. Stinebring, Astron. Zh. 75, 866 (1998) [Astron Rep. 42, 766 (1998)].
* (18) F. G. Smith, N. C. Wright, Monthly Not. Roy. Astron. Soc. 214, 97 (1985).
* (19) B. J. Rickett, Wm. A. Coles, and J. Markkanen, Astrophys. J. 533, 304 (2000).
* (20) J. A. Phillips and A. W. Clegg, Nature 360, 137 (1992).
* (21) G. R. Huguenin, J. H. Taylor, and M. Jura, Astrophys. J. (Letters) 4, 71 (1969).
* (22) S. Johnston, L. Nicastro, and B. Koribalski, Monthly Not. Roy. Astron. Soc. 297, 108 (1998).
* (23) F. G. Smith and N. C. Wright, Monthly Not. Roy. Astron. Soc. 214, 97 (1985).
* (24) V. Balasubramanian and S. Krishnamohan, Astron. and Astrophys. 6, 35 (1985).
* (25) V. M. Malofeev, V. I. Shishov, W. Sieber, et al., Astron. and Astrophys. 308, 180 (1996).
* (26) V. I. Shishov, T. V. Smirnova and S. A. Tul’bashev, Astron. Zh. 82, 281 (2005) [Astron Rep. 49, 250 (2005)].
* (27) A. V. Pynzar’ and V. I. Shishov, Astron. Zh. 57, 1187 (1980) [Sov. Astron. 24, 685 (1980)].
|
arxiv-papers
| 2008-10-07T07:54:12
|
2024-09-04T02:48:58.164052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. V. Smirnova, V. I. Shishov",
"submitter": "Tatiana Smirnova",
"url": "https://arxiv.org/abs/0810.1118"
}
|
0810.1226
|
Dissertation
for the degree of
Doctor of Philosophy
in
Physics
Traffic Dynamics of
Computer Networks
Attila Fekete
Supervisor: Prof. Gábor Vattay, D.Sc.
Eötvös Loránd University, Faculty of Science
Graduate School in Physics
Head: Prof. Zalán Horváth, MHAS
Statistical Physics, Biological Physics and
Physics of Quantum Systems Program
Head: Prof. Jenő Kürti, D.Sc.
Department of Physics of Complex Systems
Eötvös Loránd University
Budapest, 2008
For little Bori
---
## Acknowledgments
First and foremost I would like to thank my supervisor Prof. Gábor Vattay for
his help in guiding me through my research. I also thank him for his patience
for waiting until I finally finished this thesis. I would also like to thank
Prof. Ljupco Kocarev for his kind invitation to the University of California,
which was an invaluable experience. I am also grateful to the members of the
Department of Physics of Complex Systems for their courtesy. I am deeply
indebted to Máté Maródi for many fruitful discussions and his comments on my
thesis. I would also like to express my sincerest thanks to the staff of
Collegium Budapest for the peaceful atmosphere and unflagging support.
Without the comfort, help and encouragement of my family I would not have been
able to accomplish my study. I thank my wonderful wife for her love, her
wholehearted support, for proofreading the manuscript, and for lending me her
favorite desk. I thank my little daughter Bori for the joy of being with her,
for providing me with extra energy and for proving that I do not need that
much sleep at all. I am also thankful to my parents and to my brother for
encouraging me at all times. I am also grateful to my in-laws for their
selfless assistance, especially for the continual baby-sitting. Special thanks
go to Thomas Cooper for professional proofreading.
Last but not least I would like to thank the permanent members of the “Tarokk
Department”, and my other friends for tolerating my prolonged absence from
their social life.
###### Contents
1. 1 Introduction
1. 1.1 The Internet
1. 1.1.1 The short history of the Internet
2. 1.1.2 The structure of the Internet
3. 1.1.3 Traffic on the Internet
2. 1.2 Data transport mechanisms
1. 1.2.1 The User Datagram Protocol
2. 1.2.2 The Transmission Control Protocol
2. 2 Traffic dynamics in infinite buffer
1. 2.1 The fluid approximation
2. 2.2 Preliminary results of traffic modeling
1. 2.2.1 Single session models
2. 2.2.2 Multiple session models
3. 2.2.3 The Network Simulator – ns-2
3. 2.3 The infinite-buffer network model
4. 2.4 Dynamics of a single TCP
5. 2.5 Discussion
1. 2.5.1 Local Area Networks
2. 2.5.2 Wide Area Networks
3. 2.5.3 Dynamics of parallel TCPs
6. 2.6 Conclusions
3. 3 Traffic dynamics in finite buffer
1. 3.1 The finite buffer model
2. 3.2 Dynamics of a single TCP
3. 3.3 Discussion
1. 3.3.1 The interpretation of $A(\cdot)$ and the effective loss
2. 3.3.2 Histograms and probability distributions
4. 3.4 Conclusions
4. 4 Traffic on complex networks
1. 4.1 Preliminary results of topology modeling
1. 4.1.1 The Barabási–Albert model
2. 4.1.2 Other network models
3. 4.1.3 Earlier results regarding betweenness
2. 4.2 The network model
3. 4.3 Simulation of large computer networks
1. 4.3.1 The AIMD model
2. 4.3.2 Performance of different bandwidth distribution strategies
4. 4.4 Discussion
1. 4.4.1 Master equation for cluster size and in-degree
2. 4.4.2 The solution of the master equation
3. 4.4.3 Joint distribution of cluster size and in-degree
4. 4.4.4 Distributions of cluster size and in-degree
5. 4.4.5 Conditional probabilities and expectation values
6. 4.4.6 Conditional distribution of edge betweenness
5. 4.5 Conclusions
5. 5 Concluding remarks
6. A Mathematical proofs
1. A.1 Series expansion of $L(c)\,G(x)$
2. A.2 Expansion of the Kronecker-delta function
3. A.3 The $\alpha\to 0$ limit of joint distribution $\mathbb{P}_{\tau}(n,q)$
## Chapter 1 Introduction
The objects, laws, and phenomena of Nature have been the subject of physics
for hundreds of years [1]. In the second half of the 20th century, however,
new interdisciplinary and applied branches of physics were developed that
merged a wide range of scientific disciplines with physics including economy,
biology, chemistry and geology. Most of the new branches of physics could not
have evolved as they did without a specific new technological invention,
namely computer technology, which developed independent from and parallel to
physics. With the help of computers new research methods became available,
e.g., time-series analysis, computer simulation, and data mining.
As the use of computers was spreading across the globe, computers themselves
not only became increasingly useful tools for the research community, but
their evolving network attracted growing academic interest. As a research tool
the computer become the subject of research itself. In a pioneering work by
Csabai in 1994 [2] the traffic fluctuations of the then Internet as it existed
at the time was investigated. The author found that the power spectrum of the
traffic delays is $1/f$-like, similarly to other collective phenomena, e.g.,
highway traffic. Nowadays, a new interdisciplinary science is forming to
explore and model complex networks [3], in particular the Internet.
The Internet is an exceptional example of complex networks in a number of
aspects. Firstly, the structure of complex networks is often the subject of
research. The Internet’s infrastructure makes it possible to carry out
measurements on the network cheaply and easily on an incomparable scale.
Secondly, data traffic runs in the network, which adds another level of
complexity to the system. Thirdly, the Internet is a human engineered physical
network, which matches the complexity of some biological systems.
A useful mathematical abstraction of a network is a graph, because the number
of elements of real networks is finite. However, the number of elements of
complex networks is too large for the individual consideration of each network
element. Moreover, the exact principles governing connections between
different network components are usually unknown. Therefore, one should rely
on statistical methods, specifically the tools of statistical physics, in
order to describe the structure of complex networks.
The optimization of traffic performance has great practical importance. The
data flows can be regarded as interacting dynamical systems superposed onto
the network infrastructure. The theory of dynamical systems can therefore
prove to be a useful tool for studying network traffic.
All in all the Internet is an interesting new area of academic research and
several well established tools of physics can be quite useful for studying it.
Since the Internet has many layers, a number of different components and,
moreover, it is in constant development, it would be an impossible task to
cover all aspects of its operation. Instead, I will concentrate on the
dynamical modeling of the Transmission Control Protocol (TCP), the most
important traffic regulatory algorithm of the current Internet. After the
introductory chapter where the most important concepts of the Internet are
introduced I begin my survey with the investigation of TCP operating on an
elementary network configuration: a single buffer serving a link. This
scenario comprises the building blocks of Internet traffic. I proceed further
with the refinement of the first model, and I consider the finite storage
capacity of routers in the next chapter. After the analytic and simulation
study of the previous elementary single buffer models a more complex model of
the Internet follows. Specifically, in the last chapter I examine the problem
of efficient capacity distribution in a growing tree-like network.
### 1.1 The Internet
#### 1.1.1 The short history of the Internet
The price and sheer size of the first computers restricted their applicability
in the military and academic sphere. Motivated by the military needs of the
United States in the cold war era a novel concept, the theory of packet-
switching, was proposed by the Advanced Research Projects Agency (ARPA) to
connect distant computers in a decentralized manner. The concept of packet-
switching means that, contrary to connection-based circuit-switching,
resources are not reserved for communication between host and destination, but
data is split into small datagrams which are transmitted through the network
individually. The first physical network was constructed in 1969 between four
US Universities: the University of California Los Angeles, Stanford Research
Institute, University of Utah and University of California Santa Barbara. This
small network, called ArpaNet, is commonly perceived as the origin of the
current Internet. Over the course of the following years the network grew
gradually and connected more and more universities. By 1981 the number of
hosts had grown to more than 200.
Based on ARPA’s research, and that of its successor DARPA the International
Telecommunication Union (ITU) started developing the packet-switched network
standards. In 1976 the ITU standard was approved as X.25, and provided the
basis of the international and public penetration of packet switched network
technology. Using the X.25 and related standards, a number of industrial
companies created their own networks. The most notable was the first
international packet-switched network, referred to as the International Packet
Switched Service (IPSS). In 1978 IPSS was launched in Europe and the US with
the collaboration of the British Post Office, Western Union International and
Tymnet. By 1981 it covered Europe, North America, Australia and Hong Kong. The
X.25 standard also allowed the commercial use of the network, as opposed to
ArpaNet, which being a government founded project restricted its use to
military and academic purposes.
In the first packet switched networks the network infrastructure itself
assured reliable packet transfer between hosts. This approach made it
impossible to connect different networks with different network protocols. In
order to overcome this difficulty a novel concept of internetwork protocol,
the TCP, was developed. With TCP the differences between different network
protocols were hidden and the hosts became responsible for the reliability of
the data transfer. The first specifications of TCP were given in 1974. After
several years of development and testing the TCP standards were published in
1981. This paved the way for the current Internet. Since then every subnet of
the Internet has adopted TCP. A detailed introduction to the protocol will be
presented in the next section.
The pure network infrastructure would have been useless without user
applications. The basis of many early Internet applications was Unix to Unix
Copy (UUCP), developed in 1979. The most notable services using UUCP were
electronic mail, Bulletin Board Systems (BBS) and Usenet News. At the dawn of
the Internet era the most important service was, without doubt, email. Most of
the early Internet traffic was generated by emails, but even in recent years
email constitutes a significant share of Internet traffic. BBS and Usenet
services were popular among home users with slow modem connections who did not
have direct Internet connections. Messages, news, articles, programs or data
could be uploaded and/or downloaded after the user dialed into a server. BBS
and Usenet servers then periodically exchanged data via UUCP.
By the beginning of the 1990’s BBSs and Usenet had declined in importance,
mainly due to the new information medium, the World Wide Web (WWW). The WWW
was born of the merging of the Internet and the paradigm of hypertext in the
European particle physics laboratory, the CERN. The WWW started conquering the
Internet after the debut of the Mosaic web browser in 1993.
The Mosaic browser was such an enormous success that it even affected the
development of the Internet itself. The Internet crossed the borders of the
academic and industrial research domain and opened up to the wider public. The
process was accelerated by rapid technological advances in computer technology
that made personal computers a part of people’s everyday lives. The combined
effect of the above led to the Internet boom in the 1990’s, when a whole new
industry formed around the Internet.
By now the Internet has expanded even further than computers. Internet
telephony (Voice over IP), mobile Internet (GPRS, UMTS), web cameras, wireless
networks, personal digital assistants (PDAs), and sensor networks are a few
examples of the current trends. The new technologies make both the structure
and the traffic of the Internet more and more complex. I review these issues
in the following.
#### 1.1.2 The structure of the Internet
Since the development of the Internet was not regularized by any central
authority and it has been influenced by a number of random effects the
structure of the network is highly irregular. Nevertheless, the Internet can
be divided into smaller segments, called Autonomous Systems (ASs). Each AS is
administered by a separate organization, e.g. a university, an Internet
Service Provider (ISP), or a government, and is usually organized in
hierarchical, tree-like structure. ASs are connected to one another via the
_Internet backbone_. The Internet backbone is built from high capacity links,
currently up to a couple of $10\mathrm{Gbps}$. On the other end of the
hierarchy end users connect to the network. The available bandwidth for end
users can be in the range of $56\mathrm{Kbps}$ modems to $20\mathrm{Mbps}$
business ADSL. If we consider AS as the unit of the network, and
interconnections between them as links, then we speak of AS level topology.
Internet also can be viewed on a much smaller scale consisting of two basic
components: nodes and links. Nodes are devices, e.g. computers, cell phones,
PDAs, routers, switches or hubs, and links are connections between them, e.g.
cable (Ethernet, optical fiber), radio (WiFi, Bluetooth), infrared (IrDA), or
even satellite connections. Those nodes which have multiple connections must
decide in which direction they forward the through traffic. These nodes are
usually referred to as _routers_. This detailed view is called the router
level topology.
Internet topology has been studied both on AS [5, 4] and router level [6, 7,
8, 9]. On both level the Internet can be modeled as a _graph_ from graph
theory. One of the most fundamental quantities used for describing the
structure of a graph is the degree sequence, which is to say the number of the
neighbors of nodes. It has been found that the distribution of the degree
sequence follows a power law $P(k)\sim k^{-\delta}$ on both level. The
appearance of a power law indicates the scale-free nature of a particular
object, so a graph the degree distribution of which follows power law is
called _scale-free graph_. Note that in recent years the statistical
properties of other scale-free networks have been investigated by the physics
community as well [10, 11, 12, 13]. Examples of such networks vary from social
interconnections and scientific collaborations [14] to the WWW [15].
Figure 1.1: Internet AS level topology collected between 4–17 April 2005 by
CAIDA [16]. The angular position of nodes corresponds to the geographical
longitude of the AS headquarters. The radial position is calculated from the
out-degree of ASs.
Several projects have been launched over the past decade in order to map the
Internet topology. For example, the Macroscopic Topology Measurements project
of CAIDA, a research group located at the University of California San Diego,
surveys the Internet continuously with probe packets from a couple of dozen
monitoring hosts. The visualization of the AS level map produced by CAIDA is
shown in Fig. 1.1. Rocketfuel is a Internet mapping engine, developed at the
University of Washington, which aims at discovering ISP router level
topologies [17]. The engine makes use of routing tables to focus measurements
to certain ISPs, exploits the properties of Internet Protocol (IP) routing to
eliminate redundancy, and uses data from nameservers in order to classify
routers.
It should be noted that the known Internet topology is only a sample of the
real one. The surveyed topology is obtained from measurements, mostly via a
program called traceroute. The program can discover routes between the
traceroute source and given destination hosts. Since the number of sources is
limited only a section of the real network can be visible in one experiment.
It is therefore questionable whether the observed topology resembles the
actual Internet topology. Recently it has been shown that a traceroute-based
experiment can produce strong bias towards scale-free topology [18],
especially when the number of sources is one or two. Moreover, it has been
shown that a badly designed measurement can show scale-free topology even if
the original network is regular [19].
#### 1.1.3 Traffic on the Internet
The properties of the Internet traffic are as important as the structure of
the network itself. Since the time-scale of the evolution of the network
infrastructure is much larger than the time-scale of the traffic flow the
network infrastructure can be considered as a static background behind the
traffic dynamics. In comparison with the changes in the network traffic the
dynamic changes in the network structure can be neglected.
The Internet traffic is governed by communication protocols, which can be
classified into separate abstract layers according to their functionality.
Each layer takes care of one or more separate tasks of data transfer and
handles data towards a lower or an upper layer. User applications usually
communicate with the topmost layer, whilst the lowest layer deals with the
physical interaction of the hardware.
The most important classification regarding the Internet is the TCP/IP
protocol suite [20, 21], which includes five or four layers. A more general
and detailed model is the OSI model, which includes seven layers. The concept
of layers is quite important, since it provides transparency for user
applications in a very heterogeneous environment. In order to overview the
mechanisms behind the Internet traffic let us introduce the four-layer model
of the TCP/IP suite:
* •
The topmost, fourth layer of TCP/IP suite is called _Application layer_. It
provides well-known services such as TELNET, HTTP, FTP, SSH, DNS, and SMTP.
User programs should provide data to an application layer protocol in a
suitable format.
* •
The next layer is the _Transport layer_ , which is responsible among other
things for flow control, error detection, re-transmission, and connection
handling. The two most important protocols in this layer are TCP and User
Datagram Protocol (UDP), which will be discussed in more detail in Section
1.2. They represent two conceptually quite different transport mechanisms: TCP
provides reliable, connection-based data transfer, while UDP serves as an
unreliable, connectionless, best effort transport mechanism. Other protocols
at this layer are SCTP developed for Internet telephony, and RTP designed for
real-time video and audio streaming.
* •
The following layer is referred to as the _Internet layer_. This layer solves
the problem of addressing and routing of packets. The IP and the obsolete x.25
protocols reside in this layer. IP hides the details of the network
infrastructure, and allows the interconnection of different network
architectures.
* •
Finally, the lowest layer of the TCP/IP suite is the _Network access layer_ ,
which handles physical hardware and devices. Notable examples on this layer
are the ethernet, WiFi, and modems.
In order to understand the workings of the Internet, let us take the example
of a typical Internet application: let us suppose that Alice wants to download
a file from Bob. Since Alice wants to get an exact copy of the file, she
starts an FTP session. First, the FTP protocol builds a connection between the
two computers. Then the file is split into small datagrams, which are passed
on to the TCP protocol on Bob’s computer. The TCP protocol adds a header to
the datagrams, including a sequence number, a timestamp, and some other
information which ensures reliability. Then TCP passes the datagrams on to the
IP protocol, which adds its own header. The IP header contains addressing
information. The resulting IP packet is put into the outgoing queue of Bob’s
computer. If the queue is empty, then the packet is sent to the Network
Interface Card (NIC), otherwise it has to wait until the preceding packets
have been served. The NIC card disassembles the packet into ethernet frames
and puts them onto the physical cable. The frames travel to the default router
in Bob’s network and the router’s NIC assembles them back into an IP packet.
Based on the destination address in the IP header, the router decides in which
direction the packet should be forwarded and the packet is put into the
outgoing queue of the corresponding direction. The packet is then disassembled
and transferred again over the next cable. The procedure is repeated until the
packet arrives at its final destination. The actual method of data transfer on
the _Network access layer_ can differ from the above mentioned ethernet
method. If Alice uses a dial-up connection, for instance, the last step of the
packet’s path is over a telephone line via a modem. At Alice’s computer the IP
protocol takes the packet and passes it on to the TCP protocol. The TCP
acknowledges the packet and inserts it into the missing part of the file.
Finally, when Alice’s computer has received all the pieces of the file, the
FTP protocol saves the whole file to its destination on her computer.
Although both packet-switched and connection-based data transfer are present
in the above example, the Internet is called a packet-switched network because
the _Internet layer_ , which is the fundamental core of the Internet, utilizes
solely packet-switched technology. Other layers can be either packet or
circuit-switched. Ethernet traffic is packet-switched, for example, but modem
traffic is carried through circuit-switched telephone lines. Higher level
protocols (e.g. FTP, TELNET, SSH) are usually connection oriented, too.
Let us study the _Internet layer_ in more detail. First of all, packets are
injected into the _Internet layer_ randomly by higher level protocols at
certain source nodes. Then packets are served sequentially and forwarded to
neighboring nodes by routers or, if they have arrived to their destination,
removed from the network. If a router is busy serving a packet then any
incoming packet is placed into a buffer and has to wait for serving. If the
queue in the buffer has reached the buffer’s maximum capacity then all
incoming packets are dropped until the next packet in the queue is served and
an empty space becomes available in the buffer. The event when a buffer
becomes full is called _congestion_. The above described router policy, called
_drop-tail_ , is the most wide-spread nowadays. Other router policies are also
in use. The Early Random Drop (ERD) and Random Early Detection (RED) polices,
for instance, drop incoming packets randomly before the buffer becomes fully
occupied in order to forecast possible congestion to upper level protocols.
The difference between the two policies is that the drop probability depends
on the instantaneous queue length in the former case and the average queue
length in the latter. It is possible to give priority to certain packets in
order to provide Quality of Service (QoS) for certain applications, but
routers usually serve packets in First In, First Out (FIFO) order. The serving
rate of packets depends on the actual packet size and the bandwidth of the
link after the buffer. Packets obviously suffer propagation delay during their
delivery, is a consequence of two factors: link and from queuing delay. The
former is constant for a given route, but the later varies randomly with queue
lengths along the packet’s path.
The product of the link delay and link capacity, in short the _bandwidth-delay
product_ , equals the number of packets that a link can transfer
simultaneously. If this quantity is large compared to the buffer size then the
constant link delay is the dominant constituent of the propagation delay. Wide
Area Network (WAN) links are typical examples of this. On the other hand, if
the bandwidth delay product is small compared to the buffer size then the
varying buffering delay is the dominant component. Such links can be found in
Local Area Network (LAN). We will see later that the two scenarios induce
different TCP dynamics.
It is evident that queuing theory plays an important role in the modeling of
packet-switched networks in general and the Internet in particular. However,
queuing theory has been developed much earlier than the advent of packet-
switching technology. The first motivation and important application of
queuing theory was actually a circuit-switched network, the classical
telephone system.
The properties of two quantities, namely the inter-arrival and the service
times of customers, affect the behavior of queuing systems most fundamentally.
Other quantities, e.g. the size of the customer population, the number of
operators, the system capacity etc., also have an impact on the behavior of
the system, but they do not affect the essential properties of the queuing
system. Both the inter-arrival and the service time series can be modeled by
discrete time stochastic processes. It is usually assumed that both the inter-
arrival and the service times are independent and identically-distributed
(IID) random variables. Furthermore, in the most simple case, both inter-
arrival and service times are memoryless processes, that is they are
exponentially distributed random variables. This model is called Poisson
queue, since both the number of arrivals and the number of departures in a
finite time interval follow Poisson distribution. Poisson queues have been
studied extensively and they proved to be excellent models of telephone call
centers and telephone exchange centers. Most of the arising questions
regarding Poisson queues have been answered analytically [22].
Internet traffic has been analyzed on various layers of the above TCP/IP
suite. In a pioneering work by Leland et al. [23] the authors collected and
studied several hours of ethernet traffic with $20$–$100\mu s$ resolution.
They found that autocorrelations in the captured traffic decayed slower than
exponential, that is the system has long-range memory. This result indicated
problems with Poisson queuing models for packet-switched networks, since in a
Poisson queuing system autocorrelations would fall exponentially [24].
Furthermore, it has been shown that the time series of the aggregated Ethernet
traffic is statistically self-similar, and has fractal properties. Paxson and
Floyd [25] studied the usability of Poisson models for application layer
protocols and the corresponding IP traffic. They found that, though the
traffic followed a 24-hour periodic pattern, Poisson processes with fixed
arrival rates are acceptable models for user initiated sessions (FTP, TELNET)
for intervals of one hour or less. For machine initiated sessions (SMTP,
NNTP), however, the Poisson model failed even for short time-scales.
Furthermore, packet level traffic deviated considerably from Poisson arrivals
as well. Similar evidence has been found in WWW traffic [26]. Furthermore, it
has been shown that the distribution of the packet inter-arrival times follows
power law. Feldmann et al. [27] have presented the wavelet analysis of WAN
traffic samples captured around the birth of the World Wide Web between ’90
and ’97. It has been found that as WWW traffic started dominating the network
traffic gradually different scaling behavior appeared in short- and long-time
scales. The authors concluded that TCP dynamics might be responsible for
short-time scaling and application layer traffic characteristics for long-time
scaling.
All the above properties are in strong contrast with the properties of the
Poisson queuing systems, e.g. telephone networks, where both the correlations
and the inter-arrival time distribution decay exponentially. It implies that
well developed classical models, which provide excellent descriptions of
circuit-switched traffic, are essentially useless for the description of the
Internet. New traffic models, which provide realistic synthetic traffic, were
required. A few important traffic models of the Internet will be presented in
Section 2.2.
There are several theories which explain the origins of the observed long-
range dependent traffic. One explanation can be that the observed traffic is
the superposition of individual effects which happen on separate network
layers and on very different time-scales; from several minutes of user
interaction through a couple of seconds of application response until the
microsecond-scale of network protocol operation. Further assumptions are that
heavy-tailed file size distribution [26, 28, 29], or heavy-tailed processor
time distribution is behind the phenomena. There has also been some debate on
whether the TCP protocol in itself is able to generate long-range dependent
traffic [30] or not [31]. The TCP’s exponential backoff mechanism is also a
possible source of heavy-tailed inter-arrival times [32].
### 1.2 Data transport mechanisms
The Internet is an enormous data highway between computers, where data packets
play the role of vehicles and links serve as the road system. As on normal
highways, congestions can form at bottlenecks if the capacity of a junction is
exceeded by the traffic demand.
The dynamics of the Internet traffic is governed by protocols of the
_Transport layer_. Protocols on this layer control directly the injection rate
of IP packets into the network. Almost all the Internet traffic is governed by
two protocols, namely the TCP and the UDP. Therefore, understanding the
operation of these protocols is very important from the point of view of
traffic modeling. For example, fundamental questions are how distant hosts
utilize the network infrastructure and whether they can cause persistent
traffic congestion or not.
The performance of the network can be severely degraded as a result of
persistent congestion. Congestion should therefore be avoided. Just such a
congestion collapse did indeed occur in 1986 in the early Internet, when the
useful throughput of NFSnet backbone dropped three orders of magnitude. The
cause of this collapse was the faulty design of the early TCP. Instead of
decreasing the sending rate of packets after detecting congestion, the early
TCP actually started retransmitting lost packets, which led to an increasing
sending rate and positive feedback.
#### 1.2.1 The User Datagram Protocol
UDP is a very simple protocol, which provides a procedure for applications to
send messages to other applications with a minimum of protocol mechanism [33].
Neither delivery nor duplicate protection is guaranteed by UDP. Furthermore,
no congestion control is implemented in it either. UDPrealizes an open-loop
control design, that is no feedback about a possible congestion is processed.
The principal uses of UDPare the Domain Name System (DNS), streaming audio and
video applications (e.g. VoIP, IPTV), file sharing applications, the Trivial
File Transfer Protocol (TFTP), and on-line multiplayer games, to name a few.
Since UDPlacks any congestion avoidance and control algorithm, application
level programs or network-based mechanisms are required to handle congestion.
In streaming applications, for example, users are often asked for the
bandwidth of their access link, and UDPpackets are sent with the corresponding
fixed rate. Since UDPdoes not have any feedback mechanism congestion collapse
of the network due to UDP network overload is unlikely. However, aggressive
network utilization should be avoided, because it can block other protocols,
mainly TCP.
#### 1.2.2 The Transmission Control Protocol
The TCP protocol is complementary to the UDPprotocol in many sense. Contrary
to UDP, TCP is connection oriented, it guarantees in-order delivery and
duplicate protection, congestion control and avoidance. In addition, TCP is a
closed-loop design which can process feedback from packet delivery.
Accordingly, TCP is a much more complex design than UDP. In this section we
present an overview of TCP.
Among the applications using TCP are the WWW, email, Telnet, File Transfer
Protocol (FTP), Secure Shell (ssh), to name a few. Since these applications
are responsible for most of the current Internet traffic TCP is the most
dominant transport protocol at the moment. Accordingly, understanding the
workings of the TCP protocol has great importance in traffic modeling.
Since TCP is connection oriented, it does not start sending data immediately,
like UDP. Rather it uses a three-way handshake for connection establishment.
If the connection establishment phase is successful the data transfer phase
follows. Finally, when all the data has been sent, the connection is
terminated in the final phase. The connection establishment and termination
phases are usually short and involve only negligible amount of data compared
to the data transfer phase. I will therefore focus solely on the main phase,
neglecting the other two phases.
In the data transfer phase the TCP receiver acknowledges every arrived packet
by an acknowledgment packet (ACK). The ACK contains the sequence number of the
last data packet arrived in order. If a data packet arrives out of order, then
the receiver sends a duplicate ACK, that is an ACK with the same sequence
number as the previous one. Duplicate ACKs directly notify the TCP sender
about an out-of-order packet.
If all the packets are lost beyond a certain sequence number, then duplicate
ACK cannot notify the sender about packet losses. In order to recover from
such a situation, the TCP sender manages a retransmission timer. The delay of
the timer, the retransmission timeout (RTO), is updated after each arriving
ACK. The TCP sender measures the round-trip time (RTT), the elapsed time
between the departure of a packet and the arrival of the corresponding ACK.
The updated value of the RTO is calculated from the smoothed RTT, and the RTT
variation as defined in [34].
Packets are acknowledged after RTT time period from packet departure if the
transmission is successful. The data transfer would be very inefficient if the
TCP sender waited for the ACK of the last packet before it sent the next
packet. On the other hand, sending packets all at once would cause congestion.
In order to reach optimum performance without causing congestion, TCP manages
two sliding windows with the associated variables. On the sender side the
congestion window (cwnd) limits the allowed number of unacknowledged packets.
This way a cwnd number of packets is transmitted on average during a round-
trip time period. Since cwnd is used directly for congestion control it is
changed dynamically.
The other variable, the receiver’s advertised window (rwnd), is managed on the
receiver side. Rwnd is the size of a receiver buffer which can store out-of-
order packets temporally. The value of rwnd is included in every ACK, though
it usually does not change. Although the limit of the unacknowledged packets
is the minimum of cwnd and rwnd, the later is usually large enough not to
affect data transfer in practice. Rwnd therefore plays a much less important
role than cwnd. For the sake of simplicity I will assume that rwnd equals
infinity. Accordingly $\min(\textit{cwnd},\textit{rwnd})$ will be replaced
with cwnd in all the equations below where applicable. Let us keep in mind,
however, that this is an approximation.
Internet’s packet-switched technology implies that there are no reserved
resources for TCP. This approach is also called _best effort_ delivery.
Moreover, the Internet lacks any central management authority. Accordingly,
TCP does not have precise information about its fair share of the network
bandwidth in the ever-changing network conditions. In the previous section we
have seen that buffers are able to store excess traffic temporarily, but
pockets are dropped when a buffer becomes full. Flow control, the alteration
of rate at which packets are sent in order to get a fair share of the network
bandwidth without causing severe congestion, is one of the most important
tasks of the TCP. This goal is achieved by the continuous adjustment of the
congestion window and eventually the rate at which packets are sent.
Several TCP variants have been developed in recent years in order to enhance
its performance in different environments [35]. These variants differ mainly
in the congestion avoidance algorithm. The core concept, however, is the same
in all TCP variants and has not changed significantly since its first
specification in 1974\. The classical TCP variants (e.g. Tahoe, Reno) try to
find the fair bandwidth share by the following method: for every successfully
transmitted and lost packet they increase and decrease their sending rate,
respectively. This method is based on the observation that a packet loss is
most likely the result of a congestion event. Note that these TCP variants
obviously cause temporary congestions in the network in the long run. More
recent variants often try to detect upcoming congestions beforehand via
explicit congestion notifications (ECN) from routers or by detecting
increasing queuing delays from RTT fluctuations (e.g. Fast TCP).
I discuss the Reno TCP variant in more detail below, since currently this is
the most widespread variant in use. Its congestion control mechanism includes
the following algorithms: _slow start_ , _congestion avoidance_ , _fast
recovery_ , and _fast retransmission_ [36]. In figure 1.2 the schematic
development of cwnd due the above congestion control algorithms is shown.
There are two slow start periods at the beginning of the plot. This is
possible due to the wrong initial estimate of the slow start threshold
(ssthresh). After the value of ssthresh has been set to approximately half of
the maximum window the fast recovery, fast retransmission (FR/FR) algorithms
are able to take care of the upcoming packet losses. Note the small steps both
in the slow start and the congestion avoidance phase. The steps are due to the
bursty departure of packets.
Figure 1.2: Schematic plot of the development of the congestion window. The
hypothetical network can handle 20 packets simultaneously, denoted by the
dotted line. Cwnd might overrun this limit, because congestion is detected
only after RTT latency. Note the small plateaus both in the slow start and
congestion avoidance phase, which are due to the bursty arrival of packets.
##### Slow start and congestion avoidance
The core of the TCP congestion control mechanism is the slow start and the
congestion avoidance algorithms. A state variable, the ssthresh, is used to
determine whether the slow start or the congestion avoidance algorithm is used
to control data transmission. When cwnd exceeds ssthresh the slow start ends,
and TCP enters congestion avoidance. Ssthresh is recalculated when congestion
is detected by the following formula:
$\textit{ssthresh}=\max(\textit{cwnd}/2,2).$ (1.1)
The slow start algorithm is used at the beginning of data transfer to probe
the network and determine the available capacity. Slow start is used after
repairing losses detected by the retransmission timer as well. In slow start
phase TCP begins sending at most two packets, which is a “slow start” indeed.
Despite what the name might suggest, however, the growth of the packet sending
rate in this phase is quite fast actually: the cwnd is increased by one for
every ACK. This way the sending rate is doubled in every RTT, which means
exponential growth in time.
In congestion avoidance phase cwnd is increased by one every RTT period. This
implies linear growth in time, which is a much more moderate development than
the exponential growth in slow start. One common approximating formula for
updating cwnd after every non-duplicate ACK is:
$\textit{cwnd}\to\textit{cwnd}+\frac{1}{\textit{cwnd}}.$ (1.2)
This formula is not precisely linear in time, but the advantage of this
formula is that no auxiliary state variable is required for its application.
##### Fast retransmit and fast recovery
The packet sending rate is reduced drastically at the beginning of each slow
start phase. Although the slow start algorithm restores cwnd to ssthresh at an
exponential rate, its application might cause unnecessary performance
deterioration. In order to circumvent slow start algorithm when possible, fast
retransmit and fast recovery algorithms were introduced to the Reno version of
TCP in 1990 [37].
The fast retransmit algorithm uses the arrival of three duplicate ACKs as an
indication that a packet has been lost. After the arrival of the third
duplicate ACK the sender retransmits the missing segment without waiting for
the retransmission timer to expire. TCP does not enter slow start after fast
retransmission, but instead starts the fast recovery algorithm. Skipping slow
start is possible because each duplicate ACK indicates that a packet has been
removed from the network. Therefore, newly sent packets do not stress the
network further.
After fast retransmission ssthresh is set according to Eq. (1.1). In addition,
cwnd is halved,
$\textit{cwnd}\to\frac{\textit{cwnd}}{2}$ (1.3)
and for each duplicate ACK a new segment is sent if possible. After the first
non-duplicate ACK cwnd is set to ssthresh again, and TCP returns to congestion
avoidance. Note that slow start might be forced when cwnd is small and
duplicate ACKs are not accessible. Furthermore, if more than one packet is
lost within one RTT time period, then the FR/FR algorithms may not recover
from the loss either, and TCP can enter slow start algorithm instead. However,
if the packet loss rate is low and cwnd is large enough, then the slow start
algorithm is used only at the beginning of the TCP session, and cwnd is
governed in an additive increase, multiplicative decrease (AIMD) manner by the
congestion avoidance and FR/FR algorithms, respectively.
The idea behind the AIMD rule comes from the following simple control
theoretical arguments [38, 39]. In general, the control of the $\lambda$th
TCP’s cwnd can be given by
$w_{\lambda}(t_{i+1})=f(w_{\lambda}(t_{i}),y(t_{i}))$ where $f(w,y)$ is the
control function, which depends on the feedback (e.g. an ACK) from the system
$y(t_{i})\in\\{-,+\\}$, and the last value of the window $w_{\lambda}(t_{i})$.
The feedback is binary: $+$ and $-$ indicates whether to increase or decrease
traffic demand, respectively. If we restrict our study to control functions,
which are linear in $w_{\lambda}(t_{i})$, then we obtain
$w_{\lambda}(t_{i+1})=a_{y(t_{i})}+b_{y(t_{i})}w_{\lambda}(t_{i}),$ (1.4)
where the coefficients $a_{\pm}$ and $b_{\pm}$ are constants. It is obvious
that the control equation (1.4) is additive if $b_{\pm}=1$, and multiplicative
if $b_{\pm}\neq 1$. The most important special cases of the possible control
algorithms are collected in Table 1.1. A feasible control algorithm must
satisfy two important criteria: _convergence to efficiency_ and _fairness_.
Efficiency in this context means maximum possible usage of the available
resources and fairness means equal share of the bottleneck capacity. These
criteria give constraints on the coefficients $a_{\pm}$ and $b_{\pm}$. It has
been shown in [39] that the convergence to efficiency and fairness is provided
by the constraints $a_{+}>0$, $b_{+}\geq 1$, and $a_{-}=0$, $0\leq b_{-}<1$.
Moreover, it has been shown that the convergence is fastest, when $b_{+}=1$.
Therefore, the additive increase, multiplicative decrease control, which is
implemented in TCP, is the optimal control algorithm.
| $b_{+}=1$ | $b_{+}>1$
---|---|---
| $a_{+}>0$ | $a_{+}=0$
$b_{-}=1$ | Additive increase | Multiplicative increase
$a_{-}<0$ | Additive decrease | Additive decrease
$0<b_{-}<1$ | Additive increase | Multiplicative increase
$a_{-}=0$ | Multiplicative decrease | Multiplicative decrease
Table 1.1: Possible control algorithms with a linear control function.
##### The backoff mechanism
Normally in slow start or in congestion avoidance mode, the TCP estimates the
RTT and its variance from time stamps placed in ACKs. In some cases the
retransmission timer might underestimate RTT at the beginning of the data
transfer, and the retransmission timer might expire before the first ACK would
arrive back to the TCP sender. In order to avoid the persistent expiration of
the retransmission timer the so-called Karn’s algorithm [40] is applied.
According to the algorithm, if the retransmission timer expires before the
first ACK would return, then the value of the RTO is doubled. If the timer
expires again, then the timer is doubled repeatedly a maximum six consecutive
times. Since there is a definite ambiguity in estimating RTT from a
retransmitted packet the ACKs of two consecutive sent packets should arrive
back successfully in order for the TCP to estimate the RTT again and go back
to the slow start mode.
A similar situation might occur if the packet loss rate is high. In that case,
consecutive packets can be lost and the TCP might enter the backoff state,
even if RTT might actually be smaller than the retransmission timer. Since the
delay between packet departure is doubled, the effective bandwidth is halved
after each backoff step. TCP can reduce its packet sending rate with this
method below one packet per RTT.
## Chapter 2 Traffic dynamics in infinite buffer
In this chapter I study the TCP dynamics on an idealized single buffer network
model where the probability that a packet is lost at the buffer is negligible
compared to other sources of packet loss. The case of a semi-bottleneck buffer
when the size of the buffer is limited will be discussed in Chapter 3. First,
I introduce the important fluid approximation of TCP congestion window
dynamics in Section 2.1. In recent years many aspects of the TCP congestion
avoidance phase have been clarified. The most important results of the
literature are reviewed in Section 2.2. I define the network model under study
in Section 2.3. My results on the analytic study of the TCP congestion window
dynamics are presented in Section 2.4. The discussion of the model is given in
Section 2.5. Finally, I summarize my results in Section 2.6.
### 2.1 The fluid approximation
The equations of motion (1.1)–(1.3) are defined at ACK arrivals. The state
variables are therefore changed in discrete steps at discrete time intervals
(Fig. 1.2), often referred to as “in ACK time”. Note that ”ACK time” dynamics
is an essential, inherent property of TCP, because it is defined in the TCP
design and does not depend on the network environment where TCP is used.
In practice the discrete-time equations of TCP dynamics can be approximated
very well by continuous-time equations. Between two consecutive packet losses
the congestion window is changed according to the fluid “ACK time” equation
(1.2), that is
$\frac{dW}{dt_{\textrm{ACK}}}=\frac{1}{W}.$ (2.1)
Since the arrival of ACK packets is not uniform in time, the ACK and real time
averages of important quantities, for instance the throughput, are usually
different. It would be difficult and rather impractical to transform the
dynamics of the state variables from ACK to real time exactly. A usual
approximation is that the arrival rate of ACKs is estimated by the number of
packets in flight, that is the congestion window $W$ divided by the round trip
time $R$:
$\frac{dt_{\textrm{ACK}}}{dt}=\frac{W}{R},$ (2.2)
From the above equations one can obtain
$\frac{dW}{dt}=\frac{1}{R}.$ (2.3)
which is the fluid approximation of the congestion window dynamics in real
time. Although this real time approximation of TCP dynamics is often
sufficient, I will point out its defects. I will also present a roundabout
solution to the problems based on the fundamental “ACK time” dynamics of TCP.
Figure 2.1: The TCP congestion window evolution under deterministic packet
loss and constant round-trip time. The congestion window $W$ varies linearly
between the maximum value $W_{m}$ and its half, $W_{m}/2$.
As a simple example of the fluid model let us calculate the average
throughput, the transmitted data per unit of time, of a single TCP over a
lossy link [41]. Let us suppose that the round-trip time is constant and the
packet loss is deterministic. Considering these assumptions the congestion
window changes at a constant rate between consecutive packet loss events as
(2.3). The window is halved after each packet loss event. The window evolution
shown in Fig. 2.1 is therefore a periodic sawtooth in the interval
$[RW_{m}/2,RW_{m}]$ and in the range of $[W_{m}/2,W_{m}]$. The length of a
cycle is $RW_{m}/2$. The number of transmitted packets in a cycle equals the
integral of the congestion window for one period: $N=\frac{3}{8}W_{m}^{2}$.
Since in each cycle one packet is lost, the packet loss probability can be
expressed as $p=1/N$. Therefore, the average throughput $\bar{X}$ can be given
by
$\bar{X}=\frac{PN}{R\frac{W_{m}}{2}}=\frac{P/p}{R\sqrt{\frac{2}{3p}}}=\frac{P}{R}\frac{c_{0}}{\sqrt{p}}$
(2.4)
where $P$ is the size of the data packets and $c_{0}=\sqrt{\frac{3}{2}}$ is a
constant. The resulting formula, often referred to as the “ _inverse square-
root law_ ”, expresses the impact of a network on TCP dynamics. The formula
establishes a connection between throughput, an important characteristic of
TCP, and packet loss probability, an attribute of the network on which TCP
operates. The formula becomes inaccurate for large $p$, because multiple
packet losses, which force TCP into the neglected slow start phase, are more
probable in this case. The effect of multiple losses on different TCP variants
is diverse, so the validity range of the formula depends on the TCP variant
under consideration.
### 2.2 Preliminary results of traffic modeling
#### 2.2.1 Single session models
The simple model given above can be extended considerably in a number of
aspects. In a paper by Altman et al. [42] the TCP throughput for generic
stationary congestion sequence was studied. The model extends the previous
deterministic loss model to arbitrarily correlated loss sequences. The model
is based on the following difference equation
$X_{n+1}=\alpha S_{n}+\beta X_{n},$ (2.5)
where $X_{n}$ is the value of the throughput just prior to the arrival of loss
signal at $T_{n}$, $S_{n}=T_{n+1}-T_{n}$ is the time interval between
consecutive losses, and $\alpha$ and $\beta$ are the linear growth rate and
multiplicative decrease factor, respectively. From the time average of the
throughput the following loss formula was derived
$\bar{X}=\frac{P}{R\sqrt{p}}\sqrt{\frac{1+\beta}{2\left(1-\beta\right)}+\frac{1}{2}\hat{C}(0)+\sum_{k=1}^{\infty}\beta^{k}\hat{C}(k)},$
(2.6)
where
$\hat{C}(k)=\left(\mathbb{E}\left[S_{n}S_{n+k}\right]-\mathbb{E}\left[S_{n}\right]^{2}\right)/\mathbb{E}\left[S_{n}\right]^{2}$
is the normalized autocorrelation function of the loss interval process
$\left(S_{n}\right)_{n\in\mathbb{N}}$. The derived formula, applied for
uncorrelated Poisson process with $C(k)=0$ and $\beta=1/2$, provides the same
result as (2.4). A correlated loss interval scenario was modeled with
Markovian Arrival Process and the average TCP throughput was expressed with
the infinitesimal generator of the arrival process. Furthermore, the authors
derived bounds for the throughput in case of limited congestion window
evolution and discussed the effects of timeouts.
Padhye et al. [43] have studied the steady-state throughput of TCP Reno when
packet loss is detected via both duplicate ACKs and timeouts, and the
throughput is limited by the receiver’s window in more detail . The
probability of timeout was estimated by the packet loss probability and the
congestion window. It was shown that for small packet loss the timeout
probability can be approximated by $\min(1,3/W)$ and a very comprehensive loss
formula has been derived.
The performance of two classic TCP versions, namely Tahoe and Reno, has been
analyzed by Lakshman and Madhow [44] when the bandwidth-delay product of the
bottleneck link is large compared to the buffer size. The authors estimated
the average throughput for both slow start and congestion avoidance phases
with deterministic and independent random losses.
In a paper by Ott et al. [45] the stationary probability distribution of the
congestion window was calculated for constant packet loss probability. The
authors mapped the “ACK time” point process to a continuous “subjective time”
process by the mapping
$W(t)=\sqrt{p}W_{\left\lfloor\frac{t}{p}\right\rfloor}$, where both the time
and the state space of the discrete process is rescaled in order to obtain a
well behaved process. It was shown that for $p\to 0$ the rescaled process
$W(t)$ behaves as
$\displaystyle\frac{dW(t)}{dt}$ $\displaystyle=\frac{\alpha}{W(t)^{m}}$
$\displaystyle\text{if }t\neq\tau_{k}$ (2.7) $\displaystyle
W\left(t^{+}\right)$ $\displaystyle=\beta W\left(t^{-}\right)$
$\displaystyle\text{if }t=\tau_{k}$ (2.8)
where $\tau_{k}$ are the points of a Poisson process with intensity $\lambda$,
$\alpha>0$ and $0<\beta<1$ are the linear growth rate and multiplicative
decrease factor, respectively, $m\geq 0$, and lastly $t^{-}$ and $t^{+}$
denote the limit to $t$ from the left and from the right, respectively. The
parameter values for TCP congestion avoidance algorithm are $\beta=1/2$ and
$m=1$.
The stationary complementary distribution function of the process $W(t)$ has
been given in the following series expansion form:
$\bar{F}_{W}(w)=\sum_{k=0}^{\infty}R_{k}\left(c\right)\exp\left(-\frac{\eta
c^{-k}}{m+1}w^{m+1}\right),$ (2.9)
where $\eta=\lambda/\alpha$, $c=\beta^{m+1}$ and for $|c|<1$
$\begin{split}R_{k}(c)&=\frac{1}{L(c)}\frac{(-1)^{k}c^{-\frac{1}{2}k(k+1)}}{\left(1-c\right)\left(1-c^{2}\right)\dots\left(1-c^{k}\right)},\\\
L(c)&=\prod_{k=1}^{\infty}\left(1-c^{k}\right).\end{split}$ (2.10)
“ACK”, “subjective” and real time averages and other moments of the congestion
window were calculated and an inverse square root loss formula was derived.
The model has been extended for state dependent packet loss probability in
[46]. State dependent loss models the interaction of the TCP with ERD queuing
policy, where the packet drop probability is a function of the instantaneous
queue length. It is also applicable to RED routers, where the drop probability
depends on the average queue length. An iterative solution for the probability
distribution function of the congestion window was derived. The authors found
good agreement between the derived distribution and computer simulations.
An in-depth analysis of RED queuing dynamics was presented in [47]. The time
dependent congestion window development was modeled with the stochastic
differential equation
$dW_{i}(t)=\frac{dt}{a_{i}+q(t)/C}-\frac{W_{i}(t)}{2}dN_{i}(t),$ (2.11)
where $a_{i}$ denote the fix propagation delay of the bottleneck link, $C$ is
its capacity, $q(t)$ is the queue length at time $t$ and $N_{i}(t)$ is a
Poisson process with a rate that varies in time. The above equations were
transformed to a system of delayed ordinary differential equations in order to
obtain the dynamics of the expectation of the congestion window. The
expectation value of the queue length $\bar{q}(t)$ and the RED estimate of the
average queue length $\bar{x}(t)$ were approximated by two further
differential equations. As a result, the authors obtained $N+2$ coupled
equations for the same number of unknown variables
$\left(\bar{W}_{i}(t),\bar{q}(t),\bar{x}(t)\right)$. The equations were solved
numerically and were compared to computer simulations. The authors pointed out
the importance of the sampling frequency of the smoothed queue length
estimate. A high frequency sampling might cause unwanted oscillations in the
system, while a low frequency sampling can increase the initial overshoot of
the average instantaneous queue length.
#### 2.2.2 Multiple session models
The above papers considered only a single TCP connection. In real computer
networks, however, a number of TCPs might compete for the network resources.
In particular the traffic of TCP sources may flow, in a parallel fashion,
through a common link. Web browsing represents a good example of parallel TCP,
as up to four parallel TCP sessions are started at each page download.
A possible result of TCP interaction can be that parallel TCPs are
synchronized. The underlying reason for synchronization is TCP’s delayed
reaction for congestion events, which keeps drop-tail bottleneck buffers
congested for about an RTT time period. This temporary congestion can induce
further packet losses in competing TCPs. Based on this phenomenon Lakshman and
Madhow [44] supposed that the congestion window development of parallel TCPs
is synchronized in the stationary congestion avoidance regime. The authors
also took into consideration that the bottleneck buffer of large bandwidth-
delay product connections can be either under- or over-utilized. The
congestion window development was therefore split into two phases accordingly.
The authors found a fixed point solution of both the duration and the average
congestion window of the two phases. Finally, the average throughput of each
individual connection was estimated from the window size divided by the round-
trip time.
TCP synchronization is disadvantageous, since it causes performance
degradation. However, this effect appears only in drop-tail queuing systems.
Active queue management, such as RED and ERD, alleviate the problem of
synchronization. A paper by Altman et al. [48] compared the synchronization
model to one in which only one of the parallel TCPs loses a packet at a
congestion event. The probability that a specific connection is affected was
proportional to the throughput of the particular flow. This drop policy models
RED routers. The stationary distribution of the discretized congestion window
at congestion instants was calculated. The average throughput was estimated
from the calculated window distribution via a semi-Markov process. The authors
compared their results with simulations of a RED buffer and found that their
asynchronous model surpasses the synchronous model presented in [44].
Another typical effect in multiple TCP scenario is the bias against
connections with long round-trip times [49]. This effect is the fundamental
consequence of TCP dynamics, and it is not affected by the queue management
policy. The phenomenon can be explained qualitatively by the following simple
arguments. The growth rate of the congestion window is inversely proportional
to the round-trip time $R$. The average congestion window is therefore
inversely proportional to $R$ as well. Furthermore, the average throughput
$\bar{X}$ can be related to the average congestion window $\bar{W}$ by
Little’s law from the queuing theory: $\bar{X}=\bar{W}/R$. Therefore, the
throughput is approximately proportional to $1/R^{\alpha}$, with $\alpha=2$.
The exponent obtained from measurements has been shown to fall in the range
$1\leq\alpha<2$ due to the queuing delay ignored in the above arguments [44].
Floyd and Jacobson [50] have shown that small changes in the round-trip time
might cause large differences in the throughput of different parallel TCP
flows. Specifically, packets of certain TCPs can be dropped tendentiously due
to a phase-effect, causing an utterly unfair bandwidth distribution. Changing
the relative phase of arriving packets at the bottleneck link by slightly
modifying the round-trip time can completely rearrange the bandwidth share of
different TCP connections. Random effects, such as random fluctuations in the
round-trip time or RED queuing policy, also alleviate the phase-effect.
#### 2.2.3 The Network Simulator – ns-2
New models, algorithms, and analytical calculations should be validated
against experiments. Without doubt the most authentic data can be obtained
from Internet measurements, but the deployment of a measurement infrastructure
can be quite expensive and is still very limited. Moreover, models often use
simplifications which make them difficult to compare with real Internet data.
Network simulators, on the other hand, provide “laboratory” environments,
where every parameter of the network and the traffic can be precisely
controlled. Therefore, network simulators are important tools in the hands of
researchers endeavoring to carry out well controlled experiments.
One of the most widely used network simulators in the research community is
the Network Simulator—ns-2111The next major version of the simulator, ns-3, is
under active development. [51]. A short overview of the simulator is given
next, since several analytic and numeric results of this thesis have been
validated by ns-2.
The ns-2 simulator mimics every component of a real network, e.g. links,
routers, queues, protocols, applications and so on. The network traffic is
simulated at packet level, which is to say the course of every packet is
followed from its injection into the network until its removal from it. The
packet-level simulation of network traffic makes the simulator very realistic,
so fine details of the simulated network traffic can be observed. The major
disadvantage of a packet-level simulation is the considerable amount of
computing power that it requires.
The ns-2 simulator is event-driven, that is every component might schedule
events into a virtual calendar. The simulator’s scheduler runs by selecting
the next earliest event for execution. During processing of events further
events can be scheduled into the calendar. This event-based mechanism can also
be observed in every part of the simulator, for example in the handling of
data packets in ns-2. Data packets do not actually travel between virtual
nodes in the simulator, but rather are scheduled for processing at different
network elements instead. For example, when a packet is put onto a link for
transmission the link object in the simulator only schedules the packet for
the queue of the next node on the other end of the link.
The core of the simulator has been written in C++, but it also has an OTcl
scripting programming interface, the object oriented extension of Tcl. The C++
core offers fast execution of the simulator. However, average users do not
need to deal with C++ code in order to run simulations under ns-2. All the
network elements have been bound to objects in OTcl, so complex scenarios can
be built up simply and easily by writing short OTcl scripts.
### 2.3 The infinite-buffer network model
A very simple model of an access router connected to a complex network
consists of a buffer and a link, shown in Fig. 2.2. In this regard the link is
not a real connection between routers, but rather a virtual one. The influence
of the network on the traffic using the access router is modeled with a few
parameters of the virtual link: a fixed propagation delay $D$, bandwidth $C$,
and packet loss probability $p$. This probability represents the chance of
link and hardware failures [52], incorrect handling of arriving packets by
routers, losses and time variations due to wireless links in the path of the
connection [44], the likelihood of congestion in the instantaneous bottleneck
buffer, and the effect of RED and ERD queuing policies. In this chapter I
assume that the buffer is large enough that no packet loss occurs in it.
Figure 2.2: Idealized network model with infinite buffer capacity. Other parts
of the network are modeled with link delay $D$, bandwidth $C$, and packet loss
probability $p$.
Two practically important limits in respect of the role of the access buffer
are: a) LANtraffic, when the bandwidth delay product of the link is small,
only a few packets can be out in the link and the buffer is never empty; and
b) WANtraffic, when the bandwidth delay product of the link is large, packets
are in the link and the buffer is mostly empty. From now on I will refer to
systems with small and large bandwidth delay products as LAN and WAN,
respectively.
As I mentioned earlier in Section 1.2.2, the Internet traffic is governed
mostly by TCP. I will therefore neglect UDP traffic in my simple network model
and will only study the behavior of TCP dynamics.
In realistic networks many TCP sources [53] may share the resources of the
access network. The difficulty of describing the parallel TCP dynamics lies in
the interaction of individual TCP flows. It is obvious that the number of
packets in the network injected by one of the TCP sessions affects the
networking environment of the others. In particular, it contributes to the
round trip times and packet losses felt by the other TCPs. Since the
congestion window controls the maximum number of unacknowledged packets,
understanding its distribution is crucial to describe the interaction.
While an exact treatment of nonlinear interacting systems (such as this one)
is not possible in general, very efficient methods, motivated mostly by
interacting physical systems, have been developed. One of the most established
methods is the mean field approximation. In this approximation each subsystem
operates independently in an “averaged” (or mean) environment. The average
environment is calculated from the behavior of the subsystems. Finally, a
fixed point of the system has to be found where the “mean” environment and the
environment averaged over the independent subsystems coincide. This way we
obtain a self-consistent solution which provides an approximate but quite
accurate description of each subsystem.
In the case of computer networks each TCP plays the role of a subsystem, while
the environment is the round trip time. First, the congestion window
distribution of a TCP is calculated by assuming a given packet loss and round
trip time. Next, the mean round trip time is calculated using the window
distribution. Finally, a fixed point value of the round trip time is
determined.
### 2.4 Dynamics of a single TCP
For studying the behavior of interacting TCPs by mean field approximation one
should know the behavior of a single TCP first. In this section I carry out
the analysis of a single TCP with the use of the fluid approximation of TCP
dynamics, presented in Section 2.1.
Recall that between two consecutive losses the congestion window is governed
by the continuous time differential equation (2.3):
$\frac{dW}{dt}=\frac{1}{R(W)},$ (2.12)
where $W\in[0,\infty[$ is the congestion window, and $R(W)$ is the round-trip
time, which might depend explicitly on the value of the congestion window.
Consider, for example, a typical LAN scenario with a single TCP where the link
delay is small and packet delay is caused mostly by buffering. The congestion
window counts the number of unacknowledged packets, and these packets can be
found on the link and in the buffer. At each packet-shift time unit a packet
is shifted from the buffer into the link. The round-trip time of a freshly
sent packet will be the time it should wait for the shifting of all previously
sent packets in the system, which is in turn measured by the congestion window
$R(W)=WP/C$.
An idealized congestion window process is shown in Fig. 2.3(b), while a
simulated congestion window sequence can be seen on Fig. 2.3(a) for
comparison. Numerical simulations were executed by Network Simulator (ns-2),
introduced in Sec. 2.2.3. Note the small plateaus in the simulations after
each cycle of the congestion window process. These plateaus are the result of
the FR/FR algorithms. First, I will ignore the effect of the FR/FR algorithms,
and I will consider their influence later.
(a) ns-2 simulation of LAN
(b) Ideal fluid model of LAN
Figure 2.3: Comparison of ns-2 simulations with the fluid approximation of the
congestion avoidance process of TCP/Reno. Note the small plateaus in the
simulations due to the FR/FR algorithm after each period.
In order to include more general—even hypothetical—TCP dynamics in my model
the round-trip time is written in the following form:
$R(W)=\alpha^{-1}W^{m}$ (2.13)
with $m\geq 0$ and $\alpha>0$. Note that this notation includes the “ACK time”
dynamics of TCP as well.
The fluid equation (2.12) can be written now as
$\frac{dW}{dt}=\frac{\alpha}{W^{m}}$, which can be rearranged into
$\frac{dW^{m+1}}{dt}=\alpha\left(m+1\right).$ (2.14)
It is obvious that between two packet loss events the solution of this
differential equation is
$W^{m+1}(t)=W^{m+1}(\tau_{i})+\alpha\left(m+1\right)\left(t-\tau_{i}\right),$
(2.15)
where $\tau_{i}$ denote the instant of the $i$th packet loss. At $\tau_{i}$
the transformation
$W(\tau_{i}^{+})=\beta W(\tau_{i}^{-})$ (2.16)
is executed, where $W(\tau_{i}^{-})$ and $W(\tau_{i}^{+})$ are the congestion
windows immediately before and after the time of packet loss, and $0<\beta<1$.
The actual value of $\beta$ is $1/2$ in most TCP variants.
Let $W_{i}=W(\tau_{i}^{+})$ denote the congestion window immediately _after_
the $i$th packet loss, $\delta_{i}=\tau_{i+1}-\tau_{i}$ the length of the time
interval between two losses, and $c=\beta^{m+1}$ hereafter. Since TCP is
assumed to detect packet losses instantaneously in the fluid model,
$W_{i+1}^{m+1}$ can be written as
$W_{i+1}^{m+1}=\left[\beta
W(\tau_{i+1})\right]^{m+1}=cW_{i}^{m+1}+\alpha\left(m+1\right)c\,\delta_{i}.$
(2.17)
By repeated application of (2.17) one can show that the value of the
congestion window immediately after the $N$th packet loss is
$W^{m+1}_{N}=c^{N}W^{m+1}_{0}+\alpha\left(m+1\right)\sum_{k=0}^{N-1}c^{k+1}\delta_{N-k-1}.$
(2.18)
For $N\rightarrow\infty$ the initial value $W_{0}$ becomes insignificant and
the sequence of congestion window values after the packet losses
(${W_{\textrm{a.l.}}}$) can be expressed as
${W_{\textrm{a.l.}}}^{m+1}=\lim_{N\to\infty}W^{m+1}_{N}=\alpha\left(m+1\right)\sum_{k=0}^{\infty}c^{k+1}\delta_{k}.$
(2.19)
Note that the indexing of the $\delta_{i}$ sequence is reversed. This was
allowed since, as I will show below, every $\delta_{i}$ had the same
statistical properties. The reversed indexing was necessary since the infinite
sum would have been meaningless without it.
In the LAN case one packet is shifted out of the buffer in each time unit.
Therefore, in the fluid approximation the times between packet losses,
$\delta_{i}$, are independent exponentially distributed variables. The WAN
scenario is slightly different. Since there is no queue in the buffer there
are periods when no packet leaves the buffer (see the small horizontal steps
in Figure 2.6(a)), packets cannot be lost in those intervals. However, I
assume first that times between losses are exponentially distributed in the
WAN scenario as well—
$f_{\delta_{i}}(x)=\lambda e^{-\lambda x},$ (2.20)
where $1/\lambda$ is the average time between losses—and I will improve the
model later. For the LAN case $\lambda=pC/P$, since one packet is shifted from
the buffer in $P/C$ packet-shift time. Combining (2.19) and (2.20) one can
obtain the distribution of ${W_{\textrm{a.l.}}}^{m+1}$:
$f_{{W_{\textrm{a.l.}}}^{m+1}}(w)=\int\limits_{0}^{\infty}\dotsi\int\limits_{0}^{\infty}\delta\left(w-\alpha\left(m+1\right)\sum_{k=0}^{\infty}c^{k+1}x_{k}\right)\prod_{i=0}^{\infty}f_{\delta_{i}}(x_{i})\,dx_{i},$
(2.21)
where $\delta()$ is the delta distribution. In reality only the distribution
of generic values of the congestion window can be measured. Therefore, their
distribution has to be derived as well. Here I show that this can be done
analytically.
In general, between losses the congestion window is the sum of two random
variables $W^{m+1}={W_{\textrm{a.l.}}}^{m+1}+\alpha\left(m+1\right)\tau$,
where $\tau$ is a uniformly distributed random variable in the _random_
interval $[0,\delta_{i}]$. To obtain the probability distribution of the
congestion window at an arbitrary moment we have to derive the distribution of
the random variable $\tau$ as well. Its distribution can be derived as
follows: $\tau$ is distributed uniformly on each interval with length $\rho$,
assuming $\rho$ is given. This statement can be expressed mathematically with
the conditional distribution
$f_{\tau}(t\mid\rho=x)=\frac{1}{x}\chi_{[0,x]}(t),$ (2.22)
where $\chi_{H}(w)$ denotes the indicator function of set
$H\subset\mathbb{R}$. Furthermore, the probability of selecting a random
interval is proportional to the length of the given interval and the
distribution of the random variable $\delta_{i}$. The proportional factor can
be deduced from the normalization condition of the probability distribution:
$f_{\rho}(x)=\frac{xf_{\delta_{i}}(x)}{\int_{0}^{\infty}xf_{\delta_{i}}(x)\,dx}=\frac{x}{\mathbb{E}[\delta_{i}]}f_{\delta_{i}}(x)=\lambda
xf_{\delta_{i}}(x)$ (2.23)
Finally, the desired distribution follows from the total probability theorem:
$\begin{split}f_{\tau}(t)&=\int_{0}^{\infty}f_{\tau}(t\mid\rho=x)f_{\rho}(x)\,dx\\\
&=\lambda\int\limits_{0}^{\infty}\lambda e^{-\lambda
x}\int\limits_{0}^{x}\delta(t-y)\,dy\,dx=\lambda\int\limits_{t}^{\infty}\lambda
e^{-\lambda x}\,dx=\lambda e^{-\lambda t}\end{split}$ (2.24)
In order to calculate the distribution function of a general value of the
congestion window we apply the method of the Laplace transform. As is known,
the Laplace transform of the density function of the sum of independent random
variables is the product of the Laplace transform of their density functions.
The Laplace transform of (2.21) with respect to ${W_{\textrm{a.l.}}}^{m+1}$
can be defined as
$\hat{f}_{{W_{\textrm{a.l.}}}^{m+1}}(s)=\int_{0}^{\infty}e^{-sw}f_{{W_{\textrm{a.l.}}}^{m+1}}(w)\,dw$.
This can be easily evaluated and we get
$\begin{split}\hat{f}_{{W_{\textrm{a.l.}}}^{m+1}}(s)&=\prod_{k=0}^{\infty}\left[\lambda\int\limits_{0}^{\infty}\exp\left(-s\alpha\left(m+1\right)\beta^{m\left(k+1\right)}x_{k}-\lambda
x_{k}\right)\,dx_{k}\right]\\\
&=\prod_{k=1}^{\infty}\frac{\lambda}{\lambda+\alpha\left(m+1\right)c^{k}s}.\end{split}$
(2.25)
The Laplace transform of the distribution of $\alpha\left(m+1\right)\tau$ is
given by
$\hat{f}_{\alpha\left(m+1\right)\tau}(s)=\int\limits_{0}^{\infty}\frac{\lambda}{\alpha\left(m+1\right)}e^{-\frac{\lambda}{\alpha\left(m+1\right)}t}e^{-st}\,dt=\frac{\lambda}{\lambda+\alpha\left(m+1\right)s}.$
(2.26)
Therefore, the Laplace transform of the generic
$W^{m}={W_{\textrm{a.l.}}}^{m}+\alpha\left(m+1\right)\tau$ distribution is the
infinite product
$\hat{f}_{W^{m+1}}(s)=\hat{f}_{{W_{\textrm{a.l.}}}^{m+1}}(s)\,\hat{f}_{\alpha\left(m+1\right)\tau}(s)=\prod_{k=0}^{\infty}\frac{\lambda}{\lambda+\alpha\left(m+1\right)c^{k}s}.$
(2.27)
Furthermore, we can rewrite the Laplace transform as a sum of partial
fractions
$\hat{f}_{W^{m+1}}(s)=\frac{\lambda}{\alpha\left(m+1\right)}\sum_{k=0}^{\infty}\frac{h_{k}(c)}{\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}+s},$ (2.28)
where the coefficients $h_{k}(c)$ can be obtained from the residues of the
poles of $\hat{f}_{W^{m}}(s)$:
$\displaystyle h_{k}(c)$ $\displaystyle=\operatorname*{Res}_{-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}}\frac{\alpha\left(m+1\right)}{\lambda}\hat{f}_{W^{m+1}}(s)$
$\displaystyle=\lim_{s\to-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}}\frac{\alpha\left(m+1\right)}{\lambda}\left(s+\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}\right)\prod_{l=0}^{\infty}\frac{\lambda}{\lambda+\alpha\left(m+1\right)c^{l}s}$
$\displaystyle=\frac{1}{c^{k}}\prod_{\begin{subarray}{c}l=0\\\ l\neq
k\end{subarray}}^{\infty}\frac{1}{1-c^{l-k}}=\frac{1}{c^{k}L(c)}\prod_{l=1}^{k}\frac{1}{1-c^{-l}},$
(2.29) where $\displaystyle L(c)$
$\displaystyle=\prod_{l=1}^{\infty}\left(1-c^{l}\right).$ (2.30)
It follows evidently from this formula that the relative strength of
successive terms eventually decreases exponentially fast, when $k$ is large
enough:
$\frac{h_{k+1}(c)}{h_{k}(c)}=-c^{k}\frac{1}{1-c^{k+1}}\approx-c^{k}\quad\text{for
$k\gg-\frac{1}{\log c}$},$ (2.31)
therefore, only a small number of constants should be used for numerical
purposes.
We can perform a term-by-term inverse Laplace transform on (2.28). The density
function of $W^{m}$ can be given by
$f_{W^{m+1}}(w)=\frac{\lambda}{\alpha\left(m+1\right)}\sum_{k=0}^{\infty}h_{k}(c)\exp\left(-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}w\right).$ (2.32)
The distribution of the congestion window is given by a simple variable
transformation
$f_{W}(w)=\frac{\lambda}{\alpha}w^{m}\sum_{k=0}^{\infty}h_{k}(c)\exp\left(-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}w^{m+1}\right).$ (2.33)
Finally, the complementary cumulative distribution
$\bar{F}_{W}(w)=\int_{w}^{\infty}f_{W}(w^{\prime})\,dw^{\prime}$ can be given
by
$\bar{F}_{W}(w)=\sum_{k=0}^{\infty}c^{k}h_{k}(c)\exp\left(-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}w^{m+1}\right).$ (2.34)
Note that the above formulas do not change when $\lambda$ and $\alpha$ are
varied, but the $\lambda/\alpha$ ratio is kept fixed. Furthermore, the weight
of the $k$th term in the probability distribution is $c^{k}h_{k}(c)$, so the
error induced by truncating terms above a threshold index can be estimated
precisely: $1-\sum_{k=0}^{K}c^{k}h_{k}(c)$.
Compare the results (2.32) – (2.34) with (2.9) – (2.10). The calculation has
reproduced the results of Ott et al. [45] with a slight difference in the
notation. However, I have not supposed in my derivation that $p\to 0$, as Ott
et al. did. Furthermore, I have shown explicitly that $\tau$ is exponentially
distributed, which is missing in the previous derivation.
The moments of $W$ can be calculated from (2.32) as
$\begin{split}\mathbb{E}[W^{r\left(m+1\right)}]&=\int_{0}^{\infty}w^{r}f_{W^{m+1}}(w)dw=\frac{\lambda}{\alpha\left(m+1\right)}\sum_{k=0}^{\infty}h_{k}(c)\int_{0}^{\infty}w^{r}e^{-\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}w}dw\\\
&=\left(\frac{\alpha\left(m+1\right)}{\lambda}\right)^{r}\sum_{k=0}^{\infty}c^{\left(1+r\right)k}h_{k}(c)\int_{0}^{\infty}z^{r}e^{-z}dz\\\
&=\left(\frac{\alpha\left(m+1\right)}{\lambda}\right)^{r}\Gamma\left(1+r\right)\sum_{k=0}^{\infty}c^{\left(1+r\right)k}h_{k}(c),\end{split}$
(2.35)
where $r>0$. The variable transformation $z=\frac{\lambda
c^{-k}}{\alpha\left(m+1\right)}w$ was executed in the first integral, and the
integral definition of the Gamma function
$\Gamma(x)=\int_{0}^{\infty}z^{x-1}e^{-z}dz$ was applied in the second
equation. If $r=n$ is an integer the moments can be given in closed form. For
this end let us find the series expansion of the Laplace transform of
$W^{m+1}$. Observe—following Ott et al.—that the product form of the Laplace
transform (2.27) satisfies the following functional equation:
$\left(1+\frac{\alpha\left(m+1\right)}{\lambda}s\right)\hat{f}_{W^{m+1}}(s)=\hat{f}_{W^{m+1}}(cs).$
(2.36)
Differentiate $n$ times both sides of this functional equation with respect to
$s$:
$\left(1+\frac{\alpha\left(m+1\right)}{\lambda}s\right)\hat{f}^{(n)}_{W^{m+1}}(s)+n\frac{\alpha\left(m+1\right)}{\lambda}\hat{f}^{(n-1)}_{W^{m+1}}(s)=c^{n}\hat{f}^{(n)}_{W^{m+1}}(cs).$
(2.37)
Since the $n$th derivative of the Laplace transform at $s=0$ is related to the
moments as
$\mathbb{E}[W^{n\left(m+1\right)}]=(-1)^{n}\hat{f}^{(n)}_{W^{m}}(0)$, we find
the following recurrence relation:
$\left(1-c^{n}\right)\mathbb{E}[W^{n\left(m+1\right)}]=n\frac{\alpha\left(m+1\right)}{\lambda}\mathbb{E}[W^{\left(n-1\right)\left(m+1\right)}].$
(2.38)
This recursive equation with the initial condition $\mathbb{E}[W^{0}]=1$
immediately yields
$\mathbb{E}[W^{n\left(m+1\right)}]=n!\left(\frac{\alpha\left(m+1\right)}{\lambda}\right)^{n}\prod_{k=1}^{n}\frac{1}{1-c^{k}}.$
(2.39)
### 2.5 Discussion
#### 2.5.1 Local Area Networks
I will now confirm the validity of the above results by numerical simulations.
In this section I study the LAN scenario, that is when the bandwidth-delay
product of the link is much smaller than the size of the buffer. As I have
pointed out earlier the parameters of the ideal LAN scenario are $m=1$,
$c=\beta^{2}=1/4$, and $\alpha^{-1}\lambda=p$. The first nine numerical values
of $h_{k}(c)$ are shown in Table 2.1. It can be seen that the coefficients
converge to zero so quickly that it is sufficient to keep the first five terms
in practical calculations.
Table 2.1: $h_{k}\left(\beta^{m+1}\right)$ coefficients with $\beta=1/2$ and $m=1$ $h_{0}$ | $1$ | .$4523536$ | $h_{3}$ | $-3$ | .$2786819\cdot 10^{-2}$ | $h_{6}$ | $1$ | .$9643078\cdot 10^{-9}$
---|---|---|---|---|---|---|---|---
$h_{1}$ | $-1$ | .$9364715$ | $h_{4}$ | $5$ | .$1430305\cdot 10^{-4}$ | $h_{7}$ | $-4$ | .$7959661\cdot 10^{-13}$
$h_{2}$ | $5$ | .$1639241\cdot 10^{-1}$ | $h_{5}$ | $-2$ | .$0109601\cdot 10^{-6}$ | $h_{8}$ | $2$ | .$9272701\cdot 10^{-17}$
The mean of the congestion window can be calculated from (2.35) with the
parameter $r=\frac{1}{m+1}=\frac{1}{2}$:
$\mathbb{E}[W]=\sqrt{\frac{2}{p}}\frac{\sqrt{\pi}}{2}\sum_{k=0}^{\infty}\frac{h_{k}(1/4)}{8^{k}}\approx\frac{1.5269}{\sqrt{p}},$
(2.40)
which gives the well known inverse square-root formula. The second moment can
be obtained exactly from (2.39) with $n=1$:
$\mathbb{E}[W^{2}]=\frac{2}{p}\frac{1}{1-1/4}=\frac{8}{3p},$ (2.41)
therefore the standard deviation is approximately
$\sigma[W]=\sqrt{\mathbb{E}[W^{2}]-\mathbb{E}[W]^{2}}\approx\frac{0.5790}{\sqrt{p}}.$
(2.42)
(a) Average vs. mean
(b) Empirical standard deviation vs. standard deviation
Figure 2.4: Empirical mean and standard deviation of the congestion window as
the function of the corresponding theoretical values in the range of loss
probabilities $p=0.1\%$–$5\%$.
In Figure 2.4 the empirical mean and standard deviation of the congestion
window is plotted as the function of the theoretical values (2.40) and (2.42),
respectively. The model predicts measurement points on the diagonals, shown
with dotted lines. We can see that the empirical standard deviation agrees
well with the theoretical values, but the average congestion window is
systematically smaller than predicted. The linear fit $f(x)=x+b$ of the data
points gives an estimate for the average shift $b=-1.4141\pm 0.0611$.
The most important source of error is that FR/FR algorithms have been
neglected in my idealized model, but the simulator does use these algorithms.
The small plateaus appearing in the congestion window after each cycle produce
bias towards the smaller window values.
In a refined model let us consider the FR/FR algorithms as well. Denote
$\tilde{W}$ the fluid approximation of the extended congestion window process,
which can operate in either congestion avoidance (CA) or FR/FR mode. TCP
remains in FR/FR mode until the ACK of a retransmitted packet reaches the
sender, that is the round-trip time _before_ the FR/FR mode started:
$R(\beta^{-1}W)$. Furthermore, the probability that a plateau forms in the
window interval $[w,w+dw]$ equals the probability that the window is reduced
to the given interval after a packet loss: $f_{{W_{\textrm{a.l.}}}}(w)\,dw$.
The form of the distribution $f_{{W_{\textrm{a.l.}}}}(w)$ might be derived by
direct calculation, but it can also be found by a simple argument: in
stationary state of TCP—since the packet loss process is memoryless—packet
loss can occur at every congestion window value with the same probability,
supposing that the window has reached the given value. In other words the
value of the “after loss” congestion window is
${W_{\textrm{a.l.}}}=\beta{W_{\textrm{b.l.}}}=\beta W$. Accordingly, the
distribution of the “after loss” window is
$f_{{W_{\textrm{a.l.}}}}(w)\,dw=f_{W}(\beta^{-1}w)\,d\beta^{-1}w$. On
condition that TCP is in FR/FR mode the probability distribution of the
congestion window is
$f_{\tilde{W}}(w\mid\text{TCP\ in FR/FR
mode})=\frac{R(\beta^{-1}w)\beta^{-1}f_{W}(\beta^{-1}w)}{\mathbb{E}[R(W)]}$
(2.43)
On the other hand, the window distribution in the congestion avoidance mode
can clearly be given by $f_{\tilde{W}}(w\mid\text{TCP\ in CA mode})=f_{W}(w)$.
Now only the probabilities of the CA and FR/FR modes are required. Since each
congestion avoidance phase is followed by a FR/FR mode, probabilities of the
different modes are proportional to the average length of the corresponding
mode. The mean length of a congestion avoidance mode is evidently the average
time between two packet losses: $\mathbb{E}[\delta_{i}]=1/\lambda$. The
average length of a FR/FR period, on the other hand, is simply the average
length of the plateaus: $\mathbb{E}[R(W)]$. This implies that
$\displaystyle\mathbb{P}(\text{TCP\ in CA mode})$
$\displaystyle=\frac{1/\lambda}{1/\lambda+\mathbb{E}[R(W)]},\quad\text{and}$
(2.44) $\displaystyle\mathbb{P}(\text{TCP\ in FRFR mode})$
$\displaystyle=\frac{\mathbb{E}[R(W)]}{1/\lambda+\mathbb{E}[R(W)]}.$ (2.45)
Therefore, the probability distribution of the congestion window extended by
the FR/FR algorithm is
$f_{\tilde{W}}(w)=\frac{f_{W}(w)+\frac{\lambda}{\alpha}\beta^{-\left(m+1\right)}w^{m}f_{W}(\beta^{-1}w)}{1+\frac{\lambda}{\alpha}\mathbb{E}[W^{m}]}$
(2.46)
where (2.13), the definition of $R(W)$ has been substituted. This formula is
the main result of this section. In Figure 2.5 the histogram of the congestion
window simulated with ns-2 and (2.46) are compared in the range of loss
probabilities $p=0.1\%$–$5.0\%$. We can see an almost perfect match between
theory and simulation. In order to illustrate the improvement of the formula
(2.46) on (2.33), I plotted $f_{W}(w)$ with dotted lines for comparison. I
must stress here that there are no tunable parameters in (2.46) and no
parameter fit has been made.
(a) Loss rate $p=10^{-3}$
(b) Loss rate $p=5\cdot 10^{-3}$
(c) Loss rate $p=0.01$
(d) Loss rate $p=0.02$
(e) Loss rate $p=0.04$
(f) Loss rate $p=0.05$
Figure 2.5: Histograms and theoretical distributions of congestion windows in
LAN. Network parameters are $C=256\,kb/s$, $P=1500\,\mathit{byte}$, and
$D=0\,s$.
Let us calculate the moments of $\tilde{W}$:
$\mathbb{E}[{\tilde{W}}^{k}]=\frac{\mathbb{E}[W^{k}]+\beta^{k}\frac{\lambda}{\alpha}\mathbb{E}[W^{m+k}]}{1+\frac{\lambda}{\alpha}\mathbb{E}[W^{m}]}.$
(2.47)
As an important special case we can calculate the correction of the FR/FR
algorithms to the mean of the congestion window:
$\mathbb{E}[\tilde{W}-W]=-\frac{\lambda}{\alpha}\frac{\mathbb{E}[W]\mathbb{E}[W^{m}]-\beta\mathbb{E}[W^{m+1}]}{1+\frac{\lambda}{\alpha}\mathbb{E}[W^{m}]}$
(2.48)
If the formula (2.35) is substituted into the above equation one can obtain
the dependence of the correction on $\lambda/\alpha$. Specifically, for $m=1$:
$\mathbb{E}[\tilde{W}-W]\approx-\frac{0.9981}{1+1.5269\sqrt{\frac{\lambda}{\alpha}}}$.
Interestingly, the correction tends to a constant in the small loss limit:
$\lim_{\lambda/\alpha\to 0}\mathbb{E}[\tilde{W}-W]\approx-0.9981$. In the
range of loss probabilities $p=\lambda/\alpha=10^{-4}$–$5\cdot 10^{-2}$,
investigated by simulations, the correction to the congestion window average
is between $-0.8659$ and $0.9831$, which is less than observed in Fig. 2.4(a).
The remaining discrepancy comes from the difference between the continuous and
the fluid value of $W$. In the simulation the congestion window is not only
halved its integer part is also taken. This discrepancy accounts for
approximately $-0.5$ unit shift on average. The slow start mechanism, which
becomes more and more dominant as the loss probability increases, also makes
the small window values more probable. However, these effects are beyond the
scope of the applied fluid model.
#### 2.5.2 Wide Area Networks
I turn now to the WAN scenario, where buffering delay is very small compared
to the link delay. A typical congestion window sequence is shown in Figure 2.6
with $D=1s$ and $p=0.01$.
(a) ns-2 simulation of WAN
(b) Details of the cwnd in the bounded box
Figure 2.6: The congestion avoidance process of TCP in WAN setup. The global
congestion window development is seemingly linear, but the detailed plot on
Figure 2.6(b) shows a different picture. The globally linear growth is
composed of alternating idle and LAN-like active periods. See the discussion
in the text.
The applicability of the developed model depends on two crucial factors: the
validity of the exponential inter-loss distribution (2.20) and the validity of
(2.13), the dependence of round-trip time on the congestion window. The
difficulty of the WAN scenario is that—as I mentioned earlier—there are
periods when no packet leaves the buffer. This effect corrupts the validity of
both assumptions. Packets cannot be lost during idle periods, so the inter-
loss time distribution deviates from exponential distribution. Furthermore, if
we assume that the round-trip time is constant, $R(W)=2D$, then the solution
of the equation of motion (2.15) predicts linear congestion window
development, which corresponds to $m=0$ in the model. However, a typical
sequence of congestion windows, displayed in Fig. 2.6(b), shows step-like
growth instead. Another difficulty is that even if the inter-arrival times can
be approximated with an exponential distribution, the connection between the
packet loss probability $p$ and the parameter of the distribution $\lambda$ is
unknown.
Given these concerns I approach the congestion window development in a WAN
network in a manner different from simple linear growth, the model applied
exclusively in the literature. First of all let us investigate the window
development in congestion avoidance mode in more detail. The fine structure of
the congestion window is shown in Fig. 2.6(b). During the active period of
TCP, when ACKs are arriving back to the sender (in “ACK time”), the congestion
window is being increased the same way as in a LAN network. The difference
from LAN in a WAN scenario is that an idle period follows with a constant
congestion window. Since $W$ packets are transferred in an active period, the
length of an active period is $W/\alpha$. The following idle period is
$2D-W/\alpha$ long, because the total length of an active and the succeeding
idle period is precisely one round-trip time, $2D$.
Let $W^{*}$ denote the congestion window idle periods included. If the
plateaus corresponding to idle periods are approximated as if they were
blurred evenly on the active periods, then—analogously to the FR/FR mode in
LAN—the conditional distribution of the congestion window in idle mode of TCP
can be formulated as
$f_{W^{*}}(w\mid\text{TCP\ in IDLE
mode})=\frac{\frac{2D-w/\alpha}{w/\alpha}f_{W}(w)\,\Theta(2D-w/\alpha)}{\mathbb{E}\left[\frac{2D-W/\alpha}{W/\alpha}\Theta(2D-W/\alpha)\right]}.$
(2.49)
Only the probabilities of idle, CA and FR/FR modes are required. The
probability of each mode is proportional to the average time TCP spends in the
particular mode. Considering the idle mode, the average length of a plateau in
one ACK increment is
$\mathbb{E}\left[\frac{2D-W/\alpha}{W/\alpha}\Theta(2D-W/\alpha)\right]$.
Moreover, the window is increased $\mathbb{E}[\delta_{i}]=1/\lambda$ times in
one loss cycle on average. Accordingly, the probability of idle mode is
$\mathbb{P}(\text{TCP\ in IDLE mode})=\frac{\mathbb{E}\left[\frac{2\alpha
D-W}{W}\Theta(2\alpha
D-W)\right]/\lambda}{1/\lambda+\mathbb{E}[R(W)]+\mathbb{E}\left[\frac{2\alpha
D-W}{W}\Theta(2\alpha D-W)\right]/\lambda}.$ (2.50)
The probabilities of CA and FR/FR modes in Eqs. (2.44) and (2.45) should be
modified proportionately. As a result we obtain
$f_{W^{*}}(w)=\frac{f_{W}(w)+\frac{\lambda}{\alpha}\beta^{-\left(m+1\right)}w^{m}f_{W}(\beta^{-1}w)+\frac{2\alpha
D-w}{w}f_{W}(w)\,\Theta(2\alpha D-w)}{\bar{F}_{W}(2\alpha
D)+\frac{\lambda}{\alpha}\mathbb{E}[W^{m}]+2\alpha
D\,\mathbb{E}\left[\frac{1}{W}\Theta(2\alpha D-W)\right]}$ (2.51)
for the congestion window distribution, where I used that
$\mathbb{E}[\Theta(2\alpha D-W)]=\int_{0}^{2\alpha
D}f_{W}(w)\,dw=1-\bar{F}_{W}(2\alpha D)$. The truncated expectation of $1/W$
can be obtained similarly to (2.35) with $r=-\frac{1}{m+1}$, but one should
include the incomplete Gamma function
$\Gamma(z,x)=\int_{x}^{\infty}x^{z-1}e^{-x}\,dx$ as well:
$\begin{split}&\mathbb{E}\left[\frac{1}{W}\Theta(2\alpha
D-W)\right]=\int_{0}^{2\alpha D}\frac{1}{w}f_{W}(w)\,dw\\\
=&\mathbb{E}\left[\frac{1}{W}\right]-\left(\frac{\lambda}{\alpha\left(m+1\right)}\right)^{\frac{1}{m+1}}\sum_{k=0}^{\infty}\Gamma\left(\frac{m}{m+1},\frac{2\lambda
Dc^{-k}}{m+1}\right)c^{\frac{mk}{m+1}}h_{k}(c).\end{split}$ (2.52)
The moments of $W^{*}$ can be given easily:
$\mathbb{E}\bigl{[}{W^{*}}^{k}\bigr{]}=\frac{\mathbb{E}[W^{k}]+\frac{\lambda}{\alpha}\beta^{k}\mathbb{E}[W^{m+k}]+\mathbb{E}\left[\left(2\alpha
D\,W^{k-1}-W^{k}\right)\Theta(2\alpha D-W)\right]}{\bar{F}_{W}(2\alpha
D)+\frac{\lambda}{\alpha}\mathbb{E}[W^{m}]+2\alpha
D\,\mathbb{E}\left[\frac{1}{W}\Theta(2\alpha D-W)\right]}\,.$ (2.53)
The distribution and moments of the ideal WAN scenario can be obtained from
(2.51) and (2.53) in the $\alpha D\to\infty$ limit:
$\displaystyle\lim_{\alpha D\to\infty}f_{W^{*}}(w)$
$\displaystyle=\frac{1}{w\mathbb{E}\left[\frac{1}{W}\right]}f_{W}(w)$ and
$\displaystyle\lim_{\alpha D\to\infty}\mathbb{E}\bigl{[}{W^{*}}^{k}\bigr{]}$
$\displaystyle=\frac{\mathbb{E}[W^{k-1}]}{\mathbb{E}\left[\frac{1}{W}\right]}.$
(2.54)
Note that the formula (2.51) at $D=0$ reduces to (2.46), derived earlier for
an ideal LAN scenario with CA and FR/FR modes. Note further that (2.51) is
applicable not only for the ideal WAN or LAN scenarios, but also for _for the
most generic configuration_. Moreover, I would like to emphasize that the
parameters of the model can be obtained from the intrinsic “ACK time” dynamics
of TCP, and _no parameter fitting is necessary_. Specifically, for TCP/Reno
the parameters are $m=1$, $\lambda/\alpha=p$, $\beta=1/2$ and $2\alpha D$ is
the bandwidth-delay product measured in packet units.
(a) Loss rate $p=10^{-4}$
(b) Loss rate $p=5\cdot 10^{-4}$
(c) Loss rate $p=0.001$
(d) Loss rate $p=0.005$
(e) Loss rate $p=0.01$
(f) Loss rate $p=0.05$
Figure 2.7: Histograms and theoretical distributions of congestion windows in
WAN. The bandwidth-delay product is $2DC/P=170.67$, measured in packets. Note
that this value falls in the bulk of the distribution at loss rate
$p=10^{-4}$, which means a transient between the ideal LAN and WAN.
In order to verify (2.51) I carried out simulations. The link parameter
$2\alpha D$ has been set to $170$ packets and the packet loss has been varied
in the range of $p=10^{-4}-5\cdot 10^{-2}$. Simulation results are shown in
Fig. 2.7. I have plotted the contribution of the active periods to the
theoretical distribution with dotted lines for comparison. A transient between
the ideal LAN and WAN network configuration can be observed at $p=10^{-4}$,
since the parameter $2\alpha D=170$ falls in the bulk of the distribution. An
excellent fit can be seen at small loss probabilities and a small discrepancy
can be detected in the mid-range $10^{-3}<p<10^{-2}$. For probabilities
$p>0.01$ the neglected slow start mechanism becomes more and more significant.
As a result the theoretical distribution deviates from the measured histogram
more markedly.
#### 2.5.3 Dynamics of parallel TCPs
Now, I am in the position to extend my results for parallel TCPs. Since (2.51)
involves only the intrinsic TCP dynamics, the model parameters $m=1$,
$\lambda/\alpha=p$, and $\beta=1/2$ are the same for parallel and a single
TCP. The propagation delay parameter $2\alpha D$ might change, however,
because the interaction of different TCPs might alter the idle periods
experienced by the individual TCPs.
The parallel TCPs operate in WAN environment until the number of packets in
the network, that is the sum of the congestion windows
$\sum_{i=1}^{N}W^{*}_{i}$, is less than $2\alpha D$. Let us consider one of
the parallel TCPs and denote its congestion window by $W^{*}_{n}$. The length
of the idle periods felt by the selected TCP is $2D-W^{*}_{n}/\alpha$ in the
WAN case and the propagation delay is independent of the states of the
different TCPs. The congestion window distribution of each individual TCP can
therefore be given by (2.51). In Fig. 2.8(a) I show the congestion window
histogram of one out of two parallel TCPs. The link delay is $D=2s$, large
enough to leave the buffer empty. As a comparison I also show the histogram of
a single TCP and the theoretical distribution function for the same network
configuration. It is apparent that the two histograms are almost identical and
the discrepancy between the theoretical distribution and the measured
histogram remains almost the same for parallel TCPs as for a single one.
(a) WAN with $D=2s$ ($2\alpha D=85.33$).
(b) LAN with $D=0s$ ($N\mathbb{E}[W^{*}]\approx 36.55$).
Figure 2.8: Illustration for the mean field approximation for two parallel
TCPs. The congestion window histogram of one of two parallel TCPs is shown.
The histogram of a single TCP is displayed for comparison. Network parameters
are $C=256\,\textit{kb/s}$, $P=1500\,\textit{byte}$ and $p=5\cdot 10^{-3}$.
In the LAN scenario, when the sum of the congestion windows is larger than
$2\alpha D$, a queue starts forming in the buffer and the buffering delay
becomes significant. The selected TCP suffers
$\sum_{i=1}^{N}W^{*}_{i}/\alpha-W^{*}_{n}/\alpha$ long idle periods, caused by
intermediate packets sent by the rest of the TCPs. Thereby, the dynamics of
the TCPs becomes coupled and they cannot be handled as being independent any
more.
In an attempt to solve this problem I am going to use the mean field theory,
that is I suppose that TCPs are independent and they feel only the average
influence of other TCPs. For a large number of TCPs the sum of congestion
windows will fluctuate around its average
$\mathbb{E}\left[\sum_{i=1}^{N}W^{*}_{i}\right]=N\mathbb{E}[W^{*}]$ and the
deviation from this average will be of order $\sim\sqrt{N}$. For sufficiently
large $N$ the relative size of fluctuations will decay as $\sim 1/\sqrt{N}$.
Therefore, for large $N$ it is reasonable to replace the sum of congestion
windows with its average. In this approximation each TCP operates in a WAN-
like environment, since they feel a constant delay as in WAN. So we can apply
the corresponding results of WAN. We simply have to replace all occurrences of
$2\alpha D$ in (2.51) with $N\mathbb{E}[W^{*}]$, the mean field approximation
of the sum of the congestion windows.
The self-consistent mean field solution for $\mathbb{E}[W^{*}]$ can be
obtained from (2.53). The occurrences of $2\alpha D$ have to be replaced with
$N\mathbb{E}[W^{*}]$ again, and the fixed point solution for
$N\mathbb{E}[W^{*}]$ should be found. The simplest method for finding the
fixed point solution is to iterate (2.53): start with a good estimate of the
mean field solution, calculate the next estimate with the equation and replace
the new value to the right hand side of the equation. This process should be
repeated until the desired precision is achieved. A good initial value for the
iteration is the mean congestion window in the $\alpha D\to\infty$ limit
(2.54), because many parallel TCPs ($N\gg 1$) are close to the ideal WAN
scenario.
In Fig. 2.8(b) the congestion window histogram of one out of two parallel TCPs
is presented in a LAN environment, when the link delay is $D=0s$. The mean
field approximation of the distribution function shows an excellent fit. The
histogram of a single TCP in the same network configuration is also plotted
with the corresponding theoretical distribution. The two histograms are rather
different but both theoretical distributions are close to the corresponding
empirical values. Fig. 2.9 shows a similar LAN scenario with one of 20
parallel TCPs.
Figure 2.9: Congestion window histogram of one of 20 parallel TCPs. The
histogram of a single TCP is displayed for comparison. Network parameters are
$C=256\,\textit{kb/s}$, $P=1500\,\textit{byte}$ and $p=10^{-3}$. The mean
field solution of (2.53) is $N\mathbb{E}[W^{*}]\approx 827.75$, close to an
ideal WAN scenario.
### 2.6 Conclusions
In this chapter I analyzed the congestion window distributions of TCP in a
standalone, infinite-buffer network model. I derived new analytical formulas
for the distribution of generic congestion window values, which take into
consideration not only the congestion avoidance mode, but also the fast
retransmit/fast recovery modes of TCP. My novel approach for modeling WAN
configuration made it possible to describe TCP traffic with all model
parameters at hand; no parameter fitting is necessary. Moreover, I presented
analytic calculations not only for ideal LAN and WAN scenarios, but also for
intermediate network configurations, where the queuing and link delays are
comparable. The mean field theory has been applied for parallel TCP traffic.
My analytic calculations were verified against direct simulations. The
analytic results fit the histograms I received from the simulations when the
packet loss probability is small as well. Discrepancies between the analytic
results and simulations become stronger when the packet loss probability
increases, however. The differences mostly come from the neglected slow start
mode of TCP and the fluid approximation of the discrete time congestion window
process. The main virtue of my work is that it provides an analytic
description of TCP traffic in more detail than previous works, without the
need to adjust parameters empirically.
## Chapter 3 Traffic dynamics in finite buffer
In the previous chapter I assumed that the common buffer under investigation
was _not_ a bottleneck buffer. The model describes the dynamics of TCP in the
presence of external packet loss quite accurately. However, packet loss in
current networks is generated predominantly by overloaded buffers. This is an
inherent property of TCP congestion control mechanism since TCP increases its
packet sending rate until packet loss occurs in one of the buffers along the
route between the source and the destination. In the literature little or no
progress has been made towards an understanding of the detailed mechanism of
packet loss in IP networks.
In this chapter I give a detailed mathematical description of the packet loss
mechanism. In Section 3.1 the refined network model is defined. I investigate
the dynamics of TCP in the presence of a finite buffer in Section 3.2. I
discuss my model in Section 3.3, where I will derive analytic formulas for the
packet loss and the congestion window distribution. The new formulas and
distributions are validated by direct simulation. Finally, I conclude this
chapter in Section 3.4.
### 3.1 The finite buffer model
My extended network model is very similar to the model I studied in the
preceding chapter with the decisive difference that the buffer size $B$ is
finite now (Fig. 3.1). The remaining part of the network is—as in the previous
chapter—modeled by a fixed delay $D$, constant bandwidth or link capacity $C$
and random loss probability of $p$ per packet. In my idealized network model
one TCP injects packets into the buffer.
Figure 3.1: The finite buffer model. In numerical simulations packet size
$P=1500\,\textit{byte}$ and bandwidth $C=256\,\textit{kb}/\textit{s}$ have
been fixed, and the buffer size $B$ and the packet loss probability $p$ have
been changed.
The buffer is large enough such that TCP can operate in congestion avoidance
mode, but it is finite, so that packet loss can occur in it. Furthermore, the
number of packets getting lost in the buffer is comparable with the full
packet loss, including the corrupted and lost packets in the rest of the
network.
We can estimate the parameter range where the finiteness of the buffer plays
an important role in a LAN scenario, when the link delay is negligible. The
finite buffer size limits the total congestion window achievable by TCP to
$w_{\max}\approx B$. On the other hand, I have shown in the last chapter that
external packet loss in the core network would set the average congestion
window to $\left\langle w\right\rangle\approx c/\sqrt{p}$, where $c\approx
1.5269$. If $c/\sqrt{p}\approx B$, that is $pB^{2}\approx c$ holds then the
external and the buffer loss play comparable role. A more detailed analysis
will be given in Section 3.3.1.
### 3.2 Dynamics of a single TCP
In this section I present the analysis of a single TCP operating in my finite
buffer model. I start off by the fluid equation
$\frac{dW}{dt}=\frac{1}{R(W)},$ (3.1)
similar to Eq. (2.12) of the infinite buffer model, but with the important
difference that the maximum congestion window is limited: $w\in[0,\tilde{B}]$,
where $\tilde{B}=B+2DC/P$. The round-trip time is supposed to be the same as
(2.13):
$R(W)=\alpha^{-1}W^{m},$ (3.2)
where $\alpha>0$ and $m\geq 0$. The above equations can be solved the same way
as the equations of the infinite buffer scenario and we obtain the time
development of the congestion window between losses:
$W^{m+1}(\tau)=W^{m+1}(\tau_{i})+\alpha\left(m+1\right)\left(\tau-\tau_{i}\right),$
(3.3)
where $\tau_{i}$ denotes the instant of the $i$th packet loss as before.
