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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 34
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 60773)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 34
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
$~~~$The past fifteen years witnessed an increasing interest in the theories of
matter coupled Chern-Simons (CS)gauge field theories in 2+1 dimensions. From
one
point of view, the Euclidean version of such theories can be viewed as giving
the high temperature bahaviour of 3+1 dimensional models [1]. On the other
hand, in the pioneering works [2,3] it has been shown that the introduction of
the ( P and T odd ) CS term into the Lagrangian of 2+1 dimensional QED and QCD,
leads to a very peculiar property : the gauge field splits into two parts; a
massive part (that acquires a mass in a gauge-invariant manner), and a massless
part which does not contribute to the free classical Hamiltonian, but leads
to an additional interaction among the particles. This interaction appears
also in pure CS theories [4].\\
In the work [3], it was argued that in the
non-Abelian version of CS theories, the dimensionless combination of the charge
and the stochastic parameter should be quantized. It was also shown that the
mass term provides an infrared cut-off in special covariant gauges that
renders the theory superrenormalizable. \\
Many works were devoted to the consideration of the one-loop radiative
corrections to the charge and the stochastic parameter in both the Abelian [5]
and the non-Abelian [6] theories, and a theorem [7] was set which states that
under very general conditions, there are no further radiative corrections
beyond the finite one-loop for these parameters.\\
$~~$An additional thrust into the interest in CS theories was provided by the
interesting results in the non-relativistic domain; essentially the idea of
Wilczeck that non-relativistic charged particles coupled to pure CS field can
be considered as a phenomenological approach for the description of the "bound
states " of two particles called anyons [8]. This idea found wide acceptance,
and many attempts to apply it in many interesting
condensed matter phenomena, such as the fractional quantum Hall effect, and
high temperature superconductivity were made ( see the reviews [9] and the
references therein ). CS theories also found applications in the
field-theoretic formulation of the Aharonov-Bohm effect [10,11].\\
One of the issues that received considerable interest during the past period
was the
canonical quantization of the CS models [2,3,12]. However, some interesting
points
like the canonical quantization in a Lorentz covariant gauge still need
further investigation. Path integral quantization was also considered first
-up to our knowledge- in the works [11,13] where the generating functional
was also constructed.\\
Another issue that did not receive much attention is the following:
The free transverse topological photons of the pure CS theory
are absent, while the gauge field propagator is present, and gives significant
contribution to the interaction among the particles. This issue was addressed
in the work [14], and the so called topological unitarity identities were
derived. Moreover, the issue of the quantization of the charge in non-abelian
CS theories was not discussed thoroughly beyond the discussion in the works
[2,3].
We address this point in the present work.\\
$~~$This paper is a further development of the series of works [11,13,14]. The
main
goals are, to carry out the path integral quantization and construct
the generating functional for a wide class of models involving both the
Abelian and the non-Abelian CS fields (part II), to construct the
S-matrix
operator, and to develop the Feynman rules and formulate a Wick-type theorems
for the CS field (part III), and to illustrate in details the topological
unitarity identities in general, and through a specific example ( part IV).
Part V is devoted to concluding remarks.
\section{Path Integral Quantization and the Generating Functional}
The aim of this part is to develop the path integral quantization, and to
construct the generating functional of the theory of scalar and spinor fields
interacting through the Abelian and non-Abelian CS field in 2+1 dimensions.
This can be done through two different approaches : The De Witt-Fadeev-Popov
(DFP)[15] approach, or the Batalin-Fradkin-Vilkovisky (BFV) approach [16]. The
latter was developed to quantize gauge theories with both classes of
constraints and with arbitrary constraint algebra. In our case both approaches
lead to the same result . This is a consequence of the fact that the first
class
constraints, both in the Abelian and non-Abelian cases, form a closed algebra,
and that the structure functions in the algebra of the first class constraints
are just constants, as will be demonstrated later. Therefore, we shall carry
out the path integral quantization through the simpler DFP approach, and will
prove the
equivalence of both approaches by invoking the latter in the quantization of
the theory of spinors interacting through the non-abelian CS gauge field. This
proof is very helpful in understanding the connection between the usual
canonical quantization and the BFV quantization schemes, and in the
demonstration of the appearence of the BRST operators of the theory.
\subsection{De Witt-Fadeev-Popov method}
Scalar particles:\\
We begin with the theory of charged scalar particles interacting through the
CS gauge field with the action of this gauge field given slolely by the abelian
CS term (pure CS field). Following the DFP method, we get for the generating
functional in the covariant $\alpha$-gauge the expression [11,13] :
\begin{eqnarray}
Z[J_{\mu},j,j^*]&=&Z_0^{-1}\int DA_{\mu}(x)d\varphi^*(x)D\varphi(x)\exp
\{iS_{CS}+iS_g+iS_m\nonumber\\
&+&i\int d^3x(J_{\mu}(x)A^{\mu}(x)+j^*(x)\varphi(x)+j(x)\varphi^*(x))\}
\end{eqnarray}
where
\begin{eqnarray}
Z_0&=&Z(0,0,0,)\\
S_{CS}&=&{\mu\over 2}\int d^3x \varepsilon_{\mu\nu\lambda}A^{\mu}(x)
\partial^{\nu}A^{\lambda}(x)\\
S_g&=&{-1\over 2\alpha}\int (\partial_{\mu}A^{\mu})^2d^3x\\
S_m&=&\int d^3x\left(\varphi^*(x)(D_{\mu}D^{\mu}-m^2)\varphi(x)-
\lambda(\varphi^*(x)\varphi(x))^2\right)
\end{eqnarray}
Here, $J_{\mu}(x),~j(x)$ and $j^*(x)$ are external sources, $e$ and $m$ are
respectively the charge and the mass of the scalar field, and $D_{\mu}=
(\partial_{\mu}-ieA_{\mu})$. The metric is taken as $g_{\mu\nu}=diag(1,-1,-1)$
. The Greens functions of the theory are defined as usual by varying the above
generating functional, eq.(1) with respect to the sources.
For example, the free propagator of the CS field is defined as:
\begin{eqnarray}
D_{\mu\nu}(x-x')=(-i)^2{\delta^2\over \delta J_{\mu}(x)J_{\nu}(x')}
Z[J_{\mu},j,j^*]\left.\right|_{J_{\mu}=j=j^*=e=0}
\end{eqnarray}
However, the functional (1) has an essential defect in that the path integral
over the gauge field is not mathematically well-defined . This is
because in pure CS theory, there are no transverse components of the gauge
field. The natural way to overcome this difficulty is to introduce into the
total action in the exponent of the path integral (1) the Maxwell term
\begin{eqnarray}
S_M={-1\over 4\gamma}\int d^3xF_{\mu\nu}(x)F^{\mu\nu}(x)
\end{eqnarray}
Such a term is the only gauge-invariant bilinear term in $A_{\mu}$ that
guarantees gauge-invariant regularization and renormalization of the theory.
This term, not only leads to the convergence of the path integral over $
A_{\mu}$, but also plays the role of a regularization factor since the
resulting theory becomes superrenormalizable [2,3].\\
It is necessary here to make some important remarks on the dimensions of the
parameters and the fields of the theory. We have some arbitrariness in the
choice of the dimensions of the statistical parameter $\mu$, the charge $e$ and
the factor $\gamma$ in eqs.(3),(5) and (7). However, if we require the 2+1
dimensional
matter-coupled CS theory to have some relation with the real world, and
so that it
arises after compactification on the $\sim{1\over\gamma}$ layer of QED in 3+1
dimensions [17]with the parity violating term ${\mu\over 4}\int F_{\mu\nu}
\tilde
F^{\mu\nu}d^4x$ where $\tilde F^{\mu\nu}={1\over 2}\varepsilon^{\mu\nu\lambda
\sigma} F_{\lambda\sigma}$ then the charge $e$ and the parameter $\mu$ are to
be chosen dimensionless, whereas $[A_{\mu}]= x^{-1}~,~[\varphi]=
x^{{-1\over 2}}$ and $[\gamma]= x^{-1}$. In the following, we will
adopt this convention of the dimensions \footnote{If one makes the change of
variables $A_{\mu}\to A_{\mu}'={A_{\mu}\over\sqrt{\gamma}}~,~e\to e'=e\sqrt
{\gamma}~,~\mu\to\mu'={\mu\over\gamma}$ then one gets the conventions used in
the works [2,3]}. So, after introducing the Maxwell term the generating
functional takes the form
\begin{eqnarray}
Z[J_{\mu},j,j^*]&=&Z_0^{-1}\int DA_{\mu}(x)D\varphi^*(x)D\varphi(x)
\exp\{i(S_{CS}+S_M+S_g+S_m)\nonumber\\
&+&i\int d^3x(J_{\mu}(x)A^{\mu}(x)+j^*(x)\varphi(x)+j(x)\varphi^*(x))\}
\end{eqnarray}
This can be formally written in the alternative form
\begin{eqnarray}
Z[J_{\mu},j^*,j]=Z_0^{-1}\int D\varphi^*(x)D\varphi(x)\exp(ie^2\int d^3x
{\delta^2\over \delta J_{\mu}(x)\delta J^{\mu}(x)})\nonumber\\
\times\int DA_{\mu}(x)\exp\{i(S_{CS}+S_M+S_g+\tilde S_m)\nonumber\\
+i\int d^3x(J_{\mu}(x)A^{\mu}(x)+j^*(x)\varphi(x)+\varphi^*(x)j(x))\}
\end{eqnarray}
where $\tilde S_m$ does not contain the term $e^2A_{\mu}A^{\mu}$ in eq.(5),i.e
\begin{eqnarray}
\tilde S_m=-\int d^3x(ieA_{\mu}(x)(\varphi^*(x)\partial_{\mu}\varphi(x)
-\varphi(x)\partial_{\mu}\varphi^*(x))+\lambda(\varphi^*(x)\varphi(x))^2)
\end{eqnarray}
After integrating over $A_{\mu}$ in eq.(9) we get :
\begin{eqnarray}
Z[J_{\mu},j,j^*]=Z_0^{-1}\int D\varphi^*(x)D\varphi(x)\exp\{ie^2\int d^3x
\varphi^*(x)\varphi(x){\delta^2\over\delta J_{\nu}(x)\delta J^{\nu}(x)}\}
\nonumber\\
\exp\{{i\over 2}\int d^3xd^3yI_{\mu}(x)D^{\mu\nu}(x-y)I_{\nu}(y)-\lambda\int
d^3x(\varphi^*(x)\varphi(x))^2\nonumber\\
+i\int d^3x(j^*(x)\varphi(x)+j(x)\varphi^*(x))\}
\end{eqnarray}
where
\begin{eqnarray}
I_{\mu}(x)=J_{\mu}(x)+ie\int d^3x(\varphi^*(x)\partial_{\mu}\varphi(x)-\varphi
(x)\partial_{\mu}\varphi^*(x))
\end{eqnarray}
and $D_{\mu\nu}(x-y)$ is the CS gauge field's Greens function defined by the
equation:
\begin{eqnarray}
\int d^3x'{\delta^2(S_{CS}+S_g+S_M)\over \delta A^{\mu}(x)\delta A^{\lambda}
(x')}D^{\lambda\nu}(x'-y)=g^{\nu}_{\mu}\delta^3(x-y)
\end{eqnarray}
or,
\begin{eqnarray}
[{1\over\gamma}(\Box_xg_{\mu\lambda}-\partial_{\mu}\partial_{\lambda})+{1\over
\alpha}\partial_{\mu}\partial_{\lambda}+\mu\varepsilon_{\mu\lambda\rho}
\partial_x^{\rho}]D^{\lambda\nu}(x-y)=\delta^3(x-y)g^{\nu}_{\mu}
\end{eqnarray}
The solution of eq.(14) is [2,3]:
\begin{eqnarray}
D_{\lambda\nu}(x)={1\over (2\pi)^3}\int d^3pe^{ipx}\left[-\gamma
{(g_{\lambda\nu}-{p_{\nu}p_{\lambda})\over p^2}\over (p^2-\gamma^2\mu^2+
i\epsilon)}+{i\varepsilon_{\lambda\nu\rho}
p^{\rho}\over\mu(p^2-\gamma^2\mu^2+i\epsilon)}\right.\nonumber\\
\left.-{i\varepsilon_{\lambda\nu\rho}p^{\rho}\over \mu(p^2+i\epsilon)}
-{\alpha p_{\lambda}p_{\nu}\over (p^2+i\epsilon)^2}\right]
\end{eqnarray}
We note that the above Greens function (or propagator) consists of two parts:
The first two terms describe the propagation of a real massive photon with
mass equal to $\gamma\mu$; the third term describes the propagation of a
topological massless photon, and the last term is pure gauge term. The
appearence of massive photons in a gauge-invariant manner is a well-known
peculiar property of CS theory, and is independent of coupling to matter fields
[2,3]. To show that the topological term in eq.(15) does not contribute to
the
tensor $F_{\mu\nu}$ of the gauge field, we construct the general solution of
the classical equations of motion of the field $A_{\mu}$ (eq.(14)). This is
given as :
\begin{eqnarray}
A_{\mu}(x)&=&4\pi\int Im D_{\mu\nu}(p){\it e}_{\delta}^{\nu}a^{\delta}(p)e^{ik
x}d^3p\nonumber\\
&=&{1\over 2\pi}\int d^3pe^{ipx}\left[-\gamma(({\it e}^{\delta}_{\mu}(p)
-{p_{\mu}p_{\nu}\over p^2}{\it e}^{\nu}_{\delta}(p))+{i\over\gamma\mu}
\varepsilon_{\mu\nu\rho}p^{\rho}{\it e}^{\nu}_{\delta}(p))\delta(p^2-\mu^2
\gamma^2)\right.\nonumber\\
&-&{i\over\mu}\varepsilon_{\mu\nu\rho}{\it e}^{\nu}_{\delta}(p)p^{\rho}
\delta(p^2)
-({p_{\mu}p_{\nu}\over p^2-\mu^2\gamma^2}){\it e}^{\nu}_{\delta}(p)\delta(p^2)
\nonumber\\
&+&{\alpha\over 2}p_{\mu}\left({\partial\over\partial p^{\nu}}\delta(p^2)
\right) {\it e}^{\nu}_{\delta}(p)\left.\right]a^{\delta}(p)
\end{eqnarray}
Here, $ImD_{\mu\nu}(p)$ is the imaginary part of the propagator $D_{\mu\nu}$ in
eq.(15) in the momentum space representation; $e^{\nu}_{\delta}(p), \delta=0,1
,2,$ are three mutually orthogonal polarization vectors which satisfy
$p_{\mu}e^{\mu}_{\delta}(p)=0$. This choice corresponds to the gauge
$\partial_{\mu} A^{\mu}=0$. In the general case, the free solution $A_{\mu}(x)
$ in eq.(16) represents the sum of two independent parts : The terms
proportional to $\delta(p ^2-\mu^2\gamma^2) $ correspond to a real massive
photon which contributes to the free Hamiltonian; the fourth and fifth terms
are the topological parts of the gauge field which do not contribute to the
classical free
Hamiltonian, but give non-trivial contribution to the propagator (see eq.(15))
, and the last term is merely a gauge term that can be removed by a gauge
transformation. It is easy to see that the topological part of $A_{\mu}$ does
not contribute to $F_{\mu\nu}$:
\begin{eqnarray}
F_{\mu\nu}&=&\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\nonumber\\
&=&{1\over 2\pi\mu}\int d^3pe^{ipx}\delta(p^2)a^{\delta}(p)(p_{\mu}
\varepsilon_{\nu\lambda\rho}-p_{\nu}\varepsilon_{\mu\lambda\rho})
{\it e}^{\lambda}_{\delta}p^{\rho}
\end{eqnarray}
multiplying both sides by $\varepsilon_{\sigma\mu\nu}$ we get
\begin{eqnarray}
\varepsilon^{\sigma\mu\nu}F_{\mu\nu}={1\over \mu\pi}\int d^3pe^{ipx}\delta(p^2)
a^{\delta}(p)({\it e}^{\sigma}_{\delta}(p)p^2-p_{\mu}{\it e}^{\mu}_{\delta}
p^{\sigma})=0
\end{eqnarray}
since
\begin{eqnarray}
p_{\mu}{\it e}^{\mu}_{\alpha}=0
\end{eqnarray}
As for the the massive part of the solution (16), excluding the second term
in this equation in view of (19) above, then we have for the massive part
\begin{eqnarray}
A_{\mu}(x)={-1\over 2\pi}\int d^3pe^{ipx}\gamma\left({\it e}^{\delta}_{\mu}(p)
-{i\over\mu\gamma}\varepsilon_{\mu\nu\rho}p^{\rho}{\it e}^{\nu}_{\delta}\right)
a_{\delta}(p)\delta(p^2-\mu^2\gamma^2)
\end{eqnarray}
and this gives a non-vanishing contribution to $F_{\mu\nu}$. We shall
return later to the question of quantization of this $A_{\mu}$ in connection
with the construction of the S-matrix of the theory (see part III).\\
Returning to the general expression for the Greens function of the gauge field
, we stress that formally it is possible to consider two limiting procedures
in eq.(15). First, if $\gamma\to\infty$ we obtain:
\begin{eqnarray}
\lim_{\gamma\to\infty}D_{\lambda\nu}=D^{CS}_{\lambda\nu}={-1\over(2\pi)^3}\int
d^3p
e^{ipx}\left({i\varepsilon_{\nu\lambda\rho}p^{\rho}\over\mu(p^2+i\epsilon)}
+{\alpha p_{\lambda}p_{\nu}\over (p^2+i\epsilon)^2}\right)
\end{eqnarray}
which is just the propagator of the pure CS theory. In the limit $\mu\to 0$, we
get the usual Feynman propagator in 2+1 dimensional QED for massless photons:
\begin{eqnarray}
\lim_{\mu\to 0}D_{\lambda\nu}(x)=D^M_{\lambda\nu}(x)={-\gamma\over (2\pi)^3}
\int d^3pe^{ipx}{\left(g_{\lambda\nu}-({p_{\lambda}p_{\nu}\over p^2+i\epsilon})
(1-{\alpha\over\gamma})\right)\over(p^2+i\epsilon)}
\end{eqnarray}
In both cases, we have from eq.(16) $A_{\mu}$ as
\begin{eqnarray}
\lim_{\gamma\to\infty}A_{\mu}(x)&=&{-1\over 2\pi}\int d^3pe^{ipx}\left[\left(
{i\over\mu}\varepsilon_{\mu\nu\rho}p^{\rho}-
{\alpha\over 2}p_{\mu}{\partial\over\partial p^{\nu}}\right)\delta(p^2)\right]
{\it e}^{\nu}_{\delta}(p)a^{\delta}(p)\nonumber\\
&=&A^{CS}_{\mu}
\end{eqnarray}
and
\begin{eqnarray}
\lim_{\mu\to 0}A_{\mu}(x)={-\gamma\over 2 \pi}\int d^3pe^{ipx}\left[\left(
g_{\mu\nu}+{(1-{\alpha\over\gamma})\over 2}p_{\mu}{\partial\over
\partial p^{\nu}}
\right)\delta(p^2)\right]{\it e}^{\nu}_{\delta}(p) a^{\delta}(p)
\end{eqnarray}
The limits are to be taken after renormalization,
recalling that in our theory one has only the finite one-loop correction to
$\mu$ (or $\gamma$) [5-7].
Spinor CS theory:
$~~~$Let us consider now the theory of spinor particles interacting through the
CS gauge field. The DFP method gives the following expression for the
generating functional in this case [14]:
\begin{eqnarray}
Z[J_{\mu},\eta,\bar\eta]=Z_0^{-1}\int DA_{\mu}(x)D\bar\psi(x)D\psi(x)\exp\{
iS_{CS}+iS_M+iS_g+iS_{\psi}\nonumber\\
+i\int d^3x(J_{\mu}(x)A^{\mu}(x)+\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x))\}
\end{eqnarray}
where $Z_0=Z(0,0,0)~;~S_{CS},S_g$ and $S_M$ are defined by eqs.(3),(4) and
(7) respectively, and
\begin{eqnarray}
S_{\psi}=\int d^3x\bar\psi(x)(iD\!\!\!\!/-m)\psi(x)
\end{eqnarray}
where,
\begin{eqnarray}
D\!\!\!\!/=D_{\mu}\gamma^{\mu}~~~~~~,~~~D_{\mu}=(\partial_{\mu}-ieA_{\mu})
\end{eqnarray}
and the Dirac matrices are defined as
\begin{eqnarray}
\gamma_0=\sigma_3~~~,~~\gamma_i=i\sigma_i~~~~,i=1,2
\end{eqnarray}
where $\sigma$'s are the Pauli spin matrices. The $\gamma$-matrices satisfy
\begin{eqnarray}
\{\gamma_{\mu},\gamma_{\nu}\}=2g_{\mu\nu}~~~,~~\gamma_{\mu}\gamma_{\nu}=
g_{\mu\nu}-i\varepsilon_{\mu\nu\lambda}\gamma^{\lambda}
\end{eqnarray}
$\psi(x)$ and $\bar\psi(x)=\psi(x)^{\dagger}\gamma_0$ are the two-component
Grassmann spinors, $\eta$ and $\bar \eta$ are Grassmann sources. Integrating
over $A_{\mu}(x)$ in eq.(25) we get :
\begin{eqnarray}
Z[J_{\mu},\bar\eta,\eta]=Z_0^{-1}\int D\bar\psi(x)D\psi(x)\exp\{{i\over 2}
\int d^3xd^3y\tilde I_{\mu}(x)D^{\mu\nu}(x-y)\tilde I_{\nu}(y)\nonumber\\
+i\int d^3x(\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x))\}
\end{eqnarray}
where
\begin{eqnarray}
\tilde I_{\mu}(x)=J_{\mu}(x)+e\bar\psi(x)\gamma_{\mu}\psi(x)
\end{eqnarray}
and $D_{\mu\nu}(x-y)$ is the bare CS field propagator which is the same as in
the scalar case, eq.(15). Here also, as in the scalar case, one can consider
the limits (after renormalization) $\gamma\to\infty$ and $\mu\to 0$ to get the
propagators of pure CS field and 2+1 dimensional QED respectively.\\
Non-Abelian CS theory:
$~~$The path integral quantization of theories with the non-Abelian CS gauge
field is a bit more complicated than the Abelian one, so we consider it in some
detail. We start with the theory of the gauge field without coupling to matter
, i.e CS gluodynamics, defined by the Lagrangian
\begin{eqnarray}
{\cal L}={\cal L}_M+{\cal L}_{CS}
\end{eqnarray}
${\cal L}_M$ is the usual Yang-Mills Lagrangian in 2+1 dimensions,
\begin{eqnarray}
{\cal L}_M&=&{-1\over 2\gamma}tr\left(F_{\mu\nu}(x)F^{\mu\nu}(x)\right)
\nonumber\\
F_{\mu\nu}&=&\partial_{\mu}A_{\nu}(x)-\partial_{\nu}A_{\mu}(x)+g[A_{\mu}(x),
A_{\nu}(x)]
\end{eqnarray}
${\cal L_{CS}}$ is the non-Abelian CS term
\begin{eqnarray}
{\cal L}_{CS}=-\mu\varepsilon^{\mu\nu\lambda}tr(A_{\mu}(x)\partial_{\nu}
A_{\lambda}(x)+{2i\over 3}gA_{\mu}(x)A_{\nu}(x)A_{\lambda}(x))
\end{eqnarray}
The gauge group is $SU(N)$. In matrix notation
\begin{eqnarray}
A_{\mu}=A_{\mu}^at^a~~~~;~~~~F_{\mu\nu}=F_{\mu\nu}^at^a
\end{eqnarray}
The $t^a$'s are antihermitian matrices in the fundamental representation of
the group
\begin{eqnarray}
[t^a,t^b]=if^{abc}t_c~~~~,~~~tr(t^at^b)={1\over 2}\delta^{ab}
\end{eqnarray}
$f^{abc}$ are the structure constants of the $SU(N)$ group.\\
To see the difference of the non-Abelian case from the Abelian one, consider
a general gauge transformation
\begin{eqnarray}
A_{\mu}(x)\to U^{-1}(A_{\mu}(x)-{i\over g}\partial_{\mu})U
\end{eqnarray}
${\cal L}_M$ is gauge-invariant, ${\cal L}_{CS}$ is not [2,3];
\begin{eqnarray}
\int d^3x{\cal L}_{CS}\to\int d^3x{\cal L}_{CS}-{i\mu\over g}\int d^3x
\varepsilon^{\mu\nu\lambda}\partial_{\mu}tr\left((\partial_{\nu}U)U^{-1}
A_{\lambda}\right)\nonumber\\
+{8\pi^2\mu\over g}iw
\end{eqnarray}
where
\begin{eqnarray}
w={1\over 24g\pi^2}\int d^3x\varepsilon^{\mu\nu\lambda}tr\left[(U^{-1}
\partial_{\mu}U)(U^{-1}\partial_{\nu}U)(U^{-1}\partial_{\lambda}U)\right]
\end{eqnarray}
If we suppose that at $||x||=\sqrt{x_0^2+\vec x^2}\to\infty~,~A_{\mu}\to 0$
faster than ${1\over ||x||}$ then the second term in (38) vanishes. The last
term
, however, coincides in {\it euclidean\/} space, with the so called homotopy
class
or winding number, and is equal to $0,\pm 1,\pm 2,...$ . This result follows
from the fact that if
\begin{eqnarray}
U(x)_{||x||\to\infty}\to 1,
\end{eqnarray}
then 3-dimensional space can be mapped onto $S_3$; for $SU(2)$ group $U(x)$
realizes the mapping $S_3\to S_3$ and the winding number is equal to the
degree of mapping $S_3$ to the $SU(2)$ group. On the classical level, the
gauge-transformation (37) results in
\begin{eqnarray}
S_{CS}\to S_{CS}+constant
\end{eqnarray}
It is clear that this constant does not influence the equations of motion or
any physical quantity.\\
Now, we use the Fadeev-Popov trick to quantize the theory. Formally, the
vacuum functional of the theory is
\begin{eqnarray}
Z_0=N\int DA_{\mu}\exp i\{S_M+S_{CS}\}
\end{eqnarray}
where $N$ is a normalization factor that will be defined later. Introducing
into the formal equation (42) the identity operator in a general Lorentz
covariant gauge
\begin{eqnarray}
I=\bigtriangleup(A)\int D\mu(G)\delta(\partial^{\mu}A_{\mu}^G-f(x))
\end{eqnarray}
where $D\mu(G)$ is the measure of the $SU(N)$ group, and
\begin{eqnarray}
A_{\mu}^G=U^{-1}(A_{\mu}-{i\over g}\partial_{\mu})U~~~~U\in G.
