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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 18
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 71201)
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 18
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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\section{Introduction}
The remarkable correspondence between the singular vector in Virasoro $(r,s)$--Kac module and the Jack polynomial indexed by the rectangular diagram with partition $(r^s)$ \cite{MY,SSAFR,AMOSa} has been extended to the superconformal case in \cite{DLM_jhep}. In that context, singular vectors are represented by sums of Jack superpolynomials (sJack). The main difference between the two cases is thus that the one-to-one correspondence of the former situation is lost in the latter one.
\footnote{In that regard, we should point out the amazing observation made in \cite{BBT} which is that a superconformal singular vector can be represented by a single (rank 2) Uglov polynomial \cite{Uglov}. These polynomials are the specialization of the Macdonald polynomials \cite{Mac} at $q=t=-1$. Notice that these do not involve any anticommuting variables. As it will be reviewed below, the polynomial representation of states is obtained via a free-field representation. In the superconformal case, the algebra generators are expressed in terms of a free fermion and a free boson. The Uglov-polynomial representation follows once the
free fermion is bosonized \cite{CFT}.
Note that in this construction, the differential-operator representation of the super-Virasoro generators,
as well as the one-to-one correspondence between states and symmetric superpolynomials, are both lost. Understanding the connection between the Uglov polynomials and the sJacks is a puzzling issue.}\\
However, the rectangle rule still holds in disguise: the contributing terms are indexed by superpartitions which are self-complementary (in a sense which differs slightly between the Neveu-Schwarz (NS) and the Ramond (R) sectors) and such that the superpartition glued to its $\pi$-rotated version (and slightly modified in the NS sector) fills a rectangle with $r$ columns and $s$ rows. This is a severe constraint on the allowed terms. The example presented in \cite{DLM_jhep} illustrating the strength of this restriction is the NS singular vector at level 33/2 which has 11 terms, while there are 1687 states at that level (and thus, the same number of superpolynomials in a state representation expanded in a generic basis).\\
For the NS sector, a closed-form expression in terms of sJacks has been displayed in \cite{DLM_jhep}. However, we failed to obtain a similar result in the R case. Indeed, the latter is much more difficult to cope with. To illustrate the additional level of complexity, we simply note that the highest-level R singular vector that we have generated is for $(r,s)=(5,6)$, hence at level 15 (thus a level lower than the previous NS example), and it contains 86 contributing terms (out of a total of 1472 possible states). This signals a radical increase in complexity. It can be traced back to the R version of the rectangle condition, which is weaker that in the NS case and allows thus for more terms. \\
But, somewhat surprisingly, a closed-form expression has actually been found. Its exposition is the aim of this work.\footnote{We recall that until the publication of \cite{DLM_jhep,BBT}, the only singular vectors with known closed-form expressions were those with either $r=1$ or $s=2$ (see \cite{BsA} and \cite{Watts} for the NS and R sectors, respectively).} It is presented in Section 3. The formula has been tested up to level $rs/2=15$, inclusively. The next section is a review of relevant results from \cite{DLM_jhep}. The required background on superpartitions and sJacks is collected in Appendix A.\\
\iffalse
Let us end this introduction with a few remarks. We recall that until the publication of \cite{DLM_jhep}, the only singular vectors with known closed-form expressions were those with either $r=1$ or $s=2$ (see \cite{BsA} and \cite{Watts} for the NS and R sectors, respectively). Very recently, another general formula has been conjectured in \cite{BBT}. This formula represents the $(r,s)$ singular vector in the NS sector as the Uglov polynomial indexed by the rectangular partition $(r^s)$. At first sigh, this formula is simpler than ours since only one polynomial is required to represent a singular vector. However, the Uglov polynomials, being purely bosonic quantities, allow the connection with the fermionic modes only via the (non-linear) bosonization of these modes. The obvious one-to-one correspondence between the state in the superconformal Verma module and the space of symmetric superpolynomials, which is a clear advantage of our approach, is thus lost in the formula of \cite{BBT}. Of course, both approaches must be related and complementary. We hope to clarify this point in a forthcoming article.