An idealized congestion window process with $m=1$ can be seen in Figure
3.2(b)for a LAN network. In order to validate my model I implemented it in
ns-2. Simulation results of the congestion window process can be seen in
Figure 3.2(a) for comparison, with $C=256\mathit{{kb}/s}$,
$P=1500\mathit{byte}$, $B=50$, $D=0s$ and $p=0.0008$ parameter values
($pB^{2}=2$). Equation (3.3) gives a reasonably good description of the window
development. The effect of discrepancies will be discussed in Section 3.3.2.
(a) ns-2 simulation of LAN
(b) Ideal fluid model of WAN
Figure 3.2: The congestion avoidance process of TCP/Reno in the case of finite
buffer, obtained form ns-2 simulations . The idealized fluid approximation of
the congestion window is also shown for comparison.
Furthermore, $\delta_{i}$, the elapsed time between consecutive packet losses
_occurring at the external link_ , are supposed to be independent,
exponentially distributed random variables with mean $1/\lambda$ and
probability distribution
$f_{\delta_{i}}(x)=\lambda\exp(-\lambda x),\quad\forall i\in\mathbb{N}.$ (3.4)
Note the important memoryless property of the exponential distribution. It
means that if a certain length of time has elapsed since a packet loss then
the probability distribution of the time interval remaining until the next
packet loss is still given by (3.4) regardless of the elapsed time.
I now derive the formula that connects consecutive window values before
losses. Let us denote by $W_{i}=W(\tau^{-})$ the window value immediately
_before_ the $i^{\textrm{th}}$ loss event and its distribution by
$f_{W_{i}}(w)$. If the value of the random variable $\delta_{i}$ is small
enough, the next $W_{i+1}$ can be obtained from (3.3). However, if
$\delta_{i}$ is so large that the window would grow above the upper limit
$\tilde{B}$ then packets will be dropped at the buffer, and the $W_{i+1}$ will
be set to $\tilde{B}$. Accordingly, a mapping can be given that connects the
consecutive $W_{i}$ values:
$W_{i+1}=T_{\delta_{i}}(W_{i})=\begin{cases}\left(cW_{i}^{m+1}+\alpha\left(m+1\right)\delta_{i}\right)^{\frac{1}{m+1}}&\text{if
$\delta_{i}<\frac{\tilde{B}^{m+1}-cW^{m+1}_{i}}{\alpha\left(m+1\right)}$,}\\\
\hfil\tilde{B}&\text{if
$\delta_{i}\geq\frac{\tilde{B}^{m+1}-cW^{m+1}}{\alpha\left(m+1\right)}$.}\end{cases}$
(3.5)
In this manner, the time elapsed until the next packet loss might be smaller
than $\delta_{i}$ if $\delta_{i}$ is too large. Due to the property of the
distribution (3.4) noted above, at the succeeding application of (3.5) the
next $\delta_{i+1}$ time interval can be drawn from distribution (3.4) again.
The next “before loss” window distribution $f_{W_{i+1}}(w)$ can now be
calculated by the Perron – Frobenius operator, $\mathcal{L}$, of the mapping:
$\begin{split}f_{W_{i+1}}(w)&=\mathcal{L}T_{\delta_{i}}(W_{i+1})=\int\limits_{0}^{\tilde{B}}\int\limits_{0}^{\infty}\delta\left(w-T_{x}(w^{\prime})\right)\,f_{W_{i}}(w^{\prime})\,f_{\delta_{i}}(x)\,dx\,dw^{\prime}\end{split}$
(3.6)
where $\delta()$ is the Dirac-delta distribution and I averaged over the
distribution (3.4). After substituting (3.5) into (3.6) we have to consider
the condition
$0\leq\delta_{i}=\frac{W^{m+1}_{i+1}-cW^{m+1}_{i}}{\alpha\left(m+1\right)}$.
This provides us with $W_{i}<W_{i+1}/\beta$ which should be taken into account
in the upper boundary of the first integral. The integration in $x$ can be
carried out:
$\begin{split}f_{W_{i+1}}(w)&=\frac{\lambda}{\alpha}w^{m}\,e^{-\frac{\lambda
w^{m+1}}{\alpha\left(m+1\right)}}\int\limits_{0}^{\min\left(\tilde{B},w/\beta\right)}f_{W_{i}}(w^{\prime})\,e^{\frac{\lambda
c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}\,dw^{\prime}\\\
&+\delta(w-\tilde{B})\,e^{-\frac{\lambda\tilde{B}^{m+1}}{\alpha\left(m+1\right)}}\int\limits_{0}^{\tilde{B}}f_{W_{i}}(w^{\prime})\,e^{\frac{\lambda
c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}\,dw^{\prime}.\end{split}$ (3.7)
Let ${W_{\textrm{b.l.}}}=\lim_{n\to\infty}W_{n}$ denote the stationary limit
of the “before loss” window sequence. Its stationary distribution,
$f_{{W_{\textrm{b.l.}}}}(w)$, is the fixed point solution of (3.7). For
finding the fixed point solution observe that for any probability distribution
$f_{W_{i}}(w)$ the transformed one, $f_{W_{i+1}}(w)$, will contain a Dirac-
delta term $\delta(w-\tilde{B})$ because of the second term of (3.7). I
therefore use the following ansatz for the stationary distribution
$f_{{W_{\textrm{b.l.}}}}(w)=A(\lambda/\alpha,\tilde{B})\,\delta(w-\tilde{B})+\phi(w),$
(3.8)
where $\phi:[0,\tilde{B}]\to\mathbb{R}$ is a continuous regular function and
$A(\lambda/\alpha,\tilde{B})$ is a constant. The delta function represents
those points where the packet loss occurs in the buffer and the value of the
pre-loss window is $\tilde{B}$. The constant $A(\lambda/\alpha,\tilde{B})$
represents the probability that a packet gets lost in the buffer, and it might
depend on the external loss $\lambda/\alpha$ and buffer size $\tilde{B}$. I am
going to present the detailed interpretation of $A(\lambda/\alpha,\tilde{B})$
in Subsection 3.3.1.
Applying the probe function (3.8) in (3.7) and separating the regular and
$\delta(w-\tilde{B})$ terms we obtain
$\displaystyle A(\lambda/\alpha,\tilde{B})$
$\displaystyle=e^{-\frac{\lambda\tilde{B}^{m+1}}{\alpha\left(m+1\right)}}\int\limits_{0}^{\tilde{B}}\phi(w^{\prime})\,e^{\frac{\lambda
c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}\,dw^{\prime}+A(\lambda/\alpha,\tilde{B})\,e^{-\frac{\lambda\left(1-c\right)\tilde{B}^{m+1}}{\alpha\left(m+1\right)}},$
(3.9) $\displaystyle\phi(w)$
$\displaystyle=\frac{\lambda}{\alpha}w^{m}\,e^{-\frac{\lambda
w^{m+1}}{\alpha\left(m+1\right)}}\int\limits_{0}^{\min\left(\tilde{B},w/\beta\right)}\phi(w^{\prime})\,e^{\frac{\lambda
c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}\,dw^{\prime}$
$\displaystyle+A(\lambda/\alpha,\tilde{B})\frac{\lambda}{\alpha}w^{m}\,e^{-\frac{\lambda\left(w^{m+1}-c\tilde{B}^{m+1}\right)}{\alpha\left(m+1\right)}}\Theta(w-\beta\tilde{B}),$
(3.10)
where $\Theta(x)$ is the Heaviside step function. Notice that for
$w\in]\beta\tilde{B},\tilde{B}]$ the upper limit of the first integral is
independent of $w$ and the Heaviside function equals $1$. The functional form
of the unknown function $\phi(w)$ on this interval can therefore be resolved.
Only the value of the definite integral—which is a constant—should be
determined. If we look for a solution on the adjacent interval
$]\beta^{2}\tilde{B},\beta\tilde{B}]$ we can see that the upper bound of the
first integral falls in the range $]\beta\tilde{B},\tilde{B}]$, where the
functional form of the unknown function was previously found. Again, only the
value of a definite integral is to be found. Repeating these steps recursively
one can see that the solution for the integral equation (3.10) would be
simplified if one looked for the solution on disjoint intervals
$]\beta^{n+1}\tilde{B},\beta^{n}\tilde{B}]$, $n\in\mathbb{N}$. Accordingly,
let us define the following functions:
$\displaystyle\phi_{n}(w)=\phi(w)\,\chi_{]\beta^{n+1}\tilde{B},\beta^{n}\tilde{B}]}(w),$
(3.11) $\displaystyle
S_{n}(s)=\int\limits_{0}^{\beta^{n}\tilde{B}}\phi(w)\,e^{-s\frac{w^{m+1}}{\left(m+1\right)}}\,dw,$
(3.12) where $\chi_{H}(w)$ denotes the indicator function of set
$H\subset\mathbb{R}$, and the constant $\displaystyle
I_{n}=S_{n}(-c\lambda/\alpha)=\int\limits_{0}^{\beta^{n}\tilde{B}}\phi(w)\,e^{\frac{\lambda
c{w}^{m+1}}{\alpha\left(m+1\right)}}\,dw.$ (3.13)
By applying the newly introduced definition of $I_{0}$ in (3.9) we clearly
have
$A(\lambda/\alpha,\tilde{B})=\frac{e^{-\frac{\lambda\tilde{B}^{m+1}}{\alpha\left(m+1\right)}}}{1-e^{-\frac{\lambda\left(1-c\right)\tilde{B}^{m+1}}{\alpha\left(m+1\right)}}}I_{0}.\\\
$ (3.14)
This formula with the above mentioned properties of (3.10) in the interval
$]\beta\tilde{B},\tilde{B}]$ provides us with
$\phi_{0}(w)=\frac{\lambda}{\alpha}\frac{w^{m}e^{-\frac{\lambda
w^{m+1}}{\alpha\left(m+1\right)}}}{1-e^{-\frac{\lambda\left(1-c\right)\tilde{B}^{m+1}}{\alpha\left(m+1\right)}}}I_{0}.$
(3.15)
Furthermore, for $n\in\mathbb{N},n>0$ the recursion
$\displaystyle\phi_{n}(w)=\frac{\lambda}{\alpha}w^{m}e^{-\frac{\lambda
w^{m+1}}{\alpha\left(m+1\right)}}\biggl{(}I_{n}+\int\limits_{\beta^{n}\tilde{B}}^{w/\beta}\phi_{n-1}(w^{\prime})\,e^{\frac{\lambda
c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}\,dw^{\prime}\biggr{)}$ (3.16)
can be derived easily, since the Heaviside function in (3.10) is identically
zero if $w\in[0,\beta\tilde{B}[$. In order to apply the above recursion one
should know constants $I_{n}$, which in turn can be obtained from functions
$S_{n}(s)$. If we insert (3.10) into the definition of $S_{n}(s)$ then we get
$\displaystyle S_{0}(s)$ $\displaystyle=\frac{1}{1+\alpha
s/\lambda}\left[S_{0}(sc)+A(\lambda/\alpha,\tilde{B})\,E\left(s\frac{\tilde{B}^{m+1}}{m+1}\right)\right]$
(3.17) where $E(x)=e^{-cx}-e^{-x}$, and for all $n\in\mathbb{N},n>0$
$\displaystyle S_{n}(s)$ $\displaystyle=\frac{1}{1+\alpha
s/\lambda}\left(S_{n-1}(sc)-e^{-\left(s+\frac{\lambda}{\alpha}\right)\frac{c^{n}\tilde{B}^{m+1}}{\left(m+1\right)}}I_{n-1}\right).$
(3.18)
Using (3.14) and (3.15) as initial conditions the recursive expressions (3.16)
and (3.18) can be solved. In order to start the iteration the value of the
initial condition $A(\lambda/\alpha,\tilde{B})$, or equivalently
$I_{0}=S_{0}(-c\lambda/\alpha)$ is needed. In the interest of finding $I_{0}$
I calculate the function $S_{0}(s)$ next. If we suppose that $S_{0}(s)$ is
continuous at $s=0$ then, using (3.17), it can be proven by induction that
$\begin{split}S_{0}(s)&=S_{0}(0)\prod_{k=0}^{\infty}\frac{1}{1+sc^{k}\alpha/\lambda}\\\
&+A(\lambda/\alpha,\tilde{B})\sum_{k=0}^{\infty}E\left(sc^{k}\frac{\tilde{B}^{m+1}}{\left(m+1\right)}\right)\prod_{l=0}^{k}\frac{1}{1+sc^{l}\alpha/\lambda},\end{split}$
(3.19)
where I have used that $\lim_{N\to\infty}S_{0}(sc^{N})=S_{0}(0)$ for
$c\in[0,1[$. Furthermore,
$S_{0}(0)=\int\limits_{0}^{\tilde{B}}\phi(w)\,dw=1-A(\lambda/\alpha,\tilde{B})$,
because $f_{{W_{\textrm{b.l.}}}}(w)$ is normalized.
The function $S_{0}(s)$ is bounded because it is defined via the definite
integral of the regular function $\phi(w)$. The pole at $s=-\lambda/\alpha$ on
the right hand side of (3.17) must therefore be canceled by the subsequent
factor:
$S_{0}(-c\lambda/\alpha)+A(\lambda/\alpha,\tilde{B})E\left(-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}\right)=0.$
(3.20)
In addition, $S_{0}(-c\lambda/\alpha)$ can be obtained from (3.19). As a
result,
$-A(\lambda/\alpha,\tilde{B})E\left(-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}\right)=\left(1-A(\lambda/\alpha,\tilde{B})\right)\prod_{k=0}^{\infty}\frac{1}{1-c^{k+1}}\\\
+A(\lambda/\alpha,\tilde{B})\sum_{k=0}^{\infty}E\left(-\frac{\lambda}{\alpha}\frac{c^{k+1}\tilde{B}^{m+1}}{m+1}\right)\prod_{l=0}^{k}\frac{1}{1-c^{l+1}}$
(3.21)
is acquired. We can express $A(\lambda/\alpha,\tilde{B})$ now as
$A(\lambda/\alpha,\tilde{B})=\frac{1}{1-L(c)\,G\left(\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}\right)},$
(3.22)
where $L(c)$ has been defined earlier in (2.29), and
$G(x)=\sum_{k=0}^{\infty}E(-c^{k}x)\prod_{l=1}^{k}\frac{1}{1-c^{l}}$ (3.23)
with the convention that the empty product equals $1$. Note that in (3.22) the
parameters appear only in the $\lambda\tilde{B}^{m+1}/\alpha$ combination.
This expression is the control parameter in my model. Systems in which
external packet losses and buffer sizes differ, but the
$\lambda\tilde{B}^{m+1}/\alpha$ product is the same, are similar in the sense
that they can be described with the same constant
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$.
### 3.3 Discussion
#### 3.3.1 The interpretation of $A(\cdot)$ and the effective loss
Now I present a brief explanation of the meaning of
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ and highlight its importance.
First I calculate the average time elapsed between two packet-loss events.
Remember that the inter-loss times _on the link_ $\delta_{i}$ are IID random
variables with exponential distribution. However, the buffer can induce extra
packet losses. If the congestion window was ${W_{\textrm{b.l.}}}$ at the
previous packet loss then the maximum inter-loss time is clearly
$\frac{\tilde{B}^{m+1}-c{W_{\textrm{b.l.}}}^{m+1}}{\alpha\left(m+1\right)}$,
at which time the buffer becomes congested. The exponential distribution of
$\delta_{i}$ is truncated above this upper limit, and the probability that
$\delta_{i}$ exceeds this limit is concentrated at the maximum inter-loss
time. Consequently, the conditional probability distribution that a packet
gets lost at either the buffer or the link after $\delta^{\prime}$ time,
supposing that the value of the congestion window was ${W_{\textrm{b.l.}}}$ at
the previous packet loss, can be written as
$\begin{split}f_{\delta^{\prime}\mid{W_{\textrm{b.l.}}}}(x,w)&=\delta\left(x-\frac{\tilde{B}^{m+1}-cw^{m+1}}{\alpha\left(m+1\right)}\right)e^{-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}-cw^{m+1}}{m+1}}\\\
&+\lambda e^{-\lambda
x}\left[1-\Theta\left(x-\frac{\tilde{B}^{m+1}-cw^{m+1}}{\alpha\left(m+1\right)}\right)\right],\end{split}$
(3.24)
With the help of the total probability theorem and (3.24) the average inter-
loss time can be given by
$\begin{split}\mathbb{E}[\delta^{\prime}]&=\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}x\,f_{\delta^{\prime}\mid{W_{\textrm{b.l.}}}}(x,w)\,f_{{W_{\textrm{b.l.}}}}(w)\,dx\,dw\\\
&=\int\limits_{0}^{\infty}\frac{1}{\lambda}\left(1-e^{-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}-cw^{m+1}}{m+1}}\right)f_{{W_{\textrm{b.l.}}}}(w)\,dw\\\
&=\frac{1-A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}}{\lambda},\end{split}$
(3.25)
where (3.9) has been used for replacing the last integral.
The meaning of this simple expression becomes clearer if we recognize that
$\lambda^{\prime}=1/\mathbb{E}[\delta^{\prime}]$ is the total packet loss
rate—link and buffer losses combined. Therefore,
$\frac{\lambda}{\lambda^{\prime}}=1-A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)},$
(3.26)
which can be interpreted as the ratio of the number of packets that are lost
at the link and the total amount of lost packets. Similarly,
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ is _the ratio of the number
of packets that are lost at the buffer $N_{\mathrm{buffer}}$ and the total
packet loss $N_{\mathrm{total}}$_. The possibility that this ratio can be
estimated from my model is the main result of this section. This
interpretation and the exact knowledge of the form of
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ allows us to treat buffer-
losses as if they were link-losses. It also makes it possible to calculate the
total loss along a multi-buffer, multi-link route.
According to (3.26) the measured $1-\lambda/\lambda^{\prime}$ expression
should be equal to $A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ and it
should not depend on $\lambda$ and $\tilde{B}$ separately, but only on the
$\lambda\tilde{B}^{m+1}/\alpha$ product. In order to verify (3.26) I carried
out a number of simulations with different $\lambda$ and $\tilde{B}$ parameter
values in the $1\leq\lambda\tilde{B}^{m+1}/\alpha\leq 10$ parameter range for
both LAN and WAN network configurations.
The parameter settings of the present model are the same as those of the
infinite-buffer model in the previous chapter. In particular, for LAN
scenarios: $m=1$, $\beta=1/2$ and $\lambda/\alpha=p$. Setting the value of the
new parameter $\tilde{B}$ requires extra care, however. When comparing
simulations and the formula (3.26) we have to take into account that in
reality the system can store more packets than the actual buffer size. For
example, the receiver is processing one packet, and even if the link delay is
zero, one acknowledgment packet is traversing back to the sender during the
file transfer, increasing the maximum number of unacknowledged packets in the
system by two. Moreover, TCP detects packet loss one RTT later than it
actually happens, causing overshoot of the maximum window. The difference
between simulation and fluid approximation can also cause some discrepancy. In
other words, TCP behaves as if the buffer would be bigger than it really is.
The effect of this behavior can be observed in Fig. 3.2(a) where the
congestion window occasionally exceeds the buffer size $B$.
Table 3.1: Fitted $b$ parameter values for different buffer sizes $B$. The average value is $\bar{b}=2.5354$. $B$ | $b$
---|---
30 | 2.5045
40 | 2.5790
50 | 2.6798
60 | 2.4997
70 | 2.4143
In order to treat this problem I assumed that we have to set the congestion
window limit to $\tilde{B}=B+b_{L}$, where $b_{L}$ has been fitted for
different buffer sizes $B$. The fitted values of $b_{L}$ can be found in Table
3.1. It can be seen that $b_{L}$ is constant and practically independent of
$B$. Based on simulation results I set $b_{L}$ to its average value
$\bar{b}_{L}=2.5354$.
Simulation results are shown in Fig. 3.3, where I compare the theoretical
formula for $A\left(p\tilde{B}^{2}\right)$ and the ratio
$N_{\mathrm{buffer}}/N_{\mathrm{total}}$ measured by ns-2.
$N_{\mathrm{buffer}}$ and $N_{\mathrm{total}}$ are the number of packets
dropped at the buffer and the total number of dropped packets, respectively.
In the simulated parameter range I obtained an almost perfect match.
Figure 3.3: Comparison of the theoretical function
$A\left(p(B+\bar{b}_{L})^{2}\right)$ and the measured ratio
$N_{\mathrm{buffer}}/N_{\mathrm{total}}$ obtained from numerical simulations
in various LAN configurations. The control parameter $p\tilde{B}^{2}/2$ has
been varied in the range of $1$ and $6$, at various buffer sizes between
$B=30$ and $70$.
Now I turn to the WAN scenario. Notice that the time could have been replaced
with “ACK time” in the previous arguments concerning
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ and the effective loss. In
addition idle periods affect neither the number of packets dropped at the
buffer nor the number of packets lost at the link. Therefore,
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ is basically related to the
intrinsic “ACK time” dynamics of TCP. Consequently, the ratio
$N_{\mathrm{buffer}}/N){\mathrm{total}}$ in a WAN scenario should be equal to
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ with the intrinsic parameters
of TCP dynamics $m=1$, $\beta=1/2$ and $\lambda/\alpha=p$.
In an ideal WAN network the buffer size would be zero. However, in reality the
buffer size $B$ must be set to a positive number, otherwise packet bursts
cannot go through the buffer and TCP shows pathological behavior. If the size
of the buffer is smaller than the maximum value of the slow start threshold
then the slow start mechanism can have a serious impact on the number of
packets lost at the buffer. Indeed, sudden bursts of packets of the slow start
mode might cause further congestions at the buffer, which, in turn, might
induce another slow start. This cascade of slow starts lasts until the slow
start threshold is reduced below the size of the buffer.
In order to demonstrate this phenomenon I carried out simulations with such a
parameter setting that $2DC/P=60$. The buffer size was $B=3,10$, and $30$.
Simulation results are shown in Fig 3.4, where I compared the theoretical
formula $A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ and the measured loss
ratio $N_{\mathrm{buffer}}/N_{\mathrm{total}}$. Data points deviate from the
theoretical curve considerably when $B=3$. Deviation from the theory is less
for $B=10$ than for $B=3$, but it is still significant for larger values of
the control parameter. Finally, the measured data points fit
$A\bigl{(}\lambda\tilde{B}^{m+1}/\alpha\bigr{)}$ almost perfectly when $B=30$.
Figure 3.4: The ratio $N_{\mathrm{buffer}}/N_{\mathrm{total}}$, obtained from
ns-2 simulations, is plotted as the function of $p\tilde{B}^{2}/2$. The
theoretical function $A(x)$ is shown for comparison. Below $B\approx\alpha D$
the buffer cannot handle packet bursts produced by the slow start algorithm,
therefore excess packet drops appear at the buffer. Dotted lines connecting
data points at $B=3$ and $10$ are guides to the eye.
At the end of this subsection I estimate the effective loss $\lambda^{\prime}$
in the $\lambda\tilde{B}^{m+1}/\alpha\to\infty$ and
$\lambda\tilde{B}^{m+1}/\alpha\to 0$ limits. The first is the infinite buffer
case, when packets get lost only on the link. It is evident from (3.23) that
in the $x\to\infty$ limit the $-e^{x}$ term dominates $G(x)$. Therefore,
$A(\lambda/\alpha,\tilde{B})\approx
e^{-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}/L(c)$ if
$\lambda\tilde{B}^{m+1}/\alpha\gg 1$, which implies that
$\lambda^{\prime}\approx\lambda\left(1+\frac{e^{-\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}}{L(c)}\right).$
(3.27)
The fraction of packets dropped at the buffer decreases at an exponential rate
as the control parameter $\lambda\tilde{B}^{m+1}/\alpha$ increases.
The second case is the “extreme” bottleneck buffer limit, when packets only
get lost in the buffer. From (3.26) and (3.22) it follows that
$\lambda^{\prime}=\lambda\left(1-\frac{1}{L(c)\,G\left(\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}\right)}\right).$
(3.28)
With the series expansion of $L(c)\,G(x)$, derived in Appendix A.1, we can
write
$\begin{split}\lambda^{\prime}&=\lambda\left(1+\frac{1}{1-c}\frac{\alpha}{\lambda}\frac{m+1}{\tilde{B}^{m+1}}-\frac{1+c}{2}+\mathcal{O}\left(\frac{\lambda}{\alpha}\tilde{B}^{m+1}\right)\right)\\\
&=\frac{m+1}{1-c}\frac{\alpha}{\tilde{B}^{m+1}}+\frac{1-c}{2}\lambda+\mathcal{O}\left(\frac{\lambda^{2}}{\alpha}\tilde{B}^{m+1}\right),\end{split}$
(3.29)
when $\lambda^{2}\tilde{B}^{m+1}/\alpha\ll 1$. In particular, in an ideal LAN
scenario
$p^{\prime}=\frac{8}{3B^{2}}+\frac{3}{8}p+\mathcal{O}\left(p^{2}B^{2}\right)$
(3.30)
holds for the effective packet loss probability
$p^{\prime}=\lambda^{\prime}/\alpha$ in the $p^{2}B^{2}\ll 1$ limit. The first
order approximation of this formula has been calculated in [41] for the same
bottleneck scenario. This is a further indication that my calculation is
correct. Since I obtained (3.30) as a limit of my model, my work can be viewed
as a generalization of previous studies.
#### 3.3.2 Histograms and probability distributions
I continue in this section with the derivation of the congestion window
distribution from Eqs. (3.14)–(3.18). It is easy to see that the piecewise
solution of (3.16) on the disjoint intervals can be written in the form
$\phi_{n}(w)=\frac{\lambda}{\alpha}w^{m}\sum_{k=0}^{n}h_{n,k}\,e^{-\frac{\lambda}{\alpha}\frac{c^{-k}}{m+1}w^{m+1}}.$
(3.31)
Note that functional form of (3.31) is the same as (2.33) in the infinite
buffer scenario. Substituting (3.31) into (3.16) we can derive recursive
formulas for the constants $h_{n,k}\equiv h_{n,k}(\lambda/\alpha,\tilde{B})$.
The constants might depend on the parameters $\lambda/\alpha$ and $\tilde{B}$
as I denoted explicitly. After the substitution we acquire
$\displaystyle\phi_{n+1}(w)$
$\displaystyle=\frac{\lambda}{\alpha}w^{m}\sum_{k=0}^{n+1}h_{n+1,k}\,e^{-\frac{\lambda}{\alpha}\frac{c^{-k}}{m+1}w^{m+1}}$
$\displaystyle=\frac{\lambda}{\alpha}w^{m}e^{-\frac{\lambda}{\alpha}\frac{w^{m+1}}{m+1}}\left(I_{n+1}+\frac{\lambda}{\alpha}\sum_{k=0}^{n}h_{n,k}\int\limits_{\beta^{n+1}\tilde{B}}^{w/\beta}{w^{\prime}}^{m}e^{-\frac{\lambda}{\alpha}\frac{\left(c^{-k}-c\right){w^{\prime}}^{m+1}}{m+1}}\,dw^{\prime}\right)$
$\displaystyle=\frac{\lambda}{\alpha}w^{m}e^{-\frac{\lambda}{\alpha}\frac{w^{m+1}}{m+1}}\left(I_{n+1}+\sum_{k=0}^{n}\frac{h_{n,k}}{c^{-k}-c}e^{-c^{n+1}\left(c^{-k}-c\right)\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}\right)$
$\displaystyle-\frac{\lambda}{\alpha}w^{m}\sum_{k=1}^{n+1}\frac{h_{n,k-1}}{c^{-k+1}-c}e^{\frac{\lambda}{\alpha}\frac{c^{-k}}{m+1}w^{m+1}}$
(3.32)
It can be seen that after the recursive step in (3.32) only the required
$\textit{const}\times e^{-\frac{\lambda}{\alpha}\frac{c^{-k}}{m+1}w^{m+1}}$
type terms appear. Comparing the coefficients on both sides term by term we
receive the following equations:
$\displaystyle h_{0,0}$
$\displaystyle=A(\lambda/\alpha,\tilde{B})\,e^{\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}$
(3.33) $\displaystyle h_{n+1,0}$
$\displaystyle=I_{n+1}+\sum_{k=0}^{n}\frac{h_{n,k}}{c^{-k}-c}e^{-c^{n+1}\left(c^{-k}-c\right)\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}$
(3.34) $\displaystyle h_{n+1,k}$
$\displaystyle=\frac{h_{n,k-1}}{c-c^{-k+1}}=\frac{h_{n-k+1,0}}{c^{k}}\prod_{l=1}^{k}\frac{1}{1-c^{-l}}=L(c)\,h_{k}(c)\,h_{n-k+1,0},$
(3.35)
where $h_{k}(c)$ and $L(c)$ are defined in (2.29) and (2.30). In order to
complete the system of recursive equations we have to provide constants
$I_{n}$. The constant $I_{0}$ can be obtained from (3.14):
$\displaystyle I_{0}$
$\displaystyle=A(\lambda/\alpha,\tilde{B})\left(e^{\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}-e^{\frac{\lambda}{\alpha}\frac{c\tilde{B}^{m+1}}{m+1}}\right),$
(3.36) while for $n\in\mathbb{N}$ (3.13) can be applied: $\displaystyle
I_{n+1}$
$\displaystyle=I_{n}-\int\limits_{\beta^{n+1}\tilde{B}}^{\beta^{n}\tilde{B}}\phi_{n}(w)\,e^{\frac{\lambda}{\alpha}\frac{cw^{m+1}}{m+1}}\,dw$
$\displaystyle=I_{n}-\sum_{k=0}^{n}\left(e^{-c^{n+1}\left(c^{-k}-c\right)\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}-e^{-c^{n}\left(c^{-k}-c\right)\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}}\right)\frac{h_{n,k}}{c^{-k}-c}$
$\displaystyle=I_{n}-\sum_{k=0}^{n}E\left(c^{n}\left(c^{-k}-c\right)\frac{\lambda}{\alpha}\frac{\tilde{B}^{m+1}}{m+1}\right)\frac{h_{n,k}}{c^{-k}-c}$
(3.37)
Although the number of the coefficients is infinite, we can use the first few
in practice. Since the smallest congestion window value is $1$, no more than
$\log_{2}B$ number of $\phi_{n}(w)$ functions are relevant and the inequality
$k\leq n$ implies that $k$ is also limited. Furthermore, it is obvious from
(3.35) that for every $n$ the absolute value of $h_{n,k}$ decays very quickly
as $k$ increases, so $h_{n,k}\approx 0$ can be supposed if $k\gtrsim 3$.
So far I calculated analytically the distribution of the “before loss” values
of the congestion window. In practice the distribution of the congestion
window at an arbitrary moment is relevant. I calculate this distribution
$f_{W}(w)$ here. We can basically repeat the same arguments as in Sec. 3.2. In
general, between losses, the congestion window is developing according to
(3.3), where $\tau$ is a uniformly distributed random variable on the _random_
interval $[0,\rho]$. The conditional distribution of $\tau$—supposing that
$\rho$ is given—is $f_{\tau}(t\mid\rho=x)=\frac{1}{x}\chi_{[0,x]}(t)$. The
distribution of $\rho$ is
$f_{\rho}(x)=\frac{x}{\mathbb{E}[\delta_{i}]}f_{\delta_{i}}(x)$, similarly to
the infinite buffer scenario. Thus, the distribution of the congestion window
at an arbitrary moment can be given by the following transformation
$\displaystyle f_{W}(w)$
$\displaystyle=\int\limits_{0}^{\tilde{B}}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\delta(w-T_{t}(w^{\prime}))\,f_{\tau}(t\mid\rho=x)\,f_{\rho}(x)\,f_{{W_{\textrm{b.l.}}}}(w^{\prime})\,dt\,dx\,dw^{\prime}$
$\displaystyle=\frac{1}{\mathbb{E}[\delta_{i}]}\int\limits_{0}^{\tilde{B}}\int\limits_{0}^{\infty}\int\limits_{0}^{x}\delta(w-T_{t}(w^{\prime}))\,f_{\delta_{i}}(x)\,f_{{W_{\textrm{b.l.}}}}(w^{\prime})\,dt\,dx\,dw^{\prime},$
(3.38)
where
$T_{\tau}({W_{\textrm{b.l.}}})=\left[{W_{\textrm{b.l.}}}^{m+1}+\alpha\left(m+1\right)\tau\right]^{\frac{1}{m+1}}$
is the forward mapping of the congestion window from ${W_{\textrm{b.l.}}}$ to
$W$, $\tau$ time later. The integration in variable $t$ can be carried out:
$\int\limits_{0}^{x}\delta(w-T_{t}(w^{\prime}))\,dt=\frac{w^{m}}{\alpha}\Theta\left(x-\frac{w^{m+1}-c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}\right)\left[1-\Theta\left(w^{\prime}-\frac{w}{\beta}\right)\right].$
(3.39)
The $w^{m}/\alpha$ term is from the inverse-Jacobi of $T_{t}(w^{\prime})$, and
the Heaviside functions correspond to the range of integration
$t=\frac{w^{m+1}-c{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}\in[0,x]$.
Therefore, (3.38) can be written as follows:
$\begin{split}f_{W}(w)&=\frac{1}{1-A(\lambda/\alpha,\tilde{B})}\frac{\lambda}{\alpha}w^{m}\\!\\!\\!\\!\int\limits_{0}^{\min(\tilde{B},w/b)}\\!\\!\\!\\!\\!\\!\int\limits_{\frac{w^{m+1}-{w^{\prime}}^{m+1}}{\alpha\left(m+1\right)}}^{\infty}\\!\\!\\!\\!\\!\\!\lambda
e^{-\lambda x}\,f_{{W_{\textrm{b.l.}}}}(w^{\prime})\,dx\,dw^{\prime}\\\
&=\frac{1}{1-A(\lambda/\alpha,\tilde{B})}\frac{\lambda}{\alpha}w^{m}e^{-\frac{\lambda}{\alpha}\frac{w^{m+1}}{m+1}}\\!\\!\\!\\!\\!\\!\int\limits_{0}^{\min(\tilde{B},w/b)}\\!\\!\\!\\!\\!\\!f_{{W_{\textrm{b.l.}}}}(w^{\prime})e^{\frac{\lambda}{\alpha}\frac{c{w^{\prime}}^{m+1}}{m+1}}\,dw^{\prime}\end{split}$
(3.40)
where I have used (3.25). The implicit definition of
$f_{{W_{\textrm{b.l.}}}}(w)$ given in (3.7) and (3.8) yields that
$f_{W}(w)=\frac{\phi(w)}{1-A(\lambda/\alpha,\tilde{B})}$ (3.41)
It can be seen that the final distribution is proportional to the regular part
of the “before loss” distribution, so I can apply my earlier results given in
(3.33) – (3.37) again. There is no Dirac-delta distribution in (3.41).
In the $\tilde{B}\to\infty$ limit the derived formula (3.41) should converge
to (2.33), the distribution derived for the infinite buffer scenario in the
previous chapter. Let me confirm that my result is consistent with the
infinite buffer case. I showed before that $A(\lambda/\alpha,\tilde{B})\approx
e^{-\frac{\lambda}{\alpha}\frac{\tilde{B}}{m+1}}/L(c)$ if
$\lambda\tilde{B}^{m+1}/\alpha\gg 1$. Therefore,
$A(\lambda/\alpha,\tilde{B})\to 0$ if $\tilde{B}\to\infty$, which means that
$f_{W}(w)=\lim_{n\to\infty}\phi_{n}(w)$. Furthermore, (3.36) and (3.37) imply
that $\lim_{\tilde{B}\to\infty}I_{n}=1/L(c)$ for all $n\in\mathbb{N}$. Using
these results one can see from (3.33) – (3.35) that
$\lim_{n\to\infty}\lim_{\tilde{B}\to\infty}h_{n,k}=h_{k}(c)$ for all
$k\in\mathbb{N}$.
(a) $pB^{2}/2=0$
(b) $pB^{2}/2=0.5$
(c) $pB^{2}/2=1$
(d) $pB^{2}/2=2$
(e) $pB^{2}/2=3.5$
(f) $pB^{2}/2=5$
Figure 3.5: Comparison of simulation results and theoretical model at buffer
size $B=60$. ${f_{W}}_{n}(w),\;n\in\mathrm{N}$ denote piecewise solutions of
the congestion window distribution (3.41).
##### Local Area Networks
In order to verify my results I carried out simulations with ns-2. I have
applied my results for both LAN and WAN networks. Let us consider the LAN
scenario first, where the model parameters are $m=1$, $\beta=1/2$,
$\lambda/\alpha=p$ and the effective buffer size is $\tilde{B}=B$. I compare
the simulation results and my model in Fig. 3.5 at
$pB^{2}/2=0.0,0.5,1.0,2.0,3.5$ and $5.0$ parameter values and at $B=60$ buffer
size. The link capacity $C=256\textit{kb/s}$, link delay $D=0s$ and packet
size $P=1500\textit{byte}$ were fixed in the study of LAN and only buffer size
$B$ and loss probability $p$ were varied.
For the interpretation of the simulation results I consider the effect of the
FR/FR algorithms as well. I showed in the previous chapter that in the case of
an infinite buffer the effect of FR/FR algorithms can be taken into
consideration by the modified distribution (2.46). The finite buffer case can
be handled similarly, with two minor adjustments. Firstly, the packet loss
rate $\lambda$ should be replaced by the total loss rate
$\lambda^{\prime}=\frac{\lambda}{1-A}$, because plateaus of the FR/FR mode
appear after packet losses happening at the buffer, too. Secondly, the
distribution of the “after loss” congestion window
$f_{{W_{\textrm{a.l.}}}}(w)$ should be calculated directly from
${W_{\textrm{a.l.}}}=\beta{W_{\textrm{b.l.}}}$ now:
$f_{{W_{\textrm{a.l.}}}}(w)=f_{{W_{\textrm{b.l.}}}}(\beta^{-1}w)\beta^{-1}$.
Accordingly, $f_{W}(\beta^{-1}w)$ have to be replaced by
$f_{{W_{\textrm{b.l.}}}}(\beta^{-1}w)=A(\lambda/\alpha,\tilde{B})\,\delta(\beta^{-1}w-\tilde{B})+\phi(\beta^{-1}w)$
in (2.46). After a variable transformation in the delta distribution we obtain
$f_{\tilde{W}}(w)=\frac{f_{W}(w)+\frac{\lambda}{\alpha}\tilde{B}^{m}\frac{A(\lambda/\alpha,\tilde{B})}{1-A(\lambda/\alpha,\tilde{B})}\delta(w-\beta\tilde{B})+\frac{\lambda}{\alpha}\beta^{-\left(m+1\right)}w^{m}f_{W}(\beta^{-1}w)}{1+\frac{\lambda}{\alpha}\frac{\mathbb{E}[W^{m}]}{1-A(\lambda,\alpha,\tilde{B})}},$
(3.42)
where (3.41) has been used implicitly.
The most distinct consequence of FR/FR algorithms is the sharp peak in the
middle of the histograms in Fig. 3.5. In analytic formula (3.42) the peak is
represented by a Dirac-delta distribution. The delta-distribution has been
scattered over a finite region in Fig. 3.5 in order to be comparable with the
peaks in the numerical histograms. The derived analytic expression shows very
impressive agreement with the numerical simulations. The slight discrepancy at
larger packet loss probabilities comes from the differences between the fluid
model and the packet level simulation, discussed in Section 2.5.1.
##### The effect of link delay
(a) $p\tilde{B}^{2}/2=0.5$
(b) $p\tilde{B}^{2}/2=2.5$
Figure 3.6: Quantile-quantile plot of the congestion window in network
configurations with different buffer size $B$ and bandwidth-delay product
$2\alpha D$, but with the same effective buffer size $\tilde{B}=B+2\alpha
D=90$.
I assumed in the previous analysis that the link delay is zero. My results can
be applied as approximations for situations where the link delay is non-zero,
but the probability that the buffer is empty is negligible. In addition to the
buffer, $2\alpha D$ number of packets and acknowledgments can be found on the
link where $\alpha=C/P$. $C$, $D$ and $P$ are the link capacity, the link
delay and the packet size respectively. The congestion window limit in this
situation must be set to the total number of packets $\tilde{B}=B+2\alpha D$
in the system and the link can be treated as a part of the buffer. This can be
verified with simulations. In my simulation scenario $2\alpha D=15$ and
$2\alpha D=30$ number of TCP and ACK packets could be on the link. The buffer
size was set to $B=75$ and $B=60$ respectively, so that the effective buffer
size $\tilde{B}=90$ was the same. In Fig. 3.6 quantile-quantile plots of the
congestion window are shown. Percentiles of the congestion window are plotted
at the given link delay and buffer size combinations as the function of the
percentiles of cwnd in an ideal LAN scenario. Data was obtained from
simulations at two different control parameter values. It can be seen that
data points are close to the diagonal, drawn by dotted lines. This implies
that the data points are from the same distribution when the link delay is
zero and when it is small, but not zero. Some deviation from the diagonal can
only be observed at the lower quantiles of $p\tilde{B}^{2}/2=2.5$, when
$B=60$, $\alpha D=30$, because the buffer is occasionally empty in this case.
##### Wide Area Networks
(a) $p\tilde{B}^{2}/2=0$
(b) $p\tilde{B}^{2}/2=0.5$
(c) $p\tilde{B}^{2}/2=1$
(d) $p\tilde{B}^{2}/2=2$
(e) $p\tilde{B}^{2}/2=3.5$
(f) $p\tilde{B}^{2}/2=5$
Figure 3.7: Comparison of simulation results and the theoretical model. The
link could carry maximal $2\alpha D=60$ number of TCP and ACK packets, and the
buffer could store $B=3$ packets. ${f_{W}}_{i}(w),\;i\in\mathbb{N}$ denote the
piecewise solutions of the congestion window distribution.
(a) $p\tilde{B}^{2}/2=0$
(b) $p\tilde{B}^{2}/2=0.5$
(c) $p\tilde{B}^{2}/2=1$
(d) $p\tilde{B}^{2}/2=2$
(e) $p\tilde{B}^{2}/2=3.5$
(f) $p\tilde{B}^{2}/2=5$
Figure 3.8: Comparison of simulation results and the theoretical model. The
link could carry maximal $2\alpha D=60$ number of TCP and ACK packets, and the
buffer could store $B=30$ packets. ${f_{W}}_{i}(w),\;i\in\mathbb{N}$ denote
the piecewise solutions of the congestion window distribution.
In a WAN scenario buffering delay is small compared to the link delay. As I
noted in the last section, however, it cannot be set to zero, because in a
packet level simulator packet bursts appear inevitably. Accordingly, the link
delay was so large that it could carry $2\alpha D=60$ TCP and ACK packets
simultaneously. Furthermore, two buffer size values $B=3$ and $30$ were
selected for numerical simulations. The analytical formula for the congestion
window distribution can be obtained from the modification of (2.51) for the
FR/FR algorithms, analogously to the LAN scenario. Other parameters of the WAN
model are $m=1$, $\lambda/\alpha$, and $\beta=1/2$.
The theoretical distributions and histograms obtained from ns-2 simulations
can be seen in Fig. 3.7 at $B=3$, which is close to the ideal WAN scenario.
The external loss rate was varied in the $0\leq p\tilde{B}^{2}/2\leq 5$ range.
All other parameters were fixed. One can see that the histogram deviates from
the theoretical distribution even for small values of the control parameter.
The non-zero probability in the histogram that the congestion window is $1$
implies that the slow start mechanism is responsible for the discrepancy. In
Fig. 3.8 empirical histograms are compared with the theoretical distribution
at $B=30$, which is an intermediate configuration between LAN and WAN. The
effect of slow start mode is much less significant than at $B=3$.
The main source of error is the macroscopic probability of slow start mode.
The other observable difference from experiments comes from the slight
discrepancy in the position of the Dirac-delta and the finite peak in the
histogram. Despite these errors my model agrees with simulations for small
loss probabilities and gives a qualitatively correct description of the WAN
situation for larger ones.
### 3.4 Conclusions
In this chapter I investigated the TCP congestion avoidance algorithm in
networks where the finite buffer size limits the maximal achievable congestion
window size. The most important development I accomplished in this study is
that the total loss felt by TCP, including the buffer and the external packet
loss, can be predicted from the network parameters, namely the length of the
buffer and the probability of external packet loss. This formula makes it
possible to calculate the total loss along a multi-buffer, multi-link route.
The presented analytical expression, $A(x)$, can be computed numerically
without difficulty and the total loss can be calculated by a simple formula. I
also showed that $A(x)$ and the coefficients which appear in the probability
distributions depend only on a certain combination of the parameters. This
combination is the control parameter in my model. Networks with the same
control parameters are equivalent in the sense that the same portion of the
total packet loss occurs at the buffer, and the coefficients are the same in
the distribution function.
In addition, I derived the stationary probability distribution of the
congestion window process analytically in LAN, in WAN, and in general
situations. New types of congestion window distributions are discovered when
the packet loss in the buffer is large compared to other sources of packet
loss. These are different from the usual Gaussian-type single humped
distributions and my findings can help to develop a qualitative classification
of window distributions. I validated my calculations with computer simulations
and I showed that my analysis agrees with the simulations properly. I also
pointed out the limits of my model. More specifically, I demonstrated that the
effect of the slow start mechanism becomes significant if the buffer size is
small or the packet loss probability is large.
## Chapter 4 Traffic dynamics on complex networks
The focus of the previous chapters was on TCP dynamics. The model of network
topology was very basic, consisting merely of one buffer and one link. All
details of the network topology were concentrated into a few parameters of the
link, namely the link delay, bandwidth and packet loss probability. These
effective parameters could be tuned freely in the model. However, we do not
know yet how these parameters should be adjusted in a complex network of
thousands of nodes.
Since finite buffers naturally induce packet losses a long TCP session
eventually achieves an equilibrium at a certain loss probability. For a fixed
network configuration and a system of TCP connections, therefore, packet loss
probabilities are determined by the steady state of network traffic. The
steady state of the system is heavily influenced by the allocation of the
network resources, especially the link capacity.
In this chapter I study what the optimum distribution of link capacity is in
certain types of evolving networks when the local structure of the network is
known. The motivation behind this problem is that the Internet is basically
being developed locally. In my model I suppose that optimum link capacity is
proportional to the mean traffic demand of the particular link. In a
homogeneous network the average traffic demand, in turn, is proportional to
the expected number of flows that utilize a particular link. Since routing of
packets in computer networks can be supposed to be via the shortest path
between end nodes it follows that the distribution of shortest paths is a
matter of importance.
The main subject of my investigation is the “ _betweenness_ ” of links, which
is to say the number of shortest paths that pass over a link. Note that edge
betweenness is essential not only in the case of the Internet, but in other
complex networks too. For instance, edge betweenness can measure the
“importance” of relationships in social networks or the probability of
discovering an edge during a network survey. Until recently, however, less
attention has been paid to edge betweenness.
The probability distribution of edge betweenness gives a rough statistical
description of links and it characterizes the network as a whole. Therefore,
it is an important tool for an overall description of links in complex
networks. However, if the local structure of the network is known—as I suppose
in my model—then the probability distribution of edge betweenness under the
condition of the local property provides a much finer description of links
than the total distribution. Therefore, I will aim at the _conditional_
distribution of edge betweenness.
I restrict my model to trees, that is to connected loopless graphs. The
simplicity of trees allows analytic results for edge betweenness, since the
shortest paths in trees are unique between any pair of nodes. Although trees
are special graphs, a number of real networks can be modeled by trees or by
tree-like graphs with only a negligible number of shortcuts. Important
examples of such networks are the ASs in the Internet [7].
As a model of evolving scale-free trees I consider the Barabási–Albert (BA)
model extended with initial attractiveness [55, 54]. The scaling properties of
the network can be finely tuned with initial attractiveness. Note that in the
limit of initial attractiveness to infinity the network loses its scale-free
nature and becomes similar to a classical Erdős–Rényi (ER) network with
$p_{ER}=2/N$. Therefore, scale-free and non-scale free networks can be
compared within one model. For the sake of simplicity the infinite limit of
initial attractiveness is referred to as the “ER limit” hereafter.
The rest of this chapter is organized as follows. Important results of the
literature concerning network modeling are collected in Section 4.1. In
Section 4.2 a short introduction to the construction of BA trees is given.
Simulations of large scale complex networks are presented in Section 4.3 to
illustrate the importance of optimum capacity distribution. My results are
presented in Section 4.4. In particular, a master equation for the joint
distribution of cluster size and in-degree of a specific edge is derived and
solved in Section 4.4.1 and Section 4.4.2, respectively. The total joint
distribution of cluster size is calculated in Section 4.4.3. The marginal and
conditional distributions of cluster size and in-degree are derived in Section
4.4.4 and Section 4.4.5, respectively. In Section 4.4.6, the conditional
distribution of edge betweenness follows. Finally, I summarize my work in
Section 4.5.
### 4.1 Preliminary results of topology modeling
In the early 1960’s Erdős and Rényi introduced random graphs that served as
the first mathematical model of complex networks [56]. In their model the
number of nodes is fixed and connections are established randomly. In one
variant of the ER model every node pair is connected independently with
probability $p_{\mathrm{ER}}(N)$. The probability depends on the size of the
network in such a way that the average degree of nodes is fixed: $\langle
k\rangle=p_{\mathrm{ER}}N=\mathrm{const}$. It is obvious that the distribution
of the degree of any edge is binomial, which tends to Poissonian distribution
in the $N\to\infty$ limit. Several interesting properties of the ER model are
well understood, including the relative size of the giant component, the
threshold of connectivity, etc. Although the ER model leads to rich theory, it
fails to predict the power law distributions observed in scale-free networks.
#### 4.1.1 The Barabási–Albert model
Barabási and Albert proposed a more suitable evolving model of scale-free
networks [57, 58]. The BA model is also based on random graph theory, but it
involves two key principles in addition: a) _growth_ , that is, the size of
the network is increasing during development; and b) _preferential
attachment_ , that is, new network elements are connected to higher degree
nodes with higher probability. In the original BA model every new node
connects to the core network with a fixed number of links $m$ and the
probability of attachment is proportional to the degree of nodes. The above
rules can be translated into the following approximating fluid equation, which
describes the time evolution of the degree of a particular vertex: $\partial
k_{i}(t)/\partial t=k_{i}/2t$. The solution yields
$k_{i}(t)=m\left(t/t_{i}\right)^{0.5}$, where $t_{i}$ is the time instant when
the $i$th vertex was added to the network. The degree distribution can be
given, supposing that new nodes are added uniformly in time, by:
$\mathbb{P}\left[k_{i}(t)<k\right]=\mathbb{P}\left[t_{i}>m^{2}t/k^{2}\right]=1-m^{2}t/k^{2}\left(t+m_{0}\right)$,
where $m_{0}$ is the number of initial vertices. The probability density can
be obtained from
$\mathbb{P}(k)=\partial\mathbb{P}\left[k_{i}(t)<k\right]/\partial k$. The
stationary solution finally gives
$\mathbb{P}(k)=\frac{2m^{2}}{k^{3}}.$ (4.1)
The BA model explained successfully the observed scale–free nature of many
networks by the “rich-gets-richer” phenomenon. However, the model was too
simple to fit most measured quantities of the real Internet. For example, the
degree scaling-exponent in (4.1) is $\delta_{\mathrm{BA}}=3$ which is in
contrast with the exponent $\delta=2.15$–$2.2$ observed in Internet
measurements [6, 59]. The BA model was later refined by a number of other
authors. Dorogovtsev and Mendes [60] studied the aging of nodes. The authors
extended the BA preferential attachment rule so that attachment probability
was proportional not only to the degree, but also to $(t-t_{i})^{-\nu}$, a
power law function of age, where $\nu$ is a tunable parameter. It has been
shown analytically and by simulation that the scale-free structure of the
network disappears if $\nu>1$. Moreover, an implicit equation was derived
between the scaling exponent of the degree distribution and $\nu$ for
$-\infty<\nu<1$. The influence of exponentially fast aging on global and local
clustering, degree–degree correlation and the diameter of the network was
analyzed by Zhu et al. [61].
A continuum model was developed by Albert and Barabási [62] for the study of
the effect of edge rewiring and appearance of new internal edges. In the
extended model three operations are incorporated: a) $m$new edges are created
with probability $p$, b) $m$existing edges are rewired with probability $q$;
and c) a new node is connected to the network with $m$ new links with
probability $1-p-q$. In every step a node is chosen randomly first if a)
applies and a random link of this node is removed if b) applies. In case of c)
the new node is chosen. Then a new link is established between the selected
node and another one which selected with the following preferential attachment
rule:
$\Pi(k_{i})=\frac{k_{i}+1}{\sum_{j}\left(k_{i}+1\right)}.$ (4.2)
The above procedure is repeated $m$ times.
The authors have observed a transition from a scale-free regime to an
exponential regime in the $(p,q)$ phase space. The transition takes place on
the line $q_{t}=\min\left[1-p,\left(1-p+m\right)/\left(1+2m\right)\right]$. In
the scale-free regime, where $q<q_{t}$, the connectivity distribution has a
generalized power-law form:
$\mathbb{P}(k)\propto\left(k+A(p,q,m)\right)^{-\gamma(p,q,m)},$ (4.3)
where $A(p,q,m)=\left(p-q\right)\frac{2m\left(1-q\right)}{1-p-q}+1+p-q$ and
$\gamma(p,q,m)=3-2q+\frac{1-p-q}{m}$. In the limit $p=q=0$ the model reduces
to the scale-free model investigated in [57]. It can be seen that the scaling
exponent $\gamma$ changes continuously with $p$, $q$, and $m$ in the range of
$2$ to $\infty$.
The classic BA model has been extended with initial attractiveness by
Dorogovtsev et al. [54]. More specifically, the probability that a new node is
connected to a given site is proportional to $A_{i}=A+q_{i}$, where $A\geq 0$
is called the _initial attractiveness_ and $q_{i}$ is the in-degree of node
$i$. The probability distribution of the connectivity,
$\mathbb{P}(q)=\left(1+a\right)\frac{\Gamma\left[\left(m+1\right)a+1\right]}{\Gamma(ma)}\frac{\Gamma(q+ma)}{\Gamma\left[q+2+\left(m+1\right)a\right]},$
(4.4)
was derived from a Master-equation approach where $a=A/m$ and $m$ is the
number of links starting from every new node, as in the BA model. In the
special case $a=1$ the model reproduces the original BA model with
$A_{i}=k_{i}=q_{i}+m$ and the solution (4.4) reduces to
$\mathbb{P}(k)=\frac{2m\left(m+1\right)}{k\left(k+1\right)\left(k+2\right)}.$
(4.5)
Compare this result with (4.1), which comes from a fluid approach. The two
expressions converge in the $k\to\infty$ limit, but the constant factors are
different. For $ma+q\gg 1$ the expression (4.4) takes the form
$\mathbb{P}(q)\propto\left(q+ma\right)^{-\left(2+a\right)}$, that is the
scaling exponent $\gamma=2+a$ can be tuned in the range of $2$ to $\infty$,
similarly to the previous model.
The time evolution of the average connectivity has also been derived. It has
been found that
$\bar{q}(t,t_{i})\propto\left(t_{i}/t\right)^{-1/\left(1+a\right)}$ for $t\gg
t_{i}$. The scaling exponent of the average connectivity of an old node is
therefore $\beta=1/\left(1+a\right)$. It follows that scaling exponents
$\gamma$ and $\beta$ satisfy the following scaling relation:
$\beta\left(\gamma-1\right)=1.$ (4.6)
The authors have shown that (4.6) is universal, since the above scaling
relation can be derived in the case of more general conditions.
Growing random networks with non-linear attractiveness have been studied in
[64, 63]. It has been found that scale-free connectivity distribution can be
observed only if the attractiveness kernel is asymptotically linear. The
authors confirmed the above findings indicating that the scaling exponent
depends on the details of the attachment probability and can be tuned in the
range of $2$ and $\infty$. Furthermore, the authors showed that if the
attractiveness is sub-linear then the connectivity distribution decays at an
exponential rate, while if the kernel grows more quickly than linearly then
almost all nodes are connected to a single node.
#### 4.1.2 Other network models
Other mechanisms have been proposed for the formation of scale-free networks.
Evans and Saramäki [65] studied the following simple algorithm: new vertices
are connected to the end of one or more $l$-length random walk processes.
Several variations for this general algorithm have been considered: fixed or
variable length random walks, a fixed or random number of connecting edges,
different distributions for the starting vertex of the random walk process,
edge- or vertex-wise restart of random walks, and uniform or weighted random
walks on the graph. The authors argued that a random walk process is a more
realistic mechanism than preferential attachment, since the random walk uses
only the local properties of a network.
Goh et al. [66] proposed the following stochastic model for the evolution of
Internet topology: the size of the network increased exponentially,
$N(t)=N(t_{0})e^{\alpha t}$ and the connectivity of each node is changed
according to the random process
$k_{i}(t+1)=k_{i}(t)\left[1+g_{0,i}+\xi_{i}(t)\right],$ (4.7)
where $g_{0,i}$ are constants and $\xi_{i}(t)$ are assumed to be independent
white noise processes representing fluctuations with mean zero and correlation
function
$\left\langle\xi_{i}(t)\xi_{j}(t^{\prime})\right\rangle=\sigma_{0,j}^{2}\delta(t-t^{\prime})\delta_{i,j}$.
The authors showed that in a homogeneous case, that is when $g_{0}=g_{i,0}$
and $\sigma_{0}=\sigma_{i,0}$, the connectivity distribution of the network
approximately follows a power law with exponent
$\gamma=1-\frac{g_{\mathrm{eff}}}{\sigma_{\mathrm{eff}}^{2}}+\frac{\sqrt{g_{\mathrm{eff}}^{2}+2\alpha\sigma_{\mathrm{eff}}^{2}}}{\sigma_{\mathrm{eff}}^{2}},$
(4.8)
where $g_{\mathrm{eff}}\approx g_{0}-\sigma_{0}^{2}/2$,
$\sigma_{\mathrm{eff}}^{2}\approx\sigma_{0}^{2}$. Links are removed randomly
when the degree $k_{i}$ decreases and internal edges are created according to
a preferential attachment rule when the degree $k_{i}$ increases. The model
includes an adaptation mechanism in which links are only rewired to nodes with
larger connectivity. The parameters of the model have been fitted to real AS
level Internet topology. The authors have demonstrated that their model fits
the degree-degree correlation and clustering coefficient of the real Internet
better than previous models.
In a paper by Li et al. [9] the authors argued that the technological
constraints of router design should be considered as the driving force behind
the development of the Internet. They pointed out that the possible
bandwidth–degree combinations are restricted to a technologically feasible
region for every router. In particular, large bandwidth links are connected to
low degree routers and as the degree increases router capacity must be
fragmented among more and more links. A heuristic degree-preserving rewiring
algorithm has been proposed by the authors in order to take the above
technology constraint into consideration: a small number of low degree nodes
are chosen to serve as core routers first, and other high degree nodes hanging
from the core routers are selected as access routers next. Finally, the
connections among gateway routers are adjusted in such a way that their
aggregate bandwidth to core nodes becomes almost uniform. The resulting
_Heuristically Optimal Topology (HOT)_ has been compared with other commonly
used topology generators, e.g. BA preferential attachment network, and general
random graph model. Performance metrics and random graph-based likelihood
metrics have been defined to compare different topologies, which are the
realizations of the same degree distribution. It has been shown that the
overall network performance of the HOT topology surpasses the performance of
other random networks. At the same time, the “designed” HOT topology is very
unlikely to be obtained from random graph models, according to the defined
likelihood metric. The authors concluded that their first-principles approach
combined with engineered design should replace random topology generators in
the future.
#### 4.1.3 Earlier results regarding betweenness
Node betweenness has been studied recently by Goh et al. [67] who argued that
it follows power law in scale-free networks, and the exponent $\delta\approx
2.2$ is independent from the exponent of the degree distribution as long as
the degree exponent is in the range $2<\gamma\leq 3$. The authors analyzed
both static and evolving networks, directed and undirected graphs as well as a
real network of collaborators in neuroscience. Their conjecture is based on
numerical experiments. However, Barthélemy [68] presented counter-examples to
the universal behavior and demonstrated that the important exponent is the
scaling exponent of betweenness as the function of connectivity
$\eta=\left(\delta-1\right)\left(\gamma-1\right)$ instead. In a reply [69] the
authors argued that universality is still valid for a restricted class of
tree-like, sparse networks.
Szabó et al. [70] used rooted deterministic trees to model scale-free trees.
The authors have modeled BA networks with a uniform branching process in a
mean-field approximation. They obtained that the branching process is
$b(l)=\frac{1}{2}\frac{\ln N}{l}$ on a layer at distance $l>0$. The number of
nodes $n(l)$ at distance $l$ was approximated by a non-normalized Gaussian. It
has been found that the number of shortest paths going through a node at
distance $l$ from the root node is $L(l)=\frac{\mathrm{const}}{n(l)}$,
independent of the branching process $b(l)$. Finally, the authors showed that
node betweenness, which includes shortest paths originating to and from nodes
in excess of edge load, follows a power-law decay with a universal exponent of
$-2$. The same scaling exponent has been found experimentally by Goh et al.
[67] for scale-free trees.
A rigorous proof of the heuristic results of [70] has been presented by
Bollobás and Riordan [71]. The authors showed that the number of shortest
paths through a random vertex is
$\mathbb{P}(L=l)=\frac{2N-1}{\left(2l+1\right)\left(2l+3\right)},$ (4.9)
where $N$ is the size of the network and $l\in\mathbb{N}$. Furthermore, the
distribution of the length of the shortest paths has been precisely
calculated. The asymptotic limit of the distribution was proved to be normal
with mean and variance increasing as $\log N$.
### 4.2 The network model
The concepts of graph theory are used throughout my analysis, so I will define
briefly the terminology I use first. A graph consists of _vertices_ (nodes)
and _edges_ (links). Edges are ordered or un-ordered pairs of vertices,
depending on whether an ordered or un-ordered graph is considered,
respectively. The _order_ of a graph is the number of vertices it holds, while
the _degree_ of a vertex counts the number of edges adjacent to it. _Path_ is
also defined in the most natural way: it is a vertex sequence, in which any
two consecutive elements form an edge. A path is called a _simple path_ if
none of the vertices in the path are repeated. Any two vertices in a _tree_
can be connected by a unique simple path. The graph is called connected if for
any vertex pair there exists a path which starts from one vertex and ends at
the other.
The construction of the network proceeds in discrete time steps. Let us denote
time with $\tau\in\mathbb{N}$, and the developed graph with
$G_{\tau}=\left(V_{\tau},E_{\tau}\right)$, where $V_{\tau}$ and $E_{\tau}$
denote the set of vertices and the set of edges at time step $\tau$,
respectively. Initially, at $\tau=0$, the graph consists only of a single
vertex without any edges. Then, in every time step, a new vertex is connected
to the network with a single edge. The edge is _directed_ , which emphasizes
that the two sides of the edge are not symmetric. The newly connected node,
which is the source of the edge, is always “younger” than the target node. The
term “younger node of a link” is used in this sense below. Note that the
initial vertex is different from all the others, since it has only incoming
connections; I refer to it as the _root vertex_.
The target of every new edge is selected randomly from the present vertices of
the graph. The probability that a new vertex connects to an old one is
proportional to the attractiveness of the old vertex $v$, defined as
$A(v)=a+q,$ (4.10)
where parameter $a>0$ denotes the initial attractiveness and $q$ is the in-
degree of vertex $v$. It has been shown in [54] that the in-degree
distribution is asymptotically
$\mathbb{P}(q)\simeq\left(1+a\right)\frac{\Gamma(2a+1)}{\Gamma(a)}\left(q+a\right)^{-\left(2+a\right)}$.
I will improve this result and derive the exact in-degree distribution below.
Note that in the special case $a=0$ the attractiveness of every node is zero
except of the root vertex. It follows that every new vertex is connected to
the initial vertex in this case, which corresponds to a star topology. The
special case $a=1$ practically returns the original BA model. Indeed, except
for the root vertex, the attractiveness of every vertex becomes equal to its
degree if $a=1$; this is exactly the definition of the attractiveness in the
BA model [57]. Finally, if $a\to\infty$, then preferential attachment
disappears in the limit, and the model tends to a Poisson-type graph, similar
to an ER graph.
The attractiveness of sub-graph $S$ is the sum of the attractiveness of its
elements:
$A(S)=\sum_{v^{\prime}\in S}A(v^{\prime}).$ (4.11)
I refer to a connected sub-graph as a _cluster_. The attractiveness of cluster
$C$ can be given easily:
$A(C)=\left(1+a\right)|C|-1,$ (4.12)
where $|C|$ denotes the size of the cluster. It is obvious that the overall
attractiveness of the network at time step $\tau$ is
$A(V_{\tau})=\left(1+a\right)\left(\tau+1\right)-1.$ (4.13)
### 4.3 Simulation of large computer networks
Before I continue with the analytic study of betweenness I would like to
illustrate the effects of different capacity allocation strategies in large
computer networks and demonstrate the importance of finding an optimum
strategy. To this end I carried out large scale computer simulations. Since
packet level simulations of large networks are practically impossible, because
of their huge computational requirements, I implemented a fluid model of the
network traffic based on the AIMD model, introduced below.
#### 4.3.1 The AIMD model
Baccelli and Hong [72] have developed the AIMD model for $N$ parallel TCP
flows utilizing a common bottleneck buffer. The synchronization of the TCP
flows could be tuned in the range of complete synchronization and complete
randomness. The acronym AIMD stands for _additive increase, multiplicative
decrease_. The name refers to the basic governing principle behind TCP
congestion avoidance algorithm, and it emphasizes that the details of the slow
start and the FR/FR algorithms are neglected in the model. Let $T_{n}$ denote
the $n$th congestion time, $\tau_{n+1}=T_{n+1}-T_{n}$ the elapsed time between
two consecutive congestion events, and $X_{n}^{(i)}$ the throughput of $i$th
flow _after_ the $n$th congestion event. If instantaneous throughput is
approximated by its average, then the throughput can be related to the
congestion window $W_{n}^{(i)}$ by the following equation:
$X_{n}^{(i)}=W_{n}^{(i)}P/R^{(i)}$, where $R^{(i)}$ is the RTT of the $i$th
TCP flow, and $P$ is the size of the data packets, as above. The evolution of
the throughput can be given by
$X_{n+1}^{(i)}=\left[\left(1-\xi_{n+1}^{(i)}\right)+\beta^{(i)}\xi_{n+1}^{(i)}\right]\left(X_{n}^{(i)}+\frac{\alpha^{(i)}P}{R^{(i)}}\tau_{n+1}\right),$
(4.14)
where $\alpha^{(i)}$ and $\beta^{(i)}$ are the linear growth rate and the
multiplicative decrease factor of the congestion window, respectively, and
$\xi_{n}^{(i)}$ are random variables, independent in $n$, which take the value
$1$ if the $i$th TCP flow experiences packet loss at the $n$th congestion
event, and $0$ otherwise. Congestion occurs at the bottleneck, supposing
negligible or zero buffer capacity, when the total throughput reaches the
capacity of the bottleneck link $C$. Accordingly, $\tau_{n+1}$ can be
calculated from following fluid equation:
$\sum_{i=1}^{N}\left(X_{n}^{(i)}+\frac{\alpha^{(i)}P}{R^{(i)}}\tau_{n+1}\right)=C.$
(4.15)
The variable $\tau_{n+1}$ can be eliminated from (4.14) and (4.15), which
leads to
$X_{n+1}^{(i)}=\gamma_{n+1}^{(i)}\left(\rho^{(i)}C+X_{n}^{(i)}-\rho^{(i)}\sum_{j=1}^{N}X_{n}^{(j)}\right),$
(4.16)
where $\gamma_{n}^{(i)}=\left(1-\xi_{n}^{(i)}\right)+\beta^{(i)}\xi_{n}^{(i)}$
and
$\rho^{(i)}=\frac{\alpha^{(i)}/R^{(i)}}{\sum_{j=1}^{N}\alpha^{(j)}/R^{(j)}}$.
The system of recursive equations (4.16) can also be given in a simpler
stochastic matrix form:
$\mathbf{X}_{n+1}=\mathbf{A}_{n+1}\cdot\mathbf{X}_{n}+\mathbf{B}_{n+1},$
(4.17)
where $\left(\mathbf{B}_{n}\right)_{i}=\gamma_{n}^{(i)}\rho^{(i)}C$ and
$\left(\mathbf{A}_{n}\right)_{ij}=\gamma_{n}^{(i)}\left(\delta_{ij}-\rho^{(i)}\right)$,
and $\delta_{ij}$ is the Kronecker-delta symbol.
The interaction of the competing flows is taken into account by the
synchronization rate, $r_{n}^{(i)}=\mathbb{E}\left[\xi_{n}^{(i)}\right]$. Note
that $\xi_{n}^{(i)}$ are not independent at a given $n$ for $i=1\dots N$,
since at a congestion event a minimum of one TCP flow must experience packet
loss. If $\xi_{n}^{(i)}$ are generated independently with
$P\left(\xi_{n}^{(i)}=1\right)=\pi_{n}^{(i)}$, but those realizations are
discarded where $\sum_{i=1}^{N}\xi_{n}^{(i)}=0$, then the synchronization rate
can be expressed with the following conditional probability:
$r_{n}^{(i)}=P\left(\xi_{n}^{(i)}=1\Biggm{|}\sum_{i=1}^{N}\xi_{n}^{(i)}\neq
0\right)=\frac{\pi_{n}^{(i)}}{1-\prod_{j=1}^{N}\left(1-\pi_{n}^{(i)}\right)}.$
(4.18)
For the special case $N=1$, for example, it is evident that $r_{n}\equiv 1$.
Let us consider a simple homogeneous situation, where $\alpha^{(i)}=\alpha$,
$\beta^{(i)}=\beta$, $R^{(i)}=R$, and $r_{n}^{(i)}=r_{n}$. It is obvious that
$\rho^{(i)}=1/N$ in this case. Moreover, it can easily be shown that the
expectation of the steady state throughput is
$\mathbb{E}\left[\mathbf{X_{\infty}}\right]=\mathbb{E}\left[\mathbf{B_{\infty}}\right]$,
that is
$\mathbb{E}\left[X^{(i)}\right]=\mathbb{E}\left[\gamma\right]\frac{C}{N}=\left[1-\left(1-\beta\right)r\right]\frac{C}{N}$
(4.19)
for all $i$. The above formula predicts the degradation of the throughput as
the synchronization grows. This is in good agreement with simulations and
measurements. The expected time between consecutive congestion events can also
be obtained:
$\mathbb{E}\left[\tau\right]=\left(1-\beta\right)\frac{CRr}{\alpha NP}.$
(4.20)
The “$1/\sqrt{p}$” formula can be derived from the extended AIMD model as
well. The functional form of the formula is
$X=\frac{P}{R}\frac{\sqrt{2f(r,N)}}{\sqrt{p}}.$ (4.21)
The precise form of $f(r,N)$ is rather complicated. However, it has been shown
that $\lim_{N\to\infty}f(r,N)=1-r/4$. This implies that for large $N$ the
constant factor $c_{0}=\sqrt{2f(r,N)}$ varies in the range
$[\sqrt{3/2},\sqrt{2}]$ with the synchronization rate.
The authors have presented a wavelet and an auto-correlation analysis for
traces of the AIMD model for a large number of TCP connections. They concluded
that the trajectory of the aggregated throughput shows multi-fractal scaling
properties on short time scales and the wavelet and auto-correlation methods
give consistent fractal dimensions.
The single-buffer AIMD model can be generalized straightforwardly for
numerical simulations of more complex networks. One only needs to apply (4.15)
for each link and find the minimum of possible congestion events:
$\tau_{n+1}=\min_{e\in E}\frac{C_{e}-\sum_{i\in I_{e}}X_{n}^{(i)}}{\sum_{i\in
I_{e}}\frac{\alpha^{(i)}P}{R^{(i)}}},$ (4.22)
where $I_{e}$ denotes the set of flows which utilizing link $e\in E$. The
flows of the congested buffer are handled the same way as in the original
single buffer AIMD model. The remaining flows in the network develop
undisturbed until the next possible congestion event.
#### 4.3.2 Performance of different bandwidth distribution strategies
In this section different bandwidth distribution scenarios are compared using
a fluid simulator based on the above AIMD model. The underlying network
topology is the same in all scenarios: a scale-free network generated
according to the extended BA model introduced in Section 4.2. The parameter,
which controls the number of new links in the model, is set to $m=1$, that is
the resulting network is a tree. The scaling parameter is set to $a=1$ for
numerical purposes. In simulations link capacities are normalized in such a
way that the average capacity is the same in all scenarios.
Table 4.1: Link capacity and performance in case of different strategies. The assigned capacity is proportional to the quantity displayed in the second column, where $q_{A}$, $q_{B}$ denote the in-degrees of the nodes which compose a particular link, and $L_{e}$ denotes edge betweenness. Strategy | $C_{e}$ | $Q[b/s]$
---|---|---
Uniform | $\propto 1$ | 740.79
Maximum | $\propto\max(q_{A},q_{B})$ | 2391.94
Minimum | $\propto\min(q_{A},q_{B})$ | 6574.69
Product | $\propto q_{A}\cdot q_{B}$ | 5279.5
Mean field | $\propto L_{e}$ | 11284.6
The rules of different strategies are presented in Table 4.1. The uniform
scenario, when the capacity is the same for every link, is regarded as a
reference. It can be considered the worst case scenario, when no information
is available about the details of the network. On the contrary, the mean field
strategy—when the link capacity is proportional to the edge betweenness—is a
global optimum. Minimum, maximum and product strategies are a couple of naive
attempts to take the local structure of the network into account. Note that
only one global information the normalizing factor for the average capacity is
required. In the later three cases the more connection a link possesses, the
more capacity is allocated for the particular link. The difference between the
three strategies is whether they prefer loosely, moderately or highly
connected links, compared with the mean field allocation strategy.
The capacity range that different strategies are more likely to prefer can be
easily determined by the complementary distribution of capacities, shown in
Fig. 4.1. The average capacity is set to $\left\langle
C\right\rangle=10^{5}$[b/s] for all cases. The distribution of the uniform
strategy is clearly degenerated since only one capacity value is possible in
this scenario. The maximum strategy prefers the lower bandwidths at the cost
of a cutoff at about $10^{6}b/s$ capacity. The minimum strategy also prefers
lower bandwidths at the cost of high bandwidths, but no cutoff exists. The
complementary distribution of minimum strategy resembles the mean field
distribution with a different scaling exponent. The product strategy prefers
the mid-range of bandwidth, and it underestimates both the low and the high
capacity range, compared to the mean field strategy.
Figure 4.1: Comparison of the complementary CDF of link capacity is shown for
different bandwidth distribution strategies on log-log plot. Data is obtained
from 10 realizations of $N=10^{4}$ node networks. Average capacity is set to
$\bar{C}=10^{5}[b/s]$ for every network. The following scenarios are
considered: uniform (pentagons), maximum (diamonds), minimum (triangles),
product (circles), and mean field (squares).
In order to compare different strategies one needs an ordering between them.
Performance, the average throughput of TCPs, provides a natural ordering
between different strategies. Let us define the performance of individual TCPs
first as the time average of their throughput $X^{(i)}(t)$:
$Q^{(i)}=\left\langle
X^{(i)}(t)\right\rangle_{t}=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}X^{(i)}(u)\,du.$
(4.23)
The global performance of a strategy is then the mean performance of the TCPs
operating in the network
$Q=\frac{1}{N_{\textrm{TCP}}}\sum_{i=1}^{N_{\textrm{TCP}}}Q^{(i)}.$ (4.24)
The locations of TCP sources and destinations are distributed homogeneously in
my numerical simulations. The length of a simulation is such that every TCP
connection experiences $100$ congestion epochs on average. Network
performances obtained from simulations are shown in Table 4.1 for the
different bandwidth distribution strategies. The table shows that mean field
bandwidth allocation strategy is almost twice as effective as the second,
“minimum strategy”, and it is more than twice as good as the “product
strategy”. The performance of a network with maximum bandwidth distribution
strategy is about one fifth the performance of the same network when mean
field strategy is used. Moreover, the performance of uniform scenario is even
less then one third of the second worst, “maximum strategy”.
A more detailed picture can be gotten from the distribution of TCP-wise
performance $F(Q^{(i)})$. Simulation results of the cumulative distribution
function (CDF) of TCP performance are shown in Figure 4.2 for the above
mentioned bandwidth allocation strategies. The performance of mean field
strategy is clearly the best. The bulk of the distribution is concentrated to
a relatively narrow performance interval, that is most of TCPs can operate at
almost the same, high performance level. The performance distribution of the
next two best performing strategies, the minimum and the product, is very
similar below their median. Above the median the minimum strategy performs
better even though large capacity links are preferred less than the product
strategy. It follows that the whole bandwidth range must be taken into
consideration in any bandwidth distribution strategy to reach the optimum
network performance. The performance of the maximum strategy is considerably
worse than the previous two, mainly due to the sharp cutoff in the capacity
distribution. Finally, the uniform bandwidth distribution is the worst of all:
its performance is just a few percent of the mean field scenario’s
performance. The network where this strategy is applied is heavily congested,
since the bottlenecks form in the core of the network.
Figure 4.2: Comparison of the CDF of TCP performance is shown for different
bandwidth distribution strategies on normal-log plot. Data is obtained from 10
realizations of $N=10^{4}$ node networks with scaling parameter $\alpha=1/2$.
Average capacity is set to $\bar{C}=10^{5}[b/s]$ for every network. Simulation
lasted for $100N$ congestion epochs. The following scenarios are considered:
uniform (pentagons), maximum (diamonds), minimum (triangles), product
(circles), and mean field (squares).
In summary, the selection of inadequate bandwidth allocation strategy can
degrade the overall performance of the network considerably. In the following
sections I discuss analytically how additional local information could be used
to allocate capacity to links properly. Beforehand, I introduce the network
model investigated.
### 4.4 Discussion
It is my aim to derive the probability distribution distribution of edge
betweenness in evolving scale-free trees, under the condition that the in-
degree of the “younger” node of any randomly selected link is known. For the
sake of simplicity I consider the in-degree of the “younger” node only.
Whether a node is “younger” than another node or not can be defined uniquely
in evolving networks, since nodes attach to the network sequentially. Note
that the in-degree is considered instead of total degree for practical reasons
only. The construction of the network implies that the in-degree is less than
the total degree by one for every “younger” node.
To obtain the desired conditional distribution I calculate the exact joint
distribution of cluster size and in-degree for a _specific_ link first. Then,
the joint distribution of a _randomly selected_ link is derived, which is
comparable with the edge ensemble statistics obtained from a network
realization. The exact marginal distributions of cluster size and in-degree
follow next. After that, I give the distribution and mean of cluster size
under the condition that in-degree is known. For the sake of completeness the
conditional in-degree distribution is presented as well. Finally, the
distribution and mean of edge betweenness is derived under the condition that
the corresponding in-degree is known. Note that all of my analytic results are
_exact even for finite networks_ , which is valuable since the real networks
are often much smaller than the valid range of asymptotic formulas. Moreover,
_exact results for unbounded networks_ are provided as well.
#### 4.4.1 Master equation for the joint distribution of cluster size and in-
degree
Let us consider the size of the network $N$, an arbitrary edge $e$, which
connected vertex $v$ to the graph at time step $\tau_{e}>0$, and let us denote
by $C$ the cluster that has developed on vertex $v$ until $\tau>\tau_{e}$
(Fig. 4.3). The calculation of betweenness of the given edge is
straightforward in trees, since the number of shortest paths going through the
given edge, that is the betweenness of the edge, is obviously
$L=|C|\left(N-|C|\right)$. Therefore, it is sufficient to know the size of the
cluster on the particular edge to obtain edge betweenness.
Figure 4.3: Schematic illustration of the evolving network at time $\tau$.
Vertex $v$, connected to the network at $\tau_{e}$, denotes the root of
cluster $C$. Variables $q$ and $n=|C|-1$ denote the in-degree of vertex $v$
and the number of nodes in $C$ without $v$ (marked by circles), respectively.
The development of cluster $C$ can be regarded as a Markov process. The states
of the cluster are indexed by $\left(n,q\right)$, where $n=|C|-1$ denotes the
number of vertices in cluster $C$ without $v$. The in-degree of vertex $v$ is
denoted by $q$. Transition probabilities can be obtained from the definition
of preferential attachment:
$\displaystyle W_{\tau,n,q}$ $\displaystyle=\frac{A\left(C_{\tau}\setminus
v\right)}{A\left(V_{\tau}\right)}=\frac{n-\alpha q}{\tau+1-\alpha}$ (4.25)
$\displaystyle W^{\prime}_{\tau,q}$
$\displaystyle=\frac{A\left(v\right)}{A\left(V_{\tau}\right)}=\frac{\alpha
q+1-\alpha}{\tau+1-\alpha},$ (4.26)
where $\alpha=1/\left(1+a\right)\in\left]0,1\right]$ and $W_{\tau,n,q}$
denotes the transition probability $\left(n,q\right)\to\left(n+1,q\right)$,
and $W^{\prime}_{\tau,q}$ denotes the transition probability
$\left(n,q\right)\to\left(n+1,q+1\right)$, respectively.
The Master-equation, which describes the Markov process, follows from the fact
that cluster $C$ can develop to state $\left(n,q\right)$ obviously in three
ways: a new vertex can be connected
1. 1.
to cluster $C$ but not to vertex $v$, and the cluster was in state
$\left(n-1,q\right)$,
2. 2.
to vertex $v$, and the cluster was in state $\left(n-1,q-1\right)$, or
3. 3.
to the rest of the network, and the cluster was in state $\left(n,q\right)$.
Therefore, the conditional probability $\mathbb{P}_{\tau}(n,q\mid\tau_{e})$
that the developed cluster on edge $e$ is in state $\left(n,q\right)$
satisfies the following Master-equation:
$\displaystyle\mathbb{P}_{\tau}(n,q\mid\tau_{e})$
$\displaystyle=W_{\tau-1,n-1,q}\,\mathbb{P}_{\tau-1}(n-1,q\mid\tau_{e})$
$\displaystyle+W^{\prime}_{\tau-1,q-1}\mathbb{P}_{\tau-1}(n-1,q-1\mid\tau_{e})$
$\displaystyle+\left[1-W_{\tau-1,n,q}-W^{\prime}_{\tau-1,q}\right]\mathbb{P}_{\tau-1}(n,q\mid\tau_{e}),$
(4.27)
Since the process starts with $n=0$, $q=0$ at $\tau=\tau_{e}$, the initial
condition of the above Master equation is
$\mathbb{P}_{\tau_{e}}(n,q\mid\tau_{e})=\delta_{n,0}\delta_{q,0}$, where
$\delta_{i,j}$ is the Kronecker-delta symbol.
#### 4.4.2 The solution of the master equation
After substituting the above transition probabilities into (4.4.1), the
following first order linear partial difference equation is obtained:
$\displaystyle\left(\tau-\alpha\right)\,\mathbb{P}_{\tau}(n,q\mid\tau_{e})$
$\displaystyle=\left(n-1-\alpha
q\right)\mathbb{P}_{\tau-1}(n-1,q\mid\tau_{e})$ $\displaystyle+\left(\alpha
q+1-2\alpha\right)\mathbb{P}_{\tau-1}(n-1,q-1\mid\tau_{e})$
$\displaystyle+\left(\tau-n-1\right)\mathbb{P}_{\tau-1}(n,q\mid\tau_{e}),$
(4.28)
Let us seek a particular solution of (4.28) in product form:
$f(\tau)\,g(n)\,h(q)$. The following equation is obtained after substituting
the probe function into (4.28):
$\displaystyle\left(\tau-\alpha\right)\frac{f(\tau)}{f(\tau-1)}-\tau$
$\displaystyle=\left(n-1-\alpha q\right)\frac{g(n-1)}{g(n)}-n-1$
$\displaystyle+\left(\alpha
q+1-2\alpha\right)\frac{g(n-1)}{g(n)}\frac{h(q-1)}{h(q)}.$ (4.29)
The above partial difference equation can be separated into a system of three
ordinary difference equations. The solutions of the separated equations are:
$\displaystyle f(\tau)$
$\displaystyle=\frac{\Gamma(\tau+\lambda_{1})}{\Gamma(\tau-\alpha+1)},$ (4.30)
$\displaystyle g(n)$
$\displaystyle=\frac{\Gamma(n+\lambda_{2})}{\Gamma(n+\lambda_{1}+1)},$ (4.31)
$\displaystyle h(q)$
$\displaystyle=\frac{\Gamma(q+1/\alpha-1)}{\Gamma(q+\lambda_{2}/\alpha+1)},$
(4.32)
where $\lambda_{1}$ and $\lambda_{2}$ are separation parameters.
The solution of (4.4.1), which fulfills the initial conditions, is constructed
from the linear combination of the above particular solutions:
$\mathbb{P}_{\tau}(n,q\mid\tau_{e})=\sum_{\lambda_{1},\lambda_{2}}C_{\lambda_{1},\lambda_{2}}\,f(\tau)\,g(n)\,h(q),$
(4.33)
where $C_{\lambda_{1},\lambda_{2}}$ coefficients are independent of $\tau$,
$n$ and $q$.
To obtain coefficients $C_{\lambda_{1},\lambda_{2}}$, the initial condition of
(4.4.1) is expanded on the bases of $g(n)$ and $h(q)$. The detailed
calculation is presented in Appendix A.2.
The solution of (4.4.1) is
$\displaystyle\mathbb{P}_{\tau}(n,q\mid\tau_{e})$
$\displaystyle=\frac{\Gamma(\tau-\tau_{e}+1)}{\Gamma(\tau_{e})\,\Gamma(n+1)}\frac{\Gamma(\tau-n)}{\Gamma(\tau-\tau_{e}-n+1)}$
$\displaystyle\times\frac{\Gamma(\tau_{e}+1-\alpha)}{\Gamma(\tau+1-\alpha)}\frac{\Gamma(q+1/\alpha-1)}{\Gamma(1/\alpha-1)}\,\Phi_{\alpha}(n,q)$
(4.34)
where
$\Phi_{\alpha}(n,q)=\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\left(-\alpha
k\right)_{n}$ and $\left(x\right)_{n}\equiv\Gamma(n+x)/\Gamma{(x)}$ denotes
Pochhammer’s symbol. Note that $\mathbb{P}_{\tau}(n,q\mid\tau_{e})\neq 0$ iff
$0\leq q\leq n\leq\tau-\tau_{e}$. The conditions $0\leq q$ and
$n\leq\tau-\tau_{e}$ are obvious, since $1/\Gamma(k)=0$ by definition if $k$
is a negative integer or zero. Furthermore, the condition $q<n$ can easily be
seen if $\Phi_{\alpha}(n,q)$ is transformed into the following equivalent
form:
$\Phi_{\alpha}(n,q)=\frac{1}{q!}\frac{d^{n}}{dz^{n}}z^{n-1}\left(1-z^{-\alpha}\right)^{q}\bigr{|}_{z=1}$.
This result coincides with the fact that the size of a cluster $n$ cannot be
less than the corresponding number of in-degrees $q$.
#### 4.4.3 Joint distribution of cluster size and in-degree
Equation (4.34) provides the conditional probability that a particular edge
which was connected to the network at $\tau_{e}$ is in state
$\left(n,q\right)$ at $\tau>\tau_{e}$. In a fully developed network, however,
the time when a particular edge is connected to the network is usually not
known. Moreover, the development of an individual link is usually not as
important as the properties of the link ensemble when it has finally
developed. Therefore, we are more interested in the total probability
$\mathbb{P}_{\tau}(n,q)$, that is the probability that a randomly selected
edge is in state $\left(n,q\right)$ at $\tau$, than the conditional
probability (4.34). The total probability can be calculated with the help of
the total probability theorem:
$\mathbb{P}_{\tau}(n,q)=\sum_{\tau_{e}=1}^{\tau}\mathbb{P}_{\tau}(n,q\mid\tau_{e})\,\mathbb{P}_{\tau}(\tau_{e}),$
(4.35)
where $\mathbb{P}_{\tau}(\tau_{e})$ is the probability that a randomly
selected edge was included into the network at $\tau_{e}$. According to the
construction of the network one edge is added to the network at every time
step, therefore $\mathbb{P}_{\tau}(\tau_{e})=1/\tau$. The following formula
can be obtained after the above summation has been carried out:
$\mathbb{P}_{\tau}(n,q)=\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\left(2-\alpha\right)_{n+1}}\,\Phi_{\alpha}(n,q),$
(4.36)
where $0<\alpha\leq 1$. In star topology, that is when $\alpha=1$, the joint
distribution $\mathbb{P}_{\tau}(n,q)$ evidently degenerates to
$\mathbb{P}_{\tau}(n,q)=\delta_{n,0}\,\delta_{q,0}$.
The ER limit of joint distribution can be obtained via the $\alpha\to 0$ limit
of (4.36) (see Appendix A.3 for details):
$\displaystyle\lim_{\alpha\to 0}\mathbb{P}_{\tau}(n,q)$
$\displaystyle=\frac{\tau+1}{\tau}\sum_{k=q-1}^{n-1}\left(-1\right)^{k+n-1}\frac{\binom{k}{q-1}S_{n-1}^{\left(k\right)}}{\Gamma(n+3)}$
(4.37)
where $0<q\leq n<\tau$ and $S_{n}^{\left(m\right)}$ denote the Stirling
numbers of the first kind. Note that for the special case $n=q=0$ the ER limit
is $\lim_{\alpha\to 0}\mathbb{P}_{\tau}(0,0)=\frac{\tau+1}{2\tau}$.
The above formulas have been verified by extensive numerical simulations. The
joint empirical cluster size and in-degree distribution has been compared with
the analytic formula (4.36) for $\alpha=1/2$ in Fig 4.4. Figures 4.4(a) and
4.4(b) represent intersections of the joint distribution with cutting planes
of fixed in-degrees and cluster sizes, respectively. The figures confirm that
the empirical distributions, obtained as relative frequencies of links with
cluster size $n$ and in-degree $q$ in 100 network realizations, are in
complete agreement with the derived analytic results.
(a) Joint distribution of cluster size and in-degree as the function of
cluster size.
(b) Joint distribution of cluster size and in-degree as the function of in-
degree.
Figure 4.4: Joint empirical distribution of cluster size and in-degree at
$\alpha=1/2$ (symbols), and analytic formula (4.36) (solid lines) are compared
on double-logarithmic plot. Simulation results have been obtained from $100$
realizations of $10^{5}$ size networks.
Equation (4.36) is the fundamental result of this section. The derived
distribution is exact for any finite value of $\tau$, that is for any finite
BA trees. This result is valuable for modeling a number of real networks where
the size of the network is small compared to the relevant range of cluster
size or in-degree. If the size of the network is much larger than the relevant
range of cluster size or in-degree then it is practical to consider the
network as infinitely large, that is to take the $\tau\to\infty$ limit. For
the above joint distributions (4.36) and (4.37) the $\tau\to\infty$ limit is
evident, since the $\tau$ dependent prefactors obviously tend to $1$ if the
size of the networks grows beyond every limit.
#### 4.4.4 Distributions of cluster size and in-degree
I have derived the joint probability distribution of the cluster size and the
in-degree in the previous section. In many cases it is sufficient to know the
probability distribution of only one random variable, since the information on
the other variable is either unavailable or not needed. It is also possible
that the one dimensional distribution is especially necessary, for example for
the calculation of a conditional distribution in Section 4.4.5.
The one dimensional (marginal) distributions $\mathbb{P}_{\tau}(n)$ and
$\mathbb{P}_{\tau}(q)$ can be obtained from joint distribution
$\mathbb{P}_{\tau}(n,q)$ as follows:
$\displaystyle\mathbb{P}_{\tau}(n)$
$\displaystyle=\sum_{q=0}^{n}\mathbb{P}_{\tau}(n,q),$
$\displaystyle\mathbb{P}_{\tau}(q)$
$\displaystyle=\sum_{n=q}^{\tau-1}\mathbb{P}_{\tau}(n,q).$
After substituting (4.36) into the above formulas the following expressions
are obtained:
$\mathbb{P}_{\tau}(n)=\frac{\tau+1-\alpha}{\tau}\frac{1-\alpha}{\left(n+1-\alpha\right)\left(n+2-\alpha\right)}.$
(4.38)
if $0\leq n<\tau$ and $\mathbb{P}_{\tau}(n)=0$ if $n\geq\tau$. Furthermore,
$\displaystyle\mathbb{P}_{\tau}(q)$
$\displaystyle=\frac{\tau+1-\alpha}{\tau}\frac{1}{\alpha}\frac{\left(1/\alpha-1\right)_{1/\alpha}}{\left(q+1/\alpha-1\right)_{1/\alpha+1}}$
$\displaystyle-\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\left(2-\alpha\right)_{\tau}}\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\tau}}{\alpha k+2-\alpha}.$ (4.39)
if $0\leq q<\tau$ and $\mathbb{P}_{\tau}(q)=0$ otherwise. Rice’s method [73]
has been applied to evaluate the first term of $\mathbb{P}_{\tau}(q)$ in
closed form.
The ER limit of the marginal cluster size distribution can obviously be
obtained from (4.38) at $\alpha=0$. Furthermore, the ER limit of the marginal
in-degree distribution can be derived analogously to the limit of the joint
distribution, shown in Appendix A.3:
$\lim_{\alpha\to
0}\mathbb{P}_{\tau}(q)=\frac{\tau+1}{\tau}\frac{1}{2^{q+1}}+\frac{\tau+1}{\tau}\frac{1}{\Gamma(\tau+2)\Gamma(q)}\frac{d^{q-1}}{d\alpha^{q-1}}\frac{\left(1+\alpha\right)_{\tau-1}}{2-\alpha}\Biggr{|}_{\alpha=0}\\!\\!\\!.$
(4.40)
If the size of the network grows beyond every limit, that is if
$\tau\to\infty$, then the marginal distributions become much simpler:
$\displaystyle\mathbb{P}_{\infty}(n)$
$\displaystyle=\frac{1-\alpha}{\left(n+1-\alpha\right)\left(n+2-\alpha\right)}$
(4.41) $\displaystyle\mathbb{P}_{\infty}(q)$
$\displaystyle=\frac{1}{\alpha}\frac{\left(1/\alpha-1\right)_{1/\alpha}}{\left(q+1/\alpha-1\right)_{1/\alpha+1}}$
(4.42) $\displaystyle\lim_{\alpha\to 0}\mathbb{P}_{\infty}(q)$
$\displaystyle=\frac{1}{2^{q+1}}.$ (4.43)
The asymptotic behavior of the cluster size and in-degree distributions differ
significantly. The tail of the cluster size distribution follows power law
with exponent $2$ either in BA or ER network, independently of $\alpha$.
However, we learned that the tail of the in-degree distribution follows power
law with exponent $1/\alpha+1=2+a$ in BA networks, and it falls exponentially
in ER topology, which agrees with the well known results of previous works
[56].
It is worth noting that the mean cluster size diverges logarithmically as the
size of the network tends to infinity:
$\mathbb{E}_{\tau}\left[n\right]=\sum_{n=0}^{\tau-1}n\,\mathbb{P}_{\tau}(n)=\left(1-\alpha\right)\ln\tau+\mathcal{O}\left(1\right).$
(4.44)
The expectation value of the in-degree, however, obviously remains finite:
$\mathbb{E}_{\tau}\left[q\right]=\frac{\tau}{\tau+1}<1$, and
$\mathbb{E}_{\infty}\left[q\right]=1$ if the size of the network is infinite.
Moreover, the variance of the in-degree can also be given exactly when the
size of the network grows beyond every limit:
$\mathbb{E}_{\infty}\left[\left(q-1\right)^{2}\right]=\frac{2}{\left|1-2\alpha\right|}.$
(4.45)
This result implies that the fluctuations of the in-degree diverge in a
boundless network, if $\alpha=1/2$, that is in the classical BA model.
My analytic results have been verified with computer simulations. Since
cumulative distributions are more suitable to be compared with simulations
than ordinary distributions I matched the corresponding complementary
cumulative distribution function (CCDF) against simulation data. The CCDF of
cluster size,
$\bar{F}_{\tau}(n)=\sum_{n^{\prime}=n}^{\tau-1}\mathbb{P}_{\tau}(n^{\prime})$
can be calculated straightforwardly:
$\bar{F}_{\tau}(n)=\frac{\tau+1-\alpha}{\tau}\frac{1-\alpha}{n+1-\alpha}-\frac{1-\alpha}{\tau},$
(4.46)
where $0\leq n<\tau$ and $0\leq\alpha\leq 1$. The CCDF of in-degree,
$\bar{F}_{\tau}(q)=\sum_{q^{\prime}=q}^{\tau-1}\mathbb{P}_{\tau}(q^{\prime})$
is more complex, however:
$\displaystyle\bar{F}_{\tau}(q)$
$\displaystyle=\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{1/\alpha}}{\left(q+1/\alpha-1\right)_{1/\alpha}}-\frac{1-\alpha}{\tau}$
$\displaystyle+\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\left(2-\alpha\right)_{\tau}}\sum_{k=0}^{q-2}\frac{\left(-1\right)^{k}}{k!\left(q-2-k\right)!}\frac{\left(1-\alpha-\alpha
k\right)_{\tau-1}}{\left(k+1/\alpha\right)\left(k+2/\alpha\right)}$ (4.47)
where $0\leq q<\tau$ and $0<\alpha\leq 1$. If the size of the network grows
beyond every limit, then the CCDFs are the following:
$\displaystyle\bar{F}_{\infty}(n)$
$\displaystyle=\frac{1-\alpha}{n+1-\alpha},$
$\displaystyle\bar{F}_{\infty}(q)$
$\displaystyle=\frac{\left(1/\alpha-1\right)_{1/\alpha}}{\left(q+1/\alpha-1\right)_{1/\alpha}},$
(4.48)
where $0\leq n$, $0\leq q$ and $0<\alpha<1$.
Figure 4.5: Figure shows comparison of empirical CCDFs of cluster size
distributions (points) with analytic formula (4.46) (lines) on logarithmic
plots, at $\alpha=0$, and $1/2$. Empirical distributions have been obtained
from $10$ realizations of $N=10^{6}$ size networks.
Figure 4.6: Figure shows comparison of empirical CCDFs of in-degree
distributions (points) with analytic formula (4.47) (lines) on logarithmic
plots, at $\alpha=0$, $1/3$, $1/2$, and $2/3$. Empirical distributions have
been obtained from $10$ realizations of $N=10^{6}$ size networks. Inset:
Comparison at $\alpha=0$ on semi-logarithmic plot.