\end{eqnarray}
Eq.(43) defines the Fadeev-Popov determinant $\bigtriangleup (A)$.\\
We know that in perturbation theory we can forget about the Gribov ambiguity
[18] and consider only contributions to the functional integral from elements
near the identity of the group $G$;
\begin{eqnarray}
U=1+i\lambda(x)+O(\lambda^2)~~~~~;~~~~\lambda=\lambda^at^a
\end{eqnarray}
where $\lambda^a(x)$ is infinitesimally small for all x. This means that in
DFP method we
must consider gauge transformations which belongs to the zero homotopy class
for which $w=0$
since $\lambda(x)$ must go to zero when $||x||\to\infty$ \footnote{In the
general case when $U=e^{i\tau^a\lambda^a(x)}$ and $||x||\to\infty~,~\lambda(x)
=\sqrt{(\lambda^a)^2}\to 2\pi n$ where $n$ is the winding number.}.
Substituting the identity operator
(43) into the expression (42), we get after the conventional manipulations
\begin{eqnarray}
Z_0=N\Omega(G)\int DA_{\mu}(x)D\bar{\cal C}(x)D{\cal C}(x)\exp\{i(S_M+S_{CS}
+S_g)\}
\end{eqnarray}
Here $\Omega(G)$ is the infinite group volume, and
\begin{eqnarray}
S_g=\int d^3xtr\left({-1\over 2\alpha}(\partial_{\mu}A^{\mu}(x))^2+
\partial_{\mu}\bar{\cal C}^a(x)(D^{\mu ab}{\cal C}^b(x))\right)
\end{eqnarray}
where ${\cal C}(x)$ and $\bar {\cal C}(x)$ are the well-known Fadeev-Popov
ghosts that are
scalar Grassmann fields, and
\begin{eqnarray}
D_{\mu}^{ab}=\partial_{\mu}\delta^{ab}+gf^{abc}A_{\mu}^c(x)
\end{eqnarray}
Thus, the generating functional of the theory is now given by the expression:
\begin{eqnarray}
Z[J_{\mu},\eta,\bar\eta]=Z^{-1}(0,0,0)\int DA_{\mu}(x)D\bar{\cal C}(x)D{\cal C}
(x)\exp\{i(S_M+S_{CS}+S_g)\nonumber\\
+i\int d^3x(J_{\mu}^a(x)A^{\mu}_a(x)+\bar\eta^a(x){\cal C}^a(x)+\bar{\cal C}^a
(x)\eta^a(x))\}
\end{eqnarray}
here,
\begin{eqnarray}
Z(0,0,0)=Z(J_{\mu},\bar\eta,\eta)\left.\right|_{J_{\mu}=\bar\eta=\eta=0}
\end{eqnarray}
With the above generating functional, the expectation value of any observable
is well-defined. For example
\begin{eqnarray}
\left<\hat{\cal O}(\hat A,\hat{\bar{\cal C}},\hat{\cal C})\right>&=&
{\cal O}({-i\delta\over\delta J^a_{\mu}(x)},{-i\delta^l\over\delta\eta^b(x)},
{i\delta^l\over\delta\eta^c(x)})Z[J_{\mu},\bar\eta,\eta]\left.\right|_{J_{\mu}=
\bar\eta=\eta=0}\nonumber\\
&=&Z^{-1}(0,0,0)\int DA_{\mu}D\bar{\cal C}(x)D{\cal C}(x)
{\cal O}(A,\bar{\cal C},{\cal C})\nonumber\\
&\times&\exp\{i(S_M+S_{CS}+S_g)\}
\end{eqnarray}
This expression does not change under any gauge transformation of the total
action .\\
$~~~$ We have seen above that the exponent in the generating functional of the
theory contains after quantization the term $S_g$ that violates
gauge-invariance.
However, the total action in the exponent preserves invariance under a special
class of BRST [19] gauge supertransformations
\begin{eqnarray}
A_{\mu}^a(x)\to A_{\mu}^a(x)+(D_{\mu}{\cal C}(x))^a\epsilon\\
{\cal C}^a(x)\to{\cal C}^a(x)-{1\over 2}f^{abc}{\cal C}^b(x){\cal C}^d(x)
\epsilon\\
\bar{\cal C}^a(x)\to\bar{\cal C}^a(x)+{1\over\alpha}(\partial_{\mu}A^{\mu a}(x)
)\epsilon
\end{eqnarray}
where $\epsilon$ is an $x$-independent Grassmann parameter $(\epsilon^2=0)$;
\begin{eqnarray}
\{\epsilon,\bar{\cal C}\}_+=\{\epsilon,{\cal C}\}_+=0=\{\epsilon,A_{\mu}(x)\}_-
\end{eqnarray}
It is well-known that $S_g$ does not change under these transformations. If we
formally write down the transformation law of $A_{\mu}(x)$ in the form
\begin{eqnarray}
A_{\mu}'=U^{-1}(A_{\mu}-{i\over g}\partial_{\mu})U
\end{eqnarray}
where
\begin{eqnarray}
U=\exp\{it^a{\cal C}^a\epsilon\}=1+it^a{\cal C}^a\epsilon
\end{eqnarray}
then under this transformation
\begin{eqnarray}
S_{CS}\to S_{CS}+{8\pi^2\mu^2\over g^2}iw=S_{CS}
\end{eqnarray}
since $w=0$ because $\epsilon^2=0$.\\
It is very important to stress that the BRST-invariance of the CS theory
ensures satisfying all the Ward-Fradkin-Takahishi-Slavnov-Taylor identities
[20], and therefore the gauge-invariant renormalizability of the theory [21].
We thus come to the conclusion that in the framework of perturbation
theory, it is not necessary to quantize the charge in CS gluodynamics. The
same holds also if coupling to matter is introduced into the theory as well.\\
$~~$If one introduces spinor field into the theory (CS quantum chromodynamics
(CSQCD)), then it is straight forward to generalize the generating functional
eq.(49) to this case. The resulting expression is
\begin{eqnarray}
Z[J_{\mu}^a,\bar\eta,\eta]=Z_0^{-1}\prod_a\int DA_{\mu}^a(x)D\bar\psi(x)
D\psi(x)\exp\{i(S_{CS}+S_M+S_g+\tilde S_{\psi})\nonumber\\
+i\int d^3x(J_{\mu}^a(x)A^{\mu}_a(x)+\bar\eta(x)\psi(x)+\bar\psi(x)\eta(x))\}
\end{eqnarray}
Here $S_{CS},S_M$ and $S_g$ were defined earlier,eqs.(3),(4) and (7),and
\begin{eqnarray}
\tilde S_{\psi}=\int d^3x\bar\psi_i(x)(\partial\!\!\!/+eA\!\!\!/(x)-m)_{ij}
\psi_j(x)
\end{eqnarray}
$i,j=1,...,N$ above are the color indices of the $SU(N)$ group in the
fundamental
representation.It is straight forward to write down the Feynman propagator of
the non-Abelian gauge field; it will differ from
the abelian one only by the appearence of color indices viz.
\begin{eqnarray}
D_{\mu\nu}^{ab}(x)=\delta^{ab}D_{\mu\nu}(x)
\end{eqnarray}
$~~~$Before leaving this subsection, we would like to emphasize that starting
from the generating functionals for the various models that have been
considered so far, one can construct all the propagators and the primitive
vertices, and thus develop the Feynman rules for perturbation theory. For
example, the Feynman propagator for the scalar field is given as
\begin{eqnarray}
G(x-y)&=&i{\delta^2Z[J_{\mu},j,j^*]\over\delta j(x)\delta j^*(y)}\left.\right|_
{J_{\mu}=j=j^*=e=0}\nonumber\\
&=&{1\over (2\pi)^3}\int d^3p{e^{ip(x-y)}\over p^2-m^2+i\epsilon}
\end{eqnarray}
Similarly, we have for the spinor propagator from eq.(59)
\begin{eqnarray}
S(x-y)&=&(-i)^2{\delta^l\delta^rZ\over\bar\eta(x)\eta(y)}\left.\right|_{J_
{\mu}=\eta=\bar\eta=e=0}\nonumber\\
&=&{1\over (2\pi)^3}\int d^3p{e^{ip(x-y)}\over p\!\!\!/-m}
\end{eqnarray}
\subsection{Path Integral Quantization Of Pure Chern-Simons Quantum
Chromodynamic by Batalin-Fradkin-Vilkovisky
Method}
Here, we shall show how to construct the generating functional of the theory
of spinors coupled to the non-Abelian CS field by the BFV method. We shall
consider however, a theory where the gauge field kinetic action is given solely
by the non-abelian CS term, i.e pure CSQCD. This will have a more complicated
constraint structure than the one with the Maxwell term (eq.(7)) included.The
BFV quantization makes the BRST symmetry of the theory, that
is generated by the operator $\Omega$ introduced below, more transparent. We
start with the classical action:
\begin{eqnarray}
S=S_{CS}+S_{\psi}
\end{eqnarray}
where $S_{CS}$ and $S_{\psi}$ are given by eqs.(3) and (26). The action
can be written in a more transparent form:
\begin{eqnarray}
S_{CS}&=&-{\mu\over 2}\int d^3x(A_0^a(x)\varepsilon_{ij}F^{ija}+
\varepsilon_{ij}\dot A^{ia}(x)A^{ja}(x)+{g\over 3}f^{abc}
\varepsilon_{\mu\nu\lambda}A^{\mu}_a(x)A^{\nu}_b(x)A^{\lambda}_c(x))\\
S_{\psi}&=&\int d^3x\left( \bar\psi(x)(i\gamma_0\partial_0-
i\vec\gamma.\vec\nabla -m)\psi(x)-gA_{\mu}(x)\bar\psi(x)\gamma^{\mu}
\psi(x)\right)
\end{eqnarray}
The canonical momenta of the theory turn out to be all primary constraints:
\begin{eqnarray}
\pi_i^a&=&{\delta{\cal L}\over\delta\dot A^{ia}}={-\mu\over 2}
\varepsilon_{ij}
A^{ja}~~~~~;~~\theta_i^a\equiv \pi_i^a+{\mu\over 2}\varepsilon_{ij}A^{ja}
\approx 0\\
\pi_{\psi}&=&{\delta^r{\cal L}\over\delta\dot\psi}=i\psi^{\dagger}~~~~~~~~~~~~
~~;~~
\theta_3\equiv\pi_{\psi}-i\psi^{\dagger}\approx 0\\
\pi_{\psi^{\dagger}}&=&{\delta^l{\cal L}\over\delta\dot\psi^{\dagger}}=0
~~~~~~~~~~~~~~;~~\theta_4\equiv\pi_{\psi^{\dagger}}\approx 0\\
\pi_0^a&=&{\delta{\cal L}\over\delta\dot A_0^a}=0~~~~~~~~~~~~~~ ;~~
G^a\equiv\pi_0^a\approx 0
\end{eqnarray}
The standard Poisson brackets are:
\begin{eqnarray}
\{\psi(x),\pi_{\psi}(y)\}&=&\{\psi^{\dagger}(x),\pi_{\psi^{\dagger}}(y)\}
=\delta(\vec x-\vec y)\\
\{A_{\mu}^a(x),\pi_{\nu}^b(y)\}&=&g_{\mu\nu}\delta^{ab}\delta(\vec x-\vec y)
\end{eqnarray}
$\theta_i,\theta_3$ and $\theta_4$ are second class constraints, while $G^a$
is first class. The presence of the second class constraints motivates one to
define the Dirac brackets [22] using these constraints. These can be worked
out easily, and the ones that differ from the Poisson bracket are:
\begin{eqnarray}
\{\psi(x),\psi^{\dagger}(y)\}_D&=&i\delta(\vec x-\vec y)\\
\{A_i^a(x),A_j^b(y)\}_D&=&{-1\over\mu}\delta^{ab}\varepsilon_{ij}\delta(\vec x
-\vec y)\\
\{A_i^a(x),\pi_j^b(y)\}_D&=&{1\over 2}g_{ij}\delta^{ab}\delta(\vec x-\vec y)
\end{eqnarray}
The Hamiltonian assumes the form
\begin{eqnarray}
{\cal H}&=&{\cal H}_0+A_0^aT^a\nonumber\\
&=&\bar\psi(x)(i\vec\gamma.\vec\nabla+m)\psi(x)-\vec A(x).\vec J(x)\nonumber\\
&+&A_0^a(x)\left(\right.
J_0^a(x)+{\mu\over 2}\varepsilon_{ij}F^{ija}
+g{\mu\over 2}f^{abc}\varepsilon
_{ij}A^{ib}(x)A^{jc}(x)\left.\right)
\end{eqnarray}
${\cal H}_0$ is the Hamiltonian on the constraint surface, and $T^a$ is a
first class constraint that is the analogue of Gauss' law constraint in QCD
, and here also it is the generator of the gauge symmetry.
$A_0$ appears here, as is the case in QED and QCD, as a Lagrange multiplier.
The first class constraint $T^a$ can be seen to satisfy the algebra :
\begin{eqnarray}
\{T^a(x),T^b(y)\}_D&=&-gf^{abc}T^c\delta(x-y)\approx 0\\
\{T^a(x),{\cal H}(y)\}_D&=&0
\end{eqnarray}
The BFV quantization method, in attempting to maintain Lorentz covariance and
the unitarity of the S-matrix expands the phase space of the theory by making
the Lagrange multiplier of the theory dynamical, and introducing new (ghost)
degrees of freedom whose statistics are oppositte to the first class
constraints of the theory. In our case we will have two pairs of these ghosts
which are Grassmann fields ;
\begin{eqnarray*}
({\cal C}^a,\bar {\cal P}^a)~~~~;~~~~({\cal P}^a,\bar{\cal C}^a).
\end{eqnarray*}
Therefore, our canonical variables become now
\begin{eqnarray}
Q^A=(A_i^a,\psi,\psi^{\dagger},A_0^a,{\cal C}^a,{\cal P}^a)\\
P^A=(\pi_i^a,\pi_{\psi},\pi_{\psi^{\dagger}},\pi_0^a,\bar{\cal P}^a,
\bar{\cal C}^a)
\end{eqnarray}
Generally, the BFV method introduces the so called complete Hamiltonian [16]
that enters into the expresion of the generating functional, which is
defined as
\begin{eqnarray}
{\cal H}^{comp}={\cal H}_0+\{\Psi,\Omega\}_D
\end{eqnarray}
$\Psi$ is the gauge fermion of the theory and contains all the gauge degrees of
freedom. $\Omega$ is the BRST charge of the theory, and satisfies :
\begin{eqnarray}
\{\Omega,{\cal H}\}_D&=&0\\
\{\Omega,\Omega\}_D&=&0
\end{eqnarray}
Generally, ${\cal H_0},\Psi$ and $\Omega$ are found as expansions in powers of
the ghost fields by solving eqs. (83) and (84) above. However,
in our case, due to the simplicity of the algebra of the constraints, we get
${\cal H}_0$ to zeroth order, $\Psi$ to first order and $\Omega$ to second
order in the ghost fields. Thus
\begin{eqnarray}
\Psi&=&\bar{\cal C}^a\chi^a+\bar{\cal P}^aA_0^a\\
\Omega&=&\pi_0^b{\cal P}^b+T^b{\cal C}^b-{1\over 2}\bar{\cal P}_bf^{bcd}
{\cal C}^d{\cal C}^c
\end{eqnarray}
where $\chi^a$ is a gauge-fixing function
\begin{eqnarray}
\chi_i^a=\partial_iA^{a}_i-f^a(x)
\end{eqnarray}
The vacuum functional of the theory is given now by the expression
\begin{eqnarray}
Z_0=N\int D\mu(Q,P)\exp i\{\int d^3x(P_A\dot Q^A-{\cal H}^{comp})\}
\end{eqnarray}
where $P_A$ and $Q_A$ are given in eqs.(79) and (80), and
\begin{eqnarray}
D\mu(Q,P)=DA_i^a DA_0^a D\psi D\bar\psi D\pi_{\psi} D\pi_{\psi^{\dagger}}
D\pi_i^a D\pi_0^a D{\cal C}^a D\bar{\cal C}^a D{\cal P}^a D\bar{\cal P}^a
\nonumber\\
\times\delta(\pi_i^a+{\mu\over 2}\varepsilon_{ij}A^{ja})\delta(\pi_{\psi}-i\psi
^{\dagger})\delta(\pi_{\psi^{\dagger}}\left(Ber||\{\theta_l,\theta_m\}||\right)
^{1\over 2}
\end{eqnarray}
$Ber$ is the superdeterminant, or the Berezinian, which is introduced here due
to the presence of the fermionic degrees of freedom.
Integrating over the matter and gauge momenta and over $\pi_0^a,\bar{\cal P}^b$
and ${\cal P}^a$ we get
\begin{eqnarray}
Z_0=N\int DA_{\mu}D\psi D\bar\psi D{\cal C}D\bar{\cal C}\delta(\dot A_0^a(x)-
\partial_iA_i^a(x)+f^a(x))\nonumber\\
\times \exp i\{\int d^3x({\cal L}_{cl}-\bar{\cal C}^a(\partial_\mu D^{\mu ab}
{\cal C}^b))\}
\end{eqnarray}
where
\begin{eqnarray}
{\cal L}_{cl}=i\bar\psi(\partial\!\!\!/-m)\psi-A_{\mu}^aJ^{\mu a}-{\mu\over 2}
A_0^a\varepsilon_{ij}F^{ija}-{\mu\over 2}\varepsilon_{ij}\dot A^{ia}A^{ja}
\nonumber\\
-g{\mu\over 2}f^{abc}A_0^a\varepsilon_{ij}A^{ib}A^{jc}
\end{eqnarray}
and,
\begin{eqnarray}
\bar {\cal C}^a\partial_\mu D^{\mu}_{ ab}{\cal C}^b=\bar{\cal C}^a
(\partial_{\mu}
(\delta_{ab}\partial^{\mu}-f^{acb}A^{\mu}_{c})){\cal C}^b
\end{eqnarray}
The above expression -upon including external sources- coincides with the
generating functional eq.(59) without the Maxwell term .\\
\section{The S-Matrix Operator}
$~~~$Although the generating functionals of the theory, eqs.(11),(30) and (59)
contain all the information of the theory, and can be used to derive the
scattering amplitudes, it is more convenient to either introduce the path
integral representation of the S-matrix of the theory, or to construct the
S-matrix opertator. The latter is particularly convenient for the investigation
of the imaginary parts of the Greens functions, Feynman diagrams and the
scattering matrix elements, or generally speaking, for the investigation of the
unitarity of the theory. We shall first construct the S-matrix operator of the
pure CSQED, and then generalize the results to the other cases. A peculiar
property of the pure CSQED is the absence of real topological photons,
although the propagator and its imaginary part exist (see eqs.(21) and (23)
for example). As for the operator $\hat A_{\mu}(x) $; we note that
canonical quantization in covariant gauges allows one to introduce (as in QED)
operators for the scalar as well as the longitudinal components of
$A_{\mu}(x)$, and it can be proven that due to the canonical commutation
relations, the equation for the propagator
\begin{eqnarray}
D_{\mu\nu}=-i\langle T \hat A_{\mu}(x)\hat A_{\nu}(y)\rangle\nonumber
\end{eqnarray}
coincides with the classical equation (14) and have the same solution, eq.(15).
However, in the case of pure CS theory, this topological photon does not
contribute to the physical states of the Hilbert space, which can be defined
as usual; $\partial_{\mu}\hat A_{\mu}^+|phys\rangle=0$.