\fi
\section{Fundamental Correspondences}
\subsection{Representation of Ramond states as symmetric superpolynomials}
Let us first review the connection between symmetric superpolynomials and states in R highest-weight modules. Recall that in the R sector of the super-Virasoro algebra \cite{BKT,FQS,Dorr}\begin{align}\label{svir}
&[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}n(n^2-1)\delta_{n+m,0}\nonumber\\
&[L_n,G_k]=\left(\frac{n}{2}-k\right)G_{n+k}\nonumber\\
&\{G_k,G_l\}=2 L_{k+l}+\left(k^2-\frac{1}{4}\right)\frac{c}{3}\delta_{k+l,0},
\end{align}
$G$ indices are integers: $k,l \in\mathbb{Z}$. We denote by $|h{\rangle}$ the positive-chirality\footnote{In other words,
$|h{\rangle}\equiv |h{\rangle}^+$ and the negative chirality-sector is built from the highest-weight state $|h{\rangle}^-=G_0|h{\rangle}^+$. } highest-weight vector of conformal dimension $h$, i.e.,
\begin{equation} L_{n}|h{\rangle}=0=G_n|h{\rangle}\quad \forall n>0\qquad \text{and}\qquad L_{0}|h{\rangle}=h|h{\rangle}.
\end{equation} The highest-weight module $\mathscr{M}$ is generated by all states of the form
\begin{equation} G_{ -\La_1 }\cdots G_{ -\La_m }L_{-\La_{m+1}}\cdots L_{-\La_\ell} |h{\rangle}
\end{equation}
with the conditions
\begin{equation} \label{eqspart} \La_1>\ldots> \La_m\geq 0,\qquad \La_{m+1}\geq \ldots \geq \La_{\ell}> 0,
\qquad \text{with}\qquad m\geq 0,\qquad \ell \geq 0.\end{equation}
The last equation simply means that $\La=(\La_1,\ldots,\La_m;\La_{m+1},\ldots,\La_\ell)$ is a superpartition of fermionic degree $m$ (cf. Section \ref{A1}).
The state $|\chi{\rangle}$ is a singular vector if $G_n|\chi{\rangle}=0$ and $L_n|\chi{\rangle}=0$ for all $n>0$. All these constraints are consequences of the following two conditions:
\begin{equation} \label{condsingvecR} G_1|\chi{\rangle}=0\qquad\text{and}\qquad L_1|\chi{\rangle}=0.
\end{equation}
To explore singular vectors, it is sufficient to focus on Kac modules (e.g., see \cite{Dorr}). Recall that a R highest-weight module is a Kac module whenever the central charge $c$
and the conformal dimension $h$ are related via the parametrization:
\begin{equation} \label{hrsR}
c=\frac{15}{2}-3\left(t+\frac{1}{t}\right)\qquad\text{and}\qquad h_{r,s}=\frac{t}{8}(r^2-1)+\frac{1}{8t}(s^2-1)-\frac{1}{4}(rs-1)+\frac{1}{16},
\end{equation}
where $r$ and $s$ are positive integers such that $r-s$ is odd, while $t$ is a complex number.
In such a module, there is a singular vector at level $rs/2$.
The relation between singular vectors and symmetric polynomials goes through the free-field representation.
In the R sector, it is described in terms of the free-field modes $a_n$ and $b_n$, with $n\in\mathbb{Z}$, together with the vacuum charge operator $\pi_0$:
\begin{equation} \label{Ralgfock}
[a_n,a_m]=n\delta_{n+m,0} \, ,\qquad [a_0,\pi_0]=1\, ,\qquad \{b_n,b_m\}= \delta_{n+m,0}\,.
\end{equation}
We define a one-parameter family of highest-weight states as $|\eta{\rangle}\equiv e^{\eta \pi_0}|0{\rangle}$ satisfying
\begin{equation} a_0|\eta{\rangle}=\eta |\eta {\rangle},\qquad a_n|\eta{\rangle}=0 \quad\text{and}\quad b_n|\eta
{\rangle}=0,\qquad\forall\;n>0.
\end{equation}
This allows us to introduce the Fock space $\mathscr{F}$ with highest weight $|\eta{\rangle}$ over the superalgebra \eqref{Ralgfock}: it is generated by all states off the form
\begin{equation}b_{ -\La_1 }\cdots b_{ -\La_m }a_{-\La_{m+1}}\cdots a_{-\La_\ell} |{\eta}{\rangle}
\end{equation}
where the labeling of the states satisfies \eqref{eqspart}. The following expressions yield a representation of the R sector on $\mathscr{F}$:
\begin{align} \label{ffrR} L_n&=-\gamma(n+1)a_n+\frac{1}{2}\sum_{k\in\mathbb{Z}}:a_ka_{n-k}:+ \frac{1}{4}\sum_{k\in\mathbb{Z}}\big(n-2k+\frac{1}{2}\big):b_kb_{n-k}:\\
G_n&=-2\gamma\big(n+\frac{1}{2}\big)b_n+\sum_{k\in\mathbb{Z}}a_kb_{n-k} ,
\end{align}where $\gamma$ is related to the central charge via $c=\tfrac{3}{2}-12 \gamma^2$.