Comparisons of analytic CCDF of cluster size (4.46) and empirical
distributions are shown in Figure 4.5 for $\alpha=0$, $1/3$, $1/2$, and $2/3$.
Experimental data has been collected from $10$ realizations of $10^{6}$ node
networks. Figure 4.5 shows that simulations fully confirm my analytic result.
On Figure 4.6 analytic formula (4.47) and the empirical CCDFs of in-degree,
obtained from the same $10^{6}$ node realizations, are compared. Note the
precise match of the simulation and the theoretical distribution on almost the
whole range of data. Some small discrepancy can be observed around the low
probability events. This deviation is caused by the aggregation of errors on
the cumulative distribution when some rare event occurs in a finite network.
#### 4.4.5 Conditional probabilities and expectation values
In the previous sections exact joint and marginal distributions of cluster
size and in-degree were analyzed for both finite and infinite networks. All
these distributions provide general statistics of the network. In this section
I proceed further, and I investigate the scenario when the “younger” in-degree
of a randomly selected link is known. I ask the cluster size distribution
under this condition, that is the conditional distribution
$\mathbb{P}_{\tau}(n\mid q)$. The results of the previous sections are
referred to below to obtain the conditional probability distribution, and
eventually the conditional expectation of cluster size. For the sake of
completeness, the conditional distribution and expectation of in-degree are
also given at the end of this section.
The conditional cluster size distribution can be given by the quotient of the
joint and the marginal in-degree distributions by definition:
$\mathbb{P}_{\tau}(n\mid
q)=\frac{\mathbb{P}_{\tau}(n,q)}{\mathbb{P}_{\tau}(q)}.$ (4.49)
The exact conditional distribution for any finite network can be obtained
after substituting (4.36) and (4.4.4) into the above expression. For a
boundless network the conditional distribution takes the simpler form:
$\mathbb{P}_{\infty}(n\mid
q)=\alpha\frac{\left(2/\alpha-1\right)_{q+1}}{\left(2-\alpha\right)_{n+1}}\Phi_{\alpha}(n,q),$
(4.50)
where $0\leq q\leq n$. If $n\gg 1$, then $\mathbb{P}_{\infty}(n\mid
q)\sim\alpha\left(2/\alpha-1\right)_{q+1}/n^{3}+\mathcal{O}\left(1/n^{4}\right)$,
that is the conditional cluster size distribution falls faster than the
ordinary cluster size distribution. It follows that the mean of the
conditional cluster size distribution will not diverge like the mean of the
ordinary distribution.
What is the expected size of a cluster under the condition that the in-degree
of its root is known? For practical reasons, I do not calculate
$\mathbb{E}_{\tau}\left[n\mid q\right]$ directly, but I calculate
$\mathbb{E}_{\tau}\left[n+2-\alpha\mid q\right]=\mathbb{E}_{\tau}\left[n\mid
q\right]+2-\alpha$ instead:
$\mathbb{E}_{\tau}\left[n+2-\alpha\mid
q\right]=\frac{1}{\mathbb{P}_{\tau}(q)}\sum_{n=q}^{\tau-1}\left(n+2-\alpha\right)\mathbb{P}_{\tau}(n,q).$
(4.51)
Since
$\left(n+2-\alpha\right)\mathbb{P}_{\tau}(n,q)=\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\left(2-\alpha\right)_{n}}\Phi_{\alpha}(n,q)$,
the above summation can be given similarly to the marginal distribution
$\mathbb{P}_{\tau}(q)$ in (4.4.4):
$\displaystyle\sum_{n=q}^{\tau-1}\left(n+2-\alpha\right)\mathbb{P}_{\tau}(n,q)$
$\displaystyle=\frac{\tau+1-\alpha}{\tau}\frac{1/\alpha-1}{q+1/\alpha-1}$
$\displaystyle-\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\left(2-\alpha\right)_{\tau-1}}\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\tau}}{\alpha k+1-\alpha}$
After replacing the above sum in $\mathbb{E}_{\tau}\left[n\mid q\right]$, the
following equation can be obtained:
$\mathbb{E}_{\tau}\left[n+2-\alpha\mid
q\right]=\left(1-\alpha\right)\frac{\left(q+1/\alpha\right)_{1/\alpha}}{\left(1/\alpha-1\right)_{1/\alpha}}G_{\tau}(q),$
(4.52)
where
$G_{\tau}(q)=\frac{\displaystyle
1-\frac{\left(1/\alpha-1\right)_{q+1}}{\left(1-\alpha\right)_{\tau}}\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\tau}}{k+1/\alpha-1}}{\displaystyle
1-\frac{\left(2/\alpha-1\right)_{q+1}}{\left(2-\alpha\right)_{\tau}}\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\tau}}{k+2/\alpha-1}}.$ (4.53)
The identity $\lim_{\tau\to\infty}G_{\tau}(q)\equiv 1$ implies that
$G_{\tau}(q)$ involves the finite scale effects, and the factors preceding
$G_{\tau}(q)$ give the asymptotic form of
$\mathbb{E}_{\tau}\left[n+2-\alpha\mid q\right]$:
$\mathbb{E}_{\infty}\left[n+2-\alpha\mid
q\right]=\left(1-\alpha\right)\frac{\left(q+1/\alpha\right)_{1/\alpha}}{\left(1/\alpha-1\right)_{1/\alpha}}.$
(4.54)
It can be seen that the expectation of cluster size, under the condition that
the in-degree is known, is finite in an unbounded network. This stands in
contrast to the unconditional cluster size, discussed in the previous section,
which diverges logarithmically as the size of the network grows beyond every
limit.
Figure 4.7: Figure shows the average cluster size as the function of the in-
degree $q$, obtained from 100 realizations of $10^{5}$ size networks.
Simulation data has been collected at $\alpha=0$, $1/3$, $1/2$, and $2/3$
parameter values. Analytical result (4.52) of conditional expectation
$\mathbb{E}_{\tau}\left[n\mid q\right]$ is shown with continuous lines.
In the ER limit the expected conditional cluster size becomes
$\lim_{\alpha\to 0}\mathbb{E}_{\infty}\left[n+2\mid q\right]=2^{q+1}.$ (4.55)
The fundamental difference between the scale-free and non-scale-free networks
can be observed again. In the scale-free case the expected conditional cluster
size asymptotically grows with the in-degree to the power of $1/\alpha$, while
in the latter case it grows exponentially. On Figure 4.7 the exact analytic
formula (4.52) is compared with simulation results at $\alpha=0$, $1/3$,
$1/2$, and $2/3$. The simulations clearly justify my analytic solution.
Let us briefly investigate the opposite scenario, that is when the cluster
size is known and the statistics of the in-degree are sought under this
condition. The conditional distribution can be obtained from the combination
of Eqs. (4.36), (4.38) and the definition
$\mathbb{P}_{\tau}(q\mid
n)=\frac{\mathbb{P}_{\tau}(n,q)}{\mathbb{P}_{\tau}(n)}.$ (4.56)
The conditional expectation of in-degree can be acquired by the same technique
as the conditional expectation of cluster size. Let us calculate
$\mathbb{E}_{\tau}\left[q+1/\alpha-1\mid n\right]=\mathbb{E}_{\tau}\left[q\mid
n\right]+1/\alpha-1$ (4.57)
instead of $\mathbb{E}_{\tau}\left[q\mid n\right]$ directly:
$\displaystyle\mathbb{E}_{\tau}\left[q+1/\alpha-1\mid n\right]$
$\displaystyle=\frac{1}{\mathbb{P}_{\tau}(n)}\sum_{q=0}^{n}\left(q+1/\alpha-1\right)\mathbb{P}_{\tau}(n,q)$
$\displaystyle=\frac{\Gamma(2-\alpha)}{\alpha}\left(n+1-\alpha\right)_{\alpha},$
(4.58)
where $0\leq n<\tau$. Note that the conditional expectation of in-degree is
independent of $\tau$, that is of the size of the network. In the ER limit the
expectation of the in-degree becomes
$\lim_{\alpha\to 0}\mathbb{E}_{\tau}\left[q\mid n\right]=\Psi(n+1)+\gamma,$
(4.59)
where $\Psi(x)=\frac{d}{dx}\ln\Gamma(x)$ denotes the digamma function, and
$\gamma=-\Psi(1)\approx 0.5772$ is the Euler–Mascheroni constant.
Asymptotically the expectation of the in-degree in a scale-free tree grows
with the cluster size to the power of $\alpha$, while in a ER tree it grows
only logarithmically, since $\Psi(n+1)=\log n+\mathcal{O}\left(1/n\right)$.
Therefore, conditional in-degree and conditional cluster size are mutually
inverses _asymptotically_. Figure 4.8 shows the analytic solution (4.58) and
simulation data at $\alpha=0$, $1/3$, $1/2$, and $2/3$ parameter values.
Simulation data has been collected from 100 realizations of $10^{5}$ size
networks.
Figure 4.8: Figure shows the average in-degree as the function of the cluster
size $n$, obtained from 100 realizations of $10^{5}$ size networks. Simulation
data has been collected at $\alpha=0$, $1/3$, $1/2$, and $2/3$ parameter
values. Analytical result (4.58) of conditional expectation
$\mathbb{E}_{\tau}\left[q\mid n\right]$ is shown with continuous lines.
#### 4.4.6 Conditional distribution of edge betweenness
Using the results of the previous sections, I am finally ready to answer the
problem which motivated my work, that is the distribution of the edge
betweenness under the condition that the in-degree of the “younger” node of
the link is known. As I noted at the beginning of Section 4.4.1, the edge
betweenness can be expressed with cluster size:
$L=\left(n+1\right)\left(\tau-n\right).$ (4.60)
Therefore, conditional edge betweenness can be given formally by the following
transformation of random variable $n$:
$\mathbb{P}_{\tau}(L\mid
q)=\sum_{n=0}^{\tau-1}\delta_{L,\left(n+1\right)\left(\tau-n\right)}\mathbb{P}_{\tau}(n\mid
q).$ (4.61)
Obviously, $\mathbb{P}_{\tau}(L\mid q)$ is non-zero only at those values of
$L$, where (4.60) has an integer solution for $n$. If
$n_{L}=\frac{\tau-1}{2}-\sqrt{\frac{\left(\tau+1\right)^{2}}{4}-L}$ (4.62)
is such an integer solution of the quadratic equation (4.60), and
$L\neq\left(\tau+1\right)^{2}/4$, then
$\mathbb{P}_{\tau}(L\mid q)=\mathbb{P}_{\tau}(n_{L}\mid
q)+\mathbb{P}_{\tau}(\tau-1-n_{L}\mid q).$ (4.63)
If $L=\left(\tau+1\right)^{2}/4$ is integer, then $\mathbb{P}_{\tau}(L\mid
q)=\mathbb{P}_{\tau}(n_{L}\mid q)$.
The conditional expectation of edge betweenness can be obtained from (4.60):
$\mathbb{E}_{\tau}\left[L\mid q\right]=\tau\mathbb{E}_{\tau}\left[n+1\mid
q\right]-\mathbb{E}_{\tau}\left[\left(n+1\right)n\mid q\right].$ (4.64)
Therefore, for the exact calculation of $\mathbb{E}_{\tau}\left[L\mid
q\right]$ the first and the second moment of the conditional cluster size
distribution are required. The first moment, that is the mean, has been
derived in the previous section. In order to calculate the second moment let
us use the technique I have developed in the previous sections. Let us
consider:
$\mathbb{E}_{\tau}\left[\left(n+2-\alpha\right)\left(n+1-\alpha\right)\mid
q\right]=\frac{\tau+1-\alpha}{\tau}\frac{\left(1/\alpha-1\right)_{q}}{\mathbb{P}_{\tau}(q)}\sum_{n=q}^{\tau-1}\frac{\Phi_{\alpha}(n,q)}{\left(2-\alpha\right)_{n-1}}.$
(4.65)
We shall be cautious when the summation for $n$ is evaluated. The $k=1$ term
in
$\Phi_{\alpha}(n,q)=\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\left(-\alpha
k\right)_{n}$ must be treated separately to avoid a divergent term:
$\displaystyle\sum_{n=q}^{\tau-1}\frac{\Phi_{\alpha}(n,q)}{\left(2-\alpha\right)_{n-1}}$
$\displaystyle=\frac{1-\alpha}{\left(q-1\right)!}\left[\alpha\Psi(\tau-\alpha)-\alpha\Psi(1-\alpha)-\Psi(q)-\gamma\right]$
$\displaystyle-\frac{1}{\alpha}\frac{1}{\left(2-\alpha\right)_{\tau-2}}\sum_{k=2}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\tau}}{k-1}$
The exact formula for $\mathbb{E}_{\tau}\left[L\mid q\right]$ can be obtained
straightforwardly, after (4.52) and the above expressions have been
substituted into (4.64).
Let us consider the scenario when the size of the network tends to infinity.
Equation (4.60) implies that edge betweenness diverges as $\tau\to\infty$,
therefore $L$ should be rescaled for an infinite network. From the asymptotics
of the digamma function
$\Psi(\tau-\alpha)=\ln\tau+\mathcal{O}\left(1/\tau\right)$ it follows that
$\mathbb{E}_{\tau}\left[\left(n+2-\alpha\right)\left(n+1-\alpha\right)\mid
q\right]$ grows only logarithmically, slower than the linear growth of
$\tau\mathbb{E}_{\tau}\left[n+2-\alpha\mid q\right]$. Therefore, edge
betweenness asymptotically grows linearly as the size of the network grows
beyond every limit. Let us rescale edge betweenness
$\Lambda_{\tau}=\frac{L(\tau)}{\tau+1}$ (4.66)
and let us consider the limit
$\Lambda=\lim_{\tau\to\infty}\Lambda_{\tau}=n_{\Lambda}+1$. The CCDF of the
rescaled edge betweenness can be given by
$\bar{F}_{\infty}(\Lambda\mid
q)=\lim_{\tau\to\infty}\sum_{n=n_{\Lambda_{\tau}}}^{\tau-1-n_{\Lambda_{\tau}}}\mathbb{P}_{\tau}(n\mid
q)=\frac{1}{\mathbb{P}_{\infty}(q)}\sum_{n=\Lambda-1}^{\infty}\mathbb{P}_{\infty}(n,q).$
(4.67)
When the summation has been carried out, the following equation is obtained:
$\bar{F}_{\infty}(\Lambda\mid
q)=\frac{\left(2/\alpha-1\right)_{q+1}}{\left(2-\alpha\right)_{\Lambda-1}}\sum_{k=0}^{q}\frac{\left(-1\right)^{k}}{k!\left(q-k\right)!}\frac{\left(-\alpha
k\right)_{\Lambda-1}}{k+2/\alpha-1},$ (4.68)
where $q+1\leq\Lambda$. If $1<q\ll\Lambda$, then only the first term of the
sum should be taken into account, and it is easy to see that
$\bar{F}_{\infty}(\Lambda\mid
q)=\frac{\alpha^{2}\left(1-\alpha\right)}{2\Gamma(2/\alpha-1)}\frac{q^{2/\alpha}}{\Lambda^{2}}+\mathcal{O}\left(1/\Lambda^{2+\alpha}\right).$
(4.69)
It can be seen that the scaling exponent $-2$ is independent of $\alpha$. The
above asymptotic formula has been obtained for infinite networks. The same
power law scaling can be observed in finite size networks as (4.69) if
$\Lambda_{\tau}\ll\tau$. However, $\bar{F}_{\tau}(\Lambda_{\tau}\mid q)\equiv
0$ if $\Lambda_{\tau}>\tau$ in finite networks, therefore asymptotic formula
(4.69) evidently becomes invalid if $\Lambda_{\tau}\approx\tau$.
It is obvious that as the size of the network grows larger and larger,
asymptotic formula (4.68) becomes more and more accurate. One can ask how fast
this convergence is. From elementary estimations of
$\bar{F}_{\tau}(\Lambda_{\tau}\mid q)$ one can show that for fixed
$\Lambda_{\tau}$:
$\bar{F}_{\tau}(\Lambda_{\tau}\mid q)=\bar{F}_{\infty}(\Lambda_{\tau}\mid
q)-\left(1-\bar{F}_{\infty}(\Lambda_{\tau}\mid
q)\right)\frac{\alpha^{2}\left(1-\alpha\right)}{2}\frac{1}{\tau^{2}}+\mathcal{O}\left(1/\tau^{2+\alpha}\right),$
(4.70)
that is corrections to the asymptotic formula decrease with $\tau^{-2}$ for
large $\tau$.
Figure 4.9: Figure shows CCDF of edge betweenness under the condition that the
in-degree $q$ is known. Empirical CCDF has been obtained from $100$
realizations of $N=10^{4}$ and $N=10^{5}$, and $10$ realizations of $N=10^{6}$
size networks at $\alpha=1/2$ parameter value. Continuous lines show analytic
result of infinite network limit (4.68).
Figure 4.10: Figure shows average edge betweenness under the condition that
the in-degree $q$ is known as the function of $q$ on log-log plot. Numerical
data has been collected from $100$ realizations of $N=10^{4}$ and $N=10^{5}$,
and $10$ realizations of $N=10^{6}$ size networks at $\alpha=1/2$ parameter
value. Inset shows the same scenario at $\alpha=0$ parameter value on semi-
logarithmic plot. Continuous lines show analytic results of the infinite
network limit (4.71) and (4.72).
On Figure 4.9 comparison of analytic formula (4.68) with simulation results is
presented for $q=1$ and $q=2$. The empirical CCDF of rescaled edge
betweenness, under the condition that in-degree $q$ is known, is shown for
$10^{4}$, $10^{5}$, and $10^{6}$ size networks, at $\alpha=1/2$ parameter
value. The empirical CCDFs of rescaled edge betweenness evidently collapse to
the same curve for different size networks, and they coincide precisely with
my analytic result.
The expectation of the rescaled edge betweenness under the condition that in-
degree $q$ is known can be given by $\mathbb{E}_{\infty}\left[\Lambda\mid
q\right]=\mathbb{E}_{\infty}\left[n_{\Lambda}+1\mid q\right]$. Using (4.54)
and (4.55) I receive
$\displaystyle\mathbb{E}_{\infty}\left[\Lambda\mid q\right]$
$\displaystyle=\left(1-\alpha\right)\frac{\left(q+1/\alpha\right)_{1/\alpha}}{\left(1/\alpha-1\right)_{1/\alpha}}-1+\alpha,$
(4.71) $\displaystyle\lim_{\alpha\to 0}\mathbb{E}_{\infty}\left[\Lambda\mid
q\right]$ $\displaystyle=2^{q+1}-1.$ (4.72)
One can see that $\mathbb{E}_{\infty}\left[\Lambda\mid q\right]\sim
q^{1/\alpha}$ for $q\gg 1$ if $\alpha>0$ and
$\mathbb{E}_{\infty}\left[\Lambda\mid q\right]\sim e^{q}$ for $q\gg 1$ if
$\alpha\to 0$.
Analytic results (4.71) and (4.72), and simulation data are shown in Figure
4.10 at $\alpha=1/2$ and $\alpha=0$ parameter values. Numerical data has been
collected from the same $10^{4}$, $10^{5}$, and $10^{6}$ size networks as
above. As the size of the network grows a larger and larger range of the
rescaled empirical data collapses to the same analytic curve. On the high
degree region some discrepancy can be observed due to the finite scale
effects.
Finally, let us note that the precise unconditional distribution of edge
betweenness
$\mathbb{P}_{\tau}(L)=\sum_{n=0}^{\tau-1}\delta_{L,\left(n+1\right)\left(\tau-n\right)}\mathbb{P}_{\tau}(n)$
can be obtained from (4.38) as well. Furthermore, CCDF of the unconditional
betweenness $\bar{F}_{\tau}(L)=\sum_{n=n_{L}}^{\tau-
n_{L}-1}\mathbb{P}_{\tau}(n)$ can be derived in closed form:
$F_{\tau}^{c}(L)=\frac{\tau+1-\alpha}{\tau}\frac{\left(1-\alpha\right)\left(\tau-2n_{L}\right)}{\left(n_{L}+1-\alpha\right)\left(\tau-
n_{L}+1-\alpha\right)}.$ (4.73)
For the sake of simplicity I have assumed during my calculations that in-
degrees of the “younger” nodes are provided. However, it is possible that even
though both in-degrees of every link are known, we cannot distinguish them
from each other, that is we cannot tell which is the “younger” node. How could
I extend my results to this scenario? Let us consider a new edge when it is
connected to the network. The in-degree of the new node is obviously $0$. The
in-degree of the other node, to which the new node is connected, is equal to
or larger than one. Due to preferential attachment the larger the in-degree is
the faster it grows. Even if preferential attachment is absent, the growth
rate of every in-degree is the same. Therefore, it is expected that the
initial deficit in the in-degree of the “younger” node grows or remains at the
same level during the evolution of the network. It follows that it is a
reasonable approximation to substitute the in-degree of the “younger” node $q$
with $q_{\text{min}}=\min(q_{1},q_{2})$ in my formulas.
### 4.5 Conclusions
A typical network construction problem is to design network infrastructure
without wasting precious resources at places where not needed. An appropriate
design strategy is to allocate network resources proportionally to the
expected traffic. In a mean field approximation the expected traffic is
proportional to the number of shortest paths going through a certain network
element, that is the betweenness.
The precise calculation of all the betweenness requires complete information
on the network structure. In real life, however, the number of shortest paths
is often impossible to tell because the structure of the network is not fully
known. One of the practical results of this chapter is that the expectation of
edge betweenness can be estimated precisely when only limited local
information on network structure—the in-degree of the “younger” node—is
available.
Another difficulty of network design is that the size of real networks is
finite. Moreover, the size of real networks is often so small that asymptotic
formulas can be applied only with unacceptable error. The other important
novelty of my results is that the derived formulas are exact even for finite
networks, which allows better design of realistic finite size networks.
Various statistical properties of evolving random trees have been investigated
in this chapter. I have focused on the cluster size, the in-degree and the
edge betweenness. I have considered the $m=1$ case of the BA model extended
with initial attractiveness for modeling random trees. Initial attractiveness
allows fine tuning of the scaling parameter. Moreover, in the limit of the
tuning parameter $\alpha\to 0$ the applied model tends to a non-scale-free
structure, which is in many aspects similar to the classical ER model. I was
therefore able to investigate both the scale-free and the non-scale-free
scenario within the same framework.
I also presented conditional expectations of cluster size and in-degree for
both finite and unbounded networks. I have found that asymptotically the
conditional cluster size grows with in-degree to the power of $1/\alpha$ and
the conditional in-degree grows with cluster size to the power of $\alpha$,
respectively. The ER limit has been discussed as well. I have shown that the
conditional cluster size grows exponentially and the conditional in-degree
grows logarithmically when $\alpha\to 0$.
I have derived the distribution of edge betweenness under the condition that
the corresponding in-degree is known. I have found that the conditional
expectation of edge betweenness grows linearly with the size of the network.
For the analysis of unbounded networks I have defined the rescaled edge
betweenness $\Lambda$, and derived its distribution and expectation under the
condition that in-degree $q$ is provided. My analytic results have been
verified at different network sizes and parameter values by extensive
numerical simulations. I have demonstrated that numerical simulations fully
confirm my analytic results.
## Chapter 5 Concluding remarks
The study of complex networks has evolved considerably in recent years. An
interesting example of complex networks is the Internet, which has become part
of everyday life. Two important aspects of the Internet, namely the properties
of its topology and the characteristics of its data traffic, have attracted
growing attention of the physics community. My thesis has considered problems
of both aspects.
In the introduction I briefly presented an overview of the basic components of
Internet structure and traffic. The workings of the Transmission Control
Protocol (TCP), the primary algorithm governing traffic in the current
Internet, were discussed in more detail, since they are the main focus of the
first part of my analysis. Most of the terminologies I use in this thesis were
also defined here.
In the next chapter I studied the stochastic behavior of TCP in an elementary
network scenario consisting of a standalone infinite-sized buffer and an
access link. This simple model might constitute the building blocks of more
complex Internet traffic. I calculated the stationary distribution of the
stochastic congestion window process, which regulates the traffic of TCP. My
analysis not only considered the ideal congestion window dynamics, but also
included the effect of the fast recovery and fast retransmission (FR/FR)
algorithms of TCP. Furthermore, I showed that my model can be extended further
analytically to involve the effect of link propagation delay, characteristic
of Wide Area Networks. Various moments of the congestion window process were
calculated. An important achievement is that all the parameters are at hand in
the entire model, and no parameter fitting is necessary. I also applied the
mean field approximation to describe many parallel TCP flows. My analytic
results were validated against packet level numerical simulations and the
simulations agreed to a high degree with the analytic formulas I derived.
I continued my thesis with the investigation of finite-sized semi-bottleneck
buffers, where packets can be dropped not only at the link, but also at the
buffer. I demonstrated that the behavior of the system depends only on a
certain combination of the parameters. Moreover, an analytic formula was
derived that gives the ratio of packet loss rate at the buffer to the total
packet loss rate. This formula makes it possible to treat buffer-losses as if
they were link-losses. I considered the effect of the FR/FR algorithms and I
calculated the probability distribution of the congestion window for both
Local and Wide Area Network scenarios. I showed that a sharp peak might appear
in the window distribution due to the FR/FR mode of TCP. My analytical results
matched numerical simulations properly in the case of large buffer sizes and
small packet loss probabilities. In the opposite range of parameters, however,
the slow start mechanism of TCP plays a more important role and it cannot be
neglected completely from the precise description of the congestion window
dynamics. Nevertheless, my calculations gave qualitatively correct results in
these cases as well. Hopefully, my methods, developed in this chapter, can be
applied later for modeling the slow start mechanism.
In the last part of my thesis I studied computer networks from a structural
perspective. The scaling exponent of the node connectivity could be tuned in
the network model that I investigated. In addition, the non-scale-free limit
of the node connectivity could also be investigated. I demonstrated through
fluid simulations that the distribution of resources, specifically the link
bandwidth, has a serious impact on the global performance of a computer
network. Then I analyzed the distribution of edge betweenness in a growing
scale-free tree under the condition that a local property, the in-degree of
the “younger” node of an arbitrary edge, is known in order to find an optimum
distribution of link capacity. The derived formula is exact even for finite-
sized networks. I also calculated the conditional expectation of edge
betweenness, rescaled for infinite networks. My analytic results were compared
to numerical simulations that confirmed my calculations appropriately.
## Appendix A Mathematical proofs of the applied identities
### A.1 Series expansion of $L(c)\,G(x)$
In this section the series expansion of $L(c)G(x)$ in $x$ is derived, where
$L(c)$ is defined in (2.30) and $G(x)$ in (3.23). We prove two lemmas first:
Lemma 1 For $c\in\mathbb{R}$, $c\neq 1$
$\sum_{k=0}^{N}c^{k}\prod_{l=1}^{k}\frac{1}{1-c^{l}}=\prod_{k=1}^{N}\frac{1}{1-c^{k}}.$
(A.1)
###### Proof.
We prove the lemma by induction for $N$. Indeed, for $N=1$ the formula is
evidently true: $1+\frac{c}{1-c}=\frac{1}{1-c}$. As the induction hypothesis
suppose that the formula is true for $N$. Then
$\displaystyle\sum_{k=0}^{N+1}c^{k}\prod_{l=1}^{k}\frac{1}{1-c^{l}}$
$\displaystyle=\sum_{k=0}^{N}c^{k}\prod_{l=1}^{k}\frac{1}{1-c^{l}}+c^{N+1}\prod_{l=1}^{N+1}\frac{1}{1-c^{l}}$
$\displaystyle=\prod_{l=1}^{N}\frac{1}{1-c^{l}}+c^{N+1}\prod_{l=1}^{N+1}\frac{1}{1-c^{l}}$
(A.2)
$\displaystyle=\left(1-c^{N+1}\right)\prod_{l=1}^{N+1}\frac{1}{1-c^{l}}+c^{N+1}\prod_{l=1}^{N+1}\frac{1}{1-c^{l}}=\prod_{l=1}^{N+1}\frac{1}{1-c^{l}}.$
∎
Lemma 2 For $c\in[0,1[$
$\prod_{k=1}^{n-1}\left(1-c^{k}\right)=\sum_{k=0}^{\infty}c^{kn}\prod_{l=k+1}^{\infty}\left(1-c^{l}\right).$
(A.3)
###### Proof.
This lemma is proven by induction for $n$. For $n=1$ Lemma 1 proves the
formula. Indeed, multiply both sides of (A.1) by
$\prod_{k=1}^{N}\left(1-c^{k}\right)$. If $c\in[0,1[$ then we can take the
$N\to\infty$ limit, which provides exactly the formula to be proven.
As the induction hypothesis let us suppose that the formula is true for $n$.
Then
$\displaystyle\prod_{k=1}^{n}\left(1-c^{k}\right)$
$\displaystyle=\left(1-c^{n}\right)\prod_{k=1}^{n-1}\left(1-c^{k}\right)=\prod_{k=1}^{n-1}\left(1-c^{k}\right)-c^{n}\sum_{k=0}^{\infty}c^{kn}\prod_{l=k+1}^{\infty}\left(1-c^{l}\right)$
$\displaystyle=\prod_{k=1}^{n-1}\left(1-c^{k}\right)-\sum_{k=0}^{\infty}c^{\left(k+1\right)n}\prod_{l=k+1}^{\infty}\left(1-c^{l}\right)$
$\displaystyle=\prod_{k=1}^{n-1}\left(1-c^{k}\right)-\sum_{k=0}^{\infty}c^{kn}\left(1-c^{k}\right)\prod_{l=k+1}^{\infty}\left(1-c^{l}\right)$
$\displaystyle=\sum_{k=0}^{\infty}c^{k\left(n+1\right)}\prod_{l=k+1}^{\infty}\left(1-c^{l}\right),$
(A.4)
which proves the formula for $n+1$. ∎
Theorem For $x\in\mathbb{R}$ and $c\in[0,1[$
$L(c)\,G(x)=-\sum_{n=1}^{\infty}\frac{1}{n!}\prod_{l=1}^{n}\left(1-c^{l}\right)x^{n}.$
(A.5)
###### Proof.
From the series expansion of the exponential function it follows that
$E(x)=e^{-cx}-e^{-x}=-\sum_{n=1}^{\infty}\left(-1\right)^{n}\frac{1-c^{n}}{n!}x^{n}$.
Let us substitute this expression into the definition of
$G(x)=\sum_{k=0}^{\infty}F(-c^{k}x)\prod_{l=1}^{k}\frac{1}{1-c^{l}}$ :
$\begin{split}L(c)\,G(x)&=-\prod_{l=1}^{\infty}\left(1-c^{l}\right)\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\left(-1\right)^{n}\frac{1-c^{n}}{n!}\left(-c^{k}x\right)^{n}\prod_{l=1}^{k}\frac{1}{1-c^{l}}\\\
&=-\sum_{n=1}^{\infty}\frac{1-c^{n}}{n!}\sum_{k=0}^{\infty}c^{kn}\prod_{l=k+1}^{\infty}\left(1-c^{l}\right)x^{n}\\\
&=-\sum_{n=1}^{\infty}\frac{1}{n!}\prod_{l=1}^{n}\left(1-c^{l}\right)x^{n},\end{split}$
(A.6)
where we applied the Lemma 2 in the last equation in order to prove the
theorem. ∎
### A.2 Expansion of the Kronecker-delta function
We have seen that the general solution of Eq. (4.4.1) is
$\mathbb{P}_{\tau}(n,q\mid\tau_{e})=\sum_{\lambda_{1},\lambda_{2}}C_{\lambda_{1},\lambda_{2}}\,f(\tau)\,g(n)\,h(q),$
(A.7)
and the initial condition is
$\mathbb{P}_{\tau_{e}}(n,q\mid\tau_{e})=\delta_{n,0}\,\delta_{q,0}$, where
$\delta_{n,m}=\begin{cases}1,&\textrm{if $n=m$},\\\ 0,&\textrm{if $n\neq
m$}\end{cases}$ (A.8)
is the Kronecker-delta function, and $n$ and $m$ are integers. Coefficients
$C_{\lambda_{1},\lambda_{2}}$ are calculated in this section. First we show
that
$\delta_{n,0}=\sum_{k=0}^{n}\frac{\left(-1\right)^{k}}{k!}\frac{1}{\Gamma(n-k+1)}.$
(A.9)
Note that we can consider $m=0$ without any loss of generality, since
$\delta_{n,m}\equiv\delta_{n-m,0}$.
If $n<0$, then the summand in (A.9) is indeed zero by definition. If $n>0$,
then
$\sum_{k=0}^{n}\frac{\left(-1\right)^{k}}{k!}\frac{1}{\Gamma(n-k+1)}=\frac{1}{n!}\sum_{k=0}^{n}\binom{n}{k}\left(-1\right)^{k}=0$
(A.10)
follows from the binomial theorem. Finally, for $n=0$,
$\sum_{k=0}^{0}\frac{\left(-1\right)^{k}}{k!}\frac{1}{\Gamma(-k+1)}=\frac{\left(-1\right)^{0}}{0!}\frac{1}{\Gamma(1)}=1.$
(A.11)
Coefficients $C_{\lambda_{1},\lambda_{2}}$ can be obtained from the term by
term comparison of
$\mathbb{P}_{\tau_{e}}(n,q\mid\tau_{e})=\sum_{\lambda_{1},\lambda_{2}}C_{\lambda_{1},\lambda_{2}}f(\tau_{e})\,g(n)\,h(q)$
with the expansion of the initial condition $\delta_{n,0}\,\delta_{q,0}$,
shown above. One can easily confirm with the help of identity
$f(n)\delta_{n,0}\equiv f(0)\delta_{n,0}$ that the same terms appear on both
sides, if $\lambda_{1}=-k_{1}$, and $\lambda_{2}=-\alpha k_{2}$, and
coefficients $C_{k_{1},k_{2}}$ are the following:
$C_{k_{1},k_{2}}=\frac{\left(-1\right)^{k_{1}+k_{2}}}{k_{1}!\,k_{2}!}\frac{\Gamma(\tau_{e}+1-\alpha)}{\Gamma(\tau_{e}-k_{1})}\frac{1}{\Gamma(-\alpha
k_{2})}\frac{1}{\Gamma(1/\alpha-1)}.$ (A.12)
Finally, to obtain (4.36) the summation for $k_{1}$ can be carried out
explicitly:
$\displaystyle\sum_{k_{1}=0}^{n}\frac{\left(-1\right)^{k_{1}}}{k_{1}!\,\Gamma(n-k_{1}+1)}\frac{\Gamma(\tau-
k_{1})}{\Gamma(\tau_{e}-k_{1})}=\frac{\Gamma(\tau-\tau_{e}+1)}{\Gamma(n+1)\Gamma(\tau_{e})}\frac{\Gamma(\tau-n)}{\Gamma(\tau-\tau_{e}-n+1)}$
### A.3 The $\alpha\to 0$ limit of joint distribution
$\mathbb{P}_{\tau}(n,q)$
In this section we prove that the ER limit of the joint probability
$\mathbb{P}_{\tau}(n,q)$ is (4.37).
Theorem Let us consider $\mathbb{P}_{\tau}(n,q)$ as defined in (4.36), where
$0<q<n<\tau$ are integers. Then the following limit holds:
$\lim_{\alpha\to
0}\mathbb{P}_{\tau}(n,q)=\frac{\tau+1}{\tau\,\Gamma(n+3)}\sum_{k=q-1}^{n-1}\left(-1\right)^{n-1-k}S_{n-1}^{\left(k\right)}\binom{k}{q-1}.$
(A.13)
###### Proof.
First, let us note that $\Phi_{\alpha}(n,q)$ in (4.36) can be rewritten in the
following equivalent form:
$\Phi_{\alpha}(n,q)=\alpha\sum_{k=0}^{q-1}\frac{\left(-1\right)^{k}\left(1-\alpha-\alpha
k\right)_{n-1}}{k!\left(q-1-k\right)!}$. Next, Pochhammer’s symbol
$\left(1/\alpha-1\right)_{q}$ is replaced with its asymptotic form:
$\left(1/\alpha-1\right)_{q}=1/\alpha^{q}\left(1+\mathcal{O}\left(\alpha\right)\right)$.
After the obvious limits have been evaluated the following equation is
obtained:
$\lim_{\alpha\to
0}\mathbb{P}_{\tau}(n,q)=\frac{\tau+1}{\tau\,\Gamma(n+3)}\lim_{\alpha\to
0}\frac{\sum_{k=0}^{q-1}\frac{\left(-1\right)^{k}\left(1-\alpha-\alpha
k\right)_{n-1}}{k!\left(q-1-k\right)!}}{\alpha^{q-1}}.$ (A.14)
The above limit, by definition, can be substituted with $q-1$ order
differential at $\alpha=0$, if all the lower order derivates of the sum are
zero at $\alpha=0$. Indeed,
$\displaystyle\lim_{\alpha\to
0}\frac{\sum_{k=0}^{q-1}\frac{\left(-1\right)^{k}\left(1-\alpha-\alpha
k\right)_{n-1}}{k!\left(q-1-k\right)!}}{\alpha^{q-1}}$
$\displaystyle=\frac{1}{m!}\frac{d^{m}}{d\alpha^{m}}\sum_{k=0}^{q-1}\frac{\left(-1\right)^{k}\left(1-\alpha-\alpha
k\right)_{n-1}}{k!\left(q-1-k\right)!}\Biggr{|}_{\alpha=0}$
$\displaystyle=\frac{1}{m!}\frac{d^{m}\left(1+\alpha\right)_{n-1}}{d\alpha^{m}}\Biggr{|}_{\alpha=0}\sum_{k=0}^{q-1}\frac{\left(-1\right)^{k}\left(-k-1\right)^{m}}{k!\left(q-1-k\right)!},$
where the sum is $0$ if $m<q-1$ and $1$ if $m=q-1$. Therefore, the limit can
be transformed to
$\lim_{\alpha\to
0}\mathbb{P}_{\tau}(n,q)=\frac{\tau+1}{\tau\,\Gamma(n+3)}\frac{1}{\left(q-1\right)!}\frac{d^{q-1}\left(1+\alpha\right)_{n-1}}{d\alpha^{q-1}}\biggr{|}_{\alpha=0}.$
(A.15)
Finally, let us consider the power expansion of Pochhammer’s symbol:
$\left(x\right)_{m}=\sum_{k=0}^{m}\left(-1\right)^{n-k}S_{m}^{\left(k\right)}x^{k}$,
where $S_{m}^{\left(k\right)}$ are the Stirling numbers of the first kind. The
expansion formula has been applied at $x=1+\alpha$ and $m=n-1$, which implies
$\displaystyle\lim_{\alpha\to 0}\mathbb{P}_{\tau}(n,q)$
$\displaystyle=\frac{\tau+1}{\tau\,\Gamma(n+3)}\sum_{k=q-1}^{n-1}\frac{\left(-1\right)^{n-1-k}S_{m}^{\left(k\right)}}{\left(q-1\right)!}\frac{d^{q-1}\left(1+\alpha\right)^{k}}{d\alpha^{q-1}}\biggr{|}_{\alpha=0}$
$\displaystyle=\frac{\tau+1}{\tau\,\Gamma(n+3)}\sum_{k=q-1}^{n-1}\left(-1\right)^{n-1-k}S_{n-1}^{\left(k\right)}\binom{k}{q-1}.$
∎
## Glossary
## References
* Simonyi [1078] Károly Simonyi. _A fizika kultúrtörténete_. Gondolat Kiadó, Budapest, 1078. (Hung.).
* Csabai [1994] István Csabai. $1/f$ noise in computer network traffic. _J. Phys. A: Math. Gen._ , 27:L417–L421, 1994.
* Barabási [2002] Albert-László Barabási. _Linked: the new science of networks_. Perseus Pub., Cambridge, MA, April 2002.
* Siganos et al. [2003] Georgos Siganos, Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos. Power laws and the AS-level Internet topology. _IEEE/ACM Trans. Net._ , 11(4):514–524, August 2003.
* Pastor-Satorras et al. [2001] Romualdo Pastor-Satorras, Alexei Vázquez, and Alessandro Vespignani. Dynamical and correlation properties of the Internet. _Phys. Rev. Lett._ , 87(25):258701, December 2001\.
* Faloutsos et al. [1999] Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos. On power-law relationships of the Internet topology. _SIGCOMM Comput. Commun. Rev._ , 29(4):251–262, October 1999.
* Caldarelli et al. [2000] G. Caldarelli, R. Marchetti, and L. Pietronero. The fractal properties of internet. _Europhys. Lett._ , 52(4):386–391, September 2000\.
* Yook et al. [2002] Soon-Hyung Yook, Hawoong Jeong, and Albert-László Barabási. Modeling the Internet’s large-scale topology. In _Proc. Natl. Acad. Sci. U.S.A._ , volume 99 of _Appl. Phys. Sci._ , page 13382, Washington, DC, October 2002. Natl. Acad. Sci.
* Li et al. [2004] Lun Li, David Alderson, Walter Willinger, and John Doyle. A first-principles approach to understanding the Internet’s router-level topology. _SIGCOMM Comput. Commun. Rev._ , 34(4):3–14, August 2004.
* Strogatz [2001] Steven H. Strogatz. Exploring complex networks. _Nature_ , 410:268–276, March 2001.
* Albert and Barabási [2002] Réka Albert and Albert-László Barabási. Statistcal mechanics of complex networks. _Rev. Mod. Phys._ , 74:47–97, January 2002.
* Dorogovtsev and Mendes [2002] Sergei N. Dorogovtsev and Jose F. F. Mendes. Evolution of networks. _Adv. Phys._ , 51:1079–1187, 2002.
* Newman [2003] Mark E. J. Newman. The structure and function of complex networks. _SIAM Rev._ , 45(2):167–256, May 2003.
* Newman [2001] Mark E. J. Newman. Scientific collaboration networks. i. ii. _Phys. Rev. E_ , 64:016131, 016132, June 2001.
* Barabási et al. [2000] Albert-László Barabási, Réka Albert, and Hawoong Jeong. Scale-free characteristics of random networks: the topology of the world-wide web. _Physica A_ , 281(1):69–77, June 2000.
* CAI [1995] Ipv4 internet topology map, 1995. URL http://www.caida.org/
analysis/topology/as_core_network/pics/ascoreApr2005.png.
* Spring et al. [2002] Neil Spring, Ratul Mahajan, and David Wetherall. Measuring isp topologies with rocketfuel. _SIGCOMM Comput. Commun. Rev._ , 32(4):133–145, October 2002.
* Lakhina et al. [2003] Anukool Lakhina, John W. Byers, Mark Crovella, and Peng Xie. Sampling biases in IP topology measurements. In _INFOCOM’03_ , volume 1, pages 332–341, New York, NY, USA, April 2003. IEEE.
* Achlioptas et al. [2005] Dimitris Achlioptas, Aaron Clauset, David Kempe, and Cristopher Moore. On the bias of traceroute sampling: or, power-law degree distributions in regular graphs. In _STOC ’05: Proc. 37th Ann. ACM Symp. Theory of Computing_ , pages 694–703, New York, NY, USA, 2005. ACM.
* Comer [1988] Douglas E. Comer. _Internetworking with TCP/IP: principles, protocols, and architecture_. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1988.
* Socolofsky and Kale [1991] Theodore John Socolofsky and Claudia Jeanne Kale. A TCP/IP tutorial. RFC 1180, Internet Engineering Task Force, SRI International, January 1991. URL http://www.ietf.org/rfc/rfc1180.txt.
* Cooper [1981] Robert B. Cooper. _Introduction to Queuing Theory (2nd Ed.)_. North Holland, New York, 1981.
* Leland et al. [1994] Will E. Leland, Murad S. Taqqu, Walter Willinger, and Daniel V. Wilson. On the self-similar nature of Ethernet traffic (extended version). _IEEE/ACM Trans. Net._ , 2(1):1–15, February 1994\.
* Morse [1955] Philip M. Morse. Stochastic properties of waiting lines. _J. Opns. Res. Soc. Am._ , 3(3):255–261, August 1955.
* Paxson and Floyd [1995] Vern Paxson and Sally Floyd. Wide-area traffic: The failure of poisson modeling. _IEEE/ACM Trans. Net._ , 3(3):226–244, June 1995\.
* Crovella and Bestavros [1997] Marc E. Crovella and Azer Bestavros. Self-similarity in world wide web traffic: evidence and possible causes. _IEEE/ACM Trans. Net._ , 5(6):835–846, December 1997.
* Feldmann et al. [1998] Anja Feldmann, Anna C. Gilbert, Walter Willinger, and Thomas G. Kurtz. The changing nature of network traffic: scaling phenomena. _SIGCOMM Comput. Commun. Rev._ , 28(2):5–29, April 1998.
* Willinger et al. [1997] Walter Willinger, Murad S. Taqqu, Robert Sherman, and Daniel V. Wilson. Self-similarity through high-variability: statistical analysis of ethernet LAN traffic at the source level. _IEEE/ACM Trans. Net._ , 5(1):71–86, February 1997.
* Willinger et al. [1998] Walter Willinger, Vern Paxson, and Murad S. Taqqu. Self-similarity and heavy tails: structural modeling of network traffic. In Robert J. Adler, Raisa E. Feldman, and Murad S. Taqqu, editors, _A practical guide to heavy tails: statistical techniques and applications_ , pages 27–53. Birkhauser Boston Inc., Cambridge, MA, USA, 1998\. ISBN 0-8176-3951-9.
* Veres and Boda [2000] András Veres and Miklós Boda. The chaotic nature of TCP congestion control. In _INFOCOM’00_ , volume 3, pages 1715–1723. IEEE, March 2000.
* Figueiredo et al. [2005] Daniel R. Figueiredo, Benyuan Liu, Anja Feldmann, Vishal Misra, Don Towsley, and Walter Willinger. On TCP and self-similar traffic. _Perform. Eval._ , 61(2–3):129–141, July 2005\.
* Guo et al. [2000] Liang Guo, Mark Crovella, and Ibrahim Matta. TCP congestion control and heavy tails. Tech. report, Comput. Sci. Dept., Boston, MA, USA, 2000.
* Postel [1980] Jon Postel. User datagram protocol. RFC 768, Internet Engineering Task Force, SRI International, August 1980\. URL http://www.ietf.org/rfc/rfc768.txt.
* Paxson and Allman [2000] Vern Paxson and Mark Allman. Computing TCP’s retransmission timer. RFC 2988, Internet Engineering Task Force, SRI International, November 2000. URL http://www.ietf.org/rfc/rfc2988.txt.
* Barakat et al. [2000] Chadi Barakat, Eitan Altman, and Walid Dabbous. On TCP performance in a heterogenous network: A survey. _IEEE Communications Magazine_ , 38(1):40–46, January 2000. Extended version: INRIA Research Report RR-3737, July, 1999.
* Allman et al. [1999] Mark Allman, Vern Paxson, and Jon Postel. TCP congestion control. RFC 2581, Internet Engineering Task Force, SRI International, April 1999\. URL http://www.ietf.org/rfc/rfc2581.txt.
* Jacobson [1990] Van Jacobson. Modified TCP congestion avoidance algorithm. end2end-interest mailing list, April 1990. ftp://ftp.isi.edu/end2end/end2end-interest-1990.mail.
* Jain et al. [1988] Raj Jain, K. K. Ramakrishnan, and Dah-Ming Chiu. _Congestion avoidance in computer networks with a connectionless network layer_ , pages 140–156. Artech House, Inc., Norwood, MA, USA, 1988.
* Chiu and Jain [1989] Dah-Ming Chiu and Raj Jain. Analysis of the increase and decrease algorithms for congestion avoidance in computer networks. _Comput. Netw. ISDN Syst._ , 17(1):1–14, 1989\.
* Karn and Partridge [1991] Phil Karn and Craig Partridge. Improving round-trip time estimates in reliable transport protocols. _ACM Trans. Comput. Syst._ , 9(4):364–373, November 1991.
* Mathis et al. [1997] Matt Mathis, Jeffrey Semke, Jamshid Mahdavi, and Teunis Ott. The macroscopic behavior of the TCP congestion avoidance algorithm. _SIGCOMM Comput. Commun. Rev._ , 27(3):67–82, July 1997.
* Altman et al. [2000a] Eitan Altman, Konstantin Avrachenkov, and Chadi Barakat. A stochastic model of TCP/IP with stationary random losses. _SIGCOMM Comput. Commun. Rev._ , 30(4):231–242, October 2000a.
* Padhye et al. [1998] Jitendra Padhye, Victor Firoiu, Don Towsley, and Jim Kurose. Modeling TCP throughput: a simple model and its empirical validation. _SIGCOMM Comput. Commun. Rev._ , 28(4):303–314, October 1998.
* Lakshman and Madhow [1997] T. V. Lakshman and Upamanyu Madhow. The performance of TCP/IP for networks with high bandwidth-delay products and random loss. _IEEE/ACM Trans. Net._ , 5(3):336–350, June 1997\.
* Ott et al. [1996] Teunis J. Ott, Joop H. B. Kemperman, and Matt Mathis. The stationary behaviour of ideal TCP congestion avoidance. In _DIMACS Workshop on Performance of Realtime Applications on the Internet_ , November 1996.
* Misra and Ott [1999] Archan Misra and Teunis J. Ott. The window distribution of idealized TCP congestion avoidance with variable packet loss. In _IEEE Infocom: 18th Ann. Joint Conf. IEEE Comp. Comm. Soc._ , volume 3, pages 1564–1572, New York, NY, USA, March 1999. IEEE.
* Misra et al. [2000] Vishal Misra, Wei-Bo Gong, and Don Towsley. Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. _SIGCOMM Comput. Commun. Rev._ , 30(4):151–160, October 2000.
* Altman et al. [2000b] Eitan Altman, Chadi Barakat, Emmanuel Laborde, Patric Brown, and Denis Collange. Fairness analysis of TCP/IP. In _Proc. 39th IEEE Conf. Decision and Control_ , volume 1, pages 61–66, 2000b.
* Floyd [1991] Sally Floyd. Connections with multiple congested gateways in packet-switched networks part 1: One-way traffic. _SIGCOMM Comput. Commun. Rev._ , 21(5):30–47, October 1991.
* Floyd and Jacobson [1991] Sally Floyd and Van Jacobson. Traffic phase effects in packet-switched gateways. _SIGCOMM Comput. Commun. Rev._ , 21(2):26–42, 1991.
* ns2 [1995] The Network Simulator – ns-2, 1995. URL http://www.isi.edu/nsnam/ns/.
* Bolot [1993] J. Bolot. End-to-end packet delay and loss behavior in the Internet. In _SIGCOMM’93_. ACM, 1993.
* Misra et al. [1999] Archan Misra, Teunis J. Ott, and John Baras. The window distribution of multiple TCPs with random loss queues. In _GLOBECOM’99_ , volume 3, pages 1714–1726. IEEE, December 1999\.
* Dorogovtsev et al. [2000] Sergei N. Dorogovtsev, Jose F. F. Mendes, and Alexander N. Samukhin. Structure of growing networks with preferential linking. _Phys. Rev. Lett._ , 85(21):4633, November 2000\.
* Szymański [1987] Jerzy Szymański. On a nonuniform random recursive tree. In Micha Karonski and Zbigniew Palka, editors, _Random Graphs ’85_ , volume 144 of _North–Holland Math. Stud._ , pages 297–305. North–Holland, 1987. Based on Lectures Presented at the 2nd International Seminar on Random Graphs and Probabilistic Methods in Combinatorics, August 5-9, 1985.
* Erdős and Rényi [1960] Paul Erdős and Alfréd Rényi. On the evolution of random graphs. _Publ. Math. Inst. Hung. Acad. Sci._ , 5:17, 1960.
* Barabási and Albert [1999] Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. _Science_ , 286:509–512, October 1999.
* Barabási et al. [1999] Albert-László Barabási, Réka Albert, and Hawoong Jeong. Mean-field theory for scale-free random networks. _Physica A_ , 272(1):173–187, October 1999.
* Vázquez et al. [2002] Alexei Vázquez, Romualdo Pastor-Satorras, and Alessandro Vespignani. Large-scale topological and dynamical properties of the Internet. _Phys. Rev. E_ , 65(6):066130, June 2002.
* Dorogovtsev and Mendes [2000] Sergei N. Dorogovtsev and Jose F. F. Mendes. Evolution of networks with aging of sites. _Phys. Rev. E_ , 62(2):1842–1845, August 2000\.
* Zhu et al. [2003] Han Zhu, Xinran Wang, and Jian-Yang Zhu. Effect of aging on network structure. _Phys. Rev. E_ , 68(5):056121, November 2003.
* Albert and Barabási [2000] Réka Albert and Albert-László Barabási. Topology of evolving networks: Local events and universality. _Phys. Rev. Lett._ , 85:5234, December 2000.
* Krapivsky and Redner [2001] Paul L. Krapivsky and Sidney Redner. Organization of growing random networks. _Phys. Rev. E_ , 63:066123, May 2001.
* Krapivsky et al. [2000] Paul L. Krapivsky, Sidney Redner, and Francois Leyvraz. Connectivity of growing random networks. _Phys. Rev. Lett._ , 85(21):4628, November 2000\.
* Evans and Saramäki [2005] T. S. Evans and J. P. Saramäki. Scale-free networks from self-organization. _Phys. Rev. E_ , 72(2):026138, August 2005.
* Goh et al. [2002] K.-I. Goh, B. Kahng, and D. Kim. Fluctuation-driven dynamics of the Internet topology. _Phys. Rev. Lett._ , 88(10):108701, March 2002\.
* Goh et al. [2001] K.-I. Goh, B. Kahng, and D. Kim. Universal behavior of load distribution in scale-free networks. _Phys. Rev. Lett._ , 87(27):278701, December 2001\.
* Barthélemy [2003] Marc Barthélemy. Comment on “Universal behavior of load distribution in scale-free networks”. _Phys. Rev. Lett._ , 91(18):189803, Oct 2003.
* Goh et al. [2003] K.-I. Goh, C.-M. Ghim, B. Kahng, and D. Kim. Goh et al. reply:. _Phys. Rev. Lett._ , 91(18):189804, Oct 2003.
* Szabó et al. [2002] Gábor Szabó, Mikko Alava, and János Kertész. Shortest paths and load scaling in scale-free trees. _Phys. Rev. E_ , 66:026101, August 2002.
* Bollobás and Riordan [2004] Béla Bollobás and Oliver Riordan. Shortest paths and load scaling in scale-free trees. _Phys. Rev. E_ , 69:036114, March 2004.
* Baccelli and Hong [2002] François Baccelli and Dohy Hong. AIMD, fairness and fractal scaling of TCP traffic. In _INFOCOM’02_ , volume 1, pages 229–238. IEEE, June 2002.
* Odlyzko [1995] A. M. Odlyzko. Asymptotic enumeration methods. In Roland Lewis Graham, Martin Götschel, and László Lovász, editors, _Handbook of Combinatorics_ , volume 2, pages 1063–1229. Elsevier, Amsterdam, December 1995.
Summary
The study of complex networks has evolved considerably in recent years. An
interesting example of complex networks is the Internet, which has become part
of everyday life. Two important aspects of the Internet, namely the properties
of its topology and the characteristics of its data traffic, have attracted
growing attention of the physics community. My thesis has considered problems
of both aspects.
First I studied the stochastic behavior of TCP, the primary algorithm
governing traffic in the current Internet, in an elementary network scenario
consisting of a standalone infinite-sized buffer and an access link. I
calculated the stationary distribution of the stochastic congestion window
process, which regulates the traffic of TCP. My analysis not only considered
the ideal congestion window dynamics, but also included the effect of the fast
recovery and fast retransmission (FR/FR) algorithms. Furthermore, I showed
that my model can be extended further to involve the effect of link
propagation delay, characteristic of WAN. An important achievement is that no
parameter fitting is necessary in my model. I also applied the mean field
approximation to describe many parallel TCP flows. After having been validated
against packet level numerical simulations, my analytic results agreed almost
perfectly.
I continued my thesis with the investigation of finite-sized semi-bottleneck
buffers, where packets can be dropped not only at the link, but also at the
buffer. I demonstrated that the behavior of the system depends only on a
certain combination of the parameters. Moreover, an analytic formula was
derived that gives the ratio of packet loss rate at the buffer to the total
packet loss rate. This formula makes it possible to treat buffer-losses as if
they were link-losses. I calculated the probability distribution of the
congestion window for both LAN and WAN scenarios. I showed that a sharp peak
might appear in the window distribution due to the FR/FR mode of TCP. My
analytical results matched numerical simulations properly.
In the last part of my thesis I studied computer networks from a structural
perspective. I demonstrated through fluid simulations that the distribution of
resources, specifically the link bandwidth, has a serious impact on the global
performance of the network. Then I analyzed the distribution of edge
betweenness in a growing scale-free tree under the condition that a local
property, the in-degree of the “younger” node of an arbitrary edge, is known
in order to find an optimum distribution of link capacity. The derived formula
is exact even for finite-sized networks. I also calculated the conditional
expectation of edge betweenness, rescaled for infinite networks.
Összefoglalás
Az elmúlt években a komplex hálózatok kutatása rendkívül sokat fejlődött. A
komplex hálózatok egyik legérdekesebb példája a mára a mindennapi élet részévé
vált _internet_. Az internet két fontos területe – a topológiájának
tulajdonságai és a rajta folyó adatforgalom jellemzői – iránt az utóbbi időben
a fizikusok körében is egyre nagyobb az érdeklődés. Dolgozatomban az említett
két terület néhány kérdését vizsgáltam.
Elsőként a jelenlegi internet legfontosabb forgalomszabályozó algoritmusának,
a TCP-nek a sztochasztikus viselkedését tanulmányoztam egy elemi hálózati
konfigurációban, mely egy egyedülálló bufferből, és egy hozzá kapcsolódó
vezetékből állt. Meghatároztam a TCP-forgalmat szabályozó torlódási ablaknak a
stacionárius eloszlását. Vizsgálatomban nem csupán az ideális torlódási ablak
dinamikát tekintettem, hanem figyelembe vettem a FR/FR (fast recovery/fast
retransmission) algoritmusok hatását is. Megmutattam továbbá, hogy a modellem
hogyan általánosítható úgy, hogy a vezetékeken fellépő csomagkésleltetést is
figyelembe vegye. Fontos eredmény, hogy nem szükséges ismeretlen paramétert
illenszteni a modellben. A modellt párhuzamosan működő TCP-k leírására
átlagtér-közelítésben alkalmaztam. Az analítikus eredményeket csomag-szintű
numerikus szimulációkkal összevetve rendkívül jó egyezést kaptam.
A disszertációt véges méretű bufferek vizsgálatával folytattam, ahol a
csomagok nem csak a vezetéken, hanem a bufferben is elveszhetnek. Megmutattam,
hogy a rendszer viselkedése csupán a paraméterek egy bizonyos kombinációjától
függ. Levezettem továbbá egy analítikus formulát, mely megadja a bufferben
eldobott, és a vezetéken elveszett csomagok arányát. Ez a formula lehetővé
teszi, hogy a bufferben történő csomagvesztéseket úgy tekintsük, mintha az a
vezetéken történt volna. Kiszámoltam a torlódási ablak eloszlását mind lokális
(LAN), mind tág (WAN) hálózati környezetben. Megmutattam, hogy az FR/FR
algoritmusok miatt egy éles csúcs jelenik meg az eloszlásban. Az analítikus
eredmények jól egyeztek a szimulációkkal.
Dolgozatom utolsó részében szerkezeti szempontból vizsgáltam komplex
számítógépes hálózatokat. Folyadékközelítésű szimulációkkal bemutattam, hogy
az erőforrások, különösképpen a vezetékek sávszélességének elosztása,
jelentősen befolyásolja a hálózat összteljesítményét. Ezután annak érdekében,
hogy az él-kapacitások optimális elrendezését meghatározzam, az él-köztesség
eloszlását vizsgáltam növekvő skála-független fában azzal a feltétellel, hogy
tetszőleges él ,,fiatalabb” csúcsának bejövő fokszáma ismert. A levezetett
formula még véges méretű hálózatokra is egzakt. Végül megadtam a végtelen
hálózatokra átskálázott él-köztesség feltételes várható értékét.
|
arxiv-papers
| 2008-10-07T15:54:06
|
2024-09-04T02:48:58.175381
|
{
"license": "Public Domain",
"authors": "Attila Fekete",
"submitter": "Attila Fekete",
"url": "https://arxiv.org/abs/0810.1226"
}
|
0810.1481
|
# An Evidential Path Logic for Multi-Relational Networks
Marko A. Rodriguez
Center for Non-Linear Studies
Los Alamos National Laboratory
Los Alamos, New Mexico 87545
&Joe Geldart
Department of Computer Science
University of Durham
South Road, Durham, DH1 3LE
###### Abstract
> Multi-relational networks are used extensively to structure knowledge.
> Perhaps the most popular instance, due to the widespread adoption of the
> Semantic Web, is the Resource Description Framework (RDF). One of the
> primary purposes of a knowledge network is to reason; that is, to alter the
> topology of the network according to an algorithm that uses the existing
> topological structure as its input. There exist many such reasoning
> algorithms. With respect to the Semantic Web, the bivalent, monotonic
> reasoners of the RDF Schema (RDFS) and the Web Ontology Language (OWL) are
> the most prevalent. However, nothing prevents other forms of reasoning from
> existing in the Semantic Web. This article presents a non-bivalent, non-
> monotonic, evidential logic and reasoner that is an algebraic ring over a
> multi-relational network equipped with two binary operations that can be
> composed to execute various forms of inference. Given its multi-relational
> grounding, it is possible to use the presented evidential framework as
> another method for structuring knowledge and reasoning in the Semantic Web.
> The benefits of this framework are that it works with arbitrary, partial,
> and contradictory knowledge while, at the same time, it supports a tractable
> approximate reasoning process.
Knowledge structures are used to represent facts about the world. The most
common formal data structure to represent knowledge is the network. With
respect to symbolic knowledge representation, the multi-relational network
(also known as an edge labeled graph or semantic network) is widely used. A
multi-relational network is composed of a set of vertices and a family of edge
sets, where each edge set has a different nominal, or categorical, label.
Formally, a multi-relational network can be represented as $M=(V,\mathbb{E})$,
where $V$ is the set of vertices and
$\mathbb{E}=\\{E_{0},E_{1},\dots,E_{m}\subseteq(V\times V)\\}$ is the family
of directed edge sets. In recent years, perhaps the most popular instance of a
multi-relational data structure for knowledge representation is the Resource
Description Framework (?) of the Semantic Web initiative (?)111Other formal
models of RDF include a bipartite graph (?) and hypergraph (?)
representation.. An edge in an RDF network is called a statement, or triple,
as it is composed of a subject, predicate, and object. For example, suppose
the statement $(i,k,j)$. This statement denotes that $i$ is related to $j$ by
means of an $k$-type relationship. Given the previous definition of $M$, this
is equivalent to the directed edge $(i,j)\in E_{k}$. A particular instance of
a statement is $({\texttt{marko}},{\texttt{coauthor}},{\texttt{joe}})$. This
statement denotes that Marko has a coauthorship relationship to Joe. Languages
such as the RDF Schema (RDFS) and the Web Ontology Language (OWL) impose a set
of constructs that serve to structure knowledge in a particular manner. The
particularities of such a structure are used by an RDFS or OWL reasoner to
infer new statements that can be added to the RDF network. The statements
inferred by such reasoners are bivalent in that they are either true or false
and moreover, their truth value is monotonic as it does not change once it has
been asserted.
While RDFS and OWL are common languages in the Semantic Web, the flexibility
of RDF can easily support other knowledge structures and reasoning algorithms.
The purpose of this article is to present a non-bivalent, non-monotonic,
evidential logic and reasoner for multi-relational networks that leverages
many of the ideas from Non-Axiomatic Logic (NAL) (?) and the Non-Axiomatic
Reasoning System (NARS) (?). The philosophical foundation of an evidential
logic is that no statement is inherently true or false and that a statement
only maintains levels of evidence to support or negate its claim. The notion
of
> experience-grounded semantics [is where] the truth value of a judgment
> indicates the degree to which the judgment is supported by the system’s
> experience. Defined in this way, truth value is system-dependent and time-
> dependent. Different systems may have conflicting opinions, due to their
> different experiences. (?)
The typical metaphor in an evidential logic system is that of an agent that
perceives the world, represents its perceptions in an internal knowledge
structure, and reasons on that structure to infer new knowledge (?). Moreover,
it is assumed that this agent has limited computational resources in terms of
both space and time and thus, does not maintain an objective knowledge
structure nor does it necessarily have the ability to reason across its entire
subjective knowledge structure. In other words, the agent has only so much
information that it can store and process at any one time. This notion is
known as the Assumption of Insufficient Knowledge and Insufficient Resources
(AIKIR). Non-axiomatic logic is contrasted to axiomatic logic, where truth is
bivalent, is defined independent of the time and space requirements necessary
to derive it, can be reasoned from a finite set of premises, and where all
reasoning produces true, immutable conclusions.
The evidential logic presented in this article forms an algebraic ring over a
multi-relational network (i.e. the knowledge structure) equipped with two
binary operations (i.e. the atoms of the inferencing algorithms). Given the
logic’s multi-relational formulation, it is possible to comfortably represent
this structure in RDF and thus, on the Semantic Web. The primary contribution
of this article is the application of evidential logics to multi-relational
networks and the formulation of an algebraic evidential reasoner.
## Evidence in an Inheritance Network
With evidential logics, there does not exist an objective boolean truth value
for every question that can be asked of the world as the world is not reasoned
on directly (?). What is reasoned on is the agent’s internal knowledge
structure. For the agent, knowledge is gained as new evidence from the world
is discovered (either through direct perception or through communication) or
as knowledge is inferred given the agent’s internal reasoning system. The Non-
Axiomatic Reasoning System (NARS) is an example of an evidential reasoning
system (?). The data structure proposed for NARS version 2.2 is a directed
evidence network denoted $G=(V,E,\lambda)$, where $V$ is a set of vertices,
$E\subseteq(V\times V)$ is a set of directed “inheritance” edges, and
$\lambda:E\rightarrow\langle[0,1],[0,1]\rangle$ maps each edge to an evidence
tuple. For example, an edge is denoted
$(i,\langle w^{+},w^{-}\rangle,j),$
where $i$ is the tail of the inheritance edge, $j$ is the head of the
inheritance edge, and $\langle w^{+},w^{-}\rangle$ is the evidence tuple for
that edge. Moreover, $\langle w^{+},w^{-}\rangle\equiv\lambda(i,j)$. The
meaning of $(i,\langle w^{+},w^{-}\rangle,j)$ is that there is $w^{+}$
positive evidence supporting the claim that $i$ isA $j$ and $w^{-}$ negative
evidence that $i$ isA $j$. Another interpretation of this edge is that, upon
examination, it appears, to the agent, that $i$ has $w^{+}$ properties in
common with $j$ and $w^{-}$ properties not in common with $j$. This idea is
also expressed as $i$ being the intension of $j$ and $j$ being the extension
of $i$ (?).
In NARS, the evidence tuple of an edge is revised by means of external
perception or internal reasoning. The act of perceiving $i$ and $j$ in the
external world will return evidence of their relationship and thus, increment
$w^{+}$ or $w^{-}$ accordingly. With respect to internal reasoning, there are
four syllogisms that can be applied to the agent’s internal knowledge network:
deduction, induction, abduction (?), and exemplification. For deduction,
$(i,\langle w^{+}_{1},w^{-}_{1}\rangle,j),(j,\langle
w^{+}_{2},w^{-}_{2}\rangle,k)\rightarrow(i,\langle
w^{+}_{3},w^{-}_{3}\rangle,k).$
For induction,
$(i,\langle w^{+}_{1},w^{-}_{1}\rangle,j),(i,\langle
w^{+}_{2},w^{-}_{2}\rangle,k)\rightarrow(j,\langle
w^{+}_{3},w^{-}_{3}\rangle,k).$
For abduction,
$(i,\langle w^{+}_{1},w^{-}_{1}\rangle,j),(k,\langle
w^{+}_{2},w^{-}_{2}\rangle,j)\rightarrow(i,\langle
w^{+}_{3},w^{-}_{3}\rangle,k).$
Finally, there is a less widely used fourth syllogism known as exemplification
(?; ?). Exemplification is defined as
$(i,\langle w^{+}_{1},w^{-}_{1}\rangle,j),(j,\langle
w^{+}_{2},w^{-}_{2}\rangle,k)\rightarrow(k,\langle
w^{+}_{3},w^{-}_{3}\rangle,i).$
The values for $\langle w^{+}_{3},w^{-}_{3}\rangle$ depend upon the specific
inference rules of the the evidential reasoner. In (?), it is explicitly
stated that the rules presented are not set in stone, but rather subject to
revision themselves as more is understood about the design of evidential
systems.
The evidence tuple $\langle w^{+},w^{-}\rangle$ of an edge can be transformed
into a normalized “truth value” consisting of a new tuple $\langle
f,c\rangle\in\langle\emptyset\cup[0,1],\emptyset\cup[0,1]\rangle$. The first
component $f$ is the frequency of positive evidence and is defined as
$f=\frac{w^{+}}{w^{+}+w^{-}}.$
The second component $c$ is the confidence in the stability of $f$ as $k$-more
observations are made and is defined as
$c=\frac{w^{+}+w^{-}}{(w^{+}+w^{-}+k)}.$
The parameter $k\in\mathbb{R}_{0}^{+}$ is a user-defined, system constant. In
words, as more positive evidence accumulates relative to negative evidence,
$f$ increases towards $1$. As more total evidence accumulates relative to some
constant $k$, $c$ increases towards $1$. If there is no evidence, then the
edge has an $f$-component of $\emptyset$ which means that the relationship is
unknown. Finally, hard “truth” can be modeled with an evidence tuple of
$\langle 1,0\rangle$ with $k=0$ and thus, an $fc$-tuple of $\langle
1,1\rangle$.
The contribution of this article is to extend the aforementioned evidential
logic framework to multi-relational networks composed of both inheritance and
non-inheritance edges. Moreover, this article contributes an algebraic ring
formulation of evidential reasoning which situates the reasoner within well
understood mathematics. From this multi-relational foundation, evidential
reasoning can be comfortably executed in the RDF-rich world of the Semantic
Web.
## Evidential Reasoning using Path Expressions
A path algebra to map a multi-relational network to a single-relational form
was originally presented in (?). The motivation behind the algebra was to
provide a formal means by which the large class of single-relational network
analysis algorithms could be applied to multi-relational networks in a
meaningful way. With respect to this article, the binary operations of $+$ and
$\cdot$ are updated so as to work with evidence tuples.
A multi-relational, evidence network is defined as
$M=(V,\mathbb{E}=\\{E_{0},E_{1},\ldots,E_{m}\subseteq(V\times V)\\},\lambda)$,
where $V$ is a set of vertices, $\mathbb{E}$ is a family of edge sets, and
$\lambda:E_{k}\rightarrow\langle[0,1],[0,1]\rangle$ maps each edge in
$E_{k}:k\leq m$ to an evidence tuple. The algebraic formulation of the
presented evidential path algebra operates on an $n\times n\times m$ tensor
representation of this network (?)222The tensor representation can also be
thought of as a set of $m$ adjacency matrix “slices”, where each matrix has
its own label.. The evidence tensor $\mathcal{A}$ is defined as
$\mathcal{A}^{k}_{i,j}=\begin{cases}\lambda(i,j)&\text{if }(i,j)\in E_{k}\\\
\langle 0,0\rangle&\text{otherwise}.\end{cases}$
The two $n$ dimensions represent the vertices and the single $m$ dimension
represents the various edge labels. Thus, there is a one-to-one mapping
between a multi-relational, evidence network and an evidence tensor. The
entries of the tensor denote the amount of positive ($w^{+}$) and negative
($w^{-}$) evidence for the edge $(i,j)\in E_{k}$, where $\langle 0,0\rangle$
denotes no evidence when no such edge exists. Inference on this tensor can be
accomplished through the two binary operations
$+:\langle[0,1],[0,1]\rangle\times\langle[0,1],[0,1]\rangle\rightarrow\langle[0,1],[0,1]\rangle$
and
$\cdot:\langle[0,1],[0,1]\rangle\times\langle[0,1],[0,1]\rangle\rightarrow\langle[0,1],[0,1]\rangle.$
The function rules of these operations are
$\langle w_{1}^{+},w_{1}^{-}\rangle+\langle
w_{2}^{+},w_{2}^{-}\rangle=\langle(w_{1}^{+}+w_{2}^{+}),(w_{1}^{-}+w_{2}^{-})\rangle$
and
$\langle w_{1}^{+},w_{1}^{-}\rangle\cdot\langle
w_{2}^{+},w_{2}^{-}\rangle=\langle(w_{1}^{+}\cdot w_{2}^{+}),(w_{1}^{+}\cdot
w_{2}^{-}+w_{1}^{-}\cdot w_{2}^{+}+w_{1}^{-}\cdot w_{2}^{-})\rangle.$
These two operations form an algebraic ring over the evidence tensor (i.e.
multi-relational, evidence network). The binary operation $+$ is associative,
has an identity of $\langle 0,0\rangle$, and is commutative. The binary
operation $\cdot$ is associative with an identity of $\langle 1,0\rangle$. The
operation $+$ supports the notion that evidence (from two independent
experiences/inferences) can be summed together (?). The operation $\cdot$
supports the notion that positive evidence can be multiplied to form new
positive evidence and that conflicting and negative evidence accounts for
negative evidence.
The next subsections will formalize the various syllogisms on inheritance
edges in a multi-relational network and then the following subsection will
discuss the application of these operations to any arbitrary path through an
evidence tensor. Note that what is presented is a set of operations that will
operate on $n\times n$ matrix “slices” of the evidence tensor. In this
respect, the presented operations are “global” computations and thus, are
computationally inefficient and contrary to AIKIR. However, these operations
can be implemented as “local” computations using various methods such grammar
walks (?) in specific areas of the network because evidential reasoning does
not require a network-wide computation.
### Inferring Inheritance Evidence
Inheritance relations are handled as a special case situaion when reasoning in
a multi-relational network. Example inheritance predicates in an RDF network
are rdf:type and rdfs:subClassOf. Figure 1 presents a simple example network
that will be used to demonstrate the inference rules of deduction, induction,
abduction, and exemplification.
Figure 1: An an inheritance network.
#### Deductive Inheritance
Deduction is defined as a two step “walk” on the inheritance component of an
evidential network. A two step walk can be computed by squaring a matrix.
Suppose a standard square $\\{0,1\\}$-matrix denoted $\mathbf{A}$, where
$\mathbf{A}_{i,j}=1$ if there is an edge between vertex $i$ and $j$, and $0$
otherwise. The $2^{\text{nd}}$ power of this matrix, as defined by ordinary
matrix multiplication, $\mathbf{A}\mathbf{A}$, will yield a new matrix where
entry $(\mathbf{A}\mathbf{A})_{i,j}$ denotes the total number of paths of
length $2$ starting from vertex $i$ and ending on vertex $j$ (?). With respect
to an evidence tensor, determining the product of two adjacency matrix
“slices”, will yield a new adjacency matrix where the entry $(i,j)$ denotes
the total amount of deductive evidence supporting $(i,j)$.
Evidential matrix multiplication is defined as ordinary matrix multiplication,
but respective of the rules of $\cdot$ and $+$. Thus,
$\left(\mathcal{A}^{k}\mathcal{A}^{k^{\prime}}\right)_{i,j}=\sum_{l\in
V}\mathcal{A}_{i,l}^{k}\cdot\mathcal{A}_{l,j}^{k^{\prime}}:k,k^{\prime}\leq
m.$
In the degenerate case where all positive evidence is $1$ and all negative
evidence is $0$, such that
$\mathcal{A}^{k}_{i,j}=\begin{cases}\langle 1,0\rangle&\text{if }(i,j)\in
E_{k}\\\ \langle 0,0\rangle&\text{otherwise},\end{cases}$
evidential matrix multiplication will set $w^{+}$ to the total number of paths
from vertex $i$ to vertex $j$ and $0$ to $w^{-}$. In this form, the evidential
path algebra yields results that are isomorphic to the original formulation of
the path algebra in (?).
In Figure 2, deduction, as defined by
$\mathcal{A}^{{\texttt{isA}}}\mathcal{A}^{{\texttt{isA}}}$, infers four new
edges. Note that the evidence tuples are not presented in order to preserve
diagram clarity.
Figure 2: Deduction in an inheritance network.
#### Inductive Inheritance
Induction is the process of generalization given instances. In order to
compute induction in an inheritance region of an evidence tensor, it is
important to take the converse transpose of an adjacency matrix “slice”. The
operation of taking a statement like “a scholar is a writer” and reversing it
to derive the statement “a writer is a scholar” is known as taking the
converse of the statement. With respect to determining the evidence for the
converse of a statement, all positive evidence for a “scholar is a writer” is
considered positive evidence that a “writer is a scholar”. However, all
negative evidence for “a scholar is a writer” should not be considered
negative evidence for a “writer is a scholar” as there is no evidence in the
original statement for writers not being scholars. Such statement converses
can be expressed using the evidential algebra. For standard matrices, the
transpose of a matrix is defined as
$(\mathbf{A}^{\top})_{i,j}=\mathbf{A}_{j,i}$ and denotes reversing the
direction of an edge (i.e. taking the converse of a statement). However, for
evidential edges, the converse transpose of an evidential matrix is defined as
$\hat{\mathcal{A}}^{k}_{i,j}=\langle\gamma^{+}(\mathcal{A}^{k}_{j,i}),0\rangle,$
where $\gamma^{+}:\langle[0,1],[0,1]\rangle\rightarrow[0,1]$ maps an evidence
tuple to its first component (i.e. $w^{+}$ positive evidence). This operation
ensures that the converse of an evidence tuple maintains no negative evidence.
In Figure 3, induction, as defined by
$\mathcal{A}^{{\texttt{isA}}}\hat{\mathcal{A}}^{{\texttt{isA}}}$, infers two
new isA edges between journalist and scholar. As stated previous, and to
stress the point, the negative evidence for these two new evidence tuples is
$0$.
Figure 3: Induction in an inheritance network.
#### Abductive Inheritance
Abduction is the reverse of induction. In Figure 4, abduction, as defined by
$\hat{\mathcal{A}}^{{\texttt{isA}}}\mathcal{A}^{{\texttt{isA}}}$, infers two
new isA edges between person and author. Similar to induction, a converse
transpose will generate no negative evidence for these two new evidence
tuples.
Figure 4: Abduction in an inheritance network.
#### Exemplative Inheritance
Exemplary inheritance paths can be determined by the multiplication of two
converse transpose matrices. In Figure 5, exemplification, as defined by
$\hat{\mathcal{A}}^{{\texttt{isA}}}\hat{\mathcal{A}}^{{\texttt{isA}}}$, infers
four new evidence tuples.
Figure 5: Exemplification in an inheritance network.
This subsection presented the syllogisms of deduction, induction, abduction,
and exemplification and their use in an inheritance region of an evidence
tensor. Note that this region may account for more than a single $m$-dimension
as many labels can have an similar meaning to isA (e.g. similarTo,
equivalentTo, implies, etc.). The next section will discuss reasoning using
arbitrary paths through a multi-relational evidence network and thus, for
those paths that may not necessarily contain isA edges.
### Inferring Non-Inheritance Evidence
A multi-relational network may be composed of various types of relationships.
Figure 6 diagrams an example multi-relational network that will be referred to
in the examples of this subsection333There is nothing that prevents the
network in Figure 1 to be merged with the network in Figure 6 (e.g. marko and
joe are both scholars). However, for diagram clarity, this is not
represented..
Figure 6: A multi-relational knowledge network.
The network in Figure 6 is composed of a reference to this article, the
denoted authors of this article, and two citations from this article to other
articles. In a bivalent logic, these statements are true because they exist.
However, in scholarly publishing it is rare, nearly impossible, for two people
to “equally” write an article together. While ideas are shared and drafts are
written, read, and edited, the article’s final form is always a biased
reflection of the approach of some authors over others. With respect to the
statements diagrammed in Figure 6, what is the evidence for these statements?
The following descriptions are provided to expose, for each edge presented
above, how much supporting or detracting evidence there is for $i$’s $m$-type
relationship to $j$:
$({\texttt{marko}},{\texttt{wrote}},{\texttt{this\\_article}})$:
* •
$w^{+}$: notation, writing style, diagram style, american spelling
* •
$w^{-}$: logic, reasoning, citations, philosophy
$({\texttt{marko}},{\texttt{wrote}},{\texttt{path\\_article}})$:
* •
$w^{+}$: notation, writing style
* •
$w^{-}$: algebra, no diagrams
$({\texttt{joe}},{\texttt{wrote}},{\texttt{this\\_article}})$:
* •
$w^{+}$: logic, reasoning, citations, philosophy, rings
* •
$w^{-}:$ notation, writing style, diagram style, american spelling
$({\texttt{this\\_article}},{\texttt{cites}},{\texttt{path\\_article}})$:
* •
$w^{+}$: (?) citation, rings
* •
$w^{-}$: graph notation, philosophy, only a single algebra citation
$({\texttt{this\\_article}},{\texttt{cites}},{\texttt{nars\\_article}})$:
* •
$w^{+}$: (?) citation, evidential notation, syllogisms
* •
$w^{-}$: path expressions, semantic web, rdf, owl, rdfs
The presented positive and negative evidence “metadata” (e.g. writing style,
american spelling, citation patterns, etc.) can be represented in a multi-
relational network. From this multi-relational encoding, it is possible,
through automated means, to infer new evidence or revise existing evidence in
the network with prescribed path expressions. In other words, a region of the
network can provide further supporting and/or detracting evidence for another
region of the network. In order to demonstrate two examples of this, the
multi-relational network in Figure 6 will be used in this section to
1. 1.
infer new independent evidence supporting the claims that marko and joe wrote
this_article according to the notion that self-citations are positive evidence
supporting authorship, and
2. 2.
infer new evidence that marko and joe have a coauthorship edge between them.
#### Self-Citation Paths
Suppose that the article citation evidence in Figure 6 was experienced by the
agent (e.g. a repository feed) and added to its existing internal network. At
the point of insertion, it is possible to revise the evidence tuple for wrote
based upon the idea that self-citations in an article are considered evidence
for authorship of that article (?). Furthermore, in order to ensure
independent evidence, assume that the current evidence for wrote in Figure 6
was not determined using self-citation information444Refer to (?) for the
definition and importance of independent evidence in evidential logics.. Given
this scenario, the following inference rule will update all wrote evidence
according to inferred self-citation evidence:
$\mathcal{A}^{{\texttt{wrote}}}_{(t+1)}=\left(\left(c(\mathcal{A}^{{\texttt{wrote}}}_{(t)})\mathcal{A}^{{\texttt{cites}}}_{(t)}{\mathcal{A}^{{\texttt{wrote}}}_{(t)}}^{\top}\right)\circ\mathbf{I}\right)+\mathcal{A}^{{\texttt{wrote}}}_{(t)},$
where $t\in\mathbb{N}^{+}_{0}$ is the current time step,
$c(\mathcal{A}^{{\texttt{wrote}}})$ “clips” the evidence in
$\mathcal{A}^{{\texttt{wrote}}}$, $\circ$ is the entry-wise Hadamard
multiplication operation555For review, Hadamard entry-wise multiplication is
defined as
$\mathbf{A}\circ\mathbf{B}=\left[\begin{array}[]{ccc}\mathbf{A}_{1,1}\cdot\mathbf{B}_{1,1}&\cdots&\mathbf{A}_{1,j}\cdot\mathbf{B}_{1,j}\\\
\vdots&\ddots&\vdots\\\
\mathbf{A}_{i,1}\cdot\mathbf{B}_{i,1}&\cdots&\mathbf{A}_{i,j}\cdot\mathbf{B}_{i,j}\\\
\end{array}\right].$ , and $\mathbf{I}$ is the evidential identity matrix
$\mathbf{I}_{i,j}=\begin{cases}\langle 1,0\rangle&\text{if }i=j\\\ \langle
0,0\rangle&\text{otherwise}.\end{cases}$
In words, the self-citation inference rule states that evidence for
$\mathcal{A}^{{\texttt{wrote}}}$ can be modulated by the total evidence for
the path that goes from an author, to their written articles, to the articles
that those articles cite, and then finally, to the authors of those cited
articles. However, in order to ensure that those cited authors are the
original author from the start of the path (i.e. self-citation), it is
important to filter on the identity matrix $\mathbf{I}$. Hadamard entry-wise
multiplication is used to apply a matrix filter to a path. Note that the
transpose of an evidence matrix, not the converse transpose of the evidence
matrix, is used when taking the converse of a non-inheritance statement. The
reason for this is that the positive and negative evidence for the statement
“marko wrote this article” is the same for “this article was written by
marko”. Next, inferred evidence for wrote must be independent of the evidence
used to calculate it. Thus, $\mathcal{A}^{{\texttt{wrote}}}$ is mapped to a
$(\langle 1,0\rangle,\langle 0,0\rangle)$-matrix using the clip $c$ operation,
where
$c(\mathcal{A}^{k})_{i,j}=\begin{cases}\langle 1,0\rangle&\text{if
}\mathcal{A}^{k}_{i,j}\neq\langle 0,0\rangle\\\ \langle
0,0\rangle&\text{otherwise}.\end{cases}$
Finally, the total evidence for the self-citation path is summed with the
current evidence for the wrote edge. Thus, the agent has used self-citations
as further, revising evidence for wrote.
#### Coauthorship Paths
Two people are considered coauthors if they have both written an article
together. The evidence for coauthorship is determined by the total evidence
across all their jointly written articles. In its algebraic form, the evidence
for coauthor can be determined by
$\mathcal{A}^{{\texttt{coauthor}}}_{(t+1)}=\left(\left(\mathcal{A}^{{\texttt{wrote}}}_{(t)}{\mathcal{A}^{{\texttt{wrote}}}_{(t)}}^{\top}\right)\circ
n(\mathbf{I})\right)+\mathcal{A}^{{\texttt{coauthor}}}_{(t)},$
where $n(\mathbf{I})$ “nots” the evidential identity matrix such that such
that every $\langle 1,0\rangle$ is a $\langle 0,0\rangle$ and every $\langle
0,0\rangle$ is a $\langle 1,0\rangle$. The reason for the $n(\mathbf{I})$
filter is that to represent a coauthor path from a person to their authored
papers and then to other authors of those papers, the path must exclude the
original author as an author is not a coauthor of themselves. In other words,
it must filter out the identity evidence matrix. The two inferred coauthor
edges between marko and joe are diagrammed in Figure 7.
Figure 7: Coauthoring in an inheritance network.
The article (?) provides an in-depth review of different inferences that can
be made with arbitrary paths, various filters, and how the theorems of the
general path algebra can be applied to derive equivalent, yet more
computationally efficient paths. The examples presented in (?) can be applied
to an evidence tensor as long as the definitions of $+$ and $\cdot$, as
defined in this article, are respected.
## Conclusion
Reasoning with axiomatic logics is computationally expensive (?; ?). With
respect to the Semantic Web, and with the integration capabilities brought
forth by the Linked Data initiative, such reasoning is intractable. The
original assumption driving the development of NARS is the Assumption of
Insufficient Knowledge and Insufficient Resources (AIKIR) (?). Given space and
time constraints, an agent cannot reason over the entire Semantic Web, and
potentially, not even over its entire internal knowledge network. The benefit
of the inheritance-based syllogisms are that they do not require a global
analysis of the knowledge network, can be executed independently of each
other, and at the their core, are very simple and computationally efficient.
With respect to inferencing with arbitrary path expressions, the efficiency is
dependent on the length of the path and the number of applied filters. It is
important to note that the matrix formalism presented in this article is very
much intractable as the best known algorithm to compute ordinary matrix
multiplication is approximately $O(|V|^{2.807})$ (?). As stated previously,
this matrix model can be approximated using various techniques such as grammar
walk algorithms which do not compute the inferences over the entire network,
but instead, on local subgraphs (i.e. paths starting from particular vertices)
(?). With evidential logic, such walks can be executed when resources are
available and only in those areas of the knowledge network where it is deemed
necessary (e.g. $f\sim 0.5$ and/or low $c$ areas).
Finally, to actually represent a multi-relational, evidence network in RDF and
on the Semantic Web, some form of reification can be used. A popular technique
is the quad-form of a “triple” where a statement maintains a fourth component
known as a named graph (?). With reification, statements can be attached to
statements and thus, a $\langle w^{+},w^{-}\rangle$ evidence tuple can be
assigned to an RDF statement.
This article presented a non-axiomatic evidential logic that can be
implemented within the constructs of RDF and thus, can be used as an
evidential reasoning system in the Semantic Web. The benefit of this system is
that it works with arbitrary, partial, and contradictory knowledge while, at
the same time, in a non-matrix implementation, supports a tractable
approximate reasoning process.
## Acknowledgements
Vadas Gintautas provided useful insight during the development of these ideas.
This work was funded by a Director’s Fellowship from the Los Alamos National
Laboratory, EPSRC grant EP/D504376/1, and British Telecom plc.
## References
* [Berners-Lee, Hendler, and Lassila 2001] Berners-Lee, T.; Hendler, J. A.; and Lassila, O. 2001\. The Semantic Web. Scientific American 34–43.
* [Bochenski 1970] Bochenski, I. M. 1970\. A History of Formal Logic. New York, USA: Chelsea Publishing Company.
* [Carroll et al. 2005] Carroll, J. J.; Bizer, C.; Hayes, P.; and Stickler, P. 2005\. Named graphs, provenance and trust. In The Fourteenth International World Wide Web Conference (WWW05), 613–622. Chiba, Japan: ACM Press.
* [Chartrand 1977] Chartrand, G. 1977\. Introductory Graph Theory. Dover.
* [Dau 2006] Dau, F. 2006\. RDF as graph-based, diagrammatic logic. Lecture Notes in Computer Science 4203\.
* [Donini 2002] Donini, F. 2002\. The Descripton Logic Handbook. Cambridge University Press. chapter Complexity of Reasoning, 101–141.
* [Fensel and van Harmelen 2007] Fensel, D., and van Harmelen, F. 2007\. Unifying reasoning and search to web scale. IEEE Internet Computing 11(2):94–96.
* [Hayes and Gutierrez 2004] Hayes, J., and Gutierrez, C. 2004\. Bipartite graphs as intermediate model for RDF. In Proceedings of the International Semantic Web Conference ISWC04, 47–61.
* [Kolda, Bader, and Kenny 2005] Kolda, T. G.; Bader, B. W.; and Kenny, J. P. 2005\. Higher-order web link analysis using multilinear algebra. In Proceedings of the Fifth IEEE International Conference on Data Mining ICDM’05. IEEE.
* [Morale and Serodio 2006] Morale, A. A. M., and Serodio, M. E. V. 2006\. A directed hypergraph model for RDF. In Simperl, E.; Diederich, J.; and Schreiber, G., eds., Proceedings of the Knowledge Web PhD Symposium (KWEPSY07).
* [Patzig 1968] Patzig, G. 1968\. Aristotle’s Theory of The Syllogism. Boston, Massachusetts: D. Reidel Publishing Company.
* [Rodriguez and Shinavier 2008] Rodriguez, M. A., and Shinavier, J. 2008\. Exposing multi-relational networks to single-relational network analysis algorithms. Technical Report LA-UR-08-03931, Los Alamos National Laboratory.
* [Rodriguez, Bollen, and Van de Sompel 2009] Rodriguez, M. A.; Bollen, J.; and Van de Sompel, H. 2009\. Automatic metadata generation using associative networks. ACM Transactions on Information Systems 27(2).
* [Rodriguez 2008] Rodriguez, M. A. 2008\. Grammar-based random walkers in semantic networks. Knowledge-Based Systems 21(7):727–739.
* [Strassen 1969] Strassen, V. 1969\. Gaussian elimination is not optimal. Numerische Mathematik 13:354–356.
* [Wang 1993] Wang, P. 1993\. Non-axiomatic reasoning system (version 2.2). Technical Report 75, Center for Research on Concepts and Cognition at Indiana University.
* [Wang 1994] Wang, P. 1994\. From inheritance relation to non-axiomatic logic. International Journal of Approximate Reasoning 11:281–319.
* [Wang 2004a] Wang, P. 2004a. Cognitive logic versus mathematical logic. In Proceedings of the Third International Seminar on Logic and Cognition.
* [Wang 2004b] Wang, P. 2004b. Experience-grounded semantics: a theory for intelligent systems. Cognitive Systems Research 6(4):282–302.
* [Wang 2006] Wang, P. 2006\. Rigid Flexibility. Springer.
|
arxiv-papers
| 2008-10-08T17:49:15
|
2024-09-04T02:48:58.196541
|
{
"license": "Public Domain",
"authors": "Marko A. Rodriguez, Joe Geldart",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/0810.1481"
}
|
0810.1665
|
# Dipole oscillations of confined lattice bosons in one dimension
Simone Montangero NEST-CNR-INFM and Scuola Normale Superiore, I-56156 Pisa,
Italy Institut für Quanteninformationsverarbeitung, Universität Ulm, D-89069
Ulm, Germany Rosario Fazio NEST-CNR-INFM and Scuola Normale Superiore,
I-56156 Pisa, Italy International School for Advanced Studies (SISSA),
I-34014 Trieste, Italy Peter Zoller Institute for Theoretical Physics,
University of Innsbruck, A-6020 Innsbruck, Austria Institute for Quantum
Optics and Quantum Information, A-6020 Innsbruck, Austria Guido Pupillo
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria Institute for Quantum Optics and Quantum Information, A-6020
Innsbruck, Austria
###### Abstract
We study the dynamics of a non-integrable system comprising interacting cold
bosons trapped in an optical lattice in one-dimension by means of exact time-
dependent numerical DMRG techniques. Particles are confined by a parabolic
potential, and dipole oscillations are induced by displacing the trap center
of a few lattice sites. Depending on the system parameters this motion can
vary from undamped to overdamped. We study the dipole oscillations as a
function of the lattice displacement, the particle density and the strength of
interparticle interactions. These results explain the recent experiment C. D.
Fertig et al., Phys. Rev. Lett. 94, 120403 (2005).
Recent experiments with cold atoms pezze04 ; kinoshita06 ; stoeferle04 ;
fertig05 have provided realizations of non-equilibrium quantum many-body
systems, allowing to address a number of fundamental questions. For example,
the integrability of a many-body system has been demonstrated in Ref.
kinoshita06 , via the inhibition of thermalization in a one-dimensional Bose
gas, which opened the way to theoretical studies of the relaxation dynamics of
non-equilibrium many-body systems kollath07 . The dynamics of non-integrable
systems has been recently explored experimentally in Refs. stoeferle04 ;
fertig05 using interacting cold bosonic atoms trapped in an array of one-
dimensional optical lattices and confined by a parabolic potential. Dipole
oscillations were induced by displacing the center of the parabolic potential,
and the dipole dynamics was studied by monitoring the position of the center
of mass. A sudden transition from a regime of undamped motion to a regime of
strongly damped motion was observed on increasing the lattice depth. Since
damping of the center of mass oscillations is due to excitations in the
optical lattice, the results obtained in stoeferle04 ; fertig05 have provided
precious diagnostic of the dynamical correlations of the many-body system, and
thus have stimulated considerable theoretical interest Polkovnikov ;
RuostekoskiPRL05 ; PupilloNJP06 .
Good agreement with the experimental results in fertig05 has been obtained in
the regimes of very weak RuostekoskiPRL05 and very strong interactions
PupilloNJP06 , where mean-field and extended fermionization techniques apply.
However, it remains a fundamental challenge to understand the dipole dynamics
in the regime of intermediate interactions, where the sudden localization
transition occurs and the subtleties of one-dimensional (1D) correlations do
not allow (semi-)analytical treatments. With the aim to provide a
comprehensive explanation of the experiment of Fertig et al. fertig05 , in
this letter we study the dipole oscillations by means of a numerically exact
time-dependent density-matrix-renormalization-group technique (tDMRG), see
also Clark08 . We find very good agreement with the experimental results in
the interesting regime of intermediate interactions. These results demonstrate
that time-dependent numerical simulations with tDMRG have reached the same
accuracy of current experiments with cold gases in the strongly correlated
regime and thus represent a unique theoretical tool for quantitative
comparisons and predictions for experiments in the cold atoms context.
The experiment in fertig05 was performed in a parameter regime where the use
of the following Bose-Hubbard Hamiltonian is microscopically justified
Jaksch98
$\displaystyle H$ $\displaystyle=$
$\displaystyle-J\sum_{j}(b_{j}^{\dagger}b_{j+1}+\mbox{h.c.})+\Omega\sum_{j}[j+\delta(t)]^{2}n_{j}$
(1) $\displaystyle+$ $\displaystyle\frac{U}{2}\sum_{j}n_{j}(n_{j}-1).$
The first term on the r.h.s. of Eq.(1) describes the tunneling of bosons
between neighboring sites with rate $J$ ($j$ labels the sites on the lattice).
The second term is the parabolic potential with curvature $\Omega$;
$\delta(t)$ is a sudden displacement of the trap center,
$\delta_{0}(t)=\delta\;\Theta(t)$ (with $\Theta(t)$ the Heaviside function),
and $n_{j}=b^{{\dagger}}_{j}b_{j}$ is the density operator with bosonic
creation (annihilation) operators $b^{{\dagger}}_{j}$ ($b_{i}$). The last term
is the onsite contact interaction with energy $U$ Jaksch98 , (we set
$\hbar=1$).
Figure 1: Relevant density distributions [panels (a-c)], see text, and
snapshots of the corresponding dipole dynamics [panels (d-f)]. (a-b) Density
distribution for $N=11$ and 15 particles, respectively, for
$\Omega/J=0.05623$. In each panel, the dashed and solid lines are $U/J=1$ and
20, respectively. The solid line in panel (b) corresponds to a Mott insulator.
(c) Density distribution for $N=23$, $\Omega/J=0.4$, and $U/J=1$ and 20
(dashed and solid lines, respectively). The solid line corresponds to a cake-
like structure. (d-f) Snapshots of the density distribution for the cases
(a-c), with $U/J=20$ and $\delta=4$, at times $tJ=0,30,40$ and 50\. The
dynamics of a few atoms in the Mott and cake-like configurations is frozen,
however, residual oscillations can persist in the latter, see text.