Thus, starting from this result,
one can unambigously formulate rules for the construction of any
matrix elements of the different products of this operator. In this sense,
one can formulate some kind of Wick theorem for the operators of the
topological CS photon. The S-matrix operator for scalar pure CSQED has been
construsted in the works [11,13], and those for spinor CSQED in [14]. Here,
we would like to
elaborate on the construction given in these references.\\
In pure CSQED, the S-matrix operator formally has the same form as that in
2+1 dimensional QED,
\begin{eqnarray}
\hat S=T\exp \{iS_{int}(\hat{\bar\psi},\hat\psi,\hat A)\}
\end{eqnarray}
here,
\begin{eqnarray}
S_{int}(\hat{\bar\psi},\hat\psi,\hat A)=\int d^3x :e(\hat{\bar\psi}\gamma^{\mu}
\hat A_{\mu}\hat\psi):
\end{eqnarray}
where ": :" means normal ordering, and $\hat\psi(x)$ and $\hat{\bar\psi}(x)$
operators are given as
\begin{eqnarray}
\hat\psi(x)&=&\int {d^3p\over (2\pi)}\sqrt{{m\over E_{\vec p}}}[b
(\vec p)u(p)e^{-ipx}+d^{\dagger}(\vec p)v(p)e^{ipx}]\\
\hat{\bar\psi(x)}&=&\int {d^3p\over (2\pi)}\sqrt{{m\over E_{\vec p}}}[b^
{\dagger}(\vec p)\bar u(p)e^{ipx}+d(\vec p)\bar v(p)e^{-ipx}],
\end{eqnarray}
$E_{\vec p}=\sqrt{\vec p^2+m^2}$ and $b(\vec p)~(d(\vec p))$ and
$b^{\dagger}
(\vec p)~(d^{\dagger}(\vec p))$ are respectively the annihilation and creation
operators of particles (antiparticles) satisfying the usual anticommutation
relations :
\begin{eqnarray}
[b(\vec p),b^{\dagger}(\vec p')]_+=[d(\vec p),d^{\dagger}(\vec p')]_+=\delta(
\vec p-\vec p')
\end{eqnarray}
The two-component spinors $u(p), v(p)$ are respectively the
positive and negative energy solutions of the free Dirac equation in (2+1)
dimensions, with the properties:
\begin{eqnarray}
(\hat p-m)u(p)&=&(\hat p+m)v(p)=0\\
\bar u(p)u(p)&=&-\bar v(p)v(p)=1\\
\bar u(p)v(p)&=&\bar v(p)u(p)=0\\
u(p)\bar u(p)&=&{p\!\!\!/+m\over 2m}\\
v(p)\bar v(p)&=&{p\!\!\!/-m\over 2m}
\end{eqnarray}
$~$Let us next our attention to the operator $\hat A_{\mu}$: Using its above
mentioned properties, we can formulate
the following rules of the matrix elements of its
products : 1) The vacuum expectation value of the products and the
T-products
of only an even number of the operators $A_{\mu}$ is nonvanishing, and reduces
respectively to the sum of the vacuum expectation values of the product and
the T-product of two field operators defined as:
\begin{eqnarray}
\langle 0|T(\hat A_{\mu}(x)\hat A_{\nu}(y))|0\rangle&=&-iD_{\mu\nu}(x-y)\\
\langle 0|\hat A_{\mu}(x)\hat A_{\nu}(y)|0\rangle&=&-iD_{\mu\nu}^+
(x-y)\nonumber\\
&=&-i\int {d^3p\over(2\pi)^3}\left[\left({i\over\mu}\varepsilon_{\mu\nu
\lambda}p^{\lambda}-{\alpha\over 2}p_{\mu}{\partial\over \partial^{\nu}}\right)
\delta(p^2)\right]\theta(p_0) e^{ip(x-y)}
\end{eqnarray}
where $D_{\mu\nu}(x-y)$ is given by eq.(21).
For example, for four operator product we have:
\begin{eqnarray}
\langle 0|T(\hat A_{\mu}(x)\hat A_{\nu}(y)\hat A_{\lambda}(z)\hat A_{\delta}(u)
|0\rangle&=&(-i)^2\{D_{\mu\nu}(x-y)D_{\lambda\delta}(z-u)+D_{\mu\lambda}(x-z)
\nonumber\\
& &D_{\nu\delta}(y-u)+D_{\mu\delta}(x-u)D_{\nu\lambda}(y-z)\}
\end{eqnarray}
and so on. 2) All the matrix elements between physical states of the normal
product
of any number of the field operators $A_{\mu}$ are equal to zero
. However, the vacuum expectation value of the product of
the normal products of equal number of these operators only is different
from zero. For example:
\begin{eqnarray}
\langle 0|:\hat A_{\mu}(x)\hat A_{\nu}(y)::\hat A_{\lambda}(z)\hat A_{\delta}
(u):|0\rangle=\nonumber\\
(-i)^2\{D^+_{\mu\lambda}(x-z)D^+_{\nu\sigma}(y-u)+D^+_{\mu\sigma}(x-u)D_{\nu
\lambda}^+(y-z)\}
\end{eqnarray}
and so on.\\
Thus, the above rules are the same as the Wick rules except that we take into
account the absence of physical states with free topological photons
(other than the
vacuum state !). Therefore, we make now the following observation: All the
Feynman rules of the theory are identical to those of QED given that one
replaces
the Maxwell propagator in internal lines by the CS propagator, and excludes
diagrams with external photon lines. In mathematical language, the above rules
mean that the total set of physical states in the total Hilbert space of the
theory
does not contain states with real free topological photons \footnote{The
absence of the real topological free photons can be seen most generally from
the fact that the CS term does not contribute to the free classical
Hamiltonian due to
its independence of the metric tensor $g_{\mu\nu}$ in curved space-time.}
, but only the
physical states of particles and antiparticles. The interesting consequences
and applications of these statements will be considered in part IV.\\
Consider now the more general case of MCSQED, where the propagator is given by
eq.(15) and the free field solutions of the classical equations of motion by
eq.(16). This solution consists of two parts : massive physical part and
massless topological part. The canonical quantization of the massive part in
the $\alpha=0$ gauge can be carried out, and gives the following representation
for the physical massive part $\hat A_{\mu}^m(x)$ of the operator
$\hat A_{\mu}\equiv \hat A_{\mu}^m+\hat A_{\mu}^{CS}$\footnote{The details of
the canonical quantization, which is very similar to the Gupta-Bluer
quantization will be published in another paper.}
\begin{eqnarray}
\hat A_{\mu}^m(x)={-1\over 2\pi}\int d^3p e^{ipx}\gamma\left({\it e}^
{\delta}_{\mu}(p)-{i\over\mu\gamma}\varepsilon_{\mu\nu\rho}p^{\rho}{\it e}^
{\nu\delta}(p)\right)\delta(p^2-\mu^2\gamma^2)a_{\delta}(p)
\end{eqnarray}
The S-matrix in this case looks formally the same as (93), but the Wick
theorem is now the usual one
\begin{eqnarray}
\langle 0|T\hat A_{\mu}(x)\hat A_{\nu}(y)|0\rangle&=&-iD_{\mu\nu}(x-y)+:
\hat A_{\mu}(x)\hat A_{\nu}(y):\\
\langle 0|\hat A_{\mu}(x)\hat A_{\nu}(y)|0\rangle&=&-iD_{\mu\nu}^+(x-y)+:\hat
A_{\mu}(x)\hat A_{\nu}(y):
\end{eqnarray}
\begin{eqnarray}
\langle 0|:\hat A_{\mu}(x)\hat A_{\nu}(y)::\hat A_{\lambda}(z)\hat A_{\sigma}
(u):|0\rangle=\nonumber\\
(-i)^2\{D_{\mu\lambda}^+(x-z)D_{\nu\sigma}^+(y-u)&+&D_{\mu\sigma}^+(x-u)
D_{\nu\lambda}^+(y-z)\}\nonumber\\
-i\{D_{\mu\lambda}^+(x-z)\langle 0|:\hat A_{\nu}(y)\hat A_{\sigma}(u):|0\rangle
&+&D_{\mu\sigma}^
+(x-u)\langle 0|:\hat A_{\nu}(y)\hat A_{\lambda}(z):|0\rangle\nonumber\\
+D_{\nu\lambda}^+(y-z)\langle 0|:\hat A_{\mu}(x)\hat A_{\sigma}(u):|0\rangle
&+&D_{\nu\sigma}^+(y-w)\langle 0|:\hat A_{\mu}(x)\hat A_{\lambda}(z):|0\rangle
\}\nonumber\\
+\langle 0|:\hat A_{\mu}(x)\hat A_{\nu}(y)\hat A_{\lambda}(z)\hat A_{\sigma}(u)
:|0\rangle& &
\end{eqnarray}
and so on, where $D_{\mu\nu}(x-y)$ is given by eq.(15).Only one important
exception exists : Any matrix element of the normal product of the operators
$A_{\mu}$ reduces to that of the normal product of the massive operators
$A^m_{\mu}$;
\begin{eqnarray}
\langle f|:A_{\mu_1}(x_1)...A_{\mu_n}(x_n):|i\rangle=\langle f|:A_{\mu_1}^m
(x_1)...A_{\mu_n}^m(x_n):|i\rangle
\end{eqnarray}
Here, $|i\rangle$ and $|j\rangle$ are two arbitrary physical states of the
total
Hilbert space of the theory. Now the total set of physical states includes, in
addition to spinor particles, real massive photons, but never the topological
massless photons. \\
The generalization of the S-matrix operator to scalar or spinor pure CSQCD is
straight forward now. For the spinor case, the generating functional is given
by eq.(90). The S-matrix will have the form
\begin{eqnarray}
S=T\exp\{i\int d^3x\left[-\mu\varepsilon^{\mu\nu\lambda}tr({2i\over 3}e:\hat
A_{\mu}(x)\hat A_{\nu}(x)\hat A_{\lambda}(x):)-{1\over 2\alpha}tr
(:2e\hat F_{\mu\nu}(x)
[\hat A^{\mu}(x),\hat A^{\nu}(x)]_-\right.\nonumber\\
+e[\hat A_{\mu}(x),\hat A_{\nu}(x)]_-^2:)
\left.+e:\partial^{\mu}\hat{\bar{\cal C}}^a(x)f^{abc}\hat A_{\mu}^b(x)\hat
{\cal C}^c(x):+e:\hat{\bar\psi}(x)\gamma_{\mu}\hat A^{\mu}(x)\hat\psi(x) :
\right]\}
\end{eqnarray}
The Wick-type theorem for the operators $\hat\psi,\hat{\bar\psi},\hat{\cal C},
\hat{\bar{\cal C}}$ is as usual.
As for the $\hat A_{\mu}^a$ operator, we have the same rules
as in the Abelian case, except that the Greens function will have now an
additional kronecker delta in the color indices.
\section{Topological Unitarity Identities}
In this part we are going to investigate the consequences of the peculiar
property of the CS theories, namely the absence of real topological photons in
spite of the presence of the propagator and the many-particle Greens function
of the gauge field that contribute to the interaction of the particles quantum
mechanically ( It is well-known that on the classical level, the CS field do
not contribute to the interaction of the particles !). We will see that the
above property of the CS theories leads upon imposing the unitarity condition
on the theory to very interesting topological unitarity identities. These
identities have been derived in the work [14]. Here, we essentially follow the
development in this reference, however, we discuss in more details how do these
identities hold in the general case when the Maxwell term is present
along with the CS term.\\
We consider first the case of CSQED. The propagator is given by eq.(21)
, and the S-matrix operator is given by eq.(92). As
we have mentioned above, the absence of the real CS photons means that the
complete set of physical vector states in the total Hilbert space of the
theory does
not contain these topological particles. To investigate the consequences of
this fact, we introduce the $\hat T$-matrix :
\begin{eqnarray}
\hat S=1-i\hat T
\end{eqnarray}
where $\hat S$ is the S-matrix operator (the energy-momentum conserving
$\delta$-function has been suppressed). The unitarity of the S-matrix operator
leads to the well-known relation:
\begin{eqnarray}
i\left(\hat T^{\dagger}-\hat T\right)=\hat T\hat T^{\dagger}=2Im\hat T
\end{eqnarray}
For arbitrary non-diagonal ($|i\rangle\not =|f\rangle$) on-shell matrix
elements between two physical states of the total Hilbert space, we can write
the two equivalent relations
\begin{eqnarray}
2Im\langle f|\hat T|i\rangle=\langle f|\hat T\hat T^{\dagger}|i\rangle
\end{eqnarray}
and,
\begin{eqnarray}
2Im\langle f|\hat T|i\rangle=\sum_n\langle f|T|n\rangle\langle n|T^
{\dagger}|i\rangle
\end{eqnarray}
where in eq.(115) we have inserted the complete set of physical states
{$|n\rangle$} which does not contain the states of the topological photon, but
only the states of charged particles. From eq.(115) we see that in a given
order
of perturbation theory, the Feynman diagrams that contribute to the imaginary
part on the l.h.s can not have intermediate on-shell topological photon lines
because {$|n\rangle$} are physical states. On the other hand,however,
investigating eq.(114) in the framework of perturbation theory, we can see
that
diagrams with intermediate on-shell photon lines do appear since the vacuum
expectation value of the product of the normal products of equal number of
the operator $A_{\mu}$ (see eq.(105)) does not vanish as a consequence of the
non-vanishing of the imaginary part of the photon propagator. Therefore,
demanding the consistency of eqs. (114) and (115) leads to the important
conclusion that in a given order of perturbation theory, the gauge-invariant
sum of the imaginary parts of the Feynman diagrams with a given number of
intermediate on-shell photon lines is equal to zero. The vanishing of this sum
of the imaginary parts does not mean the vanishing of the sum of the real part
, or the vanishing of the imaginary part of each distinct diagram. As a rule,
the sum of such diagrams will not vanish and will give contribution to the
process involved. Moreover, each diagram in this sum will be an analytic
function of invariant variables. The imaginary part of a distinct diagram will
vanish only if the diagram is gauge-invariant. These arguments will be
demonstrated later when we consider a specific example below.\\
Now, we illustrate these unitarity identities by an explicit example. Consider
the case of scattering of a fermion-antifermion pair in one loop order in pure
CSQED. The S-matrix of this theory is given by eq.(93).
The gauge-invariant Feynman diagrams with intermediate CS topological photon
lines are shown in figure 1 below. The analytic expression for the imaginary
part of each of these diagrams is
\begin{eqnarray}
A_a={2g^4\over (2\pi )^3}\int d^3kd^3k'\left(\delta ^+(k^2)\delta ^+(k'^2)
\delta (p+q-k-k')G_{\mu\lambda}(k)G_{\nu\sigma}(k') \right . \nonumber \\
\times\frac{\bar v(q)\gamma ^\nu (p\!\!\!/-
k\!\!\!/+m)\gamma ^\mu u(p)\bar u(p')\gamma ^\lambda (p'\!\!\!\!/-k\!\!\!/+m)
\gamma
^\sigma v(q')}{((p-k)^2-m^2+i\epsilon)((p'-k)^2-m^2+i\epsilon)} \left .\right),
\end{eqnarray}
\begin{eqnarray}
A_b={2g^4\over (2\pi )^3}\int d^3kd^3k'\left(\delta ^+(k^2)\delta ^+(k'^2)
\delta
(p+q-k-k')G_{\mu\lambda}(k)G_{\nu\sigma}(k') \right . \nonumber \\
\times {\bar v(q)\gamma ^\nu (k\!\!\!/-
q\!\!\!/+m)\gamma ^\mu u(p)\bar u(p')\gamma ^\sigma (p'\!\!\!\!/-k\!\!\!/+m)
\gamma ^\lambda
v(q')\over ((k-q)^2-m^2+i\epsilon )((p'-k)^2-m^2+i\epsilon)}\left .\right).
\end{eqnarray}
where
$G_{\mu\nu}(k)=\varepsilon_{\mu\nu\lambda}k^\lambda$,
and $\delta^+(k^2)=\theta(k_0)\delta(k^2)$. For simplicity, we restrict
ourselves to the case of forward scattering in which case the imaginary part
of these diagrams give their contribution to the total cross-section of the
process. As was shown in [14],a lengthy calculation gives (an overall
irrelevant multiplicative constant has been suppressed)
\begin{eqnarray}
A_a=-\int d^3k\delta^+(k^2)\left(1+{p.k\over m^2}+{q.k\over p.k}\right)=-A_b
\end{eqnarray}
or,
\begin{eqnarray}
A_a+A_b=0
\end{eqnarray}
The same result can be obtained in the case of non-forward scattering too.
This example demonstrates the unitarity identities in the one-loop order.\\
It is not difficult to generalize the unitarity identities to the case when
Maxwell-type terms are present. In such cases, one must divide the total gauge
field propagator in eq.(15) or (62) into two parts (in the $\alpha=0$ gauge
for
example ): physical massive part and topological massless part. The operator
$\hat A_{\mu}$ in the exponent of the S-matrix in eq.(12) can be viewed as the
sum
of two parts too: the massive physical ($\sim \delta(\gamma^2\mu^2-k^2)$), and
the massless topological part ($\sim\delta(k^2)$). States of the massive photon
will appear now in the total Hilbert space of the theory. So, imposing the
unitarity condition on this S-matrix in the sense of eqs.(114) and (115) will
lead to the appearence of the topological unitarity identities in this case
too.\\
For example if we consider the diagrams with two intermediate photon lines
in the one-loop fermion-antifermion scattering, we get the two unitarity
identities illustrated diagramatically in figure 2 ( the lines with $\times$
represent the topological part of the gauge field propagator).
The first identity means that
the sum of the four diagrams (which is gauge-invariant) with one on-shell
intermediate topological photon line is zero. The second identity means the
same for the diagrams with two intermediate on-shell topological lines.\\
The identities developed above can be also shown to hold outside the
framework of
perturbation theory. That they should hold in the non-Abelian case as well,
could be demonstrated without too much difficulty. \\
\section{Concluding Remarks}
In this paper, we have shown that the covariant path integral
quantization of the theories of scalar and spinor particles interacting
through the Abelian and non-Abelian pure CS gauge fields, is mathematically
ill-defined due to the absence of the transverse components of these gauge
fields.
To define the path integral, it is necessary to introduce into the classical
action the Maxwell or Maxwell-type (in the non-Abelian theory) term that is
the only
bilinear term in the gauge field that does not violate the gauge-invariance of
the action. This term also guarantees the gauge-invariant regularization and
renormalization of the theory, which becomes then superrenormalizable [2,3].\\
The generating functionals of the various models considered were constructed,
and seen to be formally the same as those of QED (or QCD) in 2+1 dimensions,
with the substitution of the CS gauge field propagator for the photon (or
gluon) propagator. The CS propagator in these models is seen to consist of two
parts: the first part is the propagator of a real massive photon (gluon) which
contributes to the classical free Hamiltonian, and its states appear in the
Hilbert
space of the total set of physical states of the system. The second part is
that of the
topological massless photon which does not contribute to the free Hamiltonian,
but leads to additional (in comparison with QED or QCD) interaction between the
charged particles. The general solution for the free gauge field, when
constructed in a covariant gauge, was therefore seen to consist of a massive
part, and a massless topological part.\\
Taking the limits $\gamma\to\infty$ and $\mu\to 0$ of the propagators and the
general solutions of the gauge fields (see eqs.(21)-(24) ) after
renormalization,
which is possible due to the finite renormalization of these parameters [5-7],
we get respectively pure CSQED and QED in 2+1 dimensions.\\
We carried out very carefully the path integral quantization of some models
with the non-Abelian CS field by the De Witt-Fadeev-Popov and the
Batalin-Fradkin-Vilkovisky methods, and showed that it is not necessary to
quantize the dimensionless charge of the theory. First, in the DFP approach we
use gauge transformations which have zero winding number since the integral
over the gauge group takes into account only the contributions near the
identity element of the group (these elements of the group have zero winding
numbers).Also, the action in the exponent (after path integral quantization)
is not gauge
but BRST-invariant, and due to the Grassmann nature of the BRST transformation
, one gets a zero winding number too !. The BFV approach gives the same
BRST-invariant expression for the action in the exponent of the path integral
expression.
Finally, the definition of the generating functional (see eqs.(49),(59)) shows
that for any gauge transformation, the terms proportional to the winding
number in the path integral expression for the expectation value of any
observable are
cancelled due to the normalization of the generating functional (see the
footnote in the work [3] about the argument of J.Schonfeld in this respect).
It is well-known that the existence of BRST-invariance in renormalizable gauge
theories guarantees the implementation of Ward-Fradkin-Takahashi-Slavnov
-Taylor identities, and gauge-invariant renormalization of the theory. This
invariance, in turn does not require the quantization of the charge.\\
Unfortunately, a path integral representation of the S-matrix is not
available for
theories with the pure CS field. This is because the "in" and "out" limits of
the transverse part of
the pure CS gauge field do not exist. In the general case, when the
Maxwell-type
term is included in the action, such a representation can be constructed, and
this will depend only on the "in" and "out" solutions of the massive part of
the gauge field. We constructed in the general case, the S-matrix
operator for all the Abelian and non-Abelian models, and showed that
this operator gives the correct expression for all the Feynman
diagrams of the theory, and formally differs from the usual case of QED and QCD
in 2+1 dimensions only by a specific type of Wick theorem for the gauge field.
\\
Starting from this S-matrix operator, we have shown that the requirement of
the unitarity of the S-matrix leads to topolgical unitarity identities that
were derived in [14]. These identities demand that at
each order of perturbation theory, the gauge-invariant sum of the imaginary
parts of the Feynman diagrams with a given number of intermediate on-shell
topological photon
lines should vanish. These identities were illustrated by some examples in
the Abelian case. The importance of these identities stems
from the fact that they, not only provide additional check of the
gauge-invariance of the theory, but also highly facilitates the perturbative
gauge-invariant calculations of Feynman diagrams. It is also possible to get
strong restrictions on the dependence on invariant variables of the
gauge-invariant sum of the real parts of the Feynman diagrams for which the
gauge-invariant sum of the imaginary parts vanishes (on account of the
analytic properties of Feynman diagrams in the momentum space representation).
{\bf Acknowledgements:}\\
We thank I.Tyutin and M.Soloviev from the Lebedev Institute of
Physics for helpful discussions.\\
\newpage
|
{
"timestamp": "1996-04-16T18:26:50",
"yymm": "9604",
"arxiv_id": "hep-th/9604087",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/9604087"
}
|
\section{Introduction} \label{Int}
Let $S$ be a real submanifold of $\C^N$ of real codimension $m,$
assume that $S$ is generic i.e. that $T_p(S)+i T_p(S)= \C^N,$
for every $p \in S.$
($T_p(S)$ is the tangent space to $S$ at $p$). In the study of the
polynomial hulls of such submanifolds, as well as in the study of
holomorphic extendability of their C-R functions, it is of particular
importance the set of analytic discs attached to the submanifold.
Let $\Delta$ be the unit disc $ |\zeta| < 1 $ in $\C.$
A continuous map $f: \bar{\Delta} \rightarrow \C^N$ is said to be an
analytic disc attached to $S,$ if $f(\de \Delta) \subseteq S$ and $f$ is
holomorphic on $\Delta.$
Let $\alpha$ be a positive real which is not an integer,
it is convenient to consider analytic discs attached to $S$ which are in
$C^{\alpha}(\bar{\Delta},\C^N).$
If $S$ is given by the zero set of a maximal rank defining map
$\rho : \Omega \rightarrow \R^m,$ ($\Omega$ being an open neighborhood of
$S$ in $\C^N$), and $f_0 : \bar{\Delta} \rightarrow \C^N $ is an analytic
disc attached to $S,$ then the set $M$ of such analytic discs is
the zero set of the map $ f \rightarrow \rho \circ f.$
Let $\ONalpha$ be the space of $C^{\alpha}$ maps from $\de \Delta$ to
$\C^N$ which extends holomorphically to $\bar{\Delta}.$
The defining map of $M$ goes from $\ONalpha$ to $\Calpha,$ and its
differential is given by
\[ w \rightarrow Re \left(\frac{\de \rho}{\de z} \circ f_0 (w(\zeta))
\right) \]
Because of the implicit function theorem, the set $M$ is a manifold,
as soon as its differential is onto.
Whenever $M$ is a manifold, its tangent space at $f_0$ is given by
\[ \{ w \in \ONalpha : Re \left(\frac{\de \rho}{\de z}(f_0(\zeta))w
\right)=0 \}.\]
Same procedure can be carried over in the case we want to look at the set of
analytic discs with a fixed point at the boundary. In other words the set
$ M_q= \{ f \in M : f(1)=q \},$ where $q=f_0(1).$
In \cite{Ce}, \cite{Fo}, \cite{Gl}, \cite{Oh}, and \cite{Trep}, sufficient
condition are given for either $M$ or $M_q$ to be manifolds in a neighborhood
of $f_0.$
\vskip .1cm
To gather informations on the polynomial
hull of $S,$ it is obviously interesting to describe the union of the images of
$f(\Delta)$ for $f$ in $M $ and $M_q.$ Or, which is the same because of
the action on $M$ of the group of authomorphims of $\Delta,$ the images of
the evaluation map $\mu_{0} : M \rightarrow \C^N $ given by $\mu_{0}(f)=f(0).$
To derive informations on
the evaluation maps, one might study the image of their differentials
(i.e. the maps themselves), restricted to the tangent space to $M $ and
$M_q.$ This will tell us when to apply the open mapping theorem.
\vskip .1cm
Similarly, the evaluation maps at the boundary,
$\mu_{1}(f)=f(1)$ on $M,$ and $\mu_{-1}(f)=f(-1)$ on $M_q,$ are
related to the local extendability of $C-R$ functions on $S,$
see \cite{Tu}.
\vskip .1cm
In this paper we will give a sufficient criteria for $M_q$ to be a
manifold (more general then the one stated in \cite{Ce}). We will also
describe the images of the differentials of the above evaluation
maps, restricted to the suitable tangent spaces.
We will
determine their dimensions. (Some work in this direction was previously made
in \cite{Ce}, \cite{Eisen}, \cite{Pa}, and \cite{Pa2}).
In particular we show that, for $x \in \Delta,$ the dimension of the image of
the differential of $ {\mu_x}_{|M}$ and $ {\mu_x}_{|M_q}$ are independent
on $x.$ The same is true as $x$ varies in $\de \Delta.$
More, we prove that, as $x$ varies in $\bar{\Delta}-1,$ the complex
span of the image of the differential of ${\mu_{x}}_{|M_q}$
defines a $C^{\alpha}$ vector bundle which is
holomorphic on $\Delta.$ Moreover its restriction to $\de \Delta$ contains
the complex tangent bundle to $S$ as a subbundle.
We will then apply all this, as outlined above,
to find the images of the evaluation maps themselves.
For example we show that, if every point of S is minimal, then there is an
open dense subset $\Omega$ of $M,$ invariant under the action of the
authomorphism group of $\Delta,$ such that the restriction to $\Omega$
of the evaluation $\mu_0$ is an open map.
In the present work, we will make heavy use of the results of \cite{Trep}
about the operator
\[ U_A(w)= Re \left(\frac{\de \rho}{\de z}(f_0(\zeta))w \right) \]
from $\ONalpha$ to $\Calpha,$ whose kernel is the tangent space to $M$ at
$f_0.$
What was proved in \cite{Trep}, was that the study of the above operator
can be reduced to the study of its continuous $L^2$ extension which is much
easier to deal with. We will mimic this procedure and apply it to the
restriction of $U_A$ to suitable subspaces of $\ONalpha.$
The Theorems about the evaluation maps will follow fairly easily.
\vskip .1cm
In the last two sections we consider the special case, interesting and
substantially simpler, of totally real submanifolds, and of hypersurfaces.
\vskip .1cm
The Author would like to thank Prof. Claudio Rea and Doctor Patrizia Rossi,
for introducing him to the subject, and for many stimulating discussions.
\section{Preliminaries} \label{prel}
Let $\alpha$ be a positive real which is not an integer, write
$\alpha= k + \beta,$ where $k$ is a non negative integer and $0 < \beta < 1.$
Denote by $C^{\alpha}$ the space of maps which are continuously differentiable
up to order k, and such that all their derivatives of order k, are Holder
maps with Holder coefficient $\beta.$ We assume $\beta > 1/2.$
Let $\Delta$ be the disc $ |\zeta| < 1$ in $\C.$
Let $\ONalpha$ be the space of $C^{\alpha}$ maps from $\de \Delta$ to
$\C^N$ which extends holomorphically to $\bar{\Delta}.$
Denote with Re z and Im z respectively the real and the imaginary part
of a complex number $z.$
Let $f_0$ be an analytic disc attached to a generic $C^2$ submanifold $S$ of
real codimension $m.$
Then $f_0^*(T(S))$ is a vector bundle over $\de \Delta.$
If $f_0^*(T(S))$ is trivial over $\de \Delta,$ (orientable case),
there exists an open
neighborhood U of the graph of $f_0$ in $\de \Delta \times \C^N,$ and a
$C^2$ map $\rho: U \rightarrow \R^m$ such that for every fixed $x \in \de
\Delta$ the map $Y \rightarrow \rho(x,Y)$ is a defining map of $S.$
If $f_0^*(T(S))$ is non trivial, then $(f_0 \circ sq)^*(T(S))$ is.
Here $sq(\zeta)= \zeta^2,$ see \cite{Trep}.
All we say in this paper can be easily carried over to the non
orientable case, by composing with the map $sq$ and modifying the function
spaces accordingly, as in \cite{Trep}. However we will for simplicity confine
ourself to the orientable case. So for us every analytic disc attached to
S is an orientable one.
Observe that the map from $\ONalpha$ to $C^{\alpha}(\de \Delta,\R^m)$ given by
$ f \rightarrow \rho \circ f$ is $C^1.$ (see appendix in \cite{Gl}.)
Since S is generic, then the matrix
$ \frac{\de \rho}{\de z}(\zeta,f_0(\zeta))$ has maximal rank $m$ for
every $\zeta \in \de \Delta.$
We will consider this problems sketched in the Introduction from a slightly
more general point of view:
Let $A : \de \Delta \rightarrow {\cal M}(m \times N, \C)$ a
$C^{\alpha}$ map, where ${\cal M}(m \times N, \C)$ denotes the space of
$ m \times N $ complex matrices. Assume that $ m \leq N $ and that $A(\zeta)$
has maximal rank $m$ for every $ \zeta \in \de \Delta.$
(In the case of an analytic disc $f_0$ the map $A$ is given by
$A(\zeta)= \frac{\de \rho}{\de z}(\zeta,f(\zeta)).$
Let
\[ { \cal L}_A= \{ w \in { \cal O}^N : Re(A w)=0 \ \mbox{\ on \ } \de \Delta
\}. \]
and let
\[ \EA= \{ \gamma \in C^{\alpha}( \de \Delta, \R^N) \]
\[ \mbox{such that} \ \gamma^t A \ \mbox{extends holomorphically to}
\ \overline{\Delta} \}. \]
Let $ \EAC \ \mbox{be the complexification of} \ \EA,$
that is
\[ \EAC= \{ \delta \in C^{\alpha}( \de \Delta, \C^N) : Re \delta
\ \mbox{and \ } Im \delta \ \mbox{ \ belong to \ } \EA \}. \]
For a given $ x \in \overline{\Delta}. $ Define
the evaluation map
\[{\mu}_x :{\cal O}^N \rightarrow \C^N \]
as the value in $x$ of the holomorphic extension of $w.$
Define
\[ {\Psi}_x : \EAC \rightarrow \C^N \
\mbox{as the evaluation at $x$ of the holomorphic extension of $\delta^t A.$
}\] In other words
\[\begin{array}{l} {\Psi}_x(\delta)= \frac{1}{2 \pi i} \int_{\de \Delta}
\frac{\delta^t A(\zeta)}{\zeta - x} d \zeta \\
\mbox{for $x \in \Delta$ and} \\
\Psi_x(\delta)= \delta^t(x) A(x) \\
\mbox{for $ x \in \de \Delta.$} \end{array} \]
By the results in \cite{Pa2} the space $\EA$ has a finite
dimension $d.$ The number $d$ will be called defect of $A.$
Let $ \gamma_1 \ldots \gamma_{d} $ be a base of $\EA$ over $\R,$
so $ \gamma_1 \ldots \gamma_{d} $ is also a base of
$ \EAC \ \mbox{ \ over \ } \C.$
Given $x \in \overline{\Delta},$ if $\EA \neq 0 $ define $P(x)$ as the
matrix
having rows ${\Psi}_x(\gamma_1), \ldots {\Psi}_x(\gamma_{d}).$ If
$\EA=0$ let $P(x)$ be the zero matrix in ${\cal M}(N \times N, \C).$
Given $x \in \bar{\Delta,}$
denote by $(K_A)_x$ the kernel of the map ${\Psi}_x$ and by $(N_A)_x$
the kernel of its restriction to $\EA.$ We will be especially interested in
$(K_A)_0$ and $(N_A)_0$ which will be simply denoted by $K_A$ and $N_A$
respectively.
Let $l_x$ be the real dimension of the image of
${{\Psi}_x}_{|{\cal E}_A} $, and let $r_x$ be the complex
dimension of the image of ${\Psi}_x$, i.e. the rank of $P(x).$
Finally let $Q_A$ be the subspace of $\EAC$ given by
\[ Q_A = \{ \delta \in \EAC : \bar{\delta} = - \bar{\zeta} \delta. \} \]
Note that if $\delta \in Q_A,$ then $\delta^t A = - \zeta \bar{\delta}^t
A,$ therefore the holomorphic extension of $\delta^t A$ vanishes at $0,$
i.e. $Q_A \subseteq K_A.$
We can relate $Q_A$ and $\EA,$ via the following
\begin{lemma}
Let $\gamma$ be in $\EA$ such that $\gamma(1)=0,$ denote by $\delta$
the map
\[ \delta= \frac{\gamma}{1-\bar{\zeta}}. \] Then $\delta$
is in $C^{\alpha}(\de \Delta,\C^m), $ more precisely $\delta \in Q_A.$
Viceversa, if $\delta$ is in $Q_A,$ then $(1- \bar{\zeta}) \delta$ is in
$\EA$ and it vanishes at $1.$ \label{division} \end{lemma}
\begin{proof}
Given $ \delta \in Q_A, $ then $(1- \bar{\zeta}) \delta$ is real and
vanishes at $1.$ Moreover $(1- \bar{\zeta}) \delta^t A= \delta^t A -
\frac{1}{\zeta} \delta^t A $ which extends holomorphically to
$\bar{\Delta}$ since $Q_A \subseteq K_A \subseteq \EAC.$
Viceversa
Let $\gamma \in \EA$ such that $\gamma(1)=0.$
Since $ \beta > 1/2 $ we can write $A=(A',A")$ where $A'$ has
values in $GL(m,\C),$ see \cite{Trep}. We have the factorization
$- {A'}^{-1} \bar{A'}= \Theta^{-1} \Lambda \bar{\Theta}$
where $ \Theta $ is a holomorphic map from $\bar{\Delta}$ to
$GL(m,\C),$ and $\Lambda$ is a diagonal matrix of the form
\[ \Lambda = \left( \begin{array}{llcr} \zeta^{k_1} & 0 & \ldots & 0 \\
0 & \zeta^{k_2} & \ldots & 0 \\ \ldots & \ & \ & \ \\
0 & \dots & \ & \zeta^{k_m} \\ \end{array} \right). \]
Here $k_1 \ldots k_m$ are suitable integers, called partial indices of
$A,$ (Birkhoff factorization), See \cite{Gl}, \cite{Pa} and \cite{Ve}.
Now $\gamma^t A'$ extends holomorphically to $\bar{\Delta}.$
Set $\gamma^t A'= u^t,$ then $ u^t {A'}^{-1} $ is real, i.e.
$ - u^t \Theta^{-1} \Lambda \bar{\Theta}= \bar{u^t}.$
By setting $v^t = i u^t \Theta^{-1},$ we find $v_j= \bar{v_j} \zeta^{-k_j}$
on $\de \Delta,$ for $1 \leq j \leq m.$ In particular each component $v_j$
is a polynomial vanishing at $1.$ Therefore we have $v_j=(1- \zeta) w_j $
with $w_j$ polynomial. We then obtain
\[\bar{\delta}= \frac{\gamma}{1-\zeta}= -i ({{A'}^t})^{-1} \Theta^{-1} w. \]
Where $w$ is the vector with components $w_j.$ Hence $\delta$ and
$\bar{\delta}$ are in $C^{\alpha}(\de \Delta,\C^m).$
Let $f$ be the holomorphic extension of
$ \gamma^t A $ to $\bar{\Delta}.$ Let
$f_n = \frac{f}{1-\zeta + 1/n}.$ The maps
$f_n$ are holomorphic in $\bar{\Delta}$ and converge
uniformly to $ \frac{f}{1-\zeta} = \bar{\delta}^t A$ on $\de \Delta.$
By the maximum principle the sequence $f_n$ converges uniformly to a
holomorphic map on $\bar{\Delta}$ which is an extension of $\bar{\delta}^t A.$
Since $\gamma$ is real, $ \delta = -\zeta \bar{\delta}.$
Hence $\delta^t A $ extends holomorphically to $\bar{\Delta}, $ therefore
$ \delta \in Q_A.$ \end{proof}
\begin{lemma}
The space $Q_A$ is maximal totally real in $K_A.$ \label{totreal}
\end{lemma}
\begin{proof}
Clearly $Q_A \cap i Q_A =0.$ Given $\delta \in K_A$ we have
$\delta = \delta' +i \delta" $ with
$\delta'= \frac{ \delta + \zeta \bar{\delta}}{2},$
and $\delta"= \frac{ \delta - \zeta \bar{\delta}}{2i}.$ \end{proof}
\begin{remark}
There exist two
natural embeddings of $N_A$ into $Q_A.$
One is given by $ \gamma \rightarrow (1- \zeta) \gamma $ and its image
coincides with the set of elements in $Q_A$ vanishing at $1.$
The other is given by $ \gamma \rightarrow i (1+ \zeta) \gamma $
and its image coincides with the set of elements in $Q_A$ vanishing at $-1.$
The proof is as in Lemma \ref{division}. \label{division2} \end{remark}
\begin{proposition}
\[ \begin{array}{l} a) \ \mbox{The numbers $r_x$ are independent} \\
\mbox{on $x$ for $x$ in $\bar{\Delta}.$}
\ \mbox{Let $r$ be their common value.}
\\
b) \ \mbox{The numbers $l_x$ are independent on $x$ for $x$ in
$\Delta.$ Let $l$ be their common value.} \\
c) \ \mbox{We have $l_x= r$ for every $x \in \de \Delta,$}
\end{array} \] \label{ranks} \end{proposition}
\begin{proof}
Proof of a)
We will show that $r_1=r_0.$
Given any $x \in \Delta,$ let $\sigma_x$ the authomorphism of the disc
sending $0$ to $x$ and keeping $1$ fixed, it will be sufficient to apply
the above equality to $A \circ \sigma_x$ to show that $r_x$ is
independent on $x$ for $x$ in $\Delta.$ By a further
application to $A \circ \sigma,$ where $\sigma$ is an rotation, we conclude
that $r_x$ is independent on $x \in \bar{\Delta.}$
Since $A(1)$ has maximal rank m, then an element $\gamma$ in
$\EAC$ is in the kernel of $\Psi_{1}$ if and only if
$\gamma= \gamma_1 +i \gamma_2$ with $\gamma_1$ and $ \gamma_2$ in
$\EA$ vanishing at $1.$ It follows from Lemma \ref{division},
and Lemma \ref{totreal}, that the kernel of $\Psi_{1}$ has real dimension
$d- r_0,$ i.e. $r_1 =r_0.$
So we proved part a). Part c) follows directly from the fact that $A(x)$
has rank $m$ for $x \in \de \Delta.$
Let us prove b)
Given $x$ in $\Delta$ different from $0,$ let
\[ t(\zeta)=\frac{\bar{x}{\zeta}^2 -(|x|^2+1)\zeta +x}{\zeta}, \] the function
$t(\zeta)$ has only one pole in $\bar{\Delta}$ and it is a simple pole at
$0.$ It also has only one zero in $\bar{\Delta}$ and it is a simple zero at $x.$
Moreover $t(\zeta)$ is real on $\de \Delta.$
Hence the map $ \gamma \rightarrow t(\zeta) \gamma(\zeta)$ defines a linear
isomorphism from $(N_A)_0$ onto $(N_A)_x.$ \end{proof}
\begin{remark}
If $\gamma$ is in ${{\cal E}_A}$ and $w$ is in $ { \cal L}_A$ then the
map
\[ x \rightarrow {\Psi}_x(\gamma) \mu_x(w) \]
is holomorphic in $\Delta$ and it is purely imaginary on $\de \Delta.$
Hence it is a purely imaginary constant.
\label{pearing} \end{remark}
\begin{proposition}
The integers $r$ and $l$ have the following properties
\[ \begin{array}{l} a) \ 0 \leq r \leq min(l,m), \ l \leq min(d,2r),
\ r=0 \ \mbox{if and only if} \ d=0 \\
b) \ \mbox{ \ If \ } r=m \ \mbox{then \ } \\
\LA= \{ w \in \ONalpha : \mbox{such that $P(x)w(x)$ is a purely imaginary
constant} \}. \\
c) \ \mbox{If the map A is associated to a small analytic disc} \\
\mbox{attached to a generic submanifold S, then $r=d.$} \end{array} \]
\label{properties} \end{proposition}
\begin{proof}
The inequalities $ r \leq l \leq 2 r,$ and $l \leq d,$
follows immediately from the definitions.
If $r=0$ then $d=0$ because of Proposition \ref{ranks}.
Fix a point $ x \in \de \Delta, $ and a base
$\gamma_1, \ldots \gamma_{d}$ of $\EA.$
Let $M(x)$ be the matrix having the vectors
$\gamma_1(x), \ldots \gamma_{d} (x),$ as rows,
so $P(x)= M(x)A(x).$
Since $A(x)$ has rank m, the matrix $M(x)$ must have rank $r,$
hence $r \leq m.$
Assume now that $r=m.$ Given $w \in \LA,$ we know from Remark
\ref{pearing} that $P w$ is a purely imaginary constant.
Fix viceversa a map $ w \in \ONalpha$ such that $Pw$ is a purely imaginary
constant.
We have that $M(x)(Re A(x)w(x))=0,$ for $x \in \de \Delta.$
Since $ r=m,$ then $M(x)$ defines a 1-1 linear map from $\R^{d}$ into
$\R^m.$ Therefore $w \in \LA.$
We now assume that A is associated to a small analytic disc $f$
attached to a generic submanifold $ S \subseteq \C^N.$
Take $ U \subseteq \C^m$ and $ V \subseteq \C^{N-m},$
neighborhoods of the origin.
Denote with $z_1=x +i y,$ the points of $U,$ and with $z_2$ the points of
$V.$ Assume that the image of $f$ is contained in $U \times V.$
Assume moreover the existence of a map
$ h: \{ |z_1| < \rho , \ |y| < \rho \} \rightarrow \R^m$ with the following
properties :
$h({\bf 0})=dh_{{\bf 0}}=0,$ and $x- h(z_2,y)$ is a defining map of
$S.$ Such map always exists if we choose $U$ and $V$ small enough.
\cite{Tu}, and \cite{Pa2}.
In such conditions there exists a map $G: \de \Delta \rightarrow
GL(m,\R)$ such that $ \gamma \in \EA $ if and only if $\gamma^t=X^t G$
with $X$ constant vector in $\R^m$ belonging to the space
\[ V_{f}= \{ X \in \R^m : X^t G(h_{z_2} \circ f) \
\mbox{extends holomorphically to $\overline{\Delta}$} \}. \]
It is proved in \cite{Pa2} that the space $V_f$ has dimension $d.$
Let Q be a matrix having as rows the vectors of a base of $V_{f}.$
Then we can choose
$P= Q G ( I+i h_y) \circ f $
If we take a small enough $U,$ we can assume that $ ( I+i h_y) \circ f (0)$
is invertible, hence $P(0)$ has rank $d.$ \end{proof}
\begin{proposition}
We have $r=l$ if and only if they both equal $d.$
\label{grasm1} \end{proposition}
\begin{proof}
Let $D_x \subseteq \C^d $ be the complex $r$ dimensional image of $P(x).$
Let
$R_x= D_x \cap (i \R^{d}).$ The set of vectors $c \in
\C^N$ such that $P(x)\bar{c} \in R_x$ is the orthogonal space in
$\R^{2N}$ to the rows of the matrix $P(x),$ so it has real codimension $l.$
The Kernel of $P(x)$ has real codimension $2r,$ hence $R_x$
has real dimension $2r-l.$ So if $r=l,$ then $R_x$ has real
dimension $r.$
Now $R_x$ is a totally real subspace of the $r$ dimensional complex space
$D_x.$
It follows that $D_x =R_x+iR_x.$
Let $G(\C,r,d)$ be the Grasmanian of the complex $r$ subspaces of
$\C^{d},$ and let $G(\R,r,d)$ be the analogous real Grasmanian.
The map from $G(\R,r,d)$ sending the subspace $\Sigma$ to
$\Sigma + i \Sigma,$ identifies $G(\R,r,d)$
with a maximal totally real submanifold of $G(\C,r,d).$
Let $\theta: \Delta \rightarrow G(\C,r,d),$ be the holomorphic map
sending $x$ to $D_x.$
Our hypotheses implies that the image of $\theta$ is contained in
$G(\R,r,d).$ Hence $\theta $ is constant.