The sudden displacement on the trap center causes dipole oscillations of the
bosons which can be analyzed experimentally by monitoring the time evolution
of the Center Of Mass (COM) $x_{\rm com}=\sum_{j}j\langle n_{j}\rangle/N$,
with $N$ the number of particles. The experiment of Ref. fertig05 was
performed on a array of one-dimensional optical lattices where the number of
particles in each 1D lattice varied from $N\simeq 80$ to zero. Thus, in order
to provide a comprehensive and quantitative comparison with the experimental
data, here we analyze the dipole dynamics as a function of $\delta$, $U/J$,
and the number of bosons $N$. We find that overdamped motion can occur as a
function of $\delta$ for arbitrarily small interactions, Fig. 2, while in
general sizeable interactions tend to extend the parameter region where
localization occurs SmerziPRL . For a given $\Omega/J$ damping is found to
depend exponentially on $U/J$, and to be favored for small $N$. Figure 3(a),
where the damping rate is shown as a function of the interaction and the
number of bosons and, most important, Fig. 3(b), where we compare our numerics
with the experimental data finding very good agreement in the intermediate
range of interactions, allow for a new explanation of the experiment of Ref.
fertig05 , based on the role of lattices with different $N$.
Figure 2: Center of mass position as a function of time for the cases of Fig.
1(a-c) and $U/J=1$. The displacements $\delta$ are indicated in the figure.
The critical displacement $\delta_{\rm c}$ equals $\delta_{\rm c}=6$ and 2 in
panels (a-b) and (c), respectively.
Three regimes are of interest for the dipole dynamics [see Fig. 1]: $a$) for
$4J\gtrsim\Omega(N/2)^{2}$ the density distribution is Gaussian or Thomas-
Fermi-like for $4J\gg U$ and $4J\simeq U$, respectively, and for $U\gg 4J$
onsite densities are smaller than one; $b$) for $U>\Omega(N/2)^{2}>4J$ a Mott
insulator with one particle per site is formed at the trap center; $c$) for
$\Omega(N/2)^{2}>U>4J$ a shell structure is formed with a density $1<n_{j}\leq
2$ at the trap center, surrounded by a Mott-insulator with one particle per
site. All the situations above occur in the experiment, since $N$ varies from
one lattice to another. Therefore, in the following we are first interested on
the dynamics of model systems as those in Fig. 1, which exemplify all three
cases $a$), $b$) and $c$) above while still allowing for an extensive analysis
in terms of all parameters $N,\Omega/J$ and $U/J$, and then we address the
experiment of Ref. fertig05 in the most interesting regime $U/J\gtrsim 4$.
The results presented below have been obtained by means of a tDMRG algorithm
with a second order Trotter expansion of $H$, and time-steps $0.01J$ daley04 .
We take advantage of the conserved total number of particles $N$ projecting on
the corresponding subspace; the truncated Hilbert space dimension is up to
$m=100$, while the allowed number of particles per site is $D=5$. All results
below are found to be independent of this choice.
We first focus on the dipole dynamics as a function of the trap displacement
$\delta$, in the regime of weak interactions. In this regime, mean-field
theory predicts a sudden transition between undamped and overdamped motion via
a dynamical instability at a critical displacement $\delta_{\rm
c}\simeq\sqrt{2J/\Omega}$ SmerziPRL . This value for $\delta_{\rm c}$ can be
understood by employing the exact solution of Eq. (1) in the non-interacting
limit AnaPRA05 . For energies $E\lesssim 4J$ the single-particle eigenstates
of $H(t=0)$ are harmonic-oscillator-like modes extended around the center of
the parabolic trap. However, for $E>4J$ particles are Bragg-scattered by the
lattice, and perform Bloch-like-oscillations centered far from the trap center
RigolPRA . The particle localization corresponds to the population of these
latter high-energy modes, which becomes significant for displacements
$\delta\gtrsim\delta_{\rm c}$, AnaPRA05 . Our numerical results in the limit
of weak interactions are shown in Fig. 2(a-c), where dipole oscillations of
the center of mass $x_{\rm com}$ are shown as a function of time $t$, for
different values of the displacement $\delta$. In the simulations, as initial
condition we use the ground-state wavefunction of the undisplaced potential,
shifted by $\delta$ lattice sites. On increasing $\delta$, the dynamics
changes from undamped to damped, and the particles oscillate around the trap
center. On increasing further the displacement [$\delta\gtrsim 5$ in panels
(a-b)] the oscillations are overdamped, and the COM slowly drifts towards the
trap center or clings to the borders of the trap [case with $N=23$ of panel
$c)$]. This behavior corresponds to the localization transition predicted by
mean-field theory. However, Fig. 2 shows that quantum fluctuations, properly
accounted for by the tDMRG, smear out the transition into a smooth crossover
between the undamped and the overdamped regimes.
Having established a connection with known results in the mean-field regime,
we now present exact results for the particle localization in the interesting
case of stronger interactions $U/J\gtrsim 1$ and $\delta\lesssim\delta_{\rm
c}$. We first focus on model systems and fix $\delta_{\rm c}=6$ and the
displacement $\delta=1<\delta_{\rm c}$, such that for small interactions
$U/J\lesssim 1$ the dynamical instability discussed above does not occur, e.g.
for $U/J=1$ the dipole oscillations are undamped for all $N$, see Figs.
2(a)-(b). The dipole dynamics is then studied as a function of the ratio
$U/J$. In particular, Fig. 3(a) shows the damping rate $\Gamma$ of the dipole
oscillations as a function of $U/J$ for $N=11,15$ and 28 [exemplifying cases
$a),b)$ and $c)$ above]. Here, $\Gamma$ is calculated using the expression for
underdamped oscillations $x_{\rm com}(t)=e^{-\Gamma t}[1-\cos(\Omega
t+\phi_{0})]+y_{0}$, with $\Gamma$, $\phi_{0}$ and $y_{0}$ fitting parameters.
Three key observations are in order. i) The damping rate increases
exponentially with $U/J$ for intermediate interaction strengths $2\lesssim
U/J\lesssim 6$, a result which is not captured by mean-field, and is
significantly larger than what predicted using phase-slip techniques, valid
for $U\lesssim 1$ Polko ; AnaPRA05 . ii) Eventually for large enough
interactions ($U/J\sim 6$) the oscillations are overdamped for all $N$. We
find that for the cases $N=15$ and $28$, this overdamping corresponds to the
formation of a Mott-state and a cake-structure as in Fig. 1(b) and (c),
respectively. In particular, for $N=15$ the particle localization occurs for
$U/J\approx 4$, a value remarkably close to the superfluid/Mott-insulator
quantum phase transition in an homogeneous lattice at commensurate filling and
zero current. That is, the results for $\delta<\delta_{\rm c}$ naturally
interpolate between the finite-current dynamical instability and the zero-
current quantum phase transition SmerziPRL . iii) Despite the Mott-formation
for large $N$, for a given $U/J$ the damping $\Gamma$ is actually larger for
smaller N, such that for $N=11$ the dynamics is frozen already for $U/J<4$. In
the following we show that this has crucial consequences for the
interpretation of the results of Ref. fertig05 in the most interesting regime
of interactions $U/J\sim 4$.
In the experiment of Ref. fertig05 , the decay of dipole oscillations was
studied as a function of the optical lattice depth $V_{0}$ for a fixed
displacement $\delta=8$, finding damping already for weak lattices
$V_{0}/E_{R}>0.5$, with $E_{R}$ the recoil energy. The experimental data are
shown as black dots in Fig. 3(b) as a function of $V_{0}$ in the range
$2\lesssim V_{0}/E_{R}\lesssim 5$, where the use of Eq. (1) is justified
Jaksch98 ; AnaPRA05 , corresponding to the interesting regime of interactions
$3\lesssim U/J\lesssim 8$. For $V_{0}/E_{R}=3$ and $V_{0}/E_{R}>3$ the value
of the damping rate $\Gamma$ has been extracted using formulas appropriate for
underdamped and overdamped motion, respectively fertig05 . The most
interesting experimental finding shown in Fig. 3(b) is the measurement of an
abrupt transition from a weakly damped regime to an overdamped regime for a
lattice depth $V_{0}/E_{R}\simeq 3$, where the damping rate $\Gamma$ of the
dipole oscillations increases by more than an order of magnitude. The physical
mechanism behind this apparent transition has proven elusive.
Figure 3: (a) Numerical results for the damping rate $\Gamma$ of the dipole
oscillations vs $U/J$ for a fixed displacement $\delta=1<\delta_{\rm c}$, with
$\delta_{\rm c}=6$ ($\Omega/J=0.05623$) and $N=11,15$ and 28 [cases (a-c) in
the text]; (b) Damping rate $\Gamma$ for the experiment of Ref. fertig05 vs
$U/J$ and the lattice depth $V_{0}/E_{R}$. The experimental data, and the
numerical results for $N=45,80$ are the black dots, the red squares and the
green diamonds, respectively.
In Fig. 3(b) the experimental results are compared to our numerical results
for $N=80$ and 45, green diamonds and red squares, respectively. The value
$N=80$ has been chosen since it corresponds to the number of particles in the
central 1D lattice of the array in the experiment, which is the most largely
populated with $\langle n_{j}\rangle>1$ for all $U/J$, as in Fig. 1(c).
Conversely, the case $N=45$ exemplifies case (b), with $\langle
n_{j}\rangle\lesssim 1$ for $U/J\gtrsim 4$. The figure shows a very good
agreement between the numerical and the experimental results in the entire
region $2\lesssim V_{0}/E_{R}\lesssim 5$ ($3\lesssim U/J\lesssim 8$). However,
the case $N=80$ slightly underestimates the damping around $V_{0}/E_{R}\simeq
4$, while the agreement for $N=45$ is almost perfect. For $V_{0}/E_{R}\gtrsim
5$ all numerical results fall inside the experimental errorbars, however, the
case $N=45$ shows a strong damping, while the case $N=80$ falls in the middle
of the experimental errorbars. The explanation of the results above stems from
the observation that in the experiment $\delta_{\rm c}$ varies between
$\delta_{\rm c}\sim 18$ and 15 for $3\lesssim V_{0}/E_{R}\lesssim 5$, and thus
$\delta<\delta_{\rm c}$ for all lattice depths. We can then use the results
for the model systems of Fig. 3(a) to explain the experimental findings. That
is: i) the transition observed experimentally at $V_{0}/E_{R}\simeq 3$ is
actually a crossover, where the 1D systems with the lowest number of particles
tend to localize first, in agreement with the discussion of Fig. 3(a). ii) For
$V_{0}/E_{R}\gtrsim 5$, the dynamics of particles in the 1D systems with
$\langle n_{j}\rangle\leq 1$ ($N=45$ in the simulations) is completely frozen,
and the overall mobility of the cloud is due to residual oscillations in
lattices with higher onsite density. This latter observation is in agreement
with the results of Ref. PupilloNJP06 , where it is shown that for
$V_{0}/E_{R}>5$ the damping rate observed in the experiment is well reproduced
by the results for $N=80$. We notice that numerical results for $N=80$
consistent with ours have been recently reported in Clark08 , however the
focus here is on a comprehensive explanation of the experiment fertig05 .
The different behaviors of $\Gamma$ for $N=45$ and 80 and $U/J>4$ can be
modeled as follow. In the low-density case with $N=45$ the tendency to
localization is explained by noting that interactions broaden the spatial
width of the atom cloud, until the onsite density falls below one [see also
Fig. 1(a-b)]. In this case, the low-energy physics maps into that of an
extended cloud of non-interacting fermions, with single-band Hamiltonian
AnaPRA05
$\displaystyle\tilde{H}_{1}(t)=-J\sum_{<i,j>}c^{\dagger}_{i}c_{j}+\Omega\sum_{j}[j-\delta(t)]^{2}c^{\dagger}_{j}c_{j},$
with $c_{j}$ and $c^{\dagger}_{j}$ fermionic operators. For large enough
displacements $\delta$, the fermions largely occupy localized modes of the
single-particle spectrum discussed above, and the COM remains frozen. The
dynamics of interacting particles at large density, e.g. $N=80$ in Fig. 3(b),
can be modeled starting from the case of largest interactions $U/J\gg 1$,
where the density profile has a cake-like structure, Fig. 1(c). This situation
is well described by an extended fermionization model PupilloNJP06 ;
PupilloPRA06 ; Popp06 , where Eq. (1) is replaced by an effective Hamiltonian
with two coupled Fermi bands separated by an energy $U$ Popp06
$\displaystyle\tilde{H}_{2}(t)=-J\sum_{<i,j>}[c^{\dagger}_{i}c_{j}+2d^{\dagger}_{i}d_{j}+\sqrt{2}(c^{\dagger}_{i}d_{j}+d^{\dagger}_{i}c_{j})]$
$\displaystyle+\sum_{j}(\Omega[j+\delta(t)]^{2}c^{\dagger}_{j}c_{j}+\\{\Omega[j+\delta(t)]^{2}+U\\}d^{\dagger}_{j}d_{j}),$
(2)
with the operators $c_{j}$, $c^{\dagger}_{j}$ and $d_{j}$, $d^{\dagger}_{j}$
referring to the lower and higher energy bands of width $4J$ and $8J$,
respectively. Oscillations in this limit are due to the dynamics of the
(delocalized) $d_{j}$-fermions of Eq. (2) in the higher-energy band, while
$c_{j}$-fermions are frozen in a (band) insulator. Observing these residual
oscillations thus corresponds to probing the superfluidity of bosons with two-
particles per site in a homogeneous lattice, in a local-density-approximation
sense Fisher . This picture, valid for $U/J\gg 1$ PupilloNJP06 ; Popp06 , can
be extended to gain a qualitative insight in the dependence of the dipole
oscillations on interactions for $4\lesssim U/J\lesssim 10$. In fact,
neglecting the parabolic potential, in this regime the model of Eq. (2)
suggests that the spectrum is continuum, since the gap $U$ between the two
Fermi bands is smaller than their total width. It is thus plausible that
Bloch-like oscillations of the particles are here suppressed, and transport
restored. However, for $U\gtrsim 12J$ the energy spectrum develops a gap again
around $4J$, and thus transport in the lower-energy band is inhibited.
Residual current is then due to delocalized particles in the higher-energy
band, as explained above. We notice that this picture is consistent with our
numerical findings for $U/J>5$ in Fig. 3(b).
In conclusion, we have explained the experiment in fertig05 in the most
interesting regime of intermediate interactions. The very good agreement
between experimental and tDMRG results demonstrates the latter as a unique
tool for quantitative comparisons with cold gases experiments in the strongly
correlated regime in one dimension.
Discussions with A.M. Rey, C.J. Williams and C.W. Clark are gratefully
acknowledged. This work was supported by OLAQUI, NAMEQUAM, FWF, MURI, EUROSQIP
and DARPA and developed using the DMRG code released within the PwP project
(www.dmrg.it).
## References
* (1) L. Pezzè et al., Phys. Rev. Lett. 93, 120401 (2004); N. Strohmaier et al., ibid. 99, 220601 (2007); J. Mun, et al., ibid. 99, 150604 (2007); L. E. Sadler et al., Nature 443, 312 (2006).
* (2) T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).
* (3) T. Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004).
* (4) C. D. Fertig et al., Phys. Rev. Lett. 94, 120403 (2005).
* (5) M. Rigol et al., Nature 452, 854-858 (2008); Phys. Rev. Lett. 98, 050405 (2007); C. Kollath et al., ibid. 98, 180601 (2007); S.R. Manmana et al., ibid. 98, 210405 (2007).
* (6) A. J. Daley et al., J. Stat. Mech.: Theor. Exp. P04005 (2004); S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004); U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005); G. De Chiara et al. J. Comput. Theor. Nanosci. 5, 1277 (2008).
* (7) A. Polkovnikov and D.-W. Wang, Phys. Rev. Lett. 93, 070401 (2004); M. Rigol et al., ibid. 95, 110402 (2005); J. Gea-Banacloche et al., Phys. Rev. A 73, 013605 (2006); A.V. Ponomarev and A.R. Kolovsky, Laser Phys. 16, 367 (2006); M. Snoeck and W. Hofstetter, Phys. Rev. A 76, 051603(R) (2007).
* (8) J. Ruostekoski and L. Isella, Phys. Rev. Lett. 95, 110403 (2005).
* (9) G. Pupillo et al., New J. Phys. 8, 161 (2006).
* (10) Recently a complementary analysis appeared in I. Danshita and C. W. Clark, arXiv:0807.2898 (2008) (see text).
* (11) D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998).
* (12) A. Smerzi et al., Phys. Rev. Lett. 89, 170402 (2002); E. Altman et al., ibid. 95, 020402 (2005).
* (13) A. M. Rey et al., Phys. Rev. A 72, 033616 (2005).
* (14) M. Rigol and A. Muramatsu, Phys. Rev. A 70, 031603 (2004); C. Hooley and J. Quintanilla, Phys. Rev. Lett. 93, 080404 (2004).
* (15) A. Polkovnikov et al., Phys. Rev. A 71, 063613 (2005); D. McKay et al., Nature 453, 76 (2008).
* (16) H. Ott et al., Phys. Rev. Lett. 93, 120407 (2004); A. V. Ponomarev et al., ibid. 96, 050404 (2006).
* (17) A. Buchleitner and A.R. Kolovsky, Phys. Rev. Lett. 91, 253002 (2003).
* (18) G. Pupillo, C.J. Williams, and N.V. Prokof’ev, Phys. Rev. A 73, 013408 (2006).
* (19) M. Popp et al., New J. Phys. 8, 164 (2006).
* (20) M.P.A. Fisher et al., Phys. Rev. B 40, 546 (1989).
|
arxiv-papers
| 2008-10-09T14:45:40
|
2024-09-04T02:48:58.205405
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Simone Montangero, Rosario Fazio, Peter Zoller, Guido Pupillo",
"submitter": "Simone Montangero",
"url": "https://arxiv.org/abs/0810.1665"
}
|
0810.1991
|
# A Global Physician-Oriented Medical Information System
Axel Boldt Department of Mathematics
Metropolitan State University
St. Paul, MN 55106, USA Axel.Boldt@metrostate.edu and Michael Janich B2 F5
Vantage Park
22 Conduit Rd
Mid Levels
Hong Kong michael@janich.com
(Date: 8 October 2008)
###### Abstract.
We propose an Internet-based, free, world-wide, centralized medical
information system with two main target groups: practicing physicians and
medical researchers. After acquiring patients’ consent, physicians enter
medical histories, physiological data and symptoms or disorders into the
system; an integrated expert system can then assist in diagnosis and
statistical software provides a list of the most promising treatment options
and medications, tailored to the patient. Physicians later enter information
about the outcomes of the chosen treatments, data the system uses to optimize
future treatment recommendations. Medical researchers can analyze the
aggregate data to compare various drugs or treatments in defined patient
populations on a large scale.
###### Key words and phrases:
patient records; internet; diagnostic assistance; treatment recommendations;
open source software
## 1\. Introduction
The two main tasks performed by physicians are:
* •
diagnosing a disorder, based on presented symptoms, the patient’s medical
history and features, and ordered tests; and
* •
choosing the most appropriate treatment or medication for a given disorder and
a given patient.
In this article we propose a medical information system which aims to assist
physicians in both these tasks.
The ever increasing number of recognized diseases, combined with an explosion
in the number of marketed medications, poses formidable challenges to the
practicing physician. Many physicians rely mainly on four information sources:
their often outdated text books and lecture notes, a small selection of
medical journals, possibly biased informational material from pharmaceutical
companies, and their personal or anecdotal experiences. All of these are less
than ideal, and we maintain that they can and should be supplemented by more
rational decision aids based on modern data mining technology.
The core idea is simple: an expert system and treatment database that
physicians access over the Internet. After acquiring informed consent, they
enter a patient’s physiological data, symptoms and test results and the expert
system aids in diagnosis or recommends further tests. Once the disorder is
identified, the system recommends those treatments or medications with the
highest success probability for that particular patient. The physician chooses
a treatment and later records the outcome in the database. This outcome data
is used by the system to improve future treatment recommendations.
The system is designed to be used by physicians world wide, with special
regard for those working in developing countries. The use will be free of
charge and will only require an ordinary modem-speed connection to the
internet, something that is now available at reasonable cost in most
countries, often via mobile phones.
Since cost is often an important criterion when choosing a treatment or
medication, the system’s recommendations will be accompanied by cost estimates
specific to the physician’s location.
The two components of the system, diagnostic expert system and treatment
recommendation engine, use the same underlying patient database but are
logically independent and the system can go online as soon as one of the two
is fully functional. For example, physicians can eschew the expert system
altogether, rely on their own diagnostic skills and directly ask for treatment
recommendations for a particular patient’s disorder.
Apart from aiding physicians in their decision processes and thus improving
care and lowering cost on a global scale, the collected data will also provide
a rich resource for medical researchers. It will be possible to easily
determine the effectiveness of treatments and drugs in various patient
populations defined by combinations of characteristics such as age, sex,
ethnic group, pre-existing conditions, lifestyle, or any of a large number of
physiological measurements recorded by the system. In addition, the system
will be immediately useful by helping to compare new, expensive and heavily
marketed medications with older, more established alternatives. Evaluation of
alternative treatments like acupuncture or herbal remedies, heavily used all
over the world but rarely studied in a rigorous manner, will also become much
easier. Newly emerging epidemics will be detected in real time, much earlier
than is possible today. Lastly, it is likely that mining the database will
uncover rare but severe side effects of established medications that have so
far escaped detection.
## 2\. Detailed Description
### 2.1. Recruitment of physicians
The system will be open to all licensed practicing physicians world-wide.
Apart from the costs of a regular Internet connection, use of the system will
be free. Rather than attempting to contact all physicians directly, or to
advertise in relevant publications, it is hoped that the national medical
associations of the various countries can be recruited to promote the system
to their members and provide them with authorized access codes. This approach
has three advantages over a more direct marketing campaign:
* •
it is significantly cheaper;
* •
verification of physician’s credentials is done by the organizations best
suited for the task;
* •
physicians extend a natural goodwill bonus to communications from their
respective medical association.
### 2.2. Physicians’ interaction with the system
Physicians are provided with access codes (passwords). They will normally
interact with the system by accessing the project’s website and logging in
with their name and password. The website will follow W3-standards and carry
few graphics in order to be easily accessable through slow modem connections
and simple devices, including mobile devices. It will be designed to make the
most common interactions simple and fast.
When accessing the system for the first time, new users are directed to a
tutorial about the system’s features and are required to agree to a set of
rules, mainly pertaining to patients’ informed consent and privacy (discussed
below in section 2.3).
In the standard use case, after authentication physicians are presented with a
list of their patients and past treatments and are invited to input outcome
data for these treatments in a quick and simple manner. They are then able to
view their patients’ records, create new records, use the expert system to
receive diagnostic assistance, or use the statistical database to get
treatment/medication recommendations. The top-rated treatments will be
presented along with estimates of their success probability and their cost.
The system further facilitates the physician’s decision process by providing
easy access to relevant background information, such as as reviews from the
Cochrane Library ([2]) and medical guidelines from the clearinghouse
responsible for the physician’s location ([1], [5]).
Signs, symptoms and diseases will be entered using the well-known ICD-10
classification system ([3]), and information about medications will be
accessable through the standard Anatomical Therapeutic Chemical Classification
System ([4]). It is to be expected that these systems will not be entirely
adequate for our purposes and will have to be modified and extended to a
certain degree.
In addition to the direct web-based interface, the system will also provide an
XML-based interface, allowing for the easy data exchange with physicians’
other software applications, for instance with their standard patient
management software or with their systems of interfacing with health insurers.
All patient information data transport will employ a layer of encryption to
avoid data being viewed or tampered with by unauthorized third parties.
### 2.3. Patient identifiers and privacy issues
The system will obey the world’s strictest privacy laws, like those commonly
found in countries of the European Union. This requires in particular that
* •
data is collected with full consent and can only be used for the express
purposes given in the consent statement; and
* •
patients retain the right to review their data and to have it removed at any
time.
Physicians are required to obtain informed consent before they may enter
patients’ data into the system. Physicians are not allowed to reject treatment
of patients who do not wish to participate in the system.
No personally identifiable information is ever entered into the database: no
names, no exact birthdates, no addresses etc. Instead, every patient is
identified by a PatientID, a simple number. The physician provides the patient
with their PatientID, so that the patient can authorize other physicians to
access their data if they so choose. Only physicians explicitly authorized by
a patient may access the records associated with that patient’s PatientID;
physicians are required to keep proof of this authorization on file.
Physicians will typically keep a record of all their patients’ PatientIDs on
their own computer.
It is possible and even to be expected that patients’ names together with
corresponding PatientIDs will occasionally fall into the wrong hands, for
instance if malware is installed on a physician’s computer, their office is
broken into or a patient’s private PatientID note is stolen. Since only
physicians associated with the PatientID may access the corresponding record,
the risk of data leaks is actually lower than in the current situation, where
all patient data would have been stored on the doctor’s compromised computer
rather than in the online database.
Physicians, unlike patients, do not remain anonymous and are recorded with
full name and address in the database. Further, all interactions of physicians
with the system are logged. This allows to trace and identify fraudulent use
by non-physicians.
During scientific analysis the database will only be queried in the aggregate
and no individual record will be accessed in its entirety. It is however
important to note that this does not completely protect against abuses; for
instance a query of the form ”what percentage of adult black males living in
Luxembourg and standing less than 1.6 m tall are HIV positive”, even though it
uses the database only in the aggregate, can still reveal very private
information about a small number of identifiable individuals. It is therefore
necessary that scientific analyses be reviewed ahead of time; see section 2.5
below for more details on this process.
### 2.4. Software
To maximize transparency and to encourage others to submit bug fixes and
feature enhancements, the project will make use of existing Open Source
software whenever possible and all software written by the project will be
published under an Open Source license ([7]).
Diagnostic expert systems have been developed before ([8]; [6] for a
comprehensive annotated list), and it is unrealistic to expect the project to
duplicate that work. Instead, existing expert systems will be evaluated and
the most appropriate one will be licensed and extended. Most modern systems of
this type employ Bayesian networks, and these would benefit tremendously from
the statistical knowledge stored in the patient database.
The treatment recommendation engine will have to be written from scratch,
using existing algorithms that have been developed for data mining
applications in marketing; given a database containing customer features and
outcomes of past marketing strategies, these algorithms can predict the most
promising marketing strategy taylored to a given customer. (See [9] for a
survey of data mining algorithms.) The algorithm will have to account for the
fact that information about patients is not necessarily complete; for example,
only some patient records will contain a recent blood glucose level.
As is usual in data mining applications, the comparatively low quality of data
collection procedures is compensated for by the large quantity of data. A
variety of different heuristics can be used to approach the recommendation
problem and a final algorithm can only be chosen after empirical evaluation.
As mentioned above, the system requires information about the prices of
treatments and medications in the various countries. It is possible and
desirable that the collection and maintenance of this data be carried out in
collaboration with the various national medical associations. The problem is
non-trivial, since drug prices can vary widely even within countries.
### 2.5. Scientific analysis of the collected aggregate data
For reasons outlined in section 2.3 above, it is necessary that all research
proposals be reviewed ahead of time. Credentialed medical researchers may
submit research proposals explaining the study’s rationale, accompanied by the
software that is to analyze the database. After review of proposal and
software, the software is run against the live database and the results are
returned to the researcher. Research proposal, analysis software and results
are made public to ensure that negative results are not suppressed. The entire
process is free of charge for the researcher.
Toy systems with the same database structure as the live system (but without
real-world data) are provided to researchers, so that their analysis software
can be tested and debugged ahead of time. The use of free software, as
described in section 2.4 above, will hopefully result in a rich ecosystem of
analysis software freely shared among researchers.
Every statistical analysis of the system s database needs to take into account
that the system does not and cannot guarantee that different PatientIDs always
correspond to different patients. Furthermore, conclusions based on analysis
of the database are of course valid only for the subpopulation of patients who
are cared for by participating physicians and who have given consent to
participating in the system; this may not be a representative sample. External
studies comparing this subpopulation to the general patient population will be
highly desirable.
As with all data mining applications, correlations discovered in the data will
have to be confirmed by traditional randomized trials.
### 2.6. Organizational structure and funding
The system will be developed, deployed and run by a non-profit organization,
to be set up in a jurisdiction that grants tax-free status to such
organizations and that has strong privacy protection laws. The organization
will be assisted by a board of external advisors.
To maintain neutrality and to avoid unduly influences on physicians decision
processes, the system’s website will not carry any advertisings. The project
will be financed completely by donations and grants. These may come from
individuals, companies (especially health insurers), charity foundations, or
national or international health organizations. It is possible that an
organization such as the NIH’s National Library of Medicine can be convinced
to host and run the system. An alternative model of funding, especially once
the system is established and accepted, would have the governments of rich
participating countries pay a (small) set amount per patient.
## 3\. Conclusion and Outlook
The proposed system, once fully implemented, will improve world-wide medical
care in a number of important ways:
* •
practicing physicians, even in developing countries, will receive easy and
free access to a medical expert system that can assist in diagnosis;
* •
physicians receive promising treatment options, tailored to the patient, based
on past collected outcome data;
* •
inclusion of price data for treatments and medications allows physicians to
choose the most cost-effective option in any given situation;
* •
with patients’ consent, medical histories stored in the database are easily
transferrable from one physician to another;
* •
creative use of the collected aggregate data will allow medical researchers to
identify subpopulations of patients that respond particularly well to a
certain medication or treatment;
* •
comparisons of new drugs with established generic medications for the same
condition become straightforward.
In the future, the system can be extended to incorporate patients’ genotype
data, thereby representing an important step towards the longstanding goal of
truly personalized medicine.
In addition to these quite concrete and immediate benefits, we would like to
express our hope that widespread adoption of the system will cause the
profession of physician to evolve: from a passive container of knowledge about
symptom-disease correlations and disease-treatment success probabilities to an
active partner of the patient who reassures, explains disorders and
treatments, inquires and provides advice about the patient s life
circumstances and in general maximizes the placebo effect in every way
possible. This, we believe, will ultimately turn out to be one of the main
benefits of the proposed system: the placebo effect, long considered a quirky
nuisance by western medicine, will return to its rightful place at the center
of the healer s work. Traditional systems of medicine will have much to teach
to physicians freed from the more mundane tasks of their profession.
## References
* [1] G. Ollenschl ger _et al._ Improving the quality of health care: using international collaboration to inform guideline programmes by founding the Guidelines International Network (G-I-N). _Quality & safety in health care._ December 2004; 13(6):455-60.
* [2] The Cochrane Collaboration. http://www.cochrane.org (accessed 27 September 2008).
* [3] World Health Organization. _The International Statistical Classification of Diseases and Health Related Problems ICD-10, Second Edition_ , 2004
* [4] WHO Collaborating Centre for Drug Statistics Methodology. _Anatomical Therapeutic Chemical Classification System._ 2008\.
http://www.whocc.no/atcddd/ (accessed 27 September 2008).
* [5] Hatsek A, Young O, Shalom E, Shahar Y. DeGeL: a clinical-guidelines library and automated guideline-support tools. _Studies in health technology and informatics_ 2008;139:203-12.
* [6] Judith Federhofer. _Medical Expert Systems. Doctor’s Silent Partners._
http://www.computer.privateweb.at/judith/index.html (accessed 31 December
2006, now only available through http://www.archive.org/).
* [7] Andres M. St. Laurent. _Understanding Open Source and Free Software Licensing._ O’Reilly Media, 2004.
* [8] E. Coiera. _The Guide to Health Informatics (2nd Edition)._ Arnold, London, October 2003.
* [9] David Hand, Heikki Mannila, Padhraic Smyth. _Principles of Data Mining._ MIT Press, 2001
|
arxiv-papers
| 2008-10-11T02:01:45
|
2024-09-04T02:48:58.216757
|
{
"license": "Public Domain",
"authors": "Axel Boldt and Michael Janich",
"submitter": "Axel Boldt",
"url": "https://arxiv.org/abs/0810.1991"
}
|
0810.2081
|
# Capabilities of the CMS detector for studies of hard probes in heavy ion
collisions at the LHC
(for the CMS Collaboration)
D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State
University, Moscow, Russia
E-mail
###### Abstract:
The capabilities of the CMS experiment to study properties of hot and dense
QCD-matter created in heavy ion collisions at the CERN Large Hadron Collider
with the perturbative processes (so-called ”hard probes”) are presented.
Detailed studies from complete simulations of the CMS detectors in Pb+Pb
collisions at $\sqrt{s}=5.5$ TeV per nucleon pair are presented in view of two
hard probes: quarkonium and $\gamma$-jet production.
## 1 Introduction
The study of the fundamental theory of the strong interaction (Quantum
Chromodynamics, QCD) in new, unexplored extreme regimes of super-high
densities and temperatures is one of the primary goals of the modern high
energy physics. The experimental and phenomenological study of multiple
particle production in ultrarelativistic heavy ion collisions is expected to
provide valuable information on the (thermo)dynamical behaviour of QCD matter
in the form of a quark-gluon plasma (QGP), as predicted by lattice QCD
calculation. A detailed description of the potential of CMS to carry out a
series of representative Pb-Pb measurements has been presented in [1].
Heavy ion observables accessible to measurement with CMS include:
* •
“Soft” probes [2]: global particle and energy rapidity densities, elliptic
flow and spectra of low transverse momentum hadrons. These observables are
mostly sensitive to the space-time evolution of the system once thermalization
has set in and thus carry information about the thermodynamical properties of
the produced QCD matter.
* •
“Hard” probes: quarkonia, heavy quarks, jets, $\gamma$-jet and high-$p_{T}$
hadrons, which are produced with high transverse momenta $p_{T}$ or large
masses $M$ (much greater than the typical QCD scale of confinement: $p_{T}$,
$M$ $\gg$ $\Lambda_{\rm QCD}=200$ MeV). The hard probes production cross
sections can be described in the framework of perturbative QCD theory. Such
hard particles are produced in the very early stages of the evolution of the
system and thus are potentially affected by final-state interactions as they
traverse the produced medium. Modifications with respect to the “vacuum QCD”
spectra and cross-sections measured in proton-proton collisions, provide
direct information on the dynamical and transport properties of the system:
initial parton densities, transport coefficient of the medium, critical energy
density.
In this contribution, the detailed study for a complete simulation of the CMS
detectors in Pb+Pb collisions at $\sqrt{s}=5.5$ TeV per nucleon pair are
presented in view of two hard probes: quarkonium and $\gamma$-jet production.
The results of similar studies for high-$p_{T}$ hadrons are presented during
this Workshop in another talk [3].
## 2 CMS detector
CMS is a general purpose experiment at the LHC designed to explore the physics
at the TeV energy scale [4]. Since the CMS detector subsystems have been
designed with a resolution and granularity adapted to cope with the extremely
high luminosities expected in the proton-proton running mode, CMS can also
deal with the large particle multiplicities anticipated for heavy-ion
collisions. A detailed description of the detector elements can be found in
the corresponding Technical Design Reports [5, 6, 7, 8]. The central element
of CMS is the magnet, a 13 m long, 6 m diameter, high-field solenoid (a
uniform 4 T field) with an internal radius of $\approx 3$ m.
The tracker covers the pseudorapidity region $|\eta|<2.5$ and is composed of
two different types of detectors: silicon pixels and silicon strips. The pixel
detector consists of three barrel layers located at 4, 7, 11 cm from the beam
axis with granularity $150\times 150$ $\mu$m2 and two forward layers with
granularity $150\times 300$ $\mu$m2 located at the distances of 34 and 43 cm
in z-direction from the center of detector. Silicon strip detectors are
divided into inner and outer sections and fill the tracker area from 20 cm to
110 cm (10 layers) in the transverse direction and up to 260 cm (12 layers) in
longitudinal direction.
The hadronic (HCAL) and electromagnetic (ECAL) calorimeters are located inside
the coil (except the forward calorimeter) and cover (including the forward
calorimeter) from $-$5.2 to 5.2 in pseudorapidity. The HF calorimeter covers
the region $3<|\eta|<5.2$.
The CMS muon stations cover the pseudorapidity region $|\eta|<2.4$ and consist
of drift tube chambers (DT) in the barrel region (MB), $|\eta|<1.2$, cathode
strip chambers (CSCs) in the endcap regions (ME), $0.9<|\eta|<2.4$, and
resistive plate chambers (RPCs) in both barrel and endcaps, for $|\eta|<2.1$.
The RPC detector is dedicated to triggering, while the DT and CSC detectors,
used for precise momentum measurements, also have the capability to self-
trigger up to $|\eta|<2.1$.
Figure 1: CMS coverage for tracking, calorimetry, and muon identification in
pseudo-rapidity ($\eta$) and azimuth ($\phi$). The size of a jet with cone
$R=0.5$ is also depicted for comparison.
Note that CMS is the largest acceptance detector at the LHC (Fig. 1) with
unique detection capabilities also in the very forward hemisphere with the
CASTOR (5.1 $<|\eta|<$ 6.6) and the Zero-Degree (ZDCs, $|\eta_{neut}|>$ 8.3)
calorimeters [9].
Another key aspect of the CMS hard probe capabilities for heavy ion physics is
its unparalleled high-level-trigger (HLT) system running on a filter farm with
an equivalent of ${\cal O}$(104) 1.8 GHz CPU units, yielding few tens of
Tflops [10]. The HLT system is powerful enough to run “offline” algorithms on
every single Pb+Pb event delivered by the level-1 trigger, and select the
interesting events while reducing the data stream from an average 3 kHz L1
input/output event rate down to 10–100 Hz written to permanent storage. The
resulting enhanced statistical reach for hard probes is a factor of $\times$20
to $\times$300 larger, depending on the signal, than for the min-bias (MB)
trigger.
## 3 Simulation and analysis of J/$\psi$ and $\Upsilon$ production
One of the important hard probes at the LHC will be the production of heavy-
quark bound states, which should be suppressed in QGP due to colour screening
[11]. An intriguing phenomenon is the “anomalously” strong suppression of the
J/$\psi$-meson yields, observed in Pb+Pb collisions at SPS [12, 13]. Although
the interpretation of this phenomenon as a result of the formation of a QGP is
quite plausible, alternative explanations have also been put forward, such as
rescattering on co-moving hadrons. The surprisingly similar amount of J/$\psi$
suppression observed at SPS and RHIC energies [14] is not yet fully
understood. Further information on the nature of quarkonia suppression in hot
and dense QCD-matter will come from the results at the LHC at the much higher
temperatures accessible. CMS features the best dimuon mass resolution of any
LHC detector, leading to a clean separation of the various quarkonia states
and an improved signal over background ratio. This fact will open up a unique
opportunity to study the threshold dissociation behaviour of the whole
bottomonium family ($\Upsilon$, $\Upsilon`$, $\Upsilon``$) together with the
charmonium one. Since various quarkonium states are predicted to melt at
different medium temperatures, scan of corresponding suppression factors will
serve as an effective QCD-matter “thermometer”.
Event generator HIJING [15] with full GEANT4-based simulation of the tracking
of secondaries and simulated detector response were used in the analysis
presented. The J/$\psi$ and $\Upsilon$ acceptances on CMS are shown as a
function of $p_{T}$ in Fig. 2 for two $\eta$ ranges: full detector
($|\eta|<2.4$) and central barrel ($|\eta|<0.8$). Because of their relatively
small mass, low momentum J$/\psi$’s ($p<4$ GeV/$c$) are mostly not accepted:
their decay muons do not have enough energy to traverse the calorimeters and
coil, and are absorbed before reaching the muon chambers. The J/$\psi$
acceptance increases with $p_{T}$, flattening out at $\sim$ 15% for $p_{T}>12$
GeV/$c$. The $\Upsilon$ acceptance starts at $\sim\,$40% at $p_{T}=0$ GeV/$c$
and remains constant at $\sim$ 15% (full detector) or 5% (barrel only) for
$p_{T}>4$ GeV/$c$. The $p_{T}$-integrated acceptance is about 1.2% for the
J/$\psi$ and 26% for the $\Upsilon$, assuming the input theoretical
distributions.
In the central barrel of the CMS detector, the dimuon reconstruction
efficiency remains above 80% for all multiplicities whereas the purity
decreases slightly with increasing multiplicities $dN_{ch}/d\eta$ but also
stays above 80% even at as high as $dN_{ch}/d\eta|_{\eta=0}=6500$. If (at
least) one of the muons is detected in the endcaps, the efficiency and purity
drop due to stronger reconstruction cuts. Nevertheless, for the
$dN_{ch}/d\eta|_{\eta=0}\approx 2000$ multiplicity realistically expected in
central Pb+Pb at LHC, the efficiency (purity) remains above 65% (90%) even
including the endcaps.
At the $\Upsilon$ mass, the dimuon mass resolution for muon pairs in the
central barrel, $|\eta|<0.8$, is 54 MeV/$c^{2}$. In the full pseudorapidity
range, the dimuon mass resolution is about 1% of the quarkonium mass: 35
MeV/$c^{2}$ at the J/$\psi$ mass and 86 MeV/$c^{2}$ at the $\Upsilon$ mass.
There is a slight dependence of the mass resolution on the event multiplicity.
Increasing the multiplicity from $dN/d\eta=0$ to 2500 degrades the mass
resolution of the reconstructed $\Upsilon$ from 86 to 90 MeV/$c^{2}$.
Figure 2: J/$\psi$ (top) and $\Upsilon$ (bottom) CMS acceptances (convoluted
with trigger efficiencies) as a function of $p_{T}$, in the full detector
($|\eta|<2.4$, solid line) and only in the central barrel ($|\eta|<0.8$,
dashed line).
Figure 3: Dimuon mass distributions within $|\eta|<2.4$ for Pb+Pb events with
$dN_{ch}/d\eta|_{\eta=0}=5000$ (top) and $dN_{ch}/d\eta|_{\eta=0}=2500$
(bottom) in the J/$\psi$ (left) and $\Upsilon$ (right) mass regions. The main
background contributions are also shown: $h$, $c$ and $b$ stand for $\pi+K$,
charm, and bottom decay muons, respectively.
Fig. 3 shows the opposite-sign dimuon mass distributions, for the high and low
multiplicity cases and full acceptance ($|\eta|<2.4$). The different quarkonia
resonances appear on top of a continuum due to the various sources of decay
muons: $\pi+K$, charm and bottom decays. Assuming that the CMS trigger and
acceptance conditions treat opposite-sign and like-sign muon pairs equally,
the combinatorial like-sign background can be subtracted from the opposite-
sign dimuon mass distribution, giving us a better access to the quarkonia
decay signals. The statistics of J/$\psi$ and $\Upsilon$,
$\Upsilon^{{}^{\prime}}$ and $\Upsilon^{{}^{\prime\prime}}$ with both muons in
$|\eta|<2.4$ region expected in one month of data taking are 140000, 20000,
5900 and 3500 correspondingly for the multiplicity
$dN_{ch}/d\eta|_{\eta=0}=5000$ and 180000, 25000, 7300, 4400 for the
multiplicity $dN_{ch}/d\eta|_{\eta=0}=2500$. The signal-to-background ratios
are 0.6, 0.07 for J/$\psi$ and $\Upsilon$’s for $dN_{ch}/d\eta|_{\eta=0}=5000$
correspondingly and 1.2, 0.12 for $dN_{ch}/d\eta|_{\eta=0}=2500$. The signal-
to-background ratio (the number of events) collected in one month for the
dimuons in J/$\psi$ and $\Upsilon$ mass regions with both particles in
$|\eta|<0.8$ region are 2.75 (12600) and 0.52 (6000) for
$dN_{ch}/d\eta|_{\eta=0}=5000$ and 4.5 (11600), 0.97 (6400) for
$dN_{ch}/d\eta|_{\eta=0}=2500$. The background and reconstructed resonance
numbers are in a mass interval $\pm\sigma$, where $\sigma$ is the mass
resolution.
These quantities have been calculated for an integrated luminosity of 0.5 nb-1
assuming an average luminosity $\cal L$ = $4\times 10^{26}$ cm-2s-1 and a
machine efficiency of 50%. The expected statistics are large enough to allow
further offline analysis for example in correlation with the centrality of the
collision or the transverse momentum of the resonance.
## 4 Photon-tagged jet production
Another important hard probe very sensitive to the initial conditions of the
produced QCD-matter in heavy-ion collisions, is the QCD jet production. It is
expected that final state in-medium interactions should reduce the energy of
the jet partons (“jet quenching”) and result in medium-modified jet
fragmentation [16]. Recent RHIC data on high-pT hadron production [17] are
consistent with jet quenching predictions. However full event-by-event
reconstruction of jets in heavy ion collisions is rather complicated in the
lower energy RHIC experiments. In CMS, large transverse momentum probes can be
isolated experimentally from the soft particle background of the collision. In
particular, full jet reconstruction and high-pT particle reconstruction in the
high multiplicity environment of a Pb+Pb collision are possible [1]. At the
LHC, the production rates for jet pairs with transverse energy $E_{T}>50$ GeV
are several orders of magnitude larger than at RHIC. Thus, high statistics
systematic studies will be possible in a controlled perturbative regime, far
beyond the limits of RHIC. The one of the important jet-related observables
accessible to study in heavy ion collisions at CMS will be the $\gamma$-jet
(and $Z$-jet) channel provides a very clean means to determine medium-modified
parton fragmentation functions (FFs) [18]. Since the prompt $\gamma$ is not
affected by final-state interactions, its transverse energy ($E_{T}^{\gamma}$)
can be used as a proxy of the away-side parton energy ($E_{T}^{\rm jet}\approx
E_{T}^{\gamma}$) before any jet quenching has taken place in the medium. The
FF, i.e. the distribution of hadron momenta, $1/{N_{\rm jets}}\,dN/dz$,
relative to that of the parent parton $E_{T}^{\rm jet}$, can be constructed
using $z=p_{T}/E_{T}^{\gamma}$ or, similarly, $\xi=-\ln
z=\ln(E_{T}^{\gamma}/p_{T})$, for all particles with momentum $p_{T}$
associated with the jet.
Figure 4: Generated (histogram) and reconstructed (points) fragmentation
functions as a function of $z$ (left) and $\xi$ (right) for quenched partons.
Statistical errors correspond to an integrated luminosity of 0.5 nb-1. The
estimated systematic error is represented as the shaded band. Figure 5:
Generated (histogram) and reconstructed (points) ratio of quenched (PYQUEN)
and unquenched (PYTHIA) fragmentation functions as a function of $\xi$.
Full CMS simulation-reconstruction studies of the $\gamma$-jet channel have
been carried out [19], where the isolated $\gamma$ is identified in ECAL
($R_{isol}$ = 0.5), the away-side jet axis ($\Delta\phi_{\gamma-jet}>$ 3 rad)
is reconstructed in ECAL+HCAL, and the momenta of hadrons around the jet-axis
($R_{jet}<$ 0.5) are measured in the tracker. The event generators PYTHIA [20]
(non-quenched partons) and PYQUEN [21] (quenched partons) were used to
simulate signal $\gamma$+jet events, and HYDJET [21] was used to model the
underlying (background) heavy ion event (also without or with jet quenching).
For this study, the $10$% most central Pb+Pb collisions were selected by the
impact parameter of the lead nuclei, yielding an average mid-pseudorapidity
density of about 2400 (2200) charged particles in the quenched (unquenched)
HYDJET background. In total, 4000 $\gamma$-jet events in the CMS acceptance
for $E_{T}^{\gamma}>70$ GeV and $\mid{\eta^{\gamma}}\mid<2$ and about 40000
(125000) QCD background events for the quenched (unquenched) case are
simulated. This corresponds to the expected yields for one running year of
Pb+Pb data taking with an integrated luminosity of 0.5 nb-1. The working point
for this analysis was set to 60% signal efficiency, leading to a background
rejection of about 96.5%, and to a signal-to-background ratio of $4.5$ for
$0-10$% central quenched Pb+Pb.
The obtained FFs for photon-jet events with $E_{T}^{\gamma}>70$ GeV — after
subtraction of the underlying-event tracks using a R = 0.5 cone transverse to
the jet — are shown in Fig. 4 for central quenched Pb+Pb collisions. Medium
modified FFs are measurable with high significance (the systematic
uncertainties being dominated by the low jet reconstruction efficiency for
$E_{T}^{\rm jet}$ = 30–70 GeV) in the ranges $z<$ 0.7 or 0.2 $<\xi<$ 5\. The
overall capability to measure the medium-induced modification of jet
fragmentation functions in the $\gamma$+jet channel can be illustrated by
comparing the fully reconstructed quenched fragmentation function to the
unquenched MC truth distribution (Fig. 5).
## 5 Summary
With its large acceptance, nearly hermetic fine granularity hadronic and
electromagnetic calorimetry, and good muon and tracking systems, CMS is an
excellent device for the study of hard probes (such as quarkonia, jets,
photons and high-pT hadrons) in heavy ion collisions at the LHC.
## 6 Acknowledgments
The author wishes to express the gratitude to the members of CMS Collaboration
for providing the materials and the organizers of the Workshop “High-pT
Physics at LHC” for the warm welcome and the hospitality. The author also
gratefully acknowledge support from Russian Foundation for Basic Research
(grants No 08-02-91001 and No 08-02-92496) and Grants of President of Russian
Federation (No 1007.2008.2 and No 1456.2008.2).
## References
* [1] D. d’Enterria (ed.) et al. (CMS Collaboration), _J. Phys._ G 34 (2007) 2307.
* [2] F. Sikler (CMS Collaboration), in these Proceedings.
* [3] K. Krajczar (CMS Collaboration), in these Proceedings.
* [4] A. de Roeck (ed.) et al.(CMS Collaboration), _J. Phys._ G 34 (2007) 995.
* [5] _CMS HCAL Technical Design Report_ , CERN/LHCC 97-31, 1997.
* [6] _CMS MUON Technical Design Report_ , CERN/LHCC 97-32, 1997.
* [7] _CMS ECAL Technical Design Report_ , CERN/LHCC 97-33, 1997.
* [8] _CMS Tracker Technical Design Report_ , CERN/LHCC 98-6, 1998.
* [9] M. Albrow et al. (CMS/TOTEM Collaborations), CERN-LHCC-2006-039/G-124, 2006.
* [10] G. Roland et al. (CMS Collaboration), _J. Phys._ G 34 (2007) S733.
* [11] T. Matsui and H. Satz, _Phys. Lett._ B 178 (1986) 416\.
* [12] B. Alessandro et al. (NA50 Collaboration), _Eur. Phys. J._ C 39 (2005) 335.
* [13] B. Alessandro et al. (NA50 Collaboration), _Eur. Phys. J._ C 49 (2007) 559.
* [14] A. Adare et al. (PHENIX Collaboration), _Phys. Rev. Lett._ 98 (2007) 232301.
* [15] M. Gyulassy and X.-N. Wang, _Comput. Phys. Commun._ 83 (1994) 307.
* [16] R. Baier, D. Schiff and B.G. Zakharov, _Annual Rev. Nucl. Part. Sci._ 50 (2000) 37.
* [17] _RHIC White Papers_ , _Nucl. Phys._ A 757 (2005) 28.
* [18] F. Arleo et al. _JHEP_ 0411 (2004) 009.
* [19] C. Loizides (CMS Collaboration), arXiv:0804.3679.
* [20] T. Sjostrand, S. Mrenna and P. Skands, _JHEP_ 0605 (2006) 026\.
* [21] I.P. Lokhtin and A.M. Snigirev, _Eur. Phys. J._ C 45 (2006) 211\.
|
arxiv-papers
| 2008-10-12T10:31:44
|
2024-09-04T02:48:58.222925
|
{
"license": "Public Domain",
"authors": "I.P.Lokhtin (for the CMS Collaboration)",
"submitter": "Igor Lokhtin P.",
"url": "https://arxiv.org/abs/0810.2081"
}
|
0810.2201
|
# Lightest Scalar Resonances and the Dynamics of the $\gamma\gamma\to\pi\pi$
Reactions
N.N. Achasov achasov@math.nsc.ru G.N. Shestakov shestako@math.nsc.ru
Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics,
630090, Novosibirsk, Russia
###### Abstract
The high-statistics Belle data on the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$
and $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reactions have been jointly
analyzed. The main dynamical mechanisms of these reactions for energies below
1.5 GeV have been revealed. It has been shown that the direct coupling
constants of the $\sigma(600)$ and $f_{0}(980)$ resonances with a
$\gamma\gamma$ pair are small and that the $\sigma(600)$ $\to$ $\gamma\gamma$
and $f_{0}(980)$ $\to$ $\gamma\gamma$ decays are four-quark transitions due
primarily to $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ loop mechanisms, respectively.
The role of the chiral shielding of the $\sigma(600)$ resonance is emphasized.
The widths of the $f_{0}(980)$ $\to$ $\gamma\gamma$ and $\sigma(600)$ $\to$
$\gamma\gamma$ decays averaged over the resonance mass distributions, as well
as the width of the $f_{2}(1270)$ $\to$ $\gamma\gamma$ decay, are estimated as
$\langle\Gamma_{f_{0}\to\gamma\gamma}\rangle_{\pi\pi}\approx 0.19$ keV,
$\langle\Gamma_{\sigma\to\gamma\gamma}\rangle_{\pi\pi}\approx 0.45$ keV, and
$\Gamma_{f_{2}\to\gamma\gamma}(m^{2}_{f_{2}})\approx 3.8$ keV.
###### pacs:
12.39.-x, 13.40.-f, 13.75.Lb
The investigation of the lightest scalar resonances $\sigma(600)$,
$\kappa(800)$, $a_{0}(980)$, and $f_{0}(980)$ is one of the main goals of
nonperturbative QCD, because the elucidation of their nature is important for
understanding both the physics of confinement and the means of the breaking of
the chiral symmetry at low energies, which are the main consequences of QCD
for hadron physics. The nontrivial nature of these states is commonly
accepted. In particular, there is plenty of evidence of their four-quark
$(q^{2}\bar{q}^{2})$ structure (see, e.g., [1] and references therein). One of
these evidences is the suppression of the production of the $a_{0}(980)$ and
$f_{0}(980)$ resonances in the $\gamma\gamma$ $\to$ $\pi^{0}\eta$ and
$\gamma\gamma$ $\to$ $\pi\pi$ reactions, respectively, which was predicted
more than 25 years ago [2] and observed in the experiment [3]. The problem of
the mechanisms of the production of the $\sigma(600)$, $f_{0}(980)$, and
$a_{0}(980)$ resonances in the $\gamma\gamma$ collisions is closely associated
with the problem of their internal quark structure. This explains the long-
term theoretical and experimental interest in the $\gamma\gamma$ $\to$
$\pi\pi$ reactions at low energies. Recently, the Belle Collaboration obtained
new data on the cross sections for the $\gamma\gamma\to\pi^{+}\pi^{-}$ [4] and
$\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ [5] reactions with statistics two orders
of magnitude larger than all previous experiments and revealed a pronounced
signal from the $f_{0}(980)$ resonance [4,5]. The preceding indications of the
production of the $f_{0}(980)$ resonance in the $\gamma\gamma$ collisions were
much less definite [6–8]. The signal from the $f_{0}(980)$ resonance appears
to be small, which is in good agreement with the prediction of the four-quark
model [1,2].
In this paper, we report the results of the investigation of the main
dynamical mechanisms of the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ and
$\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reactions on the basis of the analysis
of the Belle data [4,5] and our previous investigations of the physics of the
scalar mesons in the $\gamma\gamma$ collisions [2,9–13].
The Belle data on the cross sections for the $\gamma\gamma$ $\to$
$\pi^{+}\pi^{-}$ and $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reactions obtained
for invariant mass $\sqrt{s}$ of the $\pi\pi$ systems from 0.8 to 1.5 GeV are
shown in Fig. 1, where the data of other groups [6–8] are also shown for
$\sqrt{s}$ from $2m_{\pi}$ to 0.85 GeV. All existing data correspond to the
incomplete solid angle of the detection of the final pions such that
$|\cos\theta|\leq 0.6$ and $|\cos\theta|\leq 0.8$ for the production of the
$\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ pairs, respectively, where $\theta$ is
the polar angle of the pion emission in the cms of the initial photons. The
pronounced peaks attributed to the production of the $f_{0}(980)$ and
$f_{2}(1270)$ resonances are observed in the cross sections for both
reactions. The background under these peaks is nearly absent in the
$\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ channel. On the contrary, the resonances
in the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ channel are seen against a large
smooth background, which is primarily attributed to the mechanism of the
charged one-pion exchange [11–16]. The pure Born cross section for the
$\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ process at $|\cos\theta|\leq 0.6$, the
total cross section $\sigma^{\mbox{\scriptsize{Born}}}$ =
$\sigma^{\mbox{\scriptsize{Born}}}_{0}$ \+
$\sigma^{\mbox{\scriptsize{Born}}}_{2}$, and the cross sections
$\sigma^{\mbox{\scriptsize{Born}}}_{0}$ and
$\sigma^{\mbox{\scriptsize{Born}}}_{2}$, where the subscript ($\lambda$ = $0$
or 2) is the absolute value of the difference between the helicities of the
initial photons, are shown in Fig. 1a for comparison. Owing to the Low theorem
and chiral symmetry, the one-pion Born contribution should dominate near the
threshold of the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ reaction. As seen in
Fig. 1a, this expectation does not contradict the near-threshold data;
however, these data were obtained with large errors. The cross section
$\sigma^{\mbox{\scriptsize{Born}}}_{0}$ decreases rapidly with an increase in
$\sqrt{s}$, so that the contribution $\sigma^{\mbox{\scriptsize{Born}}}_{2}$
dominates completely in $\sigma^{\mbox{\scriptsize{Born}}}$ at $\sqrt{s}>$ 0.5
GeV (see Fig. 1a). Note that the contributions from the $S$ and
$D_{\lambda=2}$ partial waves dominate in the region $\sqrt{s}<$ 1.5 GeV in
$\sigma^{\mbox{\scriptsize{Born}}}_{0}$ and
$\sigma^{\mbox{\scriptsize{Born}}}_{2}$, respectively. These partial Born
contributions are strongly modified due to the strong interaction between
pions in the final state, because the $\pi\pi$ interaction at $\sqrt{s}<$ 1.5
GeV is strong only in the $S$ and $D$ waves. The inclusion of the final-state
interaction in the $S$-wave Born amplitudes of the $\gamma\gamma$ $\to$
$\pi^{+}\pi^{-}$ (and $\gamma\gamma$ $\to$ $K^{+}K^{-}$) reaction leads to
certain predictions for the $S$-wave amplitude of the $\gamma\gamma$ $\to$
$\pi^{0}\pi^{0}$ reaction.
Figure 1: Cross sections for the $\gamma\gamma\to\pi^{+}\pi^{-}$ and
$\gamma\gamma\to\pi^{0}\pi^{0}$ reactions. Only statistical errors are shown
for the Belle data [4,5]. The curves in panel (a) are described in the main
text and on the figure. The curves in panel (b) are the approximations of the
data on the $\gamma\gamma\to\pi^{0}\pi^{0}$ reaction. Figure 2: Angular
distributions in the $\gamma\gamma\to\pi^{0}\pi^{0}$ reaction. The Belle
experimental data are taken from [5]. The vertical straight line
$|\cos\theta|$ = 0.8 is the boundary of the region available for the
measurements. The solid lines are the approximations.
Figure 2 shows the Belle experimental data for the angular distributions in
the $\gamma\gamma\to\pi^{0}\pi^{0}$ reaction [5]. They are excellently
reproduced by the simple two-parametric expression
$|a|^{2}+|b\,d^{2}_{20}(\theta)|^{2}$, where $d_{\lambda 0}^{l}(\theta)$ is
the $d$ function [3] and $l$ is the orbital angular momentum of the final
$\pi\pi$ system. Therefore, the cross section for the $\gamma\gamma$ $\to$
$\pi^{0}\pi^{0}$ at $\sqrt{s}<$ 1.5 GeV is described by contributions only
from the $S$ and $D_{2}$ partial waves [17].
Thus, let us consider a model for the helicity, $M_{\lambda}$, and partial,
$M_{\lambda l}$, amplitudes of the $\gamma\gamma$ $\to$ $\pi\pi$ reaction,
where the electromagnetic Born contributions from point-like charged $\pi$ and
$K$ exchanges modified in the $S$ and $D_{2}$ waves by strong final-state
interactions, as well as the contributions due to the direct interaction of
the resonances with photons (see also [11,13]), are taken into account:
$\displaystyle
M_{0}(\gamma\gamma\to\pi^{+}\pi^{-};s,\theta)=M^{\mbox{\scriptsize{Born}}}_{0}(s,\theta)+\widetilde{I}_{\pi^{+}\pi^{-}}(s)\,T_{\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}}(s)+\widetilde{I}_{K^{+}K^{-}}(s)\,T_{K^{+}K^{-}\to\pi^{+}\pi^{-}}(s)+M^{\mbox{\scriptsize{direct}}}_{\mbox{\scriptsize{res}}}(s)\,,$
(1) $\displaystyle
M_{2}(\gamma\gamma\to\pi^{+}\pi^{-};s,\theta)=M^{\mbox{\scriptsize{Born}}}_{2}(s,\theta)+80\pi
d^{2}_{20}(\theta)M_{\gamma\gamma\to f_{2}(1270)\to\pi^{+}\pi^{-}}(s),$ (2)
$\displaystyle
M_{0}(\gamma\gamma\to\pi^{0}\pi^{0};s,\theta)=M_{00}(\gamma\gamma\to\pi^{0}\pi^{0};s)$
$\displaystyle=\widetilde{I}_{\pi^{+}\pi^{-}}(s)\,T_{\pi^{+}\pi^{-}\to\pi^{0}\pi^{0}}(s)+\widetilde{I}_{K^{+}K^{-}}(s)\,T_{K^{+}K^{-}\to\pi^{0}\pi^{0}}(s)+M^{\mbox{\scriptsize{direct}}}_{\mbox{\scriptsize{res}}}(s)\,,$
(3) $\displaystyle
M_{2}(\gamma\gamma\to\pi^{0}\pi^{0};s,\theta)=5d^{2}_{20}(\theta)M_{22}(\gamma\gamma\to\pi^{0}\pi^{0};s)=80\pi
d^{2}_{20}(\theta)M_{\gamma\gamma\to f_{2}(1270)\to\pi^{0}\pi^{0}}(s)\,.$ (4)
Here, $M_{0}^{\mbox{\scriptsize{Born}}}(s,\theta)$ =
$(32\pi\alpha/s)/[1-\rho^{2}_{\pi^{+}}(s)\,\cos^{2}\theta]$ and
$M_{2}^{\mbox{\scriptsize{Born}}}(s,\theta)$ =
$8\pi\alpha\,\rho^{2}_{\pi^{+}}(s)\,\sin^{2}\theta/[1-\rho^{2}_{\pi^{+}}(s)\,\cos^{2}\theta]$
are the Born helicity amplitudes of the $\gamma\gamma\to\pi^{+}\pi^{-}$
reaction, $\rho_{\pi^{+}}(s)$ = $(1-4m^{2}_{\pi^{+}}/s)^{1/2}$, and $\alpha$ =
1/137. The function $\widetilde{I}_{\pi^{+}\pi^{-}}(s)$ at $s\geq
4m^{2}_{\pi^{+}}$ has the form
$\widetilde{I}_{\pi^{+}\pi^{-}}(s)=8\alpha\left\\{\frac{m^{2}_{\pi^{+}}}{s}\left[\pi+i\ln\frac{1+\rho_{\pi^{+}}(s)}{1-\rho_{\pi^{+}}(s)}\right]^{2}-1\right\\},$
$\mbox{Im}\widetilde{I}_{\pi^{+}\pi^{-}}(s)$ =
$\rho_{\pi^{+}}(s)M_{00}^{\mbox{\scriptsize{Born}}}(s)$, and
$\widetilde{I}_{K^{+}K^{-}}(s)$ at $s\geq 4m^{2}_{K^{+}}$ is obtained from
$\widetilde{I}_{\pi^{+}\pi^{-}}(s)$ by changing $m_{\pi^{+}}$ to $m_{K^{+}}$
and $\rho_{\pi^{+}}(s)$ to $\rho_{K^{+}}(s)$ = $(1-4m^{2}_{K^{+}}/s)^{1/2}$;
$\rho_{K^{+}}(s)$ $\to$ $i|\rho_{K^{+}}(s)|$ if 0 $<s<$ $4m^{2}_{K^{+}}$. The
functions $\widetilde{I}_{\pi^{+}\pi^{-}}(s)$ and
$\widetilde{I}_{K^{+}K^{-}}(s)$ are the amplitudes of the triangle diagrams
$\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ $\to$ $\sigma$, $f_{0}$ and
$\gamma\gamma$ $\to$ $K^{+}K^{-}$ $\to$ $\sigma$, $f_{0}$ (and other scalar
resonances); $T_{\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}}(s)$,
$T_{\pi^{+}\pi^{-}\to\pi^{0}\pi^{0}}(s)$, $T_{K^{+}K^{-}\to\pi^{+}\pi^{-}}(s)$
= $T_{K^{+}K^{-}\to\pi^{0}\pi^{0}}(s)$ = $T_{\pi^{+}\pi^{-}\to K^{+}K^{-}}(s)$
are the $S$wave amplitudes of the corresponding reactions;
$T_{\pi^{+}\pi^{-}\to\pi^{+}\pi^{-}}(s)$ = $[2T^{0}_{0}(s)+T^{2}_{0}(s)]/3$
and $T_{\pi^{+}\pi^{-}\to\pi^{0}\pi^{0}}(s)$ =
$2[T^{0}_{0}(s)-T^{2}_{0}(s)]/3$, where $T^{I}_{0}(s)$ =
$\\{\eta^{I}_{0}(s)\exp[2i\delta^{I}_{0}(s)]-1\\}/[2i\rho_{\pi^{+}}(s)]$ are
the amplitudes, $\delta^{I}_{0}(s)$ are the phases, and $\eta^{I}_{0}(s)$ are
the inelasticity factors of the $S$ wave $\pi\pi$ scattering in the channels
with isospin $I$ = 0 and 2. Really, $\eta^{0}_{0}(s)$ = 1 up to the threshold
of the $K\bar{K}$ channel. For this reason, $T_{\pi^{+}\pi^{-}\to
K^{+}K^{-}}(s)=e^{i\delta^{0}_{0}(s)}|T_{\pi^{+}\pi^{-}\to K^{+}K^{-}}(s)|$ at
$4m_{\pi}^{2}$ $<s<$ $4m^{2}_{K}$ [11,13,18]. We also set $\eta^{2}_{0}(s)$ =
1 at all $s$ values under consideration and take $\delta^{2}_{0}(s)$ from
[19]. Expressions (1) and (3) imply that the amplitudes
$T_{\pi^{+}\pi^{-}\to\pi\pi}(s)$ and $T_{K^{+}K^{-}\to\pi\pi}(s)$ in
$\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ $\to$ $\pi\pi$ and $\gamma\gamma$ $\to$
$K^{+}K^{-}$ $\to$ $\pi\pi$ rescattering loops are on-mass-shell amplitudes.
Note that the unitarity condition are satisfied in the model under
consideration [13].
The parametrization of the amplitudes $T^{0}_{0}(s)$ and
$T_{K^{+}K^{-}\to\pi^{0}\pi^{0}}(s)$, which was used in the joint analysis of
the data on the $\pi^{0}\pi^{0}$ mass spectrum in the $\phi$ $\to$
$\pi^{0}\pi^{0}\gamma$ decay, $\pi\pi$ scattering at $2m_{\pi}<$ $\sqrt{s}<$
1.6 GeV, and $\pi\pi$ $\to$ $K\bar{K}$ reaction, was described in detail in
[18]. This parametrization is based on the concept that the amplitude
$T^{0}_{0}(s)$ must include the contribution from mixed $\sigma(600)$ and
$f_{0}(980)$ resonances and the contribution from the background, which has a
large negative phase due to the chiral symmetry; the latter contribution
shields (hides) the $\sigma(600)$ resonance [1,18,20]. Formulas (1) and (3)
transfer the effect of the chiral shielding of the $\sigma(600)$ resonance
from the $\pi\pi$ scattering to the $\gamma\gamma$ $\to$ $\pi\pi$ amplitudes.
If this shielding were absent, then the $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$
cross section (see Fig. 1b) would be about 100 nb rather than 10 nb due to the
$\pi^{+}\pi^{-}$ loop mechanism of the $\sigma(600)$ $\to$ $\gamma\gamma$
decay [12]. According [18],
$T^{0}_{0}(s)=T^{\pi\pi}_{B}(s)+e^{2i\delta^{\pi\pi}_{B}(s)}T^{\pi\pi}_{\mbox{\scriptsize{res}}}(s)\,,$
$T_{K^{+}K^{-}\to\pi^{+}\pi^{-}}(s)=e^{i[\delta^{\pi\pi}_{B}(s)+\delta^{K\bar{K}}_{B}(s)]}T^{K\bar{K}\to\pi\pi}_{\mbox{\scriptsize{res}}}(s)$
and
$T^{\pi\pi}_{B}(s)=\\{\exp[2i\delta^{\pi\pi}_{B}(s)]-1\\}/[2i\rho_{\pi^{+}}(s)]\,,$
where $\delta^{\pi\pi}_{B}(s)$ and $\delta^{K\bar{K}}_{B}(s)$ are the phases
of the elastic $S$-wave background in the $\pi\pi$ and $K\bar{K}$ channels
with $I$ = 0, respectively. The amplitudes of the $\sigma(600)-f_{0}(980)$
resonance complex have the form [13,18]
$T^{\pi\pi}_{\mbox{\scriptsize{res}}}(s)=(\eta^{0}_{0}(s)\exp[2i\delta_{\mbox{\scriptsize{res}}}(s)]-1)/[2i\rho_{\pi^{+}}(s)]=3\,\frac{g_{\sigma\pi^{+}\pi^{-}}\Delta_{f_{0}}(s)+g_{f_{0}\pi^{+}\pi^{-}}\Delta_{\sigma}(s)}{32\pi[D_{\sigma}(s)D_{f_{0}}(s)-\Pi^{2}_{f_{0}\sigma}(s)]}\,,$
(5) $T^{K\bar{K}\to\pi\pi}_{\mbox{\scriptsize{res}}}(s)=\frac{g_{\sigma
K^{+}K^{-}}\Delta_{f_{0}}(s)+g_{f_{0}K^{+}K^{-}}\Delta_{\sigma}(s)}{16\pi[D_{\sigma}(s)D_{f_{0}}(s)-\Pi^{2}_{f_{0}\sigma}(s)]}\,,\\\
$ (6)
$M^{\mbox{\scriptsize{direct}}}_{\mbox{\scriptsize{res}}}(s)=s\,e^{i\delta^{\pi\pi}_{B}(s)}\,\frac{g^{(0)}_{\sigma\gamma\gamma}\Delta_{f_{0}}(s)+g^{(0)}_{f_{0}\gamma\gamma}\Delta_{\sigma}(s)}{D_{\sigma}(s)D_{f_{0}}(s)-\Pi^{2}_{f_{0}\sigma}(s)}\,,$
(7)
where $\Delta_{f_{0}}(s)$ =
$D_{f_{0}}(s)g_{\sigma\pi^{+}\pi^{-}}+\Pi_{f_{0}\sigma}(s)g_{f_{0}\pi^{+}\pi^{-}}$,
$\Delta_{\sigma}(s)=D_{\sigma}(s)g_{f_{0}\pi^{+}\pi^{-}}+\Pi_{f_{0}\sigma}(s)g_{\sigma\pi^{+}\pi^{-}}$,
and
$\delta^{0}_{0}(s)=\delta^{\pi\pi}_{B}(s)+\delta_{\mbox{\scriptsize{res}}}(s)$.
The expressions presented in [18] were used for $\delta^{\pi\pi}_{B}(s)$,
propagators $1/D_{\sigma}(s)$ and $1/D_{f_{0}}(s)$ of the $\sigma(600)$ and
$f_{0}(980)$ resonances, respectively, and the matrix element of the
polarization operator $\Pi_{f_{0}\sigma}(s)$. The values of the parameters in
the strong amplitudes ($m_{\sigma}$, $g_{\sigma\pi^{+}\pi^{-}}$,
$g_{f_{0}K^{+}K^{-}}$, etc.) correspond to variant 1 from Table 1 in [18].
Thus, according to Eqs. (1), (3), and (7), the $\sigma(600)$ $\to$
$\gamma\gamma$ and $f_{0}(980)$ $\to$ $\gamma\gamma$ decays are described by
the triangle $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ loop diagrams (the resonances
$\to$ $\pi^{+}\pi^{-}$, $K^{+}K^{-}$ $\to$ $\gamma\gamma$), which correspond
to the four-quark transitions [12,13], and by the direct coupling constants of
the resonances with the photons $g^{(0)}_{\sigma\gamma\gamma}$ and
$g^{(0)}_{f_{0}\gamma\gamma}$ [9–14].
The amplitudes of the production of the $f_{2}(1270)$ resonance in Eqs. (2)
and (4), $M_{\gamma\gamma\to
f_{2}(1270)\to\pi^{+}\pi^{-}}(s)=M_{\gamma\gamma\to
f_{2}(1270)\to\pi^{0}\pi^{0}}(s)$, have the form
$\sqrt{s}\,G_{2}(s)\sqrt{2\Gamma_{f_{2}\to\pi\pi}(s)/3}\Bigm{/}[m^{2}_{f_{2}}-s-i\sqrt{s}\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(s)]\,,$
where
$G_{2}(s)=\sqrt{\Gamma^{(0)}_{f_{2}\to\gamma\gamma}(s)}+i\frac{M_{22}^{\mbox{\scriptsize{Born}}}(s)}{16\pi}\sqrt{\frac{2}{3}\rho_{\pi^{+}}(s)\Gamma_{f_{2}\to\pi\pi}(s)}\,,$
$\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(s)$ =
$\Gamma_{f_{2}\to\pi\pi}(s)+\Gamma_{f_{2}\to K\bar{K}}(s)+\Gamma_{f_{2}\to
4\pi}(s)\,.$
By definition
$\Gamma_{f_{2}\to\gamma\gamma}(s)=|G_{2}(s)|^{2}\qquad\mbox{and}\qquad\Gamma^{(0)}_{f_{2}\to\gamma\gamma}(s)=\frac{m_{f_{2}}}{\sqrt{s}}\Gamma^{(0)}_{f_{2}\to\gamma\gamma}(m^{2}_{f_{2}})\frac{s^{2}}{m^{4}_{f_{2}}}\,.$
Here, the factor $s^{2}$ and the factor $s$ in Eq. (7) appear due to the gauge
invariance. The second term in $G_{2}(s)$ corresponds to the $f_{2}(1270)$
$\to$ $\pi^{+}\pi^{-}$ $\to$ $\gamma\gamma$ transition with real pions in the
intermediate state and ensures the satisfaction of the Watson theorem for the
$\gamma\gamma$ $\to$ $\pi\pi$ amplitude with $\lambda$ = $l$ = 2 and $I$ =0
below the first inelastic threshold. This term makes a small contribution
(less than 6%) to $\Gamma_{f_{2}\to\gamma\gamma}(m^{2}_{f_{2}})$ [13]. It is
commonly accepted that the quark-antiquark transition $q\bar{q}$ $\to$
$\gamma\gamma$, i.e., the $\Gamma^{(0)}_{f_{2}\to\gamma\gamma}(m^{2}_{f_{2}})$
contribution dominates in the $f_{2}(1270)$ $\to$ $\gamma\gamma$ decay. As
shown in [12,13] and noted below, the situation for the scalar mesons is
opposite.
The leading contribution to $\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(s)$
comes from the partial decay width $f_{2}(1270)$ $\to$ $\pi\pi$,
$\Gamma_{f_{2}\to\pi\pi}(s)=\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(m^{2}_{f_{2}})B(f_{2}\to\pi\pi)\frac{m^{2}_{f_{2}}}{s}\frac{q^{5}_{\pi^{+}}(s)}{q^{5}_{\pi^{+}}(m^{2}_{f_{2}})}\frac{D_{2}(q_{\pi^{+}}(s)R_{f_{2}})}{D_{2}(q_{\pi^{+}}(m^{2}_{f_{2}})R_{f_{2}})}\,,$
where $D_{2}(x)$ = $1/(9+3x^{2}+x^{4})$, $q_{\pi^{+}}(s)$ =
$\sqrt{s}\rho_{\pi^{+}}(s)/2$, $R_{f_{2}}$ is the interaction radius, and
$B(f_{2}$ $\to$ $\pi\pi)$ = 0.847. Small contributions from $\Gamma_{f_{2}\to
K\bar{K}}(s)$ and $\Gamma_{f_{2}\to 4\pi}(s)$ are the same as in [13]. The
parameter $R_{f_{2}}$ [4–8,13] controls the relative shape of the wings of the
$f_{2}(1270)$ resonance and is important particulary for the approximation of
the data with small errors.
We use the following notation and normalizations for the cross sections
$\sigma(\gamma\gamma\to\pi^{+}\pi^{+};|\cos\theta|\leq
0.6)\equiv\sigma=\sigma_{0}+\sigma_{2}\qquad\mbox{and}\qquad\sigma(\gamma\gamma\to\pi^{0}\pi^{0};|\cos\theta|\leq
0.8)\equiv\tilde{\sigma}=\tilde{\sigma}_{0}+\tilde{\sigma}_{2}\,,$
where
$\sigma_{\lambda}=\frac{\rho_{\pi^{+}}(s)}{64\pi
s}\int^{0.6}_{-0.6}|M_{\lambda}(\gamma\gamma\to\pi^{+}\pi^{+};s,\theta)|^{2}d\cos\theta\qquad\mbox{and}\qquad\tilde{\sigma}_{\lambda}=\frac{\rho_{\pi^{+}}(s)}{128\pi
s}\int^{0.8}_{-0.8}|M_{\lambda}(\gamma\gamma\to\pi^{0}\pi^{0};s,\theta)|^{2}d\cos\theta\,.$
First, we consider the approximation of the data only on the cross section for
the $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reaction (see Fig. 1b); as mentioned
above, the background situation in this channel is more pure than in the
$\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ one. The solid line in Fig. 1b, which
well describes these data, corresponds to the following model parameters:
$m_{f_{2}}$ = 1.269 GeV,
$\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(m^{2}_{f_{2}})$ = 0.182 GeV,
$R_{f_{2}}$ = 8.2 GeV-1, $\Gamma_{f_{2}\to\gamma\gamma}(m_{f_{2}})$ = 3.62
keV, $m_{f_{0}}$ = 0.969 GeV, $g^{(0)}_{\sigma\gamma\gamma}$ = 0.536 GeV-1,
and $g^{(0)}_{f_{0}\gamma\gamma}$ = 0.652 GeV-1. The approximation indicates
that the direct constants $g^{(0)}_{\sigma\gamma\gamma}$ and
$g^{(0)}_{f_{0}\gamma\gamma}$ are small in agreement with the prediction in
[2]: $\Gamma^{(0)}_{\sigma\to\gamma\gamma}(m^{2}_{\sigma})$ =
$|m^{2}_{\sigma}g^{(0)}_{\sigma\gamma\gamma}|^{2}/(16\pi m_{\sigma})$ = 0.012
keV and $\Gamma^{(0)}_{f_{0}\to\gamma\gamma}(m^{2}_{f_{0}})$ =
$|m^{2}_{f_{0}}g^{(0)}_{f_{0}\gamma\gamma}|^{2}/(16\pi m_{f_{0}})$ = 0.008
keV. In turn, this indicates the dominance of the $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$ loop mechanisms of the coupling of $\sigma(600)$ and $f_{0}(980)$
with photons. Indeed, according to estimates [11,12], the width of the
$\sigma(600)$ $\to$ $\pi^{+}\pi^{-}\to\gamma\gamma$ decay through the
$\pi^{+}\pi^{-}$ loop mechanism is approximately 1–1.75 keV in the region 0.4
$<\sqrt{s}<$ 0.5 GeV [12], and the width of the $f_{0}(980)$ $\to$
$K^{+}K^{-}$ $\to$ $\gamma\gamma$ decay through the $K^{+}K^{-}$ loop
mechanism after averaging over the resonance mass distribution is
approximately 0.15–0.2 keV [11].
However, such an approximation of the $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$
cross section leads to a contradiction with the data for $\gamma\gamma$ $\to$
$\pi^{+}\pi^{-}$ (see the solid line for $\sigma$ = $\sigma_{0}$ \+
$\sigma_{2}$ in Fig. 1a). This is associated with a large Born contribution to
$\sigma_{2}$ and a strong constructive (destructive) interference of this
contribution with the contribution from the $f_{2}(1270)$ resonance at
$\sqrt{s}<$ $m_{f_{2}}$ ($\sqrt{s}>$ $m_{f_{2}}$). Note that these
contributions are absent in $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reaction.
The problem of the joint description of the data for the $\gamma\gamma$ $\to$
$\pi^{+}\pi^{-}$ and $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reactions was
pointed out in [13], where the solution of this problem was proposed. The
situation can be significantly corrected by multiplying the $\gamma\gamma$
$\to$ $\pi^{+}\pi^{-}$ Born amplitudes for point particles,
$M_{\lambda}^{\mbox{\scriptsize{Born}}}(s,\theta)$, by the common suppressing
form factor $G(t,u)$ [7,8,10,13,16,21], where $t$ and $u$ are the normal
Mandelstam variables for the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ process. To
demonstrate this, we use the following expression proposed in [21]:
$G(t,u)=\frac{1}{s}\left[\frac{m^{2}_{\pi^{+}}-t}{1-(u-m^{2}_{\pi^{+}})/x^{2}_{1}}+\frac{m^{2}_{\pi^{+}}-u}{1-(t-m^{2}_{\pi^{+}})/x^{2}_{1}}\right]\,,$
where $x_{1}$ is the free parameter. This ansatz is acceptable in the physical
region of the $\gamma\gamma\to\pi^{+}\pi^{-}$ reaction. Changing $m_{\pi^{+}}$
to $m_{K^{+}}$ and $x_{1}$ to $x_{2}$, we also obtain the form factor for the
Born amplitudes of the $\gamma\gamma$ $\to$ $K^{+}K^{-}$reaction. The solid
lines for the cross sections $\sigma$ = $\sigma_{0}$ \+ $\sigma_{2}$ and
$\tilde{\sigma}$ = $\tilde{\sigma}_{0}+\tilde{\sigma}_{2}$ in Figs. 3a and 3b,
respectively, demonstrate the joint approximation of the data for the
$\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ reaction in the region 0.85 $<\sqrt{s}<$
1.5 GeV and for the $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$ reaction in the
region $2m_{\pi}<$ $\sqrt{s}<$ 1.5 GeV including the form factors modifying
the Born contributions for point particles. The resulting description is more
than satisfactory, but only with the inclusion of the total (statistical and
systematic) errors in the Belle data, which are shown in Figs. 3a and 3b in
the form of shaded bands. We
Figure 3: Joint description of the data on the cross sections for the
$\gamma\gamma\to\pi^{+}\pi^{-}$ and $\gamma\gamma\to\pi^{0}\pi^{0}$ reaction.
The shaded bands correspond to the Belle data [4,5] with the statistical and
systematic errors (errors are added quadratically). The curves are described
in the main text and on the figures; $\tilde{\sigma}^{\mbox{\scriptsize{
Born}}}_{2}$ in panel (a) is the Born cross section for the
$\gamma\gamma\to\pi^{+}\pi^{-}$ reaction with the inclusion of the form
factor.
believe that this treatment is justified. The statistical errors of two Belle
measurements are small so that it is impossible to obtain the formally
acceptable $\chi^{2}$ values in the joint approximation of the
$\pi^{+}\pi^{-}$ and $\pi^{0}\pi^{0}$ data without the inclusion of the
systematic errors. The lines in Figs. 3a and 3b correspond to the parameters
$m_{f_{2}}$ = 1.272 GeV,
$\Gamma^{\mbox{\scriptsize{tot}}}_{f_{2}}(m^{2}_{f_{2}})$ = 0.196 GeV,
$R_{f_{2}}$ = 8.2 GeV-1, $\Gamma_{f_{2}\to\gamma\gamma}(m_{f_{2}})$ = 3.84
keV, $m_{f_{0}}$ = 0.969 GeV, $g^{(0)}_{\sigma\gamma\gamma}$ = -0.049 GeV-1
($\Gamma^{(0)}_{\sigma\to\gamma\gamma}(m^{2}_{\sigma})$ is negligible),
$g^{(0)}_{f_{0}\gamma\gamma}$ = 0.718 GeV-1
($\Gamma^{(0)}_{f_{0}\to\gamma\gamma}(m^{2}_{f_{0}})$ $\approx$ 0.01 keV),
$x_{1}$ = 0.9 GeV and $x_{2}$ = 1.75 GeV. A comparison of Figs. 1b and 3b
shows that the effect of the form factors on the cross section for the
$\gamma\gamma\to\pi^{0}\pi^{0}$ reaction is weak in contrast to the cross
section for the $\gamma\gamma\to\pi^{+}\pi^{-}$ (see Figs. 1a and 3a). We
emphasize that our conclusions on the mechanisms of the two-photon decays
(productions) of the $\sigma(600)$ and $f_{0}(980)$ resonances remain valid.
It is interesting to consider the $\gamma\gamma$ $\to$ $\pi^{+}\pi^{-}$ cross
section attributed only to the resonance contributions, i.e.,
$\sigma_{\mbox{\scriptsize{res}}}(\gamma\gamma\to\pi^{+}\pi^{-};s)=\frac{\rho_{\pi^{+}}(s)}{32\pi
s}|\widetilde{I}^{\mbox{\
\scriptsize{ff}}}_{\pi^{+}\pi^{-}}(s)\,e^{2i\delta^{\pi\pi}_{B}(s)}T^{\pi\pi}_{\mbox{\scriptsize{res}}}(s)+\widetilde{I}^{\mbox{\
\scriptsize{ff}}}_{K^{+}K^{-}}(s)T_{K^{+}K^{-}\to\pi^{+}\pi^{-}}(s)+M^{\mbox{\scriptsize{direct}}}_{\mbox{\scriptsize{res}}}(s)|^{2}\,,$
[see Eqs. (1) and (5)–(7)]. Here, the superscript ff means that the functions
$\widetilde{I}(s)$ are obtained with the inclusion of the form factors [10].
The cross section $\sigma_{\mbox{\scriptsize{res }}}(\gamma\gamma$ $\to$
$\pi^{+}\pi^{-};s)$ has a pronounced peak near 1 GeV from the $f_{0}(980)$
resonance, which is due primarily to the contribution from the $\gamma\gamma$
$\to$ $K^{+}K^{-}$ $\to$ $\pi^{+}\pi^{-}$ transition. Following [9,11], we
determine the width of the $f_{0}(980)$ $\to$ $\gamma\gamma$ decay averaged
over the resonance mass distribution in the $\pi\pi$ channel:
$\langle\Gamma_{f_{0}\to\gamma\gamma}\rangle_{\pi\pi}=\int\limits_{0.8\mbox{\,\scriptsize{GeV}}}^{1.1\mbox{\,\scriptsize{GeV}}}\frac{3s}{8\pi^{2}}\,\sigma_{\mbox{\scriptsize{res}}}(\gamma\gamma\to\pi^{+}\pi^{-};s)\,d\sqrt{s}\,.$
(8)
This quantity is an adequate characteristic of the coupling of the
$f_{0}(980)$ resonance with a $\gamma\gamma$ pair [11]. For the present joint
approximation, $\langle\Gamma_{f_{0}\to\gamma\gamma}\rangle_{\pi\pi}$
$\approx$ 0.19 keV. Accepting that $2m_{\pi}<$ $\sqrt{s}<$ 0.8 GeV is the
region of the wide $\sigma(600)$ resonance, we obtain
$\langle\Gamma_{\sigma\to\gamma\gamma}\rangle_{\pi\pi}$ $\approx$ 0.45 keV by
analogy with Eq. (8).
Note that the contributions from the $\omega(782)$ and $h_{1}(1170)$ exchanges
to the $S$-wave amplitude of the $\gamma\gamma$ $\to$ $\pi^{0}\pi^{0}$
reaction have opposite signs and cancel each other.
This work was supported in part by the RFFI Grant No. 07-02-00093 from the
Russian Foundation for Basic Research and by the Presidential Grant No.
NSh-1027.2008.2 for Leading Scientific Schools.
## References
* (1) N.N. Achasov, in Proc. of the 14th International Seminar QUARKS-2006, Repino, St. Peterburg, 2006, Ed. by S.V. Demidov, V.A. Matveev, V.A. Rubakov, and G.I. Rubtsov (INR RAS, Moscow, 2007), p. 37.
* (2) N.N. Achasov et al., Phys. Lett. 108B, 134 (1982); Z. Phys. C 16, 55 (1982); Z. Phys. C 27, 99 (1985).
* (3) W.-M. Yao et al., J. Phys. G 33, 1 (2006).
* (4) T. Mori et al., Phys. Rev. D 75, 051101 (2007); J. Phys. Soc. Jap. 76, 074102 (2007).
* (5) S. Uehara et al., arXiv: 0805.3387 [hep-ex].
* (6) H. Marsiske et al., Phys. Rev. D 41, 3324 (1990).
* (7) J. Boyer et al., Phys. Rev. D 42, 1350 (1990).
* (8) H.J. Behrend et al., Z. Phys. C 56, 381 (1992).
* (9) N.N. Achasov and G.N. Shestakov, Z. Phys. C 41, 309 (1988).
* (10) N.N. Achasov and G.N. Shestakov, Mod. Phys. Lett. A 9, 1351 (1994); Yad.Fiz. 55, 2999 (1992) [Sov. J. Nucl. Phys. 55, 1677 (1992)].
* (11) N.N. Achasov and G.N. Shestakov, Phys. Rev. D 72, 013006 (2005); Yad.Fiz. 69, 1545 (2006) [Phys. At. Nucl. 69, 1510 (2006)].
* (12) N.N. Achasov and G.N. Shestakov, Phys. Rev. Lett. 99, 072001 (2007).
* (13) N.N. Achasov and G.N. Shestakov, Phys. Rev. D 77, 074020 (2008).
* (14) G. Mennessier, Z. Phys. C 16, 241 (1983).
* (15) R.P. Johnson, Ph.D. thesis, SLAC-Report-294, 1986.
* (16) D. Morgan and M.R. Pennington, Phys. Lett. B 192, 207 (1987); Z. Phys. C 37, 431 (1988); Phys. Lett. B 272, 134 (1991).
* (17) We verify that, if the approximation includes the contribution from the $D_{\lambda=0}$ wave; i.e., if the expression $|a+c\,d^{2}_{00}(\theta)|^{2}+|b\,d^{2}_{20}(\theta)|^{2}$ with four free parameters is used to describe the angular distributions, it is impossible to reliably determine the absolute value and relative phase of the amplitude $c$, because this expression does not provide any noticeable improvement to the approximation. Note that the so-called SD and SDG approximations used in [5] lead to the choice of the nonphysical values of the parameters and, as a result, to negative differential cross sections (see Fig. 7 in [5]). For this reason, they cannot provide a reliable tool for the partial wave analysis.
* (18) N.N. Achasov and A.V. Kiselev, Phys. Rev. D 73, 054029 (2006); Yad.Fiz. 70, 2005 (2007) [Phys. At. Nucl. 70, 1956 (2007)].
* (19) N.N. Achasov and G.N. Shestakov, Phys. Rev. D 67, 114018 (2003); Yad.Fiz. 67, 1380 (2004) [Phys. At. Nucl. 67, 1355 (2004)].
* (20) N.N. Achasov and G.N. Shestakov, Phys. Rev. D 49, 5779 (1994); Yad.Fiz. 56, No. 9, 206 (1993) [Phys. At. Nucl. 56, 960 (1993)].
* (21) M. Poppe, Int. J. Mod. Phys. A 1, 545, (1986).
|
arxiv-papers
| 2008-10-13T10:31:59
|
2024-09-04T02:48:58.230746
|
{
"license": "Public Domain",
"authors": "N.N. Achasov, G.N. Shestakov",
"submitter": "Georgii Shestakov",
"url": "https://arxiv.org/abs/0810.2201"
}
|
0810.2368
|
# Structure of Neutron Star with a Quark Core
G.H. Bordbar 111Corresponding author222E-mail : Bordbar@physics.susc.ac.ir, M.
Bigdeli and T. Yazdizadeh
Department of Physics, Shiraz University, Shiraz 71454, Iran333Permanent
address
and
Research Institute for Astronomy and Astrophysics of Maragha,
P.O. Box 55134-441, Maragha, Iran
###### Abstract
The equation of state of de-confined quark matter within the MIT bag model is
calculated. This equation of state is used to compute the structure of a
neutron star with quark core. It is found that the limiting mass of the
neutron star is affected considerably by this modification of the equation of
state. Calculations are carried out for different choices of the bag constant.
## 1 Introduction
Neutron stars (NS) are among the densest of massive objects in the universe.
They are ideal astrophysical laboratories for testing theories of dense matter
physics and provide connections among nuclear physics, particle physics and
astrophysics. The maximum mass of a neutron star is a subject that several
theoretical astrophysicists have tried to compute it. Below a certain maximum
mass, degeneracy pressure prevents an object collapse into a black hole. To
calculate the maximum mass, we require enough information about the
composition of the star. Different compositions lead to different equations of
state (EOS). When nuclear matter is compressed to densities so high that the
nucleon cores substantially overlap, one expects the nucleons to merge and
undergo a phase transition to de-confined quark matter. Such a system could be
realized in two possible ways: (a) complete strange quark matter stars (b)
neutron stars with a core of quark matter. Glendenning [1] has shown that a
proper construction of the nucleon-quark phase transition inside neutron stars
implies the coexistence of nucleon matter and quark matter over a finite range
of pressure. This has the effect that a core, or a spherical shell, of a mixed
quark-nucleon phase can exist inside neutron stars.
In this work, we calculate the structure of the neutron star with a quark core
and also strange star, and compare our results with our pervious works in
which we investigated the NS structure without a quark core [2].
## 2 Quark Matter Equation of State
For the de-confined quark phase, within the MIT bag model [3], the total
energy density is the sum of a non-perturbative energy shift B (the bag
constant) and the kinetic energy for non-interacting massive quarks of flavors
$f$ with mass $m_{f}$ and Fermi momentum
$k_{F}^{(f)}=(\pi^{2}\rho_{f})^{\frac{1}{3}}$ where $\rho_{f}$ is the quarks
density of flavor f:
$\displaystyle\varepsilon_{Q}=\frac{3}{8}\frac{m^{4}c^{5}}{\pi^{2}\hbar^{3}}\left[x\sqrt{x^{2}+1}(2x^{2}+1)-\sinh^{-1}x\right]+\frac{3\hbar
c}{2\pi^{2}}(\pi^{2}\rho)^{\frac{4}{3}}+B,$ (1)
where $x={\hbar k_{F}}/{mc}$ , $\rho_{s}=\rho_{d}=\rho_{u}=\rho$, $\rho$ is
baryon density, and $\varepsilon_{Q}={E}/{V}$. We assume in this work that u
and d quarks are massless and the s quark has a mass equal to $m=150MeV$. The
bag constant B, can be interpreted as the difference between the energy
densities of the non-interacting quarks and interacting ones, which has a
constant value such as $B=55$ and $90MeV$ in the initial model of MIT.
Inclusion of perturbative interaction among quarks introduces additional terms
in the thermodynamic potential [4]. We try to determine a range of possible
values for B by using the experimental data obtained at the CERN SPS [5].
According to the analysis of those experiments, the quark-hadron transition
takes place at about seven times normal nuclear matter energy density (
$\varepsilon_{0}=156MeVfm^{-3}$). We assume a density dependent B. In the
literature there are attempts to understand the density dependence of B [6, 7,
8]; however, currently the results are highly model dependent and no definite
picture has come out yet. Therefore, we attempt to provide effective
parameterizations for this density dependence. Our parameterizations are
constructed in such a way that at asymptotic densities B has some finite value
$B_{\infty}$:
$\displaystyle
B(\rho)=B_{\infty}+(B_{0}-B_{\infty})\exp\left[-\beta(\frac{\rho}{\rho_{0}})^{2}\right]\cdot$
(2)
The parameter $B_{0}=B(\rho=0)$ has constant which is assumed to be
$B_{0}=400$ in this work. and $\beta$ is numerical parameter usually equal to
$\rho_{0}\approx 0.17fm^{-3}$, the normal nuclear matter density. $B_{\infty}$
depends only on the free parameter $B_{0}$. In order to fix $B_{\infty}$, we
proceed in the following way:
Firstly, we use the equation of state (EOS) of asymmetric hadronic matter
characterized by a proton fraction $x_{p}=0.4$ and the $UV_{14}+TNI$
potential. By assuming that the hadron-quark transition takes place at the
energy density $\varepsilon=1100MeVfm^{-3}$, we find that hadronic matter
baryon density is $\rho_{t}=0.98fm^{-3}$(transition density) and at values
lower than it the quark matter energy density is higher than that of nuclear
matter, while with increasing baryon density the two energy densities become
equal at this density and after that the nuclear matter energy density remains
always higher. Eq. (1) for quark matter with two flavors u and d reduces to:
$\varepsilon_{Q}=\frac{3\hbar
c}{4}\pi^{\frac{2}{3}}\left[\rho_{u}^{\frac{4}{3}}+\rho_{d}^{\frac{4}{3}}\right]+B,$
(3)
where $\rho_{d}=2\rho_{u}=2\rho$.
Secondly, we determine $B_{\infty}=8.99$ by putting quark energy density and
hadronic energy density equal to each other
$(\varepsilon_{Q}=\varepsilon|_{\rho=\rho_{t}})$.
Finally, we calculate the EOS for the three flavors quark matter using
$P=\rho\frac{\partial\varepsilon_{Q}}{\partial\rho}-\varepsilon_{Q}\cdot$ (4)
## 3 Mixed Phase
The hadron-quark phase transition takes place within a range of baryon density
values. In other words, the fraction of space occupied by quark matter
smoothly increases from zero to unity when eventually the last nucleons
dissolve into quarks. In this case, we have a mixture of the hadron, quark and
electron background in the system. Glendenning s construction [1] describes a
global division of the baryon number between the two phases. The equilibrium
conditions, in the case where the geometry of droplets is neglected, are those
for bulk systems. The neutron star matter is assumed to be stable and charge
neutral. Thermal effects are not expected to play any important role in
neutron star cores. We neglect such effects and put the temperature $T=0$. The
equilibrium conditions for the quark matter droplet to coexist with the
nucleon medium are that pressure and chemical potentials in both phases
coincide. We choose pressure as an independent variable. The coexistence
requires that (Gibbs conditions):
$\mu_{N}^{n}(P)=\mu_{N}^{q}(P),$ (5)
and
$\mu_{P}^{n}(p)=\mu_{P}^{q}(P),$ (6)
where $\mu_{N}^{n}$ and $\mu_{N}^{q}$ are the neutron chemical potential in
the nucleon phase (NP) and the quark phase (QP), respectively. Similarly,
$\mu_{P}^{n}$ and $\mu_{P}^{q}$ are the proton chemical potential in the
respective phases. The strange quark and lepton chemical potentials are
dictated by the conditions of weak equilibrium
$\mu_{s}=\mu_{d},$ (7)
and
$\mu_{d}-\mu_{u}=\mu_{l},$ (8)
where $\mu_{l}$ is lepton chemical potential.