In other words, there exist $r$ independent vectors $v_1, \ldots,
v_{r}$ in
$\R^{d},$ generating over $\C$ the image of $P(x)$ for every $x$ in
$\Delta,$ hence, for every $x$ in $\de \Delta.$
But the image of $P(x)$ coincides with the image of $M(x)$ as soon as $x$
is in $\de \Delta.$
Therefore, there exist $r$ maps, $ \lambda_1, \ldots, \lambda_{r}$
from $\de \Delta$ to $\R^m,$ such that every map
$\gamma_j$ of a base of $\EA$ is a linear combination with constant real
coefficients of the maps $ \lambda_1, \ldots \lambda_{r}.$ Hence
$d \leq r.$ We conclude by invoking part a) of Lemma \ref{properties}.
\end{proof}
\begin{remark}
Since $A$ has rank $m$ in $\de \Delta,$
Proposition \ref{ranks} implies that
$ \theta: \bar{\Delta} \rightarrow G(\C,r,d)$ is an analytic disc attached
to the maximal totally real submanifold
$ G(\R,r,d). $ From the above proof we see that, whenever
$r < d,$ the map $\theta$ is non-constant. \label{grasm2} \end{remark}
Recall that a map $A$ is defined in \cite{Trep} to be regular if $N_A=0$
i.e. if ${{\Psi}_0}_{| \EA}$ is one to one.
Then it is natural to give the following
\begin{definition}
We say that A is strongly regular if the map ${\Psi}_0$ is one to one.
(i.e. if $K_A=0 $).
\end{definition}
\begin{examples}
If $ d \leq 1,$ then A is strongly regular.
If m=1, then A is strongly regular if and only if it is regular.
It follows from the results of the next section and of \cite{Trep}.
If A is the map associated to a small analytic disc, then A is strongly
regular, see part c) of Proposition \ref{properties}.
If m=N, (totally real case) then A is regular if and only if every partial
index is greater then or equal to $-1.$
\vskip .1cm
A is strongly regular if and
only if every partial index is greater then or equal to $0.$
See Section \ref{tot*real}.
See also \cite{Oh} and \cite{Ce}. \label{strongreg} \end{examples}
It follows from Proposition \ref{grasm1} that $A$ is strongly
regular if and only if $l=r.$
\section{The Operators} \label{Operator}
Let $U_A : \ONalpha \rightarrow \Calpha,$ be the operator given by
$U_A(w)= Re(A(w)).$
As we observed in the Introduction, the above operator is
important for the study of the set of analytic discs attached to a generic
submanifold of $\C^N.$
In particular Tr\'{e}preau gives in $\cite{Trep}$ important properties of the
extension of $U_A$ to $L^2(\de \Delta,\R^N).$
More precisely
let $H$ be the closure in $ L^2(\de \Delta, \C) $ of functions in $\ONalpha.$
Then $H$ is the space of functions in $ L^2(\de \Delta,\C) $ whose
Fouerier series expansion has only non-negative coefficients.
Let $\tilde{U_A} : H^N \rightarrow L^2(\de \Delta,\R^m)$ be the $L^2$
continuous extension of $U_A.$
We will state Tr\'{e}preau result (slightly differently from the statement of
his paper) in the following
\begin{theorem}
\[ \begin{array}{l} a) \ \mbox{The operator $U_A : \ONalpha \rightarrow
\Calpha$ has closed finite codimensional range.} \\
\mbox{Moreover its kernel has a closed supplementary in $\Calpha.$} \\
\mbox{In case $m=N,$ the kernel is finite dimensional.} \\
\mbox{i.e. $U_A$ is Fredholm.} \\
b) \ \mbox{The operator $\tilde{U_A} : H^N \rightarrow L^2(\de \Delta,\R^m)$
has also closed finite codimensional range.} \\
\mbox{Moreover the range $R(U_A)$ coincides with the intersection of
$R(\tilde{U_A})$ with $\Calpha.$} \\
\mbox{In particular the kernel of $\tilde{U_A}$ is in $\Calpha,$
hence it coincides with the kernel of $U_A.$} \\
c) \ \mbox{The $L^2$ orthogonal to $R(\tilde{U_A})$ is contained in
$\Calpha,$} \ \mbox{and it coincides with $N_A.$} \\
\ \mbox{It follows that the operator $U_A$ is onto if and only if $A$} \\
\mbox{is regular (i.e. $N_A=0$). } \end{array} \]
\label{treptrep} \end{theorem}
\begin{proof}
Tr\'{e}preau deals with the operator $B_A : C^{\alpha}(\de \Delta, \R^N)
\rightarrow C^{\alpha}(\de \Delta,\R^m)$ given by
$\phi \rightarrow Re(A(\phi +i T_0(\phi)))$ where $T_0$ is the Hilbert
transform normalized at 0. Let
$ \tilde{B_A}: L^2(\de \Delta,R^n) \rightarrow L^2(\de \Delta,R^m).$ The
continuous extension of $B_A.$
Let us observe that
\[U_A(w)= U_{\frac{A(\zeta)}{\zeta}}(\zeta w)=B_{\zeta^{-1}A}(Re(\zeta w)) \]
\[ Re(\zeta w)(0)= \int_0^{2 \pi} Re(\zeta w) d \theta=0. \]
Since the set
\[ L= \{ \phi \in L^2(\de \Delta,\R^m) : \int_0^{2 \pi} \phi d \theta=0 \}. \]
is closed m-codimensional, the Theorem follows from the results in
\cite{Trep}. \end{proof}
Let $U_A(1)$ be the restriction of $U_A$ to the space
\[ \ONalpha(1)= \{ w \in \ONalpha: w(1)=0 \}. \]
seen as an operator from $\ONalpha(1)$ to
\[ \Calpha(1) = \{ \phi \in \Calpha : \phi(1)=0 \}. \]
Since $U_A$ has a closed range with finite codimension, and
$\ONalpha(1)$ is closed in $\ONalpha$ with finite codimension, if
follows that $ U_A(1) $ has closed range with finite codimension as well.
We introduce
another operator $V_A $ given by
\[ V_A(w) = (1-\zeta)^{-1} U_A( (1-\zeta)w ) \]
\[ = 1/2 \left( A(\zeta)w(\zeta)- \overline{\zeta A(\zeta)w(\zeta)} \right). \]
So
\[ V_A : \ONalpha \rightarrow C^{\alpha}(\de \Delta, \C^m) \]
\begin{lemma}
Let $ w \in \ONalpha$ such that $w(1)=0,$ then the map
$\frac{w(\zeta)}{1-\zeta}$ is in $ L^2(\de \Delta, \C^N) $ and it is a
limit in $ L^2(\de \Delta, \C^N), $ of a sequence of maps in $\ONalpha.$
\label{limL2} \end{lemma}
\begin{proof}
Since $\beta > 1/2,$ then
$\frac{w(\zeta)}{1-\zeta}$ is in $ L^2(\de \Delta, \C^N). $
By Lebesque dominated convergence theorem,
it is a limit in $L^2,$ of the sequence
$ w_n= \frac{w}{1-\zeta + 1/n}.$ \end{proof}
We take the continuous $L^2$ extension
$\tilde{V_A}$ of $V_A.$
\[ \tilde{V_A} : H^N \rightarrow L^2(\de \Delta,\C^m). \]
\begin{proposition}
Let $\phi \in \Calpha$ such that $\phi(1)=0, $ then $\phi$ belongs to the
range of $U_A(1)$ if and only if $\frac{\phi(\zeta)}{1- \zeta}$ belongs to
the range of $\tilde{V_A}.$ \label{range}
\end{proposition}
\begin{proof}
Note that, since $\beta > 1/2,$ then for every $\phi$ in $\Calpha,$ with
$\phi(1)=0,$ we have that $\frac{\phi(\zeta)}{1- \zeta}$ is in
$ L^2(\de \Delta,\C^m). $
If $\phi$ is in the range of $U_A(1),$ then $\frac{\phi(\zeta)}{1-\zeta}$
is in the range of $\tilde{V_A},$ because of Lemma \ref{limL2}.
Viceversa, if $\frac{\phi(\zeta)}{1-\zeta}$
is in the range of $\tilde{V_A},$ then there exists
$u \in H^N$ such that $\tilde{U_A}((1- \zeta) u ) =
\phi.$ Set $w= (1- \zeta) u.$
Because of Theorem \ref{treptrep} $ w \in \ONalpha.$ Since
$ \frac{w(\zeta)}{1- \zeta} = u(\zeta)$ is in $ L^2(\de \Delta,\C^m),$
we must have $w(1)=0.$ \end{proof}
Note that the range of $\tilde{V_A}$ is contained in the subspace Z of
$ L^2(\de \Delta,\C^m) $ given by the maps $\phi $ such that
$(1-\zeta) \phi(\zeta)$ is real. So
\[ Z= \{ \phi \in L^2(\de \Delta,\C^m) : \bar{\phi}= -\zeta \phi \} \]
Hence $Z$ is a closed subspace of $ L^2(\de \Delta,\C^m) $ and we can regard
$\tilde{V_A}$ as an operator from $ H^N $ into $Z.$
We are going to show that the range of $\tilde{V_A}$ is closed and finite
codimensional in $Z.$
We first look at the orthogonal to the range of $\tilde{V_A}$ in $Z.$
\begin{proposition}
An element $\phi \in Z$ is $L^2$ orthogonal to the range of $\tilde{V_A}$
if and only if $\bar{\phi}$ is in $Q_A.$ In particular the orthogonal to
$R(\tilde{V_A})$ is finite dimensional. \label{ort2} \end{proposition}
\begin{proof}
An element $\phi$ in Z is orthogonal to $R(\tilde{V_A})$ if and only if
\[\int_{0}^{2 \pi} \frac{\bar{\phi}^t}{1- \zeta} Re(A(1- \zeta)w) d \theta =0\]
for every $ w \in \ONalpha.$
However $\phi= \frac{\gamma}{1- \zeta}$ with real $\gamma,$ so
\[\int_{0}^{2 \pi} \frac{\bar{\phi}^t}{1- \zeta} Re(A(1- \zeta)w) d \theta \]
\[= \int_{0}^{2 \pi}Re \left(\frac{\gamma^t}{|1- \zeta|^2}(A(1- \zeta)w)\right)d
\theta \]
\[= Re \int_{0}^{2 \pi} \left( \frac{\gamma^t}{1- \bar{\zeta}}Aw \right)
d \theta.\]
So we have that $\phi$ in Z is orthogonal to $R(\tilde{V_A})$ if and only if
\begin{equation} \int_{0}^{2 \pi} \bar{\phi^t}A(w)= 0
\ \mbox{for every $ w \in \ONalpha$} \label{bar} \end{equation}
(replace $w$ with $i w$).
Since $\bar{\phi}= - \zeta \phi,$ then formula (\ref{bar}) is equivalent to
\[ \int_{0}^{2 \pi} \bar{\phi^t}( \zeta A)(w)= 0 \]
and
\[ \int_{0}^{2 \pi} \phi^t (\zeta A)(w)= 0 \]
for every $ w \in \ONalpha.$
Theorem \ref{treptrep} (applied to $ \zeta A $) shows that $\phi \in Z$ is
orthogonal to $R(\tilde{V_A})$ if and only if
$\phi$ is in ${\cal C}^{\alpha}(\de \Delta,\C^m),$
and $\bar{\phi}^t A$ extends to a map which is
holomorphic in $\bar{\Delta}$ and vanishes at $0.$ We set
$\delta=\bar{\phi}.$ By definition
$ \bar{\delta}= -\bar{\zeta}\delta=- \frac{1}{\zeta}\delta.$ Hence
$ \bar{\delta}^t A$ extends holomorphically to $\bar{\Delta}$ as well,
i.e. $\delta \in Q_A.$
\end{proof}
\begin{lemma}
Given an integer $k,$ the map
$ p_k : H \rightarrow L^2(\de \Delta,\C) $
given by
$ w \rightarrow w + \zeta^k \bar{w},$
has a closed range in $ L^2(\de \Delta,\C).$
\label{closed} \end{lemma}
\begin{proof}
Let us identify $ L^2(\de \Delta,\C) $ with $l^2(\C)$ via the orthonormal
complete system $e^{i m \theta}, \ \mbox{with} $
$ m \in \Z.$
The map $p_k$ becomes $ a_m \rightarrow a_m + \overline{a_{k-m}}$ under the
restriction $a_m =0 $ for negative m.
The range of $p_k$ is then described by the equations $b_{k-m}= \overline{b_m}.$
\end{proof}
We then come to the
\begin{proposition}
The operator $\tilde{V_A}$ has a closed range with finite codimension in
$Z.$ \label{Vrange} \end{proposition}
\begin{proof}
Since $ \beta > 1/2 $ we can assume that $A=(A',A")$ where $A'$ has
values in $GL(m,\C),$ (see \cite{Trep}). So it is sufficient to prove
the Proposition for the case $m=N.$ In this case we have the factorization
$-{A}^{-1} \bar{A}= \Theta^{-1} \Lambda \bar{\Theta}.$ (See the proof of
Lemma \ref{division}).
We can write
\[ V_A = \frac{1}{2} A \Theta \left( (\Theta^{-1}w)+ (\zeta)^{-1} \Lambda
\overline{(\Theta^{-1}w)} \right). \]
Now multiplication by $A \Theta$ defined an automorphism of
$ L^2(\de \Delta,\C^m),$
moreover multiplication by $\Theta^{-1}$ defines an automorphism of $H^N.$
Therefore the range of $\tilde{V_A}$ is closed because of Lemma
\ref{closed} and it is of finite codimension in $Z,$ because of Lemma
\ref{ort2}. \end{proof}
We summarize the above statements in the following
\begin{theorem}
Let $\phi \in \Calpha$ with $\phi(1)=0.$ Then $\phi$ is in the range of
$U_A(1)$ if and only if
\[ \int_{0}^{2 \pi} \gamma^t(\zeta) \frac{\phi(\zeta)}{|1- \zeta|^2} d \theta=0
\] for every $\gamma$ in $\EA$ with $\gamma(1)=0.$
Alternatively if and only if
\[ \int_{0}^{2 \pi} \delta^t(\zeta) \frac{\phi(\zeta)}{1- \zeta} d \theta=0
\] for every $\delta$ in $Q_A.$ \label{Trange} \end{theorem}
\begin{proof}
It follows from Lemma \ref{range}, Proposition \ref{ort2}, Proposition
\ref{Vrange} and Lemma \ref{division}. \end{proof}
\begin{remark}
Note that the space $\{ (1- \zeta) \bar{\delta} \ \mbox{for $\delta \in Q_A$}
\}$ is a supplementary space of $R(U_A(1))$ in
$C^{\alpha}(\de \Delta,\C^N)(1).$
Moreover it can be proved in the same way as in \cite{Trep}
that the kernel of $U_A(1)$ has a closed supplementary in
$C^{\alpha}(\de \Delta,\C^N)(1).$ \label{suppl} \end{remark}
\begin{corollary}
The operator $U_A(1)$ is onto if and only if $r=d,$ i.e. if and only if
${\Psi}_0$ is one-to-one. \label{onto} \end{corollary}
\begin{proof}
It follows from Remark \ref{suppl}, and Lemma \ref{totreal}. \end{proof}
Suppose that $A_{f_0}$ is the map associated to an analytic disc $f_0$
attached to a generic submanifold $S$ in $\C^N.$
(Recall that $ f \rightarrow \rho \circ f$ is $C^1$).
Let $q=f_0(1).$ Let
$M = \{ f \in \ONalpha \ \mbox{such that }$
$ \ \mbox{$f$ extends to an analytic disc attached to $ S$} \},$
and
$M_q= \{ f \in M \ \mbox{such that $f(1)=q$} \}.$
It is proved in \cite{Trep}, by using Theorem \ref{treptrep} together with
the implicit function theorem, that if $A_{f_0}$ is regular then $M$ is a
manifold in a neighborhood of $f_0.$ With a similar proof by using corollary
\ref{onto} and Remark \ref{suppl}, we obtain the following
\begin{corollary}
If $A_{f_0}$ is strongly regular, then $M_q$ is a manifold in a neighborhood of
$f_0.$ \label{varieta'} \end{corollary}
\begin{remark}
In the totally real case $M_q$ is finite dimensional and its dimension
equal the total index k of the submanifold S.
In general (when S is totally real but A may not be strongly regular)
the number k is the index of $U_A(1).$ (See section \ref{tot*real}).
\end{remark}
\begin{remark}
More generally $M$ is a manifold in a neighborhood of $f_0$ as
long as $d-l$ is constant as $f$ varies in a
neighborhood of $f_0.$
Similarly $M_q$ is a manifold in a neighborhood of $f_0$ as
long as $d-r$ is constant as $f$ varies in a neighborhood of $f_0.$
\label{implfunc} \end{remark}
We finally need to study the operator $U_A$ restricted to the subspace
of $\ONalpha$ given by the maps vanishing at the point $1$ and at the point
$-1$ simultaneously.
Denote by $\ONalpha(1,-1)$ the space of maps in $\ONalpha$ which vanish at
$1$ and at $-1.$ Similarly denote by $C^{\alpha}(1,-1),$ the space of
$C^{\alpha}$ maps vanishing at $1$ and $-1.$
Denote by $U_A(1,-1)$ the restriction of $U_A$ to the space
$\ONalpha(1,-1)$
seen as an operator from $\ONalpha(1,-1)$ to $C^{\alpha}(1,-1).$
Note that if a map $ \phi$ is in $C^{\alpha}$ and it vanishes at $1$ and
$-1,$ then $\frac{\phi(\zeta)}{1- \zeta^2} \in L^2.$
Define
\[ {V^*}_A(w) = (1-\zeta^2)^{-1} U_A( (1-\zeta^2)w ) \]
\[= 1/2 \left( A(\zeta)w(\zeta)- \overline{\zeta^2 A(\zeta)w(\zeta)} \right).\]
So
\[ {V^*}_A : \ONalpha \rightarrow C^{\alpha}(\de \Delta \C^m) \]
and
\[ \tilde{{V^*}_A} : H^N \rightarrow L^2(\de \Delta,\C^m). \]
\begin{lemma}
Let $ w \in \ONalpha$ such that $w(1)=0$ and $w(-1)=0,$ then the map
$\frac{w(\zeta)}{1-\zeta^2}$ is in $ L^2(\de \Delta, \C^N) $ and it is a
limit in $ L^2(\de \Delta, \C^N), $ of a sequence of maps in $\ONalpha.$
\label{limL2bis} \end{lemma}
\begin{proof}
The proof is analogous to that of Lemma \ref{limL2}. \end{proof}
\begin{proposition}
Let $\phi \in \Calpha$ such that $\phi(1)=0, $ and $\phi(-1)=0,$
then $\phi$ belongs to the
range of $U_A(1,-1)$ if and only if $\frac{\phi(\zeta)}{1- \zeta^2}$
belongs to the range of $\tilde{{V^*}_A}.$ \label{range,bis}
\end{proposition}
The range of $\tilde{{V^*}_A}$ is contained in the subspace $Z^*$ of
$ L^2(\de \Delta,\C^m) $ given by the maps $\phi $ such that
$(1-\zeta^2) \phi(\zeta)$ is real. So
\[ Z^*= \{ \phi \in L^2(\de \Delta,\C^m) : \bar{\phi}= -\zeta^2 \phi \} \]
$Z^*$ is a closed subspace of $ L^2(\de \Delta,\C^m) $ and we can regard
$\tilde{{V^*}_A}$ as an operator from $ H^N $ into $Z^*.$
Let $K^*_A$ be the subset of $\EAC$ of maps $\delta$ such that the
holomorphic extension of $\delta^t A $ vanishes of order at least 2 at $0.$
Let
$ Q^*_A \subseteq K^*_A$ the subspace given by
\[ Q^*_A = \{ \delta \in K^*_A : \bar{\delta} =
-\bar{\zeta^2} \delta. \} \]
\begin{proposition}
An element $\phi \in Z^*$ is $L^2$ orthogonal to the range of $\tilde{V^*_A}$
if and only if $\bar{\phi}$ is in $Q^*_{(\zeta A)}.$
In particular the orthogonal to $R(\tilde{V^*_A})$ is finite dimensional.
\label{ort2,bis} \end{proposition}
\begin{proposition}
The operator $\tilde{V^*_A}$ has a closed range with finite codimension in
$Z^*.$ \label{Vrange,bis} \end{proposition}
\begin{lemma}
The space $Q^*_{(\zeta A)}$ is maximal totally real in $K^*_{(\zeta A)}.$
\label{totreal,bis} \end{lemma}
However, the space $Q^*_{(\zeta A)}$ is essentially $\EA,$ more precisely
we have
\begin{lemma}
The map from $\EA$ into $Q^*_{(\zeta A)}$ given by
\[ \gamma \rightarrow i \zeta \gamma \] is an isomorphism,
which extends to an isomorphism of $\EAC$ onto $K^*_{(\zeta A)}.$
\label{defect} \end{lemma}
\begin{proof}
Just observe that $\delta= i \zeta \gamma$ with real $\gamma,$ if and only
if $ \bar{\delta}=-\bar{\zeta^2} \delta.$ Moreover $ i \zeta \gamma^t
(\zeta A)$ extends to a holomorphic map vanishing of order at least 2 at
$0,$ if and only if $\gamma^t A$ extends holomorphically.
The last assertion of the Lemma follows from Lemma \ref{totreal,bis}.
\end{proof}
From the above Lemma, with a similar proof as in Theorem \ref{Trange} we
find
\begin{theorem}
Let $\phi \in \Calpha$ with $\phi(1)=\phi(-1)=0.$ Then $\phi$ is in the range
of $U_A(1,-1)$ if and only if
\[ \int_{0}^{2 \pi} \gamma^t(\zeta)\frac{ \zeta \phi(\zeta)}{1- \zeta^2}
d \theta=0 \] for every $\gamma$ in $\EA.$ \label{Trange,bis} \end{theorem}
\begin{remark}
Note that the space $\{ i \bar{\zeta}(1- \zeta^2) \gamma \ \mbox{for $\gamma
\in \EA$} \}$ is a supplementary space of $R(U_A(1,-1))$ in $C^\alpha(1,-1).$
\label{suppl,bis} \end{remark}
\section{Evaluation spaces } \label{eval}
Fix an element $y \in \de \Delta,$ and an element $ x \in \bar{\Delta},$
define
\[ \LA(y)= \{ w \in \LA : w(y)=0 \} .\]
We are interested in studying the various evaluation spaces
$ W_x= \mu_x(\LA),$ and $ W_x(y)= \mu_x(\LA(y)).$
(Recall that $\mu_x$ is the evaluation at $x$ of an element in
$\ONalpha$ and $\LA= \{ w \in \ONalpha : Re(Aw)= 0 \}.$ )
We have the following theorems:
\begin{theorem}
Given $ x \in \Delta$ we have
\[a) \ W_x = \{ c \in \C^N : Re(\Psi_x(\gamma)c)=0 \ \mbox{for every
$\gamma \in \EA$} \}. \]
In other words $W_x= \{ c \in \C^N : Re(P(x)c)=0 \}.$
So $W_x \cap i W_x$ coincides with the kernel of $P(x).$
b) The real codimension of $W_x$ in $\R^{2N}$ is $l.$
c) The complex codimension of $W_x \cap i W_x$ in $\C^N$ is $r.$
d) The complex codimension of $W_x + i W_x$ in $\C^N$ is $ l-r.$
\label{val0} \end{theorem}
\begin{proof}
We assume first that $x=0.$
A vector c in $\C^N,$
belong to $W_0$ if and only if there exists
$u \in {{\cal O}^N}_{\alpha}$
such that $\zeta u(\zeta) + c $ belongs to $\LA,$ if and only if
$-Re(A(\zeta) c)$ belongs to the range of $U_{ \zeta A}.$
Now $ \gamma \in N_{ \zeta A}$ if and only if $ \zeta \gamma^t A$ extends
holomorphically to $\bar{\Delta}$ and its extension vanishes at 0. This is
the case if and only if $\gamma \in \EA.$ It follows from Theorem
\ref{treptrep} that $c \in W_0$ if and only if
\[ \int_0^{2 \pi} \gamma^t Re(A c) d \theta = 0 \]
for every $\gamma \in \EA.$
However, since $ \gamma \in \EA,$ we have
\[ \int_0^{2 \pi} \gamma^t Re(A c)d \theta =Re(\Psi_0(\gamma)c). \]
In other words the image of $W_0$ under the conjugation map, is the
orthogonal space in $\R^{2N}$ to the space $\Psi_0(\EA).$ This proves b).