Using the semi-empirical mass formula, the energy per particle of nuclear
matter can be expressed as
$E(\rho,x)=T(\rho,x)+V_{0}(\rho)+(1-2x)^{2}V_{2}(\rho),$ (9)
where $x={\rho_{p}}/{\rho}$ is proton fraction. The kinetic energy
contribution is
$\displaystyle
T(\rho,x)=\frac{3}{5}\frac{\hbar^{2}}{2m}\left(3\pi^{2}\rho\right)^{\frac{2}{3}}\left[(1-x)^{\frac{5}{3}}+x^{\frac{5}{3}}\right]\cdot$
(10)
The functions $V_{0}$ and $V_{2}$ represent the interaction energy
contributions and we can determine them from the results of symmetric nuclear
matter $(x=\frac{1}{2})$ and pure neutron matter $(x=0)$ [9]. Using our
results for the nuclear matter with the $UV_{14}+TNI$ potential, we have
obtained the following fit for $V_{0}$ and $V_{2}$:
$\displaystyle
V_{0}(\rho)=-559.9\rho^{5}+1695.62312\rho^{4}-1946.86437\rho^{3}+1327.04\rho^{2}-411.57428\rho-0.30327,$
(11) $\displaystyle
V_{2}(\rho)=191.21\rho^{5}-626.712\rho^{4}+776.623\rho^{3}-473.40909\rho^{2}+141.8709\rho+4.75638\cdot$
(12)
From Eqs. (9) and (10), we obtain the chemical potentials of neutrons and
protons as:
$\displaystyle\mu_{N}^{n}(p)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\left(3\pi^{2}\rho\right)^{\frac{2}{3}}\left[(1-x)^{\frac{5}{3}}+x(1-x)^{\frac{2}{3}}\right]+V_{0}(\rho)+\rho
V_{0}^{{}^{\prime}}(\rho)$ (13) $\displaystyle+(1-2x)^{2}\rho
V_{2}^{{}^{\prime}}(\rho)+(1-4x^{2})V_{2}(\rho)+mc^{2},$
and
$\displaystyle\mu_{P}^{n}(p)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\left(3\pi^{2}\rho\right)^{\frac{2}{3}}\left[x^{\frac{5}{3}}+(1-x)x^{\frac{2}{3}}\right]+V_{0}(\rho)+\rho
V_{0}^{{}^{\prime}}(\rho)$ (14) $\displaystyle+(1-2x)^{2}\rho
V_{2}^{{}^{\prime}}(\rho)+(-3+8x-4x^{2})V_{2}(\rho)+mc^{2}\cdot$
The quark chemical potential with favor $f$ is
$\displaystyle\mu_{f}$ $\displaystyle=$
$\displaystyle\left[m_{f}^{2}c^{4}+\hbar^{2}c^{2}(\pi^{2}\rho_{f})^{\frac{2}{3}}\right]^{\frac{1}{2}}\cdot$
(15)
From Eqs. (1), (4), (7) and (15), we obtain the chemical potential of quark
matter:
$\displaystyle\mu_{u}$ $\displaystyle=$
$\displaystyle\left(4\pi^{2}\hbar^{3}c^{3}\left[P+B+D-\rho(\frac{\partial
D}{\partial\rho}+\frac{\partial
B}{\partial\rho})\right]-\mu_{d}^{4}\right)^{\frac{1}{4}},$ (16)
where
$D=\frac{3}{8}\frac{m^{4}c^{5}}{\pi^{2}\hbar^{3}}\left[x\sqrt{x^{2}+1}(2x^{2}+1)-\sinh^{-1}x\right]$.
We also have
$\mu_{N}^{q}=2\mu_{d}+\mu_{u},$ (17)
and
$\mu_{P}^{q}=2\mu_{u}+\mu_{d}\cdot$ (18)
By plotting $\mu_{P}$ versus $\mu_{N}$ for both nucleon and quark phases, we
can identify the cross point of two curves that satisfy the Gibbs condition.
As the chemical potentials determine the charge densities of the two phases,
the volume fraction occupied by quark matter, $\chi$, can be obtained by
exploiting the requirement of global charge neutrality:
$\chi\rho_{q}^{c}+(1-\chi)\rho_{p}-\rho_{e}=0,$ (19)
where $\rho_{q}^{c}$ is the quark charge density. The total energy density and
density of mixed phase (MP) are given by:
$\displaystyle\varepsilon_{MP}=\chi\varepsilon_{QP}+(1-\chi)\varepsilon_{NP},$
(20)
and
$\rho_{MP}=\chi\rho_{QP}+(1-\chi)\rho_{NP}\cdot$ (21)
We plot pressure versus baryon density for hadron, mixed and quark phases in
Figures 1 and 2, for bag constants $B=90$ and density dependent $B$,
respectively. It is seen that there is a mixed phase at a range of densities.
A pure quark phase is presented at densities beyond this range and a pure
hadronic phase is presented at densities below it.
## 4 Structure of Neutron Star with a Quark Core
We calculate the structure of a neutron star for various values of the central
mass density, $\varepsilon_{c}$, by using the equation of state and
numerically integrating the general relativistic equation of hydrostatic
equilibrium, Tolman-Oppenheimer-Volkoff (TOV) equation[10]. The derivation of
TOV-equation can be found in standard textbooks [11, 12, 13, 14].
For a neutron star with a quark core, we use the following equations of state:
* •
Below the density $0.05fm^{-3}$, we use the equation of state calculated by
Baym et al. [15].
* •
From this density up to the beginning point of the mixed phase, we use the
equation of state which is calculated with the $UV_{14}+TNI$ potential [2].
* •
For the range of densities in the mixed phase, we use the equation of state
which was calculated in the previous section.
* •
Beyond the end point of the mixed phase, we use the equation of state of pure
quark matter which was calculated in section 2.
Calculations are done both for constant $B=90$, and density dependent $B$.
Based on these EOS’s, we calculate the mass and radius of the NS with quark
core. The calculations are also repeated for the strange star (i.e. pure quark
matter). We plot the NS mass versus central mass energy density for B=90 and
density-dependent B, in Figures 3 and 4, respectively. The NS mass versus
radius for quark core NS and strange star are plotted for B=90 and density-
dependent B in Figures 5 and 6, respectively. For the sake of comparison, we
have also plotted our previous results of the neutron star structure without
quark core, in these figures. It is seen that there is a profound difference
between the new results for NS with a quark core and those of NS without a
quark core.
The extracted maximum mass of a NS and the corresponding radius and central
mass density for both cases B=90 and density dependent B are presented in
Tables 1 and 2, respectively. It is seen that the inclusion of the quark core
leads to a considerable reduction of the maximum mass, while the radius is not
affected appreciably. Note that the maximum mass for the NS with quark core
and B=90 is quite near to the observed maximum mass of neutron stars [16].
The maximum mass energy density versus the radial coordinate for NS without
core, NS with a quark core and strange star are plotted in Figures 7 and 8 for
B=90 and density-dependent B, respectively. It can be seen that a major part
of the core is composed of pure quark matter (about 8 Km). A layer of mixed
phase (thickness about 1.5 Km) exists between the core and a thin crust.
## 5 Summary
As we go from the center toward the surface of a neutron star, the state of
baryonic matter changes from the de-confined quark-gluon to a mixed state of
quark matter and hadronic matter, and thin crust of hadronic matter. The
transition between these states occurs in a smooth way. In order to calculate
the structure and the mass limit of neutron stars, it is important to have a
fairly accurate physical description of these states.
In this paper, we calculated the equation of state of the de-confined quark
phase within the MIT bag model. We then calculated the mixed phase of nucleons
and quarks. The equilibrium volume fractions of nucleon and quark matter were
obtained by applying the Gibbs condition. Curves were presented which showed
the dependence of pressure on the baryon density.
Our results for the equation of state were then used to calculate the
structure of a neutron star with a quark core. As usual, the Tolman-
Oppenheimer-Volkoff equation were integrated from the center to the surface of
the neutron star where the density drops to zero. Calculations were carried
out both for B=90 and a density-dependent B.
The maximum mass, radius, and central mass density of neutron stars with a
quark core and strange star were calculated and compared with the traditional
neutron star. It was found that the limiting mass decreases when the quark
core is taken into account. This brings the maximum mass closer to the
observational limits.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha, and Shiraz University Research Council.
## References
* [1] N. K. Glendenning, Phys. Rev. D46 (1992) 1274.
* [2] G. H. Bordbar, and M. Hayati, Int. J. Mod. Phys. A21 (2006) in press.
* [3] A. Chodos, R. L. Jaff, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D9 (1974) 3471.
* [4] E. Fahri, and R. L. Jeff, Phys. Rev. D30 (1984) 2379.
* [5] U. Heinz, and M. Jacobs, nucl-th/0002042
U. Heinz, hep-ph/0009170
* [6] C. Adami, and G. E. Brown, Phys. Rep. 234 (1993) 1.
* [7] xue-min Jin and B. K. Jenning, Phys. Rev. C55 (1997) 1567.
* [8] D. Blaschke,H. Grigorian, G. Poghosyan, C. D. Roberts, and S. Schmidt, Phys. Lett. B450 (1999) 207.
* [9] I. E. Lagaris and V. R. Pandharipande, Nucl. Phys A369 (1981) 470.
* [10] S. Shapiro and S. Teukolsky, Blak Holes, White Dwarfs and Neutron Stars, (Wiley, New york, 1983).
* [11] N. K. Glendenning, Compact Stars-Nuclear Physics, Particle Physics, and general Relativity, (Springer, New York, 2000).
* [12] F. Weber, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics, (Institute of Physics, Bristol, 1999).
* [13] R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity (McGraw-Hill, New York, 1965).
* [14] C. W. Misner, K. S. Theorne, and J. A. Wheeler, Gravitation (W. H. Freeman Company, New York, 1973).
* [15] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. 170 (1971) 299.
* [16] S.E. Thorsett and D. Chakrabarty, Astrophys. J. 512, 288 (1999).
Table 1: Maximum gravitational mass $(M_{\max})$, corresponding radius(R) and central mass density$(\varepsilon_{c})$ for $B=90$. star | $M_{\max}(M_{\odot})$ | R(Km) | $\varepsilon_{c}(10^{14}gr/cm^{3})$
---|---|---|---
NS | 1.98 | 9.81 | 27.17
NS+quark core | 1.57 | 9.73 | 33.27
stange star | 1.34 | 7.77 | 34.81
Table 2: Maximum gravitational mass $(M_{\max})$, corresponding radius(R)and central mass density$(\varepsilon_{c})$ for density dependent $B$. star | $M_{\max}(M_{\odot})$ | R(Km) | $\varepsilon_{c}(10^{14}gr/cm^{3})$
---|---|---|---
NS | 1.98 | 9.81 | 27.17
NS+quark core | 1.75 | 9.66 | 28.92
stange star | 1.63 | 8.2 | 28.92
Figure 1: The pressure versus baryon density for Hadron Phase (solid line),
Mixed Phase (dashed line) and Quark Phase (dotted line) for B=90.
Figure 2: The pressure versus baryon density for Hadron Phase (solid line),
Mixed Phase (dashed line) and Quark Phase (dotted line) for density dependent
B.
Figure 3: The gravitational mass versus central mass density for different
cases with bag constant B=90.
Figure 4: The gravitational mass versus central mass density for different
cases with density dependent B.
Figure 5: The mass-radius relation for different cases with bag constant B=90.
Figure 6: The mass-radius relation for different cases with density dependent
B.
Figure 7: Mass density as a function of radial coordinate for neutron star
(dotted line) neutron star with quark core (solid curve) and strange star
(dashed curve) with B=90.
Figure 8: Mass density as a function of radial coordinate for neutron star
(dotted line) neutron star with quark core (solid curve) and strange star
(dashed curve)with density dependent B.
|
arxiv-papers
| 2008-10-14T06:27:29
|
2024-09-04T02:48:58.238910
|
{
"license": "Public Domain",
"authors": "G.H. Bordbar, M. Bigdeli and T. Yazdizadeh",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/0810.2368"
}
|
0810.2391
|
# GEOMETRICALLY DERIVED TIMESCALES FOR STAR FORMATION IN SPIRAL GALAXIES
D. Tamburro, H.-W. Rix and F. Walter Max-Planck-Institut für Astronomie,
Königstuhl 17, D-69117 Heidelberg, Germany tamburro@mpia.de, rix@mpia.de,
walter@mpia.de E. Brinks Centre for Astrophysics Research, University of
Hertfordshire, College Lane, Hatfield AL10 9AB, United Kingdom
e.brinks@herts.ac.uk W.J.G. de Blok Department of Astronomy, University of
Cape Town, Private Bag X3, Rondebosch 7701, South Africa
edeblok@circinus.ast.uct.ac.za R.C. Kennicutt Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
robk@ast.cam.ac.uk M.-M. Mac Low11affiliation: also Max-Planck-Institut für
Astronomie, and Institut für Theoretische Astrophysik, Zentrum für Astronomie
der Universität Heidelberg Department of Astrophysics, American Museum of
Natural History, 79th Street and Central Park West, New York, NY 10024-5192,
USA mordecai@amnh.org
###### Abstract
We estimate a characteristic timescale for star formation in the spiral arms
of disk galaxies, going from atomic hydrogen (H I) to dust-enshrouded massive
stars. Drawing on high-resolution H I data from The H I Nearby Galaxy Survey
and 24 $\mu$m images from the Spitzer Infrared Nearby Galaxies Survey we
measure the average angular offset between the H I and 24$\mu$m emissivity
peaks as a function of radius, for a sample of 14 nearby disk galaxies. We
model these offsets assuming an instantaneous kinematic pattern speed,
$\Omega_{p}$, and a timescale, $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$, for the
characteristic time span between the dense H I phase and the formation of
massive stars that heat the surrounding dust. Fitting for $\Omega_{p}$ and
$t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$, we find that the radial dependence of
the observed angular offset (of the H I and 24 $\mu$m emission) is consistent
with this simple prescription; the resulting corotation radii of the spiral
patterns are typically $R_{\rm cor}\simeq 2.7R_{s}$, consistent with
independent estimates. The resulting values of $t_{{\rm HI}\mapsto 24\,\mu{\rm
m}}$ for the sample are in the range 1–4 Myr. We have explored the possible
impact of non-circular gas motions on the estimate of $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ and have found it to be substantially less than a factor of
2. This implies that a short timescale for the most intense phase of the
ensuing star formation in spiral arms, and implies a considerable fraction of
molecular clouds exist only for a few Myr before forming stars. However, our
analysis does not preclude that some molecular clouds persist considerably
longer. If much of the star formation in spiral arms occurs within this short
interval $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$, then star formation must be
inefficient, in order to avoid the short-term depletion of the gas reservoir.
###### Subject headings:
galaxies: evolution – galaxies: ISM – galaxies: kinematics and dynamics –
galaxies: spiral – stars: formation
††slugcomment: Accepted for publication in the AJ special THINGS issue. For a
high-resolution version visit: http://www.mpia.de/THINGS/Publications.html
## 1\. INTRODUCTION
Roberts (1969, hereafter R69) was the first to develop the scenario of spiral-
arm-driven star formation in galaxy disks. In this picture a spiral density
wave induces gravitational compression and shocks in the neutral hydrogen gas,
which in turn leads to the collapse of (molecular) gas clouds that results in
star formation. This work already pointed out the basic consequences for the
relative geometry of the dense cold gas reservoir111This paper pre-dates
observational studies of molecular gas in galaxies. and the emergent young
stars: when viewed from a reference frame that corotates with the density
wave, the densest part of the atomic hydrogen (H I) lies at the shock (or just
upstream from it), while the young stars lie downstream from the density wave.
Using H I and H$\alpha$ as the tracers of the cold gas and of the young stars,
respectively, R69 found a qualitative support in the data available at the
time. In this picture, the characteristic timescale for this sequence of
events is reflected in the typical angular offset, at a given radius, between
tracers of the different stages of spiral-arm-driven star formation.
While this qualitative picture has had continued popularity, quantitative
tests of the importance of spiral density waves as star-formation trigger (Lin
& Shu, 1964) and of the timescales for the ensuing star formation have proven
complicated. First, it has become increasingly clear that even in galaxies
with grand-design spiral arms, about half of the star formation occurs in
locations outside the spiral arms (Elmegreen & Elmegreen, 1986). Second, stars
formed from molecular clouds (not directly from H I) and very young star
clusters are dust enshrouded at first. Moreover, the actual physical mechanism
that appears to control the rate and overall location of star formation in
galaxies is the gravitational instability of the gas and existing stars (e.g.
Li, Mac Low, & Klessen, 2005). Stars only form above a critical density
(Martin & Kennicutt, 2001) which is consistent with the predicted Toomre
(1964) criterion for gravitational instability as generalized by Rafikov
(2001). Although, in galaxies with prominent spiral structure local gas
condensations are governed by magneto-rotational instabilities—spiral arms are
regions of low shear where the transfer of angular momentum is carried out by
magnetic fields (Kim & Ostriker, 2002, 2006). Obtaining high-resolution,
sensitive maps of all phases in this scenario (H I, molecular gas, dust-
enshrouded young stars, unobscured young stars) has proven technically
challenging.
If the star formation originates from direct collapse of gravitationally
unstable gas, and if the rotation curve and approximate pattern speed of the
spiral arms are known, the geometric test suggested by R69 provides a
timescale for the end-to-end (from H I to young stars) process of star
formation. Of course, there are other ways of estimating the timescales that
characterize the evolutionary sequence of the interstellar medium (ISM), based
on other physical arguments. However, other lines of reasoning have led to
quite a wide range of varying lifetime estimates as discussed below.
Offsets between components such as CO and H$\alpha$ emission in the disks of
spiral galaxies have indeed been observed (Vogel, Kulkarni, & Scoville, 1988;
Garcia-Burillo, Guelin, & Cernicharo, 1993; Rand & Kulkarni, 1990; Scoville et
al., 2001). Mouschovias, Tassis, & Kunz (2006) remarked that the angular
separation between the dust lanes and the peaks of H$\alpha$ emission found
for nearby spiral galaxies (e.g. observed by Roberts, 1969; Rots, 1975)
implied timescales of the order of 10 Myr. More recently Egusa, Sofue, &
Nakanishi (2004), using the angular offset between CO and H$\alpha$ in nearby
galaxies, derived $t_{{\rm CO}\mapsto{\rm H}\alpha}$ $\simeq 4.8$ Myr.
Observationally, the H I surface density is found to correlate well with sites
of star formation and emission from molecular clouds (Wong & Blitz, 2002;
Kennicutt, 1998). The conversion timescale of H I$\mapsto$H2 is a key issue
since it determines how well the peaks of H I emission can be considered as
potential early stages of star formation. H2 molecules only form on dust grain
surfaces in dusty regions that shield the molecules from ionizing UV photons.
Their formation facilitates the subsequent building up of more complex
molecules (e.g. Williams, 2005). Within shielded clouds the conversion
timescale H I$\mapsto$H2 is given by $\tau_{{\rm H}_{2}}\sim
10^{9}/n_{0}\;$yr, where $n_{0}$ is the proton density in cm-3 (Hollenbach &
Salpeter, 1971; Jura, 1975; Goldsmith & Li, 2005; Goldsmith, Li, & Krčo,
2007). Given the inverse proportionality with $n_{0}$, the conversion
timescale can vary from the edge of a molecular cloud ($\tau\simeq 4\times
10^{6}\;$yr, $n_{0}\sim 10^{3}$) to the central region ($\tau\sim 10^{5}\;$yr,
$n_{0}\sim 10^{4}$) where the density is higher. Local turbulent compression
can further enhance the local density, and thus decrease the conversion
timescale (Glover & Mac Low, 2007). Thus, even short cloud-formation
timescales remain consistent with the H I$\mapsto$H2 conversion timescale.
The subsequent evolution (see e.g. Beuther et al., 2007, for a review)
involves the formation of cloud cores (initially starless) and then star
cluster formation through accretion onto protostars, which finally become main
sequence stars. High-mass stars evolve more rapidly than low-mass stars. Stars
with $M\geq 5M_{\odot}$ reach the main sequence quickly, in less than 1 Myr
(Hillenbrand et al., 1993; Palla & Stahler, 1999), while they are still deeply
embedded and actively accreting. The O and B stars begin to produce an intense
UV flux that photoionizes the surrounding dusty environment within a few Myr,
and subsequently become optically visible (Thronson & Telesco, 1986).
A different scenario is suggested by Allen (2002), in which young stars in the
disks of galaxies produce H I from their parent H2 clouds by
photodissociation. According to this scenario, the H I should not be seen
furthest upstream in the spiral arm, but rather between the CO and
UV/H$\alpha$ regions. Allen et al. (1986) indeed report observation of H I
downstream of dust lanes in M83.
Several lines of reasoning, however, point toward longer star-formation
timescales and molecular cloud lifetimes, much greater than 10 Myr. Krumholz &
McKee (2005) conclude that the star-formation rate in the solar neighborhood
is low. In fact, they point out that the star-formation rate in the solar
neighborhood is $\sim 100$ times smaller than the ratio of the masses of
nearby molecular clouds to their free-fall time $M_{\rm MC}/\tau_{\rm ff}$,
which also indicates the rate of compression of molecular clouds. Individual
dense molecular clouds have been argued to stay in a fully molecular state for
about 10-15 Myr before their collapse (Tassis & Mouschovias, 2004), and to
transform about 30% of their mass into stars in $\geq 7\>\tau_{\rm ff}$ ($\sim
10^{6}\;$yr e.g. considering the mass of the Orion Nebula Cluster, ONC Tan,
Krumholz, & McKee, 2006). Large molecular clouds have been calculated to
survive 20 to 30 Myr before being destroyed by the stellar feedback by
Krumholz, Matzner, & McKee (2006). Based on observations, Palla & Stahler
(1999, 2000) argue that the star formation rate in the ONC was low
10${}^{7}\;$yr ago, and that it increased only recently. Blitz et al. (2007),
using a statistical comparison of cluster ages in the Large Magellanic Cloud
(LMC) to the presence of CO, found that the lifetime for giant molecular
clouds is 20–30 Myr.
However, other studies conclude that the timescales for star-formation are
rather short. Hartmann (2003) pointed out that the Palla–Stahler model is not
consistent with observations since most of the molecular clouds in the ONC are
forming stars at the same high rate. The stellar age or the age spread in
young open stellar clusters is not necessarily a useful constraint on the
star-formation timescale: the age spread, for example, may result from
independent and non-simultaneous bursts of star-formation (Elmegreen, 2000).
Ballesteros-Paredes & Hartmann (2007) pointed out that the molecular cloud
lifetime must be shorter than the value of $\tau_{\rm MC}\simeq 10\;$Myr
suggested by Mouschovias, Tassis, & Kunz (2006). Also subsequent star
formation must proceed very quickly, within a few Myr (Vázquez-Semadeni et
al., 2005; Hartmann, Ballesteros-Paredes, & Bergin, 2001). Prescott et al.
(2007) found strong association between 24$\;\mu$m sources and optical H II
regions in nearby spiral galaxies. This provides constraints on the lifetimes
of star-forming clouds: the break out time of the clouds and their parent
clouds is less or at most of the same order as the lifetime of the H II
regions, therefore a few Myr. Dust and gas clouds must dissipate on a
timescale no longer than 5–10 Myr.
In conclusion, all the previous studies listed aim to estimate the lifetimes
of molecular clouds or the timescale separation between the compression of
neutral gas and newly formed stars. Most of these studies are based on
observations of star-forming regions both in the Milky Way and in external
galaxies, and in all cases the derived timescales lie in a range between a few
Myr and several tens of Myr.
In this paper, we examine a new method (§ 2) for estimating the timescale to
proceed from H I compression to star formation in nearby spiral galaxies. We
compare Spitzer Space Telescope/MIPS $24\>\mu$m data from the Spitzer Near
Infrared Galaxies Survey (SINGS; Kennicutt et al. 2003) to 21 cm maps from the
H I Nearby Galaxy Survey (THINGS; Walter et al. 2008). The proximity of our
targets allows for high spatial resolution. In § 3 we give a description of
the data. The MIPS bands (24, 70 and $160\;\mu$m) are tracers of warm dust
heated by UV and are therefore good indicators of recent star-formation
activity (see for example Dale et al., 2005). We used the band with the best
resolution, $24\;\mu$m, which has been recognized as the best of the Spitzer
bands for tracing star formation (Calzetti et al., 2005, 2007; Prescott et
al., 2007); the $8\;\mu$m Spitzer/IRAC band has even higher resolution but is
contaminated by PAH features that undergo strong depletion in the presence of
intense UV radiation (Dwek, 2005; Smith et al., 2007). In § 4 we describe how
we use azimuthal cross-correlation to compare the H I and $24\;\mu$m images
and derive the angular offset of the spiral pattern. This algorithmic approach
minimizes possible biases introduced by subjective assessments. We describe
our results in § 5 where we derive $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ for
our selection of objects. Finally, we discuss the implications of our results
in § 6 and draw conclusions in § 7.
## 2\. METHODOLOGY
The main goal of this paper is to estimate geometrically the timescales for
spiral-arm-driven star formation using a simple kinematic model, examining the
R69 arguments in light of state-of-the-art data. Specifically, we set out to
determine the relative geometry of two tracers for different stages of star-
formation sequence in a sample of nearby galaxies, drawing on the SINGS and
THINGS data sets (see § 3): the 24 $\mu$m and the H I emission.
While the angular offset between these two tracers is an empirical model-
independent measurement, a conversion into a star-formation timescale assumes
(a) that peaks of the H I trace material that is forming molecular clouds, and
(b) that the peaks of the 24 $\mu$m emission trace the very young, still dust-
enshrouded star clusters, where their UV emission is absorbed and re-radiated
into the mid- to far-infrared wavelength range ($\sim$5 $\mu$m to
$\sim$500$\mu$m). The choice of these particular tracers was motivated by the
fact that they should tightly bracket the conversion process of molecular gas
into young massive stars, and by the availability of high-quality data from
the SINGS and THINGS surveys. Note that a number of imaging studies in the
near-IR have shown (e.g. Rix & Zaritsky, 1995) that the large majority of
luminous disk galaxies have a coherent, dynamically relevant spiral arm
density perturbation. Therefore, this overall line of reasoning can sensibly
be applied to a sample of disk galaxies.
We consider a radius in the galaxy disk where the spiral pattern can be
described by a kinematic pattern speed, $\Omega_{p}$, and the local circular
velocity $v_{c}(r)\equiv\Omega(r)\times r$. Then two events separated by a
time $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$will have a phase offset of
$\Delta\phi(r)=(\Omega(r)-\Omega_{p})\;\mbox{$t_{{\rm HI}\mapsto 24\,\mu{\rm
m}}$},$ (1)
where $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ denotes the time difference between
two particular phases that we will study here. If the spiral pattern of a
galaxy indeed has a characteristic kinematic pattern speed, the angular offset
between any set of tracers is expected to vary as a function of radius in a
characteristic way. Considering the chronological sequence, defining the
angular phase difference $\Delta\phi\equiv\phi_{24\,\mu{\rm m}}-\phi_{\rm HI}$
and adopting the convention that $\phi$ increases in the direction of
rotation, we expect the qualitative radial dependence plotted in Figure 1:
$\Delta\phi>0$ where the galaxy rotates faster than the pattern speed,
otherwise $\Delta\phi<0$. Where $\Omega(R_{\rm cor})=\Omega_{p}$, at the so-
called corotation radius, we expect the sign of $\Delta\phi$ to change.
In practice, the gaseous and stellar distribution is much more complex than in
the qualitative example of Figure 1, since the whole spiral network, even for
galaxies where the spiral arms are well defined such as in grand-design
galaxies, typically exhibits a full wealth of smaller scale sub-structures
both in the arms and in the inter-arm regions. The optimal method to measure
the angular offset between the two observed patterns is therefore through
cross-correlation (§ 4). We treat the timescale $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ and the present-day pattern speed $\Omega_{p}$ as global
constants for each galaxy, although these two parameters might, in principle,
vary as function of galactocentric radius. Note that we need not to rely on
the assumption that the spiral structure is quasi-stationary over extended
periods, $t\geq t_{\rm dyn}$. Even if spiral arms are quite dynamic,
continuously forming and breaking, and with a pattern speed varying with
radius, our analysis will hold approximately.
Figure 1.— Schematic geometry adopted to derive the azimuthal phase difference
$(\phi_{24\,\mu{\rm m}}-\phi_{\rm HI})(r)\equiv\Delta\phi(r)$ between the H I
and the $24\,\mu{\rm m}$ emission, with $\phi$ increasing in the direction of
rotation. The sketch shows part of a face-on galaxy rotating anti-clockwise,
with the center as indicated. The solid curved lines represent the two
components within one spiral arm, namely the H I and the heated dust. The
angular separation between the two components is exaggerated for clarity. We
measured the deprojected phase difference $\Delta\phi(r)$ at a given radius.
Inside corotation, $R_{\rm cor}$, the material is rotating faster than the
pattern speed and the $24\,\mu{\rm m}$ emission lies ahead of the H I
($\phi_{\rm HI}<\phi_{24\,\mu{\rm m}}$). At corotation the two patterns
coincide, and outside $R_{\rm cor}$ the picture is reversed since the pattern
speed exceeds the rotation of the galaxy.
## 3\. DATA
The present analysis is based on the 21 cm emission line maps, a tracer of the
neutral atomic gas for the 14 disk galaxies listed in Table 1, which are taken
from THINGS. These high-quality NRAO222The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. Very Large Array
observations provide data cubes with an angular resolution of $\simeq
6^{\prime\prime}$ and spectral resolution of 2.6 or 5.2 $\rm km\;s^{-1}$.
Since the target galaxies are nearby, at distances of 3–10 Mpc, the linear
resolution of the maps corresponds to 100–300 pc. The H I data cubes of our
target galaxies are complemented with near-IR images, which are public data.
In particular, the majority of the THINGS galaxies (including all those in
Table 1) have also been observed within the framework of the SINGS and we make
an extensive use of the $24\;\mu$m MIPS images (see § 4.2). Figure 2
illustrates our data for two of the sample galaxies, NGC 5194 and NGC 2841.
The $24\;\mu$m band image is shown in color scale, and the contours show the H
I emission map. To obtain the exponential scale length of the stellar disk
(see § 4), we use $3.6\;\mu$m Infrared Array Camera (IRAC) images when
available, otherwise we use $H$ band images taken from the Two Micron All Sky
Survey (2MASS; Jarrett et al., 2003). To check the consistency of our results,
we use CO maps from the Berkeley-Illinois-Maryland Aaaociation Survey Of
Nearby Galaxies (BIMA-SONG Helfer et al., 2003) for some of our target
galaxies.
Table 1THINGS and SINGS Target Galaxies Obj. Name | Alt. Name | $R_{25}$ (′) | $R_{s}$ (′) | Band | $i$ (∘) | P.A. (∘) | $D$ (Mpc) | $v_{\rm max}$ ($\rm km\;s^{-1}$)
---|---|---|---|---|---|---|---|---
| | (1) | (2) | (3) | (4) | (5) | (6) | (7)
NGC 2403 | | 9.98 | $1.30^{\star}$ | $H$ | 63 | 124 | 3.22 | 128
NGC 2841 | | 3.88 | $0.92^{\star}$ | 3.6 | 74 | 153 | 14.1 | 331
NGC 3031 | M81 | 10.94 | $3.63\pm 0.2$ | 3.6 | 59 | 330 | 3.63 | 256
NGC 3184 | | 3.62 | $0.92\pm 0.09$ | $H$ | 16 | 179 | 11.1 | 260
NGC 3351 | | 3.54 | $0.86\pm 0.03$ | 3.6 | 41 | 192 | 9.33 | 210
NGC 3521 | | 4.8 | $0.74\pm 0.02$ | 3.6 | 73 | 340 | 10.05 | 242
NGC 3621 | | 5.24 | $0.80^{\star}$ | $H$ | 65 | 345 | 6.64 | 144
NGC 3627 | M66 | 4.46 | $0.95^{\star}$ | 3.6 | 62 | 173 | 9.25 | 204
NGC 5055 | M63 | 6.01 | $1.16\pm 0.05$ | $H$ | 59 | 102 | 7.82 | 209
NGC 5194 | M51 | 3.88 | $1.39\pm 0.11$ | $H$ | 42 | 172 | 7.77 | 242
NGC 628 | M74 | 4.77 | $1.10\pm 0.09$ | $H$ | 7 | 20 | 7.3 | 220
NGC 6946 | | 5.35 | $1.73\pm 0.07$ | $H$ | 32.6 | 242 | 5.5 | 201
NGC 7793 | | 5.0 | $1.16\pm 0.05$ | $H$ | 50 | 290 | 3.82 | 109
NGC 925 | | 5.23 | $1.43^{\star}$ | 3.6 | 66 | 286 | 9.16 | 121
Figure 2.— The $24\;\mu$m band image is plotted in color scale for the
galaxies NGC 5194 (left) and NGC 2841 (right); the respective H I emission map
is overlayed with green contours.
## 4\. ANALYSIS
All analysis in this paper started from fully reduced images and data cubes.
On this data we carry out two main steps. First, we derive the rotation curve
$v_{c}(r)$ of the H I and the geometrical projection parameters of the galaxy
disk, and use these parameters to deproject the maps of the galaxies to face-
on orientation (see Table 1). Second, we sample the face-on maps in concentric
annuli. For each annulus we cross-correlate the corresponding pair of H I and
$24\;\mu$m fluxes, in order to derive the angular offset between the H I and
the $24\;\mu$m patterns as a function of radius.
For three of the galaxies listed in Table 1 (NGC 628, NGC 5194, and NGC 3627),
we also measure the angular offset between the CO and $24\;\mu$m emission
maps. If the ISM evolves sequentially from atomic into molecular gas, and then
subsequently initiates the formation of stars, considering the kinematics
expressed in Eq. 1, we expect the CO emission to lie in between the H I and
the $24\;\mu$m.
### 4.1. Analysis of the H I Kinematics
For each object we apply the same general approach: we first perform adaptive
binning of the H I data cube regions with low signal-to-noise (S/N) ratio
using the method described by Cappellari & Copin (2003). From the resulting
spatially binned data cubes we fit the 21 cm emission lines with a single
Gaussian profile and use the parameterization to derive (1) the line-of-sight
velocity map $v(x,y)$, given by the line centroid, and (2) the flux maps
$\mu_{0}(x,y)\equiv a(x,y)/(\sqrt{2\pi}\,\sigma(x,y))$, where $a$ and $\sigma$
are the Gaussian peak amplitude and width, respectively. Since we do not need
to derive the rotation curve with high accuracy for the purpose of this paper,
we limit our model to a co-planar rotating disk with circular orbits described
by
$v(x,y)=v_{\rm sys}+v_{c}(r)\,\sin i\cos\psi,$ (2)
where $v(x,y)$ is the observed velocity map along the line of sight (see
Begeman, 1989). For simplicity, we assume here that the orbits are circular,
though we address the issue of non-circular motions in § 5.4. By $\chi^{2}$
minimization fitting333The fitting has been performed with the mpfit IDL
routine found at the URL:
http://cow.physics.wisc.edu/$\sim$craigm/idl/fitting.html of the model
function in Eq. 2 to the observed velocity map $v(x,y)$, we obtain the
systemic velocity $v_{\rm sys}=\rm const$, the inclination $i$ and the
position angle (P.A.) of the geometric projection of the disk on to the sky.
Here, $\psi$ is the azimuthal angle on the plane of the inclined disk (not the
sky) and is a function of $i$ and P.A. The line where $\psi=0$ denotes the
orientation of the line of nodes on the receding side of the disk. The
kinematic center $(x_{0},y_{0})$ is fixed a priori and is defined as the
central peak of either the IRAC $3.6\;\mu$m or the 2MASS $H$ band image. The
positions of the dynamical centers used here are consistent with those derived
in Trachternach et al. (2008). We parameterize the deprojected rotation curve
$v_{c}$ with a four-parameter arctan-like function (e.g., Rix et al., 1997)
$v_{c}(r)=v_{0}\;(1+x)^{\beta}\;(1+x^{-\gamma})^{-1/\gamma},$ (3)
where $x=r/r_{0}$. Here, $r_{0}$ is the turn-over radius, $v_{0}$ is the scale
velocity, $\gamma$ determines the sharpness of the turnover and $\beta$ is the
asymptotic slope at larger radii.
The values for the projection parameters $i$, and P.A., the systemic velocity
$v_{\rm sys}$, and the asymptotic velocity that have been obtained applying
the approach described above, are consistent with the values reported in Table
1. From the maximum value of Eq. 3 we obtain the maximum rotational velocity
$v_{\rm max}$, which is listed for all the sample galaxies in Table 1.
### 4.2. Azimuthal Cross-Correlation
The central analysis step is to calculate by what angle $\Delta\phi$ the
patterns of H I and $24\;\mu$m need to be rotated with respect to each other
in order to best match. We use the kinematically determined orientation
parameters, $i$ and P.A., to deproject both the H I and $24\;\mu$m images to
face-on. To estimate the angular offset $\Delta\phi$ between the two flux
images at each radius, we divide the face-on images into concentric rings of
width $\sim$5′′ and extract the flux within this annulus as a function of
azimuth. We then use a straightforward cross-correlation (CC) to search for
phase lags in $f_{\rm HI}(\phi|r)$ versus $f_{24\,\mu{\rm m}}(\phi|r)$. In
general, the best match between two discrete vectors $x$ and $y$ is realized
by minimizing as a function of the phase shift $\ell$ (also defined as lag)
the quantity
$\chi^{2}_{x,y}(\ell)=\sum_{k}{\left[x_{k}-y_{k-\ell}\right]}^{2},$ (4)
where the sum is calculated over all the $N$ elements of $x$ and $y$ with
$k=0,1,2,...,N-1$. Specifically here, for a given radius $r=\hat{r}$ we
consider for all discrete values of azimuth $\phi$:
$x_{k}=f_{\rm HI}(\phi_{k}|\hat{r})\quad\mbox{and}\quad
y_{k-\ell}=f_{24\,\mu{\rm m}}(\phi_{k-\ell}|\hat{r}).$ (5)
Expanding the argument of the sum in Eq. 4 one obtains that $\chi^{2}(\ell)$
is independent of the terms $\sum_{k}x_{k}^{2}$ and $\sum_{k}y_{k-\ell}^{2}$,
and $\chi^{2}$ is minimized by the maximization of
$cc_{x,y}(\ell)=\sum_{k}\left[x_{k}\,y_{k-\ell}\right],$ (6)
which is defined as the CC coefficient. Here we used the normalized CC
$cc_{x,y}(\ell)=\frac{\sum_{k}\left[(x_{k}-\bar{x})\,(y_{k-\ell}-\bar{y})\right]}{\sqrt{\sum_{k}(x_{k}-\bar{x})^{2}\;\sum_{k}(y_{k}-\bar{y})^{2}}},$
(7)
where $\bar{x}$ and $\bar{y}$ are the mean values of $x$ and $y$ respectively.
Here the slow, direct definition has been used and not the fast Fourier
transform method. The vectors are wrapped around to ensure the completeness of
the comparison. With this definition the CC coefficient would have a maximum
value of unity for identical patterns, while for highly dissimilar patterns it
would be much less than 1. We apply the definition in Eq. 7 using the
substitutions of Eq. 5 to compute the azimuthal CC coefficient $cc(\ell)$ of
the H I and the $24\;\mu$m images. The best match between the H I and the
$24\;\mu$m signals is realized at a value $\ell_{\rm max}$ such that
$cc(\ell_{\rm max})$ has its peak value. Since the expected offsets are small
(only a few degrees) we search the local maximum around $\ell\simeq 0$. The
method is illustrated in Figure 3, which shows that $cc(\ell)$ has several
peaks, as expected due to the self-similarity of the spiral pattern.
Figure 3.— Representative examples for the determination of the azimuthal H
I–24 $\mu$m offset: shown is the cross-correlation $cc(\ell)$ of the two
functions $f_{\rm HI}(\phi_{k},\hat{r})$ and $f_{24\,\mu{\rm
m}}(\phi_{k-\ell},\hat{r})$, as a function of azimuth offset $\phi$ at a fixed
radius $\hat{r}$. The present example shows the $cc(\ell)$ profile calculated
for NGC 628 at $\hat{r}\simeq 2^{\prime}$ ($\simeq$4.2 kpc) and NGC 5055 at
$\hat{r}\simeq 2.6^{\prime}$ ($\simeq$6.4 kpc), in the left and right columns,
respectively. Top panel: the $cc(\ell)$ profile in the entire range
$[-180^{\circ},180^{\circ}]$; bottom panel: a zoom of the range
$[-30^{\circ},30^{\circ}]$. We considered an adequate range greater than the
width of the $cc(\ell)$ profile around $\ell_{\rm max}$ and interpolated
$cc(\ell)$ locally ($\sim\pm 10^{\circ}$ in the two example plots) with a
fourth-degree polynomial $p_{4}(\ell)=\sum_{n=0}^{4}a_{n}\,\ell^{n}$, and
calculated numerically the peak value $\ell_{\rm max}$. The bottom panel shows
the fit residuals overplotted around the zero level.
We consider a range that encompasses the maximum of the $cc(\ell)$ profile,
i.e. the central $\sim$100–150 data points around $\ell_{\rm max}$. This
number, depending on the angular size of the ring, is dictated by the
azimuthal spread of the spiral arms and the number of substructures (e.g.,
dense gas clouds, star clusters, etc.) per unit area. This corresponds, for
example, to a width in $\ell$ of a few tens of degrees at small radii ($\sim
1^{\prime}$), depending on the distance of the object, and a range in width of
$\ell$ decreasing linearly with the radius. We interpolate $cc(\ell)$ around
$\ell_{\rm max}$ with a fourth-degree polynomial using the following
approximation: $cc(\ell)\simeq p_{4}(\ell)=\sum_{n=0}^{4}a_{n}\,\ell^{n}$ and
calculate numerically (using the Python444http://www.python.org package
scipy.optimize) the peak value at $\ell_{\rm max}$, $p_{4}(\ell_{\rm max})$.
By repeating the procedure for all radii, the angular offset H
I$\mapsto$24$\;\mu$m results in $\Delta\phi(r)=-\ell_{\rm max}(r)$. The
direction or equivalently the sign of the lag $\ell_{\rm max}$ between two
generic vectors $x$ and $y$ depends on the order of $x$ and $y$ in the
definition of the CC coefficient. Note that $cc_{x,y}(\ell)$ in Eq. 6 is not
commutative for interchange of $x$ and $y$, being
$cc_{y,x}(\ell)=cc_{x,y}(-\ell)$. For $\ell_{\rm max}=0$ the two patterns best
match at zero azimuthal phase shift. The error bars for $\delta\ell_{\rm
max}(r)$ have been evaluated through a Monte Carlo approach, adding normally
distributed noise and assuming the expectation values of $\ell_{\rm max}$ and
$\delta\ell_{\rm max}$ as the mean value and the standard deviation,
respectively, after repeating the determination $N=100$ times.
Our analysis is limited to the radial range between low S/N regions at the
galaxy centers and their outer edges. In the H I emission maps the S/N is low
near the galaxy center, where the H I is converted to molecular H2, whereas
for the $24\;\mu$m band the emission map has low S/N near $R_{25}$ (and in
most cases already at $\sim 0.8\>R_{25}$). Regions with S/N $<3$ in either the
H I or $24\;\mu$m images have been clipped. We also ignore those points
$\ell_{\rm max}$ with a coefficient $cc(\ell_{\rm max})$ lower than a
threshold $cc\simeq 0.2$. We further neglect any azimuthal ring containing
less than a few hundred points, which occurs near the image center and near
$R_{25}$. The resulting values $\Delta\phi(r)$ are shown in Figure 4.
Figure 4.— Radial profiles for the angular offset H I$\mapsto$24$\;\mu$m for
the entire sample, obtained by sampling face-on H I and 24$\;\mu$m maps
concentric rings and cross-correlating the azimuthal profiles for each radius.
The solid line is the best-fit model to the observed data points, denoted by
squared symbols, which has been obtained by $\chi^{2}$ minimization of Eq. 8;
the solid curve intersects the horizontal axis at corotation (defined as
$\Delta\phi=0$). The solid and dashed vertical lines indicate the
$2.7\>R_{s}\simeq R_{\rm cor}$ value and error bars, derived by Kranz, Slyz, &
Rix (2003).
### 4.3. Disk Exponential Scale Length
We also determine the disk exponential scale length $R_{s}$ for our sample
using the galfit555Found at URL:
http://zwicky.as.arizona.edu/$\sim$cyp/work/galfit/galfit.html algorithm
(Peng et al., 2002). In particular, we fit an exponential disk profile and a
de Vaucouleurs profile to either the IRAC $3.6\;\mu$m or to the 2MASS $H$ band
image. As galfit underestimates the error on $R_{s}$ (as recognized by the
author of the algorithm), typically $\delta R_{s}/R_{s}<1$%, we therefore also
use the IRAF task ellipse (Jedrzejewski, 1987) to derive the radial surface
brightness profile and fit $R_{s}$. After testing the procedure on a few
objects, we note only small differences (of the order of the error bars in
Table 1) when deriving $R_{s}$ from the $H$ band and the $3.6\;\mu$m band.
## 5\. RESULTS
### 5.1. Angular Offset
With the angular offset $\Delta\phi(r)\equiv\langle\phi_{24\,\mu{\rm
m}}-\phi_{\rm HI}\rangle(r)$, where $\phi$ increases in the direction of
rotation, and the rotation curve $v_{c}(r)$ for each radial bin, we can
rewrite Eq. 1 as
$\Delta\phi(r)=\left(\frac{v_{c}(r)}{r}-\Omega_{p}\right)\;\times\;t_{{\rm
HI}\mapsto 24\,\mu{\rm m}},$ (8)
where $\Omega(r)\equiv v_{c}/r$. Since $\Omega(r)>\Omega_{p}$ inside the
corotation radius $R_{\rm cor}$, and $\Omega(r)<\Omega_{p}$ outside
corotation, we expect $\Delta\phi(r)>0$ for $r<R_{\rm cor}$ and
$\Delta\phi(r)<0$ for $r>R_{\rm cor}$. At corotation, where $\Delta\phi(R_{\rm
cor})=0$, the two components H I and $24\;\mu$m should have no systematic
offset. We assume $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and $\Omega_{\rm p}$ to
be constant for any given galaxy, and that all the spirals are trailing, since
the only spiral galaxies known to have a leading pattern are NGC 3786, NGC
5426 and NGC 4622 (Thomasson et al., 1989; Byrd, Freeman, & Buta, 2002). By
$\chi^{2}$ fitting the model prediction of Eq. 8 to the measured angular
offsets $\Delta\phi(r)$ in all radial bins of a galaxy, we derive best-fit
values for $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and $\Omega_{\rm p}$.
The $\Delta\phi(r)$ data and the resulting best-fits are shown in Figure 4,
with the resulting best-fit values listed in Table 2. In Figure 4 we plot for
all objects the radial profile of the angular offsets $\Delta\phi(r)$. The
solid line represents the best fit model proscribed by Eq. 8. The square
symbols in the plot represent the fitted data points from § 4.2. Looking at
the ensemble results in Figure 4, two points are noteworthy: (1) the geometric
offsets are small, typically a few degrees and did need high-resolution maps
to become detectable, (2) the general radial dependence follows overall the
simple prescription of Eq. 8 quite well.
Table 2Characteristic Timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and Pattern Speed $\Omega_{p}$ Resulting from a $\chi^{2}$ fit of the Observed Angular Offset via Equation 1 Obj. Name | Alt. Name | $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ | $\Omega_{p}$ | $R_{\rm cor}/R_{s}$ | $\Omega_{p}$
---|---|---|---|---|---
| | [Myr] | [km s-1 kpc-1] | | [km s-1 kpc-1]
NGC 2403 | | $1.4\pm 0.5$ | $30\pm 4$ | $2.8\pm 0.3$ |
NGC 2841 | | $4.4\pm 0.5$ | $42\pm 2$ | $2.8\pm 0.1$ |
NGC 3031 | M81 | $0.5\pm 0.3^{\dagger}$ | $27\pm 13^{\dagger}$ | $2.3\pm 1.4$ | $24^{\rm a}$
NGC 3184 | | $1.8\pm 0.4$ | $38\pm 5$ | $2.3\pm 0.5$ |
NGC 3351 | M95 | $2.2\pm 0.3$ | $38\pm 3$ | $2.3\pm 0.4$ |
NGC 3521 | | $2.9\pm 0.4$ | $32\pm 2$ | $3.6\pm 0.2$ |
NGC 3621 | | $2.3\pm 1.3^{\dagger}$ | $31\pm 11^{\dagger}$ | $2.8\pm 1.0$ |
NGC 3627 | M66 | $3.1\pm 0.4$ | $25\pm 4$ | $3.0\pm 0.5$ |
NGC 5055 | M63 | $1.3\pm 0.3$ | $20\pm 5$ | $3.8\pm 1.5$ | $30-40^{\rm b}$
NGC 5194 | M51 | $3.4\pm 0.8$ | $21\pm 4$ | $1.5\pm 0.6$ | $38\pm 7^{\rm c}$, $40\pm 8^{\rm d}$
NGC 628 | M74 | $1.5\pm 0.5$ | $26\pm 3$ | $2.2\pm 0.4$ | $32\pm 2^{\rm e}$
NGC 6946 | | $1.3\pm 0.3^{\dagger}$ | $36\pm 4^{\dagger}$ | $1.7\pm 0.4$ | $39\pm 9^{\rm c}$, $42\pm 6^{\rm d}$
NGC 7793 | | $1.2\pm 0.5^{\dagger}$ | $40\pm 10^{\dagger}$ | $2.5\pm 0.9$ |
NGC 925 | | $5.7\pm 1.6$ | $11\pm 1.0$ | $2.1\pm 0.2$ | $7.7^{\rm f}$
### 5.2. $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and $R_{\rm cor}$
Because $\Delta\phi(r)$ is consistent with (and follows) the predictions of
the simple geometry and kinematics in Eq. 8, the procedure adopted here turns
out to be an effective method to derive the following: (1) the time lag
$t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$, which should bracket the timescale
needed to compress the molecular gas, trigger star formation, and heat the
dust; it therefore represents also an estimate for the lifetime of star-
forming molecular clouds; (2) the kinematic pattern speed $\Omega_{p}$ of the
galaxy spiral pattern and, equivalently, the corotation radius $R_{\rm cor}$.
We now look at the ensemble properties of the resulting values for $t_{{\rm
HI}\mapsto 24\,\mu{\rm m}}$ and $R_{\rm cor}$. The scatter of the individually
fitted $\Delta\phi$ points is significantly larger than their error bars (as
shown in Fig. 4), which may be due to the galactic dynamics being more complex
than our simple assumptions. For example, the pattern speed may not be
constant over the entire disk or there may be multiple corotation radii and
pattern speeds, as, for instance, found by numerical simulations (Sellwood &
Sparke, 1988) and observed in external galaxies (Hernández et al., 2005).
Even though a considerable intrinsic scatter characterizes $\Delta\phi(r)$ for
most of our sample galaxies, and the error bars of $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ listed in Table 2 are typically $>15\,$%, the histogram of
the characteristic timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ in Figure 5
shows overall a relatively small spread for a sample of 14 galaxies of
different Hubble types: the timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$
occupy a range between 1 and 4 Myr for almost all the objects.
Figure 5.— Histogram of the timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$
derived for the 14 sample galaxies listed in Table 1 from the fits in Figure 4
(also listed in Table 2). The timescales range between 1 and 4 Myr for almost
all galaxies.
The solid curve in Figure 4, representing the prescription of Eq. 8,
intersects the horizontal axis at the corotation radius $R_{\rm cor}$, which
can be formally derived by inverting Eq. 8 at $\Delta\phi=0$. We report in
Table 2 the ratio between $R_{\rm cor}$ and the exponential scale length
$R_{s}$ for each object and show this result in Figure 6. Comparisons of our
pattern speed $\Omega_{p}$ measurements with other methodologies (e.g.
Tremaine & Weinberg, 1984) are listed in Table 2. The differences with our
results may arise since the Tremaine–Weinberg method assumes the continuity
condition of the tracer, which may break down for the gas as it is easily
shocked, it changes state, and it is converted into stars (Hernández et al.,
2005; Rand & Wallin, 2004), or it can be obscured by dust (Gerssen &
Debattista, 2007). Comparison of the observed non-axisymmetric motions with
hydrodynamical models based on the actual stellar mass distribution (Kranz,
Slyz, & Rix, 2003) have found a characteristic value of $R_{\rm
cor}/R_{s}\simeq 2.7\pm 0.4$ for a sample of spirals. We plot this range with
dashed vertical lines in Figure 4, whereas the solid curve in each panel
denotes the actual fit value of $R_{\rm cor}/R_{s}$ for each object. Figure 6
illustrates the interesting fact that the corotation values found in the
present paper, $R_{\rm cor}/R_{s}\simeq 2.7\pm 0.2$, agree well with the
completely independent estimates that Kranz, Slyz, & Rix (2003) derived by a
different approach and for a different sample. Therefore, a generic value for
the pattern speed of the dominant spiral feature of $R_{\rm cor}/R_{s}\simeq
2.7\pm 0.4$ seems robust.
Figure 6.— The best fit for the spiral arm corotation radius (in units of
exponential scale radii) obtained by inverting the best-fit model of Eq. 1 at
$\Delta\phi(r)=0$, and evaluating $R_{s}$ via a bulge-disk decomposition on
either $H$ band or $3.6\;\mu$m band images. The solid and dashed horizontal
lines represent the $R_{\rm cor}/R_{s}=2.7\pm 0.4$ value found by Kranz, Slyz,
& Rix (2003). The galaxies in the plot are sorted by asymptotic rotation
velocity (see Table 1), giving no indication of a correlation between
dynamical mass and the $R_{\rm cor}/R_{s}$ ratio.
### 5.3. Comparison With CO Data
If the basic picture outlined in the introduction is correct, then the
molecular gas traced by the CO, as an intermediate step in the star-formation
sequence, should lie in between and have a smaller offset from the H I than
the $24\;\mu$m does, but in the same direction. To check this qualitatively,
we retrieve the BIMA-SONG CO maps (Helfer et al., 2003) for the galaxies NGC
628, NGC 5194, and NGC 3627. For comparison, we derive the angular offset
between the CO emission and the $24\;\mu$m, applying the same method described
above for the H I. The results are plotted in Figure 7. The scarcity of data
points (e.g., NGC 628) and their scatter, which is typically larger than the
error bars, make estimates of $t_{{\rm CO}\mapsto 24\,\mu{\rm m}}$ and $R_{\rm
cor}$ rather uncertain. Therefore, we simply focus on $\Delta\phi_{\rm
CO-24}(r)$ versus $\Delta\phi_{\rm{\sc Hi}-24}(r)$, which is shown in Figure
7. This figure shows that the values of $\Delta\phi_{\rm CO-24}$ all lie
closer to zero than the values of $\Delta\phi_{\rm{\sc Hi}-24}$.
Figure 7.— Comparison of the angular offsets obtained for H
I$\mapsto$24$\;\mu$m (squares) and CO$\mapsto$24$\;\mu$m (triangles) for the
galaxies NGC 628, NGC 5194, and NGC 3627. CO maps are taken from the BIMA-SONG
survey. The solid vertical line in each panel indicates the position of
corotation as obtained by $\chi^{2}$ fitting of Eq. 1 for the H I. These
results are qualitatively consistent with a temporal star-formation sequence H
I$\mapsto$CO$\mapsto$24$\;\mu$m.
Hence this check shows that the peak location of the molecular gas is
consistent with the evolutionary sequence where the H I represents an earlier
phase than the CO. In this picture the H I has a larger spatial separation
with respect to the hot dust emission, except at corotation, where the three
components are expected to coincide. However, it is clear that higher
sensitivity CO maps are needed to improve this kind of analysis.
### 5.4. Analysis of Non-Circular Motions
So far we have carried out an analysis that is based on the assumption of
circular motions. We quantify here non-circular motions and determine to what
extent their presence affects the estimate of the timescales $t_{{\rm
HI}\mapsto 24\,\mu{\rm m}}$, which scale with $\Delta\phi$. In the classic
picture (e.g., R69), the radial velocity of the gas is reversed around the
spiral shock, so that the material is at nearly the same galactocentric radius
before and after the shock. Gas in galaxies with dynamically important spiral
arms does not move on circular orbits, though. Shocks and streaming motions
transport gas inwards, and gas orbits undergo strong variations of direction.
If a continuity equation for a particular gas phase applies, it implies that
in the rest frame of the spiral arm, the change of relative velocity
perpendicular to the arm $v_{\perp}$ is proportional to the arm to pre-arm
mass flux ratio (for a recent illustration in M51 see Shetty et al., 2007,
hereafter S07). For example, for an orbit passing through an arm with mass
density contrast of 10, $v_{\perp}$ would drop by the same factor, producing a
net inward deflection of the orbit. We do note that the gas continuity
equation may not actually be valid, since stars may form, or the gas may
become molecular or ionized. Non-circular motions could modify the simplified
scheme of Fig. 1. If the orbit is inward bound near the arm, the path of the
material between the H I and the 24$\;\mu$m arm components is larger than that
previously assumed for circular orbits. We reconsider the scheme of Fig. 1 for
a non-circular orbit in the frame corotating with the spiral pattern as
illustrated in Fig. 8. Here, the material moves not along a line at constant
radius $r$, but along a line proceeding from a larger radius $r+{\rm d}r$,
specifically from the point $A$ (see Fig. 8) on the H I arm toward the point
$B$ on the 24$\;\mu$m arm, where the two parallel horizontal lines denote the
galactocentric distances $r+{\rm d}r$ and $r$. The material departs from $A$
with an angle $\alpha$ inwards (if the material were to proceed instead from
an inner radius, then $\alpha$ is directed outwards). If $\alpha$ is large,
the measurement of the spatial shift $\overline{A^{\prime}B}\simeq
r\,\Delta\phi^{\prime}$ in the simple scheme of circular (deprojected) rings
no longer represents the actual value of the spatial offset
$\overline{AB}\propto\Delta\phi$, but rather is only a lower limit.
Consequently, the measurement of the timescale $t_{{\rm HI}\mapsto 24\,\mu{\rm
m}}$ would be underestimated, since $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$
$\propto\Delta\phi$ from Eq. 8. From the geometry of the triangle
$ABA^{\prime}$ in Fig. 8, and since
$\overline{A^{\prime}B}\propto\Delta\phi^{\prime}$ and
$\overline{AB}\propto\Delta\phi$, it can be straightforwardly shown that
$\Delta\phi(r)=\Delta\phi^{\prime}(r)\>\frac{\cos\beta}{\cos(\beta+\alpha)},$
(9)
where
$\tan\alpha=\left.\frac{-v_{R}}{v_{\phi}^{\prime}}\right|_{r+{\rm d}r},$ (10)
$v_{R}(r)$ and $v_{\phi}^{\prime}(r)=v_{\phi}-\Omega_{p}\,r$ are the radial
and tangential velocity components of the gas in the arm frame, respectively,
and $\beta$ is the H I arm pitch angle defined by
$\tan\beta=\left.\frac{{\rm d}\phi_{\rm arm}}{{\rm d}r}\right|_{r},$ (11)
with $\beta\rightarrow 90^{\circ}$ for a tightly wound spiral. Note that the
radial and the azimuthal dependences of $v_{R}$ and $v_{\phi}^{\prime}$ are
anchored to each other at the spiral arms through Eq. 11. It follows that if
the radial and tangential components of the gas velocity, and the pitch angle
of the H I spiral arms are known, the actual value of $\Delta\phi$ can be
calculated by applying the correction factor
$k(r)\equiv\frac{\cos\beta}{\cos(\beta+\alpha)}$ (12)
to the directly measured quantity $\Delta\phi^{\prime}$. Note from Fig. 8 that
if $\alpha\rightarrow 90-\beta$, then $k(r)\rightarrow\infty$, but in this
case also $\Delta\phi^{\prime}\rightarrow 0$, resulting in a finite value for
$\Delta\phi$ through Eq. 9, and for the timescale, since $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ $\propto\Delta\phi$.
Figure 8.— Accounting for non-circular motions in the frame corotating with
the spiral pattern, the two horizontal lines denoted with $r$ and $r+{\rm d}r$
represent two galactocentric distances. Gas (and the resulting young stars)
move along the line $\overline{AB}\propto\Delta\phi$, proceeding from the mean
location of dense H I to the mean 24$\;\mu$m peak, both denoted with solid
lines, where $\beta$ is the H I pitch angle. The gas velocity vector is at an
angle $\alpha$ with respect to the circle at radius $r+{\rm d}r$. The line
$\overline{A^{\prime}B}$ denotes the (biased) measure of $\Delta\phi^{\prime}$
at constant radius $r$, which we actually estimate in our data analysis, and
which is shorter than the actual shift value $\overline{AB}\propto\Delta\phi$
by a factor of $\cos\beta/\cos(\beta+\alpha)$, as explained in the text.
#### 5.4.1 Methodology
In these circumstances, if spiral arms in galaxies are dynamically affecting
the gas kinematics, then it is sensible to test the effects of non-circular
motions especially in strong arm spiral galaxies from our sample. We select,
NGC 5194 (M51), NGC 628, and NGC 6946. Therefore, we examine the extreme case,
calculating along a narrow region along spiral arms how much the streaming
motions affect our angular offset measurements, thus the derivation of the
timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$. Subsequently, we generalize
the procedure accordingly with our specific methodology by averaging these
effects over the product of H I $\times$ 24 $\mu$m fluxes, which is the
weighting function of the cross-correlation.
The radial and tangential velocity components are difficult to separate
unambiguously. The radial component is accurately measured along the minor
axis, where however information on the azimuthal component is lost. The
opposite occurs along the major axis. Therefore, we follow the prescriptions
for non-circular streaming motions analysis from § 3 of S07, to separate the
velocity components $v_{R}$ and $v_{\phi}$ from the observed H I velocity
field $v_{\rm obs}$, adopting
$v_{\rm obs}(R,\phi)=v_{\rm
sys}+(v_{R}(R,\phi)\,\sin\phi+v_{\phi}(R,\phi)\,\cos\phi)\>\sin i$ (13)
as a generalization of Eq. 2. Knowing $v_{R}$ and $v_{\phi}$, then we obtain
the geometry of the orbits using Equations 10 and 11. Specifically, the H I
column density and velocity map are deprojected and resampled into a polar
coordinate system ($R,\phi$) for simplicity, so that the azimuthal phase of
the spiral arms can be described by a logarithmic spiral $\psi=\phi_{\rm
arm}-\phi_{0}=\ln(R_{\rm arm}/R_{0})$, for a given fiducial radius $R_{0}$. We
extract the observed velocity $v_{\rm obs}$ along logarithmic lines as
illustrated in Fig. 9, which best represent the arm phase. Though these
logarithmic lines are drawn by eye, on top of the spiral arms, we obtain
reasonable results, e.g., for the case of NGC 5194 we find a logarithmic slope
of 26∘, similar to the 21∘ found by S07. Assuming $v_{R}$ and $v_{\phi}$ to be
constant along equal arm phases, we fit Eq. 13 to the observed velocities
extracted at each arm phase $\psi\in[\phi_{\rm arm},\phi_{\rm arm}+2\pi]$. The
fitting gives $v_{R}$ and $v_{\phi}$ as a function of $\psi$ and therefore as
a function of radius through Eq. 11 (see also S07). Provided the angles
$\alpha$ and $\beta$ from Eq. 10 and Eq. 11, we straightforwardly calculate
the actual value of the angular offset $\Delta\phi$ given
$\Delta\phi^{\prime}$ from circular orbits assumption using Eq. 9 and Eq. 12.
Figure 9.— The figure illustrates logarithmic spirals of the form
$\psi=\phi_{\rm arm}-\phi_{0}=\ln(R_{\rm arm}/R_{0})$ where the observed
velocity $v_{\rm obs}$ is extracted in order to be fitted to Eq. 13 and obtain
the $v_{R}$ and $v_{\phi}$ velocity components (see § 5.4). The top and bottom
panels represent the projection into a polar coordinates system ($R,\phi$) of
the observed H I line-of-sight velocity field and the H I column density map,
respectively, for the galaxy NGC 5194. The solid lines represent logarithmic
spiral arms with phases $\psi=\phi_{\rm arm}$ and $\psi=\phi_{\rm arm}\pm
180$∘. In the top panel, the gray scale image of the velocity field indicates
velocity values from $-90$ $\rm km\;s^{-1}$ (dark) to $+90$ $\rm km\;s^{-1}$
(light). The velocity map in figure is not deprojected for inclination
effects, which are instead taken into account in Eq. 13. The coordinate
$\phi=0$ represents the kinematic position angle of the galaxy.
Consistently with Gómez & Cox (2002) and S07, we find in the three considered
galaxies that the locations of the spiral arms coincide with a net drop-off of
the tangential velocity and negative radial velocity, possibly indicating that
near the arms the orbits bend inwards. We then calculate, using Eq. 12, the
correction $k(r)$ at the position of the arms, indicated in Fig. 9, where we
expect the largest variations for $v_{R}$ and $v_{\phi}$. Even so, we find for
the three considered galaxies that $k(r)$ results near unity for all radii,
except where $v_{\phi}^{\prime}$ approaches zero, which does not necessarily
coincide with corotation, but rather where the division in Eq. 10 diverges and
the error bars are large. In particular, for NGC 5194, where
$|\alpha|<20^{\circ}$ for all $r$, the correction $k(r)$ ranges between 0.7
and 1.2, and for both NGC 628 and NGC 6946 $k(r)$ ranges between 0.9 and 1.5,
as indicated in Fig. 10. After calculating the corrected offsets $\Delta\phi$
from Eq. 9, and fitting Eq. 8 to the values $\Delta\phi$, we find that the
timescale $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and the pattern speed
$\Omega_{p}$ do not change significantly—the differences are below the error
bars for the three galaxies. Note that the data points where
$v_{\phi}^{\prime}\simeq 0$, that is where Eq. 10 diverges, are not excluded
from the fit. The results of these fits are listed in Table 3 and plotted in
Fig. 10. We also find that the radial displacements,
${\rm d}r\simeq r\,\tan[\Delta\phi^{\prime}(r)]\,k(r)\,\sin[\alpha(r)],$ (14)
are typically as small compared to the radial steps of $\Delta\phi^{\prime}$,
i.e. $|{\rm d}r|<70$ pc for all $r$ for NGC 5194. In § 4.2 we calculated
$\Delta\phi^{\prime}$ through Eq. 7, hence not only within the spiral arms,
but also as intensity-weighted mean across all the azimuthal values. If we
were to calculate the angle-averaged value $\Delta\phi^{\prime}$ from the
average $\langle v_{R}\rangle$ and $\langle v_{\phi}\rangle$ weighted by the
product H I $\times$ 24$\;\mu$m, which is the weighting function of the cross-
correlation, we find $\alpha\simeq 0$ for all radii and a correction $k(r)$
even closer to unity than over a region limited to the arms. With this
approach we obtain $0.95<k(r)<1.05$ for NGC 5194.
Table 3Characteristic Timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and Pattern Speed $\Omega_{p}$ Resulting from a $\chi^{2}$ Fit to the Angular Offsets $\Delta\phi$ After Correction for Non-circular Motions Following the Prescriptions in Section 5.4 Obj. Name | Alt. Name | $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ | $\Omega_{p}$
---|---|---|---
| | [Myr] | [km s-1 kpc-1]
NGC 5194 | M51 | $3.3\pm 0.6$ | $20\pm 3$
NGC 628 | M74 | $1.4\pm 0.5$ | $26\pm 3$
NGC 6946 | | $1.1\pm 0.3$ | $36\pm 4$
Figure 10.— Correction factor $k(r)$ from Eq. 12, as applied to the angular
offset measurement for the galaxies NGC 5194, NGC 628, and NGC 6946. Bottom
panels: the solid curve represents the correction factor
$k(r)=\cos\beta/\cos(\beta+\alpha)$ as a function of radius. Top panels: the
squares denote the angular offset measurements $\Delta\phi^{\prime}$ of Fig. 4
calculated assuming circular orbits, the circles denote $\Delta\phi$ after
correction, and the solid curve represents the model fit of Eq. 8 to the
corrected offset values.
After exploring the two extreme cases, (1) simple model of circular orbits and
(2) streaming motions near spiral arms for galaxies with prominent spiral
structure—where these effect are supposed to be the largest—we find that the
implied timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ do not vary
significantly. By estimating the streaming motions in three galaxies from our
data set, we find that the correction $k(r)$ which we must apply to the
angular offset measurements in the scheme of circular orbits is generally near
unity. Non-circular motions do not greatly affect the offset measurements
$\Delta\phi$ for the galaxies with the most prominent spiral arms of our data
set, where we expect indeed the highest deviations from circular orbits,
suggesting that, since $t_{{\rm HI}\mapsto 24\,\mu{\rm
m}}$$\;\propto\Delta\phi$, the timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$
will not vary by more than a factor of 1.5.
However, these conclusions should be viewed cautiously, since (1) we are
assuming that $v_{R}$ and $v_{\phi}$ can be obtained by fitting Eq. 13, and
(2) we do not account for beam effects. While the presence of non-circular
motions produces apparent radial variations of inclination and position angle,
our estimates are suggesting that these effects do not influence much the
determination of the timescales. Yet, tidal interactions could ensue physical
variations of inclination and position angle, thus of the actual geometry of
the observed velocities. Although we note that among the sample galaxies only
NGC 3031 (M81), NGC 3627, NGC 5055, and NGC 5194 are affected by tidal
interaction. Moreover, we do not consider extra-planar motions due to the
implied numerical difficulties. For instance, to include a vertical velocity
component $v_{z}$ into Eq. 13 would introduce a degeneracy while fitting
$v_{R}$, $v_{\phi}$, and $v_{z}$, and a degeneracy in the geometry of the
motions, rendering the estimates of the timescales uncertain. However, after
subtracting a circular orbit model from the observed velocity field of the
galaxy NGC 3184, an inspection of the velocity residuals reveals deviations
from circular motions of $\sim$5–10 $\rm km\;s^{-1}$ amplitude, and about zero
near the spiral arms. Considering that NGC 3184 is nearly face-on ($i=16$∘),
then vertical motions should not exceed 5–10 $\rm km\;s^{-1}$. The beam
deconvolution, on the other hand, would enlarge the uncertainties on
separating $v_{R}$ and $v_{\phi}$, but we rely on a resolution size limit
which is far below the typical thickness of a spiral arm and allows us to
resolve fine sub-structures within the arms. We also rely on a large number of
data points—several hundred to several thousand depending on the
galactocentric radius.
## 6\. DISCUSSION
By analyzing the angular offsets between H I and $24\;\mu$m in the context of
a simple kinematic model, we found short timescales, $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$, as summarized in the histogram of Figure 5 and in Table 2.
The implied characteristic timescales for almost all sample galaxies lie in
the range 1–4 Myr. This result sets an upper limit to the timescale for
massive star formation under these circumstances, since we are observing the
time lag between two phases: $a)$ the atomic gas phase, which subsequently is
compressed into molecular clouds and forms clusters of young, embedded,
massive stars, and $b)$ the warm dust phase, produced by heating from young
stars, whose UV radiation is reprocessed by the dust into the mid-IR, as
observed at 24$\;\mu$m. For the few objects where there are suitable CO data,
we checked that this geometric picture also holds for the molecular phase.
### 6.1. Timescales Derived from Pattern Offsets
Egusa, Sofue, & Nakanishi (2004) used the same angular offset technique to
compare CO and H$\alpha$ emission. They report timescales $t_{{\rm
CO}\mapsto{\rm H}\alpha}$ $\simeq 4.8$ Myr for the galaxy NGC 4254 (not in our
sample). The H$\alpha$ traces a later evolutionary stage than the warm dust
emission, which can indicate the presence of a young cluster still enshrouded
by dust. Therefore, the H$\alpha$ and the dust emission are expected to be
separated by the time needed to remove the dusty envelope, though they are
observed to be spatially well correlated (Wong & Blitz, 2002; Kennicutt,
1998). Prescott et al. (2007) found a strong association between 24 $\mu$m
sources and optical H II regions in SINGS galaxies. Also infrared sources
located on top of older, UV-bright, clusters that do not have H$\alpha$
emission are rare. Since Prescott et al. (2007) suggest that the break out
time from dust clouds is short ($\sim$1 Myr), we do not expect a strong
offset. Egusa, Sofue, & Nakanishi (2004) derived the angular offset by
subjective assessment of the separation of the intensity peaks. They report
that this may be a source of systematic errors, since they cannot detect by
eye, the angular phase differences less than a certain threshold and in turn,
their results would possibly be an upper limit. Given these considerations,
the timescales derived in this paper are likely consistent with the
conclusions of Egusa, Sofue, & Nakanishi (2004).
Rots (1975) and Garcia-Burillo, Guelin, & Cernicharo (1993) found time lags of
$\sim$10 Myr for M81 and M51, respectively. In particular, Rots applied the
angular offset method to the dust lanes and H$\alpha$. This may be in part
problematic since the dust absorption in the optical bands only traces the
presence of dust and it is unrelated to the warm dust emission due to star-
formation onset. Garcia-Burillo et al. measured the spatial separation
projected on the sky between CO and H$\alpha$ and not the azimuthal offset.
Allen (2002) has argued that H I is a photodissociation product of UV shining
on molecular gas, so it should be seen between the CO and UV/H$\alpha$
regions. Allen et al. (1986) observed H I between spiral arm dust lanes and H
II regions. However, Elmegreen (2007) points out that there is no time delay
between dust lanes and star formation: dust lanes may only represent a heavy
visual extinction effect and may not be connected to star-formation onset. Our
finding that CO is situated between H I and hot dust (i.e. Fig. 7) stands in
conflict with the predictions of the model proposed by Allen (2002).
The evolutionary timescales of the ISM phases, especially for star formation,
are not well constrained. The observational results of the last few decades,
arrive at different and, in some cases, controversial conclusions. The
discrepancies might in part be attributed to effects of limited resolution
(see § 6.3). This suggests that higher resolution and sensitivity maps,
especially for the CO emission, are needed to improve the presented technique
in the future.
#### 6.1.1 Photodissociation of H2
We argue that illumination effects of UV radiation shining on molecular and
dusty regions cannot photodissociate molecules in order to produce the
observed peaks of H I—which correspond to typical surface densities of several
solar masses per pc2. First, the mean free path of the UV photons is
remarkably short, typically $\sim$100 pc. The presence of dust, particularly
abundant in disk galaxies, is the main source of extinction in particular
within spiral arms, where we observe the peak of dust emission. Second, none
of the galaxies from our data set presents prominent nuclear activity, whose
UV flux could ionize preferentially the inner surfaces of the molecular
clouds. Also, we exclude that UV radiation from young stellar concentrations
can ionize preferentially one side of the clouds causing H I and CO emissions
to lie offset with respect to each other, which we instead interpret as due to
an evolutionary sequence. In fact, if the light from young stars effectively
produces H I by photodissociation of H2, then the neutral to molecular gas
fraction is expected to increase with star-formation rate per unit area,
$\Sigma_{\rm SFR}$, as a consequence of the increasing UV radiation flux. Yet,
the H I to H2 ratio decreases with increasing $\Sigma_{\rm SFR}$, as also
shown, for example, in Kennicutt et al. (2007) for the galaxy M51. Moreover,
the H I density does not vary much as a function of $\Sigma_{\rm SFR}$
(Kennicutt, 1998).
### 6.2. Pattern Speeds
The results of the fits in Figure 4 are consistent with the existence of a
kinematic pattern speed for the considered galaxies, and are suggesting that
the spiral pattern must be metastable at least over a few Myr or, strictly
speaking, quasi-stationary. If the apparent spiral structure seen in young
stars were produced by stochastic self-propagating star formation (e.g. Gerola
& Seiden, 1978; Seiden & Gerola, 1982) and shear, without an underlying
coherent mass perturbation, then presumably we would not observe the
systematic radial variation of the offsets, seen in Figure 4, in particular
not that the offset changes sign at $R\simeq 2-3\,R_{\rm exp}$. Our results,
however, do not exclude the stochastic star-formation mechanism to occur,
rather that this is not the dominant trigger of star formation. The spiral
pattern might be the manifestation of full wealth of modes of propagating
density waves, which are continuously forming and dissolving gas clouds and
structures, where, i.e. the azimuthal modes for a grand-design spiral are
dominant at low orders (e.g., $m=2,3,4$). If the spiral structure is quasi-
stationary, then it can be characterized by an instantaneous pattern speed to
first order—at least over a timescale much shorter than the orbital time. This
is opposed to density waves dynamically driven by bars or interaction with
companions, which could last a few orbital times. Note, however, that our
analysis holds approximately in both possible cases.
Some of the fits in Figure 4 do not accurately mirror the trend of the
observed data points, where the scatter is so large that it renders the
interpretation problematic. In particular, we could designate as a bad fit the
results for the galaxies NGC 3031, NGC 3621, NGC 7793, and NGC 6946 (Table 2).
We also note that the pattern speed obtained in this paper for the galaxies
NGC 5055 and NGC 5194 disagree with previous independent measurements (e.g.
Thornley & Mundy, 1997; Zimmer et al., 2004). The fit for NGC 5055 presents
large error bars due to the large scatter in the azimuthal offsets (see Figure
4), which may explain the differences. For NGC 5194 the large difference
between our result and the Tremaine–Weinberg method prediction could be due,
as mentioned in § 5.2, to the assumption of continuity for the gas. In fact,
Zimmer et al. (2004) find a pattern speed $\Omega_{p}^{Z}\simeq 38$ $\rm
km\;s^{-1}$ kpc-1 that is much faster than the pattern speed predicted by the
hydrodynamical models from Kranz et al. (2003), $\Omega_{p}^{K}\simeq 12$ $\rm
km\;s^{-1}$ kpc-1, using $R_{\rm cor}\simeq 2.7\,R_{s}$ and $R_{s}\simeq
1.4^{\prime}$. Moreover, the presence of large variations of the azimuthal
offsets as a function of radius with respect to the smoother fitted curves of
Figure 4 suggests that the pattern speed may not be constant over the entire
disk. Instead, the spiral pattern could be described by more than one pattern
speed, implying that the delay time is not exactly the same in all parts of an
individual galaxy, and not necessarily the same in all the considered
galaxies. However, the offsets measurements and the implied pattern speeds and
timescales could be statistically pointing toward regions of high H I and 24
$\mu$m fluxes when using the weighting of Equation 7, therefore toward high
$\Sigma_{\rm SFR}$, ensuing the timescales to be comparable in all cases.
### 6.3. Is Star Formation Triggered by Spiral Waves?
Addressing to our results (§ 5 and Fig. 4), a further aspect emerging from our
analysis concerns the same general behavior displayed by an heterogeneous
sample of galaxies going from grand-design (e.g. NGC 5194) to flocculent
morphologies (e.g. NGC 2841 and NGC 5055, see also Fig. 3 for an example
comparison). Visual examination of the IR band images for our sample galaxies
(e.g. at 3.6 $\mu$m) indicates that almost all have two-arm or multi-arm
coherent spiral arms; for none of these galaxies the spiral structure is so
chaotic as to be characterized as flocculent. Moreover, both types of galaxies
from our sample display comparable integrated star-formation rates. Grand-
design density waves are not likely to be the primary trigger of star
formation, since a substantial portion of stars are also formed in the inter-
arm regions (Elmegreen & Elmegreen, 1986). The concentration of young stellar
populations near prominent spiral arms is rather an effect of kinematics
(Roberts, 1969). Grand-design and flocculent galaxies exhibit the same
intrinsic self-similar geometry (Elmegreen, Elmegreen, & Leitner, 2003). The
difference between these two types of galaxies is only dictated by a different
distribution of azimuthal modes. Large scales structures (low-order modes) are
the dominant features in grand design. However, the short timescales suggest
that the physical scales where star formation occurs are rather small, since
this time delay cannot exceed the free-fall time. Weak density waves, those
described by high-order azimuthal modes, not necessarily grand-design modes,
are likely to facilitate the growth of super-critical structures which end in
star-forming events. In galaxies morphologically classified as flocculent and
grand design, the mechanism that triggers star formation must be the
same—gravitational instability. Direct compression of gas clouds is indeed
able to locally trigger star formation.
The sizes of the structures of active star formation, however, are not
representative of the timescales involved. Star-forming regions are organized
hierarchically according to the small-scale turbulent motions of gas and
stars, where the dynamical time varies as a function of the local scale sizes
(Efremov & Elmegreen, 1998; Ballesteros-Paredes et al., 1999). The improved
accuracy of recent observational techniques, e.g. THINGS and SINGS, allows us
to observe smaller and smaller structures, which evolve thus more rapidly and
are characterized by shorter timescales. Stellar activity and turbulence limit
the lifetime of molecular clouds by causing the destruction of the parent
cloud and the cloud dispersal, respectively. This lifetime, typically $10-20$
Myr, drops by a factor of $\sim$10 for the star-forming clouds. Moreover, star
formation begins at high rate in only a few Myr, appearing in large structures
of O-B complexes—beads on a string—of few hundred pc scales, and remaining
active for $\sim$30–50 Myr, but with a gradually decreasing star-forming rate,
until the complete quenching (Elmegreen, 2007). The timescales measured here
refer to the very initial phase of star formation, specifically when star-
formation rate is the highest—traced by 24 $\mu$m peak emission. The time
delay $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ needs not to represent the average
time difference between these two phases, especially if there is a gradually
declining tail of star-formation activity, taking longer than $t_{{\rm
HI}\mapsto 24\,\mu{\rm m}}$. If much of the star formation were actually
occurring within our short $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ estimate, then
star formation would have to be inefficient to avoid conflicts with the short-
term depletion of the gas reservoir.
### 6.4. Can We Rule Out Timescales of 10 Myr?
Since the 24$\;\mu$m emission traces the mass-weighted star-formation
activity, the timescale $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ measures the time
for star clusters to form from H I gas, but it does not show that all
molecular clouds live only $\sim$2 Myr. It might show that the bulk of massive
stars that form in disks has emerged from molecular clouds that only lived
$\sim$2 Myr, though there could still be molecular clouds that live an order
of magnitude longer. The full molecular cloud lifetimes estimated for the LMC
correspond to $\sim$10 Myr (Mizuno et al., 2001; Yamaguchi et al., 2001),
while star clusters are formed from molecular clouds in only few Myr. Yet,
more recent observations suggest much slower evolution. In particular, Blitz
et al. (2007) propose molecular cloud lifetimes for the LMC as long as 20–30
Myr. In the Milky Way this timescale is estimated to be a few Myr (e.g.
Hartmann, Ballesteros-Paredes, & Bergin, 2001), but the examined cloud
complexes, such as Taurus and Ophiuchus, have low star-formation rate and
masses more than an order of magnitude lower than those studied by Blitz et
al. (2007). Thus, it remains possible that our results and these previous
arguments are all consistent. At high density star formation also occurs at
higher rate. Blitz et al. (2007) also find that the timescales for the
emergence of the first H II regions traced by H$\alpha$ in molecular clouds is
$t_{\rm HII}\sim 7\;$Myr, which does not exclude that $t_{\rm HII}\geq t_{{\rm
HI}\mapsto 24\,\mu{\rm m}}$, but it could be problematic with the short break
out suggested by Prescott et al. (2007), since the time delay which is
required between the onset of star formation (as traced by 24$\;\mu$m from
obscured H II regions) and the emergence of H$\alpha$ emission would need to
be large.
In conclusion, we note that the timescale $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$
results from a $\chi^{2}$ fit to all the data, and it is treated as a global
constant individually for each galaxy, so it is not a function of radius.
Globally, all the characteristic timescales $t_{{\rm HI}\mapsto 24\,\mu{\rm
m}}$ listed in Table 2 are $\leq$4 Myr, except one single case (NGC 925). The
error bars are also relatively small: $<$1 Myr for the majority of the cases.
These results clearly exclude characteristic timescales $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ of the order of $\sim$10 Myr, even for the highest value
recorded in our data set which is NGC 925.
### 6.5. Theoretical Implications
The short timescale found here between the peak of H I emission and the peak
of emission from young, dust-enshrouded stars has implications for two related
theoretical controversies. First is the question of whether molecular clouds
are short-lived, dynamically evolving objects (Ballesteros-Paredes, Hartmann,
& Vázquez-Semadeni 1999, Hartmann et al. 2001; Elmegreen 2000; Ballesteros-
Paredes & Hartmann 2007; Elmegreen 2007) or quasi-static objects evolving over
many free-fall times (Matzner, 2002; Krumholz, Matzner, & McKee, 2006). The
second, related question is what the rate-limiting step for star formation in
galaxies is: formation of gravitationally unstable regions in the H I that can
collapse into molecular clouds (Elmegreen, 2002; Kravtsov, 2003; Li et al.,
2005, 2006; Elmegreen, 2007), or formation of dense, gravitationally unstable
cores within quasi-stable molecular clouds (Krumholz & McKee, 2005; Krumholz,
Matzner, & McKee, 2006; Krumholz & Tan, 2007).
The short timescales found here for the bulk of massive star formation in
regions of strong gravitational instability appears to support the concept
that molecular cloud evolution occurs on a dynamical time once gravitational
instability has set in, and that the rate-limiting step for star formation is
the assembly of H I gas into gravitationally unstable configurations. Our work
does not, however, address the total lifetime of molecular gas in these
regions, as we only report the separation between the peaks of the emission
distributions. Molecular clouds may well undergo an initial burst of star
formation that then disperses fragments of molecular gas that continues star
formation at low efficiency for substantial additional time (Elmegreen, 2007).
Averaging over the efficient and inefficient phases of their evolution might
give the overall low average values observed in galaxies (e.g. Krumholz & Tan,
2007).
## 7\. CONCLUSIONS
We have derived characteristic star-formation timescales for a set of nearby
spiral galaxies, using a simple geometric approach based on the classic
Roberts (1969) picture that star formation occurs just downstream from the
spiral pattern, where gas clouds have been assembled into super-critical
configurations. This derived timescale, $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$,
refers to the processes from the densest H I, to the molecular phase, to
enshrouded hot stars heating the dust. The analysis is based on high-
resolution 21 cm maps from THINGS, which we combined with $24\;\mu$m maps from
SINGS. We assume that the observed spiral arms have a pattern speed
$\Omega_{p}$. Given the rotation curve, $v_{c}(r)=r\,\Omega(r)$, this allows
us to translate angular offsets at different radii between the H I flux peaks
and the $24\;\mu$m flux peaks in terms of a characteristic time difference.
At each individual point along the spiral arm we found considerable scatter
between the H I and $24\;\mu$m emission peaks. However, for each galaxy we
could arrive at a global fit, using a cross-correlation technique, and derive
two characteristic parameters, $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$ and
$R_{\rm cor}$. For our 14 objects we found the general relation $R_{\rm
cor}=(2.7\pm 0.2)\,R_{s}$, which is consistent with previous studies (e.g.
Kranz, Slyz, & Rix, 2003) and, more importantly, we found $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ to range between 1 and 4 Myr. Even when accounting for
uncertainties, at the highest peak of star-formation rate timescales as long
as $t_{{\rm HI}\mapsto 24\,\mu{\rm m}}$$\;\sim 10\,$Myr, which have been
inferred from other approaches, do not appear consistent with our findings. At
least for the case of nearby spiral galaxies, our analysis sets an upper limit
to the time needed to form massive stars (responsible for heating the dust) by
compressing the (atomic) gas. Therefore, it points to a rapid procession of
star formation through the molecular-cloud phase in spiral galaxies. If star
formation really is as rapid as our estimate of $t_{{\rm HI}\mapsto
24\,\mu{\rm m}}$ suggests, it must be relatively inefficient to avoid the
short-term depletion of gas reservoirs.
We thank Henrik Beuther, Mark Krumholz, Adam Leroy, and Eve Ostriker for
useful discussions and suggestions. We are also grateful to the anonymous
referee, whose comments helped us to improve the manuscript. The work of
W.J.G.d.B. is based upon research supported by the South African Research
Chairs Initiative of the Department of Science and Technology and National
Research Foundation. E.B. gratefully acknowledges financial support through an
EU Marie Curie International Reintegration Grant (Contract No. MIRG-
CT-6-2005-013556). M.-M.M.L. was partly supported by US National Science
Foundation grant AST 03-07854, and by stipends from the Max Planck Society and
the Deutscher Akademischer Austausch Dienst. This research has made use of the
NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
## References
* Allen (2002) Allen, R. J. 2002, in ASP Conf. Ser. 276, Seeing Through the Dust: The Detection of HI and the Exploration of the ISM in Galaxies (San Francisco, CA: ASP), 288
* Allen et al. (1986) Allen, R. J., Atherton, P. D., & Tilanus, R. P. J. 1986, Nature, 319, 296
* Ballesteros-Paredes & Hartmann (2007) Ballesteros-Paredes, J., & Hartmann, L. 2007, RevMexA&A, 43, 123
* Ballesteros-Paredes et al. (1999) Ballesteros-Paredes, J., Hartmann, L., & Vázquez-Semadeni, E. 1999, ApJ, 527, 285
* Begeman (1989) Begeman, K. G. 1989, A&A, 223, 47
* Beuther et al. (2007) Beuther, H., Churchwell, E. B., McKee, C. F., & Tan, J. C. 2007, Protostars and Planets V (Hawaii), 165
* Blitz et al. (2007) Blitz, L., Fukui, Y., Kawamura, A., Leroy, A., Mizuno, N., & Rosolowsky, E. 2007, Protostars and Planets V (Hawaii), 81
* Byrd et al. (2002) Byrd, G., Freeman, T., & Buta, R. 2002, BAAS, 34, 1116
* Calzetti et al. (2005) Calzetti, D., et al. 2005, ApJ, 633, 871
* Calzetti et al. (2007) Calzetti, D., et al. 2007, ApJ, 666, 870
* Cappellari & Copin (2003) Cappellari, M., & Copin, Y. 2003, MNRAS, 342, 345
* Dale et al. (2005) Dale, D. A., et al. 2005, ApJ, 633, 857
* de Blok et al. (2008) de Blok, W.J.G, et al. 2008, AJ, in press
* Dwek (2005) Dwek, E. 2005, AIP Conf. Proc. 761: The Spectral Energy Distributions of Gas-Rich Galaxies: Confronting Models with Data, 103
* Efremov & Elmegreen (1998) Efremov, Y. N., & Elmegreen, B. G. 1998, MNRAS, 299, 588
* Egusa et al. (2004) Egusa, F., Sofue, Y., & Nakanishi, H. 2004, PASJ, 56, L45
* Elmegreen (2000) Elmegreen, B. G. 2000, ApJ, 530, 277
* Elmegreen (2002) Elmegreen, B. G. 2002, ApJ, 577, 206
* Elmegreen (2007) Elmegreen, B. G. 2007, ApJ, 668, 1064
* Elmegreen & Elmegreen (1986) Elmegreen, B. G., & Elmegreen, D. M. 1986, ApJ, 311, 554
* Elmegreen et al. (2003) Elmegreen, B. G., Elmegreen, D. M., & Leitner, S. N. 2003, ApJ, 590, 271
* Elmegreen et al. (1998) Elmegreen, B. G. , Wilcots, E., & Pisano, D. J. 1998, ApJ, 494, L37
* Garcia-Burillo et al. (1993) Garcia-Burillo, S., Guelin, M., & Cernicharo, J. 1993, A&A, 274, 123
* Gerola & Seiden (1978) Gerola, H., & Seiden, P. E. 1978, ApJ, 223, 129
* Gerssen & Debattista (2007) Gerssen, J., & Debattista, V. P. 2007, MNRAS, 378, 189
* Glover & Mac Low (2007) Glover, S. C. O., & Mac Low, M.-M. 2007, ApJ, 659, 1317
* Goldsmith & Li (2005) Goldsmith, P. F., & Li, D. 2005, ApJ, 622, 938
* Goldsmith et al. (2007) Goldsmith, P. F., Li, D., & Krčo, M. 2007, ApJ, 654, 273
* Gómez & Cox (2002) Gómez, G. C., & Cox, D. P. 2002, ApJ, 580, 235
* Hartmann (2003) Hartmann, L. 2003, ApJ, 585, 398
* Hartmann et al. (2001) Hartmann, L., Ballesteros-Paredes, J., & Bergin, E. A. 2001, ApJ, 562, 852
* Helfer et al. (2003) Helfer, T. T., Thornley, M. D., Regan, M. W., Wong, T., Sheth, K., Vogel, S. N., Blitz, L., & Bock, D. C.-J. 2003, ApJS, 145, 259
* Hernández et al. (2004) Hernández, O., Carignan, C., Amram, P., & Daigle, O. 2004, Penetrating Bars Through Masks of Cosmic Dust, ASSL, 319, 781
* Hernández et al. (2005) Hernández, O., Wozniak, H., Carignan, C., Amram, P., Chemin, L., & Daigle, O. 2005, ApJ, 632, 253
* Hillenbrand et al. (1993) Hillenbrand, L. A., Massey, P., Strom, S. E., & Merrill, K. M. 1993, AJ, 106, 1906
* Hollenbach & Salpeter (1971) Hollenbach, D., & Salpeter, E. E. 1971, ApJ, 163, 155
* Jarrett et al. (2003) Jarrett, T. H., Chester, T., Cutri, R., Schneider, S. E., & Huchra, J. P. 2003, AJ, 125, 525
* Jedrzejewski (1987) Jedrzejewski, R. I. 1987, MNRAS, 226, 747
* Jura (1975) Jura, M. 1975, ApJ, 197, 575
* Kennicutt (1998) Kennicutt, R. C., Jr. 1998, ApJ, 498, 541
* Kennicutt et al. (2003) Kennicutt, R. C., Jr., et al. 2003, PASP, 115, 928
* Kennicutt et al. (2007) Kennicutt, R. C., Jr., et al. 2007, ApJ, in press (arXiv:0708.0922)
* Kim & Ostriker (2002) Kim, W.-T., & Ostriker, E. C. 2002, ApJ, 570, 132
* Kim & Ostriker (2006) Kim, W.-T., & Ostriker, E. C. 2006, ApJ, 646, 213
* Kranz et al. (2003) Kranz, T., Slyz, A., & Rix, H.-W. 2003, ApJ, 586, 143
* Kravtsov (2003) Kravtsov, A. V. 2003, ApJ, 590, L1
* Krumholz et al. (2006) Krumholz, M. R., Matzner, C. D., & McKee, C. F. 2006, ApJ, 653, 361
* Krumholz & McKee (2005) Krumholz, M. R., & McKee, C. F. 2005, ApJ, 630, 250
* Krumholz & Tan (2007) Krumholz, M. R., & Tan, J. C. 2007, ApJ, 654, 304
* Li et al. (2005) Li, Y., Mac Low, M.-M., & Klessen, R. S. 2005, ApJ, 620, L19
* Li et al. (2006) Li, Y., Mac Low, M.-M., & Klessen, R. S. 2006, ApJ, 639, 879
* Lin & Shu (1964) Lin, C. C., & Shu, F. H. 1964, ApJ, 140, 646
* Martin & Kennicutt (2001) Martin, C. L., & Kennicutt, R. C., Jr. 2001, ApJ, 555, 301
* Matzner (2002) Matzner, C. D. 2002, ApJ, 566, 302
* Mizuno et al. (2001) Mizuno, N., et al. 2001, PASJ, 53, 971
* Mouschovias et al. (2006) Mouschovias, T. C., Tassis, K., & Kunz, M. W. 2006, ApJ, 646, 1043
* Palla & Stahler (1999) Palla, F., & Stahler, S. W. 1999, ApJ, 525, 772
* Palla & Stahler (2000) Palla, F., & Stahler, S. W. 2000, ApJ, 540, 255
* Peng et al. (2002) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266
* Prescott et al. (2007) Prescott, M. K. M., et al. 2007, ApJ, 668, 182
* Rafikov (2001) Rafikov, R. R. 2001, MNRAS, 323, 445
* Rand & Kulkarni (1990) Rand, R. J., & Kulkarni, S. R. 1990, ApJ, 349, L43
* Rand & Wallin (2004) Rand, R. J., & Wallin, J. F. 2004, ApJ, 614, 142
* Rix et al. (1997) Rix, H.-W., Guhathakurta, P., Colless, M., & Ing, K. 1997, MNRAS, 285, 779
* Rix & Zaritsky (1995) Rix, H.-W., & Zaritsky, D. 1995, ApJ, 447, 82
* Roberts (1969) Roberts, W. W. 1969, ApJ, 158, 123
* Rots (1975) Rots, A. H. 1975, A&A, 45, 43
* Sakhibov & Smirnov (2004) Sakhibov, F. K., & Smirnov, M. A. 2004, Astron. Rep., 48, 995
* Scoville et al. (2001) Scoville, N. Z., Polletta, M., Ewald, S., Stolovy, S. R., Thompson, R., & Rieke, M. 2001, AJ, 122, 3017
* Seiden & Gerola (1982) Seiden, P. E., & Gerola, H. 1982, Fundamentals of Cosmic Physics, vol. 7, no. 3, p. 241
* Sellwood & Sparke (1988) Sellwood, J. A., & Sparke, L. S. 1988, MNRAS, 231, 25P
* Shetty et al. (2007) Shetty, R., Vogel, S. N., Ostriker, E. C., & Teuben, P. J. 2007, ApJ, 665, 1138
* Smith et al. (2007) Smith, J. D. T., et al. 2007, ApJ, 656, 770
* Tan et al. (2006) Tan, J. C., Krumholz, M. R., & McKee, C. F. 2006, ApJ, 641, L121
* Tassis & Mouschovias (2004) Tassis, K., & Mouschovias, T. C. 2004, ApJ, 616, 283
* Thomasson et al. (1989) Thomasson, M., Donner, K. J., Sundelius, B., Byrd, G. G., Huang, T. Y., & Valtonen, M. J. 1989, Ap&SS, 156, 205
* Thornley & Mundy (1997) Thornley, M. D. , & Mundy, L. G. 1997, ApJ, 484, 202
* Thronson & Telesco (1986) Thronson, H. A., Jr., & Telesco, C. M. 1986, ApJ, 311, 98
* Toomre (1964) Toomre, A. 1964, ApJ, 139, 1217
* Trachternach et al. (2008) Trachternach, C., et al. 2008, AJ, submitted
* Tremaine & Weinberg (1984) Tremaine, S., & Weinberg, M. D. 1984, ApJ, 282, L5
* Vázquez-Semadeni et al. (2005) Vázquez-Semadeni, E., Kim, J., Shadmehri, M., & Ballesteros-Paredes, J. 2005, ApJ, 618, 344
* Vogel et al. (1988) Vogel, S. N., Kulkarni, S. R., & Scoville, N. Z. 1988, Nature, 334, 402
* Walter et al. (2008) Walter, F., et al. 2008, AJ, submitted
* Westpfahl (1991) Westpfahl, D. 1991, in ASP Conf. Ser. 18, The Interpretation of Modern Synthesis Observations of Spiral Galaxies, 175
* Williams (2005) Williams, D. A. 2005, JPhCS, 6, 1
* Wong & Blitz (2002) Wong, T., & Blitz, L. 2002, ApJ, 569, 157
* Yamaguchi et al. (2001) Yamaguchi, R., et al. 2001, PASJ, 53, 985
* Zimmer et al. (2004) Zimmer, P., Rand, R . J., & McGraw, J. T. 2004, ApJ, 607, 285
|
arxiv-papers
| 2008-10-14T08:45:45
|
2024-09-04T02:48:58.244241
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Tamburro, H.-W. Rix, F. Walter, E. Brinks, W.J.G. de Blok, R.C.\n Kennicutt, and M.-M. Mac Low",
"submitter": "Domenico Tamburro",
"url": "https://arxiv.org/abs/0810.2391"
}
|
0810.2444
|
# High Performance Quantum Computing
Simon J. Devitt devitt@nii.ac.jp National Institute for Informatics, 2-1-2
Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. William J. Munro Hewlett-
Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United
Kingdom National Institute for Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,
Tokyo 101-8430, Japan. Kae Nemoto National Institute for Informatics, 2-1-2
Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan.