Part c) and part d) follow easily.
In the general case, let $\sigma$ be an authomorphism of the disc such that
$\sigma(0)=x.$ It follows from Proposition \ref{ranks} that
the numbers $r$ and $l$ relative to the map $A,$ coincide with the
corresponding numbers relative to $A \circ \sigma.$ \end{proof}
\begin{theorem}
Given $ x \in \de \Delta$ we have
\[ a) \ W_x = \{ c \in \C^N : Re(A(x)c)=0 \ \mbox{and} \
\delta^t(x)A(x)c=0 \ \mbox{ for every $\delta \in K_A $ } \}. \]
So the kernel of $A(x)$ is contained in $ W_x.$
b) The real codimension of $W_x$ in the space $Re(A(x)c)=0$ is
$l-r.$
c) We have $W_x \cap i W_x = Ker(A(x)).$
d) The complex codimension of $ W_x + i W_x$ in $\C^N$ is $ l-r.$
\label{val1} \end{theorem}
\begin{proof}
Assume first that $x=1.$
Obviously
\[ W_1 \subseteq \{ c \in \C^N : Re(A(1)c)=0 \}. \]
Now, given $c \in \C^N \ \mbox{with} \ Re(A(1)c)=0,$ we have $ c \in W_1 $
if and only if $- Re(A(\zeta)c)$ belongs to the range of $U_A(1).$
This is the case if and only if
\[ \int_0^{2 \pi} \frac{\delta^t Re(A(\zeta)c)}{1- \zeta} d \theta =0 \
\mbox{for every $ \delta \in Q_A.$} \]
By the definition of $Q_A$ this is equivalent to saying that
\[ \int_0^{2 \pi} Re \left( \frac{\delta^t A(\zeta)c}{1- \zeta}\right)d \theta
=0 \ \mbox{for every $ \ \delta$ in $ Q_A.$} \]
Let $f$ be the holomorphic extension of $\delta^t A c.$
Since $\delta= - \zeta \bar{\delta},$ then $\delta(1)$ is purely
imaginary, and, by assumption, so is $A(1)c,$ therefore $f(1)$ is real.
Moreover, since $\delta \in Q_A \subseteq K_A,$ then $f(0)=0.$
We have:
\[ \frac{1}{2 \pi} \int_0^{2 \pi} Re \left( \frac{f(\zeta)}{1- \zeta}
\right) d \theta = \]
\[\frac{1}{2 \pi} \int_0^{2 \pi} Re \left( \frac{f(\zeta)- f(1)}{1- \zeta}
\right) d \theta \]
\[+ \frac{f(1)}{2 \pi} \int_0^{2 \pi} Re \left( \frac{1}{1- \zeta} \right)
d \theta. \]
By the proof of Lemma \ref{limL2} we know that
\[\frac{1}{2 \pi} \int_0^{2 \pi} Re \left( \frac{f(\zeta)- f(1)}{1- \zeta}
\right) d \theta \]
\[ = \lim_{n \rightarrow \infty} \frac{1}{2 \pi} \int_0^{2 \pi}
Re \left( \frac{f(\zeta)- f(1)}{1- \zeta +1/n} \right) d \theta \]
\[= \lim_{n \rightarrow \infty} \frac{Re(f(0)- f(1))}{1 +1/n}= -f(1), \]
since $f(0)=0$ and $f(1)$ is real.
On the other hand
\[ \frac{f(1)}{2 \pi} \int_0^{2 \pi} Re \left( \frac{1}{1- \zeta} \right)
d \theta = \frac{f(1)}{2} \]
We have then proved that
\[ W_1 = \{ c \in \C^N : Re(A(1)c)=0 \ \mbox{and} \ \delta^t(1)A(1)c=0 \]
\[ \mbox{ for every $ \delta \in Q_A $} \}. \]
To replace $Q_A$ with $K_A$ we only need to recall that $Q_A$ is maximal totally
real in $K_A.$
As we already observed if $\delta \in Q_A,$ then $i \delta(1)$ is real.
Let $X \subseteq \R^m,$
be the orthogonal space to the image of the map
$\delta \rightarrow i \delta(1)$ for $\delta \in Q_A.$ Then because of
Remark \ref{division2},
$X$ has real codimension $d-r - (\mbox{dimension of $N_A$}) = l-r.$
Moreover $W_1={A(1)}^{-1}(iX).$ Since the kernel of
$A(1)$ has complex dimension $N-m,$ then $W_1$ has real dimension
$2N-m-l+r,$ therefore it has codimension $l- r $ in
$Ker(Re A(1)).$ This proves part b). Part c) and d) are immediate.
In the general case, let $\sigma$ be the rotation such that $\sigma(1)=x.$
Since $\sigma(0)=0,$ then $ \delta \in K_A$ if and only if
$\delta \circ \sigma \in K_{A \circ \sigma}.$ The conclusion follows.
\end{proof}
\begin{remark}
Let $ x \in \de \Delta,$ and $\gamma$ in $\EA,$ such that $\gamma(x)=0.$
Define
\[\gamma'(x)= \lim_{y \rightarrow x} \frac{\gamma(y)}{y-x}. \]
(Such limit exists and its finite because of Lemma \ref{division}).
Then part a) of Theorem \ref{val1} can be restated as follows:
\[ W_x = \{ c \in \C^N : Re(A(x)c)=0 \ \mbox{and} \]
\[ \overline{\gamma'(x)}A(x)c=0 \mbox{ for every $\gamma \in \EA $
such that $\gamma(x)=0 $} \}. \]
This follows from the proof of Theorem \ref{val1}, and from Lemma
\ref{division}. \end{remark}
For $x \in \Delta,$ let
\[ \begin{array}{l} \tilde{\Psi}_x :
{\cal E}_{(\zeta-x)A} \rightarrow \C^N \\
\mbox{be the evaluation at $x$ of the holomorphic extension of
$\delta^t(\zeta-x)A,$} \end{array} \]
In other words $ \tilde{\Psi}_x(\delta) $ is the residue at $x$ of the
meromorphic extension of $\delta^t A.$
Moreover let $r_1$ and $l_1$ and $d_1$ be the numbers $r$ and $l$ and $d$
relative to the map $\zeta A.$
\begin{theorem}
Given $ x \in \Delta$ and $y \in \de \Delta,$ we have
\[ a) \ W_x(y) = \{ c \in \C^N : Re(\tilde{\Psi}_x(\gamma)c)=0 \]
\[ \mbox{for every} \ \gamma \in {\cal E}_{(\zeta-x)A} \ \mbox{such that} \
\gamma(y)=0 \}. \]
\[ W_x(y) \cap i W_x(y) = \{ c \in \C^N : \tilde{\Psi}_x(\gamma)c=
0 \]
\[ \mbox{for every} \ \gamma \in {\cal E^{\C}}_{(\zeta-x)A} \ \mbox{such that}
\ \gamma(y)=0 \}. \]
b) The real codimension of $W_x(y)$ in $\R^{2N}$ is
$l_1 - r_1 + r.$
c) The complex codimension of $W_x(y) \cap i W_x(y)$ in $\C^N$ is
$l_1 - r_1.$
d) We have $W_x(y) + i W_x(y)=W_x \cap i W_x = Ker(P(x)).$
\label{val2} \end{theorem}
\begin{proof}
Assume first that $x=0$ and $y=1.$
Because of Lemma \ref{division}, statement a) is equivalent to
\[ W_0(1) = \{ c \in \C^N : Re(\tilde{\Psi}_0(\bar{\delta})c))=0 \]
for every $ \delta \in Q_{\zeta A} \}. $
\vskip .1cm
A vector $c$ belongs to $W_0(1)$ if and only if there exists a map
$ w \in \LA$ with $w(1)= 0 $ and $w(0)=c.$ This is the case if and only
if there exists $u$ in $\ONalpha$ such that,
\[ Re( A(\zeta u(\zeta)+c))= Re((\zeta A)(u(\zeta)+c))+
Re(A((1-\zeta)c))=0. \]
Since $w= \zeta u +c,$ we have $u(1)+c=0.$
Therefore $c$ is in $W_0(1)$ if and only if $-Re(A((1 -\zeta)c))$
belongs to the range of $U_{\zeta A}(1).$
This is so if and only if
\[ \int_0^{2 \pi} \delta^t \frac{Re(A((1 -\zeta)c))}{1- \zeta} d \theta= 0 \]
for every $\delta \in Q_{(\zeta A)}.$
Since every $\delta \in Q_{(\zeta A)}$ is of the form
$\delta= \frac{\gamma}{1- \bar{\zeta}}$ with $\gamma$ real, we find
\[ \int_0^{2 \pi} \delta^t \frac{Re(A((1 -\zeta)c))}{1- \zeta} d \theta \]
\[= Re ( \int_0^{2 \pi} \delta^t Ac d \theta) = \]
\[ Re \left( \int_0^{2 \pi} -\bar{\delta}^t(\zeta A)c d \theta \right) \]
\[=-Re(\tilde{\Psi}_0(\bar{\delta})c).\]
Therefore the image of $W_0(1)$ under the conjugation map, is the
orthogonal space in $\R^{2N}$ to $H(Q_{(\zeta A)})$ where
$ H: K_{\zeta A} \rightarrow \C^N $ is given by
\[ H(\delta)= \tilde{\Psi}_0(\bar{\delta}). \]
It follows directly from the definitions that the kernel of H is precisely
$\EAC,$ and its intersection with $Q_{\zeta A} $ is
$Q_A.$ Now $Q_{\zeta A}$ has dimension
$ d_1- r_1,$ and $Q_A$ has dimension $d-r.$
Hence the image of $H$ has dimension
$ d_1- d- r_1 +r.$ However $d$ is the dimension of
$N_{\zeta A}$ which is $ d_1- l_1.$
So the codimension of
$W_0(1)$ is $l_1- r_1 + r.$
\vskip .1cm
Similarly, since $Q_A$ is maximal totally real in $K_A,$ then
a vector $c \in \C^N,$ belongs to $W_0(1) \cap i W_0(1)$
if and only if $\tilde{\Psi}_0(\bar{\delta})c=0$ for every $\delta \in
K_A.$ Therefore the complex codimension of $W_0(1) \cap i
W_0(1)$ equals the complex dimension of $K_{\zeta A}$ minus the
complex dimension of $\EAC,$ which is $l_1 - r_1.$
From above we derive that
$W_0(1) + i W_0(1))$
has complex codimension $r.$ It follows from Remark \ref{pearing}
and Theorem \ref{val0} that
$W_0(1)+i W_0(1) \subseteq W_0 \cap i W_0.$
\vskip .1cm
Since, by Theorem \ref{val0}, these two
complex spaces have the same dimension, they must coincide.
For the general case, let $\sigma$ be the unique authomorphism of the disc
such that $\sigma(1)=y $ and $\sigma(0)=x.$
By replacing $A$ with $A \circ \sigma$ we immediately prove part c).
\vskip .1cm
Now $\gamma \circ \sigma \in {\cal E}_{ \zeta A \circ \sigma}$
if and only if $\gamma \in {\cal E}_{ \sigma^{-1} A}.$
On the other hand $\sigma^{-1}(x)=0,$ and since $\sigma$ is an authomorphism,
the function $\eta(\zeta)$ given by
\[ \eta(\zeta) = \frac{\sigma^{-1}(\zeta)}{\zeta - x} \]
for $\zeta \neq x$ and
\[ \eta(x)= \frac{\de \sigma^{-1}}{\de \zeta}_{| \zeta=x } \]
is holomorphic on $\Delta,$ $C^{\alpha}$ on $\de \Delta$ and does not have
zeros on $\bar{\Delta}.$
Therefore $ {\cal E}_{\sigma^{-1} A}= {\cal E}_{(\zeta-x) A}.$
\vskip .1cm
We are left to show that the dimension of $W_x(y)$ is independent on
$x$ and $y.$ First of all the space
\[ \{ \gamma \in {\cal E}_{(\zeta-x)A} : \gamma(y)=0 \} \]
has dimension
independent on $x$. In fact the map
$ \gamma \rightarrow t \gamma,$ (where the function
\[ t(\zeta)=\frac{\bar{x}\zeta^2 -(|x|^2+1)\zeta +x}{\zeta}, \]
was defined in the proof of Proposition \ref{ranks}), gives a linear
isomorphism between
$ \{ \gamma \in {\cal E}_{(\zeta-x)A} : \gamma(y)=0 \} $
and
$ \{ \gamma \in {\cal E}_{\zeta A} : \gamma(y)=0 \}. $
However, since for every rotation $\sigma,$ we have that
${\cal E}_{\zeta A}$ is isomorphic to $ {\cal E}_{\zeta A \circ \sigma},$
by choosing the rotation sending $1$ to $y$ we show that the dimension of
the above spaces is independent on $y$ as well.
To conclude we still need to prove the independence on $x$ and $y$ of the
dimension of the spaces
\[ \{ \gamma \in {\cal E}_{(\zeta-x)A} : \gamma(y)= \tilde{\Psi}_x(\gamma)
=0 \} \]
This space coincides with
$ \{ \gamma \in {\cal E}_{A} : \gamma(y)=0 \} $
whose dimension is independent on $y$ because of part b) of Proposition
\ref{ranks}. \end{proof}
\begin{theorem}
Given $ x \ \mbox{and} \ y \ \in \de \Delta \ \mbox{with $x \neq y$} $
we have
\[ a) \ W_x(y) = \{ c \in \C^N : Re(A(x)c)=0 \ \mbox{and} \
\gamma^t(x)A(x)c=0 \mbox{ for every $\gamma \in \EA$ } \}. \]
So the kernel of $A(x)$ is contained in $ W_x.$
b) The real codimension of $W_x(y)$ in the space $Re(A(x)c)=0$ is
$r$
c) We have $W_x(y) \cap i W_x(y) = Ker(A(x)).$
d) We have $ W_x(y) + i W_x(y)= Ker P(x).$
\label{val3} \end{theorem}
\begin{proof}
We first assume that $y=1$ and $x=-1.$ Given $c \in \C^N$ with
$Re(A(-1)c)=0,$ we have that $c$ belongs to $W_{-1}(1)$ if and only if
there exists $w \in \ONalpha$ such that $w(1)=0, \ w(-1)=c$ and
$Re(A(\zeta)w(\zeta))=0.$ Let us write $w= u + \frac{(1-\zeta)c}{2}.$
Then $c$ belongs to $W_{-1}(1)$ if and only if
$Re((1-\zeta)A(\zeta)c)$ belongs to the range of $U_A(1,-1).$
This is to say that
\[ \int_0^{2 \pi} \zeta \gamma^t \frac{Re(A((1 -\zeta)c))}{1- \zeta^2}
d \theta=0 \ \mbox{for every} \ \gamma \in \EA. \]
However
\[ \int_0^{2 \pi} \zeta \gamma^t \frac{Re(A((1 -\zeta)c))}{1- \zeta^2}
d \theta =\]
\[ \frac{1}{2} \int_0^{2 \pi} \zeta \gamma^t \frac{Ac}{1+\zeta} d \theta + \]
\[ \frac{1}{2} \int_0^{2 \pi} \zeta \gamma^t\frac{(1-\bar{\zeta})\bar{A}\bar{c}}
{1-\zeta^2} d \theta = \]
\[ \frac{1}{2} \int_0^{2 \pi} \zeta \gamma^t \frac{Ac}{1+\zeta} d \theta \]
\[-\frac{1}{2} \int_0^{2 \pi} \gamma^t \frac{\bar{A} \bar{c}}{1+\zeta} d
\theta = \]
\[=i Im \left(\int_0^{2 \pi} \zeta \gamma^t \frac{Ac}{1+\zeta} d \theta
\right). \]
Let $f$ be the holomorphic extension of $\zeta \gamma^t A c,$ then $f(0)=0$
and $Re(f(-1))=0.$ Then
\[i Im \left( \int_0^{2 \pi} \frac{f(\zeta)}{1+ \zeta} d \theta \right) =\]
\[i Im \left( \int_0^{2 \pi} \frac{f(\zeta) - f(-1)}{1+ \zeta} d \theta
\right) \]
\[+ f(-1) Re \left( \int_0^{2 \pi} \frac{1}{1+ \zeta} d \theta \right)=
- \pi f(-1). \]
Since $A(-1)$ has maximal rank, we find from part b) of
Proposition \ref{ranks}, that the space $ \{ \gamma(-1), \gamma \in \EA \}
\subseteq \R^m $ has real dimension $r$
and its orthogonal space $X$ in $\R^m,$ has real dimension $m-r.$
We conclude that
$ W_{-1}(1)= i A(-1)^{-1}(X)$ has real dimension $2N-m -r,$ so
it has codimension $r$ in $ Ker (Re(A(-1)),$ this proves part b).
Part c) follows from a). As far as d) concernes we conclude from a) and c)
that
$ W_{-1}(1)+iW_{-1}(1)$ has complex codimension $r$ in $\C^N.$
However form Remark \ref{pearing} we know that
$ W_{-1}(1)+iW_{-1}(1)$ is a subspace of $Ker P(-1)$ which also has codimension
$r$ in $\C^N.$
The general case follows by acting with the authomorphism group of the
disc. \end{proof}
\begin{proposition}
a) The assignment $x \rightarrow Ker(P(x))$ defines a complex $C^{\alpha}$
vector bundle $F$ on $\bar{\Delta}$ of rank $N-r,$ which is holomorphic on
$\Delta.$ Moreover the bundle $Ker A$ on $\de \Delta$ is a subbundle of the
restriction of $F$ to $\de \Delta.$
b) The assignment $x \rightarrow W_x \cap i W_x$ coincides with
$F$ on $\Delta.$
c) Given $y \in \de \Delta,$ the assignment
$x \rightarrow W_x(y) + i W_x(y)$ coincides with
$F$ on $\bar{\Delta}-y.$
\end{proposition}
\begin{proof}
Part a) follows from the definition of $P$ and from Proposition
\ref{ranks}.
Part b) follows from part a) of Theorem \ref{val0}.
Part c) follows from Theorem \ref{val3}. \end{proof}
There is an interesting special case
\begin{proposition}
If for the map $A$ we have $r=m,$ then $W_x(y)$ is a complex space
for every $y \in \de \Delta,$ and every $x \in \bar{\Delta}-y.$
Therefore, in this case, the bundle $F$ is a holomorphic extension to
$\bar{\Delta}$ of the bundle $Ker(A(x))$ on $\de \Delta.$
\label{r=m} \end{proposition}
\begin{proof}
If $x \in \de \Delta,$ the result follows from Theorem \ref{val3} part c).
Assume then, that $x \in \Delta,$ by acting with the authomorphism group of
the disc we may suppose $x=0,$ $y=1.$
We want to show that $W_0(1)= Ker(P(0)).$
The inclusion $W_0(1) \subseteq Ker(P(0))$ follows from Remark
\ref{pearing}. Fix $c \in Ker P(0).$
Theorem \ref{val0} part a) says that there exists $w \in \LA $ with $w(0)=c.$
Hence $Pw$ is a purely imaginary constant such that $P(0)w(0)=0,$
so $Pw \equiv 0.$ Set $\tilde{w}= (1- \zeta) w,$ then $\tilde{w}(1)=0$
and $\tilde{w}(0)=c,$ from Proposition \ref{properties} part b) we conclude
that $\tilde{w}$ belongs to $\LA.$ \end{proof}
Suppose now that $A_{f_0}$ is the map associated to an analytic disc $f_0$
attached to a generic submanifold $S$ in $\C^N.$ Let $q=f_0(1),$
Recall that, whenever $A$ is regular, the set
$M = \{ f \in \ONalpha \ \mbox{such that }$
$ \ \mbox{$f$ extends to an analytic disc attached to $ S $} \},$
is a manifold in a neighborhood of $f_0,$ and that, whenever $A_{f_0}$
is strongly regular, the set
$M_q= \{ f \in M \ \mbox{such that $f(1)=q $} \}, $
is a manifold in a neighborhood of $f_0.$
We then have the following
\begin{theorem}
a) If $ \zeta A$ is regular, that is if $\EA=0,$ then there exists a
neighborhood $U$ of $f_0$ in $M$ such that ${\mu}_0(U)$ is an open set in
$\C^N.$
b) If $ \zeta A$ is strongly regular, then there exists a
neighborhood $U$ of $f_0$ in $M_q$ such that ${\mu}_0(U)$ is an open set in
$\C^N.$ \label{openmap} \end{theorem}
\begin{proof}
a) If $\zeta A$ is regular, then the operator $U_{\zeta A}$ is
onto, in particular so is the operator $U_{A}.$ Hence
$M$ is a manifold in a neighborhood of $f_0.$
The tangent space to $M$ at $f_0$ coincides with $\LA$ and the map
${{\mu}_0}_{| \LA}$ coincide with the differential of ${{\mu}_0}_{|M}.$
It follows from Theorem \ref{val0} that such differential is onto.
We conclude by using the open mapping theorem.
The proof of part b) goes in the same way if we use Theorem \ref{onto}
\end{proof}
\begin{lemma}
Let $\Omega$ be the subset of $\ONalpha$ given by the analytic discs $f$ such
that there exists a neighborhood of f in $\ONalpha$ where the numbers
d, l and r are constant. Then $\Omega$ is open and dense in
$\ONalpha,$ moreover $\Omega \cap M$ is open and dense in $M.$
\label{locconst} \end{lemma}
\begin{proof}
Let $f$ be an analytic disc with associated map A. Recall that $d-l$ is the
dimension of the cokernel of $U_A,$ whereas $d-r$ is the dimension of the
cokernel of $U_A(1)$ and d is the dimension of the cokernel of $U_A(1,-1).$
Since the map $f \rightarrow A_f$ is continuous on $\ONalpha,$
It follows from, \cite{K} pag.235, that $d,$ $d-r,$ and $d-l$ are upper
semicontinuous functions in $\ONalpha$ with value in the set of non negative
integers. Let $ \Omega_1 $ be the set of discs where $d$ is locally constant,
$\Omega_2$ the set of discs where $d-l$ is locally costant and
$ \Omega_3$ the set of discs where $d-r$ is locally constant.