###### Abstract
The architecture scalability afforded by recent proposals of a large scale
photonic based quantum computer, utilizing the theoretical developments of
topological cluster states and the photonic chip, allows us to move on to a
discussion of massively scaled Quantum Information Processing (QIP). In this
letter we introduce the model for a secure and unsecured topological cluster
mainframe. We consider the quantum analogue of High Performance Computing,
where a dedicated server farm is utilized by many users to run algorithms and
share quantum data. The scaling structure of photonics based topological
cluster computing leads to an attractive future for server based QIP, where
dedicated mainframes can be constructed and/or expanded to serve an
increasingly hungry user base with the ideal resource for individual quantum
information processing.
Since the introduction of quantum information science in the late 1970’s and
early 1980’s, a large scale physical device capable of high fidelity quantum
information processing (QIP) has been a major and highly sought after goal.
While quantum information has lead to many extraordinary developments in
foundational quantum theory, quantum atom/optics, solid state physics and
optics many researchers world wide are still striving towards a physical
quantum computer.
The issue of computational scalability for QIP has been an intensive area of
research for not only physicists but also computer scientists, mathematicians
and network analysis and in the past decade has seen many proposals for
scalable quantum devices for a variety of quantum architectures arch . The
complexity in designing a large scale quantum computer is immense and research
in this area must incorporate complex ideas in theoretical and experimental
physics, information theory, quantum error correction, quantum algorithms and
network design. Due to the relative infancy of theoretical and experimental
QIP it has been difficult to implement theoretically scalable ideas in quantum
information theory, error correction and algorithm design into an
architectural model where the transition from 1-100 qubits to 1-100 million
qubits is conceptually straightforward.
Recent theoretical advancements in computational models for QIP has introduced
an extremely elegant pathway to realize a enormously large QIP system in
optics. Topological cluster state computing, first introduced by Raussendorf,
Harrington and Goyal Raussendorf4 has emerged as an extremely promising
computational model for QIP and integration of this model with chip-based
photon/photon gates such as the photonic module has lead to a promising
optical realization of quantum computation Devitt1 ; Devitt2 . The conceptual
scalability of the chip based topological optical computer allows, for the
first time, a grounded discussion on large scale quantum information
processing, beyond the individual computer. In this letter we take the
scalability issue one step further, examining the possible long term
implementation of topological cluster state computing with the photonic chip
and discuss what the future may hold for this architectural model of QIP.
Traditional discussions of scalability in QIP is generally limited to the
issue of constructing a single, moderately large scale quantum computer,
capable of performing non-trivial algorithms for a single user. In the case of
topological cluster state computation in optics we can consider the
possibility of client/mainframe quantum devices and start to consider the
quantum analogue of classical high performance computing, namely High
Performance Quantum Computing (HPQC) where a large, generic quantum resource
is made available to multiple clients to perform independent (or joint) QIP.
Topological cluster state computing in optics is uniquely suited to this task
for several reasons. Aside from the error correcting and resource benefits of
the topological cluster model, the basic geometric structure of the lattice
allows for multi-user computation that would be problematic when utilizing the
more traditional 2D cluster state techniques Raussendorf1 . In traditional 2D
cluster state computing, multiple users could not interact data with each
other or with a central resource core without transporting quantum information
through the cluster resource of other users. Essentially multi-user
interactions would be a Linear Nearest Neighbor (LNN) network. Moving to 3D
topological clusters convert this LNN network topology into a 2D grid,
enabling the partitioning of the cluster lattice into user regions and
resource regions. Additionally, as the lattice is carried by single photons we
can potentially integrate a mainframe model with developments in quantum
communications and entanglement distribution ent . This gives a layer of
security to the HPQC which would be difficult, if not impossible to achieve
for multi-user, matter based qubit architectures.
Here we introduce the basic framework for a potential HPQC based on
topological cluster state computing in the photonic regime [Fig. 1]. We
discuss two possible mainframe models, one where multi-user computation is
performed locally by the mainframe and another where partitions of the
mainframe lattice are sent via quantum communications channels to individual
users. We complete the discussion by providing a example of a partition
structure for the mainframe lattice which satisfies many of the necessary
components of a HPQC mainframe and give a basic estimate of the number of
photonic chips required for a massive mainframe quantum server.
Figure 1: A central mainframe HPQC would consist of a massive cluster
preparation network built from single photons sources and photonic chips. Once
the cluster is prepared, users can log on and perform individual computations
in one of two ways. A trusted mainframe model is where the user submits a
classical data stream corresponding to the measurement pattern for a quantum
algorithm. The secured quantum user has access to a high fidelity quantum
communications link between themselves and the mainframe. The alloted portion
of the global lattice is then physically routed to the user and photon
measurements are performed locally.
The first model we consider we denote the trusted mainframe model. This is
where individual users connect via classically secure data pathways and the
mainframe host is trustworthy. Each client logs onto the host and transmits a
classical data stream, corresponding to the desired quantum algorithm, to the
host (via a sequence of photon measurement bases). The mainframe will then run
the quantum algorithm locally and once the computation is complete, transmits
the resulting classical information back to the user.
This model has very substantial benefits. First, each user does not require
quantum communications channels or any quantum infrastructure locally. All
that is required is that each user compile a quantum algorithm into an
appropriate classical data stream which is sent to the mainframe.
Additionally, the host does not need to transmit any data to the user during
computation. All internal corrections to the lattice which arise due to its
preparation and error correction procedures are performed within the
mainframe. The only data which is transmitted to the user is the classical
result from the quantum algorithm. Finally, as each user independently logs on
to the system to run a quantum algorithm, the mainframe can be configured to
assign resources dynamically. If one user requires a large number of logical
qubits and if the mainframe load is low, then the host can adjust to allocate
a larger partition of the overall lattice to one individual user.
While the user/mainframe interaction of this mainframe model is identical to
classical models for high performance computing, the fact that we are working
with qubits suggests the possibility of secure HPQC. In the trusted mainframe
model the classical data stream from the user to host is susceptible to
interception (although quantum key distribution and secure data links can be
utilized to mitigate this issue) and the quantum mainframe has complete access
to both the quantum algorithm being run on the server and the results of the
computation. If sensitive computation is required we can combine the mainframe
with high fidelity communication channels to perform a secure version of HPQC
in a manner unavailable to classical distributed computing.
As the topological lattice prepared by the mainframe is photon based, we are
able to utilize high fidelity optical communications channels to physically
transmit a portion of the 3D lattice to the client. Compared with the trusted
mainframe model, this scheme has some technological disadvantages. High
fidelity quantum communication channels are required to faithfully transmit
entangled photons from the mainframe to each client. While purification
protocols could, in principal, be utilized to increase transmission fidelity,
this would be cumbersome, and given that topological models for QIP exhibit
very high thresholds (of the order of 0.1-1%) it is fair to assume that
communication channels will be of sufficient reliability when a mainframe
device is finally constructed. Secondly, each client must have access to a
certain amount of quantum technology. Specifically, a set of classically
controlled, high fidelity single photons wave-plates and detectors. This
allows each client to perform their own measurement of the photon stream to
perform computation locally.
Security arises as the quantum data stream never carries information related
to the quantum algorithm being run on the client side. As the photon stream
transmitted to the client is the 3D topological lattice generated by the
mainframe, interrogation of the quantum channel is unnecessary as the state
transmitted is globally known. Additionally, the only classical information
sent between mainframe and user is related to the initial eigenvalues of the
prepared lattice (obtained from the mainframe preparation network), no other
classical data is ever transmitted to or from the user. This implies that even
if an eavesdropper successfully taps into the quantum channel and entangles
their own qubits to the cluster they will not know the basis the user chooses
to measure in or have access to the classical error correction record. While
an eavesdropper could employ a denial of service attack, the ability to
extract useful information from the quantum channel is not possible without
access to the classical information record measured by the client.
A second benefit to the secure model is that the client has ultimate control
of whether their portion of the lattice generated by the host remains
entangled with the larger global lattice of the mainframe. Performing
$\sigma_{z}$ basis measurements on any photon within the cluster simply
disentangles it from the lattice. Hence if the mainframe transmits a partial
section of the generated lattice to the client, they simply perform
$\sigma_{z}$ basis measurements on all photons around the edge of their
partitioned allotment and they are guaranteed that neither the host and/or
other users sharing the mainframe lattice can interact their portion of the
lattice with the clients alloted section. This severing of the users sub-
lattice from the mainframe would generally be recommended. If the sub-lattice
is still linked to the mainframe, error correction procedures would need to be
co-ordinated with the mainframe and classical data continually exchanged. This
is due to the fact that error chains are able to bridge the region between
user and host when links remain in-tact.
When a user has completed their task they have the option of making their
results available to the global lattice, either to be utilized again or shared
with other users. If the client does not wish to share the final quantum state
of their algorithm, they measure all defect qubits and restore their portion
of the lattice to a defect free state. If however, they wish to make available
a non-trivial quantum state to the mainframe, then after their quantum
algorithm is completed they can cease to measure the photons on the boundary
of their allotted lattice. Once the client logs off the system, the quantum
state of the defect qubits within this lattice will remain (provided the
mainframe automatically continues measuring the sub-lattice to enact identity
operations). Consequently, at a later time, the original user may decide to
log onto the system again, or a second user may choose to log on that sub-
lattice and continue to manipulate the stored data as they see fit (note that
it is assumed that the global lattice is of sufficient size to allow for
significant error protection and hence long term information storage).
Additionally, this same methodology can be utilized to allow different users
to interact quantum states. As with the previous case, two users may decide to
perform independent, private, quantum algorithms up to some finite time and
then interact data. Each user then ceases severing the connections to the
global lattice and receives half an encoded Bell state from the mainframe,
allowing for the implementation of teleportation protocols.
Although the preparation of a large 3D cluster lattice with photonic chips has
been examined, how to partition resources for an optimal, multi-user device is
a complicated networking problem. At this stage we will simply present an
example partition structure for the resource lattice, hopefully demonstrating
some of the essential features that would be needed for this model. We will
approach this analysis with some basic numerical estimates to give an idea of
the resource costs and physical lattice sizes for a mainframe device.
Figure 2: Illustrated is an example partitioning of the global 3D lattice for
a HPQC mainframe. This global lattice measures $4000\times 500,000$ unit cells
and requires approximately $7.5\times 10^{9}$ photonic chips to prepare. If
utilized as a single cluster computer, 2.5 million logical qubits are
available with sufficient topological protection for approximately $10^{16}$
logical operations (where a logical operation is defined as the measurement of
a single unit cell).
The HPQC mainframe will consist of two regions, an outer region corresponding
to user partitions and an inner region which we will denote as scratch space.
The scratch space will be utilized to for two primary tasks. The first is to
provide logical Bell states to individual users in order to interact quantum
information, the second is to distill and provide the high fidelity logical
ancillae states $|A\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$ and
$|Y\rangle=(|0\rangle+\exp(i\pi/4)|1\rangle)\sqrt{2}$ which are needed to
enact non-trivial single qubit rotations that cannot be directly implemented
in the topological model. Purifying these states is resource intensive and as
these states are required often for a general quantum algorithm it would be
preferable to have an offline source of these states which does not consume
space on the user partitions.
It should be stressed that the size of the scratch space lattice will be
heavily dependent on the fundamental injection fidelity of these non-trivial
ancilla states and consequently the amount of required state distillation.
This illustrative partitioning of the mainframe lattice, shown in Fig. 2
allocates a scratch space of $1000\times 1000$ cells for each user region
(effectively another computer the of the same size). In general, state
distillation of ancilla states requires a large number of low fidelity qubits
and distillation cycles and users will require a purified ancilla at each step
of their computation Fowler . Therefore, the scratch space could be
significantly larger than each user partition. This does not change the
general structure of the lattice partitioning, instead the width of the
central scratch region is enlarged with user partitions still located on the
boundaries. The primary benefit of requiring the mainframe to prepare purified
ancillae is dynamical resource allocation, performed at the software level by
the mainframe. By allowing the mainframe to prepare all distilled ancillae it
is able to adjust the user/scratch partition structure to account for the
total number of users and the required preparation rate of distilled states.
Based on this partitioning of the mainframe lattice we can illustrate the
resource costs through a basic numerical estimate. As shown in Devitt2 , under
reasonable physical assumptions, a large scale topological computer capable of
approximately $10^{16}$ logical operations (where a logical operation is
defined as the measurement of a single cluster cell) requires approximately
3000 photonic chips per logical qubit, measuring $20\times 40$ cells in the
lattice. We therefore allocate each user a square region of the overall
lattice measuring 1000$\times 1000$ unit cells, containing $50\times 25$
logical qubits and requiring approximately $3.75\times 10^{6}$ photonic chips
to prepare. Additionally we consider a HPQC mainframe of sufficient size to
accommodate 1000 individual user regions of this size with a scratch space two
user regions wide and 500 user regions deep. Hence, this HPQC will need to
generate a rectangular lattice measuring $4000\times 500,000$ cells and
require of order $7.5\times 10^{9}$ photonic chips to prepare.
This may seem like a extraordinary number of devices to manufacture and
incorporate into a large scale lattice generator, but one should recognize the
enormous size of this mainframe. The partition structure is determined at the
software level, no changes to the lattice preparation network is required to
alter the structure of how the lattice is utilized. Hence, if desired, this
mainframe can be utilized as a single, extremely large, quantum computer,
containing 2.5 million logical qubits, with topological protection for
approximately $10^{16}$ operations, more than sufficient to perform any large
scale quantum algorithm or simulation ever proposed.
In conclusion, we have introduced the concept of the High Performance Quantum
Computer, where a massive 3-dimensional cluster lattice is utilized as a
generic resource for multiple-user quantum information processing. The
architectural model of 3D topological clusters in optics allows for the
conceptual scaling of a large topological cluster mainframe well beyond what
could theoretically be done with other architectures for QIP. As an example we
illustrated a possible lattice partitioning of the mainframe system. This
partition, while not optimal, shows some of the necessary structures that
would be required for multi-user quantum computing. With this partition
structure we were able to estimate the number of photonic chips required to
construct a mainframe device. The construction of approximately 7.5 billion
photonic chips leads to an extraordinary large multi-user quantum computer.
While this is certainly a daunting task, this sized computer would represent
the ultimate goal of QIP research that began in the late 1970’s.
The authors thank A.G. Fowler, N. Lütkenhaus, A. Broadbent and T. Moroder for
helpful discussions. The authors acknowledge the support of MEXT, JST, HP and
the EU project QAP.
## References
* (1) D. Kielpinski, C. Monroe, and D. J. Wineland. Nature 417, 709 (2002); J. M. Taylor _et. al._ , Nature Phys. 1, 177 (2005); L. C. L. Hollenberg, A. D. Greentree, A. G. Fowler, and C. J. Wellard. Phys. Rev. B 74, 045311 (2006); A. G. Fowler _et. al._ Phys. Rev. B 76, 174507 (2007); R. Stock and D.F.V. James, arXiv:0808.1591 (2008).
* (2) R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007); R. Raussendorf, J. Harrington and K. Goyal, New J. Phys. 9, 199 (2007); A.G. Fowler, A.M. Stephens and P. Groszkowski, arXiv:0803.0272 (2008); A.G. Fowler and K. Goyal, arXiv:0805.3202 (2008).
* (3) S.J. Devitt _et. al._ , Phys. Rev. A. 76, 052312 (2007);
* (4) S.J. Devitt _et. al._ , arXiv:0808.1782 (2008).
* (5) R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
* (6) P. Villoresi _et. al._ New. J. Phys. 10, 033038 (2008); R. Ursin _et. al._ Proc. 2008 Microgravity Sciences and Process Symposium (2008); SECOQC Project www.sqcoqc.net.
* (7) A.G. Fowler, quant-ph/0506126 (2005).
|
arxiv-papers
| 2008-10-14T15:03:52
|
2024-09-04T02:48:58.253005
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Simon J. Devitt, William J. Munro, Kae Nemoto",
"submitter": "Simon Devitt Dr",
"url": "https://arxiv.org/abs/0810.2444"
}
|
0810.2559
|
# Mutual space-frequency distribution of Gaussian signal
###### Abstract
Mutual space-frequency distribution is proposed and it is shown that Wigner
and Weyl distribution functions are only particular cases of these
distribution. Mutual distribution for Gaussian signal is analytically
obtained. The simple connection between Wigner and Weyl distributions is
established. It is shown that Wigner distribution forms as the rotational
displacement of Weyl distribution on informational diagram of conjugate
coordinates $(x;p)$ on an angle proportional to the mutual parameter $t$. The
results of direct calculations of mutual distribution for Gaussian signal in
the mutual domain are presented.
Yu. M. Kozlovskii
## 1 Introduction
The main aim of space-frequency analysis is elaboration of distributions in
order to get the information about signal simultaneously in coordinate and
frequency domains.As a rule the Fourier transform is used to receive the
signal frequency spectrum. This well-known transformation is good tool for the
analysis of signal intensity distribution in the frequency plane. Such
analysis foresees the calculation of Fourier-spectrum for constant
coordinates. Practically, we have to deal with a certain momentary value of
coordinates for which the signal and their Fourier-image are simultaneously
determinated. With the coordinate variation the appropriate conversion of
Fourier-spectrum also takes place and at once the problem of signal analysis
occurs; the latter contains frequency components that are variable in
accordance with coordinate. In such a case it is important to know the value
of coordinate at which the corresponding transformation of frequency spectrum
takes place. In order to investigate the variation of signal spectrum with the
variation of its coordinate as far back as in 60-80-ies of the previous
century was propound a new approach. It unites the information about
coordinate and frequency constituent of the signal in so-called space-
frequency representations. In such representations under consideration is a
certain mutual function of coordinate and frequency. The idea of construction
of mutual representations originates in works of E.Wigner (1932) [1], D.Gabor
(1946) [2] and J.Weyl (1932) [3]. Before the 80-ies of the previous century
tens of space-frequency representations of such case were taken under
consideration [7, 8, 10]. However, the Wigner and Weyl distributions that are
the most used for the present day remained the prerogative of quantum
mechanics and they have not have precisely expressed use. Only in 1980
T.Klasen and B. Mecklenbrauker worked out the theory of application of Wigner
distribution for space-frequency analysis of signals. Its main results were
published in the series of works under the title ”Wigner distribution - the
instrument for space-frequency analysis of signals” [4, 5, 6]. The successful
use of the Wigner distribution in the theory of signals was stipulated by its
”good” mathematical characteristics, especially by its representative
characteristics that is basic in the restoration of signal intensity
distribution.
Within the framework of the given investigation among the variety of space-
frequency representations we single out two basic distributions by Wigner and
Weyl that are widely used by the theory of signals in solving inverse physical
problems [8, 11]. The investigation of the signal characteristics takes place
on the basis of comparison with its displaced analogues. The shift within a
time results in subtraction of a specific value from the signal argument
$\rightarrow x_{\tau}(t)=x(t+\tau).$ (1)
The suitable displacement in accordance with frequency results in displacement
of argument of the Fourier-spectrum signal, what equals to multiplication by
phase multiplier in coordinate plane.
${\bf X_{\omega}}\rightarrow X_{\omega}(t)=x(t)e^{i\omega t}.$ (2)
Similar correlations are well-known from classical analysis [11]. Within the
limits of space-frequency analysis we are interested in the signals displaced
simultaneously with time and frequency, namely
$x_{-\frac{\tau}{2},-\frac{\omega}{2}}=x\left(t-\frac{\tau}{2}\right)e^{-i\omega
t/2},$ (3)
$x_{\frac{\tau}{2},\frac{\omega}{2}}=x\left(t+\frac{\tau}{2}\right)e^{i\omega
t/2}.$ (4)
We can easily calculate the value of displacement between the signals
$d({\bf x_{-\frac{\tau}{2},-\frac{\omega}{2}}},{\bf
x_{\frac{\tau}{2},\frac{\omega}{2}}})^{2}=2||x||^{2}-2\Re\left\\{{\cal
A}_{xx}(\omega,\tau)\right\\}.$ (5)
${\cal A}_{xx}(\omega,\tau)$ in this correlation plays a part of the distance
and is called a time-frequency autocorrelation function or the ambiguity
function
${\cal A}_{xx}(\omega,\tau)=\int\limits
x^{*}\left(t-\frac{\tau}{2}\right)x\left(t+\frac{\tau}{2}\right)\exp{(-i\omega
t)}dt.$ (6)
In accordance to the Parceval theory we can rewrite this correlation by
Fourier-images of displaced signals
${\cal A}_{xx}(\omega,\tau)=\frac{1}{2\pi}\int\limits
X\left(\nu-\frac{\omega}{2}\right)X^{*}\left(\nu+\frac{\omega}{2}\right)\exp{(i\omega\tau)}d\omega.$
(7)
The function in such a form was for the first time set by J. Weyl [3] and for
the present time is known in the theory of signals under the name of Weyl
distribution or the ambiguity function. Having realized the direct and inverse
Fourier-transform we get the value
${\cal
W}_{xx}(t,\nu)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}{\cal
A}_{xx}(\omega,\tau)e^{i\omega t}e^{-i\nu\tau}d\omega d\tau.$ (8)
that in explicite form has the following notation
${\cal W}_{xx}(t,\nu)=\int\limits
x^{*}\left(t-\frac{\tau}{2}\right)x\left(t+\frac{\tau}{2}\right)\exp{(-i\nu\tau)}d\tau,$
(9)
or by the Fourier-image of the function ${x}(t)$
${\cal W}_{xx}(t,\nu)=\frac{1}{2\pi}\int\limits
X\left(\nu-\frac{\omega}{2}\right)X^{*}\left(\nu+\frac{\omega}{2}\right)\exp{(i\omega
t)}d\omega.$ (10)
The function was firstly introduced by E.Wigner and is named after him - the
Wigner function of distribution or just Wigner distribution [1].
Joint space-frequency represenations are widely used not only in the theory of
signals. They have a number of practical use in different fields of physics,
geology, seismology, etc. Within the limits of the given research we are
interested in the use of such distributions in the region of representations
treatment and recognition of images. This field of the physics imposes a set
of demands that the generalized distribution of signals have to meet. For the
efficient use in the theory of representatives of space-frequency distribution
they have to be characterized by representative property; for the limiting
values of the variable $t$ they have to develop into known distributions; to
have high distributive capacity in the field of Wigner distribution as well as
in Weyl distribution; to take positive values.
Representations that the most precisely meet the demands stated in the theory
of image are basic distributions by Wigner and Weyl. These distributions also
have their own peculiarities. Unfortunately, up to the present there exists no
simple deduction about the expediency of use of this or that distribution.
There exist a number of signals for which Wigner and Weyl distributions
proceed into the negative region. In such cases these distributions are
interpreted as quasiprobable. In spite of the external resemblance of the
properties of Wigner and Weil distributions they have the peculiarity of
principle in the mechanism of renewal of the entrance signal according to the
known distribution. Wigner formalism allows renewing the signal according to
the so-called marginal distribution
$|x(t)|^{2}=\int\limits_{-\infty}^{\infty}{\cal W}_{xx}(t,\nu)d\nu.$ (11)
In the frame of Weyl formalism signal reconstruction takes place using signal
restoration scheme
$|x(t)|^{2}=\int\limits_{-\infty}^{\infty}{\cal A}_{xx}(0,\omega)e^{i\omega
t}d\omega.$ (12)
Sometimes working with the scheme of renewal according to the Weyl
distribution is to the great extent more easily (due to the integrating of its
crosscut) than in the case of marginal distribution. For the present Wigner
distribution is more commonly applied as it uses marginal distributions that
can be measured by experiment. Though, both approaches have the right to
existence.
Within the framework of the given experiment we investigate uninterrupted
transition between these distributions by means of introduction of a
generalized common function of time and frequency that depends from a variable
$t$.
For present day various types of space-frequency distributions are
successfully used for the analysis of nonstationary signals [7, 8]. Many of
such distributions are characterized by advantages as well as by disadvantages
in use in various fields of physics. Wigner and Weyl distributions are widely
used in space-frequency analysis and in particular in optical information
processing systems. Well known is a fact, that the very distributions posess
such charateristics, that are successfully used for description of many
optical systems. The spectrum of appliance of these distributions is extremely
wide. They are used, in particular, in the theory of optical lens system,
theory of communication, hydrolocation and other fields [9, 10, 11]. The
researches of last year proved the efficiency of use of space-frequency
distributions in biology and medicine; especially Wigner distribution was
successfully used for renewal of volumetric structure of objects within the
framework of optical tomography [12, 13, 14]. One of the promising
investigation directions within the space-frequency processing of signals is
studying the properties of novel space-frequency representations of the
distributions, with the aim of their further applications in different areas
of physics and medicine. Unfortunately, it often happens that some space-
frequency distributions do not meet demands raised by one or another specific
application. In this relation, many of the existing distributions need
generalization or improvements when applied to a given problem. During the
second half of the past century and the beginning of this one, a cleartendency
has been observed towards generalization of different space-frequency
distributions. The first attempt of such a generalization has been due to
L.Cohen [15] as long ago as in 1966. The author has introduced a number of
quasiprobable distributions that provide proper quantum mechanical marginal
distributions. Within the limits of this research the Wigner distribution was
examined as a separate case. The next step has been done by N. De Brujin in
1973 [16]. His work has been devoted to elaboration of theory of generalized
functions, with application concerned with Wigner and Weil distributions.
Summarizing the results of numerous investigations L.Cohen [7, 10] has
suggested to has suggested a generalized distribution involving a certain
kernel
$\displaystyle C(x,\omega,\Phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f\left(y+\frac{x_{0}}{2}\right)f^{*}\left(y-\frac{x_{0}}{2}\right)$
(13) $\displaystyle\times$ $\displaystyle\Phi(x,\omega)e^{-i(\omega
x_{0}-\omega_{0}x+\omega_{0}y)}dydx_{0}d\omega_{0}.$
Depending on the form of the kernel $\Phi(x,\omega)$ , this distribution
degenerates into one of the known distributions (Wigner, Weyl, Woodward,
Kirkwood, Page, Mark, etc.).
In the general case expression (13) describes the class of space-frequency
distribution later named Cohen’s class. Members of this class are known
distributions as well as a set of still unknown distributions, that also
satisfy all necessary requirements of existing distributions. The theory of
generalization of space-frequency distributions has been developed by also
famous specialist in the theory of signals by A. Mertins. In his monograph
”Signal Analysis”[11] he has singled out thistopic into a separate section
”General space-frequency distributions”. The author has stated that the Wigner
distribution serves as an excellent tool for space-frequency analysis as long
as a linear dependence is kept between the instantaneous coordinates
andfrequencies. Otherwise, a need in generalizing appears, whose general
principles are described in detail in the mentioned work.
Among numerous recent studies related to generalizing space-frequency
distributions, we should mention only the most typical ones. The PhD Thesis by
L. Durak ”Novel time-frequency analysis technique for deterministic signals”
[17] is one of such studies, where a close attention has been paid to
generalizing distributions and introducing their additional parameters.
Different types of generalizations of space-frequency distributions have been
thoroughly considered in the book by B. Boashash [8]. Among a number of
studies included in it, we would like to emphasize the works by R. Baraniuk
(p. 123), X.Xia (p. 223) and A. Papandreou-Suppappola (p. 643). Within the
mentioned collection, the work by G. Matz by F. Hlawatsch (p. 400) is of
particular interest in relation to the problem of distributions
generalization. It considers methodology for constructing generalized
distributions on the basis of both the Wigner distribution and the ambiguity
function (i.e. the Weyl distribution).
In the present work we try to use interlinks between the Wigner and Weyl
distributions with the aim of joining them into a single, more general
distribution. Up to date, it has been revealed that the two distributions are
related by a double Fourier transform. The results obtained by us allow
tracing transformation of one of thedistributions into the other, while
changing the distribution parameter $t$. This generalized distribution
generates a whole set of new distributions formed in the process of switching
between the basic distributions. The latter fact may be important from the
viewpoint of possible practical applications. For the present day a choice
between the Weyl and Wigner distributions remains ambiguous. Each of them has
its own scheme for reconstruction of signal intensity distribution. The scheme
adopted for the Wigner distribution includes calculating the marginal
distributions [7]. The Weyl distribution provides much simpler reconstruction
scheme,owing to simpler mathematical transformations [18, 22]. Traditionally,
the Wigner distribution has been used in a large majority of studies performed
within the field. Introduction of the mutual distribution would mean a
possibility for calculating ’mixed’states and determining the necessary
contributions of each of the limiting distributions. As stated above, there
appears a possibility for generalization of distributions concerning various
applied problems. However, only S. Chountasis has suggested the approach [19]
that enables transitions between the Wigner and Weyl distributions. Such
distributions play an important role in the analysis of phase space and,
moreover, can be immediately applied in the Wigner tomography [12, 14]..In
1999 S. Chountasis and co-authors [19, 21] have developed a general
distribution based on the Wigner formalism, which involves an additional
parameter $\theta$. This study has been performed in frame of quantum-
mechanical formalism. It allows passing the Wigner and Weyl distributions into
each other by means of changing the common parameter.
The problem of calculation of a classical analogue of this generalized
distribution remains urgent. It may be constructed based on the results [19]
or using the formalism of Weyl distribution, as has been done by the present
author when studying the properties of fractional Fourier transform [23].
Similarly to the works [18, 22], the author has employed peculiarities of
reconstruction of signal intensity based upon the Weyl distribution.
Meanwhile, it is just this reconstruction scheme is realized experimentally in
the real optical schemes [23].
## 2 Mutual distribution: basic relations
### 2.1 Theoretical statements
In the present work we propose to use a type of generalized distribution with
paramether $t$ based upon the Weyl distribution function. The use of Weyl
distribution has a peculiarity of principle in comparison with the function of
Wigner distribution, that consists of the possibility to renew the intensity
of distribution. The latter is registered experimentally at the output of the
optical system. Common distribution of two signals $f_{1}(x)$ and $f_{2}(x)$
may be written as follows [25]
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{(t)}(x;p)$ $\displaystyle=$
$\displaystyle\frac{C_{t}}{1+t}\int\int
dx_{0}d\omega_{0}\exp\left\\{i\left[x_{0}p-\omega_{0}x\right]\right\\}$ (14)
$\displaystyle\times$
$\displaystyle\exp\left\\{-i\frac{(x-x_{0})^{2}+(p-\omega_{0})^{2}}{tan(\theta/\
2)}\right\\}$ $\displaystyle\times$ $\displaystyle\int\limits
f_{1}\left(z+\frac{x_{0}}{2}\right)f_{2}^{*}\left(z-\frac{x_{0}}{2}\right)\exp{(-i\omega_{0}z)}dz.$
Constant $C_{t}$ and variable of the generalized distribution $t$ are
determined by the expressions
$C_{t}=\frac{2}{\pi}\frac{1}{1-\exp{i\theta}},\qquad t=\frac{\theta}{\pi}.$
(15)
Distribution (14) is called the mutual space-frequency distribution, or
concisely the mutual distribution. Limiting cases of ditribution (14) is Weyl
distribution (ambiguity function) (6) with the value of the parameter $t=0$
and Wigner distribution (9) with the value of parameter $t=1$. Accordingly,
two known distributions (6) and (9) have plenty of alternative distributions
and to each of them corresponds a specific value of parameter $t$.
The expression of the mutual distribution (14) may be also represented in the
simplified form using Weyl distribution (6)
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{(t)}(x;p)$ $\displaystyle=$
$\displaystyle\frac{C_{t}}{1+t}\int\int dx_{0}d\omega_{0}{\cal
A}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})$ (16) $\displaystyle\times$
$\displaystyle\exp\left\\{i\left[x_{0}p-\omega_{0}x\right]\right\\}\exp\left\\{-i\frac{(x-x_{0})^{2}+(p-\omega_{0})^{2}}{tan(\theta/\
2)}\right\\}.$
Performing the converted transformation we can render the Weyl distribution by
means of the above introduced function of the mutual distribution
$\displaystyle{\cal
A}_{f_{1}f_{2}^{*}}(x_{0}^{{}^{\prime}};\omega_{0}^{{}^{\prime}})$
$\displaystyle=$ $\displaystyle\frac{1+t}{C_{t}}\int\int dxdp{\cal
K}_{f_{1}f_{2}^{*}}^{(t)}(x;p)$ (17) $\displaystyle\times$
$\displaystyle\exp\left\\{-i\left[x_{0}^{{}^{\prime}}p-\omega_{0}^{{}^{\prime}}x\right]\right\\}\exp\left\\{i\frac{(x-x_{0}^{{}^{\prime}})^{2}+(p-\omega_{0}^{{}^{\prime}})^{2}}{tan(\theta/\
2)}\right\\}.$
Formula (17) constitutes the inverse connection between the simple ${\cal
A}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})$ and generalized ${\cal
K}_{f_{1}f_{2}^{*}}^{(t)}(x;p)$ Weyl distributions. This makes the possibility
of restoring the distribution of signal intensity according to the mutual
distribution what has not been established when using the generalized Wigner
distribution [19].
### 2.2 Representation in the terms of Wigner distribution
In order to make a comparison of the results with the existing analogues it is
indispensable to have possibility to precisely calculate the limiting cases of
the mutual distribution (17). Calculation of the limiting case $t=1$ by means
of the formula (16) may be conducted precisely, and in the case $t=0$ such
transition is not a trivia matter. To make the calculations simplier we
introduce the representation of the mutual space-frequency distribution by
means of Wigner distribution. We make use of the known identity
$\displaystyle{\cal
A}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})=\frac{1}{2\pi}\int\int d\xi d\eta{\cal
W}_{f_{1}f_{2}^{*}}(\eta;\xi)\exp(-i\omega_{0}\eta)\exp(i\xi x_{0}),$ (18)
that connects Weyl and Wigner distributions.
Placing (18) into the expression (16) and making a set of conversions we
receive a formula describing the mutual space-frequency distribution in the
terms of Wigner distribution
$\displaystyle{\cal{K}}_{f_{1}f_{2}^{*}}^{(t)}(x;p)$ $\displaystyle=$
$\displaystyle\widetilde{C}_{t}\int\int dx_{0}d\omega_{0}{\cal
W}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})\exp\left\\{-i\left[+x_{0}p+\omega_{0}x\right]\right\\}$
(19) $\displaystyle\times$
$\displaystyle\exp\left\\{i\frac{1}{4}tan\frac{\theta}{2}\left[(x-x_{0})^{2}+(p-\omega_{0})^{2}\right]\right\\},$
where the constants $\widetilde{C}_{t}$ are determined by means of the
correlation
$\widetilde{C}_{t}=\frac{-i}{\pi}\frac{1}{1-\exp{i\theta}}\tan\frac{\theta}{2}\frac{1}{1+t}.$
(20)
As it can be easily seen in the case of representation of the mutual
distribution by means of the Wigner distribution peculiarities in the point
$t=0$ and around it dissappear, however, there appear peculiarities around the
point $t=1$. Thereby the pair of representations: (16) and (19) complement one
another and completely describe mutual space-frequency distribution in the
region $t=[0,1]$.
### 2.3 Limiting cases
The aim of this work is to determine the mechanism of re-distribution between
Wigner and Weyl distributions and to investigate the peculiarities of mutual
distributions describing the region of values $t=[0,1]$. Consequently the
investigation of limiting cases (16) and (19) of mutual distribution is of
peculiar importance. Let us study the limiting cases.
_Case $t=1$._
To describe this case we make use of coordinate representation of the mutual
distribution (16). Placing $t=1$ (or $\theta=\pi$) under (16) we arrive to the
following result
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{t=1}(x;p)=\frac{1}{2\pi}\int\int
dx_{0}d\omega_{0}{\cal
A}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})\exp(ix_{0}p-i\omega_{0}x).$ (21)
Accordingly to (8) we have
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{t=1}(x;p)={\cal
W}_{f_{1}f_{2}^{*}}(x;p).$ (22)
The limiting case $t=1$ of mutual space-frequency distribution corresponds to
the function of Wigner distribution.
_Case $t=0$._
To describe this case we make use of coordinate representation of the mutual
distribution (19). Placing here $t=0$ (or $\theta=0$) we arrive to the
following result
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{t=0}(x;p)=\frac{1}{2\pi}\int\int
dx_{0}d\omega_{0}{\cal
W}_{f_{1}f_{2}^{*}}(x_{0};\omega_{0})\exp(-i\omega_{0}x+ix_{0}p).$ (23)
Accordingly to the formula connecting Wigner and Weyl distributions we obtain
$\displaystyle{\cal K}_{f_{1}f_{2}^{*}}^{t=0}(x;p)={\cal
A}_{f_{1}f_{2}^{*}}(x;p).$ (24)
Expression (24) is identical with the function of Weyl distribution. Hereby,
introduced by us distribution (14) in the values of the limiting cases of the
parameter $t$ describes known distributions (6) and (9). This distribution has
two equivalent representations: by means of the function of Wigner
distribution (19) or function of Weyl distribution (16). Having at least one
from the basic functions of distribution we can obtain the image of mutual
distribution. The object of the further investigation is to study the
peculiarities of the set of intermediate distributions when $(0<t<1)$. In
order to display the peculiarities of the mutual distribution and to visually
demonstrate the results we illustrate it on the example of Gaussian signal.
This is one of the simplest types of signals that allows us to make
calculations in the explicit form. We shall notice that similar calculations
may be also made with other signals (in peculiar with orthogonal impulse,
etc.). Choice of the Gaussian signal is related to explicit form of the common
distribution. Herewith we can trace the mechanism of transition of Weyl
distribution into Wigner distribution and vice versa.
## 3 Mutual distribution of Gaussian signal
### 3.1 Basic relations
In the work [24] have been investigated the main peculiarities of generalized
coordinate-frequency distribution on the example of Gaussian signal. Obtained
results were based on the frequent calculations of the proper distributions
that allow to properly evaluate processes of distribution. In the present work
we conduct an analytical calculation of the mutual space-frequency
distribution on the example of Gaussian signal that is represented in the
following way
$g(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp{\left(-\frac{x^{2}}{2\sigma^{2}}\right)}.$
(25)
Fourier image of these function is
$\hat{{\cal
F}}[g(x)]=\int\limits_{-\infty}^{\infty}g(x)e^{ixp}dx=\exp{\left(-\frac{p^{2}\sigma^{2}}{2}\right)}.$
(26)
Such selection form of Gaussian functions are stipulated by the fact that its
integrating according to all values $x$ results in value
$\int\limits_{-\infty}^{\infty}g(x)dx=1.$ (27)
As investigations of the mutual space-frequency distribution provides the
study of re-distribution between Wigner and Weyl distributions it is
reasonable to present the explicit form of these distributions for the case of
Gaussian signal [24].
_Weyl distribution for Gaussian signal_
${\cal
A}_{ff^{*}}(x_{0};\omega_{0})=\frac{1}{2\sqrt{\pi}\sigma}\exp{\left(-\frac{x_{0}^{2}}{4\sigma^{2}}-\frac{\omega_{0}^{2}\sigma^{2}}{4}\right)}.$
(28)
_Wigner distribution for Gaussian signal_
${\cal
W}_{ff^{*}}(x;\omega)=\frac{1}{\sqrt{\pi}\sigma}\exp{\left(-\frac{x^{2}}{\sigma^{2}}-\omega^{2}\sigma^{2}\right)}.$
(29)
It is known [7], that one of the basic characteristics of the signal is its
representativity that is the possibility to restoring the signal according to
its distribution. Schemes of renewing for Weyl and Wigner distributions are
well known. For the case of Gaussian signal they have the following form
_restoration scheme for Gaussian signal by Weyl distribution_
$|g(x)|^{2}=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}{\cal
A}_{ff^{*}}(0;\omega_{0})e^{i\omega_{0}x}d\omega_{0}=\frac{1}{2\pi\sigma^{2}}\exp{\left(-\frac{x^{2}}{\sigma^{2}}\right)}.$
(30)
_restoration scheme for Gaussian signal by Wigner distribution_
$|g(x)|^{2}=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}{\cal
W}_{ff^{*}}(x;\omega)d\omega=\frac{1}{2\pi\sigma^{2}}\exp{\left(-\frac{x^{2}}{\sigma^{2}}\right)}.$
(31)
Independently from the choice of distribution form Gaussian signal is
precisely renewed using both schemes of renewing.
We find explicit analytical form of the mutual distribution for the case of
Gaussian signal. For this reason we will place into the formula of mutual
distribution (16) the Weyl distribution of Gaussian signal (16). We obtain
expression
$\displaystyle{\cal
K}_{ff^{*}}^{t}(x;p)=\frac{C_{t}}{1+t}\frac{1}{2\sigma\sqrt{\pi}}\exp\left\\{-\frac{i}{T}(x^{2}+p^{2})\right\\}I_{1}(x,p)I_{2}(x,p),$
(32)
where
$\displaystyle I_{1}(x,p)=\int
dx_{0}\exp\left\\{ix_{0}p+\frac{i}{T}2xx_{0}-\frac{i}{T}x_{0}^{2}-\frac{x_{0}^{2}}{4\sigma^{2}}\right\\},$
(33) $\displaystyle I_{2}(x,p)=\int
d\omega_{0}\exp\left\\{-i\omega_{0}x+\frac{i}{T}2p\omega_{0}-\frac{i}{T}\omega_{0}^{2}-\frac{\omega_{0}^{2}\sigma^{2}}{4}\right\\}.$
(34)
Expressions (33) and (34) may be written in the following form
$\displaystyle
I_{1}(x,p)=\frac{\sqrt{\pi}}{\sqrt{a_{1}}}\exp\left\\{-\frac{1}{a_{1}}\left(\frac{x}{T}+\frac{p}{2}\right)^{2}\right\\},$
(35) $\displaystyle
I_{2}(x,p)=\frac{\sqrt{\pi}}{\sqrt{a_{2}}}\exp\left\\{-\frac{1}{a_{2}}\left(\frac{p}{T}-\frac{x}{2}\right)^{2}\right\\},$
(36)
where the following symboles are introduced
$\displaystyle a_{1}=\frac{T+i4\sigma^{2}}{4T\sigma^{2}},\qquad
a_{2}=\frac{T\sigma^{2}+4i}{4T},\qquad T=\tan\left(\frac{\theta}{2}\right).$
(37)
The mutual distribution of Gaussian signal (32) has the form
$\displaystyle{\cal K}_{ff^{*}}^{t}(x;p)$ $\displaystyle=$ $\displaystyle
C_{t}^{\sigma}\exp\left\\{-\frac{i}{T}\left(x^{2}+p^{2}\right)\right\\}$ (38)
$\displaystyle\times$
$\displaystyle\exp\left\\{-\frac{1}{a_{1}}\left(\frac{x}{T}+\frac{p}{2}\right)^{2}\right\\}\exp\left\\{-\frac{1}{a_{2}}\left(\frac{p}{T}-\frac{x}{2}\right)^{2}\right\\}.$
where constant
$C_{t}^{\sigma}=\frac{C_{t}}{1+t}\frac{\sqrt{\pi}}{2\sigma}\frac{1}{\sqrt{a_{1}a_{2}}}$
(39)
depends from dispersion of Gaussain distribution of signal $\sigma$ and from
the values $a_{1}$, $a_{2}$. From the values $a_{1}^{-1}$ and $a_{2}^{-1}$
depend also expressions in the index of the expression exponential curve (41).
We depict them in the form
$a_{1}^{-1}=a_{11}+ia_{12}:\qquad
a_{11}=\frac{4T^{2}\sigma^{2}}{16\sigma^{4}+T^{2}},\qquad
a_{12}=-\frac{16T\sigma^{4}}{16\sigma^{4}+T^{2}},$ (40)
$a_{2}^{-1}=a_{22}+ia_{21}:\qquad
a_{22}=\frac{4T^{2}\sigma^{2}}{16+T^{2}\sigma^{4}},\qquad
a_{21}=-\frac{16T}{16+T^{2}\sigma^{4}}.$ (41)
Placing (40) and (41) under (38) we find explicit analytical form of the
mutual distribution of the Gaussian signal (14)
$\displaystyle{\cal K}_{ff^{*}}^{t}(x;p)$ $\displaystyle=$
$\displaystyle\frac{C_{t}}{1+t}\frac{\sqrt{(}\pi)}{2\sigma}\frac{1}{\sqrt{a_{1}a_{2}}}\exp\left\\{-\frac{i}{T}\left(x^{2}+p^{2}\right)\right\\}$
(42) $\displaystyle\times$
$\displaystyle\exp\left\\{-a_{11}\left(\frac{x}{T}+\frac{p}{2}\right)^{2}-ia_{12}\left(\frac{x}{T}+\frac{p}{2}\right)^{2}\right\\}$
$\displaystyle\times$
$\displaystyle\exp\left\\{-a_{22}\left(\frac{p}{T}-\frac{x}{2}\right)^{2}-ia_{21}\left(\frac{p}{T}-\frac{x}{2}\right)^{2}\right\\}.$
We may check that expression (42) in limiting cases $t=0$ and $t=1$ changes
into Weyl and Wigner distributions respectively. In order to ascertain the
circumstance we investigate the conduct of the expression (42) in the limit
$t\rightarrow 1$. Parameter $t$ is determined according to (15) by value
$\theta$. We introduce small value $\alpha$ ($\alpha\ll 1$) and depict
$\theta$ in the form
$\theta=\pi-\alpha.$ (43)
Region of small values $\alpha$ corresponds to quasi-Wigner region of mutual
function of distribution. Then the value $C_{t}$ that is a part of
$C_{t}^{\sigma}$ in case of small values $\alpha$ can be depicted as follows
$C_{t}\approx\frac{1}{\pi}\left(1+i\frac{\alpha}{2}\right).$ (44)
Asymptotics of the values $(a_{1}a_{2})^{-1/2}$ may be easily find when use
its module $r_{12}$ and argument $\varphi_{12}=\varphi_{1}+\varphi_{2}$
$(a_{1}a_{2})^{-1/2}=r_{12}e^{i(\varphi_{1}+\varphi_{2})/2},$ (45)
where
$\displaystyle
r_{12}=\pm\frac{4T\sigma}{(16+T^{2}\sigma^{4})^{1/4}(16\sigma^{4}+T^{2})^{1/4}},$
$\displaystyle\varphi_{1}=\arctan\left(-\frac{4\sigma^{2}}{T}\right),\quad\varphi_{2}=\arctan\left(-\frac{4}{T\sigma^{2}}\right).$
(46)
In the case of small values $\alpha$ ($T\gg 1$) we have such approximate form
for the value $r_{12}$
$r_{12}=4\left(1-\frac{4}{T^{2}}\left(\sigma^{4}+\sigma^{-4}\right)\right),$
(47)
and for value $\varphi_{1}$ and $\varphi_{2}$ we find
$\varphi_{12}=\varphi_{1}+\varphi_{2}=-\frac{4}{T}\left(\sigma^{2}+\sigma^{-2}\right).$
(48)
Thereby, the constant $C_{t}^{\sigma}$ from (42) in quasi-Wigner region
$\alpha\ll 1$ has the form
$C_{t}^{\sigma(1)}=\frac{1}{\sigma\sqrt{\pi}}\left(1+\frac{i}{2}\alpha\right)\left(1-\frac{4}{T^{2}}\left(\sigma^{4}+\sigma^{-4}\right)\right)e^{-\frac{2i}{T}\left(\sigma^{2}+\sigma^{-2}\right)}.$
(49)
Index 1 in the value $C_{t}^{\sigma}$ denotes a condition $\alpha\ll 1$.
Naturally, that in the $\alpha\rightarrow 0$ we has the value
$C_{t}^{\sigma}=\frac{1}{\sigma\sqrt{\pi}},$
that precisely corresponds to the amplitude of the expression for Wigner
distribution (29).
Let us observe the coefficient $C_{t}^{\sigma}$ from (42) in the case of small
values $\theta$ $(t\rightarrow 0)$. We shall call the region of parameter $t$
values quasi-Weyl region as far as in the value $t=0$ we aquire Weyl
distribution. Similarly as in the case $t=1$, we find for small values the
following expressions
$C_{t}=\frac{i}{\pi T}(1-iT),\qquad
r_{12}=T,\qquad\varphi_{1}=\varphi_{2}=-\pi/2,\qquad
C_{t}^{\sigma(2)}=\frac{1}{2\sqrt{\pi}\sigma}.$ (50)
Such value of constant $C_{t}^{\sigma(2)}$ exactly corresponds to the
amplitude of the value from Weyl distribution of Gaussian signal (28). Thus,
the mutual distribution of Gaussian signal in limiting cases according to
amplitude precisely coincides with the known distributions of Wigner and Weyl.
Calculation of real and imaginary part of constant $C_{t}^{\sigma}$ depicted
on the Fig.1. for $C_{t}^{\sigma}$ provides an exact correspondence according
to amplitude with Wigner and Weyl distributions. In the common region appears
imaginary part $C_{t}^{\sigma}$ that is inherent only in mutual distribution.
In the limiting cases $t=0(\theta=0^{0})$ and $t=1(\theta=180^{0})$ imaginary
part dissappears what corresponds to the cases of basic distributions. Worth
mentioning is also peculiar conduct of the imaginary part of constant
$C_{t}^{\sigma}$ having maximum in the quasi-Wigner region.
Let us proceed the investigation of limiting cases of expessions placed in the
index of the exponent on a curve formula (42). As was shown above the
amplitude of mutual distribution in limiting cases coincides with the
amplitude of known distributions of Weyl and Wigner. Providing that the form
of these dirtibutions will also coincide functionally (as functions $x$ and
$p$), mutual distribution may be considered as generalization of well-known
distributions of Wigner and Weyl. Having made a number of mathematical
transformations the expression (42) aquires the following form
$\displaystyle{\cal K}_{rr^{*}}^{(t)}(x;p)$ $\displaystyle=$ $\displaystyle
C_{t}^{\sigma}\exp\left\\{-gx^{2}-fp^{2}-dxp\right\\},$ (51)
where coefficients $g$, $p$ and $d$ are certain complex values
$g=g_{0}+ig_{1},\quad f=f_{0}+if_{1},\quad d=d_{0}+id_{1}$ (52)
moreover
$\displaystyle g_{0}=\frac{\sigma^{2}}{\mu}(64+20T^{2}\sigma^{4}+T^{4}),\qquad
g_{1}=\frac{T}{\mu}(16[1-4\sigma^{4}]+T^{2}[\sigma^{4}-4]);$ $\displaystyle
f_{0}=\frac{\sigma^{2}}{\mu}(64\sigma^{4}+20T^{2}+T^{4}\sigma^{4}),\qquad
f_{1}=\frac{T\sigma^{4}}{\mu}(16[\sigma^{4}-4]+T[1-4\sigma^{4}]);$
$\displaystyle d_{0}=\frac{4T\sigma^{2}}{\mu}(1-\sigma^{4})(16-T^{2}),\qquad
d_{1}=16T^{2}\frac{1-\sigma^{8}}{\mu}.$ (53)
where the value $\mu$ has the form
$\displaystyle\mu=16^{2}\sigma^{4}+16T^{2}(1+\sigma^{8})+\sigma^{4}T^{4}$ (54)
Constant $C_{t}^{\sigma}$ from (39) may be represented as
$\displaystyle
C_{t}^{\sigma}=\frac{i}{\pi}\frac{1+e^{-i\theta}}{\sin\theta}\frac{1}{1+t}\frac{\sqrt{\pi}}{2\sigma}r_{12}e^{i(\varphi_{1}+\varphi_{2})/2},$
(55)
where
$\displaystyle r_{12}=\frac{4\sigma
T}{(16+T^{2}\sigma^{4})^{1/4}(16\sigma^{4}+T^{2})^{1/4}},$
$\displaystyle\varphi_{1}=\arctan\left(-\frac{4\sigma^{2}}{T}\right),\qquad\varphi_{2}=\arctan\left(-\frac{4}{T\sigma^{2}}\right)$
(56)
Representation (51) is an explicit form of the mutual distribution of Gaussian
signal. Let us consider asymptotic values of $g$, $f$ and $d$ in the limiting
cases $t=0$ and $t=1$.
In the case $t=0$ ($T\rightarrow 0$) from (53), (54) we find
$g_{0}=(4\sigma^{2})^{-1},\quad f_{0}=\sigma^{2}/4.$ (57)
The rest of coefficients turns into zero
$g_{1}=f_{1}=d_{0}=d_{1}=0.$ (58)
In the limiting case $t=1$ ($T\rightarrow\infty$) we receive
$g_{0}=\sigma^{-2},\quad f_{0}=\sigma^{2}.$ (59)
Other coefficients dissapear as in (58). We shall notice that imaginary part
of cross-expressions in (51) real part dissapears in $t\rightarrow 0$ as well
as in $t\rightarrow 1$.
What concerns the values $g$ and $f$, their imaginary parts also tend to zero
in the case $t=0$ and $t=1$. Thus, the very quadratical members are
responsible for forming of the distribution in limiting cases as they are
included into known Wigner and Weyl distributions, and cross-representations
that are inherent only in mutual distribution dissapear.
## 4 Signal restoration scheme in mutual distribution domain
As it is known, restoration of the signal according to Weyl distribution takes
place correspondingly to (30), and according to Wigner distribution -
accordning to (31). Above mentioned schemes of signal restoration may be
united into one formula using the mutual space-frequency distribution
suggested above. We shall introduce the value
$f_{\theta}(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\rho
w_{\theta}\left(x\sin(\theta/2)p\right)e^{ipx\cos(\theta/2)}.$ (60)
It can be easily seen, that when $\theta=0$ we arrive at Weyl renewal scheme,
and when $\theta=\pi$ \- we aquire Wigner renewal signal. Taking into
consideration (51), for the function $w_{\theta}$ we have
$\displaystyle
w_{\theta}\left(x\sin(\theta/2),p\right)=C_{t}^{\sigma}\exp\left\\{-g\sin^{2}(\theta/2)x^{2}-fp^{2}-d\sin(\theta/2)xp\right\\}.$
(61)
In the result of integration of (60) we aquire
$\displaystyle f_{\theta}(x)$ $\displaystyle=$
$\displaystyle\frac{C_{t}^{\sigma}}{2\pi}\frac{\sqrt{\pi}}{(f_{0}+if_{1})^{1/2}}e^{-x^{2}\sin\theta/2(g_{0}+ig_{1})}$
(62) $\displaystyle\times$
$\displaystyle\exp\left\\{\frac{x^{2}}{4}\frac{1}{(f_{0}+if_{1})}\left(\sin\frac{\theta}{2}(d_{0}+id_{1})-i\cos\theta/2\right)^{2}\right\\}.$
It can be easily seen that in limiting cases function $f_{\theta}(x)$
transforms into expressions (30) and (31). Taking into account expressions for
real and imaginary part of coefficients $g$, $f$ and $d$ we arrive at
$f_{\theta}(x)=\frac{1}{2\sqrt{\pi
r_{f}}}e^{-\frac{i}{2}\varphi_{f}}e^{-Gx^{2}},$ (63)
where
$r_{f}=\frac{\sigma^{2}}{\mu}\left[\left\\{64\sigma^{4}+20T^{2}+\sigma^{4}T^{4}\right\\}+T^{2}\sigma^{4}\left\\{16(\sigma^{4}-4)+T^{2}(1-4\sigma^{4})\right\\}^{2}\right]^{1/2},$
$\varphi_{f}=\arctan\left(\sigma^{2}T\frac{16(\sigma^{4}-4)+T^{2}(1-4\sigma^{4})}{64\sigma^{4}+20T^{2}+T^{4}\sigma^{4}}\right),$
$G=r_{g}\sin\frac{\theta}{2}e^{i\varphi_{g}}-\frac{1}{4}\frac{1}{r_{f}}\left\\{r_{\alpha}^{2}\sin^{2}\frac{\theta}{2}e^{i(\varphi_{f}+2\varphi_{d})}-2r_{d}\sin\frac{\theta}{2}e^{i(\varphi_{f}+\varphi_{d}+\frac{\pi}{2})}-\cos^{2}\frac{\theta}{2}e^{i\varphi_{f}}\right\\}.$
In order to make the formula shorten certain symbols are introduced
$r_{g}=\frac{\sigma^{2}}{t_{4}}\left[\left\\{64+20T^{2}\sigma^{4}+T^{4}\right\\}+\frac{T^{2}}{\sigma^{4}}\left\\{16(1-4\sigma^{4})+T^{2}(\sigma^{4}-4)\right\\}^{2}\right]^{1/2},$
$\varphi_{g}=\arctan\frac{g_{1}}{g_{0}}=\arctan\left\\{\frac{T[16(1-4\sigma^{4})+T^{2}(\sigma^{4}-4)]}{\sigma^{2}(64+20T^{2}\sigma^{4}+T^{4})}\right\\},$
$r_{d}=\frac{4T}{t_{4}}\left[\sigma^{4}(16-T^{2})(1-\sigma^{4})^{2}+16T^{2}(1-\sigma^{8})^{2}\right]^{1/2},$
$\varphi_{d}=\arctan\left(\frac{16T^{2}(1-\sigma^{8})}{4T\sigma^{2}(1-\sigma^{4})(16-T^{2})}\right)=\arctan\left(4T\frac{(1-\sigma^{8})}{\sigma^{2}(1-\sigma^{4})(16-T^{2})}\right).$
Thus, suggested restoration scheme (63) in limiting cases precisely renew
Gaussian signal (25)
$f_{(t=1)}(x)=f_{(t=0)}(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp{\left(-\frac{x^{2}}{2\sigma^{2}}\right)}.$
(64)
Thereby, side by side with known limiting cases arises possibility of signal
intensity distribution restoration in the region $t=[0..1]$. In our opinion
investigation of intensity distribution in the mentioned region is an urgent
problem, however it passes the limits of the present investigation.
## 5 Conclusions
In this work we propose a mutual space-frequency distribution as
generalization of Weyl distribution (16). The mutual distributions is
characterized by a certain parameter $t$. It is a generalization of space-
frequency representations suggested by Wigner and Weyl and comprises them as
limiting cases. In the course of the investigation it has been determined that
transition from Weyl distribution into Wigner one occurs by means of mutual
region as a curve at the information diagram of mutual coordinates $(x,p)$
Fig.2. When modificating the mutual parameter $t$ the distribution changes and
simultaneously suffers deformation for the angle propotional to parameter $t$.
Fig.3 depicts the region of mutual distribution close to Weyl distribution
(quasi-Weyl region of distribution). Fig. 3(a) illustrates Weyl distribution
of Gaussian signal that is formed from the mutual distribution (51) when the
value of mutual parameter $t=0$. Increase of value of mutual parameter till
$t=0,1$ leads to the curve of the mutual distribution at informational diagram
(Fig.3(b)). When $t=0,25$ beside curve also starts the pocess of deformation
that leads to the transformation of Weyl distribution into Wigner distribution
(Fig.3(c)). The peculiar is the value of parameter $t=0,5$. mutual
distribution in this case is placed precisely in the middle between limiting
cases of Weyl and Wigner distributions (Fig.5). In the process of increase of
the mutual parameter $t$ a transformation of Weyl distribution into Wigner
distribution takes place by means of change of the curve counterclockwise to
the mutual space-frequency distribution. In the limiting case $t=1$ from the
mutual distribution Wigner distribution is formed (Fig.4(a)). When the mutual
parameter $t$ decreases in the region of Wigner distribution the curve at the
informational diagram of joined coordinates $(x,p)$ is changed (Fig.4(b,c)).
In the region of Wigner distribution when the parameter $t$ decreases the
curve of mutual distribution is changed clockwise. When the value $t=0,5$ the
mutual distribution is formed what can be observed at Fig.5. Thereby, we come
to conclusion that in the process of changing the parameter $t$ of the mutual
distribution the Weyl and Wigner distributions move towards one another at
informational diagram and are put in equilibrium in the point $t=0,5$ (Fig.5).
It can be stated that Wigner distribution is formed as a change of curve of
Weyl distribution at informational diagram for the angle $\theta<90^{0}$.
Similarly, Weyl distribution is formed as a change of curve of Wigner
distribution in the contrary way. It should be noticed that in the mutual
region the distribution becomes complex one (Fig.6(b-d)), (Fig.7(b-d)).
However, in the known liniting cases only the real part has a contribution:
$t=0$ (Fig.6(a)),(Fig.7(a)) and $t=1$ (Fig.6(e)),(Fig.7(e)).
In this paper we propose new space-frequency distribution (16), which could
play an important role in the optical information processing schemes
describing.
## References
* [1] E.P. Wigner, Phys. Rev. 40, 749(1933).
* [2] D. Gabor, Journal of the IEE 93, 429(1946).
* [3] J. Ville, Cables et Transmissions 2A, 61(1948).
* [4] T. Claasen W. Mecklenbrauker, Philips Jour. of Research 35, 217(1980).
* [5] T. Claasen W. Mecklenbrauker, Philips Jour. of Research 35, 276(1980).
* [6] T. Claasen W. Mecklenbrauker, Philips Jour. of Research 35, 372(1980).
* [7] L.Cohen, 77, 941(1989).
* [8] B. Boashash, Time-frequency Signal Analysis and Processing (Elsevier, 2003, 743 p.).
* [9] G. Cristobal, C. Gonzalo, J. Bescos, Advances in Electronics and Electron Physics Series 80, 309(1991).
* [10] L. Cohen Time-frequency Analysis (Prentice Hall, 1995, 299 p.).
* [11] A. Mertings Signal Analysis (WileySons, 1999, 310 p.).
* [12] Ch. Kurtsiefer, T. Pfau, J. Mlynek, Nature 386, 150(1997).
* [13] G. Breitenbach, S. Schiller, J. Mlynek, Nature 387, 471(1997).
* [14] D. Smithey, M. Beck, M. Raymer, Phys. Rev. Lett. 70(9), 1244(1996).
* [15] L. Cohen, J. Math. Phys. 7, 781(1966).
* [16] N. De Bruijn, Nieuw Archief voor Wiskunde (3)(XXI), 205(1973).
* [17] L. Durak, PhD thesis, Institute of engineering and science of Bilken University, Turkey, 2003, 139 p.
* [18] M. Shovgenyuk, Yu. Kozlovskii, Dep. NAS Ukraine, 6, 92(2000).
* [19] S. Chountasis, A. Vourdas and C. Bendjaballah, Phys. Rev. A. 60(5), 3467(1999).
* [20] S. Chountasis, A. Vourdas, Phys. Rev. A. 58(2), 848(1998).
* [21] S. Chountasis, A. Vourdas, Phys. Rev. A. 58(3), 1794(1999).
* [22] M. Shovgenyuk, Preprint ICMP-92-25U, Lviv, (1992). (www.icmp.lviv.ua).
* [23] Yu.M. Kozlovskii, Ukr.J.Phys.Opt. 3, 124(2003).
* [24] Yu.M. Kozlovskii, Preprint ICMP-06-27U, Lviv, (2006). (www.icmp.lviv.ua).
* [25] Yu.M. Kozlovskii, Ukr.Phys.Journ. 3, 124(2008).
Figure 1: Real and Image parts of the Gaussian mutual distribution constant
$C_{t}^{\sigma}$.
Figure 2: Redistribution of the Gaussian mutual distribution from Weyl to
Wigner domain.
Figure 3: Rotational displaysment of the Gaussian mutual distribution in the
Weyl domain at different values of mutual parameter $t$ (Real part).
Figure 4: Rotational displaysment of the Gaussian mutual distribution in the
Wigner domain at different values of mutual parameter $t$ (Real part).
Figure 5: Mutual distribution of the Gaussian signal at the value of mutual
parameter $t=0.5$ (Real part).
Figure 6: Mutual space-frequency distribution in direction $x=0$ of Gaussian
signal at different values of mutual parameter $t$, solid line real part and
dot line image part.
Figure 7: Mutual space-frequency distribution in direction $p=0$ of Gaussian
signal at different values of mutual parameter $t$, solid line real part and
dot line image part.
|
arxiv-papers
| 2008-10-14T20:50:21
|
2024-09-04T02:48:58.259802
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yura Kozlovskii",
"submitter": "Yura Kozlovskii M",
"url": "https://arxiv.org/abs/0810.2559"
}
|
0810.2677
|
# Double-Lepton Polarization Asymmetries in $B\rightarrow K_{1}l^{+}l^{-}$
Decay in Universal Extra Dimension Model
B. B. Şirvanlı
Gazi University, Faculty of Arts and Science, Department of Physics
06100, Teknikokullar Ankara, Turkey
###### Abstract
Double-lepton polarization asymmetries for the exclusive decay $B\rightarrow
K_{1}l^{+}l^{-}$ in the Universal Extra Dimension (UED) Model is studied. It
is obtained that double-lepton polarization asymmetries are very sensitive to
the UED model parameters. Experimental measurements of double lepton
polarizations can give valuable information on the physics beyond the Standard
Model (SM).
PACS number(s):12.60.–i, 13.20.–v, 13.20.He
## 1 Introduction
The rare B-meson decays pointed out by the flavor-changing neutral currents
(FCNC) have been significant channels for acquiring knowledge on the SM
parameter and analyzing the new physics predictions. Rare B meson decays are
not allowed at the tree level in the SM and seem at loop level. By rare B
decays, one generally comprehend Cabibbo-suppressed $b\rightarrow u$
transitions or flavour-changing neutral currents (FCNC) $b\rightarrow s$ or
$b\rightarrow d$. So rare decays are significant testing basic of the SM and
take an important part in the search for new physics. The examinations of
different FCNC processes can be used to determine different fundamental
parameters of SM like elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix ,
various decay constants etc. Between testing SM the FCNC processes can be very
important for discovering indirect effects of possible TeV scale extensions of
SM. Therefore,we examine $b\rightarrow q(q=d,s)$ transitions in terms of an
effective Hamiltonian. For observing to the new physics in these decays, there
are two different ways. First of all, the differences in the Wilson
coefficients form the ones existing in the SM. And the second one the new
operator in the effective Hamiltonian which are absent in the SM. All decay
channels of B meson include many physically quantities which are very useful
testing for the SM and investigating for new physics beyond the SM. Exclusive
processes such as $B\rightarrow K(K^{*})l^{+}l^{-}$ and $B\rightarrow\gamma
l^{+}l^{-}$ decays [4, 5, 6, 7, 8] have been studied extensive in literature .
Colangelo et al. have studied $B\rightarrow K(K^{*})l^{+}l^{-}$ decays in
framework of one Universal Extra Dimension model (ACD),proposed in the Ref.
[16] and analyzed the branching ratio and forward-backward asymmetry . In
meanwhile, in the Ref. [14] the single lepton polarizations is studied for
$\mu$ for the $B\rightarrow K_{1}l^{+}l^{-}$ decay in UED model. The Branching
ratios (BR) of the Semileptonic decays $\mathcal{B}(B\rightarrow
K^{*}l^{+}l^{-})=7.8\pm 1.2\times 10^{-7}$ [2] and $\mathcal{B}(B\rightarrow
Kl^{+}l^{-})=5.5\pm 0.02\times 10^{-7}$ [1] have been measured by BELLE [1]
and BaBar [2] collaborations. It is noted that the measurement of the
polarization of the $b\rightarrow s$ decay can provide important information
about more observables. Some of the single lepton polarization asymmetries can
be too small to be observed. Since it might not provide number of observables
for control the structure of the effective Hamiltonian,we calculate to double
lepton polarization for more observables [9]. Among the different models of
physics beyond the SM, extra dimensions is very interesting models. Since the
extra dimension model contain of gravity, they give to clue on the hierarchy
problem and a connection with string theory. The model of Appelquist, Cheng
and Dobrescu (ACD) [10, 11, 20] with one universal extra dimension (UED),
where all the SM particles can propagate in the extra dimension.
Compactification of the extra dimension leads to Kaluza-Klein model in the
four-dimension. In the extra dimension model, we have extra free parameter is
$1/R$,which is inverse of the compactification radius. With the aid of $1/R$,
we can determined all the masses of the KK particles and their interactions
with SM particles. In the meanwhile, If we have not tree level contribution of
KK states to the low energy processes, KK parity is conservation in ACD model
at scale $\mu\ll 1/R$.
In this work, we study the double-lepton polarization asymmetries for the
$B\rightarrow K_{1}l^{+}l^{-}$ decay in the UED model. In section 2, we
shortly examine ACD model. In section 3, we obtain matrix element for the
$B\rightarrow K_{1}l^{+}l^{-}$ decay. In section 4, Double lepton polarization
for the $B\rightarrow K_{1}l^{+}l^{-}$ decay are calculated. Section 5 is
devoted to the numerical analysis and discussion of our results.
## 2 $B\rightarrow K_{1}l^{+}l^{-}$ Decay in ACD Model
Before calculation of the double lepton polarizations few words about the ACD
model. This model is the minimal extension of the SM to the $4+\delta$
dimensions. We consider simple case which is $\delta=1$. In the universe, we
have 3 space + 1 time dimensions and one possibility is the propagation of
gravity in whole ordinary plus extra dimensional universe. The five-
dimensional ACD model with a single UED uses orbifold compactification, the
fifth dimension $y$ that is compactified in a circle of radius $R$, with
points $y=0$ and $y=\pi R$ that are fixed points of the orbifolds [11, 12, 13,
14]. The Lagrangian in ACD model can be written as:
$\displaystyle\mathcal{L}=\int
d^{4}xdy\\{\mathcal{L}_{A}+\mathcal{L}_{H}+\mathcal{L}_{F}+\mathcal{L}_{Y}\\}$
where
$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}W^{MNa}W_{MN}^{a}-\frac{1}{4}B^{MN}B_{MN}$
$\displaystyle\mathcal{L}_{H}$ $\displaystyle=$
$\displaystyle(\mathcal{D}^{M}\phi)^{\dagger}\mathcal{D}_{M}\phi-V(\phi)$
$\displaystyle\mathcal{L}_{F}$ $\displaystyle=$
$\displaystyle\overline{\mathcal{Q}}(i\Gamma^{M}\mathcal{D}_{M})\mathcal{Q}+\overline{u}(i\Gamma^{M}\mathcal{D}_{M})u+\overline{\mathcal{D}}(i\Gamma^{M}\mathcal{D}_{M})\mathcal{D}$
$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle-\overline{\mathcal{Q}}\widetilde{Y}_{u}\phi^{c}u-\overline{\mathcal{Q}}\widetilde{Y}_{d}\phi\mathcal{D}+h.c..$
where $M$ and $N$ are the five-dimensional Lorentz indices which can run from
$0,1,2,3,5$.
$W_{MN}^{a}=\partial_{M}W_{N}^{a}-\partial_{N}W_{M}^{a}+\widetilde{g}\varepsilon^{abc}W_{M}^{b}W_{N}^{c}$
are the field strength tensor for the $SU(2)_{L}$ electroweak group,
$B_{MN}=\partial_{M}B_{N}-\partial_{N}B_{M}$ are that of the $U(1)$ group.
$\mathcal{D}_{M}=\partial_{M}-i\widetilde{g}W_{M}^{a}T^{a}-i\widetilde{g}^{{}^{\prime}}B_{M}Y$
is the covariant derivative, where $\widetilde{g}$ and
$\widetilde{g}^{{}^{\prime}}$ are the five-dimensional gauge couplings for the
$SU(2)_{L}$ and $U(1)$ groups. $\Gamma^{M}$ are five-dimensional matrices
which is $\Gamma^{\mu}=\gamma^{\mu}$ , $\mu=0,1,2,3$ and
$\Gamma^{5}=i\gamma^{5}$. $F(x_{t},y)$ is the periodic function of $y$ which
is $1/R$. It can be written as follow:
$\displaystyle F(x_{t},y)=F_{0}(x_{t})+\sum_{n=1}^{+\infty}F_{n}(x_{t},x_{n})$
where $x_{t}=\frac{m_{t}^{2}}{m_{w}^{2}}$, $x_{n}=\frac{m_{n}^{2}}{m_{w}^{2}}$
and $m_{n}=n/R$. These function can be found in [10,15].
## 3 Effective Hamiltonian for $B\rightarrow K_{1}l^{+}l^{-}$ Decay
At quark level, the exclusive $B\rightarrow K_{1}l^{+}l^{-}$ decay is
described by $b\rightarrow sl^{+}l^{-}$ transition governed by effective
Hamiltonian:
$\displaystyle{\cal H}_{eff}$ $\displaystyle=$
$\displaystyle-4\frac{G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\sum_{i=1}^{10}C_{i}(\mu)O_{i}(\mu)$
(1)
where $O_{i}$’s are local quark operators and $C_{i}$’s are Wilson
coefficients. $G_{F}$ is the Fermi constant and $V_{ij}$ are elements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix element for $B\rightarrow
K_{1}l^{+}l^{-}$ decay is obtained by $b\rightarrow sl^{+}l^{-}$ sandwiching
transition amplitude between initial and final meson states. Using effective
Hamiltonian the matrix element of the $B\rightarrow K_{1}l^{+}l^{-}$ decay
which can be written as follows:
$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha}{2\sqrt{2}\pi}V_{tb}V_{ts}^{*}\Bigg{\\{}-2m_{b}C^{eff}_{7}\overline{s}i\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b\overline{l}\gamma^{\mu}l$
(2) $\displaystyle+$ $\displaystyle
C^{eff}_{9}\overline{s}\gamma_{\mu}(1-\gamma_{5})b\overline{l}\gamma^{\mu}l+C_{10}\overline{s}\gamma_{\mu}(1-\gamma_{5})b\overline{l}\gamma^{\mu}\gamma_{5}l\Bigg{\\}}$
where $s=q^{2}$, q is the momentum transfer, $q=p_{1}+p_{2}=p_{B}-p_{K_{1}}$.
Here, $p_{1}$, $p_{2}$, $p_{B}$ and $p_{K_{1}}$ are the four-momenta of the
leptons, $B$ meson and $K_{1}$ meson respectively. Already the free quark
decay amplitude $\mathcal{M}$ contains certain long-distance effects which
usually are absorbed into a redefinition of the Wilson coefficient. These
coefficients in UED are calculated by Ref.[11] and [12] which can be written
as follows,
$\displaystyle C^{0}_{7}(\mu_{w})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}D^{{}^{\prime}}(x_{t},1/R),$ $\displaystyle
C_{9}(\mu)$ $\displaystyle=$ $\displaystyle
P^{NDR}_{0}+\frac{Y(x_{t},1/R)}{sin^{2}\theta_{w}}-4Z(x_{t},1/R)+P_{E}E(x_{t},1/R),$
$\displaystyle C_{10}$ $\displaystyle=$
$\displaystyle-\frac{Y(x_{t},1/R)}{sin^{2}\theta_{w}}$ (3)
where $P^{NDR}_{0}=2.60\pm 0.25$ and referring to leading log approximation.
Explicit expression the functions of the detail
$D^{{}^{\prime}}(x_{t},1/R),Y(x_{t},1/R)$ and $Z(x_{t},1/R)$ are calculated in
Ref.[11, 12, 16]. From Eq.(2) it follows that, for obtaining matrix element
for the $B\rightarrow K_{1}l^{+}l^{-}$ decay we need to know following matrix
elements
$\left<K_{1}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}(\gamma_{5})b\right|B(p)\right>$
and
$\left<K_{1}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B(p)\right>$.
These matrix elements in terms of form factors are parametrized as
$\displaystyle\left<K_{1}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B(p)\right>$
$\displaystyle=$ $\displaystyle
i\varepsilon^{*}_{\mu}(m_{B}+m_{K_{1}})V_{1}(s)-(p+k)_{\mu}(\varepsilon^{*}.q)\frac{V_{2}(s)}{m_{B}+m_{K_{1}}}$
(4) $\displaystyle-$ $\displaystyle
q_{\mu}(\varepsilon.q)\frac{2m_{K_{1}}}{s}[V_{3}(s)-V_{0}(s)]~{},$
$\displaystyle\left<K_{1}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B(p)\right>$
$\displaystyle=$
$\displaystyle\frac{2i\epsilon_{\mu\nu\alpha\beta}}{m_{B}+m_{K_{1}}}\varepsilon^{*\nu}p^{\alpha}k^{\beta}A(s)$
(5)
$\displaystyle\left<K_{1}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B(p)\right>$
$\displaystyle=$
$\displaystyle\Bigg{[}(m_{B}^{2}-m_{K_{1}}^{2})\varepsilon_{\mu}-(\varepsilon.q)(p+k)_{\mu}\Bigg{]}F_{2}(s)$
(6) $\displaystyle+$
$\displaystyle(\varepsilon^{*}.q)\Bigg{[}q_{\mu}-\frac{s}{m_{B}^{2}-m_{K_{1}}^{2}}(p+k)_{\mu}\Bigg{]}F_{3}(s)~{},$
$\displaystyle\left<K_{1}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma_{5}b\right|B(p)\right>$
$\displaystyle=$
$\displaystyle-i\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}k^{\beta}F_{1}(s)$
(7)
where $\varepsilon$ is the polarization vector of the $K_{1}$ meson. The form
factors entering Eq.(4) and (5) are estimated in [18, 19].
$\displaystyle V_{1}(s)$ $\displaystyle=$
$\displaystyle\frac{V_{1}(0)}{(1-s/m^{2}_{B^{*}_{A}})(1-s/m^{{}^{\prime}2}_{B^{*}_{A}})}\Bigg{(}1-\frac{s}{m_{B}^{2}-m_{K_{1}}^{2}}\Bigg{)}$
(8) $\displaystyle V_{2}(s)$ $\displaystyle=$
$\displaystyle\frac{\tilde{V_{2}}(0)}{(1-s/m^{2}_{B^{*}_{A}})(1-s/m^{{}^{\prime}2}_{B^{*}_{A}})}-\frac{2m_{K_{1}}}{m_{B}-m_{K_{1}}}\frac{V_{0}(0)}{(1-s/m^{2}_{B})(1-s/m^{{}^{\prime}2}_{B})}$
(9) $\displaystyle V_{3}(s)$ $\displaystyle=$
$\displaystyle\frac{m_{B}+m_{K_{1}}}{2m_{K_{1}}}V_{1}(s)-\frac{m_{B}-m_{K_{1}}}{2m_{K_{1}}}V_{2}(s)$
(10) $\displaystyle A(s)$ $\displaystyle=$
$\displaystyle\frac{A(0)}{(1-s/m^{2}_{B})(1-s/m^{{}^{\prime}2}_{B})}$ (11)
We can also define to the other matrix elements of the $B\rightarrow
K_{1}l^{+}l^{-}$ decay in terms of penguin form factors. Using the Ward
identities following relationship between form factors, we get
$\displaystyle F_{1}(s)$ $\displaystyle=$
$\displaystyle-\frac{(m_{b}-m_{s})}{(m_{B}+m_{K_{1}})}2A(s)$ (12)
$\displaystyle F_{2}(s)$ $\displaystyle=$
$\displaystyle-\frac{(m_{b}+m_{s})}{(m_{B}-m_{K_{1}})}V_{1}(s)$ (13)
$\displaystyle F_{3}(s)$ $\displaystyle=$
$\displaystyle\frac{2m_{K_{1}}}{s}(m_{b}+m_{s})[V_{3}(s)-V_{0}(s)]$ (14)
In order to avoid the kinematical singularity in the matrix element at $s=0$
we demand $F_{1}(0)=2F_{2}(0)$. The corresponding values at $s=0$ are given by
[14, 17, 18, 19],
$\displaystyle A(0)$ $\displaystyle=$ $\displaystyle-(0.52\pm 0.05)$
$\displaystyle V_{1}(0)$ $\displaystyle=$ $\displaystyle-(0.24\pm 0.02)$
$\displaystyle\tilde{V_{1}}(0)$ $\displaystyle=$ $\displaystyle-(0.39\pm
0.03)$ $\displaystyle V_{0}(0)$ $\displaystyle=$ $\displaystyle-(0.29\pm
0.04)$ $\displaystyle A_{1}(0)$ $\displaystyle=$ $\displaystyle(0.23\pm 0.02)$
$\displaystyle\tilde{A_{2}}(0)$ $\displaystyle=$ $\displaystyle(0.33\pm 0.05)$
(15)
Using Eq.(4),(5)(6) and (7) for the matrix element of the $B\rightarrow
K_{1}l^{+}l^{-}$ decay we set,
$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha}{2\sqrt{2}\pi}V_{tb}V_{ts}^{*}m_{B}\Bigg{\\{}\Bigg{[}A(\hat{s})\epsilon_{\mu\nu\alpha\beta}\epsilon^{*\nu}p^{\alpha}_{B}p^{\beta}_{K_{1}}-iB(\hat{s})\epsilon^{*}_{\mu}+iC(\hat{s})(\epsilon^{*}.p_{B})(p_{B}+p_{K_{1}})_{\mu}$
(16) $\displaystyle+$ $\displaystyle
iD(\hat{s})(\epsilon^{*}.p_{B})q_{\mu}\Bigg{]}(\bar{l}\gamma^{\mu}l)+\Bigg{[}E(\hat{s})\epsilon_{\mu\nu\alpha\beta}\epsilon^{*\nu}p^{\alpha}_{B}p^{\beta}_{K_{1}}-iF(\hat{s})\epsilon^{*}_{\mu}$
$\displaystyle+$ $\displaystyle
iG(\hat{s})(\epsilon^{*}.p_{B})(p_{B}+p_{K_{1}})_{\mu}+iH(\hat{s})(\epsilon^{*}.p_{B})q_{\mu}\Bigg{]}(\bar{l}\gamma^{\mu}\gamma^{5}l)\Bigg{\\}}$
where
$\displaystyle A(\hat{s})$ $\displaystyle=$
$\displaystyle-\frac{2A(\hat{s})}{1+\frac{m_{K_{1}}}{m_{B}}}C^{eff}_{9}(\hat{s})+\frac{2m_{b}}{m_{B}\hat{s}}C^{eff}_{7}F_{1}(\hat{s})$
$\displaystyle B(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\frac{m_{K_{1}}}{m_{B}})\Bigg{[}C^{eff}_{9}(\hat{s})V_{1}(\hat{s})+\frac{2m_{b}}{m_{B}\hat{s}}C^{eff}_{7}(1-\frac{m_{K_{1}}}{m_{B}})\Bigg{]}$
$\displaystyle C(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{(1-(\frac{m_{K_{1}}}{m_{B}})^{2})}\Bigg{\\{}C^{eff}_{9}(\hat{s})V_{2}(\hat{s})+\frac{2m_{b}}{m_{B}}C^{eff}_{7}\Bigg{[}F_{3}(\hat{s})+\frac{1-(\frac{m_{K_{1}}}{m_{B}})^{2}}{\hat{s}}F_{2}(\hat{s})\Bigg{]}\Bigg{\\}}$
$\displaystyle D(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{s}}\Bigg{[}\Bigg{(}C^{eff}_{9}(\hat{s})(1+\frac{m_{K_{1}}}{m_{B}})V_{1}(\hat{s})-(1-\frac{m_{K_{1}}}{m_{B}})V_{2}(\hat{s})-2\frac{m_{K_{1}}}{m_{B}}V_{0}(\hat{s})\Bigg{)}-2\frac{2m_{b}}{m_{B}}C^{eff}_{7}F_{3}(\hat{s})\Bigg{]}$
$\displaystyle E(\hat{s})$ $\displaystyle=$
$\displaystyle-\frac{2A(\hat{s})}{1+\frac{m_{K_{1}}}{m_{B}}}C_{10}$
$\displaystyle F(\hat{s})$ $\displaystyle=$
$\displaystyle(1+\frac{m_{K_{1}}}{m_{B}})C_{10}V_{1}(\hat{s})$ $\displaystyle
G(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{(1+\frac{m_{K_{1}}}{m_{B}})}C_{10}V_{2}(\hat{s})$
$\displaystyle H(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{s}}\Bigg{[}C_{10}(\hat{s})(1+\frac{m_{K_{1}}}{m_{B}})V_{1}(\hat{s})-(1-\frac{m_{K_{1}}}{m_{B}})V_{2}(\hat{s})-2\frac{m_{K_{1}}}{m_{B}}V_{0}(\hat{s})\Bigg{]}$
(17)
Having the explicit expression for the matrix element for the $B\rightarrow
K_{1}l^{+}l^{-}$ decay, the next task is the calculation its differential
decay rate. In the center of mass frame (CM) of the dileptons $l^{+}l^{-}$,
where we take $z=cos\theta$ and $\theta$ is the angle between the momentum of
the $B$ meson and that of $l^{-}$, differential decay width is found to belike
follows,
$\displaystyle\frac{d\Gamma}{d\hat{s}}(B\rightarrow
K_{1}l^{+}l^{-})=\frac{G_{F}^{2}\alpha^{2}|V_{tb}V_{ts}^{*}|^{2}}{8m_{B}^{4}\pi^{2}}\Delta$
(18)
where $\lambda=r^{2}+(-1+\hat{s})^{2}-2r(1+\hat{s})$ with
$\hat{s}=q^{2}/m_{B}^{2}$ and $r=m_{l}^{2}/m_{B}^{2}$ and
$\hat{m_{l}}=m_{l}/m_{B}$. $s=q^{2}$ is the dilepton invariant mass. The
function $\Delta$ is defined as follows:
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{2}{3}m_{B}^{2}\Bigg{\\{}2m_{B}^{4}(2\hat{m_{l}}^{2}+\hat{s})\lambda|A|^{2}+\frac{1}{r\hat{s}}(2\hat{m_{l}}^{2}+\hat{s})(r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s}))|B|^{2}$
(19) $\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}m_{B}^{4}(2\hat{m_{l}}^{2}+\hat{s})\lambda^{2}|C|^{2}-2m_{B}^{4}(4\hat{m_{l}}^{2}-\hat{s})\lambda|E|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}\hat{s}\Bigg{(}r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s})\Bigg{)}+2\hat{m_{l}}^{2}\Bigg{(}r^{2}+(-1+\hat{s})^{2}-2r(1+13\hat{s}\Bigg{)}\Bigg{]}|F|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}m_{B}^{4}\hat{s}\lambda^{2}+2\hat{m_{l}}^{2}(1+r^{2}+4\hat{s}-2\hat{s}^{2}+r(-2+4\hat{s}))\Bigg{]}|G|^{2}+\frac{1}{r}6m_{B}^{4}\hat{m_{l}}^{2}\hat{s}\lambda|H|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}2m_{B}^{2}(2\hat{m_{l}}^{2}+\hat{s})\Bigg{(}r^{3}+(-1+\hat{s})^{3}-r^{2}(3+\hat{s})-r(-3+2\hat{s}+\hat{s}^{2})\Bigg{)}Re(B^{*}C)$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}2m_{B}^{2}\lambda\Bigg{(}2\hat{m_{l}}^{2}(-1+r-2\hat{s})+\hat{s}(-1+r+\hat{s})\Bigg{)}\Bigg{]}Re(F^{*}G)$
$\displaystyle-$ $\displaystyle\frac{1}{r}12m_{B}^{2}\hat{m_{l}}^{2}\lambda
Re(F^{*}H)-\frac{1}{r}12m_{B}^{4}\hat{m_{l}}^{2}\lambda(-1+r)Re(G^{*}H)\Bigg{\\}}$
## 4 Lepton Polarization Asymmetries
Now, we would like to discuss the lepton polarizations in the $B\rightarrow
K_{1}l^{+}l^{-}$ decays. For calculation of the double lepton polarization
asymmetries, in the rest frame of $l^{+}l^{-}$, unit vectors $s_{i}^{\mp\mu}$
($i=L,T,N$) are defined as [8, 13]
$\displaystyle s_{L}^{-\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{L}^{-})=\left(0,\frac{\vec{p}_{-}}{\left|\vec{p}_{-}\right|}\right)~{},$
$\displaystyle s_{T}^{-\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{T}^{-})=\left(0,\vec{e}_{L}^{-}\times\vec{e}_{N}^{-}\right)~{},$
$\displaystyle s_{N}^{-\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{N}^{-})=\left(0,\frac{\vec{p_{K_{1}}}\times\vec{p}_{-}}{|\vec{p_{K_{1}}}\times\vec{p}_{-}|}\right)~{},$
$\displaystyle s_{L}^{+\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{L}^{+})=\left(0,\frac{\vec{p}_{+}}{\left|\vec{p}_{+}\right|}\right)~{},$
$\displaystyle s_{T}^{+\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{T}^{+})=\left(0,\vec{e}_{L}^{+}\times\vec{e}_{N}^{+}\right)~{},$
$\displaystyle s_{N}^{+\mu}$ $\displaystyle=$
$\displaystyle(0,\vec{e}_{N}^{+})=\left(0,\frac{\vec{p_{K_{1}}}\times\vec{p}_{+}}{|\vec{p_{K_{1}}}\times\vec{p}_{+}|}\right)~{}.$
(20)
where $\vec{p}_{\pm}$ and $\vec{p}_{K_{1}}$ are the three-momenta of the
leptons $l^{+}l^{-}$ and $K_{1}$ meson in the center of mass frame (CM) of
$l^{+}l^{-}$ system, respectively. The longitudinal unit vector $S_{L}$ is
boosted to the CM frame $l^{+}l^{-}$ under the Lorentz transformation:
$\displaystyle(s_{L}^{\mp\mu})_{CM}=(\frac{\left|\vec{p}_{\mp}\right|}{m_{l}},\frac{E_{l}\vec{p}_{\mp}}{m_{l}\left|\vec{p}_{\mp}\right|})~{},$
(21)
where $\vec{p}_{+}=-\vec{p}_{-}$, $E_{l}$ and $m_{l}$ are the energy and mass
of leptons in the CM frame, respectively. The transversal and normal unit
vectors $s_{T}^{\mp\mu}$, $s_{N}^{\mp\mu}$ are not changed under the Lorentz
boost. The double lepton polarization asymmetries are defined as:
$\displaystyle
P_{i}^{\mp}(s)=\frac{\frac{d\Gamma}{ds}(\vec{n}^{\mp}=\vec{e}_{i}^{\mp})-\frac{d\Gamma}{ds}(\vec{n}^{\mp}=-\vec{e}_{i}^{\mp})}{\frac{d\Gamma}{ds}(\vec{n}^{\mp}=\vec{e}_{i}^{\mp})+\frac{d\Gamma}{ds}(\vec{n}^{\mp}=-\vec{e}_{i}^{\mp})}$
(22)
where $\vec{n}^{\mp}$ is the unit vectors in the rest frame of the lepton. The
next step, we calculated double-lepton polarization asymmetries which is
define as $P_{ij}$:
$\displaystyle P_{LL}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{2}{3}m_{B}^{2}\Bigg{\\{}2m_{B}^{4}(2\hat{m_{l}}^{2}-\hat{s})\lambda|A|^{2}+\frac{1}{r\hat{s}}(2\hat{m_{l}}^{2}-\hat{s})(r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s}))|B|^{2}$
(23) $\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}m_{B}^{4}(2\hat{m_{l}}^{2}-\hat{s})\lambda^{2}|C|^{2}+2m_{B}^{4}(4\hat{m_{l}}^{2}-\hat{s})\lambda|E|^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}\hat{s}\Bigg{(}r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s})\Bigg{)}-2\hat{m_{l}}^{2}\Bigg{(}5r^{2}+5(-1+\hat{s})^{2}+2r(-5+7\hat{s}\Bigg{)}\Bigg{]}|F|^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}m_{B}^{4}\hat{s}\lambda^{2}-2\hat{m_{l}}^{2}(5+5r^{2}-4\hat{s}+2\hat{s}^{2}-2r(5+2\hat{s}))\Bigg{]}|G|^{2}+\frac{1}{r}6m_{B}^{4}\hat{m_{l}}^{2}\hat{s}\lambda|H|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}2m_{B}^{2}(2\hat{m_{l}}^{2}-\hat{s})\Bigg{(}r^{3}+(-1+\hat{s})^{3}-r^{2}(3+\hat{s})-r(-3+2\hat{s}+\hat{s}^{2})\Bigg{)}Re(B^{*}C)$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}2m_{B}^{2}\lambda\Bigg{(}2\hat{m_{l}}^{2}(-5+5r+2\hat{s})-\hat{s}(-1+r+\hat{s})\Bigg{)}\Bigg{]}Re(F^{*}G)$
$\displaystyle-$ $\displaystyle\frac{1}{r}12m_{B}^{2}\hat{m_{l}}^{2}\lambda
Re(F^{*}H)-\frac{1}{r}12m_{B}^{4}\hat{m_{l}}^{2}\lambda(-1+r)Re(G^{*}H)\Bigg{\\}}$
$\displaystyle P_{NN}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{2}{3}m_{B}^{2}\Bigg{\\{}m_{B}^{4}(-4\hat{m_{l}}^{2}+\hat{s})\lambda|A|^{2}+\frac{1}{r\hat{s}}\Bigg{(}\hat{s}\lambda+2\hat{m_{l}}^{2}(r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s}))\Bigg{)}|B|^{2}$
(24) $\displaystyle-$
$\displaystyle\frac{1}{r\hat{s}}m_{B}^{4}(2\hat{m_{l}}^{2}+\hat{s})\lambda^{2}|C|^{2}+m_{B}^{4}(4\hat{m_{l}}^{2}-\hat{s})\lambda|E|^{2}+\frac{1}{r\hat{s}}(2\hat{m_{l}}^{2}+\hat{s})\lambda|F|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}m_{B}^{4}\hat{s}\lambda^{2}+2\hat{m_{l}}^{2}(1+r^{2}+4\hat{s}-2\hat{s}^{2}+2r(-2+4\hat{s}))\Bigg{]}|G|^{2}+\frac{1}{r}6m_{B}^{4}\hat{m_{l}}^{2}\hat{s}\lambda|H|^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{r\hat{s}}2m_{B}^{2}(2\hat{m_{l}}^{2}+\hat{s})\Bigg{(}r^{3}+(-1+\hat{s})^{3}-r^{2}(3+\hat{s})-r(-3+2\hat{s}+\hat{s}^{2})\Bigg{)}Re(B^{*}C)$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}2m_{B}^{2}\lambda\Bigg{(}2\hat{m_{l}}^{2}(-1+r-2\hat{s})+\hat{s}(-1+r+\hat{s})\Bigg{)}\Bigg{]}Re(F^{*}G)$
$\displaystyle-$ $\displaystyle\frac{1}{r}12m_{B}^{2}\hat{m_{l}}^{2}\lambda
Re(F^{*}H)-\frac{1}{r}12m_{B}^{4}\hat{m_{l}}^{2}\lambda(-1+r)Re(G^{*}H)\Bigg{\\}}$
$\displaystyle P_{TT}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{2}{3}m_{B}^{2}\Bigg{\\{}m_{B}^{4}(4\hat{m_{l}}^{2}+\hat{s})\lambda|A|^{2}+\frac{1}{r\hat{s}}\Bigg{(}-\hat{s}\lambda+2\hat{m_{l}}^{2}(r^{2}+(-1+\hat{s})^{2}+2r(-1+5\hat{s}))\Bigg{)}|B|^{2}$
(25) $\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}m_{B}^{4}(2\hat{m_{l}}^{2}-\hat{s})\lambda^{2}|C|^{2}+m_{B}^{4}(4\hat{m_{l}}^{2}-\hat{s})\lambda|E|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\lambda(-10m_{l}^{2}+\hat{s})|F|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}m_{B}^{4}\hat{s}\lambda^{2}-2\hat{m_{l}}^{2}(5+5r^{2}-4\hat{s}+2\hat{s}^{2}-2r(5+2\hat{s}))\Bigg{]}|G|^{2}-\frac{1}{r}6m_{B}^{4}\hat{m_{l}}^{2}\hat{s}\lambda|H|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}2m_{B}^{2}(2\hat{m_{l}}^{2}-\hat{s})\Bigg{(}r^{3}+(-1+\hat{s})^{3}-r^{2}(3+\hat{s})-r(-3+2\hat{s}+\hat{s}^{2})\Bigg{)}Re(B^{*}C)$
$\displaystyle+$
$\displaystyle\frac{1}{r\hat{s}}\Bigg{[}2m_{B}^{2}\lambda\Bigg{(}-2\hat{m_{l}}^{2}(-5+5r+2\hat{s})+\hat{s}(-1+r+\hat{s})\Bigg{)}\Bigg{]}Re(F^{*}G)$
$\displaystyle+$ $\displaystyle\frac{1}{r}12m_{B}^{2}\hat{m_{l}}^{2}\lambda
Re(F^{*}H)+\frac{1}{r}12m_{B}^{4}\hat{m_{l}}^{2}\lambda(-1+r)Re(G^{*}H)\Bigg{\\}}$
$\displaystyle P_{LN}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{1}{r\sqrt{\hat{s}}}m_{B}^{2}\hat{m_{l}}\pi\sqrt{\lambda}\Bigg{[}(-1+r+\hat{s})Im(B^{*}F)+m_{B}^{2}\lambda
Im(C^{*}F)$ (26) $\displaystyle+$ $\displaystyle
m_{B}^{2}(-1+r)(-1+r+\hat{s})Im(B^{*}G)+m_{B}^{4}(-1+r)\lambda Im(C^{*}G)$
$\displaystyle-$ $\displaystyle
m_{B}^{2}\hat{s}(-1+r+\hat{s})Im(B^{*}H)-m_{B}^{4}\hat{s}\lambda
Im(C^{*}H)\Bigg{]}$ $\displaystyle P_{LT}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{1}{\sqrt{\hat{s}}}m_{B}^{2}\hat{m_{l}}\pi\lambda\sqrt{1-\frac{4\hat{m_{l}}^{2}}{\hat{s}}}\Bigg{(}-\frac{1}{r}(-1+r+\hat{s})|F|^{2}-\frac{1}{r}m_{B}^{4}(-1+r)\lambda|G|^{2}+2m_{B}^{2}\hat{s}Re(B^{*}E)$
(27) $\displaystyle+$ $\displaystyle
2m_{B}^{2}\hat{s}Re(A^{*}F)-\frac{1}{r}m_{B}^{2}(2+2r^{2}+\hat{s}^{2}-r(4+\hat{s}))Re(F^{*}G)+\frac{1}{r}m_{B}^{2}\hat{s}(-1+r+\hat{s})Re(F^{*}H)$
$\displaystyle+$ $\displaystyle\frac{1}{r}m_{B}^{4}\hat{s}\lambda
Re(G^{*}H)\Bigg{)}$ $\displaystyle P_{TL}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{1}{\sqrt{\hat{s}}}m_{B}^{2}\hat{m_{l}}\pi\lambda\sqrt{1-\frac{4\hat{m_{l}}^{2}}{\hat{s}}}\Bigg{(}-\frac{1}{r}(-1+r+\hat{s})|F|^{2}-\frac{1}{r}m_{B}^{4}(-1+r)\lambda|G|^{2}-2m_{B}^{2}\hat{s}Re(B^{*}E)$
(28) $\displaystyle-$ $\displaystyle
2m_{B}^{2}\hat{s}Re(A^{*}F)-\frac{1}{r}m_{B}^{2}(2+2r^{2}-3\hat{s}+\hat{s}^{2}-r(4+\hat{s}))Re(F^{*}G)+\frac{1}{r}m_{B}^{2}\hat{s}(-1+r+\hat{s})Re(F^{*}H)$
$\displaystyle+$ $\displaystyle\frac{1}{r}m_{B}^{4}\hat{s}\lambda
Re(G^{*}H)\Bigg{)}$ $\displaystyle P_{TN}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{-4m_{B}^{2}\lambda}{3r}\sqrt{1-\frac{4\hat{m_{l}}^{2}}{\hat{s}}}\Bigg{[}-m_{B}^{4}r\hat{s}Im(A^{*}E)+Im(B^{*}F)+m_{B}^{2}\Bigg{(}(-1+r+\hat{s})Im(C^{*}F)$
(29) $\displaystyle+$ $\displaystyle(-1+r+\hat{s})Im(B^{*}G)+m_{B}^{2}\lambda
Im(C^{*}G)\Bigg{)}\Bigg{]}$ $\displaystyle P_{NL}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{1}{r\sqrt{\hat{s}}}m_{B}^{2}\hat{m_{l}}\pi\lambda\Bigg{[}-(-1+r+\hat{s})Im(B^{*}F)-m_{B}^{2}\lambda
Im(C^{*}F)$ (30) $\displaystyle-$ $\displaystyle
m_{B}^{2}(-1+r)(-1+r+\hat{s})Im(B^{*}G)-m_{B}^{4}(-1+r)\lambda Im(C^{*}G)$
$\displaystyle+$ $\displaystyle
m_{B}^{2}\hat{s}(-1+r+\hat{s})Im(B^{*}H)+m_{B}^{2}\hat{s}\lambda
Im(C^{*}H)\Bigg{]}$ $\displaystyle P_{NT}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\frac{4m_{B}^{2}\lambda}{3r}\sqrt{1-\frac{4\hat{m_{l}}^{2}}{\hat{s}}}\Bigg{[}-m_{B}^{4}r\hat{s}Im(A^{*}E)+Im(B^{*}F)+m_{B}^{2}\Bigg{(}(-1+r+\hat{s})Im(C^{*}F)$
(31) $\displaystyle+$ $\displaystyle(-1+r+\hat{s})Im(B^{*}G)+m_{B}^{2}\lambda
Im(C^{*}G)\Bigg{)}\Bigg{]}$
## 5 Numerical analysis and discussion
In this section, we present our numerical results on the double lepton
polarization asymmetries for the $B\rightarrow K_{1}l^{+}l^{-}$ decays. First,
we present the values of input parameters are:
$\displaystyle
m_{B}=5.28\,GeV\,,\,m_{B^{*}}=5.32\,GeV\,,m_{K_{1}}=1.402\,GeV\,,\,$
$\displaystyle
m_{b}=4.8\,GeV\,,m_{s}=0.13\,GeV\,,m_{\mu}=0.105\,GeV\,,m_{\tau}=1.77\,GeV\,,$
$\displaystyle|V_{tb}V^{*}_{ts}|=0.04\,\,,\,\,\alpha^{-1}=137\,,\,\,G_{F}=1.17\times
10^{-5}\,GeV^{-2}\,,\,\,\tau_{B}=1.53\times 10^{-12}\,s.$ (32)
The $B\rightarrow K$ transition form factors are the main input parameters in
performing the numerical analysis, which are embedded into the expressions of
the double-lepton polarization asymmetries. For them we have used their
expression given by Eq. (8-15). The differential decay rate for $B\rightarrow
K_{1}l^{+}l^{-}$ can be defined in terms of integration on $\hat{s}$, which is
determined to the range of the $4\hat{m_{l}}^{2}\leq
s\leq(m_{B}-m_{K_{1}})^{2}$.