Let us consider for example $\Omega_1.$ Let U be any non empty open subset
of $\ONalpha$, then, since d gives a semicontinuous function, it follows that
the set of discs in U where d takes its minimum value in U, is open and non
empty, so $U \cap \Omega_1 \neq \emptyset.$ In the same way it can be shown
that $\Omega_2$ and $\Omega_3,$ are dense in M. So
$\Omega= \Omega_1 \cap \Omega_2 \cap \Omega_3$
is a dense open subset of $\ONalpha,$ in the same way it can be proved that
$\Omega \cap M$ is dense in $M.$ \end{proof}
Observe that $\Omega$ is invariant under the action of the group of
authomorphisms of the disc.
\begin{definition}
Let $p$ be a point on the submanifold S in $\C^N,$ we say that $p$ is a
minimal point if every immersed submanifold H of $S$ containing $p$ and such
that the complex tangent bundle to $H$ concides with the restriction to H of
the complex tangent bundle to S, is an open set in S. \label{min}
\end{definition}
\begin{corollary}
Let $S$ be a generic submanifold of $\C^N$ where every point is minimal,
then the set $\tilde{\Omega}$ of analytic discs attached to S with defect zero
is open and dense in $M.$ In particular $\mu_0$ restricted to
$\tilde{\Omega}$ is an open map. \label{opendense} \end{corollary}
\begin{proof}
Let $\Omega$ be the open dense subset defined in Lemma \ref{locconst},
clearly $\tilde{\Omega}, $ is contained in $\Omega \cap M.$ Viceversa let
$f_0 \in \Omega \cap M$ such that $f(0)=q,$ then by Remark \ref{implfunc},
the sets $\Omega \cap M$
and $\Omega \cap M_q$ are manifolds. Let us consider the
the map $\mu_{-1}$ restricted to $\Omega \cap M_q,$
seen as a map from $ \Omega \cap M_q$ into S. By Theorem \ref{val3} the
image of the differential of such map has a locally constant
codimension r. Therefore $ \mu_{-1}(\Omega \cap M_q) $ is an immersed
submanifold as in definition \ref{min}. Since every point of $S$ is assumed
to be minimal, it follows that $ r=0,$ hence, because of Proposition
\ref{properties} d=0. \end{proof}
\section{The case $N=m$ (totally real case)} \label{tot*real}
If we assume that $A$ is at values in $GL(N,\C),$ then we may use the Birkhoff
factorization $-A^{-1} \bar{A}= \Theta \Lambda \overline{\Theta^{-1}},$
described in Lemma \ref{division}.
We are going to determine everything we need in terms of the partial indices
$k_1, k_2, \ldots k_N.$ (See also \cite{Pa}).
We have
\[ \LA= \{ u \in \ONalpha : Au + \overline{A} \overline{u}=0 \} \]
\[ = \{ u \in \ONalpha : u + A^{-1} \overline{A} \overline{u}=0 \} \]
\[ = \{ u \in \ONalpha : \Theta^{-1} u - \Lambda \overline{
\Theta^{-1}u}=0 \} \]
\[ = \{ v \in \ONalpha : \bar{v_j}= \zeta^{-k_j}v_j \ \mbox{on $\de \Delta $}
\} \]
Here $j$ runs from $1$ to $N$ and $v_j$ is the jth component of the vector
$v= \Theta^{-1}u.$
Therefore if $k_j < 0,$ then $v_j \equiv 0,$ if $k_j=0$ then $v_j$ is a
real constant function, and if $j >0 $ then $v_j$ is a polynomial of degree
$k_j.$ It has the form $v_j= \sum_{i=0}^{k_j} a_i z^i,$ with
$a_s= \overline{a_{k_j-s}}$ for every $s$ from $0$ to $k_j.$
Let $k_{+}$ be the sum of all the positive partial indeces, and let $m_{+}$ be
their number. Let $k_{-}$ be the absolute value of the sum of the negative
partial indices, and let $m_{-}$ their number, finally let $m_0$ be the
number of partial indices equal to $0.$
So we have $dim_{\R} \LA= k_{+} + m_{+} + m_{0}.$
On the other hand
\[ \EA= \{ \gamma \in \Calpha: \gamma^t A=u \ \mbox{extends
holomorphically}\}.\]
Hence $\EA$ is isomorphic to the space
\[ \{ u \in \ONalpha : u^t A^{-1} \ \mbox{is real} \} \]
\[= \{ u \in \ONalpha : - u^t \Theta \Lambda \overline{\Theta^{-1}} =
\bar{u}^t.\} \]
\[= \{ v \in \ONalpha : \bar{v_j}= \zeta^{k_j}v_j \} \]
where $j$ runs from $1$ to $N$ and $v_j$ is the jth component of the vector
$v= i (\Theta^{-1})^t u.$
In the same way as above we find that $d= k_{-} \ + \ m_{-} \ + \ m_0.$
\vskip .1cm
If we replace A with $\zeta^{-1}A$ the index $k_j$ is replaced by $k_j+2,$
therefore the dimension of $N_A$ equals $ \sum_{k_j < 0} (|k_j|-1).$
It follows that the index of $U_A$ is $k+N,$ where $k = \sum k_j$ is the
total index, see also \cite{Oh}.
\vskip .1cm
Moreover $A$ is regular if and only if
every partial index is greater then or equal to $-1.$
\vskip .1cm
We have $l= d- \ (\mbox{dimension of $N_A$}) = 2m_{-} + m_0.$
On the other hand $(\Psi_0)(\EAC)$ has complex dimension $m_{-} + m_0 = r.$
In particular $A$ is strongly regular if and only if every partial index
is greater then or equal to $0.$
\vskip .1cm
Similarly we can compute $r_1$ and $l_1.$
\vskip .1cm
Let us look at the index of $U_A(1).$
We find $dim_{\R} \LA(1) = k_{+}. $
On the other hand the dimension of the cokernel of $U_A(1)$ is
$d-r= \sum_{k_j < 0} |k_j|,$ therefore $U_A(1)$ has index $k.$
\section{The case $m=1$ (hypersurface case)} \label{hyper}
\begin{lemma}
Assume $m=1,$
given $x \in \Delta.$
we have the following possibilities
\[ \begin{array}{l}
\mbox{ \ $d=l=r=0,$ \ hence A is regular and $W_x= \C^N.$} \\
\mbox{$d=l=r=1,$ hence A is regular and $W_x$ is a real hyperplane in
$\R^{2N}.$} \\
\mbox{$d > 1, \ l=2, \ r=1$ and A is not regular.} \\
\mbox{In this case $W_x$ is a complex hyperplane in $\C^N.$} \\
\mbox{In particular we can not have $d=2.$} \end{array} \]
\label{m2} \end{lemma}
\begin{proof}
If $d=0,$ then $l=r=0.$ If $d \neq 0,$ by the Proposition
\ref{properties} $r=1$ and $l$ is either 1 or 2.
Moreover, by
Proposition \ref{grasm1}, and Example \ref{strongreg}, $l=1,$ if and only if
$A$ is regular, if and only if $d=l.$
If we had $d=2,$ we would have $r=1.$ If it was $l=1,$ we
would have $l=r,$ so $A$ regular, so $l=d=2.$ If it was $l=2,$ we
would have $l=d,$ hence $A$ regular, and $l=r=1.$ So we found an
absurd. The other statements follow from Theorem \ref{val0}. \end{proof}
More generally we have the following
\begin{proposition}
If $m=1,$ then $d$ is either zero or an odd positive integer.
For every d which is either 0 or an odd positive integer,
there exists an analytic disc attached to an hypersurface in $\C^2$
with defect d. \label{disp} \end{proposition}
\begin{proof}
Let $2k_0$ be the smallest strictly positive even
integer such that there exists an $A$ with $m=1$ and $d=2k_0.$
From the above Lemma we know that $2k_0 > 2,$ and that $l=2,$ so
$2k_0-2$ is even and strictly positive. However
$2k_0-2$ is the defect of $ {\zeta^{-1}A}.$
This is against the minimality of $2k_0.$
Let us consider the example in \cite{Pa} given by the hypersurface
\[ S= \{ (z_1,z_2) \in \C^2 : Re(z_1^k z_2)=0, z_1 \neq 0 \} \]
and the disc $f(\zeta)= (\zeta,0)$ attached to S.
If $ k < 0$ the disc $f$ has defect 0, while if $k \geq 0,$ the disc has
defect $2k+1.$ \end{proof}
Moreover
\begin{proposition}
Assume that $m=1.$
Given $y \in \de \Delta$ and $x \in \Delta, $ we have that
$W_x(y)$ is a complex space, more precisely
\vskip .1cm
If $\EA =0,$ then $W_x(y)= \C^N.$
\vskip .1cm
If $\EA \neq 0,$ then $W_x(y)$ is a complex
hyperplane in $\C^N.$ \end{proposition}
\begin{proof}
By Proposition \ref{properties} we know that $r \leq 1.$ If $r=1$ we
conclude applying Proposition \ref{r=m}, if $r=0,$ then $\EA=0,$ and so
$\zeta A $ is regular. It follows from example \ref{strongreg} that
$ \zeta A$ is strongly regular, so $l_1= r_1$ and by
Theorem \ref{val2} we conclude. \end{proof}
\begin{proposition}
Assume that $m=1.$
Fix a point $x \in \de \Delta. $
\vskip .1cm
If $A$ is not regular, then, $W_x= Ker(A(x)).$
\vskip .1cm
If A is regular, then $W_x= Ker(Re(A(x))).$ \end{proposition}
\begin{proof}
$A$ is regular, if and only if its strongly regular if and only if
$l=r. $ (See Proposition \ref{grasm1}). We conclude using Theorem
\ref{val1}. \end{proof}
\begin{proposition}
Assume that $m=1.$
Fix two points $x$ and $y$ in $ \de \Delta $ with $x \neq y.$
\vskip .1cm
If $\EA \neq 0,$ then $W_x(y)= Ker(A(x))$
\vskip .1cm
If $\EA =0,$ then $W_x(y)= Ker(Re(A(x)).$ \end{proposition}
\begin{proof}
It follows directly from Theorem \ref{val3}. \end{proof}
Suppose now that $A_{f_0}$ is the map associated to an analytic disc $f_0$
attached to an hypersurface $S$ in $\C^N.$ Let $q=f_0(1).$
\begin{corollary}
If $\EA=0$ and $m=1,$ then there exists a neighborhood $U$
of $f_0$ in $M_q$ such that ${\mu}_0(U)$ is an open set in
$\C^N.$ \end{corollary}
\begin{proof}
Since $\zeta A$ is strongly regular if and only if it is regular, if and
only if $\EA=0,$ the result follows from Theorem \ref{openmap} \end{proof}
|
{
"timestamp": "1996-04-16T19:31:02",
"yymm": "9604",
"arxiv_id": "math/9604202",
"language": "en",
"url": "https://arxiv.org/abs/math/9604202"
}
|
\section{Introduction}
Broadly speaking, production of narrow $p$ wave pseudovector bound states
of heavy quarks $^3P_1$ with $J^{PC}=1^{++}$, like $\chi_{b1}(1p)\,,\,
\chi_{b1}(2p)$ and $\chi_{c1}(1p)$ \cite{pdg-1994}, in $e^+e^-$ collisions via
the virtual intermediate $Z$ boson ($e^+e^-\rightarrow Z\rightarrow\, ^3P_1$),
could be observed by experiment at least at $b$ and $c-\tau$ factories for
amplitudes of weak interactions grow with energy increase in this energy
regions $\propto G_FE^2$.
In present paper it is shown that the experimental investigation of this
interesting phenomenon is possible at current facilities.
In Section II partial widths and branching ratios of $^3P_1\rightarrow
e^+e^-$ decays are calculated within the Born approximation. It is shown that
$BR(\chi_{b1}(1P)\rightarrow Z\rightarrow e^+e^-)\simeq 3.3\cdot 10^{-7}$,
$BR(\chi_{b1}(2P)\rightarrow Z\rightarrow e^+e^-)\simeq 4.1\cdot 10^{-7}$ and
$BR(\chi_{c1}(1P)\rightarrow Z\rightarrow e^+e^-)\simeq 10^{-8}$. A chance to
search for the direct production of pseudovector $^3P_1$ heavy quarkonia in
$e^+e^-$ collisions ($e^+e^-\rightarrow Z\rightarrow\, ^3P_1$) at current
facilities is estimated high enough not to mention $b$ and $c-\tau$
factories.
In Section III QCD corrections to widths of decays are discussed.
Conclusion (Section IV) discusses experimental perspectives.
\section{ Widths, branching ratios, cross sections and numbers of events}
First and foremost let us calculate the $^3P_1\rightarrow Z\rightarrow
e^+e^-$ amplitude. We use a handy formalism of description of a
nonrelativistic bound state decays given in the review \cite{novikov-1978}.
The Feynman amplitude describing a free quark-antiquark annihilation into
the $e^+e^-$ pair has the form
\begin{eqnarray}
&& M(\bar Q(p_{\bar Q})Q(p_Q)\rightarrow Z\rightarrow e^+(p_+)e^-(p_-))=
\frac{\alpha\pi}{2\cos^2\theta_W\sin^2\theta_W}\frac{1}{E^2-m^2_Z}
j^{e\,\alpha}j^Q_\alpha= \nonumber\\
&&\frac{\alpha\pi}{2\cos^2\theta_W\sin^2\theta_W}\frac{1}{E^2-m^2_Z}
\bar e(p_-)[(-1+4\sin^2\theta_W)\gamma^\alpha-\gamma^\alpha\gamma_5]e(-p_+)
\cdot \nonumber\\
&& \bar Q_C(-p_{\bar Q})[(t_3-e_Q\sin^2\theta_W)\gamma_\alpha+
t_3\gamma_\alpha\gamma_5]Q^C(p_Q)
\end{eqnarray}
where the notation is quite marked unless generally accepted.
One can ignore the vector part of the electroweak electron-positron current $
j^{e\,\alpha}$ in Eq. (1) for $(1-4\sin^2\theta_W)\simeq 0.1$. As for the
electroweak quark-antiquark current $j^Q_\alpha$, only its axial-vector part
contributes to the pseudovector ($1^{++}$) quarkonia annihilation. So, the
amplitude of interest is
\begin{eqnarray}
&& M(\bar Q(p_{\bar Q})Q(p_Q)\rightarrow Z\rightarrow e^+(p_+)e^-(p_-))=
\sigma_Q\frac{\alpha\pi}{4\cos^2\theta_W\sin^2\theta_W}
\frac{1}{m^2_Z}j^{e\,\alpha}_5j^Q_{\alpha\,5}=
\nonumber\\
&& \sigma_Q\frac{\alpha\pi}{4\cos^2\theta_W\sin^2\theta_W}
\frac{1}{m^2_Z}\bar e(p_-)\gamma^\alpha\gamma_5e(-p_+)\bar
Q_C(-p_{\bar Q})\gamma_\alpha\gamma_5Q^C(p_Q)
\end{eqnarray}
where $\sigma_c=1$ and $\sigma_b=-1$. The term of order of $E^2/m_Z^2$ is
omitted in Eq. (2).
In the c.m. system one can write that
\begin{eqnarray}
&& M(\bar Q(p_{\bar Q})Q(p_Q)\rightarrow Z\rightarrow e^+(p_+)e^-(p_-))\simeq
-\sigma_Q\frac{\alpha\pi}{4\cos^2\theta_W\sin^2\theta_W}\frac{1}{m^2_Z}
j^e_{i\,5}\,j^Q_{i\,5}=\nonumber\\
&& -\sigma_Q\frac{\alpha\pi}{4\cos^2\theta_W\sin^2\theta_W}
\frac{1}{m^2_Z}\bar e(p_-)\gamma_i\gamma_5e(-p_+)\bar Q_C(-p_{\bar Q})
\gamma_i\gamma_5Q^C(p_Q)=M({\bf p})
\end{eqnarray}
ignoring the term of order of $2m_e/E$ ($j^e_{0\,5}=(2m_e/E)j^e_5$ in the c.m.
system). Hereafter the three-momentum ${\bf p}={\bf p_Q}=-{\bf p_{\bar Q}}$
in the c.m. system.
To construct the effective Hamiltonian for the $1^{++}$ quarkonium
annihilation into the $e^+e^-$ pair one expresses the axial-vector
quark-antiquark current $j^Q_{i\,5}$ in Eq. (3) in terms of two-component
spinors of quark $w^\alpha$ and antiquark $v_\beta$ using four-component
Dirac bispinors
\begin{eqnarray}
&& Q^C(p_Q)= Q^CQ(p_Q)=\frac{1}{\sqrt{2m_Q}}\, Q^C\left (\sqrt{\varepsilon
+ m_Q}\,\, w\atop {\sqrt{\varepsilon - m_Q}\,\, ({\bf n}\cdot\mbox{\boldmath
$\sigma$})w}\right)\,,\nonumber\\
&&\bar Q_C(-p_{\bar Q})=Q_C\bar Q(-p_{\bar Q})=-\frac{1}{\sqrt{2m_Q}}\,
Q_C\left(\sqrt{\varepsilon -m_Q}\,\, v({\bf n}\cdot\mbox{\boldmath $\sigma$})
\,,\,\sqrt{\varepsilon +m_Q}\,\, v\right)
\end{eqnarray}
where $Q^C$ and $Q_C$ are color spinors of quark and antiquark respectively,
${\bf n}={\bf p}/|{\bf p}|$.
As the result
\begin{equation}
j^Q_{i\,5}=\imath\frac{2\sqrt{6}}{2m_Q}\,\varepsilon_{kin}p_k\chi_n \eta_0
\end{equation}
where $\chi_n=v\sigma_nw/\sqrt{2}$ and $\eta_0=Q_CQ^C/\sqrt{3}$.
The spin-factor $\chi_i$ and the color spin-factor $\eta_0$ are contracted as
follows $\chi_i\chi_j=\delta_{ij}$ and $\eta_0\eta_0=1$.
The $^3P_1$ bound state wave function in the coordinate representation has
the form
\begin{equation}
\Psi_j(^3P_1\,,\,\mbox{\boldmath $r$}\,,\,m_A )= \frac{1}{\sqrt{2}}%
\,\eta_0\varepsilon_{jpl}\chi_p\frac{r_l}{r} \sqrt{\frac{3}{4\pi}}\,
R_P(r\,,\,m_A)
\end{equation}
where $r=|{\bf r}|$, $R_P(r,m_A)$ is a radial wave function with the
normalization $\int_0^{\infty}|R_P(r\,,\,m_A)|^2r^2dr=1$, $m_A$ is a mass of
a $^3P_1$ bound state.
For use one needs the $^3P_1$ bound state wave function in the momentum
representation. It has the form
\begin{equation}
\Psi_j(^3P_1\,,\,\mbox{\boldmath $p$}\,,\,m_A )= \frac{1}{\sqrt{2}}
\,\eta_0\varepsilon_{jpl}\chi_p (\psi_P(\mbox{\boldmath $p$}\,,\,m_A ))_l
\end{equation}
where
\begin{equation}
(\psi_P(\mbox{\boldmath $p$}\,,\,m_A ))_l=\sqrt{\frac{3}{4\pi}}\,\int \frac{
r_l}{r}R_P(r\,,\,m_A)\exp\{-\imath (\mbox{\boldmath $p$}\cdot
\mbox{\boldmath $r$)}\}d^3r\,.
\end{equation}
The amplitude of the $^3P_1$ bound state $\rightarrow e^+e^-$ annihilation
is given by
\begin{equation}
M(A_j\rightarrow e^+e^-)\equiv\int M(\mbox{\boldmath $p$}) \Psi_j(^3P_1\,,\,%
\mbox{\boldmath $p$}\,,\,m_A)\frac{d^3p}{(2\pi)^3}
\end{equation}
where $A_j$ stands for a $^3P_1$ state.
As seen from Eq. (3) to find the $M(A_j\rightarrow e^+e^-)$ amplitude one
needs to calculate the convolution and contraction of a quark-antiquark pair
axial-vector current with a $^3P_1$ bound state wave function
\begin{eqnarray}
&& \int j^Q_{i\,5}\Psi_j(^3P_1\,,\,\mbox{\boldmath $p$}\,,\,m_A)
\frac{d^3p}{(2\pi)^3}=\imath\frac{2\sqrt{6}}{m_A}\,\varepsilon_{kin}
\chi_n\eta_0\int p_k\cdot\Psi_j(^3P_1\,,\,\mbox{\boldmath $p$}\,,\,m_A)
\frac{d^3p}{(2\pi)^3}=\nonumber\\
&& \imath\delta_{ij}\frac{4}{\sqrt{3}}\,\frac{1}{m_A}
\int p_k(\psi_P(\mbox{\boldmath $p$}\,,\,m_A))_k\frac{d^3p}{(2\pi)^3}=
\delta_{ij}2\frac{3}{\sqrt{\pi}}\,\frac{1}{m_A}R_P^\prime(0\,,\,m_A)
\end{eqnarray}
where $R^\prime_P(0\,,\,m_A)=dR_P(r\,,\,m_A)/dr|_{r=0}$. Deriving Eq. (10)
we put, as it usually is, $2m_Q=m_A$ and took into account that $
R_P(r\,,\,m_A) \rightarrow rR^\prime_P(0\,,\,m_A)$ when $r\rightarrow 0$.
So,
\begin{equation}
M(A_j\rightarrow e^+e^-)=-\sigma_Q3\sqrt{\pi}\,\frac{\alpha}{2\cos^2\theta_W
\sin^2\theta_W}\frac{1}{m^2_Z}\frac{1}{m_A}R^\prime_P(0) \bar
e(p_-)\gamma_j\gamma_5e(-p_+)\,.
\end{equation}
The width of the $A\rightarrow e^+e^-$ decay
\begin{eqnarray}
&&\Gamma (A\rightarrow e^+e^-)=\frac{1}{3}\sum_{j\,e^+e^-}\int |M(A_j
\rightarrow e^+e^-)|^2(2\pi)^4\delta^4(m_A-p_--p_+)\frac{d^3p_+}
{(2\pi)^32p^0_+}\frac{d^3p_-}{(2\pi)^32p^0_-}\simeq\nonumber\\
&&\frac{1}{3}\sum_{j\,e^+e^-}\int |M(A_j\rightarrow e^+e^-)|^2
\frac{1}{8\pi}\simeq\nonumber\\
&&\frac{\alpha^2}{32}\frac{3}{\cos^4\theta_W\sin^4\theta_W}
\frac{1}{m_Z^4}\frac{1}{m^2_A}|R^\prime_P(0\,,\,m_A)|^2Sp (\hat p_+\gamma_j
\gamma_5\hat p_-\gamma_j\gamma_5)\simeq\nonumber\\
&&\frac{\alpha^2}{8}\frac{3}{\cos^4\theta_W\sin^4\theta_W}\frac{1}{m_Z^4}
|R^\prime_P(0\,,\,m_A)|^2\simeq 12.3\alpha^2\frac{1}{m_Z^4}|R^\prime_P(0\,,
\,m_A)|^2
\end{eqnarray}
where terms of order of $(2m_e/m_A)^2$ are omitted, $\sin^2\theta_W=0.225$
is put, the normalization $\bar e(p_-)e(p_-)=2m_e$ and $\bar
e(-p_+)e(-p_+)=-2m_e$ is used. Note that for the quark and antiquark we use
the normalization $\bar Q(p_Q)Q(p_Q)=1$ and $\bar Q(-p_{\bar Q})Q(-p_{\bar
Q})=-1$, see Eq. (4).
To estimate a possibility of the $\chi_{c1}(1P)$, $\chi_{b1}(1P)$ and
$\chi_{b1}(2P)$ production in $e^+e^-$ collisions it needs to estimate the
branching ratio $BR(A\rightarrow e^+e^-)$.
In a logarithmic approximation \cite{novikov-1978,barbieri-1976} the decay
of the $^3P_1$ level into hadrons is caused by the decays $^3P_1\rightarrow
g + q\bar q$ where $g$ is gluon and $q\bar q$ is a pair of light quarks:
$u\bar u,\, d\bar d,\,s\bar s$ for $\chi_{c1}(1P)$ and $u\bar u,\,d\bar
d,\,s\bar s,\, c\bar c$ for $\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ . The
relevant width \cite{novikov-1978,barbieri-1976}
\begin{equation}
\Gamma\left(^3P_1\equiv A\rightarrow gq\bar q\right)\simeq \frac{N}{3}
\frac{128}{3\pi}\frac{\alpha^3_s}{m_A^4}|R^\prime_P(0\,,\,m_A)|^2
\ln\frac{m_AR(m_A)}{2}
\end{equation}
where N is the number of the light quark flavors and $R(m_A)$ is the
quarkonium radius. Using Eqs. (12) and (13) one gets that
\begin{equation}
BR(A\rightarrow e^+e^-)\simeq \frac{3}{N}0.9\frac{\alpha^2}{\alpha_s^3}
\left (\frac{m_A}{m_Z}\right)^4\frac{1}{\ln\left(m_AR(m_A)/2\right)}\left[1-
\sum_VBR(A\rightarrow\gamma V)\right]
\end{equation}
where the radiative decays $\chi_{c1}(1P)\rightarrow\gamma J/\psi$,
$\chi_{b1}(1P)\rightarrow\gamma \Upsilon (1S)$,
$\chi_{b1}(2P)\rightarrow\gamma \Upsilon (1S)$ and
$\chi_{b1}(2P)\rightarrow\gamma \Upsilon (2S)$
are taken into account.
A convention \cite{novikov-1978} uses that $L(m_A)=
\ln\left(m_AR(m_A)/2\right)\simeq 1$ for $\chi_{c1}(1P)$, i.e. when $m_A=
3.51\,GeV$ \cite{pdg-1994}. As for $\chi_{b1}(1P)$, $m_A=9.89\,GeV$ \cite
{pdg-1994}, and $\chi_{b1}(2P)$, $m_A=10.2552\,GeV$ \cite{pdg-1994}, it
depends on the $m_A$ behavior of the quarkonium radius $R(m_A)$. For example,
the coulomb-like potential gives that $R(m_A)\sim 1/m_A$ and the logarithm
practically does not increase, $L(3.51\,GeV)\simeq L(9.89\,GeV)\simeq
L(10.2552\,GeV)\simeq 1$. Alternatively, the harmonic oscillator potential
gives $R(m_A)\sim 1/\sqrt{m_A\omega_0}$, where $\omega_0\simeq 0.3\,GeV$, that
leads to $L(9.89\,GeV)\simeq L(10.2552\,GeV)\simeq 1.5$. To be conservative
one takes $L(9.89\,GeV)=L(10.2552\,GeV)=2$.
So, putting $\alpha_s(3.51\,GeV)=0.2\,,\,BR(\chi_{c1}(1P)\rightarrow\gamma
J/\psi)=0.27$ \cite{pdg-1994}, $N=3\,,\,L(3.51\,GeV)=1\,,\,m_A=
m_{\chi_{c1}(1P)}=3.51\,GeV$ and $m_Z=91.2\,GeV$ one gets from Eq. (14) that
\begin{equation}
BR(\chi_{c1}(1P)\rightarrow e^+e^-)=0.96\cdot 10^{-8}\,.
\end{equation}
Putting $\alpha_s(9.89\,GeV)=\alpha_s(10.2552\,GeV)=0.17\,,\,
BR(\chi_{b1}(1P)\rightarrow\gamma \Upsilon (1S))=0.35 \cite{pdg-1994}\,,
BR(\chi_{b1}(2P)\rightarrow\gamma \Upsilon (1S))+
BR(\chi_{b1}(2P)\rightarrow\gamma \Upsilon (2S))=0,085+0.21=0.295
\cite{pdg-1994}\,,\, N=4\,,\,L(9.89\,GeV)=L(10.2552\,GeV) =2\,,\,m_A=
m_{\chi_{b1}(1P)}=9.89\,GeV\,,\,m_A=m_{\chi_{b1}(2P)}=10.2552\,GeV$ and
$m_Z=91.2\,GeV$ one gets from Eq. (14) that
\begin{eqnarray}
&&BR(\chi_{b1}(1P)\rightarrow e^+e^-)=3.3\cdot 10^{-7}\,,\\ \nonumber
&&BR(\chi_{b1}(2P)\rightarrow e^+e^-)=4.1\cdot 10^{-7}\,.
\end{eqnarray}
Let us discuss possibilities to measure the branching ratios under
consideration .
The cross section of a reaction $e^+e^-\rightarrow A\rightarrow out$ at
resonance peak \cite{pdg-1994}
\begin{equation}
\sigma (A)\simeq 1.46\cdot 10^{-26}BR(A\rightarrow e^+e^-)BR(A\rightarrow
out) \left(\frac{GeV}{m_A}\right)^2\,cm^2\,.
\end{equation}
So, for the $\chi_{c1}(1P)$ state production
\begin{equation}
\sigma \left(\chi_{c1}(1P)\right)\simeq 1.14\cdot 10^{-35}BR\left
(\chi_{c1}(1P) \rightarrow out\right)\,cm^2
\end{equation}
and for the production of the $\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ states
\begin{eqnarray}
&&\sigma \left(\chi_{b1}(1P)\right)\simeq 4.8\cdot
10^{-35}BR\left(\chi_{b1}(1P) \rightarrow out\right)\,cm^2\,,\\ \nonumber
&&\sigma \left(\chi_{b1}(2P)\right)\simeq 5.6\cdot
10^{-35}BR\left(\chi_{b1}(2P) \rightarrow out\right)\,cm^2\,.
\end{eqnarray}
In general, the visible cross section at the peak of the narrow resonances
like $J/\psi\,,\,\Upsilon (1S)$ and so on is suppressed by a factor of order
of $\Gamma_{tot}/\Delta E$ where $\Delta E$ is an energy spread. But,
fortunately, the $\chi_{c1}(1P)$ resonance width equal to 0.88 $MeV$ \cite
{pdg-1994} is not small in comparison with energy spreads of current
facilities, for example, $\Delta E\simeq 2\,MeV$ at BEPC (China), see \cite
{pdg-1994}. Taking into account that the luminosity at BEPC \cite{pdg-1994} is
equal to $10^{31}\,cm^{-2} s^{-1}\,,\,\Gamma_{tot}(\chi_{c1}(1P))/\Delta E
\simeq 0.44$ and the cross section of the $\chi_{c1}(1P)$ production is equal
to $1.14\cdot 10^{-35}\,cm^2$, see Eq. (18), one can during an effective year
($10^7$ seconds) working produce 501 $\chi_{c1}(1P)$ states.
Note that such a number of $\chi_{c1}(1P)$ states gives 135 (27\%) unique
decays $\chi_{c1}(1P)\rightarrow\gamma J/\psi$.
The $c-\tau $ factories (luminosity $\sim 10^{33}\,cm^{-2}s^{-1}$) could
produce several tens of thousands of the $\chi _{c1}(1P)$ states.
As for the $\chi_{b1}(1P)$ state, its width is unknown up to now \cite
{pdg-1994}. Let us estimate it using the $\chi_{c1}(1P)\,,\,J/\psi\,,\,
\Upsilon(1S)$ widths and the quark model.
In the quark model
\begin{equation}
\Gamma\left(^3S_1\equiv V\rightarrow ggg\right)=\frac{40}{81\pi}\left(\pi^2-9
\right)\frac{\alpha^3_s}{m_V^2}|R_S(0\,,\,m_V)|^2
\end{equation}
One gets from Eqs. (13) and (20) that
\begin{eqnarray}
&&\frac{\Gamma (A\rightarrow gq\bar q)}{\Gamma (V\rightarrow ggg)}=
\nonumber\\[3pt]
&&\frac{\Gamma_{tot}(A)}{\Gamma_{tot}(V)}\frac{BR(A\rightarrow hadrons)}{
[BR(V\rightarrow hadrons)- BR(V\rightarrow virtual\,\gamma\rightarrow
hadrons)]}=\nonumber\\[3pt]
&& 99.4\frac{N}{3}\left(\frac{m_V}{m_A}\right)^2
\left|\frac{R^{\prime}_P(0\,,\,m_A)}{m_AR_S(0\,,\,m_V)}\right|^2
\ln\frac{m_AR(m_A)}{2}\,.
\end{eqnarray}
So,
\begin{eqnarray}
&&\frac{\Gamma_{tot}(\chi_{b1}(1P))}{\Gamma_{tot}(\Upsilon (1S))}=
0.53\frac{\Gamma_{tot}(\chi_{c1}(1P))}{\Gamma_{tot}(J/\psi)}\left|\frac{
R^\prime_P(0\,,\,m_{\chi_{b1}(1P)})R_S(0\,,\,m_{J/\psi})}{R^\prime_P(0\,,
\,m_{\chi_{c1}(1P)})R_S(0\,,\,m_{\Upsilon (1S)})}\right|^2=\nonumber\\[6pt]
&&5.3\left|\frac{R^\prime_P(0\,,\,m_{\chi_{b1}(1P)})R_S(0\,,\,m_{J/\psi})}
{R^\prime_P(0\,,\,m_{\chi_{c1}(1P)})R_S(0\,,\,m_{\Upsilon (1S)})}\right|^2\,.
\end{eqnarray}
Calculating Eq. (22) one used data from \cite{pdg-1994},
$BR(\Upsilon (1S)\rightarrow hadrons)- BR(\Upsilon (1S)\rightarrow virtual\,
\gamma\rightarrow hadrons)=0.83$, $BR(J/\psi\rightarrow hadrons)-
BR(J/\psi\rightarrow virtual\, \gamma\rightarrow hadrons)=0.69\,,\,
\Gamma_{tot}(\chi_{c1}(1P))/\Gamma_{tot}(J/\psi)=10$, and $L(m_{b1(1P)})
/L(m_{c1(1P)})=2$ as in the foregoing.
The unknown factor in Eq. (22) depends on a model. In the Coulomb-like
potential model it is
\begin{equation}
\left|\frac{R^\prime_P(0\,,\,m_{\chi_{b1}(1P)})R_S(0\,,\,m_{J/\psi})}
{R^\prime_P(0\,,\,m_{\chi_{c1}(1P)})R_S(0\,,\,m_{\Upsilon (1S)})}\right|^2=
\left(\frac{m_{\chi_{b1}(1P)}}{m_{\chi_{c1}(1P)}}\right)^5 \left(\frac{%
m_{J/\psi}}{m_{\Upsilon (1S)}}\right)^3=6.2 \,.
\end{equation}
In the harmonic oscillator potential it is
\begin{equation}
\left|\frac{R^\prime_P(0\,,\,m_{\chi_{b1}(1P)})R_S(0\,,\,m_{J/\psi})}
{R^\prime_P(0\,,\,m_{\chi_{c1}(1P)})R_S(0\,,\,m_{\Upsilon (1S)})}\right|^2=
\left(\frac{m_{\chi_{b1}(1P)}}{m_{\chi_{c1}(1P)}}\right)^{2.5} \left(\frac{%
m_{J/\psi}}{m_{\Upsilon (1S)}}\right)^{1.5}=2.5 \,.
\end{equation}
To be conservative one takes Eq. (24). Thus one expects
\begin{equation}
\Gamma_{tot}(\chi_{b1}(1P))\simeq 13\Gamma_{tot}(\Upsilon (1S))\simeq 0.695\,
MeV\,.
\end{equation}
Let us estimate a number of the $\chi_{b1}(1P)$ states which can be produced
at CESR (Cornell) \cite{pdg-1994}. Taking into account that luminosity at
CESR is equal to $10^{32}\,cm^{-2}s^{-1}\,,\,\Delta E\simeq 6\,MeV\,,\,
\Gamma_{tot}(\chi_{b1}(1P))/\Delta E\simeq 0.12$ and the cross section of the
$\chi_{b1}(1P)$ production is equal to $4.8\cdot 10^{-35}\,cm^2$, see Eq.
(19), one can during an effective year ( $10^7$ seconds) working produce 5622
$\chi_{b1}(1P)$ states. This number of the $\chi_{b1}(1P)$ states gives 1968
(35\%) unique decays $\chi_{b1}(1P)\rightarrow\gamma\Upsilon (1S)$.
At VEPP-4M (Novosibirsk) \cite{pdg-1994} one can produce a few hundreds of
the $\chi_{b1}(1P)$ states.
As for the $b$ factories with luminosities $10^{33}\,cm^{-2}s^{-1}$ and
$10^{34}\,cm^{-2}s^{-1}$ \cite{pdg-1994}, they could produce tens and
hundreds of thousands of the $\chi_{b1}(1P)$ states.
As for the $\chi_{b1}(2P)$ state, it is impossible to estimate its width by
the considered way. The point is that the $\chi_{c1}(2P)$ state is unknown up
to now \cite{pdg-1994} (probably, this state lies above the threshold of the
$D\bar D^* + D^*\bar D$ production). But, one can express the $\chi_{b1}(2P)$
width in terms of the $\chi_{b1}(1P)$ one using the quark model, see Eq. (13),
and the experimental information \cite{pdg-1994} on the radiative decays
$A\rightarrow\gamma V$.
In the Coulomb-like potential model one gets
\begin{equation}
\Gamma_{tot}(\chi_{b1}(2P)\simeq 0.35\Gamma_{tot}(\chi_{b1}(1P)
\end{equation}
and in the harmonic oscillator potential model one gets
\begin{equation}
\Gamma_{tot}(\chi_{b1}(2P)\simeq 2.9\Gamma_{tot}(\chi_{b1}(1P)\,.
\end{equation}
As seen from Eqs. (26) and (27) the result depends strongly on a potential.
Nevertheless, even in the worst case from the standpoint of search for the
$\chi_{b1}(2P)$ state production, in the case of Eq. (26), a number of
produced $\chi_{b1}(2P)$ states is equal to 41 \% , see Eq. (19), of a number
of produced $\chi_{b1}(1P)$ states.
That is why it is reasonable to believe that there is a good chance to
search for the direct production of the $\chi_{b1}(2P)$ state as in the case
of the $\chi_{b1}(1P)$ state.
\section{QCD corrections to widths}
Let us take into account the leading radiative corrections in QCD to
the width of the annihilation of the $^3P_1$ state into the $e^+e^-$ pair.
Using the well-known result \cite{mackenzie-1981}
\begin{equation}
\Gamma \left(^3S_1\rightarrow e^+e^-\right)=4\alpha^2e^2_Q\frac{1}{m_V^2}
|R_S(0)\,m_V|^2\left(1-\frac{16}{3}\frac{\alpha_s(m_V)}{\pi}\right)
\end{equation}
one can easy get
\begin{equation}
\Gamma \left(^3P_1\rightarrow e^+e^-\right)=
\frac{\alpha^2}{8}\frac{3}{\cos^4\theta_W\sin^4\theta_W}\frac{1}{m_Z^4}
|R^\prime_P(0\,,\,m_A)|^2\left(1-\frac{16}{3}\frac{\alpha_s(m_V)}{\pi}\right)\,.
\end{equation}
Correspondingly, the correction to the logarithmic approximation Eq. (13)
must be taken into account. Beyond the logarithmic approximation the decay
of the $^3P_1$ level into hadrons is caused by the decays $^3P_1\rightarrow
g + q\bar q$ and $^3P_1\rightarrow 3g$. The relevant width
\cite{barbieri-1981}
\begin{equation}
\Gamma\left(^3P_1\equiv A\rightarrow gq\bar q +3g\right)\simeq
\frac{N}{3}\frac{128}{3\pi}\frac{\alpha^3_s}{m_A^4}|R^\prime_P(0\,,\,m_A)|^2
\left(\ln\frac{m_AR(m_A)}{2}-0.51\right)\,.
\end{equation}
So, in place of Eq. (14) one gets
\begin{eqnarray}
&&BR(A\rightarrow e^+e^-)\simeq \nonumber \\[3pt]
&&\frac{3}{N}0.9\frac{\alpha^2}{\alpha_s^3}\left (\frac{m_A}{m_Z}\right)^4
\left[\frac{1-16\alpha_s(m_V)/3\pi}{\ln\left(m_AR(m_A)/2\right)-0.51}\right]
\left[1-\sum_VBR(A\rightarrow\gamma V)\right]
\end{eqnarray
that, at least, does not lower a chance to produce pseudovector heavy
quarkonia in the $e^+e^-$ collisions by the virtual $Z$ boson.
As for replacement of Eqs. (21) and (22), one takes into account that the
leading radiative corrections in QCD to Eq. (20) give \cite{mackenzie-1981}
\begin{equation}
\Gamma\left(^3S_1\equiv V\rightarrow ggg\right)=\frac{40}{81\pi}\left(\pi^2-9
\right)\frac{\alpha^3_s}{m_V^2}|R_S(0\,,\,m_V)|^2\left[1+\frac{\alpha_s(m_V)}
{\pi}(11.2(5)-1.9N)\right]\,.
\end{equation}
Eqs. (30) and (32) lead to
\begin{eqnarray}
&&\frac{\Gamma (A\rightarrow gq\bar q+ggg)}{\Gamma (V\rightarrow ggg)}=
\nonumber\\[3pt]
&&\frac{\Gamma_{tot}(A)}{\Gamma_{tot}(V)}\frac{BR(A\rightarrow hadrons)}{
[BR(V\rightarrow hadrons)- BR(V\rightarrow virtual\,\gamma\rightarrow
hadrons)]}=\nonumber\\[3pt]
&& 99.4\frac{N}{3}\left(\frac{m_V}{m_A}\right)^2
\left|\frac{R^{\prime}_P(0\,,\,m_A)}{m_AR_S(0\,,\,m_V)}\right|^2\left[
\frac{\ln\left(m_AR(m_A)/2\right)-051}{1+(11.2(5)-1.9N)\alpha_s(m_V)/\pi }
\right]
\end{eqnarray}
and
\begin{eqnarray}
&&\frac{\Gamma_{tot}(\chi_{b1}(1P))}{\Gamma_{tot}(\Upsilon (1S))}=
2.65\left|\frac{R^\prime_P(0\,,\,m_{\chi_{b1}(1P)})R_S(0\,,\,m_{J/\psi})}
{R^\prime_P(0\,,\,m_{\chi_{c1}(1P)})R_S(0\,,\,m_{\Upsilon (1S)})}\right|^2\cdot
\nonumber\\[6pt]
&&\left[\frac{\ln\left(m_{\chi_{b1}(1P)}R(m_{\chi_{b1}(1P)})/2\right)-0.51}
{\ln\left(m_{\chi_{c1}(1P)}R(m_{\chi_{c1}(1P)})/2\right)-0.51}\right]\left[
\frac{1+5.5(5)\alpha_s(m_{J/\Psi})/\pi}{1+3.6(5)\alpha_s(m_{\Upsilon(1S)})/\pi}
\right]
\end{eqnarray}
that, at least, does not lower a chance to produce the $\chi_{b1}(1P)$ state
in the $e^+e^-$ collisions by the virtual $Z$ boson, too.
\section{conclusion}
So, the current facilities give some chance to observe the $\chi_{c1}(1P)$
state production in the $e^+e^-$ collisions and to study the production of
the $\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ states in the $e^+e^-$ collisions in
sufficient detail.
The $c-\tau$ and $b$ factories would give possibilities to study in the
$e^+e^-$ collisions the $\chi_{c1}(1P)$ state production in sufficient detail
and the production of the $\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ states in depth.
Probably, it is possible to observe the $\chi_{c1}(2P)$ state production at
the $c-\tau$ factories.
The fine effects considered above are essential not only to the
understanding of the quark model but can be used for identification of the
$\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ states because the angular momentum $J$
of the states named as $\chi_{b1}(1P)$ and $\chi_{b1}(2P)$ needs confirmation
\cite{pdg-1994}.
\acknowledgements
I would like to thank V.V. Gubin, A.A. Kozhevnikov and G.N. Shestakov for
discussions.
|
{
"timestamp": "1996-04-09T08:46:45",
"yymm": "9604",
"arxiv_id": "hep-ph/9604226",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/9604226"
}
|
"\\section{#1}}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\newcommand{\\vs}[(...TRUNCATED)
| {"timestamp":"1996-05-07T13:21:21","yymm":"9604","arxiv_id":"hep-th/9604165","language":"en","url":"(...TRUNCATED)
|
"\\section{Introduction}\n\nIn the previous workshop in Waikola, Hawaii, I presented a talk on\nsear(...TRUNCATED)
| {"timestamp":"1996-04-26T08:12:07","yymm":"9604","arxiv_id":"hep-ph/9604409","language":"en","url":"(...TRUNCATED)
|
"\\section{Introduction}\n\nDuring the last year and in preparation for the experiments to be perfor(...TRUNCATED)
| {"timestamp":"1996-06-25T16:27:25","yymm":"9604","arxiv_id":"hep-ph/9604326","language":"en","url":"(...TRUNCATED)
|
"\\section{Introduction}\n\nRecent cosmic ray observations, reported by the Fly's Eye (\\cite{Fly}) (...TRUNCATED)
| {"timestamp":"1996-04-01T21:54:42","yymm":"9604","arxiv_id":"astro-ph/9604005","language":"en","url"(...TRUNCATED)
|
"\\section*{Figure Captions\\markboth\n\t{FIGURECAPTIONS}{FIGURECAPTIONS}}\\list\n\t{Figure \\arabic(...TRUNCATED)
| {"timestamp":"1997-06-22T14:41:58","yymm":"9604","arxiv_id":"hep-th/9604003","language":"en","url":"(...TRUNCATED)
|
{"timestamp":"1998-07-28T06:56:37","yymm":"9604","arxiv_id":"math/9604235","language":"en","url":"ht(...TRUNCATED)
|
|
"\\section{Introduction}\n\\setcounter{equation}{0}\n\n\\noindent\nSqueezed states have been studied(...TRUNCATED)
| {"timestamp":"1996-04-20T12:05:44","yymm":"9604","arxiv_id":"quant-ph/9604017","language":"en","url"(...TRUNCATED)
|
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