In Fig.1, we present the dependence of the $P_{LL}$ for the $B\rightarrow
K_{1}\mu^{+}\mu^{-}$ decay as a function of $s/m_{B}^{2}$. We see that,
$P_{LL}$ in UED compatible with the SM result. Increasing $\hat{s}$, $P_{LL}$
is moderate for the low of $\hat{s}$. The effect of KK contribution in the
Wilson coefficient are consistent for $1/R=200GeV$ at low value of $\hat{s}$.
$1/R=200GeV$ value is greater than $1/R=400GeV$. In Fig.2, Double lepton
longitudinal polarization asymmetries for the $B\rightarrow
K_{1}\tau^{+}\tau^{-}$ decay is presented, From this figure is follows, UED
model prediction coincide with the SM result. One can see that the value of
the longitudinal polarization is different in the low of $\hat{s}$ for the
$B\rightarrow K_{1}\tau^{+}\tau^{-}$ decay. While $1/R=200GeV$ value is max in
the UED model, The SM result is approximately two times lower than this value.
In Fig.3, For the $B\rightarrow K_{1}\mu^{+}\mu^{-}$ decay, we analysis to the
normal polarizations. We obtained good result at the $1/R=200GeV$ in UED
model. We can see that the effect of extra dimension are very noticeable at
the small value of $\hat{s}$. When the value of $\hat{s}$ close to $0.2$, all
the value of normal polarization is coincide with each other. In
$\hat{s}=0.36$, the value of $1/R=200GeV$ is five times bigger than SM result.
But in Fig.4, for the $B\rightarrow K_{1}\tau^{+}\tau^{-}$ decay, it is
similar to the $P_{LL}$ result. In Fig.5, We examine to the transversal
polarization for the $B\rightarrow K_{1}\mu^{+}\mu^{-}$ decay. At the
$1/R=200GeV$ value, we compared to that of the SM prediction $P_{TT}$ is
larger from SM. Again, the effects of extra dimension are distinguished at the
small value of momentum transfer $\hat{s}$ where $P_{TT}$ is minimum. For the
$\hat{s}=0.53$ value, all polarization values are decreases. In Fig.6, We
analysis to transversal polarization as a function of the $\hat{s}$ for the
$B\rightarrow K_{1}\tau^{+}\tau^{-}$ decay. We observe a little contributions
from UED model, especially in the $1/R=400GeV$ value. But UED model is better
than SM in this figure. All model values come together with the SM result in
the $\hat{s}=0.53$ value. In Fig.7, we investigate $P_{LT}$ polarization. We
see that increasing $\hat{s}$, $P_{LT}$ increase until $\hat{s}=0.5GeV^{2}$.
After this value of $\hat{s}$ two models are decrease until
$\hat{s}=0.55GeV^{2}$. $(P_{LT})_{UED}=2(P_{LT})_{SM}$ at $1/R=200GeV$. So It
is also very useful for establishing new physics. In Fig.8, We show our
predictions for the $P_{TL}$ for $B\rightarrow K_{1}\tau^{+}\tau^{-}$ decay.
We get $|(P_{TL})_{UED}|>|(P_{TL})_{SM}|$. This result can serve as a good
test for discrimination of two models. The other polarizations for the
$B\rightarrow K_{1}l^{+}l^{-}$ decay, we have imaginary part and therefore
there is no interference terms between SM and UED model contributions.
In conclusion, we have studied the double-lepton polarization asymmetries in
the UED model. We obtain different double-lepton polarization asymmetries
which is very sensitive to the UED model. It has been shown that all these
physical observebles are very sensitive to the existence of new physics beyond
SM and their experimental measurements can give valuable information on it.
Acknowledgements
The author would like to thank T. M. Aliev, M. Savcı and A. Ozpineci for
useful discussions during the course of the work.
Figure 1: The dependence of the Longitudinal polarization,for $B\rightarrow
K_{1}\mu^{+}\mu^{-}$ decay, as a function of the $\hat{s}$ . Figure 2: The
dependence of the Longitudinal polarization,for $B\rightarrow
K_{1}\tau^{+}\tau^{-}$ decay, as a function of the $\hat{s}$ . .
Figure 3: The dependence of the Normal polarization,for $B\rightarrow
K_{1}\mu^{+}\mu^{-}$ decay, as a function of the $\hat{s}$ . Figure 4: The
dependence of the Normal polarization,for $B\rightarrow K_{1}\tau^{+}\tau^{-}$
decay, as a function of the $\hat{s}$ . Figure 5: The dependence of the
Transversal polarization,for $B\rightarrow K_{1}\mu^{+}\mu^{-}$ decay, as a
function of the $\hat{s}$ . Figure 6: The dependence of the Transversal
polarization,for $B\rightarrow K_{1}\tau^{+}\tau^{-}$ decay, as a function of
the $\hat{s}$ . Figure 7: The dependence of the $P_{LT}$ polarization,for
$B\rightarrow K_{1}\mu^{+}\mu^{-}$ decay, as a function of the $\hat{s}$ .
Figure 8: The dependence of the $P_{TL}$ polarization,for $B\rightarrow
K_{1}\tau^{+}\tau^{-}$ decay, as a function of the $\hat{s}$ .
## References
* [1] K. Abe, et al.,Belle Collaboration Prep, hep-ex/0410006, (2004) .
* [2] B. Aubert, et al.,BaBar Collaboration Phys. Rev. Lett. 93, 081802, (2004) .
* [3] T. Mannel and S. Recksiegel, J. Phys., G 24 (1998) 979.
* [4] A. Ali, P. Ball, L. T. Handoko and G. Hiller, Phys. Rev., D 61 (2000) 074024 [arXiv : hep-ph/9910221].
* [5] T. M. Aliev, M. K. Cakmak and M.Savci, Nucl. Phys. ,B 607 (2001) 305 [arXiv : hep-ph/0009133]; T. M. Aliev, A. Ozpineci, M. Savci and C. Yuce, Phys. Rev. ,D 66 (2002) 115006 [arXiv : hep-ph/0208128].
* [6] F.Kruger and E. Lunghi, Phys. Rev. ,D 63 (2001) 014013 [arXiv : hep-ph/0008210].
* [7] S. Rai Choudhury, N. Gaur and N. Mahojan Phys. Rev.,D 66, (2002) 054003.
* [8] U. O. Yilmaz, B. B. Sirvanli and G. Turan, Nucl. Phys.,B 692 (2004) 249 [arXiv : hep-ph/0407006].
* [9] W. Bensalem, D. London, N. Sinha and R. Sinha, Phys. Rev. ,D 67 (2003) 034007.
* [10] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev ,D 64 (2001) 035002.
* [11] A. J. Buras, M. Sprander and A. Weiler, Nucl. Phys. ,B 660 (2003) 225.
* [12] A. J. Buras, A. Poschenrieder, M. Sprander and A. Weiler, Nucl. Phys. ,B 678 (2004) 455.
* [13] T. M. Aliev, M. Savci and B. B. Sirvanli, Eur. Phys. J.,C 52,(2007) 375-382 [arXiv : hep-ph/0608143].
* [14] I. Ahmed, M. A. Paracha and M. J. Aslam Eur. Phys. J.,C 54,(2008) 591-599 [arXiv : hep-ph/08020740].
* [15] A. J. Buras et al. Nucl. Phys.,B 424 (1994) 374.
* [16] P. Colangelo, F. De Fazio, R. Ferrendes, T. N. Pham, Prep : hep-ph/ 0604029,(2006).
* [17] M. A. Paracha, I. Ahmed, M. J. Aslam, Eur. Phys. J. ,C 52 (2007) 967-973 [arXiv : hep-ph/07070733].
* [18] A. H. S. Gilani, Riazuddin and T. A. Al-Aithan, JHEP,09 (2003) 065.
* [19] C. A. Dominguez, N. Paver and Riazuddin, Phys. Lett. ,B 214 (1988) 459.
* [20] A. Saddique, M. J. Aslam and C.D. L , [arXiv : hep-ph/08030192].
|
arxiv-papers
| 2008-10-15T12:01:30
|
2024-09-04T02:48:58.266897
|
{
"license": "Public Domain",
"authors": "Berin Belma Sirvanli",
"submitter": "Berin Belma Sirvanli",
"url": "https://arxiv.org/abs/0810.2677"
}
|
0810.2829
|
# Blume-Emery-Griffiths dynamics in social networks
Yao-Hui Yang Department of Mathematics and Physics, Chongqing University of
Science and Technology, Chongqing $401331$, China
###### Abstract
We introduce the Blume-Emery-Griffiths (BEG) model in a social networks to
describe the three-state dynamics of opinion formation. It shows that the
probability distribution function of the time series of opinion is a Gaussian-
like distribution. We also study the response of BEG model to the external
periodic perturbation. One can observe that both the interior thermo-noise and
the external field result in phase transition, which is a split phenomena of
the opinion distributions. It is opposite between the effect acted on the
opinion systems of the amplitude of the external field and of the thermo-
noise.
###### pacs:
02.50.-r, 87.23.Ge, 89.75.-k, 05.45.-a,
## I INTRODUCTION
Over the last few years, the study of opinion formation in complex networks
has attracted a growing amount of works and becomes the major trend of
sociophysics Intro-1 . Many models have been proposed, like those of Deffuant
Intro-2 , Galam Intro-3 , Krause-Hegselmann (KH) Intro-4 , and Sznajd Intro-5
. But most models in the literature consider two-state opinion agents, in
favor ($+1$) or against ($-1$) about a certain topic. In the Galam’s majority
rule and the Sznajd’s updating rule, the interaction between the agents is
randomly changed during the evolution, and the time to reach consensus is
associated with the initial traction $p$ of $+1$ state. The consensus time $T$
reaches its maximal value at $p=0.5$. In the Sznajd model, a pair of nearest
neighbors convinces its neighbors to adopt the pair opinion if and only if
both members have the same opinion. Otherwise the pair and its neighbors do
not change opinion. In the KH consensus model, the opinions between $0$ and
$1$ and a confidence bound parameter is introduced. The agent $i$ would take
the average opinion of all neighboring agents that are within a confidence
bound during the evolution. In the Deffuant model, the opinion of two randomly
selected neighboring agents $i$ and $j$ would remain unchanged, if their
opinions $\sigma_{i}$ and $\sigma_{j}$ differ by more than a fixed threshold
parameter. Otherwise, each opinion moves into the direction of the other by an
amount $\mu\times\mid\sigma_{i}-\sigma_{j}\mid$.
Additionally, complex networks have received much attention in recent years.
Topologically, a network is consisted of nodes and links. The complex network
models, such as the lattice network, the random network Intr_6 ; Intr_7 ;
Intr_8 , the small-world network Intr_9 ; Intr_10 , and the scale-free network
Intr_11 , are studied in many branches of science. It is meaningful to mention
that opinion formation models are set up in complex networks.
In the present work, we investigate the implication of a social network in a
stochastic opinion formation model. We first introduce the Blume-Emery-
Griffiths (BEG) model Intr_12 ; Intr_13 ; Intr_14 to describe the dynamics of
opinion formation, and the model of complex networks we used is social network
which is more reality. Our simulation focuses on the average opinion for
different situation. And we also simulated the system under the influence of
external field.
In the rest of this paper we will give a description of this dynamic model and
how to generate the underlying networks. In Sec.III, we show the simulation
results without external filed. In Sec. IV we present the results with the
influence of external field. The final section presents further discussion and
conclusion.
## II The model
Generally speaking, social networks include some essential characteristics,
such as short average path lengths, high clustering, assortative mixing
Model-1 ; Model-2 , the existence of community structure, and broad degree
distributions Model-3 ; Model-4 . As a result, we use Riitta Toivonen’s social
network model in our present work Model-5 . This network is structured by two
processes: $1)$ attachment to random vertices, and $2)$ attachment to the
neighborhood of the random vertices, giving rise to implicit preferential
attachment. These processes give rise to essential characteristics for social
networks. The second process gives rise to assortativity, high clustering and
community structure. The degree distribution is also determined by the number
of edges generated by the second process for each random attachment.
Figure 1: Degree distribution of networks with $N=10000$. Result is averages
over $20$ simulation runs. The number of initial contacts is distributed as
$p(n_{init}=1)=0.25$, $p(n_{init}=2)=0.75$, and the number of secondary
contacts from each initial contact $n_{2nd}\sim U[0,3]$.
In this paper, the network is grown from a chain with $10$ nodes. The number
of initial contacts is distributed as $p(n_{init}=1)=0.25$,
$p(n_{init}=2)=0.75$, and the number of secondary contacts from each initial
contact $n_{2nd}\sim U[0,3]$ (uniformly distributed between $0$ and $3$). The
total number of nodes in the social network structure is $N=10000$. The degree
distribution of simulated networks is displayed in Fig. 1. We note that the
degree distributon $P(k)$ is a power-law functional form and a peak around the
degree $k=5$, also that consistent with real world observations Intr_11 ;
Model-6 .
Now, we consider a system with $N$ agents, which is represented by nodes on a
social network. For each node, we consider three states which are represented
by $+1$, $0$, and $-1$. A practical example could be the decision to agree
$\sigma_{i}(t)=+1$, disagree $\sigma_{i}(t)=-1$, or neutral $\sigma_{i}(t)=0$.
The states are updated according to the stochastic parallel spin-flip dynamics
defined by the transition probabilities
$Prob\left(\sigma_{i,t+1}=s^{\prime}|\sigma_{N}(t)\right)=\frac{\exp\left\\{-\beta\epsilon_{i}\left[s^{\prime}|\sigma_{N}(t)\right]\right\\}}{\sum_{s}\exp\left\\{-\beta\epsilon_{i}[s|\sigma_{N}(t)]\right\\}}$
(1)
where $s,s^{\prime}\in\\{+1,0,-1\\}$, and $\beta=a/T$, $a$ represents the
active degree of system, defined as $a=\left<\sigma_{N}^{2}(t)\right>$. The
energy potential $\epsilon_{i}\left[s|\sigma_{N}(t)\right]$ is defined by
$\epsilon_{i}\left[s|\sigma_{N}(t)\right]=-sh_{i}\left(\sigma_{N}(t)\right)-s^{2}\theta_{i}\left(\sigma_{N}(t)\right),$
(2)
where the following local field in node $i$ carries all information
$\displaystyle h_{N,i}(t)$ $\displaystyle=$ $\displaystyle\sum_{j\neq
i}J_{ij}\sigma_{j}(t),$ $\displaystyle\theta_{N,i}(t)$ $\displaystyle=$
$\displaystyle\sum_{j\neq i}K_{ij}\sigma_{j}^{2}(t).$
Here, we define coupling $J_{ij}$ and $K_{ij}$ are positive numbers less than
or equal to $1$, and with Gaussian distribution. $h_{N,i}(t)$ represents the
time dependent interaction strengths between the node $i$ and his $n_{i}$
nearest neighboring nodes. $\theta_{N,i}(t)$ instead the strengths of feedback
and $T$ is interior thermo-noise. So the average opinion is defined by
$r(t)=\frac{1}{N}\sum_{j=1}^{N}\sigma_{j}(t).$ (3)
## III Simulation results
Figure 2: (a) Time series of average opinion with the total time steps is
$t=10000$, (b) the distribution functions $P(R)$, and (c) the autocorrelation
function $c(\tau)$. The parameters used in the simulation are
$p(n_{init}=1)=0.95$, $N=10000$, $T=1.0$ and $L=10000$. The parameter $J_{ij}$
and $K_{ij}$ are positive numbers which are not larger than $1$ in whole
simulations. All the results in this paper are obtained over $20$ realizations
of the social networks.
At first we investigate the time series of average opinion, as illustrated in
Fig. 2(a). It shows there exists the fluctuation around the average opinion
$r=0$. In order to compare the fluctuation of different scales, the time
series have been normalized according to
$R(t)=\left(r(t)-\left<r(t)\right>_{\tau}\right)/\delta\left(r(t)\right),$
where $\left<r(t)\right>_{\tau}$ and $\delta(r(t))$ denote the average and the
standard deviation over the period considered, respectively. In Fig. 2(b), we
present the distribution functions $P(R)$ associated with the time series. It
is clear that this function $P(R)$ is a Gaussian form.
We calculate the autocorrelation function $c(\tau)$ of our model. For a time
series of $L$ samples, $r(t)$ for $t=1,2,\ldots,L$, $c(\tau)$ is defined by
$c(\tau)=\frac{\sum_{t=1}^{L-\tau}(r(t)-\bar{r})(r(t+\tau)-\bar{r})}{\sum_{t=1}^{L-\tau}(r(t)-\bar{r})^{2}},$
(4)
where $\tau$ is the time delay and $\bar{r}$ represents the average over the
period under consideration. Fig. 2(c) shows the result of autocorrelation
function of our model. It is found that $c(\tau)$ decreases rapidly in very
small rang of $\tau$. It means the system has short-time memory effects. As is
now well known, the stock market has nontrivial memory effects simulation-2 .
For example, the autocorrelation funciton of Dow Jones (DJ), also in the small
rang of $\tau$, decreases rapidly from $1$ to $0$. From this point, perhaps
our model is helpful to understand the financial markets.
## IV The influence of external field
Figure 3: Time series of the average opinion with different values of
amplitude $A=0.08$, $0.12$, $0.16$, $0.22$, $0.28$, $0.32$. Parameters are
$T=1.0$, $\omega=\pi/3$, and $\varphi=0$.
In order to explore what phenomena maybe happen to system under the influence
of external field. We add a period external field to the energy potential
$\epsilon_{i}$,
$\epsilon_{i}\left[s^{\prime}|\sigma_{N}(t)\right]=-sh_{i}\left(\sigma_{N}(t)\right)-s^{2}\theta_{i}\left(\sigma_{N}(t)\right)-s\left[A\cos(\omega
t+\varphi)\right],$ (5)
where $A$ is the amplitude of period external field, $\omega$ is frequency and
$\varphi$ denotes the initial phase of external field.
We investigate the effect of amplitude $A$ by fixing other parameters. In Fig.
3 we plot the time series of the average opinion $r(t)$ under different values
of $A$. It is obvious that the distribution functions have a remarkable change
with increasing $A$. With increasing strength of external field, the average
opinion comes into several discrete parts. For small amplitude $A=0.02$,
$P(R)$ is still a Gaussian form. When $A=0.08$, it begins to appear two
fluctuation around nonzero symmetric values of average opinions. Then, four
nonzero average opinions appear at $A=0.16$. Note that the intervals among the
discrete average opinions increase with increase in the strength $A$ of
external fields. Fig. 3 gives the process from two wave crests to four
independent parts. And the average opinion of the whole system will jump from
one part to the other parts at all times.
Figure 4: (a) The distribution functions $P(R)$ of average opinion time series
under different amplitudes $A$. Parameters are $T=1.0$, $\omega=\pi/3$, and
$\varphi=0$. (b) $P(R)$ for different frequencies $\omega$. Parameters are
$A=0.06$, $\varphi=\pi/2$, and $T=1.0$.
In Fig. 4, we present the distribution function $P(R)$ of the average opinion.
Again, it is easy to verify that the average opinions oscillate among serval
separate symmetric nonzero values under the external periodic driving force
[see Fig. 4(a)]. A similar oscillation behavior is observed for simulation on
the influence of the frequency $\omega$ which is shown in Fig. 4(b). Noted
that $P(R)$ for the frequency $\omega=\pi/3$ is same to the case for
$\omega=2\pi/3$, and the same distribution is observed between $\omega=\pi/6$
and $\omega=5\pi/6$. But there are distinct difference for $\omega=0$ and
$\omega=\pi/2$. It indicates a possible period $\pi$ in the case of fixed
other parameters.
Figure 5: The distribution functions $P(R)$ of average opinion time series
under different initial phases $\varphi$. Parameters are $A=0.16$,
$\omega=\pi/3$, and $T=1.0$.
Fig. 5 shows the distribution functions $P(R)$ of average opinion time series
for different initial phases $\varphi$. For $\varphi=0$, the average opinion
vibrates among four symmetric nonzero values. When $\varphi$ increases to
$\pi/2$, clearly, the average opinion comes into a $3$-value oscillation.
Additionally, note that the distribution functions is almost same for
$\varphi=0$ and $\varphi=\pi$ (or $\varphi=\pi/2$ and $\varphi=3\pi/2$).
Again, one can conjecture $P(R)$ is a $\pi$-period behavior. We also observe
the system’s average opinion time series only have two types of distribution
functions in different values of initial phases $\varphi$.
Figure 6: The distribution functions $P(R)$ of average opinion time series
under different interior thermo-noises $T$. The parameters used in the
simulation are $A=0.16$, $\omega=\pi/3$, and $\varphi=0$.
Another important parameter for the systems is the interior thermo-noise $T$.
We explore its effects with (or without) external fields. It is found that
there is not remarkable influence on the system without external field.
Contrarily, in the case of external field, $P(R)$ shows a similar oscillation
with it in Fig. 4(a) (see Fig. 6). Note that their influences are opposite. In
Fig. 6, with increasing $T$ the forms of $P(R)$ transform from four-peak to
two-peak gradually, and merge into only one-peak at last. At the same time,
the average opinion $r$ is expanded from some separate regions to the whole
more expansive scale for larger $T$.
By comparing the Fig. 4(a) with the Fig. 6, it is clear that the amplitude $A$
and interior thermo-noise $T$ have opposite effects acting on the systems. It
looks like a couple of contradictory parameters, even though both lead to the
split phenomena of the distribution of average opinion $P(R)$ and the nonzero
average $R$.
It exists similar behaviors in the Ising ferromagnetic systems. In Ising
model, the order-disorder transition is a second order transition. It will be
a non-zero magnetization $\pm|M_{sp}|$ for a finite system. There is a nonzero
probability for ever that the system from near $+|M_{sp}|$ to near
$-|M_{sp}|$, and vice versa external-1 . In our model under the influence of
external field, it is also observed the phenomena of phase transition caused
by $T$ (or by $A$), which is similar to the Ising paramagnetic-
antiferromagnetic transition.
As discussed above, the energy potential increases with increasing $T$, and
the system’s entropy becomes larger (more disordered). But the external field
tends to restrict the disordered effects in the system and reduces the
disordered strength into several separate regions.
## V Conclusion
In the present work we introduce Blume-Emery-Griffiths model on opinion
formation with three-state. Considering the characters of real social systems,
we construct a social network to link between agents. In this BEG model, each
person’s opinion is influenced not only by his specific local information from
his neighbors but also by the average opinion of the whole network.
Moreover, we focus on the behaviors of BEG systems under external
perturbation. The simulation results show that this system is sensitive to the
external field. As discussed in Sec. III, the parameters in the external
periodic perturbation, such as amplitude $A$, initial phase $\varphi$, and
frequency $\omega$, have obvious impacts on the opinion systems. Besides, the
effect of the amplitude $A$ or interior thermo-noise $T$ is similar to the
Ising paramagnetic-antiferromagnetic transition, and the influence acted on
systems from $A$ and $T$ is opposite.
## References
* (1) C. Borghesi, and S. Galam, Phys. Rev. E 73 (2006) 066118.
* (2) G. Deffuant, D. Neau, and F. Amblard, Adv. Complex Syst. 3 (2000) 87.
* (3) S. Galam, J. Stat. Phys. 61, (1990) 943; S. Galam, Physica A 238 (1997) 66.
* (4) R. Hegselmann and U. Krause, J. Artif. Societies Social Simulation 5, (3) (2002) paper 2 (jasss.soc.surrey.ac.uk).; U. Krause, Soziale Dynamiken mit vielen interakteuren. Eine Problemskizze, in: U. Krause, M. Stockler, eds.), Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren, Bremen University, January 1997, pp. 37–51.
* (5) K. Sznajd-Weron and J. Sznajd, Int. J. Mod. C 11 (2000) 1157.
* (6) P. Erdos and A. Renyi, Publ. Math. 6 (1959) 290.
* (7) P. Erdos and A. Renyi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.
* (8) P. Erdos and A. Renyi, Bull. Inst. Int. Stat. 38 (1961) 343.
* (9) D. J. Watts and S. H. Strogatz, Nature 393 (1998) 440.
* (10) M. E. J. Newman and D. J. Watts, Phys. Lett. A 263 (1999) 341.
* (11) A. -L. Barabasi and R. Albert, Science 286 (1999) 509.
* (12) M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A 4 (1971) 1071.
* (13) R. David, C. Dominguez, and E. Korutcheva, Phys. Rev. E 62 (2000) 2620.
* (14) D. Bollé, I. Pérez Castillo, and G. M. Shim, Phys. Rev. E 67 (2003) 036113.
* (15) M. E. J. Newman, Phys. Rev. Lett. 89 (2002) 208701.
* (16) M. E. J. Newman and J. Park, Phys. Rev. E 68 (2003) 036122.
* (17) L. A. N. Amaral, A. Scala, M. Barthélémy, and H. E. Stanley, Prol. Natl. Acad. Sci. USA 97 (2000) 11149.
* (18) M. Boguña, R. Pastor-Satorras, A. Diaz-Guilera, and A. Arenas, Phys. Rev. E 70 (2004) 056122.
* (19) R. Toivonen, J.-P. Onnela, J. Saramäki, J. Hyvönen, and K. Kaski, Physica A 371, (2006) 851.
* (20) A. Grönlund, and P. Holme, Phys. Rev. E 70 (2004) 036108.
* (21) R. Y. You and Z. Chen, Chinese J. Comput. Phys. 21 (2004) 341.
* (22) M. Bartolozzi, D. B. Leinweber, and A. W. Thomas, Phys. Rev. E 72 (2005) 046113.
* (23) K. Binder and D. W. Heermann, Monte Carlo simulation in statistical physics: an introduction, Springer-Verlag, Berlin, 2002.
|
arxiv-papers
| 2008-10-16T01:05:03
|
2024-09-04T02:48:58.276594
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yao-Hui Yang",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/0810.2829"
}
|
0810.2841
|
# Liquid behavior of hot QGP in the finite temperature field theory
Hui Liu tliuhui@jnu.edu.cn Physics Department, Jinan University,
Guangzhou(510632), P.R.China Defu Hou Jiarong Li Institute of Particle
Physics, Central China Normal University, Wuhan(430079), P.R.China
###### Abstract
In this paper, we compare the dispersion relations of hard thermal loop and
complete one loop. It is shown that in the dynamical screening regime, the
completely one-loop calculation presents a prominent threshold frequency,
below which no pure imaginary mode survives. This phenomenon is responsible
for the oscillatory static in-medium potential and ultimately results in a
damping oscillation of the radial distribution function. We consider this
typical shape is the footprint of liquid QGP.
###### pacs:
12.38.Mh,11.10.Wx
## I Introduction
The experiments of ultra-relativistic heavy ion collision at RHIC provide us a
platform to study the quark-gluon plasma(QGP) signal as well as its novel
properties. One of those surprises the scientists is the low viscous flow. At
Au+Au 200GeV collision, the elliptic flow $v_{2}$ can be well fitted by an
ideal hydrodynamics up to 2GeV of the transverse momentastar ; phenix , which
implies a perfect fluid behavior. This perfect behavior of QGP makes people
consider it in a liquid stateThoma2 ; Peshier , with the temperature slightly
above the critical temperature $T_{c}$. How to understand such a good liquid
of QGP is a fundamental problem that attracts much attention. Some ideals and
methods came from other fields, for example the AdS/CFT correspondence from
the superstring theory and the physics of strongly coupled QED plasma. For
more details, please refer to the report of E. Shuryak in RefShuryak and
references therein.
In this paper, we try to investigate the radial distribution function of
liquid QGP in the framework of finite temperature theory. Hard Thermal
Loop(HTL) approximation and HTL resummation scheme were widely used in thermal
field theory when discussing measurable medium effects such as Debye
screening, collective modes, particle energy loss so on and so forth. The HTL
physics were proved reliable in the temperature limit. For example, it can
represent the correct collective modes in hot plasmaLeBellac . The boson and
fermion damping rates obtained in the HTL resummation scheme are positive and
gauge invariant even in the non-abelian systemPisarski ; Braaten . However,
although the HTL has this and that good qualities, it has its own
restrictions. The HTL approximation as well as the corresponding resummation
scheme request the high temperature limit which is not a trivial condition for
a real system like the QGP at $1\sim 2T_{c}$. This temperature is obvious not
reaching the high temperature limit so that the HTL scheme might be doubtful.
To avoid such suspicion, one can adopt complete one loop scheme instead of
HTL.
In this paper, we will start with QED plasma, comparing the dispersion
relations of HTL and complete one loop, demonstrating their distinct screening
behaviors. Then we will turn to the quark-gluon plasma, calculating the static
in-medium inter-quark potential and the radial distribution function. The
damping oscillatory radial distribution function suggests the QGP might be in
a liquid state. Finally, we will discuss the general factors that decide the
state of matter, pointing out a possible way to study the properties of QGP
liquid.
## II dispersion relation
Dispersion relation is a basic relation of many-particle system which carries
essential physical information. A slight difference between dispersion
relations may indicate totally different physics. In this section, we will
compute the QED dispersion relations at HTL and completely one-loop level
respectively. One will see the distinct dispersion curves in both dispersion
regime and dynamic screening regime.
The dispersion relation is defined as the energy-moment relation at the pole
of full boson propagator,
$\displaystyle\omega^{2}-q^{2}-\Pi_{L}(\omega,q)=0$ (1)
$\displaystyle\omega^{2}-q^{2}-\Pi_{T}(\omega,q)=0$ (2)
where $\Pi_{L}(\omega,q)$ and $\Pi_{T}(\omega,q)$ are the longitudinal and
transverse components of boson polarization tensor respectively. In this paper
we just take the longitudinal dispersion relation as an example and study the
color-electric properties of hot plasma.
In HTL approximation,
$\Pi_{L}^{\mbox{\tiny HTL}}(\omega,q)=-\frac{4\pi\alpha
T^{2}}{3}\left[1-\frac{\omega}{2q}\ln\left(\frac{\omega+q}{\omega-q}\right)\right]$
(3)
where $\alpha=1/137$ is the fine structure constant of QED. For a complete one
loop,
$\displaystyle\Pi_{L}^{\mbox{\tiny one-loop}}(\omega,q)$ $\displaystyle=$
$\displaystyle\frac{4\alpha}{\pi}\int^{\infty}_{0}dp\frac{p^{2}n_{f}}{E_{p}}\left[\frac{\omega^{2}-q^{2}+4E_{p}^{2}+4\omega
E_{p}}{4pq}\ln\left(\frac{\omega^{2}-q^{2}+2\omega
E_{p}+2pq+i\epsilon}{\omega^{2}-q^{2}+2\omega
E_{p}-2pq+i\epsilon}\right)\right.$ (4)
$\displaystyle+\left.\frac{\omega^{2}-q^{2}+4E_{p}^{2}-4\omega
E_{p}}{4pq}\ln\left(\frac{\omega^{2}-q^{2}-2\omega
E_{p}+2pq-i\epsilon}{\omega^{2}-q^{2}-2\omega
E_{p}-2pq-i\epsilon}\right)-2\right],$
where $E_{p}=\sqrt{p^{2}+M^{2}}$, and $M$ is the electron mass.
$n_{f}(E_{p})=(e^{\beta E_{p}}+1)^{-1}$ is the Fermi-Dirac distribution
function with $\beta=1/T$.
Inserting Eqs.(3) and (4) into Eq.(1) and figuring out the relation between
$\omega$ and $q$ numerically, one could obtain FIG.1. This figure is plotted
in not only the dispersion regime where the momentum $q$ is real, but also the
dynamic screening regime where $q$ is pure imaginary. In FIG.1 the abscissa
combines both regimes, separated by a zero line of $q=0$. The right area to
the zero line is for the common dispersion relation when the momenta are real.
The left area, on the contrary, is the dynamic screening regime for pure
imaginary momenta.
The HTL dispersion relation has been obtained and discussed in detailsWeldon .
We represent it in FIG.1 with dashed curves to compare with the complete one
loop. However we do not intend to compare the whole regime, since the two
curves in the normal dispersion regime behaves very similar. Instead, we would
like to concern about the prominent difference in the dynamical regime. In
this regime, the HTL curve reaches the abscissa, indicating a screening effect
at zero frequency referred to the well-known Debye screening. While in the
completely one-loop case, an threshold frequency shows up, below which no pure
imaginary mode survives. That is to say a real part of the momentum is
necessary and the dynamical screening described by the HTL Weldon is broken
up. Especially, in the static limit where $\omega\rightarrow 0$, the plasma is
not screened with Debye form contributed by the pure imaginary mode. Instead,
the screening oscillates due to the complex mode in the completely one-loop
calculation. We will see it in the next section.
Figure 1: Comparison of dispersion relations between HTL and completely one-
loop calculations. The dashing line denotes for the HTL calculation and the
solid line is for the completely one-loop calculation.
## III Oscillatory potential
So far the Debye screening picture has been changed in the completely one-loop
calculation based on the dispersion analysis in last section, one would like
to check the static potential and see how it will look like in the new
picture.
In the relativistic plasma, the in-medium potential is explained by the
skeleton diagram with full boson propagator, as shown in FIG.2. The shadowed
circle denotes all possible polarizations. In math language, it is
$V(r)=\frac{\alpha}{\pi r}\
\mbox{Im}\int^{\infty}_{-\infty}dq\frac{qe^{iqr}}{q^{2}-\Pi_{L}(0,q)},$ (5)
where $r$ is the distance between two arbitrary electrons. To perform the
integral in Eq.(5), one should construct a contour according to the analytic
structure of the integrand, locating all poles within the contour on complex
plane. We would like to point out here that the Eq.(5) is actually involving a
resummation scheme, because the effective boson propagator is obtained from
Dyson-Schwinger equation.
Figure 2: Diagrammatical description of in-medium interparticle potential. The
shadowed circle stands for all possible polarization patterns.
To demonstrate the general form of the potential contributed by poles, one can
first define the pole as
$q_{\mbox{\tiny pole}}=q_{r}+iq_{i},$ (6)
where $q_{r}$ and $q_{i}$ are the real and imaginary parts of the pole. With
this definition, one can perform the contour integral and find
$V(r)=\sum_{\mbox{\tiny
poles}}\frac{2\alpha}{a^{2}+b^{2}}\frac{e^{-q_{i}r}}{r}\
[a\cos(q_{r}r)+b\sin(q_{r}r)],$ (7)
where the sum includes all pole contributions. $a$ and $b$ are defined as the
real and imaginary parts of the residue,
$\left.\frac{(q^{2}-\Pi_{L})^{\prime}}{q}\right|_{q=q_{r}+iq_{i}}=a+ib,$ (8)
with the prime denoting $\partial/\partial q$.
Notice that the pole of the integral in Eq.(5) is nothing else but the point
of $\omega=0$ on the dispersion curve. Due to the appearance of the threshold
frequency in FIG.1, the potential from HTL polarization and completely one-
loop polarization may behave differently. The HTL dispersion curve extends
directly to zero frequency in the dynamic screening regime, which means the
pole is purely imaginary with $q_{r}=0$ at the static limit. More explicitly,
$\Pi_{L}^{\mbox{\tiny HTL}}(\omega\rightarrow 0,q)=-\frac{4\pi\alpha
T^{2}}{3},$ (9)
and
$V_{\mbox{\tiny HTL}}(r)\propto\frac{e^{-q_{i}r}}{r},\hskip
28.45274pt\mbox{with}\hskip 28.45274ptq_{i}=\sqrt{\frac{4\pi\alpha
T^{2}}{3}}.$ (10)
While on the completely one-loop dispersion curve, no pure imaginary solution
is found at $\omega\rightarrow 0$, which implies the pole contains both real
and imaginary parts and the static potential takes the general form of damping
oscillation shown as Eq.(7). One can find out the poles numerically by solving
the equation
$q^{2}-\frac{8\alpha}{\pi}\int^{\infty}_{0}dp\
\frac{p^{2}}{E_{p}}\left[\frac{4E_{p}^{2}-q^{2}}{4p\
q}\log\left(\frac{q-2p}{q+2p}\right)-1\right]n_{f}(E_{p})=0,$ (11)
which is Eq.(1) in the static limit ($\omega\rightarrow 0$) where the mode
$q=q_{r}+iq_{i}$.
In FIG.3 we demonstrated the oscillatory potential of QED. This damping
oscillation is qualitatively different for the monotonic Debye potential in
Eq.(10).
Figure 3: Oscillatory potential in completely one-loop calculation.
## IV radial distribution function and liquid QGP
Generally speaking, In the picture of Debye screening, the in-medium particles
are ”dressed” with the effective radii of Debye length. Therefore the
interactions among the component particles are rather weak so that the system
can be treated as the ideal gas. However, once the Debye potential is replaced
by the oscillatory potential, the ideal gas is no longer a qualified model.
Then what kind of state of matter is the oscillatory potential relevant to? To
answer this question, one must know about the typical character of each state.
To identify different states of matter, one is to distinguish the different
spacial configurations of the component particles. The so-called radial
distribution function (RDF), which is the probability of finding two particles
at a distance $r$ from each other, is introduced as a powerful tool. For
instance, particles in the gas state are completely random, so that the
possibilities of finding any two particles are almost the same. Therefore its
RDF remains constant111The monotonic increasing is due to the inaccessible
core of the component particle. as shown in FIG.4. While the particles in the
liquid state have short range order so that the possibilities of finding
nearby particles are much larger than those far particles. Accordingly, the
RDF in the liquid state will present several damping peaks along the radial
directionMarch ; Egelstaff ; Ichimaru which is also sketched in FIG.4. We
consider this damping oscillation shape as the basic characteristic of a
liquid state, in other words, if someone could obtain such kind of RDF, he may
discover the footprint of a liquid state. ThomaThoma3 calculated the RDF of
QGP in the HTL scheme, which gives the exact Debye screening, and confirmed
the negative result for identifying a liquid. In the following, we will give
up the HTL scheme and work with complete one loop.
Figure 4: Typical radial distribution functions of gas and liquid.
In the liquid state theory, one can define the RDF through
$g(r)=\exp\left[-{{V(r)}}/{T}\right]$ (12)
where $V(r)$ is nothing else but the in-medium potential of average inter-
particle forcesEgelstaff . In the classical liquid state theory, the RDF can
be obtained analytically through a certain pair potential model including the
often used Hypernetted-chain(HNC) or Percus-Yevick(PY) approximations, or
through some computer simulations like Monte Carlo or Molecular dynamicsMarch
. In this paper, we follow none of those schemes, instead, we adopt the static
in-medium potential obtained in the completely one-loop calculation referring
to the last section.
Figure 5: Gluon polarization.
As for plain QCD, the one-loop gluon polarization is determined by the
diagrams in FIG.5. Compared with QED, QCD involves the gluon self-coupling.
One can calculate the temperature-dependent polarization tensor in the
framework of thermal field theory, like what we do in the last section. We
skip the standard steps and directly present the expressions of completely
one-loop polarization tensor of QCD in the temporary axis gauge (TAG)
asKapusta2
$\displaystyle\Pi_{L}^{(a)}=\frac{8\alpha_{s}}{\pi}\int_{0}^{\infty}dp\
\frac{p^{2}}{\omega_{q}}\left[\frac{4\omega_{q}^{2}-q^{2}}{4p\
q}\log\left(\frac{q-2p}{q+2p}\right)-1\right]n_{f}(\omega_{q})$ (13)
$\displaystyle\Pi_{L}^{(b+c)}=-\frac{3\alpha_{s}}{\pi}\int_{0}^{\infty}dp\
p\left\\{4-\frac{2q^{2}}{p^{2}}+\frac{2p}{q}\left[1+\left(\frac{2p^{2}-q^{2}}{2p^{2}}\right)^{2}\right]\log\left(\frac{q+2p}{q-2p}\right)\right\\}n_{b}(p).$
(14)
$\omega_{q}=\sqrt{p^{2}+m_{q}^{2}}$ where $m_{q}$ is the quark mass.
$n_{b}(p)=(e^{\beta p}-1)^{-1}$ is the gluon distribution function. Here we
study the 2-flavor QGP. For the running coupling $\alpha_{s}$, we use the two-
loop renormalization group expressionKaczmarek
$\alpha_{s}=\left[\frac{9}{2\pi}\ln\left(\frac{T}{\Lambda}\right)+\frac{16}{9\pi}\ln\left(2\ln\left(\frac{T}{\Lambda}\right)\right)\right]^{-1},$
(15)
where $\Lambda=73$MeV for the temperature range $1\sim 2T_{c}$.
We would like to point out that although applying the linear response theory
to non-Abelian gauge theory is at the risk of gauge noninvariance, the TAG is
believed safe enough because in this gauge one can obtain the same vacuum
polarization corrected effective charge as the renormalization group
chargeKapusta2 . We hope the discussion in TAG may give at least the
qualitative features of the potential and RDF.
Adding up Eqs.(13) and (14) and inserting them into Eq.(5), one can find out
the pole numerically. Then the interquark potential (7) is obtained and so as
to the RDF considering Eq.(12). FIG.6 is the RDF of QCD plasma where we choose
two different temperatures 0.2 and 0.3GeV.222The deconfined QGP is a Coulomb-
like plasma, whose dimensionless coupling parameter is
$\Gamma=C_{\alpha}\alpha_{s}\left(\frac{3}{4\pi n}\right)^{\frac{1}{3}}/T$
where $C_{\alpha}=4/3$ is the eignvalue of the Casimir operator for quark and
antiquark, $n$ is the particle number density. For estimation, we take
$n=6.3T^{3}$ for 2-flavor QGP by considering it as a massless gasGelman ;
Thoma4 For T=0.2 and 0.3GeV, the running coupling constants are 0.5 and 0.35,
and the corresponding coupling parameters are 2.0 and 1.4, which are great
than 1, indicating a liquid state. In FIG.6, one can see clearly the damping
oscillatory behavior of the RDF, which is very similar to the typical shape of
liquid in FIG.4. This result might indicate the liquid state of hot QGP.
Furthermore, the RDF oscillation becomes weaker and weaker with the increase
of temperature, thus one may expect the QGP is approaching to an ideal gas at
the high temperature limit.
Figure 6: RDF of QCD plasma. The solid and the dotted lines are for $T$=0.2,
0.3GeV respectively.
## V discussion
In this paper, we start with the comparison of dispersion relations of HTL and
complete one loop, pointing out an important discrepancy in the dynamical
screening regime which results in the different behaviors of the static in-
medium potentials. Then we discuss the RDF of hot QGP. It appears an obvious
damping oscillation which implies the QGP might be in a liquid state.
How to deal with the interacting many-body system, especially the strongly
coupled or strongly correlated system, is a rather difficult but fundamental
problem. In principle, one can reduce the many-particle distribution function
to two- or single-particle distribution functionMohling . The RDF is actually
the two-particle distribution function. It is the basic physical quantity in
the atomic liquid theory that has been related to various kinetic and
thermodynamic observablesMarch ; Egelstaff . On one hand, the RDF is obtained
by considering certain dynamical and thermal statistical model from the
theoretical aspects. On the other hand, it can be measured through scattering
experiments in the atomic liquid. Compare the theoretical RDF and the RDF
extracted from experiments, then one can figure out deeper discipline that
rules over the phenomenon. Parallel to the classical liquid theory, the RDF in
this paper is the static two-quark distribution function with spherical
symmetry. Although we can not measure the quark distribution in QGP through
scattering experiment as we do to the atomic liquid, we can still measure the
density-density correlations, which is relevant to the Fourier transformation
of RDFThoma3 , by observing the final state distributions. We hope in this
way, the picture in our calculation can be tested by the experiments.
###### Acknowledgements.
This work is partly supported by the National Natural Science Foundation of
China under project Nos. 10747135, 10675052 and 10575043\.
## References
* (1) STAR collaboration, Phys. Rev. Lett. 90 (2003) 032301
* (2) PHENIX collaboration, Phys. Rev. Lett. 91 (2003)182301
* (3) M.H. Thoma, J. Phys. G 31, L7 (2005); Erratum, J. Phys. G 31, 539 (2005)
* (4) A. Peshier and W. Cassing, Phys. Rev. Lett. 94, 172301 (2005)
* (5) E. Shuryak, arXiv:0807.3033v1
* (6) M. Le Bellac, Thermal Field Theory (Cambridge Univ. Press, Cambridge, 1996)
* (7) R.D. Pisarski, Phys. Rev. Lett. 63, 1129 (1989)
* (8) E. Braaten and R.D. Pisarski, Phys. Rev. D, 42, 2156 (1990)
* (9) E.V. Shuryak and I. Zahed, Phys. Rev. C 70, 021901(R) (2004)
* (10) H.A. Weldon, Phys. Rev. D, 26, 1394 (1982)
* (11) N.H. March and M.P. Tosi, Introduction to Liquid State Physics (World Scientific Publishing, Singapore 2002)
* (12) P.A. Egelstaff, An Introduction to the Liquid State (Clarendon Press, Oxford 1992)
* (13) S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982)
* (14) M.H. Thoma, Phys. Rev. D 72, 094030 (2005)
* (15) J.I. Kapusta, Finite Temperature Field Theory (Cambridge Univ. Press, Cambridge, 1989)
* (16) J. Diaz Alonso, A. Pérez and H. Sivak, Nucl. Phys. A 505, 695 (1989)
* (17) J. Diaz Alonso, E. Gallego and A. Pérez, Phys. Rev. Lett. 73, 2536 (1994)
* (18) J. Kapusta and T. Toimela, Phys. Rev. D 37, 3731 (1988)
* (19) A. L. Fetter and J. D.Wakecka, Quantum Theory of Many Particle Systems (McGraw-Hill, New York, 1971)
* (20) H. Sivak, A. Pérez and J. Diaz Alonso, Prog. Theor. Phys. 105, 961 (2001)
* (21) O. Kaczmarek and F. Zantow, Phys. Rev. D 71, 114510 (2005)
* (22) B.A. Gelman, E.V. Shuryak and I. Zahed, Phys. Rev. C 74, 044908 (2006)
* (23) M.H. Thoma, J. Phys. G 31, L7 (2005); ibid 31, 539 (2005)
* (24) F. Mohling, Statistical Mechanics: Methods and Applications (Publishers Creative Services Inc., 1982)
|
arxiv-papers
| 2008-10-16T02:29:10
|
2024-09-04T02:48:58.280630
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui Liu, Defu Hou and Jiarong Li",
"submitter": "Hui Liu",
"url": "https://arxiv.org/abs/0810.2841"
}
|
0810.2985
|
# Tradeoff between Efficiency and Melting for a High-Performance
Electromagnetic Rail Gun
William C. McCorkle and Thomas B. Bahder
Army Aviation and Missile Research, Development, and Engineering Center,
Redstone Arsenal, AL 35898 USA
Email: thomas.bahder@us.army.mil
###### Abstract
We estimate the temperature distribution in the rails of an electromagnetic
rail gun (EMG) due to the confinement of the current in a narrow surface layer
resulting from the skin effect. In order to obtain analytic results, we assume
a simple geometry for the rails, an electromagnetic skin effect boundary edge
that propagates with the accelerating armature, and a current carrying channel
controlled by magnetic field diffusion into the rails. We compute the
temperature distribution in the rails at the time that the armature leaves the
rails. For the range of exit velocities, from 1500 m/s to 5000 m/s, we find
the highest temperatures are near the gun breech. After a single gun firing,
the temperature reaches the melting temperature of the metal rails in a layer
of finite thickness near the surface of the rails, for rails made of copper or
tantalum. We plot the thickness of the melt layer as a function of position
along the rails. In all cases, the thickness of the melt layer increases with
gun velocity, making damage to the gun rails more likely at higher velocity.
We also calculate the efficiency of the EMG as a function of gun velocity and
find that the efficiency increases with increasing velocity, if the length of
the gun is sufficiently long. The thickness of the melted layer also decreases
with increasing rail length. Therefore, there is a tradeoff: for rails of
sufficient length, the gun efficiency increases with increasing velocity but
the melted layer thickness in the rails also increases.
## I Introduction
Electromagnetic launch systems, such as the railgun, are based on transient
phenomena [1]. During launch, the transient involves the build-up and
penetration of a magnetic field into the surrounding metallic material. The
dynamics of magnetic field penetration into the metal rails is described by a
well-known diffusion equation [2]. The diffusion of the magnetic field leads
to a skin effect, where a large current is transported inside a narrow
channel. In a railgun that has a moving conducting armature, the effect is
called a velocity skin effect (VSE) and is believed to be one of the major
problems in limiting railgun performance [2, 3], because it leads to intense
Joule heating of the conducting materials, such as rails and armatures. To
what extent the VSE effect is responsible for limiting the performance of
solid armatures is still the subject of research [4, 5].
Recently, using a new generation of magnetic field sensors, the magnetic field
distributions caused by the VSE in the rails have been measured [6, 7]. These
experimental efforts are even more significant in light of the large
investments planned by the navy to develop EMGs and power sources for nuclear
and conventional warships [8, 9]. The motivation for this paper is the large
investment planned for EMG technology and the historical lack of understanding
of the reasons for the low endurance of the gun rails in service, sometimes
limited to one shot at maximum energies before replacement is needed.
For high-performance EMGs, in order to increase the armature velocity while
keeping the length of the rails fixed, the current pulse during firing must be
shorter and have a higher average amplitude, causing a stronger skin effect in
the rails, which leads to an increase in Joule heating of the rails. In this
paper, we show that the EMG efficiency is higher at higher velocity, but there
is increased melting of the rails, leading to a tradeoff between efficiency of
the EMG and melting of the rails due to Joule heating (during a single
firing). We do not discuss gun barrel erosion due to repeated firings [10,
11]. We also do not treat the interaction of the high temperature plasma in
the contact regions of the EMRG.
The article is organized as follows. We define our model for EMG heating in
Section II. We consider the problem of rail heating in two steps. First, in
Section III we assume a current-carrying channel described by a local skin
depth $\delta$ (along the rail) that has a simple time dependence due to
motion of the armature, which leads to a time and position-dependent current
density that causes the Joule heating. In Section IV, we use an improved
expression for the current density based on the diffusion of the magnetic
field into the rails. In Section V, we give a crude approximation to the
armature heating. In Section VI, we discuss the efficiency of the EMG and its
dependence on gun velocity and length. We show that the there is a tradeoff
between efficiency and melting the rails. Finally, we present our conclusions
in Section VII.
## II Temperature Distribution in Rails
The dynamics of an EMG can be properly described in terms of a thermodynamic
free energy that expresses the coupling of the mechanical and electromagnetic
degrees of freedom [12]. For example, for the case of an electromagnetic gun
that can be described in terms of a lumped circuit model, which has a rotor
coil with self inductance $L_{R}$ carrying current $I_{R}$, and a stator coil,
rail, and armature circuit with self inductance $L$ with current $I$, the free
energy has the form [13]
$F(I_{R},I,x,\theta)=F_{0}(T)+\frac{1}{2}L_{R}I_{R}^{2}+\frac{1}{2}L(x)I^{2}+M(\theta)\,I_{R}\,I$
(1)
where $L(x)$ depends on the $x$-position of the armature,
$L(x)=L_{0}\,+\,L^{\prime}\,\,x$ (2)
and where $L^{\prime}=dL_{S}/dx$, and $L_{0}$ is the self inductance of the
stator, rail and armature circuit when the armature is at $x=0$. The
interaction of the stator and rotor circuits is specified in terms of their
mutual inductance, $M(\theta)$, where $\theta$ is the angle of the rotor coil
with respect to the stator coil. The term $F_{0}(T)$ depends only on
temperature. Derivatives of the free energy with respect to the coordinates,
$x$ and $\theta$, give the generalized forces on the system [12]. By Newton’s
law, the acceleration of the armature, $\ddot{x}$, is given in terms of the
derivative of the free energy with respect to the coordinate,
$m\,\ddot{x}=\left({\frac{{\partial{\kern
1.0pt}F}}{{\partial\,x}}}\right)_{I_{R},I,\theta}$ (3)
where $m$ is the mass of the armature (including the payload, comprising the
launch package), leading to the well-known dynamical equation
$m\,\ddot{x}=\frac{1}{2}{\kern 1.0pt}L^{\prime}{\kern 1.0pt}I^{2}$ (4)
Equation (4) shows that the acceleration is directly proportional to the
square of the instantaneous current, $I$. As a first approximation, we assume
that the current in the rails is constant during a shot, $I(t)=I_{o}$. If the
current is constant, and the armature starts at $t=0$ at $x=0$, and moves to
the end of the gun rails at $x=\ell$ at time $t=t_{f}$, then we have the
following relations between armature mass $m$, gun length $\ell$, self
inductance per unit length of armature travel $L^{\prime}$, and mass velocity
$v$,
$\begin{array}[]{l}I_{o}=\left({\frac{{m\,}}{{L^{\prime}\,\ell}}}\right)^{1/2}\,v\\\
t_{f}=\frac{2}{{I_{o}}}\left({\frac{{m\,\ell}}{{L^{\prime}}}}\right)^{1/2}\\\
t_{o}(x)=\frac{2}{{I_{o}}}\left({\frac{{x{\kern
1.0pt}m}}{{L^{\prime}}}}\right)^{1/2}\\\ \end{array}$ (5)
The function $t_{o}(x)$ gives the time at which the mass $m$ is at position
$x$ along the rails, see section III.
In order to calculate the Joule heating in the rails, a lumped circuit model
is not sufficiently detailed. Instead, we must use a more detailed model,
where the free energy is expressed in terms of electromagnetic fields with a
spatial distribution. The problem is complicated due to the coupling of the
electromagnetic and mechanical degrees of freedom. The problem is further
complicated by the fact that the dynamics of an EMG shot is a transient effect
in time. Consequently, when Long [14] and Nearing and Huerta [15] computed
current density (and the heating in EMG rails) they assumed that the
mechanical and electromagnetic degrees of freedom are decoupled. Furthermore,
they assumed the armature was moving at a constant speed so the problem became
translationally invariant in time, thereby avoiding the complexities
associated with the initial conditions and the resulting transient effects. In
particular, it is the transient nature of the EMG shot that gives rise to a
dependence of EMG performance on rail length, see for example our Eq. (14) and
(36) for dependence of rail temperature rise and EMG efficiency, respectively,
on rail length $\ell$. Using the above stated assumptions, Long [14] and
Nearing and Huerta [15] solved for the complicated distribution of the current
density using a simplified geometric model of the rails and armature [16].
In this paper, we use a simpler approach that addresses the transient nature
of the EMG shot and allows us to get approximate analytic results for the
temperature distribution in the rails and armature of an EMG. We model the
skin effect in the rails, which limits the channel through which current can
flow. For a given total current in the rails, a narrower channel (smaller skin
depth) leads to a higher current density and results in greater Joule heating
of the rails. In contrast to Nearing and Huerta [15], our approach allows us
to discuss transient effects dependent on the length of EMG rails and how they
impact EMG performance, see for example our Eq. (14) and (36). We use the same
2-dimensional simplified geometry for the rails and armature as Nearing and
Huerta. However, we assume that the conducting armature is arbitrarily thin,
which allows us to get simple results. Corrections to such an assumption are
expected to be of order $O(a/\ell)$, where $a$ is the armature width and
$\ell$ is the length of the rails. We take the coordinate $x$ running down the
length of the rails, and we assume that the top rail occupies $y>0$ and the
bottom rail occupies $y<-b$, where $b$ is the rail separation. We take the
length of the gun from breech to muzzle to be $\ell$. We assume that the rails
have an arbitrary large thickness $w$ in the $z$-direction, see Figure 1. When
an EMG is fired, the current flows down one rail, through the conducting
armature, and up the other rail to complete the electrical circuit. The
transient response leads to a complicated distribution of eddy currents in the
rails and the armature. We assume that we can describe this effect by the
local skin effect that depends on position and time.
We take the temperature rise due to the Joule heating to be
$T(x,y,t_{f})-T_{0}=\frac{{1\,}}{{\rho\,\,{\kern
1.0pt}C}}\int\limits_{0}^{t_{f}}{\,\frac{{J^{2}(x,y,t)}}{\sigma}{\kern
1.0pt}\,dt}$ (6)
where $T(x,y,t_{f})$ is the temperature at position $(x,y)$ at time $t_{f}$,
when the armature leaves the rails, assuming the current starts at $t=0$. The
quantity $T_{0}=T(x,y,0)$ is the initial temperature at position $(x,y)$ at
time $t=0$ before the shot, $C$ is the specific heat (assumed constant up to
the melting point) of the metal rails, and $\rho$ is the density of the metal
rail. We assume the electric field $E(x,y,t)$ is linearly related to the
current density, $J(x,y,t)=\sigma\,E(x,y,t)$, where $\sigma$ is the electrical
conductivity that is independent of temperature up to the melting point. In
Eq. (6), we have neglected the heat of melting, so the temperature rise is
only valid up to the melting point of the metal rail. If latent heat of
melting $L_{Q}$ is included, then the term $-L_{Q}/C$ must be added to the
right side of Eq. (6). This additional term subtracts from the temperature
rise that may be expected when additional Joule heat is created beyond what is
required to reach the melting temperature. For Cu or Ta, this term is
significant, with value $L_{Q}/C=465\,^{\circ}{\rm C}$ and $1141\,^{\circ}{\rm
C}$, respectively. Below, we do not consider the temperature rise above the
melting point of the metal rails.
## III Current Channel with Constant Current in Rails
The moving armature carries all the current of the rails. As the armature
moves, it exposes a new plane region on the rail that carries current. Due to
the diffusive nature of the magnetic field ${\bf H}$, and the relation
${\bf J}=\textrm{curl}\,\,{\bf H}$ (7)
current does not flow uniformly in the rail [12, 17], instead the current
flows through a layer of thickness $\delta$ that increases with time $t$,
starting with $\delta=0$ as the leading edge of the armature passes a point on
the rail. Assuming a plane geometry for the rail, we can approximate the time
dependence of the skin depth to be [17]
$\delta(t)=\left({\frac{{4\,t}}{{\mu\,\sigma}}}\right)^{1/2}$ (8)
where $\mu$ is the magnetic permeability and $\sigma$ is the electrical
conductivity, and $t$ is the time elapsed since the armature has passed a
given element on the rail. The exact factor inside the square root (here we
take it to be 4) is somewhat arbitrary in defining a skin depth. (In Section
IV we remove this arbitrariness by using the solution of the time-dependent
magnetic field diffusion equation to compute the current density
distribution.) As stated in the second to last paragraph in Section II, we
assume an idealized armature that is arbitrarily thin in the $x$-direction and
we assume that the skin depth $\delta$ in the rail starts at zero thickness at
the position where the armature contacts the rails. As remarked above, this
drastic assumption is expected to have corrections of order $O(a/\ell)$ where
$a$ is the width of the armature and $\ell$ is the length of the rail. This
assumption allows us to obtain analytic results and see the dependence on a
number of parameters. As a start, we model the conducting channel by assuming
that the current density in the rail inside the skin depth $\delta$ is a
function of $x$ and $t$ but not $y$, and that the current density is zero
outside the skin depth $\delta$, see Figure 1 [18]. Furthermore, we assume
that the skin depth has zero thickness on the leading edge of the armature,
since all the current in the rails has to flow through the armature. Assuming
a current carrying channel of finite width $\delta$, at time $t$ and position
$x$ along the rail, the $y$ coordinate of the boundary of the current carrying
channel is
$y_{c}(x,t)=\left\\{{\begin{array}[]{*{20}c}{\delta(t-t_{o}(x))\,,\quad
t>t_{o}(x)}\\\ {0\,,\quad\quad\quad\quad\quad t\leq t_{o}(x)}\\\
\end{array}}\right.$ (9)
where $t_{o}(x)$ is the time the leading edge of the armature passes the
position $x$. In other words,
$\frac{{d\,t_{o}(x)}}{{d\,x}}=\frac{1}{{v(x)}}$ (10)
where $v(x)$ is the velocity of the armature when it is at point $x$, see
Figure (1).
Consider an element of volume $dV=wdx\,dy$ at position $(x,y)$. Define
$\tau(x,y)$ as the time at which the boundary of the current carrying channel
intersects this volume element. From Eq. (9) for the time-dependent boundary
of the current carrying channel, we find
$\tau(x,y)=t_{o}(x)+\frac{1}{4}\mu\sigma y^{2}$ (11)
For time $0<t\leq\tau(x,y)$ there is no current flowing through this volume
element. At time $t=\tau(x,y)$, the boundary of the current carrying channel
intersects the volume element at position $(x,y)$ and Joule heating starts.
See Figures 2, 3 and 4. Finally, at time $t=t_{f}$ the armature leaves the
rails, the circuit is broken, and there is no more Joule heating of the
element $dV$.
The current density in the rail can be written down by considering three
domain regions, see Figure 5. At position $(x,y)$ and time $t$, we take the
current density to have the form
$\begin{array}[]{l}J(x,y,t)=\\\
\quad\left\\{{\begin{array}[]{*{20}c}{\frac{{I(t)}}{{w\,\delta(t-t_{o}(x))}},}&{\quad
x<x_{o}(t)\quad{\rm{and}}\quad y\leq\delta(t-t_{o}(x))}\\\
{0,}&{x<x_{o}(t)\quad{\rm{and}}\quad y>\delta(t-t_{o}(x))}\\\
0&{x>x_{o}(t)}\\\ \end{array}}\right.\\\ \end{array}$ (12)
where $x_{o}(t)$ is the function that gives the $x$ coordinate of the armature
at time $t$, and $\delta(t)$ is given by Eq.(8).
Figure 1: The rails and conducting armature are shown for the electromagnetic
gun. At time $t=0$ the leading edge of the armature is at $x=0$. The trailing
edge of the armature leaves the rails at time $t=t_{f}$. Figure 2: At time
$t<\tau$ the skin depth boundary has not yet reached the element of volume
$dV=w\,dx\,dy$ at position $(x,y)$. Figure 3: At time $t=\tau$ the skin depth
boundary overlaps the element of volume $dV=w\,dx\,dy$ at position $(x,y)$. At
this time, current starts to flow in the element $dV$ and the temperature
starts to rise due to Joule heating. Figure 4: After the skin depth boundary
passes the element of volume $dV$, heating of the element continues until the
armature exits the rails at the time $t_{f}$. For time $t>t_{f}$, we assume
that no energy is input into the element $dV$. Figure 5: The three domain
regions are shown for the current density. The non-zero current density is
assumed to be inside the skin depth for $x<x_{o}(t)$ where $x_{o}(t)$ gives
the $x$-coordinate of the armature at time $t$.
Using Eq. (6) and assuming a constant current in the armature, $I(t)=I_{0}$,
from Eq. (6) and (12) we obtain
$\begin{array}[]{l}T(x,y,t_{f})-T_{0}=\\\
\;\frac{{\mu\,I_{0}^{2}}}{{4w^{2}\rho
C}}\;\left\\{{\begin{array}[]{*{20}c}{Log\left({\frac{{t_{f}-t_{o}(x)}}{{\tau(x,y)-t_{o}(x)}}}\right),}&{y\leq\left[{\frac{4}{{\mu\sigma}}\left({t_{f}-t_{o}(x)}\right)}\right]^{1/2}}\\\
{0,}&{y>\left[{\frac{4}{{\mu\sigma}}\left({t_{f}-t_{o}(x)}\right)}\right]^{1/2}}\\\
\end{array}}\right.\\\ \end{array}$ (13)
Equation (13) gives the temperature rise, $T(x,y,t_{f})-T_{0}$, of an element
of volume at position $(x,y)$ at time $t_{f}$ at which the armature leaves the
rails, for a constant gun current over the time interval $0<t<t_{f}$. For
realistic time-dependent currents, see Figure 3 in McCorkle [19].
Using a lump circuit model to describe the constant acceleration $\ddot{x}=a$
of the armature for a constant current $I_{0}$, we can write the position of
the armature as a function of time as $x=\frac{1}{2}at^{2}$. The time at which
the armature leaves the rails, $t_{f}$, is then related to the length of the
gun, $\ell=\frac{1}{2}at_{f}^{2}$. The time $t_{o}(x)$ at which the armature
passes coordinate $x$ is then $t_{o}(x)=(2x/a)^{1/2}$, see also Eq. (5). Using
these approximations in Eq. (13) gives the temperature rise at position
$(x,y)$
$\begin{array}[]{l}T(x,y,t_{f})-T_{0}=\frac{{\mu\,I_{0}^{2}}}{{4w^{2}\rho\,C}}\;\;\times\\\
\left\\{{\begin{array}[]{*{20}c}{Log\left[{\frac{{8\,\ell}}{{\mu\,\sigma\,v\,y^{2}}}\left({1-\left({\frac{x}{\ell}}\right)^{1/2}}\right)}\right],}&{y\,\leq\,\,\tilde{y}(x)}\\\
{0,}&{y>\,\tilde{y}(x)}\\\ \end{array}}\right.\\\ \end{array}$ (14)
where
$\tilde{y}(x)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v\,}}}\right)^{1/2}\,\left({1-\left({\frac{x}{\ell}}\right)^{1/2}}\right)^{1/2}$
(15)
and $x$ is assumed to be in the interval $0<x\leq\ell$. Note that Eq. (14)
predicts that the temperature on the surface of the rail, at $y=0$, is
infinite. This feature of the solution is well-known and is not a problem
[17]. The singularity with respect to $y$ is integrable, and therefore the
energy deposited in a thin layer near the surface is finite. We use Eq. (14)
to locate the points $(x,y)$ of the surface that reaches the melting point.
Using the values for the gun parameters and material parameters in Tables I
and II, we plot the distribution of the temperature rise in the rail,
$T(x,y,t_{f})-T_{0}$, as a function of the $y-$coordinate, for several points
along the rail given by coordinate $x$. For comparison, we also plot the
melting point of Cu and Ta, which is 1083${}^{\circ}C$ and 2996${}^{\circ}C$,
respectively. The $y-$coordinate where the curve intersects the melting point
of each metal is the depth to which melting of the rails occurs, if the EMG
gun shot occurred with rails at initial temperature $T_{0}$. As mentioned
previously, we have not taken into account the heat of melting, so these
curves are not correct for temperatures above the melting point of the rails.
Figure 6: Plot of Eq. (14), giving the temperature rise, assuming rails made
from copper, plotted vs. position $x$ along the rail. Parameters used are
given in Tables I and II. The gun velocity is taken as 3000 m/s and the inital
temperature $T_{o}=0^{o}$C. Constant current is assumed.
Figure 6 and 7 show a plots of the thickness of the melted layer assuming the
rails are made of copper and tantalum, respectively, for a constant total gun
current, which corresponds to a unifomly accelerating armature.
Figure 7: Plot of Eq. (14), giving the temperature rise, assuming rails made
from tantalum, plotted vs. position $x$ along the rail. Parameters used are
given in Tables I and II. The gun velocity is taken as 3000 m/s and the inital
temperature $T_{o}=0^{o}$C. Constant current is assumed.
Our model predicts that the thickness of the melted layer is largest at $x=0$,
at the breech of the gun rail. This is reasonable because the current at the
breech of the gun flows for the longest time, causing maximum Joule heating at
$x=0$.
The intersection of the curves in Figure 6 and 7 with the melting point of
metal (horizontal line) gives the thickness of the melted layer. The surface
inside the rails that reaches the melting temperature, $T_{melt}$, is given by
points $(x,y)$ that satisfy
$T(x,y,t_{f})-T_{0}=T_{melt}-T_{0}$ (16)
Alternatively, we can say that at position $x$ the thickness of the rail that
reaches the melting point, $y_{melt}(x)$, is given implicitly by
$T(x,y_{melt},t_{f})-T_{0}=T_{melt}-T_{0}$. For any element of volume, the
temperature rise is due to the length of time that the current was flowing
through that element. For an element at $(x,y)$, current starts to flow at
time $\tau(x,y)$. We assume that for all elements the current stops flowing at
time $t_{f}$, when the armature leaves the rails.
From Eq. (16) we solve for the thickness of the melted layer, $y_{melt}(x)$,
as a function of $x$ position along the rail
$y_{melt}(x)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v}}}\right)^{1/2}\,\;e^{-\frac{{2\,{\kern
1.0pt}w^{2}\rho{\kern 1.0pt}{\kern 1.0pt}\,L^{\prime}\,\ell\,C{\kern
1.0pt}\left({T_{{\rm{melt}}}-T_{o}}\right)}}{{\mu\,{\kern
1.0pt}m\,v^{2}}}}\,\,\left[{1-\left({\frac{x}{\ell}}\right)^{1/2}}\right]^{1/2}$
(17)
Figure 8 shows a plot of the thickness of the melted layer assuming the rails
are made of aluminum, copper, and tantalum. We used numerical values given in
Tables I and II.
Figure 8: Plot of Eq. (17), giving the melt layer thickness vs. x along the
rails for rails made from copper, tantalum and aluminum. The initial
temperature of rails is assumed to be $T_{o}=0^{o}$C. Constant current is
assumed and the gun velocity is taken to be 3000 m/s.
The thickness of the melted layer has a strong dependence on gun velocity,
which is the velocity of the armature at $x=\ell$. Figure 9 shows plots of the
melted layer thickness at $x=0$ for Cu, Ta, and Al rails vs. gun velocity.
Figure 9: Plot of Eq.(17) giving the melted layer thickness for Cu, Ta, and Al
rails at the gun breech (at x=0) vs. gun velocity, $v$. The initial
temperature of rails is assumed to be $T_{o}=0^{o}$C. Constant current is
assumed. This plot also gives an estimate of the melted layer thickness in the
armature, see discussion in Section VI.
The plots in Figure 9 show that for gun velocities above 1500 m/s, using
materials such a copper, aluminum, and tantalum, the thickness of the melted
layer in the gun rails increases rapidly with gun velocity.
TABLE I: Gun parameters. quantity | symbol | value
---|---|---
width of rails | $w$ | 0.10 m
length of rails (gun length) | $\ell$ | 10.0 m
mass of armature | $m$ | 20 kg
derivative of self inductance of rail | $L^{\prime}$ | 0.46$\times$10-6 H/m
magnetic permeability | $\mu$ | $4\pi\times 10^{-7}$ H/m
TABLE II: Material parameters for the rails. symbol | $T_{melt}$ | $\sigma$ | $\rho$ | $C$
---|---|---|---|---
| melting temp. ${}^{o}C$ | conductivity (Ohm m)-1 | density kg/m-3 | specific heat J/kg ${}^{o}C$
aluminum | 660 | 3.82 $\times$ 107 | 2720. | 950.
copper | 1084 | 5.8 $\times$ 107 | 8960. | 440.
tantalum | 2996 | 7.40$\times$106 | 16600. | 150.62
## IV Current Distribution due to Magnetic Field Diffusion
In this section we obtain the current density in the rail, $J(x,y,t)$ (to be
used in Eq. (6)), by considering the magnetic field diffusion into the rails.
We treat the diffusion of the magnetic field into the rails as diffusion into
a plane surface, with a time dependent boundary condition on the field, given
by the armature passing a surface element of the rail. Consider an arbitrarily
thin armature at position $x_{o}$ at some time $t$. For the geometry in Figure
1, the magnetic field $H_{z}$ is in the $z$-direction. For $x>x_{o}$, which is
outside the rail-armature circuit, the magnetic field $H_{z}=0$. For
$0<x<x_{o}$, which is inside the rail-armature circuit, the magnetic field has
some constant value $H_{0}$. As the (arbitrarily thin) armature passes the
point $x$ on the surface of the rail, the surface magnetic field changes from
$H=0$ to some finite value $H_{0}$, and the field starts to diffuse into the
rail. Essentially, as the armature passes the point $x$ the boundary condition
on the field $H_{z}$ on the rail surface changes from $H_{z}=0$ to
$H_{z}=H_{0}$. Considering the rail to be a plane surface, the magnetic field
diffuses into the rail according to the equation [12, 17]
$\frac{{\partial^{2}H_{z}}}{{\partial y^{2}}}-\frac{1}{\kappa}\frac{{\partial
H_{z}}}{{\partial t}}=0$ (18)
For a plane surface with magnetic field boundary condition $H_{z}=0$ for $t<0$
and $H_{z}=H_{0}$ for $t>0$, the solution for $y>0$ is
$H_{z}=H_{0}\,{\rm{erfc}}\left({\frac{y}{{2\sqrt{\kappa{\kern
1.0pt}t}}}}\right)$ (19)
where $\kappa=1/(\mu\sigma)$ is the magnetic diffusion length,
${\rm{erfc}}\left(\xi\right)$ is the complimentary error function,
${\rm{erfc}}\left(\xi\right)=1-{\rm{erf}}\left(\xi\right)$ and
${\rm{erf}}\left(\xi\right)$ is the error function given by
$erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt$
The current density associated with this magnetic field can be obtained from
Eq. (7), leading to
${\bf{J}}={\bar{J}}(y,t)\,\;{\bf{e}}_{x}=\frac{{H_{0}}}{{\sqrt{\pi{\kern
1.0pt}\kappa{\kern 1.0pt}t}}}\,\exp\left({-\frac{{y^{2}}}{{4\,\kappa\,{\kern
1.0pt}t}}}\right)\,\,{\bf{e}}_{x}$ (20)
where ${\bf{e}}_{x}$ is the unit vector in the $x$-direction, and the
auxiliary function, ${\bar{J}}(y,t)$, is defined by Eq. (20). The value of the
magnetic field is related to the total current $I_{o}$ in the rails by
$I_{o}=\int{\int{dy\,dz\,}}{\bar{J}}(y,t)=wH_{0}$ (21)
leading to $H_{0}=I_{o}/w$ where $w$ is the width of the rail in the
$z$-direction.
We use the plane surface solution to approximate the magnetic field diffusion
and resulting current distribution in the rail. As the armature sweeps past a
surface element in the rail at position $x$, the field starts diffusing into
the rail at time $t_{o}(x)$, where $t_{o}(x)$ gives the time the armature
passes point $x$. We take the current density to have the form
${\bf{J}}(x,y,z,t)=\bar{J}(y,t-t_{o}(x))\,{\bf{e}}_{x}$ (22)
when the coordinates $(x,y,z)$, and time $t$ satisfy the conditions
$0<x<x_{o}(t)\,\,\>,\,\,y>0\,\>,\,\,\,\>|z|<\frac{w}{2}\,\,\,,\,\,\;0<t<t_{f}$
(23)
and otherwise we take ${\bf{J}}(x,y,z,t)=0$. Here, $x_{o}(t)$ is the
$x$-position of the armature at time $t$, and ${\bar{J}}(y,t)$ is given by Eq.
(20). The quantity ${\bf J}(x,y,z,t))$ is an approximation to the time-
dependent current density in the rails during the EMG shot. Note that ${\bf
J}(x,y,t)$ does not depend on $z$ because we assume a large (essentially
infinite) extent in the $z$-direction.
The temperature rise is obtained from Eq. (6) and we neglect the effect of re-
distribution of heat during the time of the shot, since it takes a long time
on the time scale of a shot (which is $t_{f}$), The temperature rise
immediately after the shot is
$T(x,y,t_{f})-T_{0}=\frac{{\mu\,I_{o}^{2}}}{{\pi{\kern 1.0pt}{\kern
1.0pt}w^{2}{\kern 1.0pt}\rho{\kern 1.0pt}{\kern
1.0pt}C}}\,\,\,\Gamma\left({0,\frac{{\mu{\kern
1.0pt}\sigma}}{2}\frac{{y^{2}}}{{t_{f}-t_{o}(x)}}}\right)$ (24)
where $\Gamma(a,z)$ is the incomplete Gamma function, given by
$\Gamma(a,z)=\int_{z}^{\infty}t^{a-1}e^{-t}dt$
Equation (24) is our basic result for the temperature distribution in the
rails, and is expressed in terms of the current in the rails $I_{o}$. It is
useful to express the temperature rise in terms of the gun velocity $v$.
In order to get simple results, we assume that the acceleration is constant,
using the relations in Eq. (5) leads to the temperature rise
$T(x,y,t_{f})-T_{0}=\frac{{\mu{\kern 1.0pt}{\kern 1.0pt}m{\kern 1.0pt}{\kern
1.0pt}v^{2}}}{{\pi\,w^{2}\,\rho\,C\,L^{\prime}\,\ell}}\;\Gamma\left({0,\frac{{\mu\,\sigma\,v}}{{4\,\sqrt{\ell}}}\,\frac{{y^{2}}}{{\sqrt{\ell}-\sqrt{x}}}}\right)$
(25)
Figure 12 shows a plot of the temperature rise given by Eq. (25), for the case
of copper rails, using the parameters in Tables I and II.
Figure 10: The temperature rise is plotted vs. $y$ coordinate into the rail
for various positions $x$ along the copper rail. The time dependence of the
current is determined by the time dependence of magnetic field diffusion into
the rails. Initial temperature of the rails was assumed to be $T_{0}$ =0 C.
Parameters used are given in Tables I and II. Figure 11: Plot of the thickness
of the melted layer vs. position x along the rail, assuming rails made from
copper. Curves for three different velocities are shown. Initial temperature
of the rails was assumed to be $T_{0}$ =0 C. Parameters are those in Tables I
and II. Figure 12: Plot of the thickness of the melted layer vs. position x
along the rail, assuming rails made from tantalum. Curves for three different
velocities are shown. Initial temperature of the rails was assumed to be
$T_{0}$ =0 C. Parameters are those in Tables I and II.
In Eq. (25), the argument of the $\Gamma(\xi)$ function is large for all
values of $x$ and $y$, except for $y$ near $y=0$. For the values in Tables I
and II, the argument of the $\Gamma$ function is
$\xi=\frac{vy^{2}\mu\sigma}{4\left(\ell-\sqrt{\ell}\sqrt{x}\right)}=\frac{54663.71\,\,y^{2}}{10.-3.1622\sqrt{x}}$
(26)
Therefore, the $\Gamma(\xi)$ function for $\xi>>1$ can be approximated by its
asymptotic expansion
$\Gamma(0,\xi)=e^{-\xi}\left(\frac{1}{\xi}-\frac{1}{\left(\xi\right)^{2}}+O\left[\frac{1}{\xi}\right]^{3}\right)$
(27)
Near y=0, the behavior of the $\Gamma(\xi)$ function for $\xi<<1$ is
logarithmic
$\Gamma(0,\xi)=(-\gamma-Log[\xi])+\xi-\frac{\xi^{2}}{4}+O[\xi]^{3}`$ (28)
where $\gamma$ is Euler’s constant, $\gamma\approx 0.5772$.
As before, see Eq. (16), we use Eq.(25) to solve for the thickness of the
melted layer in the $y$-direction by setting the temperature rise equal to the
melting temperature of the metal. Figures 11 and 12 show plots of the
thickness of the melted layers verses position $x$ along the rail, for several
velocities, for rails made of copper and tantalum, respectively. From these
figures, we see that even though Ta has a much higher melting temperature than
Cu, 2996oC compared to 1084oC, respectively, the electrical conductivity of Ta
is lower, which leads to a comparable thickness for the melted layer for Ta
and Cu rails.
## V Armature Heating
We can estimate the heating of the armature from the calculations that we have
already done. We now imagine that the armature has a finite thickness.
Furthermore, depending on the EMG design, the armature material can be
different than that of the rails, and consequently, the electrical
conductivity $\sigma$ may be different. Unlike the rails, the armature
conducts current for the whole duration of the shot from $t=0$ to $t=t_{f}$.
In this sense, the whole length of the armature has current flowing in it like
the element of rail at coordinate $x=0$. The thickness of the melted layer of
the armature can be found from Eq.(17) by setting $x=0$,
$y_{melt}(0)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v}}}\right)^{1/2}\,\;e^{-\frac{{2\,{\kern
1.0pt}w^{2}\rho{\kern 1.0pt}{\kern 1.0pt}\,L^{\prime}\,\ell\,C{\kern
1.0pt}\left({T_{{\rm{melt}}}-T_{0}}\right)}}{{\mu\,{\kern 1.0pt}m\,v^{2}}}}$
(29)
Figure 9 shows a plot of the melted layer thickness of the armature verses gun
velocity, as given by Eq. (29).
## VI Efficiency and Melt Layer Thickness
During an EMG shot, the energy that is supplied to the gun appears as
projectile kinetic energy, Joule heating of the rails and armature, armature
and rail heating caused by frictional forces between rails and armature,
vibration of rails and armature, sound and light produced in surrounding air,
and electromagnetic radiation due to a time-dependent magnetic field. We
neglect all these effects except for the Joule heating of the rails and
kinetic energy of the projectile and armature.
From the current density in Eq. (22), we calculate the energy in one rail,
$Q$, that is dissipated over one shot of the EMG by integrating the Joule
heating over the time $t_{f}$ during which the armature is in contact with the
rails,
$Q=2\int{d^{3}x}\int\limits_{0}^{t_{f}}{dt}\,\frac{{J^{2}}}{\sigma}$ (30)
where the spatial integration is over all space inside one of the rails. The
factor of two is due to the fact that there are two rails, and by symmetry the
energy dissipated is twice that of one rail. Assuming constant current, for
which the time $t_{f}$ is defined in Eq. (5), assuming the conductivity
$\sigma$ is a constant, and using the expression given by Eq. (22) for the
current density derived by considering magnetic field diffusion into rails
each of length $\ell$, we find
$\begin{array}[]{c}Q=\frac{{32}}{{15}}\left({\frac{\mu}{{\pi\sigma}}}\right)^{1/2}\,\frac{{\ell^{5/4}\,\,m^{1/4}}}{{L^{\prime
1/4}\,\,w}}\,\,I_{o}^{3/2}\\\
=\,\frac{{32}}{{15}}\left({\frac{\mu}{{\pi\sigma}}}\right)^{1/2}\,\frac{m}{{w\,L^{\prime}}}\,\ell^{1/2}\,\,v^{3/2}\\\
\end{array}$ (31)
where we have assumed a constant current (armature acceleration) and used the
value of $t_{f}$ in Eq. (5). Note that $Q$ is the total amount of energy
dissipated in both rails. The two forms for $Q$ are obtained assuming the
relation between EMG velocity and current given in Eq. (5), which assumes a
constant current. Equation (31) gives the amount of energy $Q$ dissipated in
the rails due to Joule heating over the time $t_{f}$ of one shot. Equation
(31) contains some interesting physics because the total energy dissipated as
Joule heat is not proportional to $I_{o}^{2}$ as is the case for a resistor
carrying a constant current $I_{o}$ for a time $t_{f}$. If the rails acted as
a simple resistor with resistance $R$, then the amount of energy dissipated
due to Joule heating during the shot would be $I_{o}^{2}\,R\,t_{f}$. If
$t_{f}$ was a constant independent of current $I_{o}$, then the amount of heat
dissipated in the resistor would be proportional to $I_{o}^{2}$. However, in
an EMG, the time $t_{f}$ is inversely proportional to current $I_{o}$, as
given by Eq. (5), because the armature spends less time in contact with the
rails at higher current (due to higher average armature velocity at higher
current). So if the rails acted like a resistor with no skin effect, and the
time $t_{f}$ varied inversely with current $I_{o}$, then the amount of heat
dissipated in such a resistor would be proportional to $I_{o}$. However, in
the EMG model that we are considering, we have included the skin effect. With
increasing current $I_{o}$, the time $t_{f}$ is shorter and there is a
stronger skin effect at larger $I_{o}$ leading to an increased resistance $R$,
resulting in an increased Joule heat dissipation during the shot time $t_{f}$,
since all current has to flow through a thinner channel in the rails. The net
effect is that the skin effect causes an increased Joule heat over the shot,
so that for the EMG with skin effect we have $Q\sim I_{o}^{3/2}$, as given in
Eq. (31).
The ratio of armature kinetic energy divided by the Joule heat is a sort of
“thermal $q$” and is given by
$\begin{array}[]{c}q=\frac{{\frac{1}{2}mv^{2}}}{Q}\\\
=\frac{{15}}{{32}}\left({\frac{{\pi\sigma}}{\mu}}\right)^{1/2}\frac{{{\rm{w}}\,L^{\prime{\kern
1.0pt}{\kern 1.0pt}5/4}}}{{\left({m{\kern
1.0pt}\ell}\right)^{1/4}}}\,I_{o}^{1/2}\\\
=\frac{{15}}{{32}}\left({\frac{{\pi\sigma}}{\mu}}\right)^{1/2}\,{\rm{w}}\,L^{\prime}\,\left({\frac{v}{\ell}}\right)^{1/2}\\\
\end{array}$ (32)
The quantity $q$ shows that, in order to achieve a certain velocity $v$, a
larger fraction of input energy goes into the armature kinetic energy when
using shorter rails (smaller $\ell$). Figure 15 shows the dependence of $q$ on
velocity $v$ for parameters given in Table I and II for gun rails made of
copper.
Figure 13: The plot shows the thermal $q$ defined in Eq. (32) as a function of
velocity, using the parameters in Table I, assuming copper rails with
conductivity in Table II.
However, the quantity $q$ does not contain the whole story. We next consider
the EMG efficiency.
Within our model, we define the efficiency of an EMG, $\eta$, as the armature
kinetic energy, divided by the total energy
$\eta=\frac{{\frac{1}{2}mv^{2}}}{{\frac{1}{2}mv^{2}+Q+E_{mag}(\ell)}}$ (33)
We consider the total energy as a sum of three terms: the kinetic energy of
the armature $\frac{1}{2}mv^{2}$, the energy dissipated by Joule heating $Q$
in Eq. (31), and the energy stored in the magnetic field $E_{mag}(\ell)$ when
the armature is at the end of the rails at $x=\ell$. In general, the magnetic
energy $E_{mag}(x)$ is the energy stored in the magnetic field in the rails
and surrounding space when the armature is at position $x$:
$E_{mag}(x)=\frac{1}{2}\,L(x)\,I_{o}^{2}$ (34)
As previously defined in Eq. (2), we write $L(x)$ in terms of $L_{0}$ and
$L^{\prime}$, where $L_{0}$ is the self inductance of the EMG that does not
depend on armature position (and corresponds to magnetic energy
$L_{0}\,I_{o}^{2}\,/2$ stored in the power supply, surrounding space, and in
electrical leads to the rails) and $L^{\prime}$, which is the derivative of
the self inductance with respect to armature position $x$. When the armature
reaches the end of the rails, the circuit is broken and the magnetic energy
$E_{mag}(\ell)$ is dissipated in the form of an electrical arc, sound, light,
and mechanical vibration. Assuming constant current and acceleration, when the
armature is at the end of the rails at position $x=\ell$, the stored magnetic
energy is
$E_{mag}(\ell)=\frac{1}{2}\,mv^{2}\left({1+\frac{{L_{0}}}{{L^{\prime}\;\ell}}}\right)$
(35)
where we have used Eqs. (4). From Eqs. (30), (33) and (35), the efficiency of
the EMG is then
$\eta=\frac{1}{{2+\frac{{L_{0}}}{{L^{\prime}\;\ell}}+\frac{{64}}{{15\,w{\kern
1.0pt}L^{\prime}}}\left(\frac{\mu\,\ell}{\pi\sigma\,v}\right)^{1/2}\,\;\,}}$
(36)
Equation (36) shows how the efficiency depends on a number of variables. The
efficiency of an EMG increases with increasing electrical conductivity
$\sigma$, velocity $v$, and rail width $w$. The efficiency depends in a
complicated way on the rail length $\ell$. In the limit of high velocity $v$,
the efficiency of the EMG has the limiting value
$\mathop{\lim}\limits_{v\to\infty}\;\eta=\frac{1}{{2+\frac{{L_{0}}}{{L^{\prime}\;\ell}}\,}}$
(37)
The high-velocity limit of EMG efficiency depends on the ratio of the
stationary self inductance, $L_{0}$, to the dynamic part of the self
inductance of the rails, ${L^{\prime}\;\ell}$. We call ${L^{\prime}\;\ell}$
the dynamic part of the self inductance because it depends on the length of
the rails. For a given gun design, the quantities $L_{0}$ and ${L^{\prime}}$
are constants, but in principle the length of the rails, $\ell$, can be
increased to make the term $\frac{L_{0}}{L^{\prime}\;\ell}<<1$, thereby
increasing the high-velocity limit of the efficiency. Longer rails (larger
$\ell$) lead to a higher EMG efficiency at high velocity. However, for all
velocities the EMG efficiency for this model is always less than 1/2.
The denominator of the efficiency in Eq. (36) has three terms. For the
parameters in Table I and II, the third (last) term in the denominator is
$24.35/\sqrt{(}v)$, where $v$ is in units of m/s. Therefore, it is clear that
the EMG is more efficient at higher velocities. The efficiency will increase
with velocity significantly when the stationary part of the self inductance,
$L_{0}$, is smaller than the dynamic part of the self inductance,
$L^{\prime}\;\ell$, so that the term $\frac{{L_{0}}}{{L^{\prime}\;\ell}}<<1$,
which may rarely be true in real designs unless the length of the rails $\ell$
can be made sufficiently large.
Stated in another way, a longer EMG is more efficient at higher velocities,
however, the thickness of the melted layer also increases at higher
velocities. Therefore, there is a tradeoff between efficiency and melting of
the rails. Increasing the length of the rails may prevent melting, however,
this may lead to rail length that is not practical for applications.
In our simple model of an EMG, and our definition of efficiency, we have
neglected many effects such as friction between armature and rails, plasma
contacts, and many other details. These effects would make the denominator of
Eq. (36) larger, and hence would reduce the efficiency.
## VII Conclusion
We have constructed a model of the EMG based on the electrodynamics of the
launch rails and armature. Our model takes into account the skin effect in the
rails and armature, which is significant due to the short-duration and
extremely high current densities in high-performance EMGs. We used two
approaches. The first approach, in Section III we modeled the skin effect by a
constant-in-time current flowing through a channel equal to the local skin
depth along the rail, $\delta$, given by Eq. (7) at each point along the rail.
We find that a finite thickness layer of the rails, given by Eq. (17), reaches
the melting point of the metal. Although aluminum is lighter by a large
factor, we have assumed the rails are made from copper because of its
significantly higher melting point (1083 C for copper verses 660 C for
aluminum). Figures 6 and 7 show plots of the temperature rise (immediately
after a shot) versus depth into the rails, for different positions spaced one
meter apart along the rails, assuming 10 meter long rails, for copper and
tantalum rails, respectively. For all positions along the rails, immediately
after the shot and before thermal diffusion takes place, the temperature rise
is the highest at the rail surface and decreases into the interior of the
rails. For constant current, near the breech end of the gun the skin depth
exceeds 15 mm, and a layer approximately 2 mm or 3 mm thick melts, for
tantalum or copper rails, respectively, see Figure 8. The model predicts that,
immediately after a shot and before thermal energy is redistributed, the
temperature is largest near the rail surface and a thin surface melt layer
exists for any rail gun, see Eq. (17).
In the second approach, in Section IV, we obtained an approximate current
density by computing the time dependence of the diffusion of the magnetic
field into the rail surface during the EMG shot. From the magnetic field ${\bf
H}$, we obtain the current density from Eq. (7), which is used to compute the
temperature rise immediately after a EMG shot, see Figures 6, 7 and 10.
Figures 11 and 12 show the melted layer thickness we can expect for copper and
tantalum rails, for different gun velocities (due to different gun currents,
which are related to velocities by Eq. (4)). We expect this approach (Section
IV) to be our most accurate estimation of the temperature rise in the rails
and resultant melting. In both approaches, we find the thickness of the melted
layer of the rail increases rapidly with EMG velocity, see Eq. (29) and Figure
9.
Finally, in Section V, we estimated the temperature rise of the armature, and
the resulting thickness of the melted layer. The temperature rise in the
armature may be a major limiting factor in EMG design, see Figure 9.
In Section VI, we computed the efficiency of an EMG as a function of armature
velocity and gun length. In this efficiency calculation, we only considered
Joule heating, kinetic energy of the armature, and the stored magnetic energy
in the system, and we neglected all other energies, such as frictional heating
between armature and rails. We find that if the stationary part of the self
inductance is small compared to the dynamic part of the self inductance (that
of the rails), which is equivalent to long rails, the efficiency of an EMG
increases with velocity. Stated in another way, if the rails are long enough
the EMG will have an efficiency that increases with velocity. In all cases the
upper limit of efficiency is 1/2 at high velocity. However, with increasing
velocity, the thickness of the melted layer in the rails also increases, see
Figure 9. The thickness of the melted layer also decreases with rail length.
Therefore, there is a tradeoff: for sufficiently long rails, with increasing
gun velocity the EMG is more efficient, however, at higher velocity the
thickness of the melted layer is larger, which is likely to result in more
damage to the rails. Also, we may need rails that are too long for practical
applications. In our model of an EMG, we have neglected many practical
effects, such as heating and ablation of armature/projectile at high
velocities in the atmosphere.
The most important conclusion is that, given the available choice of materials
for rails and armatures, the gun has increased efficiency at higher velocity,
however, there is increased melting of the rails and damage is more likely
during each firing of high performance guns, where payload and range is
designed for (high-velocity) naval guns. This analysis was performed for
simple rail guns. Other geometries exist, although all appear to rely on the
sliding armature to carry the high currents and hence will generate a
characteristic skin effect that results in significant rail heating.
Finally, heating of the rails on a microscopic scale depends on
inhomogeneities of the metal, such as crystalline alignment and dislocations.
These inhomogeneities will lead to spatial fluctuations of the current
density. This will likely lead to a spatially inhomogeneous deposit of energy,
and local “hot spots”. Such spatial fluctuations in the case of an EMG are
likely to lead to the analogue of what is usually termed “gun barrel erosion”.
A comment made at the end of the chapter written by I. Ahmad in 1988, entitled
“The Problem of Gun Barrel Erosion: An Overview”, is still highly relevant:
“Other forms of erosion/corrosion problems might appear as a result of
advances made in the development of liquid propellant guns and electromagnetic
gun technologies. In fact, the erosion in the latter could be quite severe, as
it involves interaction of high-temperature plasma with the launch surface
causing it to partially melt at each firing event. It will be necessary to
identify materials and design methodologies to minimize these problems.” Our
paper does not include any effect of the high-temperature plasma, but it shows
the effect of Joule heating and geometric design considerations on the rails.
## Acknowledgment
This work was sponsored in part by ILIR at the AMRDEC.
## References
* [1] For the status of electromagnetic launch technology see, ”2008 14th Symposium on Electromagnetic Launch Technology”, Proceedings of a meeting held 10-13 June 2008, Victoria, British Columbia, Canada, Institute of Electrical and Electronics Engineers (IEEE).
* [2] H. E. Knoepfel, Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use. New York: Wiley, 2000.
* [3] F. Young and W. Hughes, Rail and armature current distributions in electromagnetic launchers, IEEE Trans. Magn., vol. MAG-18, no. 1, pp. 33 41, Jan. 1982.
* [4] E. M. Drobyshevski, R. O. Kurakin, S. I. Rozov, B. G. Zhukov, M. V. Beloborodyy and V. G. Latypov, The importance of three dimensions in the study of solid armature transitions in railguns, J. Phys. D, Appl. Phys., vol. 32, no. 22, pp. 2910 2917, Nov. 1999.
* [5] F. Stefani, R. Merrill, and T.Watt, Numerical modeling of melt-wave erosion in two-dimensional block armatures, IEEE Trans. Magn., vol. 41, no. 1, pp. 437 441, Jan. 2005.
* [6] M. Schneider, R. Schneider, V. Stankevic, S. Balevicius, and N. Zurauskiene, Highly local measurements of strong transient magnetic fields during railgun experiments using CMR-based sensors, IEEE Trans. Magn., vol. 43, no. 1, pp. 370 375, Jan. 2007.
* [7] M. Schneider, O. Liebfried, V. Stankevic, S. Balevicius, and N. Zurauskiene, ”Magnetic Diffusion in Railguns: Measurements Using CMR-Based Sensors”, IEEE Trans. Magn., vol. 45, no. pp. 1, Jan. 2009.
* [8] W. A. Walls, W. F. Weldon, S. B. Pratap, M. Palmer, D. Adams, “Application of electromagnetic guns to future naval platforms”, IEEE Trans. Magn., vol. 35, no. 1, pp. 262-267, Jan. 1999.
* [9] I. R. McNab, F. C. Beach, “Naval Railguns”, IEEE Trans. Magn., vol. 43, no. 1, pp. 463-468, Jan. 2007.
* [10] I. Ahmad, “The Problem of Gun Barrel Erosion: An Overview” in L. Stiefel, ed., Gun Propulsion Technology, Vol. 109 of Progress in Astronautics and Aeronautics, AIAA,Washington DC, United States, chapter 10, pp. 311-355, (1988).
* [11] I. A. Johnston, Understanding and Predicting Gun Barrel Erosion, Published by Weapons Systems Division DSTO Defence Science and Technology Organisation, PO Box 1500, Edinburgh, South Australia, Australia 5111. Available as http://dspace.dsto.defence.gov.au/dspace/bitstream/1947/4091/1/DSTO-TR-1757PR.pdf.
* [12] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, “Electrodynamics of Continuous Media”, Pergamon Press, New York, 2nd Edition, 1984.
* [13] T. B. Bahder and J. D. Bruno, “Transient Response of an Electromagnetic Rail Gun: A Pedagogical Model”, Army Research Laboratory report No. ARL-TR-1663, May 1998. See http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA345008 or http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA345008&Location=U2&doc=GetTRDoc.pdf or http://handle.dtic.mil/100.2/ADA345008.
* [14] G.C. Long, “Railgun current density distributions”, IEEE Trans. Magn. vol. 22, no. 6, pp. 1597–1602, Nov. 1986.
* [15] J. C. Nearing and M. A. Huerta, ”Skin and heating effects of railgun current”, IEEE Trans. Magn., vol. 25, no. 1, pp. 381–386, Jan. 1989.
* [16] For a more recent treatment that uses a more complicated rail geometry, see , J. D. Powell and A. E. Zielinski, ”Two-Dimensional Current Diffusion in the Rails of a Railgun”, Army Research Laboratory Technical Report ARL-TR-4618, October 2008, see http://www.arl.army.mil/www/default.cfm?Action=17&Page=239&TRAction=GetAbstract&ReportID=1683&Topic=TechnicalReports.
* [17] See sections 3.8 through 3.12 and 4.6 to 4.7 of H. Knoepfel, “Pulsed High magnetic Fields”, North-Holland Publishing Company, London, 1970.
* [18] The first part of this assumption is conservative, since a deviation from ${\frac{{\partial{\kern 1.0pt}I}}{{\partial\,x}}}$ increases the total heating, somewhat compensating for the optimistic assumption that $\frac{{\partial{\kern 1.0pt}I}}{{\partial\,y}}=\infty$ or $I=0$ at the boundary, see later discussion.
* [19] W. C. McCorkle, “Compensated Pulse Alternators to Power Electromagnetic Railguns”, p. 364, 12th IEEE International Pulsed Power Conference, Monterey, California, 1999.
* [20] M. Abromowitz and I. A. Stegun, Dover Publications, Inc. New York, 9th printing, 1965.
|
arxiv-papers
| 2008-10-16T18:53:17
|
2024-09-04T02:48:58.287255
|
{
"license": "Public Domain",
"authors": "William C. McCorkle, Thomas B. Bahder",
"submitter": "Thomas B. Bahder",
"url": "https://arxiv.org/abs/0810.2985"
}
|
0810.3238
|
# Hypergeometric functions, their $\varepsilon$ expansions and Feynman
diagrams
M. Yu. Kalmykova,b, B.A. Kniehla, B.F.L. Wardc, S.A. Yostd 111e-mail:
kalmykov@theor.jinr.ru, kniehl@mail.desy.de, BFL Ward@baylor.edu,
scott.yost@citadel.edu
a II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
b Joint Institute for Nuclear Research, $141980$ Dubna (Moscow Region), Russia
c Department of Physics, Baylor University, One Bear Place, Waco, TX 76798,
USA
d Department of Physics, The Citadel, 171 Moultrie St., Charleston, SC 29409,
USA
###### Abstract
We review the hypergeometric function approach to Feynman diagrams. Special
consideration is given to the construction of the Laurent expansion. As an
illustration, we describe a collection of physically important one-loop vertex
diagrams for which this approach is useful.
1\. Introduction. Recent interest in the mathematical structure of Feynman
diagrams has been inspired by the persistently increasing accuracy of high-
energy experiments and the advent of the LHC epoch. For stable numerical
evaluation of diagrams, a knowledge of their analytical properties is
necessary. We will review some of the progress in this area, focusing on the
hypergeometric function representation of Feynman diagrams.
Forty-five years ago, Regge proposed [1] that any Feynman diagram can be
understood in terms of a special class of hypergeometric functions satisfying
some system of differential equations so that the singularity surface of the
relevant hypergeometric function coincides with the surface of the Landau
singularities [2] of the original Feynman diagram.222 For a review of
different approaches to the analysis of the singularities of Feynman diagrams
see Ref. [3]. Based on Regge’s conjecture, explicit systems of differential
equations for particular types of diagrams have been constructed. For some
examples, the hypergeometric representation for $N$-point one-loop diagrams
has been derived in Ref. [4] via a series representation (Appell functions and
Lauricella functions appear here), the system of differential equations and
its solution in terms of Lappo-Danilevsky functions [5] has been constructed
in Ref. [6], and the monodromy structure of some Feynman diagrams has been
studied in Ref. [7].
A review of results derived up to the mid-1970’s can be found in Ref. [8]. It
was known at that time that each Feynman diagram is a function of the “Nilsson
class.” This means that the Feynman diagram is a multivalued analytical
function in complex projective space ${{\mathbb{C}}{\mathbb{P}}^{n}}$. The
singularities of this function are described by Landau’s equation. Later,
Kashiwara and Kawai showed [9] that any regularized Feynman integral satisfies
some holonomic system of linear differential equations whose characteristic
variety is confined to the extended Landau variety.
The modern technology for evaluating Feynman diagrams is based mainly on
techniques which do not explicitly use properties of hypergeometric functions,
but are based on relationships among the Feynman diagrams derived from their
internal structure.333By “internal structure,” we mean any representation
described in standard textbooks, such as Ref. [10]. It was shown, for example,
that there are algebraic relations between dimensionally regularized [11]
Feynman diagrams with different powers of propagator [12]. Tarasov showed in
1996 that similar algebraic relations could also be found relating different
dimensions of the integral [13]. The Davydychev-Tarasov algorithm [13, 14]
allows a Feynman diagram with arbitrary numerator to be transformed into a
linear combination of diagrams of the original type with shifted powers of
propagators and space-time dimension, multiplied by a linear combination of
tensors constructed from the metric tensor and external momenta. This set of
algebraic relations is analogous to contiguous relations for hypergeometric
functions.444The full set of contiguous relations for generalized
hypergeometric functions ${}_{p}F_{q}$ is found in Ref. [15].
Solving the algebraic relations among Feynman diagrams allows them to be
expressed in terms of a restricted set called “master integrals.” Such a
solution is completely equivalent to the differential reduction of
hypergeometric functions [16, 17, 18]. The technique of describing Feynman
diagrams by a system of differential equations was further extended in Ref.
[19], where it was realized that the solution of the recurrence relations can
be used to close the system of differential equations for any Feynman diagram.
This led to useful techniques for evaluating diagrams [20, 21]. Most of the
progress to date in this type of analysis has been for diagrams related to the
“Fuchs” type of differential equation, with three regular singular points
[22]555The analysis of some diagrams with four regular singularities was done
recently in Ref. [23]..
Since Feynman diagrams are often UV- or IR-divergent, it is important to also
consider the construction of the Laurent expansion of dimensionally-
regularized diagrams about integral values of the dimension (typically
$d=4-2\varepsilon$). This is called an “$\varepsilon$ expansion” of the
diagram. For practical applications, we need the numerical values of the
coefficients of this expansion. Purely numerical approaches are under
development (e.g. Ref. [24]), but this is a complex problem for many realistic
diagrams having UV and IR singularities and several mass scales.
The case of one-loop Feynman diagrams has been studied the most. The
hypergeometric representations for N-point one-loop diagrams with arbitrary
powers of propagators and an arbitrary space-time dimension have been derived
for non-exceptional kinematics666“Non-exceptional kinematics” refers to the
case where all masses and momenta are non-zero and not proportional to each
other. by Davydychev in 1991 [25]. His approach is based on the Mellin-Barnes
technique [26]. The results are expressible in terms of hypergeometric
functions with one less variable than the number of kinematic invariants.
An alternative hypergeometric representation for one-loop diagrams has been
derived recently in Ref. [28], using a difference equation in the space-time
dimension. This approach has been applied only to a set of master
integrals777The hypergeometric representations of one-loop master integrals of
propagator and vertex type have been constructed in [26, 27]., but,
fortunately, an arbitrary $N$-point function can be reduced to the set of
master integrals analytically [29, 30]. In Ref. [28], the one-loop $N$-point
function was shown to be expressible in terms of hypergeometric functions of
$N\\!-\\!1$ variables. One remarkable feature of the derived results is a one-
to-one correspondence between arguments of the hypergeometric functions and
Gram and Cayley determinants, which are two of the main characteristics of
diagrams.
Beyond one loop, a general hypergeometric representation is available only for
sunset-type diagrams with arbitrary kinematics [31], with a simpler
representation for particular kinematics [32, 33]. In all other cases beyond
one loop, master integrals have been expressed in terms of hypergeometric
functions of type ${}_{p}F_{p-1}$ [34].
The program of constructing the analytical coefficients of the
$\varepsilon$-expansion is a more complicated matter. The finite parts of one-
loop diagrams in $d=4$ dimension are expressible in terms of the Spence
dilogarithm function [35]. However, only partial results for higher-order
terms in the $\varepsilon$-expansion are known at one loop. The all-order
$\varepsilon$-expansion of the one-loop propagator with an arbitrary values of
masses and external momentum has been constructed [37] in terms of Nielsen
polylogarithms [36]. The term linear in $\varepsilon$ for the one-loop vertex
diagram with non-exceptional kinematics has also been constructed in terms of
Nielsen polylogarithms [38]. It was shown in Ref. [39] that the all-order
$\varepsilon$ expansion for the one-loop vertex with non-exceptional
kinematics is expressible in terms of multiple polylogarithms of two variables
[40].
Beyond these examples, the situation is less complete. The term linear in
$\varepsilon$ for the box diagram is still under construction. Some cases for
particular masses888In Ref. [28], box diagrams have been written in terms of
the Lauricella-Saran function $F_{S}$ of three variables, and a one-fold
integral representation was established for their all-order $\varepsilon$
expansion. However, it is not proven that this representation coincides with
multiple polylogarithms. have been analyzed [41, 42]. Many physically
interesting particular cases have been considered beyond one loop. Among these
are the $\varepsilon$ expansion of massless propagator diagrams [43] and the
sunset diagram [44].
2\. Hypergeometric Functions. Let us recall that there are several different
ways to describe special functions: (i) as an integral of the Euler or Mellin-
Barnes type; (ii) by a series whose coefficients satisfy certain recurrence
relations; (iii) as a solution of a system of differential and/or difference
equations (holonomic approach). These approaches and interrelations between
them have been discussed in series of a papers [45]. In this section, we
review some essential definitions relevant for each of these representations.
* •
Integral representation: An Euler integral has the form
$\displaystyle\Phi(\vec{\alpha},\vec{\beta},P)=\int_{\Sigma}\Pi_{i}P_{i}(x_{1},\cdots,x_{k})^{\beta_{i}}x_{1}^{\alpha_{1}}\cdots
x_{k}^{\alpha_{k}}dx_{1}\cdots dx_{k}\;,$ (1)
where $P_{i}$ is some Laurent polynomial with respect to variables
$x_{1},\cdots,x_{k}$: $P_{i}(x_{1},\cdots,x_{k})=\sum
c_{\omega_{1}\cdots\omega_{k}}x_{1}^{\omega_{1}}\ldots x_{k}^{\omega_{k}}$,
with $\omega_{j}\in\mathbb{Z}$, and $\alpha_{i},\beta_{j}\in\mathbb{C}.$ We
assume that the region $\Sigma$ is chosen such that the integral exists.
A Mellin-Barnes integral has the form
$\displaystyle\Phi\left(a_{js},b_{kr},c_{i},d_{j},\gamma,\vec{x}\right)=\int_{\gamma+i\mathbb{R}}dz_{1}\ldots
dz_{m}\frac{\Pi_{j=1}^{p}\Gamma\left(\sum_{s=1}^{m}a_{js}z_{s}+c_{j}\right)}{\Pi_{k=1}^{q}\Gamma\left(\sum_{r=1}^{m}b_{kr}z_{r}+d_{k}\right)}x_{1}^{-z_{1}}\ldots
x_{m}^{-z_{m}}\;,$ (2)
where $a_{js},b_{kr},c_{i},d_{j}\in\mathbb{R},\ \alpha_{k}\in\mathbb{C},$ and
$\gamma$ is chosen such that the integral exists. This integral can be
expressed in terms of a sum of the residues of the integrated expression.
* •
Series representation: We will take the Horn definition of the series
representation. In accordance with this definition, a formal (Laurent) power
series in $r$ variables,
$\displaystyle\Phi(\vec{x})=\sum
C(\vec{m})\vec{x}^{m}\equiv\sum_{m_{1},m_{2},\cdots,m_{r}}C(m_{1},m_{2},\cdots,m_{r})x_{1}^{m_{1}}\cdots
x_{r}^{m_{r}},$ (3)
is called hypergeometric if for each $i=1,\cdots,r$ the ratio
$C(\vec{m}+\vec{e}_{i})/C(\vec{m})$ is a rational function999A “rational
function” is any function which can be written as the ratio of two polynomial
functions. in the index of summation: $\vec{m}=(m_{1},\cdots,m_{r})$, where
$\vec{e}_{j}=(0,\cdots,0,1,0,\cdots,0),$ is unit vector with unity in the
$j^{\rm th}$ place. Ore and Sato [46] found that the coefficients of such a
series have the general form
$\displaystyle
C(\vec{m})=\Pi_{i=1}^{r}\lambda_{i}^{m_{i}}R(\vec{m})\Biggl{(}\Pi_{j=1}^{N}\Gamma(\mu_{j}(\vec{m})+\gamma_{j}+1)\Biggr{)}^{-1}\;,$
(4)
where $N\geq 0,$ $\lambda_{j},\gamma_{j}\in\mathbb{C}$ are arbitrary complex
numbers, $\mu_{j}:\mathbb{Z}^{r}\to\mathbb{Z}$ are arbitrary linear maps, and
$R$ is an arbitrary rational function. The fact that all the $\Gamma$ factors
are in the denominator is inessential: using the relation
$\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$, they can be converted to factors in
the numerator. A series of this type is called a “Horn-type” hypergeometric
series. In this case, the system of differential equations has the form
$Q_{j}\left(\sum_{k=1}^{r}x_{k}\frac{\partial}{\partial
x_{k}}\right)\frac{1}{x_{j}}\Phi(\vec{x})=P_{j}\left(\sum_{k=1}^{r}x_{k}\frac{\partial}{\partial
x_{k}}\right)\Phi(\vec{x})\;,\quad j=1,\cdots,r,$ (5)
where $P_{j}$ and $Q_{r}$ are polynomials satisfying
$\frac{C(\vec{m}+e_{j})}{C(\vec{m})}=\frac{P_{j}(\vec{m})}{Q_{j}(\vec{m})}.$
(6)
* •
Holonomic representation: A combination of differential and difference
equations can be found to describe functions of the form
$\displaystyle\Phi(\vec{z},\vec{x},W)=\sum_{k_{1},\cdots,k_{r}=0}^{\infty}\left(\Pi_{a=1}^{m}\frac{1}{z_{a}+\sum_{b=1}^{r}W_{ab}k_{j}}\right)\Pi_{j=1}^{r}\frac{x_{j}^{k_{j}}}{k_{j}!}\;,$
(7)
where $W$ is an $r\times m$ matrix. In particular, this function satisfies the
equations
$\displaystyle\frac{\partial\Phi(\vec{z},\vec{x},W)}{\partial
x_{j}}=\Phi(\vec{z}+\omega_{j},x,W)\;,\quad j=1,\cdots,r,$ (8)
$\displaystyle\frac{\partial}{\partial
z_{i}}\left(z_{i}\Phi+\sum_{j=1}^{r}W_{i}x_{j}\frac{\partial\Phi}{\partial
x_{j}}\right)=0\;,\quad i=1,\cdots,m,$ (9)
where $\omega_{j}$ is the $j^{\rm th}$ column of the matrix $W$.
3\. Construction of the all-order $\varepsilon$ expansion of hypergeometric
functions. Recently, several theorems have been proven on the all-order
$\varepsilon$ expansion of hypergeometric functions about integer and/or
rational values of parameters [33, 37, 47, 48, 49, 50, 51, 52]. For
hypergeometric functions of one variable, all three of the representations
(i)–(iii) described in the previous section are equivalent, but some
properties of the function may be more evident in one representation than
another.
In the Euler integral representation, the most important results are related
to the construction of the all-order $\varepsilon$ expansion of Gauss
hypergeometric function with special values of parameters in terms of Nielsen
polylogarithms [37]. There are several important master integrals expressible
in terms of this type of hypergeometric function, including one-loop
propagator-type diagrams with arbitrary values of mass and momentum [26], two-
loop bubble diagrams with arbitrary values of masses, and one-loop massless
vertex diagrams with three non-zero external momenta [53].
The series representation (ii) is an intensively studied approach. The first
results of this type were derived in the context of the so-called “single-
scale” diagrams [54] related to multiple harmonic sums. These results have
been extended to the case of multiple (inverse) binomial sums [57] that
correspond to the $\varepsilon$-expansion of hypergeometric functions with one
unbalanced half-integer parameter and values of argument equal to $1/4$, or
diagrams with two massive-particle cuts. Particularly impressive results
involving series representations were derived in the framework of the nested-
sum approach for hypergeometric functions with a balanced set of parameters in
Refs. [47, 48], 101010Computer realizations of nested sums approach to
expansion of hypergeometric functions are given in [55, 56]. and in framework
of the generating-function approach for hypergeometric functions with one
unbalanced set of parameters in Refs. [33, 51, 58, 59].
An approach using the iterated solution of differential equations has been
explored in Refs. [33, 49, 50, 52]. One of the advantages of the iterated-
solution approach over the series approach is that it provides a more
efficient way to calculate each order of the $\varepsilon$ expansion, since it
relates each new term to the previously derived terms, rather than having to
work with an increasingly large collection of independent sums at each order.
This technique includes two steps: (i) the differential-reduction algorithm
(to reduce a generalized hypergeometric function to basic functions); (ii)
iterative solution of the proper differential equation for the basic functions
(equivalent to iterative algorithms for calculating the analytical
coefficients of the $\varepsilon$ expansion).
An important tool for constructing the iterative solution is the iterated
integral defined as
$I(z;a_{k},a_{k-1},\ldots,a_{1})=\int_{0}^{z}\frac{dt}{t-a_{k}}I(t;a_{k-1},\ldots,a_{1})\;,$
where we assume that all $a_{j}\neq 0$. A special case of this integral,
$G_{m_{k},m_{k-1},\ldots,m_{1}}(z;a_{k},\ldots,a_{1})\equiv
I(z;\underbrace{0,\ldots,0}_{m_{k}-1\mbox{
times}},a_{k},\underbrace{0,\ldots,0}_{m_{k-1}-1\mbox{
times}},a_{k-1},\cdots,\underbrace{0,\ldots,0}_{m_{1}-1\mbox{
times}},a_{1})\;,$
where all $a_{k}\neq 0$, is related to the multiple polylogarithm [40, 61]
${\mbox{Li}}_{k_{1},k_{2},\ldots,k_{n}}\left(x_{1},x_{2},\ldots,x_{n}\right)=\sum_{m_{n}>m_{n-1}>\cdots>m_{2}>m_{1}>0}^{\infty}\frac{x_{1}^{m_{1}}}{m_{1}^{k_{1}}}\frac{x_{2}^{m_{2}}}{m_{2}^{k_{2}}}\times\cdots\times\frac{x_{n}^{m_{n}}}{m_{n}^{k_{n}}}$
(10)
by
$\displaystyle
G_{m_{n},m_{n-1},\ldots,m_{1}}\left(z;x_{n},x_{n-1},\ldots,x_{1}\right)=(-1)^{n}{\mbox{Li}}_{m_{1},m_{2},\ldots,m_{n-1},m_{n}}\left(\frac{x_{2}}{x_{1}},\frac{x_{3}}{x_{2}},\ldots,\frac{x_{n}}{x_{n-1}},\frac{z}{x_{n}}\right)\;,$
$\displaystyle{\mbox{Li}}_{k_{1},k_{2},\ldots,k_{n-1},k_{n}}\left(y_{1},y_{2},\ldots,y_{n-1},y_{n}\right)=(-1)^{n}G_{k_{n},k_{n-1},\ldots,k_{2},k_{1}}\left(1;\frac{1}{y_{n}},\ldots,\frac{1}{y_{n}\times\cdots\times
y_{1}}\right)\;.$
In Eq. (10), $k=k_{1}+k_{2}+\cdots+k_{n}$ is called the “weight” and $n$ the
“depth.” Multiple polylogarithms (10) are defined for $|z_{n}|<1$ and
$|z_{i}|\leq 1(i=1,.\cdots,n\\!-\\!1)$ and for $|z_{n}|\leq 1$ if $m_{n}\leq
2$. We mention also that multiple polylogarithms form two Hopf algebras. One
is related to the integral representation, and the other one to the series.
A particular case of the multiple polylogarithm is the “generalized
polylogarithm” defined by
${\mbox{Li}}_{k_{1},k_{2},\ldots,k_{n}}\left(z\right)=\sum_{m_{n}>m_{n-1}>\cdots>m_{1}>0}^{\infty}\frac{z^{m_{n}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{n}^{k_{n}}}={\mbox{Li}}_{k_{1},k_{2},\ldots,k_{n}}\left(1,1,\cdots,1,z\right)\;,$
(11)
where $|z|<1$ when all $k_{i}\geq 1$, or $|z|\leq 1$ when $k_{n}\leq 2$.
Another particular case is a “multiple polylogarithm of a square root of
unity,” defined as
${\mbox{Li}}_{\left(\sigma_{1},\sigma_{2},\cdots,\sigma_{n}\atop
s_{1},s_{2},\cdots,s_{n}\right)}\left(z\right)=\sum_{m_{n}>m_{n-1}>\cdots
m_{1}>0}z^{m_{n}}\frac{\sigma_{n}^{m_{n}}\cdots\sigma_{1}^{m_{1}}}{m_{n}^{s_{n}}\cdots
m_{1}^{s_{1}}}\;.$ (12)
where $\vec{s}=(s_{1},\cdots s_{n})$ and
$\vec{\sigma}=(\sigma_{1},\cdots,\sigma_{n})$ are multi-indices and
$\sigma_{k}$ belongs to the set of the square roots of unity, $\sigma_{k}=\pm
1$. This particular case of multiple polylogarithms has been analyzed in
detail by Remiddi and Vermaseren [62]111111As was pointed out by Goncharov
[40], the iterated integral as a function of the variable $z$ has been studied
by Kummer, Poincare, and Lappo-Danilevky, and was called a hyperlogarithm.
Goncharov [40] analyzed it as a multivalued analytical function of
$a_{1},\ldots,a_{k},z$. From this point of view, only the functions considered
in Ref. [63] are multiple polylogarithms of two variables..
Special consideration is necessary when the last few arguments
$a_{k-j},a_{k-j-1},\ldots,a_{k}$ in the integral $I(z;a_{1},\cdots,a_{k})$ are
equal to zero, which is called the “trailing-zero” case. It is possible to
factorize such a function into a product of a power of a logarithm and a
multiple polylogarithm. An appropriate procedure for multiple polylogarithms
of a square root of unity was described in Ref. [62] and extended to the case
of multiple polylogarithms in Ref. [64]. For the numerical evaluation of
multiple polylogarithms or its particular cases, see Ref. [64, 65]. Let us
consider the Laurent expansion of a generalized hypergeometric functions of
one variable ${}_{p}F_{p-1}(\vec{A};\vec{B};z)$ with respect to its
parameters. Such an expansion can be written as
$\displaystyle{}_{p}F_{p-1}(\vec{A};\vec{B};z)={}_{p}F_{p-1}(\vec{A_{0}};\vec{B_{0}};z)$
$\displaystyle+\sum_{m_{i},l_{j}=1}^{\infty}\Pi_{i=1}^{p}\Pi_{j=1}^{p-1}\frac{(A_{i}\\!-\\!A_{0i})^{m_{i}}}{m_{i}!}\frac{(B_{j}\\!-\\!B_{0j})^{l_{j}}}{l_{j}!}\left.\left(\frac{\partial}{\partial
A_{i}}\right)^{m_{i}}\left(\frac{\partial}{\partial
B_{j}}\right)^{l_{j}}{}_{p}F_{p-1}(\vec{A};\vec{B};z)\right|_{\begin{smallmatrix}A_{i}=A_{0i}\\\
B_{j}=B_{0j}\end{smallmatrix}}$
$\displaystyle={}_{p}F_{p-1}(\vec{A_{0}};\vec{B_{0}};z)+\sum_{m_{i},l_{j}=1}\Pi_{i=1}^{p}\Pi_{j=1}^{p-1}(A_{i}-A_{0i})^{m_{i}}(B_{j}-B_{0j})^{l_{j}}L_{\vec{A},{\vec{B}}}(z)\;,$
(13)
where ${}_{p}F_{p-1}(\vec{A};\vec{B};z)$ is a hypergeometric function defined
by
${}_{p}F_{p-1}(\vec{A};\vec{B};z)\\!=\\!\sum_{j=0}^{\infty}\frac{\Pi_{i=1}^{p}(A_{i})_{j}}{\Pi_{k=1}^{p-1}(B_{k})_{j}}\frac{z^{j}}{j!}\;$
and $(A)_{j}$ is the Pochhammer symbol: $(A)_{j}={\Gamma(A+j)}/{\Gamma(A)}$.
Our goal is to completely describe the coefficients $L_{\vec{A},{\vec{B}}}(z)$
entering the r.h.s. of Eq. (13). To reach this goal, we must first
characterize the complete set of parameters for which known special functions
suffice to express the coefficients. Beyond this, we wish to identity the
complete set of new functions which must be invented in order to express all
of the coefficients in the Laurent expansion.
The first simplification comes from the well-known fact that any
hypergeometric function ${}_{p}F_{p-1}(\vec{A}+\vec{m};\vec{B}+\vec{k};z)$ may
be expressed in terms of $p$ other functions of the same type:
$\displaystyle
R_{p+1}(\vec{A},\vec{B},z){}_{p}F_{p-1}(\vec{A}+\vec{m};\vec{B}+\vec{k};z)=\sum_{j=1}^{p}R_{j}(\vec{A},\vec{B},z){}_{p}F_{p-1}(\vec{A}+\vec{e_{k}};\vec{B}+\vec{E_{k}};z)\;,$
(14)
where $\vec{m},\vec{k},\vec{e}_{k}$, and $\vec{E}_{k}$ are lists of integers,
and the $R_{k}$ are polynomials in the parameters $\vec{A},\vec{B}$, and $z$.
In particular, we can take the function and its first $p\\!-\\!1$ derivatives
as a basis for the reduction (see Ref. [16] for the details of this approach).
Then Eq. (14) will take the form121212For simplicity, we will assume that no
difference $B_{k}-A_{j}$ is a positive integer.
$\displaystyle\widetilde{R}_{p+1}(\vec{A},\vec{B},z){}_{p}F_{p-1}(\vec{A}+\vec{m};\vec{B}+\vec{k};z)=\sum_{k=1}^{p}\widetilde{R}_{k}(\vec{A},\vec{B},z)\left(\frac{d}{dz}\right)^{k-1}{}_{p}F_{p-1}(\vec{A};\vec{B};z)\;,$
(15)
with a new polynomial $\widetilde{R}_{k}$. In this way, the problem of finding
the Laurent expansion of the original hypergeometric function is reduced to
the analysis of a set of basic functions and the Laurent expansion of a
(formally) known polynomial.
As is well known, hypergeometric functions satisfy the differential
equation131313 This equation follows from Eqs. (5) – (6), where
$P(j)=\Pi_{k=1}^{p}(A_{k}+j)$ and $Q(j)=(j+1)\Pi_{k=1}^{p-1}(B_{k}+j)$.
$\displaystyle\left[z\Pi_{i=1}^{p}\left(z\frac{d}{dz}\\!+\\!A_{i}\right)\\!-\\!z\frac{d}{dz}\Pi_{k=1}^{p-1}\left(z\frac{d}{dz}\\!+\\!B_{k}\\!-\\!1\right)\right]{}_{p}F_{p-1}(\vec{A};\vec{B};z)=0.$
(16)
Due to the analyticity of the hypergeometric function
${}_{p}F_{p-1}(\vec{A};\vec{B};z)$ with respect to its parameters
$A_{i},B_{k}$, the differential equation for the coefficients
$L_{\vec{A},{\vec{B}}}(z)$ of the Laurent expansion could be directly derived
from Eq. (16) without any reference to the series or integral representation.
This was the main idea of the approach developed in Refs. [33, 49, 50, 52,
60]. An analysis of this system of equations and/or their explicit analytical
solution gives us the analytical form of $L_{\vec{A},{\vec{B}}}(z)$. It is
convenient to introduce a new parametrization, $A_{i}\to
A_{0i}+a_{i}\varepsilon,B_{j}\to B_{0i}+b_{i}\varepsilon\;,$ where
$\varepsilon$ is some small number, so that the Laurent expansion (13) takes
the form of an “$\varepsilon$ expansion,”
${}_{p}F_{p-1}(\vec{A}+\vec{a}\varepsilon;\vec{B}+\vec{b}\varepsilon;z)={}_{p}F_{p-1}(\vec{A};\vec{B};z)+\sum_{k=1}^{\infty}\varepsilon^{k}L_{\vec{a},\vec{b},k}(z)\equiv\sum_{k=0}^{\infty}\varepsilon^{k}L_{\vec{a},\vec{b},k}(z)\;,$
where $L_{\vec{a},\vec{b},0}(z)={}_{p}F_{p-1}(\vec{A};\vec{B};z)$. The
differential operator can also be expanded in powers of $\varepsilon$:
$\displaystyle
D^{(p)}=\left[\Pi_{i=1}^{p}\left(\theta\\!+\\!A_{i}\\!+\\!a_{i}\varepsilon\right)\\!-\\!\frac{1}{z}\theta\Pi_{k=1}^{p-1}\left(\theta\\!+\\!B_{k}\\!-\\!1\\!+\\!b_{k}\varepsilon\right)\right]=\sum_{j=0}^{p}\varepsilon^{j}D_{j}^{(p-j)}(\vec{A},\vec{B},\vec{a},\vec{b},z)\;,$
(17)
where $\theta=z\frac{d}{dz}\;,$ the upper index gives the order of the
differential operator, $D_{p}^{(0)}=\Pi_{k=1}^{p}a_{k}\;,$ and
$\displaystyle D_{0}^{(p)}$ $\displaystyle=$
$\displaystyle\Pi_{i=1}^{p}\left(\theta\\!+\\!A_{i}\right)\\!-\\!\frac{1}{z}\theta\Pi_{k=1}^{p-1}\left(\theta\\!+\\!B_{k}\\!-\\!1\right)$
$\displaystyle=$
$\displaystyle\left\\{-(1\\!-\\!z)\frac{d}{dz}\\!+\\!\sum_{k=1}^{p}A_{k}\\!-\\!\frac{1}{z}\sum_{j=1}^{p-1}(B_{j}\\!-\\!1)\right\\}\theta^{p-1}\\!+\\!\sum_{j=1}^{p-1}\left[X_{j}(\vec{A},\vec{B})\\!-\\!\frac{1}{z}Y_{j}(\vec{A},\vec{B})\right]\theta^{p\\!-\\!1\\!-\\!j}\;,$
where $X_{j}(\vec{A},\vec{B})$ and $Y_{j}(\vec{A},\vec{B})$ are polynomials.
Combining all of the expansions together, we obtain a system of equations
$\sum_{r=0}^{\infty}\varepsilon^{r}\sum_{j=0}^{p}D_{j}^{(p-j)}L_{\vec{a},\vec{b},r-j}(z)=0\;,$
which could be split into following system (each order of $\varepsilon$):
$(\varepsilon^{0})~{}D_{0}^{(p)}L_{\vec{a},\vec{b},0}(z)=0\;;$
$(\varepsilon^{k},1\leq k\leq
p)~{}\sum_{r=0}^{k}D_{k}^{(p-k)}L_{\vec{a},\vec{b},k-r}(z)=0\;;$
$(\varepsilon^{k},k\geq
p+1)~{}\sum_{j=0}^{p}D_{j}^{(p-j)}L_{\vec{a},\vec{b},k-j}(z)=0\;.$ Further
simplification comes from the explicit forms of $D_{k}^{(p-k)}$ and the
polynomials $X_{j}(\vec{A},\vec{B}),Y_{j}(\vec{A},\vec{B})$ in Eq.
(Hypergeometric functions, their $\varepsilon$ expansions and Feynman
diagrams). For example, for integer values of parameters, we can put
$A_{k}=0,B_{j}=1$, so that all of the $X_{j}(\vec{A},\vec{B})$ and
$Y_{j}(\vec{A},\vec{B})$ are equal to zero. Further details can be found in
our papers, Refs. [33, 50, 51, 52, 60].
Here, we will mention some of the existing results. 141414 In the following,
we will assume that $a,b,c$ are an arbitrary numbers and $\varepsilon$ is a
small parameter.
* •
If $I_{1},I_{2},I_{3}$ are arbitrary integers, the Laurent expansions of the
Gauss hypergeometric functions
$\displaystyle{}_{2}F_{1}(I_{1}+a\varepsilon,I_{2}+b\varepsilon;I_{3}+\tfrac{p}{q}+c\varepsilon;z)\;,\quad{}_{2}F_{1}(I_{1}+\tfrac{p}{q}+a\varepsilon,I_{2}+\tfrac{p}{q}+b\varepsilon;I_{3}+\tfrac{p}{q}+c\varepsilon;z)\;,$
$\displaystyle{}_{2}F_{1}(I_{1}+\tfrac{p}{q}+a\varepsilon,I_{2}+b\varepsilon;I_{3}+c\varepsilon;z)\;,\quad{}_{2}F_{1}(I_{1}+\tfrac{p}{q}+a\varepsilon,I_{2}+b\varepsilon;I_{3}+\tfrac{p}{q}+c\varepsilon;z)$
are expressible in terms of multiple polylogarithms of arguments being powers
of $q$-roots of unity and a new variable, that is an algebraic function of
$z$, with coefficients that are ratios of polynomials.
* •
If $\vec{A},\vec{B}$ are lists of integers and $I,p,q$ are integers, the
Laurent expansions of the generalized hypergeometric functions
${}_{p}F_{p-1}(\vec{A}+\vec{a}\varepsilon,\tfrac{p}{q}+I;\vec{B}+\vec{b}\varepsilon;z)\;,\quad{}_{p}F_{p-1}(\vec{A}+\vec{a}\varepsilon;\vec{B}+\vec{b}\varepsilon,\tfrac{p}{q}+I;z)$
are expressible in terms of multiple polylogarithms of arguments that are
powers of $q$-roots of unity and a new variable that is an algebraic function
of $z$, with coefficients that are ratios of polynomials.
* •
If $\vec{A},\vec{B}$ are lists of integers, the Laurent expansion of the
generalized hypergeometric function
${}_{p}F_{p-1}(\vec{A}+\vec{a}\varepsilon;\vec{B}+\vec{b}\varepsilon;z)$
are expressible via generalized polylogarithms (11).
We should also mention the following case [48] in which the $\varepsilon$
expansion has been constructed via the nested sum approach:
If $p,q,I_{k}$ are any integers and $\vec{A},\vec{B}$ are lists of integers,
the generalized hypergeometric function
${}_{p}F_{p-1}(\\{\tfrac{p}{q}\\!+\\!\vec{A}\\!+\\!\vec{a}\varepsilon\\}_{r},\vec{I_{1}}\\!+\\!\vec{c}\varepsilon;\\{\tfrac{p}{q}\\!+\\!\vec{B}\\!+\\!\vec{b}\varepsilon\\}_{r},\vec{I_{2}}\\!+\\!\vec{d}\varepsilon;z)\;$
is expressible in terms of multiple polylogarithms of arguments that are
powers of $q$-roots of unity and the new variable $z^{1/q}$, with coefficients
that are ratios of polynomials. A hypergeometric function of this form is said
to have a zero-balance set of parameters.
We will now demonstrate some algebraic relations between functions generated
by the $\varepsilon$ expansion of hypergeometric functions with special sets
of parameters. Let us consider the analytic continuation of the generalized
hypergeometric function $~{}_{p+1}F_{p}$ with respect to the transformation
$z\to{1}/{z}$ [34]:
$\displaystyle\left(\Pi_{j=1}^{p}\frac{1}{\Gamma(b_{j})}\right)~{}_{p+1}F_{p}\left(\begin{array}[]{c|}a_{1},a_{2},\cdots,a_{p+1}\\\
b_{1},b_{2},\cdots,b_{p}\end{array}~{}z\right)=\sum_{k=1}^{p+1}\frac{\Pi_{j=1,j\neq
k}^{p+1}\Gamma(a_{j}\\!-\\!a_{k})}{\left(\Pi_{j=1,j\neq
k}^{p+1}\Gamma(a_{j})\right)\left(\Pi_{j=1}^{p}\Gamma(b_{j}\\!-\\!a_{k})\right)}$
(21) $\displaystyle\hskip
14.22636pt\times(-z)^{-a_{k}}~{}_{p+2}F_{p+1}\left(\begin{array}[]{c|}1,a_{k},1\\!+\\!a_{k}\\!-\\!b_{1},1\\!+\\!a_{k}\\!-\\!b_{2},\cdots,1\\!+\\!a_{k}\\!-\\!b_{p}\\\
1\\!+\\!a_{k}\\!-\\!a_{1},1\\!+\\!a_{k}\\!-\\!a_{2},\cdots,1\\!+\\!a_{k}\\!-\\!a_{p+1}\end{array}~{}\frac{1}{z}\right)\;,$
(24)
where none of the differences between pairs of parameters $a_{j}-a_{k}$ is an
integer.
On the r.h.s. of Eq. (24), we actually have a hypergeometric function
$~{}_{p+1}F_{p}$, since one of the parameters is always equal to unity. If we
make the replacements
$a_{j}\to\frac{r}{q}+a_{j}\varepsilon\;,\quad
b_{j}\to\frac{r}{q}+b_{j}\varepsilon$
in Eq. (24), we obtain the relation
$\displaystyle~{}_{p+1}F_{p}\left(\begin{array}[]{c|}\left\\{\frac{r}{q}+a_{j}\varepsilon\right\\}_{p+1}\\\
\left\\{\frac{r}{q}+b_{j}\varepsilon\right\\}_{p}\end{array}~{}z\right)=\sum_{s=1}^{p}c_{s}~{}_{p+1}F_{p}\left(\begin{array}[]{c|}\frac{r}{q}+\tilde{c}\varepsilon,\left\\{1+\tilde{a}_{j}\varepsilon\right\\}_{p}\\\
\left\\{1+\tilde{b}_{j}\varepsilon\right\\}_{p}\end{array}~{}\frac{1}{z}\right)\;,$
(29)
where the $c_{r}$ are constants. Another relation follows if we choose in Eq.
(24) the following set of parameters:
$a_{j}\to a_{j}\varepsilon\;,\quad j=1,\cdots,p+1\;,\quad b_{k}\to
b_{k}\varepsilon\;,\quad k=1,\cdots,p-1\;,\quad
b_{p}=\frac{r}{q}+b_{p}\varepsilon\;.$
Then we have
$\displaystyle~{}_{p+1}F_{p}\left(\begin{array}[]{c|}\left\\{a_{j}\varepsilon\right\\}_{p+1}\\\
\left\\{b_{j}\varepsilon\right\\}_{p-1},\frac{r}{q}+b_{p}\varepsilon\end{array}~{}z\right)=\sum_{s=1}^{p}\tilde{c}_{s}~{}_{p+1}F_{p}\left(\begin{array}[]{c|}1-\frac{r}{q}-\tilde{c}\varepsilon,\left\\{1+\tilde{a}_{j}\varepsilon\right\\}_{p}\\\
\left\\{1+\tilde{b}_{j}\varepsilon\right\\}_{p}\end{array}~{}\frac{1}{z}\right)\;,$
(34)
where the $\tilde{c}$ are constants. In this way, we find a proof of the
following result:
Lemma: When none of the difference between two upper parameters is an
integer, and the differences between any lower and upper parameters are
positive integers, the coefficients of the $\varepsilon$ expansion of the
hypergeometric functions
$~{}_{p+1}F_{p}\left(\begin{array}[]{l|}\vec{A}\\!+\\!\tfrac{r}{q}\\!+\\!\vec{a}\varepsilon\\\
\vec{B}\\!+\\!\tfrac{r}{q}\\!+\\!\vec{b}\varepsilon\end{array}~{}z\right)\;,~{}_{p+1}F_{p}\left(\begin{array}[]{c|}\vec{A}\\!+\\!\vec{a}\varepsilon\\\
\vec{B}\\!+\\!\vec{b}\varepsilon,I\\!+\\!\tfrac{r}{q}\\!+\\!c\varepsilon\end{array}~{}z\right)\;,~{}_{p+1}F_{p}\left(\begin{array}[]{c|}I\\!+\\!\tfrac{r}{q}\\!+\\!c\varepsilon,\vec{A}\\!+\\!\vec{a}\varepsilon\\\
\vec{B}\\!+\\!\vec{b}\varepsilon\end{array}~{}z\right)\;,$
where $\vec{A},\vec{B},\vec{a},\vec{b},c$ and $I$ are all integers, are
related to each other.
Note that none of the functions of this lemma belongs to the zero-balance
case.
4\. One-loop vertex as hypergeometric function. Let us consider now the one-
loop vertex diagram. We recall that any one-loop vertex diagram with the
arbitrary masses, external momenta and power of propagators can be reduced by
recurrence relations to a vertex-type master integral (with all powers of
propagators being equal to unity) or, in the case of zero Gram and/or Cayley
determinants, to a linear combination of propagator-type diagrams [29]. In the
case of non-zero Gram and/or Cayley determinants, the one-loop master
integrals are expressible in terms of linear combinations of two Gauss
hypergeometric functions and the Appell function $F_{1}$ [27, 28].
Figure 1: One-loop vertex-type diagrams expressible in terms of generalized
hypergeometric functions. Bold and thin lines correspond to massive and
massless propagators, respectively.
Our starting point is the hypergeometric representation for one-loop diagrams
with three arbitrary external momenta and one massive line or two or three
massive lines with an equal masses, derived in Ref. [26]. Let us consider a
one-loop vertex-type diagram, as shown in Fig. 1. Using properties of
functions of several variables [34, 67], these diagrams can be expressed in
terms of hypergeometric functions of one variable151515We are indebted to A.
Davydychev for assistance in that analysis., whose $\varepsilon$ expansions up
to weight 4 are presented in Ref. [56, 59, 66] 161616This is enough for the
calculation of two-loop corrections. and available via the web [70]. We recall
that up to weight 4, the $\varepsilon$ expansions of all master integrals
collected here are expressible in terms of Nielsen polylogarithms only. The
hypergeometric representations have been derived also in [68] for $C_{1}$ and
$C_{2}$, in [28, 67] for $C_{6}$ and in [26] for $C_{11}$. Up to the finite
part, some of these diagrams have been studied in [69]. For certain diagrams
($C_{4},C_{6},C_{9},C_{10},C_{11}$), the $\varepsilon$ expansion of the first
several coefficients was given in Ref. [42] in terms of multiple
polylogarithms of two variables. We use the notations
$j_{123}\\!=\\!j_{1}\\!+\\!j_{2}\\!+\\!j_{3}$,
$j_{mn}\\!=\\!j_{m}\\!+\\!j_{n}$ below.
We will conclude with a review of the results for special cases:
* •
The massless triangle diagram with one massless external on-shell momentum is
expressible in terms of two Gauss hypergeometric functions. This result
follows directly from a relation in Ref. [26]. The Cayley determinant vanishes
in this case.
* •
The analytical result for diagram $C_{1}$ with arbitrary powers of the
propagators is expressible in terms of a Gauss hypergeometric function with
one integer upper parameter:
$\frac{C_{1}}{i^{1-n}\pi^{n/2}}=(-m^{2})^{n/2\\!-\\!j_{123}}\frac{\Gamma\left(j_{123}\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!j_{13}\right)}{\Gamma\left(\frac{n}{2}\right)\Gamma\left(j_{2}\right)}\;{}_{2}F_{1}\left(\begin{array}[]{c|}j_{123}\\!-\\!\tfrac{n}{2},j_{1}\\\
\tfrac{n}{2}\end{array}~{}\frac{Q^{2}}{m^{2}}\right)\;.$
The differential reduction will result in one Gauss hypergeometric function.
The Cayley determinant vanishes for $C_{1}$.
* •
The diagram $C_{2}$ with arbitrary powers of propagators is expressible in
terms of two hypergeometric functions ${}_{3}F_{2}$. In this case, both the
Gram and Cayley determinants are nonzero, and the master integral is
$\displaystyle\frac{C_{2}}{i\pi^{{n}/{2}}}=-(m^{2})^{\tfrac{n}{2}-3}\Biggl{\\{}\frac{\Gamma\left(3\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!2\right)}{\Gamma\left(\frac{n}{2}\right)}\;{}_{2}F_{1}\left(\begin{array}[]{c|}1,1\\\
\tfrac{n}{2}\end{array}~{}-\frac{Q^{2}}{m^{2}}\right)$ (37)
$\displaystyle\hskip
14.22636pt+\left(-\frac{Q^{2}}{m^{2}}\right)^{\tfrac{n}{2}-2}\frac{\Gamma^{2}\left(\frac{n}{2}\\!-\\!1\right)\Gamma\left(2\\!-\\!\frac{n}{2}\right)}{\Gamma\left(n\\!-\\!2\right)}\;{}_{2}F_{1}\left(\begin{array}[]{c|}1,\tfrac{n}{2}-1\\\
n-2\end{array}~{}-\frac{Q^{2}}{m^{2}}\right)\Biggr{\\}}\;.$ (40)
* •
For diagram $C_{3}$, the result for arbitrary powers of propagators is
expressible in terms of the function ${}_{3}F_{2}$. Both the Gram and Cayley
determinants are nonzero, and the master integral is a combination of two
Gauss hypergeometric functions:
$\displaystyle\frac{C_{3}}{i\pi^{{n}/{2}}}=-(m^{2})^{\tfrac{n}{2}-3}\frac{\Gamma\left(\frac{n}{2}\\!-\\!2\right)}{\Gamma\left(n\\!-\\!3\right)}\Biggl{\\{}\frac{\Gamma\left(n\\!-\\!4\right)}{\Gamma\left(\frac{n}{2}\\!-\\!1\right)}\;{}_{2}F_{1}\left(\begin{array}[]{c|}1,1\\\
5-n\end{array}~{}\frac{Q^{2}}{m^{2}}\right)$ (43) $\displaystyle\hskip
14.22636pt+\left(-\frac{Q^{2}}{m^{2}}\right)^{\tfrac{n}{2}-2}\frac{\Gamma\left(\frac{n}{2}\\!-\\!1\right)\Gamma\left(2\\!-\\!\frac{n}{2}\right)}{\Gamma\left(3\\!-\\!\frac{n}{2}\right)}{\;}_{2}F_{1}\left(\begin{array}[]{c|}1,\tfrac{n}{2}-1\\\
3-\tfrac{n}{2}\end{array}~{}\frac{Q^{2}}{m^{2}}\right)\Biggr{\\}}\;.$ (46)
* •
The diagram $C_{4}$ with arbitrary powers of propagators is expressible in
terms of a Gauss hypergeometric function with one integer parameter:
$\frac{C_{4}}{i^{1-n}\pi^{{n}/{2}}}=\frac{\Gamma\left(j_{123}\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!j_{13}\right)\Gamma\left(n\\!-\\!j_{12}\\!-\\!2j_{3}\right)}{(-m^{2})^{j_{123}\\!-\\!\tfrac{n}{2}}\Gamma\left(n\\!-\\!j_{123}\right)\Gamma\left(\frac{n}{2}\\!-\\!j_{3}\right)\Gamma(j_{2})}\;{}_{2}F_{1}\left(\begin{array}[]{c|}j_{123}\\!-\\!\tfrac{n}{2},j_{1}\\\
\tfrac{n}{2}\\!-\\!j_{3}\end{array}~{}\frac{Q^{2}}{m^{2}}\right).$
* •
For arbitrary powers of propagators, the diagram $C_{5}$ is expressible in
terms of the Appell function $F_{1}$:
$\frac{C_{5}}{i^{1-n}\pi^{{n}/{2}}}=(-m^{2})^{\tfrac{n}{2}\\!-\\!j_{123}}\frac{\Gamma\left(j_{123}\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!j_{12}\right)}{\Gamma\left(j_{3}\right)\Gamma\left(\frac{n}{2}\right)}\;{}F_{1}\left(\left.j_{123}\\!-\\!\tfrac{n}{2},j_{1},j_{2};\tfrac{n}{2}\right|~{}\frac{Q_{1}^{2}}{m^{2}},\frac{Q_{2}^{2}}{m^{2}}\right)\;.$
When the squared external momenta are equal, $Q_{1}^{2}=Q_{2}^{2}=Q^{2}$, it
reduces to the Gauss hypergeometric function:
$\left.\frac{C_{5}}{i^{1-n}\pi^{{n}/{2}}}\right|_{Q_{1}^{2}=Q_{2}^{2}=Q^{2}}=(-m^{2})^{\tfrac{n}{2}\\!-\\!j_{123}}\frac{\Gamma\left(j_{123}\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!j_{12}\right)}{\Gamma\left(j_{3}\right)\Gamma\left(\frac{n}{2}\right)}\
{}_{2}F_{1}\left(\begin{array}[]{c|}j_{123}\\!-\\!\tfrac{n}{2},j_{12}\\\
\tfrac{n}{2}\end{array}~{}\frac{Q^{2}}{m^{2}}\right)\;.$
For $Q_{1}^{2}=Q_{2}^{2}$, the Gram determinant is zero, and when
$Q_{1}^{2}=Q_{2}^{2}=m^{2}$, the Cayley determinant is also zero.
* •
For $C_{6}$, both the Gram and Cayley determinants are nonzero, and
$\displaystyle\frac{C_{6}}{i\pi^{{n}/{2}}}=-(m^{2})^{\tfrac{n}{2}-3}\Biggl{\\{}\frac{\Gamma\left(3\\!-\\!\frac{n}{2}\right)\Gamma\left(n\\!-\\!5\right)}{\Gamma\left(n-3\right)}\
{}_{2}F_{1}\left(\begin{array}[]{c|}1,1\\\
\tfrac{7-n}{2}\end{array}~{}\frac{Q^{2}}{4m^{2}}\right)$ (49)
$\displaystyle\hskip
14.22636pt+\left(-\frac{Q^{2}}{m^{2}}\right)^{\tfrac{n}{2}-2}\frac{\Gamma^{2}\left(\frac{n}{2}\\!-\\!1\right)\Gamma\left(2\\!-\\!\frac{n}{2}\right)}{\Gamma\left(n\\!-\\!2\right)}\left(\frac{3-n}{2}\right)\
{}_{2}F_{1}\left(\begin{array}[]{c|}1,\tfrac{n}{2}-1\\\
\frac{3}{2}\end{array}~{}\frac{Q^{2}}{4m^{2}}\right)\Biggr{\\}}\;.$ (52)
* •
The diagram $C_{7}$ with arbitrary powers of propagators is expressible in
terms of the function ${}_{3}F_{2}$. For this diagram, both the Gram and
Cayley determinants are nonzero, and the master integral is
$\frac{C_{7}}{i\pi^{{n}/{2}}}=-(m^{2})^{\tfrac{n}{2}-3}\frac{\Gamma\left(\frac{n}{2}\\!-\\!1\right)\Gamma\left(3\\!-\\!\frac{n}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}\
{}_{3}F_{2}\left(\begin{array}[]{c|}1,1,3-\tfrac{n}{2}\\\
\tfrac{n}{2},2\end{array}~{}\frac{Q^{2}}{m^{2}}\right)\;.$
* •
The diagram $C_{8}$ with arbitrary powers of propagators is expressible in
terms of the function ${}_{4}F_{3}$. For this diagram, both the Gram and
Cayley determinants are nonzero. The master integral is
$\frac{C_{8}}{i\pi^{{n}/{2}}}=-(m^{2})^{\tfrac{n}{2}-3}\frac{\Gamma\left(\frac{n}{2}\\!-\\!1\right)\Gamma\left(3\\!-\\!\frac{n}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}\
{}_{3}F_{2}\left(\begin{array}[]{c|}1,3-\tfrac{n}{2},\tfrac{n}{2}-1\\\
\tfrac{n}{2},\tfrac{3}{2}\end{array}~{}\frac{Q^{2}}{4m^{2}}\right)\;.$
* •
For $C_{9}$, both the Gram and Cayley determinants are nonzero.
$\displaystyle\frac{C_{9}}{i\pi^{{n}/{2}}}$ $\displaystyle=$
$\displaystyle-(m^{2})^{\tfrac{n}{2}-3}\frac{\Gamma\left(3\\!-\\!\frac{n}{2}\right)\Gamma\left(\frac{n}{2}\\!-\\!1\right)}{\Gamma\left(\frac{n}{2}\right)}\frac{1}{Q_{1}^{2}-Q_{2}^{2}}$
(57)
$\displaystyle\times\Biggl{\\{}{}_{3}F_{2}\left(\begin{array}[]{c|}3\\!-\\!\tfrac{n}{2},1,1\\\
\tfrac{n}{2},2\end{array}~{}\frac{Q_{1}^{2}}{m^{2}}\right)Q_{1}^{2}-{}_{3}F_{2}\left(\begin{array}[]{c|}3\\!-\\!\tfrac{n}{2},1,1\\\
\tfrac{n}{2},2\end{array}~{}\frac{Q_{2}^{2}}{m^{2}}\right)Q_{2}^{2}\Biggr{\\}}\;.$
When $Q_{1}^{2}=Q_{2}^{2}$, the Gram determinant is equal to zero.
* •
For diagram $C_{10}$, the Cayley determinant vanishes, so that the diagram can
be reduced to a linear combination of one-loop propagator diagrams (see Ref.
[37]). The hypergeometric function representation is
$\displaystyle\frac{C_{10}}{i\pi^{{n}/{2}}}=-\frac{\Gamma\left(3-\frac{n}{2}\right)}{2Q^{2}(n-4)}$
$\displaystyle\hskip
14.22636pt\times\Biggl{\\{}(Q^{2}\\!+\\!m_{1}^{2}\\!-\\!m_{2}^{2})(m_{1}^{2})^{\tfrac{n}{2}-3}\
{}_{2}F_{1}\left(\begin{array}[]{c|}1,3\\!-\\!\tfrac{n}{2}\\\
\tfrac{3}{2}\end{array}~{}\frac{(Q^{2}+m_{1}^{2}-m_{2}^{2})^{2}}{4m_{1}^{2}Q^{2}}\right)$
(60) $\displaystyle\hskip
28.45274pt+(Q^{2}\\!-\\!m_{1}^{2}\\!+\\!m_{2}^{2})(m_{2}^{2})^{\tfrac{n}{2}-3}\
{}_{2}F_{1}\left(\begin{array}[]{c|}1,3\\!-\\!\tfrac{n}{2}\\\
\tfrac{3}{2}\end{array}~{}\frac{(Q^{2}-m_{1}^{2}+m_{2}^{2})^{2}}{4m_{2}^{2}Q^{2}}\right)\Biggr{\\}}\;.$
(63)
* •
For this diagram, both the Gram and Cayley determinants are nonzero. The
master integral is
$\frac{C_{11}}{i\pi^{{n}/{2}}}=-\frac{1}{2}(m^{2})^{\tfrac{n}{2}\\!-\\!3}\Gamma\left(3\\!-\\!\frac{n}{2}\right)\
{}_{3}F_{2}\left(\begin{array}[]{c|}3\\!-\\!\frac{n}{2},1,1\\\
\frac{3}{2},2\end{array}~{}\frac{Q^{2}}{4m^{2}}\right)\;.$
The all-order $\varepsilon$ expansions of $C_{11}$ is expressible in terms of
multiple polylogarithm of a square root of unity.
* •
The master integral for diagram $C_{12}$ was evaluated in Ref. [67] in terms
of a linear combination of two ${}_{3}F_{2}$ functions of the same type, as
for the diagram $C_{8}$:
$\displaystyle\frac{C_{12}}{i\pi^{\tfrac{n}{2}}}$ $\displaystyle=$
$\displaystyle-(m^{2})^{\tfrac{n}{2}-3}\Gamma\left(3\\!-\\!\frac{n}{2}\right)\frac{1}{2(Q_{1}^{2}-Q_{2}^{2})}$
(68)
$\displaystyle\times\Biggl{\\{}{}_{3}F_{2}\left(\begin{array}[]{c|}3\\!-\\!\tfrac{n}{2},1,1\\\
\tfrac{3}{2},2\end{array}~{}\frac{Q_{1}^{2}}{4m^{2}}\right)Q_{1}^{2}-{}_{3}F_{2}\left(\begin{array}[]{c|}3\\!-\\!\tfrac{n}{2},1,1\\\
\tfrac{3}{2},2\end{array}~{}\frac{Q_{2}^{2}}{m^{2}}\right)Q_{2}^{2}\Biggr{\\}}\;.$
Acknowledgments. M.Yu.K. is grateful to the Organizers of “Quark-2008” for
their hospitality and to all participants, but especially to K. Chetyrkin, A.
Isaev, A. Kataev, S. Larin and A. Pivovarov, for useful discussion. We are
indebted to A. Davydychev and O. Tarasov for a careful reading of manuscript.
M.Yu.K. is thankful to A. Kotikov, T. Huber and D. Maître for useful comments.
M.Yu.K.’s research was supported in part by BMBF Grant No. 05 HT6GUA and DFG
Grant No. KN 365/3-1. B.F.L.W.’s research was partly supported by US DOE grant
DE-FG02-05ER41399 and by NATO grant PST.CLG.980342.
## References
* [1] T. Regge, Algebraic Topology Methods in the Theory of Feynman Relativistic Amplitudes, Battelle Rencontres, 1967. Lectures in Mathematics and Physics, ed. C. M. DeWitt, J. A. Wheeler. New York: W. A. Benjamin 1968.
* [2] L.D. Landau, Nucl. Phys. 13 (1959) 181;
N. Nakanishi, Prog. Theor. Phys. 22 (1959) 128; ibid 23 (1960) 284.
* [3] R.J. Eden, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, The Analytic $S$-Matrix, Cambridge, Cambridge University Press 1966;
R. Hwa, V. Teplitz, Homology and Feynman Integrals, W.A.Benjamin, New York,
1966;
J.D. Bjorken, Doctoral dissertation, Stanford University, 1959.
* [4] D.S. Kershaw, Phys. Rev. D 8 (1973) 2708; A.C.T. Wu, Phys. Rev. D 9 (1974) 370;
K. Mano, Phys. Rev. D 11 (1975) 452.
* [5] J.A. Lappo-Danilevsky, Theory of Functions on Matrices and Systems of Linear Differential Equations (Leningrad, 1934).
* [6] G. Barucchi, G. Ponzano, J. Math. Phys. 14 (1973) 396.
* [7] G. Ponzano, T. Regge, E.R. Speer, M.J. Westwater, Commun. Math. Phys. 15 (1969) 83; ibid 18 (1970) 1; T. Regge, E.R. Speer, M.J. Westwater, Fortsch. Phys. 20 (1972) 365.
* [8] V.A. Golubeva, Russ. Math. Surv. 31 (1976) 139.
* [9] M. Kashiwara, T. Kawai, Publ. Res. Inst. Math. Sci. Kyoto 12 (1977) 131; Commun. Math. Phys. 54 (1977) 121;
T. Kawai, H.P. Stapp, Commun. Math. Phys. 83 (1982) 213.
* [10] N.N. Bogoliubov, D.V. Shirkov, Introduction to the Theory of Quantized Fields, (Wiley & Sons, New York, 1980);
C. Itzykson, J.B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
* [11] G. ’tHooft, M. Veltman, Nucl. Phys. B 44 (1972) 189.
* [12] F.V. Tkachov, Phys. Lett. B 100 (1981) 65;
K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192 (1981) 159.
* [13] O.V. Tarasov, Phys. Rev. D 54 (1996) 6479.
* [14] A.I. Davydychev, Phys. Lett. B 263 (1991) 107.
* [15] E.D. Rainville, Special Functions (MacMillan Co., New York, 1960).
* [16] M. Saito, B. Sturmfels, N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, (Springer-Verlag, Berlin, 2000).
* [17] M.Yu. Kalmykov, JHEP 0604 (2006) 056.
* [18] O.V. Tarasov, Acta Phys. Polon. B 29 (1998) 2655.
* [19] A.V. Kotikov, Phys. Lett. B 254 (1991) 158; ibid 259 (1991) 314; ibid 267 (1991) 123;
E. Remiddi, Nuovo Cim. A 110 (1997) 1435.
* [20] G. ’t Hooft, M.J.G. Veltman, Nucl. Phys. B 44 (1972) 189;
G. Rufa, Annalen Phys. 47 (1990) 6.
* [21] M. Argeri, P. Mastrolia, Int. J. Mod. Phys. A 22 (2007) 4375.
* [22] V.V Golubev, Lectures on the Analytic Theory of Differential Equations, 2nd ed. (Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950).
* [23] O.V. Tarasov, Phys. Lett. B 638 (2006) 195;
U. Aglietti, R. Bonciani, L. Grassi, E. Remiddi, Nucl. Phys. B 789 (2008) 45.
* [24] F.V. Tkachov, Nucl. Instrum. Meth. A 389 (1997) 309;
G. Passarino, Nucl. Phys. B 619 (2001) 257;
G. Passarino, S. Uccirati, Nucl. Phys. B 629 (2002) 97;
F. Jegerlehner, M.Yu. Kalmykov, O. Veretin, Nucl. Phys. B 641 (2002) 285;
A. Ferroglia, M. Passera, G. Passarino, S. Uccirati, Nucl. Phys. B 650 (2003)
162;
C. Anastasiou, A. Daleo, JHEP 0610 (2006) 031;
C. Anastasiou, S. Beerli, A. Daleo, JHEP 0705 (2007) 071;
S. Actis, G. Passarino, C. Sturm, S. Uccirati, arXiv:0809.3667;
G. Heinrich, Int. J. Mod. Phys. A 23 (2008) 1457.
* [25] A.I. Davydychev, J. Math. Phys. 32 (1991) 1052; ibid 33 (1992) 358.
* [26] E.E. Boos, A.I. Davydychev, Theor. Math. Phys. 89 (1991) 1052.
* [27] O.V. Tarasov, Nucl. Phys. Proc. Suppl. 89 (2000) 237.
* [28] J. Fleischer, F. Jegerlehner, O.V. Tarasov, Nucl. Phys. B 672 (2003) 303.
* [29] J. Fleischer, F. Jegerlehner, O.V. Tarasov, Nucl. Phys. B 566 (2000) 423;
T. Binoth, J. P. Guillet, G. Heinrich, Nucl. Phys. B 572 (2000) 361;
T. Binoth, J.P. Guillet, G. Heinrich, E. Pilon, C. Schubert, JHEP 0510 (2005)
015.
* [30] G. Passarino, M.J.G. Veltman, Nucl. Phys. B 160 (1979) 151;
A.V. Kotikov, Mod. Phys. Lett. A 6 (1991) 3133.
* [31] F.A. Berends, M. Buza, M. Böhm, R. Scharf, Z. Phys. C 63 (1994) 227.
* [32] A.I. Davydychev, “Loop calculations in QCD with massive quarks”, talk at Int. Conf. “Relativistic Nuclear Dynamics” (Vladivostok, Russia, September 1991),
D.J. Broadhurst, J. Fleischer, O.V. Tarasov, Z. Phys. C 60 (1993) 287;
A.I. Davydychev, A.G. Grozin, Phys. Rev. D 59 (1999) 054023;
F. Jegerlehner, M.Yu. Kalmykov, Nucl. Phys. B 676 (2004) 365.
* [33] M.Yu. Kalmykov, B. Kniehl, doi: 10.1016/j.nuclphysb.2008.08.022 (arXiv:0807.0567).
* [34] A. Erdelyi (Ed.), Higher Transcendental Functions, vol.1 (McGraw-Hill, New York, 1953); L.J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, Cambridge 1966).
* [35] G. ’t Hooft, M.J.G. Veltman, Nucl. Phys. B 153 (1979) 365;
A. Denner, U. Nierste, R. Scharf, Nucl. Phys. B 367 (1991) 637.
* [36] L. Lewin, Polylogarithms and associated functions (North-Holland, Amsterdam, 1981).
* [37] A.I. Davydychev, Phys. Rev. D 61 (2000) 087701;
A.I. Davydychev, M.Yu. Kalmykov, Nucl. Phys. Proc. Suppl. 89 (2000) 283; Nucl.
Phys. B 605 (2001) 266; arXiv:hep-th/0203212.
* [38] U. Nierste, D. Müller, M. Böhm, Z. Phys. C 57 (1993) 605.
* [39] A.I. Davydychev, Nucl. Instrum. Meth. A 559 (2006) 293; O.V. Tarasov, arXiv:0809.3028.
* [40] A.B. Goncharov, Proceedings of the International Congress of Mathematicians, Zurich, 1994 (Birkhäuser, Basel, 1995) Vol. 1, 2, p. 374; Math. Res. Lett. 4 (1997) 617; ibid 5 (1998) 497; arXiv:math/0103059.
* [41] J. Fleischer, T. Riemann, O.V. Tarasov, Acta Phys. Polon. B 34 (2003) 5345.
* [42] J.G. Körner, Z. Merebashvili, M. Rogal, Phys. Rev. D 71 (2005) 054028; J. Math. Phys. 47 (2006) 072302.
* [43] D.J. Broadhurst, D. Kreimer, Int. J. Mod. Phys. C 6 (1995) 519; Phys. Lett. B 393 (1997) 403; I. Bierenbaum, S. Weinzierl, Eur. Phys. J. C 32 (2003) 67; F. Brown, arXiv:0804.1660.
* [44] S. Bauberger, F. A. Berends, M. Bohm, M. Buza, Nucl. Phys. B 434 (1995) 383;
A.I. Davydychev, R. Delbourgo, J. Phys. A 37 (2004) 4871;
G. Passarino, Nucl. Phys. Proc. Suppl. 135 (2004) 265;
B.A. Kniehl et al., Nucl. Phys. B 738 (2006) 306;
D.H. Bailey et al., J. Phys. A 41 (2008) 20520;
S. Laporta, Phys. Lett. B 549 (2002) 115; arXiv:0803.1007;
P. Aluffi, M. Marcolli, arXiv:0807.1690
* [45] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Adv. Math. 84 (1990) 255;
I.M. Gel’fand, M.I. Graev, V.S. Retakh, Russian Math. Surveys 47 (1992) 1;
I.M. Gelfand, M.I. Graev, Russian Math. Surveys 52 (1997) 639; ibid 56 (2001)
615.
* [46] O. Ore, J. Math. Pure Appl. 9 (1930) 311; M. Sato, Nagoya Math. J. 120 (1990) 1.
* [47] S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys. 43 (2002) 3363.
* [48] S. Weinzierl, J. Math. Phys. 45 (2004) 2656.
* [49] Shu Oi, math.NT/0405162.
* [50] M.Yu. Kalmykov, B.F.L. Ward, S. Yost, JHEP 0702 (2007) 040.
* [51] M.Yu. Kalmykov, B.F.L. Ward, S.A. Yost, JHEP 0710 (2007) 048.
* [52] M.Yu. Kalmykov, B.F.L. Ward, S.A. Yost, JHEP 0711 (2007) 009.
* [53] A.I. Davydychev, J.B. Tausk, Nucl. Phys. B 397 (1993) 123; Phys. Rev. D 53 (1996) 7381.
* [54] N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48 (1990) 673;
D.J. Broadhurst, Z. Phys. C 47 (1990) 115; ibid 54 (1992) 599; arXiv:hep-
th/9604128;
J.M. Borwein, D.M. Bradley, D.J. Broadhurst, Electron. J. Combin. 4 (1997)
#R5;
J.A.M. Vermaseren, Int. J. Mod. Phys. A 14 (1999) 2037;
M. Bigotte, G. Jacob, N.E. Oussous, M. Petitot, Theoret. Comput. Sci. 273
(2002) 271.
* [55] S. Weinzierl, Comput. Phys. Commun. 145 (2002) 357;
S. Moch, P. Uwer, Comput. Phys. Commun. 174 (2006) 759;
T. Huber, D. Maître, Comput. Phys. Commun. 175 (2006) 122.
* [56] T. Huber, D. Maître, Comput. Phys. Commun. 178 (2008) 755.
* [57] D.J. Broadhurst, Eur. Phys. J. C8 (1999) 311;
J. Fleischer, M.Yu. Kalmykov, A.V. Kotikov, Phys. Lett. B 462 (1999) 169;
J. Fleischer, M.Yu. Kalmykov, Phys. Lett. B 470 (1999) 168;
J.M. Borwein, D.J. Broadhurst, J. Kamnitzer, Exper. Math. 10 (2001) 25;
M.Yu. Kalmykov, O. Veretin, Phys. Lett. B 483 (2000) 315;
M.Yu. Kalmykov, A. Sheplyakov, Comput. Phys. Commun. 172 (2005) 45;
M.Yu. Kalmykov, Nucl. Phys. B 718 (2005) 276;
Jianqiang Zhao, arXiv:math/0302055.
* [58] F. Jegerlehner, M.Yu. Kalmykov, O. Veretin, Nucl. Phys. B 658 (2003) 49.
* [59] A.I. Davydychev, M.Yu. Kalmykov, Nucl. Phys. B 699 (2004) 3;
M.Yu. Kalmykov, Nucl. Phys. Proc. Suppl. 135 (2004) 280.
* [60] S.A. Yost, M.Yu. Kalmykov, B.F.L. Ward, ICHEP 2008, Philadelphia, arXiv:0808.2605.
* [61] J.M. Borwein et al., Trans. Am. Math. Soc. 353 (2001) 907;
M. Waldschmidt, “Multiple polylogarithms: an introduction,” in Number theory
and discrete mathematics (Chandigarh, 2000), 1–12, (Trends Math., Birkhäuser,
Basel, 2002).
* [62] E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A 15 (2000) 725.
* [63] T. Gehrmann, E. Remiddi, Nucl. Phys. B 601 (2001) 248.
* [64] J. Vollinga, S. Weinzierl, Comput. Phys. Commun. 167 (2005) 177.
* [65] D. Maître, Comput. Phys. Commun. 174 (2006) 222; arXiv:hep-ph/0703052.
* [66] J. Fleischer, A.V. Kotikov, O.L. Veretin, Nucl. Phys. B 547 (1999) 343
* [67] A.I. Davydychev, P. Osland, L. Saks, Phys. Rev. D 63 (2001) 014022; JHEP 0108 (2001) 050.
* [68] C. Anastasiou, E.W.N. Glover, C. Oleari, Nucl. Phys. B 572 (2000) 307.
* [69] R.K. Ellis, G. Zanderighi, JHEP 0802 (2008) 002;
J.R. Andersen, T. Binoth, G. Heinrich, J.M. Smillie, JHEP 0802 (2008) 057.
* [70] M.Yu. Kalmykov, Hypergeometric functions: reduction and epsilon-expansion, http://theor.jinr.ru/$\;\widetilde{}\;$kalmykov/hypergeom/hyper.html
|
arxiv-papers
| 2008-10-20T14:15:23
|
2024-09-04T02:48:58.298844
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Yu. Kalmykov (Hamburg U., Inst. Theor. Phys. II & Dubna, JINR),\n Bernd A. Kniehl (Hamburg U., Inst. Theor. Phys. II), B.F.L. Ward (Baylor U.),\n S.A. Yost (Citadel Military Coll.)",
"submitter": "Kalmykov Mikhail",
"url": "https://arxiv.org/abs/0810.3238"
}
|
0810.3275
|
# Schrödinger Operators with
Purely Discrete Spectrum
Barry Simon Dedicated to A. Ya. Povzner
(Date: October 13, 2008)
###### Abstract.
We prove $-\Delta+V$ has purely discrete spectrum if $V\geq 0$ and, for all
$M$, $\lvert\\{x\mid V(x)<M\\}\rvert<\infty$ and various extensions.
###### Key words and phrases:
compact resolvent, Schrödinger operators
###### 2000 Mathematics Subject Classification:
47B07, 35Q40, 47N50
Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.
E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0652919 and by
Grant No. 2006483 from the United States-Israel Binational Science Foundation
(BSF), Jerusalem, Israel
To appear in the journal, Methods of Functional Analysis and Topology, volume
in memory of A. Ya. Povzner
## 1\. Introduction
Our main goal in this note is to explore one aspect of the study of
Schrödinger operators
$H=-\Delta+V$ (1.1)
which we’ll suppose have $V$’s which are nonnegative and in
$L_{\text{\rm{loc}}}^{1}({\mathbb{R}}^{\nu})$, in which case (see, e.g., Simon
[16]) $H$ can be defined as a form sum. We’re interested here in criteria
under which $H$ has purely discrete spectrum, that is,
$\sigma_{\text{\rm{ess}}}(H)$ is empty. This is well known to be equivalent to
proving $(H+1)^{-1}$ or $e^{-sH}$ for any (and so all) $s>0$ is compact (see
[10, Thm. XIII.16]). One of the most celebrated elementary results on
Schrödinger operators is that this is true if
$\lim_{\lvert x\rvert\to\infty}V(x)=\infty$ (1.2)
But (1.2) is not necessary. Simple examples where (1.2) fails but $H$ still
has compact resolvent were noted first by Rellich [11]—one of the most
celebrated examples is in $\nu=2$, $x=(x_{1},x_{2})$, and
$V(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}$ (1.3)
where (1.2) fails in a neighborhood of the axes. For proof of this and
discussions of eigenvalue asymptotics, see [12, 17, 18, 21, 22].
There are known necessary and sufficient conditions on $V$ for discrete
spectrum in terms of capacities of certain sets (see, e.g., Maz’ya [7]), but
the criteria are not always so easy to check. Thus, I was struck by the
following simple and elegant theorem:
###### Theorem 1.
Define
$\Omega_{M}(V)=\\{x\mid 0\leq V(x)<M\\}$ (1.4)
If (with $\lvert\,\cdot\,\rvert$ Lebesgue measure)
$\lvert\Omega_{M}(V)\rvert<\infty$ (1.5)
for all $M$, then $H$ has purely discrete spectrum.
I learned of this result from Wang–Wu [26], but there is much related work. I
found an elementary proof of Theorem 1 and decided to write it up as a
suitable tribute and appreciation of A. Ya. Povzner, whose work on continuum
eigenfunction expansions for Schrödinger operators in scattering situation [8]
was seminal and inspired me as a graduate student forty years ago!
The proof has a natural abstraction:
###### Theorem 2.
Let $\mu$ be a measure on a locally compact space, $X$ with $L^{2}(X,d\mu)$
separable. Let $L_{0}$ be a selfadjoint operator on $L^{2}(X,d\mu)$ so that
its semigroup is ultracontractive ([2]): For some $s>0$, $e^{-sL_{0}}$ maps
$L^{2}$ to $L^{\infty}(X,d\mu)$. Suppose $V$ is a nonnegative multiplication
operator so that
$\mu(\\{x\mid 0\leq V(x)<M\\})<\infty$ (1.6)
for all $M$. Then $L=L_{0}+V$ has purely discrete spectrum.
###### Remark.
By $L_{0}+V$, we mean the operator obtained by applying the monotone
convergence theorem for forms (see, e.g., [14, 15]) to $L_{0}+\min(V(x),k)$ as
$k\to\infty$.
The reader may have noticed that (1.3) does not obey Theorem 1 (but, e.g.,
$V(x_{1},x_{2})=x_{1}^{2}x_{2}^{4}+x_{1}^{4}x_{2}^{2}$ does). But out proof
can be modified to a result that does include (1.3). Given a set $\Omega$ in
${\mathbb{R}}^{\nu}$, define for any $x$ and any $\ell>0$,
$\omega_{x}^{\ell}(\Omega)=\lvert\Omega\cap\\{y\mid\lvert
y-x\rvert\leq\ell\\}\rvert$ (1.7)
For example, for (1.3), for $x\in\Omega_{M}$,
$\omega_{x}^{\ell}(\Omega_{M})\leq\frac{C_{\ell}}{\lvert x\rvert+1}$ (1.8)
We will say a set $\Omega$ is $r$-polynomially thin if
$\int_{x\in\Omega}\omega_{x}^{\ell}(\Omega)^{r}\,d^{\nu}x<\infty$ (1.9)
for all $\ell$. For the example in (1.3), $\Omega_{M}$ is $r$-polynomially
thin for any $M$ and any $r>0$. We’ll prove
###### Theorem 3.
Let $V$ be a nonnegative potential so that for any $M$, there is an $r>0$ so
that $\Omega_{M}$ is $r$-polynomially thin. Then $H$ has purely discrete
spectrum.
As mentioned, this covers the example in (1.3). It is not hard to see that if
$P(x)$ is any polynomial in $x_{1},\dots,x_{\nu}$ so that for no
$v\in{\mathbb{R}}^{\nu}$ is $\vec{v}\cdot\vec{\nabla}P\equiv 0$ (i.e., $P$
isn’t a function of fewer than $\nu$ linear variables), then $V(x)=P(x)^{2}$
obeys the hypotheses of Theorem 3.
In Section 2, we’ll present a simple compactness criterion on which all
theorems rely. In Section 3, we’ll prove Theorems 1 and 2. In Section 4, we’ll
prove Theorem 3.
It is a pleasure to thank Peter Stollmann for useful correspondence and Ehud
de Shalit for the hospitality of Hebrew University where some of the work
presented here was done.
## 2\. Segal’s Lemma
Segal [13] proved the following result, sometimes called Segal’s lemma:
###### Proposition 2.1.
For $A,B$ positive selfadjoint operators,
$\lVert e^{-(A+B)}\rVert\leq\lVert e^{-A}e^{-B}\rVert$ (2.1)
###### Remarks.
1\. $A+B$ can always be defined as a closed quadratic form on $Q(A)\cap Q(B)$.
That defines $e^{-(A+B)}$ on $\overline{Q(A)\cap Q(B)}$ and we set it to $0$
on the orthogonal complement. Since the Trotter product formula is known in
this generality (see Kato [6]), (2.1) holds in that generality.
2\. Since $\lVert C^{*}C\rVert=\lVert C\rVert^{2}$, $\lVert
e^{-A/2}e^{-B/2}\rVert^{2}=\lVert e^{-B/2}e^{-A}e^{-B/2}\rVert$, and since
$\lVert e^{-(A+B)/2}\rVert^{2}=\lVert e^{-(A+B)}\rVert$, (2.1) is equivalent
to
$\lVert e^{-A+B}\rVert\leq\lVert e^{-B/2}e^{-A}e^{-B/2}\rVert$ (2.2)
which is the way Segal [13] stated it.
3\. Somewhat earlier, Golden [5] and Thompson [23] proved
$\text{\rm{Tr}}(e^{-(A+B)})\leq\text{\rm{Tr}}(e^{-A}e^{-B})$ (2.3)
and Thompson [24] later extended this to any symmetrically normed operator
ideal.
###### Proof.
There are many; see, for example, Simon [19, 20]. Here is the simplest, due to
Deift [3, 4]: If $\sigma$ is the spectrum of an operator
$\sigma(C\\!D)\setminus\\{0\\}=\sigma(DC)\setminus\\{0\\}$ (2.4)
so with $\sigma_{r}$ the spectral radius,
$\sigma_{r}(C\\!D)=\sigma_{r}(DC)\leq\lVert DC\rVert$ (2.5)
If $C\\!D$ is selfadjoint, $\sigma_{r}(C\\!D)=\lVert C\\!D\rVert$, so
$C\\!D\text{ selfadjoint}\Rightarrow\lVert C\\!D\rVert\leq\lVert DC\rVert$
(2.6)
Thus,
$\lVert e^{-A/2}e^{-B/2}\rVert^{2}=\lVert
e^{-B/2}e^{-A}e^{-B/2}\rVert\leq\lVert e^{-A}e^{-B}\rVert$ (2.7)
By induction,
$\lVert(e^{-A/2^{n}}e^{-B/2^{n}})^{2^{n}}\rVert\leq\lVert
e^{-A/2^{n}}e^{-B/2^{n}}\rVert^{2n}\leq\lVert e^{-A}e^{-B}\rVert$ (2.8)
Take $n\to\infty$ and use the Trotter product formula to get (2.1). ∎
In [19], I noted that this implies for any symmetrically normed trace ideal,
${\mathcal{I}}_{\Phi}$, that
$e^{-A/2}e^{-B}e^{-A/2}\in{\mathcal{I}}_{\Phi}\Rightarrow
e^{-(A+B)}\in{\mathcal{I}}_{\Phi}$ (2.9)
I explicitly excluded the case ${\mathcal{I}}_{\Phi}={\mathcal{I}}_{\infty}$
(the compact operators) because the argument there doesn’t show that, but it
is true—and the key to this paper!
Since $C\in{\mathcal{I}}_{\infty}\Leftrightarrow
C^{*}C\in{\mathcal{I}}_{\infty}$ and $e^{-(A+B)}\in{\mathcal{I}}_{\infty}$ if
and only if $e^{-\frac{1}{2}(A+B)}\in{\mathcal{I}}_{\infty}$, it doesn’t
matter if we use the symmetric form (2.2) or the following asymmetric form
which is more convenient in applications.
###### Theorem 2.2.
Let ${\mathcal{I}}_{\infty}$ be the ideal of compact operators on some Hilbert
space, ${\mathcal{H}}$. Let $A,B$ be nonnegative selfadjoint operators. Then
$e^{-A}e^{-B}\in{\mathcal{I}}_{\infty}\Rightarrow
e^{-(A+B)}\in{\mathcal{I}}_{\infty}$ (2.10)
###### Proof.
For any bounded operator, $C$, define $\mu_{n}(C)$ by
$\mu_{n}(C)=\min_{\psi_{1}\dots\psi_{n-1}}\,\sup_{\begin{subarray}{c}\lVert\varphi\rVert=1\\\
\varphi\perp\psi_{1},\dots,\psi_{n-1}\end{subarray}}\lVert C\varphi\rVert$
(2.11)
By the min-max principle (see [10, Sect. XIII.1]),
$\lim_{n\to\infty}\,\mu_{n}(C)=\sup(\sigma_{\text{\rm{ess}}}(\lvert C\rvert))$
(2.12)
and $\mu_{n}(C)$ are the singular values if $C\in{\mathcal{I}}_{\infty}$. In
particular,
$C\in{\mathcal{I}}_{\infty}\Leftrightarrow\lim_{n\to\infty}\,\mu_{n}(C)=0$
(2.13)
Let $\wedge^{\ell}({\mathcal{H}})$ be the antisymmetric tensor product (see
[9, Sects. II.4, VIII.10], [10, Sect. XIII.17], and [19, Sect. 1.5]). As usual
(see [19, eqn. (1.14)]),
$\lVert\wedge^{m}(C)\rVert=\prod_{j=1}^{m}\mu_{j}(C)$ (2.14)
Since $\mu_{1}\geq\mu_{2}\geq\cdots\geq 0$, we have
$\lim_{n\to\infty}\,\mu_{n}(C)=\lim_{n\to\infty}\,(\mu_{1}(C)\dots\mu_{n}(C))^{1/n}$
(2.15)
(2.13)–(2.15) imply
$C\in{\mathcal{I}}_{\infty}\Leftrightarrow\lim_{n\to\infty}\,\lVert\wedge^{n}(C)\rVert^{1/n}=0$
(2.16)
As usual, there is a selfadjoint operator, $d\wedge^{n}(A)$ on
$\wedge^{n}({\mathcal{H}})$ so
$\wedge^{n}(e^{-tA})=e^{-t\,d\wedge^{n}(A)}$ (2.17)
so Segal’s lemma implies that
$\displaystyle\lVert\wedge^{n}(e^{-(A+B)})\rVert$
$\displaystyle\leq\lVert\wedge^{n}(e^{-A})\wedge^{n}(e^{-B})\rVert$
$\displaystyle=\lVert\wedge^{n}(e^{-A}e^{-B})\rVert$ (2.18)
Thus,
$\lim_{n\to\infty}\,\lVert\wedge^{n}(e^{-(A+B)})\rVert^{1/n}\leq\lim_{n\to\infty}\,\lVert\wedge^{n}(e^{-A}e^{-B})\rVert^{1/n}$
(2.19)
By (2.16), we obtain (2.10). ∎
## 3\. Proofs of Theorems 1 and 2
###### Proof of Theorem 1.
By Theorem 2.2, we need only show $C=e^{\Delta}e^{-V}$ is compact. Write
$C=C_{m}+D_{m}$ (3.1)
where
$C_{m}=C\chi_{\Omega_{m}}\qquad D_{m}=C\chi_{\Omega_{m}^{c}}$ (3.2)
with $\chi_{S}$ the operator of multiplication by the characteristic function
of a set $S\subset{\mathbb{R}}^{\nu}$.
$\lVert e^{-V}\chi_{\Omega_{m}^{c}}\rVert_{\infty}\leq e^{-m}$
and $\lVert e^{\Delta}\rVert=1$, so
$\lVert D_{m}\rVert\leq e^{-m}$ (3.3)
and thus,
$\lim_{m\to\infty}\,\lVert C-C_{m}\rVert=0$ (3.4)
If we show each $C_{m}$ is compact, we are done. We know $e^{\Delta}$ has
integral kernel $f(x-y)$ with $f$ a Gaussian, so in $L^{2}$. Clearly, since
$V$ is positive, $C_{m}$ has an integral kernel $C_{m}(x,y)$ dominated by
$\lvert C_{m}(x,y)\rvert\leq f(x-y)\chi_{\Omega_{m}}(y)$ (3.5)
Thus,
$\int\lvert C_{m}(x,y)\rvert^{2}\,d^{\nu}xd^{\nu}y\leq\lVert
f\rVert^{2}_{L^{2}({\mathbb{R}}^{\nu})}\lVert\chi_{\Omega_{m}}\rVert_{L^{2}({\mathbb{R}}_{\nu})}<\infty$
since $\lvert\Omega_{m}\rvert<\infty$. Thus, $C_{m}$ is Hilbert–Schmidt, so
compact. ∎
###### Proof of Theorem 2.
We can follow the proof of Theorem 1. It suffices to prove that
$e^{-sL_{0}}e^{-sV}$ is compact, and so, that $e^{-sL_{0}}\chi_{\Omega_{m}}$
is Hilbert–Schmidt.
That $e^{-sL_{0}}$ maps $L^{2}$ to $L^{\infty}$ implies, by the Dunford–Pettis
theorem (see [25, Thm. 46.1]), that there is, for each $x\in X$, a function
$f_{x}(\,\cdot\,)\in L^{2}(X,d\mu)$ with
$(e^{-sL_{0}}g)(x)=\langle f_{x},g\rangle$ (3.6)
and
$\sup_{x}\,\lVert f_{x}\rVert_{L^{2}}=\lVert e^{-sL_{0}}\rVert_{L^{2}\to
L^{\infty}}\equiv C<\infty$ (3.7)
Thus, $e^{-sL_{0}}$ has an integral kernel $K(x,y)$ with
$\sup_{x}\int\lvert K(x,y)\rvert^{2}\,d\mu(y)=C<\infty$ (3.8)
(for $K(x,y)=f_{x}(y)$). But $e^{-sL_{0}}$ is selfadjoint, so its kernel is
complex symmetric, so
$\sup_{y}\int\lvert K(x,y)\rvert^{2}\,d\mu(x)=C<\infty$ (3.9)
Thus,
$\int\lvert K(x,y)\chi_{\Omega_{m}}(y)\rvert^{2}\,d\mu(x)d\mu(y)\leq
C\mu(\Omega_{m})<\infty$ (3.10)
and $e^{-sL_{0}}\chi_{\Omega_{m}}$ is Hilbert–Schmidt. ∎
## 4\. Proof of Theorem 3
As with the proof of Theorem 1, it suffices to prove that for each $M$,
$e^{\Delta}\chi_{\Omega_{M}}$ is compact. $e^{\Delta}$ is convolution with an
$L^{1}$ function, $f$. Let $Q_{R}$ be the characteristic function of
$\\{x\mid\lvert x\rvert<R\\}$. Let $F_{R}$ be convolution with $fQ_{R}$. Then
$\lVert e^{\Delta}-F_{R}\rVert\leq\lVert f(1-Q_{R})\rVert_{1}\to 0$ (4.1)
as $R\to\infty$, so
$\lVert e^{\Delta}\chi_{\Omega_{M}}-F_{R}\chi_{\Omega_{M}}\rVert\to 0$ (4.2)
and it suffices to prove for each $R,M$,
$C_{M,R}=F_{R}\chi_{\Omega_{M}}$ (4.3)
is compact. Clearly, this works if we show for some $k$,
$(C_{M,R}^{*}C_{M,R})^{k}$ is Hilbert–Schmidt.
Let $D$ be the operator with integral kernel
$D(x,y)=\chi_{\Omega_{M}}(x)Q_{2R}(x-y)\chi_{\Omega_{M}}(y)$ (4.4)
Since $f$ is bounded, it is easy to see that
$(C_{M,R}^{*}C_{M,R})(x,y)\leq cD(x,y)$ (4.5)
for some constant $c$, so it suffices to show $D^{k}$ is Hilbert–Schmidt.
$D^{k}$ has integral kernel
$D^{k}(x,y)=\int D(x,x_{1})D(x_{1},x_{2})\dots
D(x_{k-1},y)\,dx_{1}\dots,dx_{k-1}$ (4.6)
Fix $y$. This integral is zero unless $\lvert x-x_{1}\rvert<2R,\dots\lvert
x_{k-1}-y\rvert<2R$, so, in particular, unless $\lvert x-y\rvert\leq 2kR$.
Moreover, the integrand can certainly be restricted to the regions $\lvert
x_{j}-y\rvert\leq 2kR$. Thus,
$\displaystyle D^{k}(x,y)$ $\displaystyle\leq
Q_{2kR}(x-y)\biggl{(}\int_{\lvert x_{j}-y\rvert\leq
2kR}\,\prod_{j=1}^{k-1}\chi_{\Omega_{M}}(x_{j})\,dx_{1}\dots
dx_{k-1}\biggr{)}\chi_{\Omega_{m}}(y)$ (4.7)
$\displaystyle=Q_{2kR}(x-y)(\omega_{y}^{2kR}(\Omega_{M})^{k-1})\chi_{\Omega_{M}}(y)$
(4.8)
by the definition of $\omega_{x}^{\ell}$ in (1.7).
Thus,
$\int\lvert D^{k}(x,y)\rvert^{2}\,d^{\nu}xd^{\nu}y\leq
C(kR)^{\nu}\int_{x\in\Omega}[\omega_{x}^{2kR}(\Omega_{M})]^{2k-2}\,d^{\nu}x$
so if $2k-2>r$ and (1.9) holds, $D^{k}$ is Hilbert–Schmidt. ∎
## References
* [1]
* [2] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335–395.
* [3] P. A. Deift, Classical Scattering Theory With a Trace Condition, Ph.D. dissertation, Princeton University, 1976.
* [4] P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267–310.
* [5] S. Golden, Lower bounds for the Helmholtz function, Phys. Rev. (2) 137 (1965), B1127–B1128.
* [6] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in “Topics in Functional Analysis,” pp. 185–195, Adv. in Math. Suppl. Stud., 3, Academic Press, New York-London, 1978.
* [7] V. Maz’ya, Analytic criteria in the qualitative spectral analysis of the Schrödinger operator, in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th birthday,” pp. 257–288, Proc. Sympos. Pure Math., 76.1, American Mathematical Society, Providence, RI, 2007\.
* [8] A. Ya. Povzner, On expansions in functions which are solutions of a scattering problem, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 360–363. [Russian]
* [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
* [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York, 1978.
* [11] F. Rellich, Das Eigenwertproblem von $\Delta u+\lambda u=0$ in Halbröhren, in “Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948,” pp. 329–344, Interscience Publishers, New York, 1948.
* [12] D. Robert, Comportement asymptotique des valeurs propres d’opérateurs du type Schrödinger à potentiel “dégénéré [Asymptotic behavior of the eigenvalues of Schrd̈inger operators with “degenerate” potential], J. Math. Pures Appl. (9) 61 (1982), 275–300 (1983).
* [13] I. Segal, Notes towards the construction of nonlinear relativistic quantum fields. III. Properties of the $C^{*}$-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390–1395.
* [14] B. Simon, Lower semicontinuity of positive quadratic forms, Proc. Roy. Soc. Edinburgh 29 (1977), 267–273.
* [15] B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385.
* [16] B. Simon, Maximal and minimal Schrödinger forms, J. Oper. Theory 1 (1979), 37–47.
* [17] B. Simon, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84–98.
* [18] B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983), 209–220.
* [19] B. Simon, Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005.
* [20] B. Simon, Ed Nelson’s work in quantum theory, in “Diffusion, Quantum Theory, and Radically Elementary Mathematics,” pp. 75–93, Mathematical Notes, 47, Princeton University Press, Princeton, NJ, 2006.
* [21] M. Z. Solomyak Asymptotic behavior of the spectrum of a Schrödinger operator with nonregular homogeneous potential, Soviet Math. Dokl. 30 (1984), 379–383; Russian original in Dokl. Akad. Nauk SSSR 278 (1984), 291–295.
* [22] H. Tamura, The asymptotic distribution of eigenvalues of the Laplace operator in an unbounded domain, Nagoya Math. J. 60 (1976), 7–33.
* [23] C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys. 6 (1965), 1812–1813.
* [24] C. J. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971/72), 469–480.
* [25] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
* [26] F.-Y. Wang and J.-L. Wu, Compactness of Schrödinger semigroups with unbounded below potentials, to appear in Bull. Sci. Math.
|
arxiv-papers
| 2008-10-17T23:52:57
|
2024-09-04T02:48:58.305310
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Barry Simon",
"submitter": "Barry Simon",
"url": "https://arxiv.org/abs/0810.3275"
}
|
0810.3298
|
# A comment to: On $3$-colorable planar
graphs without short cycles
S. Akbari b,a and Behrooz Bagheri Gh.a
###### Abstract
Let $G$ be a graph. It was proved that if $G$ is a planar graph without
$\\{4,6,7\\}$-cycles and without two $5$-cycles sharing exactly one edge, then
$G$ $3$-colorable. We observed that the proof of this result is not correct.
$a$ Department of Mathematical Sciences
Sharif University of Technology
$b$ Institute for Studies in Theoretical Physics and Mathematics (IPM)
## 1
Let $G$ be a simple graph with vertex set $G$. A planar graph is one that can
be drawn on a plane in such a way that there are no “edge crossings,” i.e.
edges intersect only at their common vertices. A $k$-face is a face whose
boundary has $k$ edges. A cycle $C$ in a planar graph $G$ is said to be
separating, if $int(C)\neq\varnothing$ and $ext(C)\neq\varnothing$, where
$int(C)$ and $ext(C)$ denote the sets of vertices located inside and outside
$C$, respectively. Let $C_{i}$ denote an $i$-cycle. A $k$-coloring of $G$ is a
mapping $c$ from $V(G)$ to the set $\\{1,\ldots,k\\}$ such that $c(x)\neq
c(y)$ for any adjacent vertices $x$ and $y$. The graph $G$ is $k$-colorable if
it has a $k$-coloring. Let $\cal G$ denote the class of planar graphs without
$\\{4,6,7\\}$-cycles and without two $5$-cycles sharing exactly one edge.
The following theorem was proved in [1].
Theorem. Let $G$ be a graph in $\cal G$ that contains $5$-cycles. Then every
proper $3$-coloring of the vertices of any $3$-face or $9$-face of $G$ can be
extended into a proper $3$-coloring of the whole graph.
The authors divided the proof of the theorem into $9$ lemmas and their proof
is by contradiction. They considered a graph $G$ with the minimum number of
vertices such that satisfies the assumptions, but the assertion of the theorem
is not true for $G$. In the proof of many lemmas they made a common mistake.
For instance in Lemma 1, they applied induction on $G\setminus int(C_{i})$,
where $C_{i}$ is a separating $i$-cycle. But $G\setminus int(C_{i})$ has no
necessarily $5$-cycle and so their proof is not correct.
## References
* [1] M. Chen, W. Wang, On $3$-colorable planar graphs without short cycles, Appl. Math. Letter (2208) 9, P. 961-965.
|
arxiv-papers
| 2008-10-20T06:16:29
|
2024-09-04T02:48:58.309398
|
{
"license": "Public Domain",
"authors": "S. Akbari and Behrooz Bagheri Gh",
"submitter": "Behrooz Bagheri ghavamabadi",
"url": "https://arxiv.org/abs/0810.3298"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.