We are now in position to formulate the correspondence between the free-field modes and the differential operators acting on the space of symmetric superpolynomials (cf. Section \ref{A2}), which space will be denoted by $\mathscr{R}$ . It reads:
\begin{equation} \label{corR} a_n\longleftrightarrow \begin{cases} \frac{(-1)^{n-1}}{\sqrt{\alpha}}p_{-n} & n<0 \\ \eta & n=0\\
{(-1)^{n-1}} n\sqrt{\alpha}\partial_n &n>0\end{cases}\qquad b_n\longleftrightarrow \begin{cases} \frac{(-1)^n}{\sqrt{2}}\tilde p_{-n} & n<0 \\ \frac{1}{\sqrt{2}}(\tilde p_0 +\tilde \partial_0) & n=0\\ (-1)^{n}\sqrt{2}\tilde \partial_n &n>0\end{cases}\, .
\end{equation}
Two points are noteworthy: the absence of $\alpha$ factors in the representation of the $b$ modes and the presence of the zero mode $b_0$ which is represented by a combination of {the fermionic polynomial $\ti p_0$} and its derivative. Eq \eqref{corR} together with the identification $|\eta{\rangle}\leftrightarrow 1$ induce the following correspondence between $\mathscr{F}$ and $\mathscr{R}$:
\begin{equation} \label{corR2} b_{-\La_1}\cdots b_{-\La_m}a_{-\La_{m+1}}\cdots a_{\La_\ell}|\eta{\rangle}\longleftrightarrow \zeta_\La p_\La\,
\qquad \, ,\end{equation}
where
\begin{equation} \zeta_\La=\frac{(-1)^{|\La|-(\ell-m)}}{2^{m/2}\alpha^{(\ell-m)/2}}.\end{equation}
We stress that the level in the Fock space is equal to the bosonic degree of the superpolynomials (namely, the sum of all parts of $\La$).
The free-field representation \eqref{ffrR} and the correspondence \eqref{corR} yield a representation of the super-Virasoro generators in the R sector as differential operators acting on the space $\mathscr{R}$ of symmetric superpolynomials. The two relevant expressions for our present purpose are
\begin{align} \label{GLpol}
\mathcal{G}_1&= {\sqrt{2}}(3\gamma-\eta) \tilde \partial_1+\sqrt{\frac{\alpha}{2}}(\tilde p_0+\tilde \partial_0)\partial_1+\sqrt{\frac{\alpha}{2}} \sum_{n\geq 1} n \tilde p_n\partial_{n+1}+\sqrt{\frac{2}{\alpha}}\sum_{n\geq 1} p_n \tilde \partial_{n+1}\nonumber\\
\mathcal{L}_1&= {\sqrt{\alpha}}(\eta-2\gamma) \partial_1- {\frac{1}{2}}(\tilde p_0+\tilde \partial_0)\tilde \partial_1-\sum_{n\geq 1} n p_n\partial_{n+1}-{\frac{1}{2}}\sum_{n\geq 1} (2n+1) \tilde p_n \tilde \partial_{n+1}.
\end{align}
We end up with the following correspondence between states of the module $ \mathscr{M}$ with highest-weight state $|h{\rangle}$ and symmetric superpolynomials:
\begin{align} \label{corrhwR}
\sum_\La c_\La G_{-\La^a}L_{-\La^s}|h{\rangle}& \longleftrightarrow \sum_\La c_\La \mathcal{G}_{-\La^a}\mathcal{L}_{-\La^s}(1)
\end{align}
where
\begin{equation} h=\frac{1}{2}{\eta}(\eta-2\gamma)+\frac{1}{16}. \end{equation}
\subsection{Singular vector as superpolynomials}
In order to apply the correspondence \eqref{corrhwR} to the $(r,s)$-{type} Kac module, we must set \cite{DLM_jhep}
\begin{equation} \label{parametsing}
t=\alpha,\qquad\gamma=\frac{1}{2\sqrt{\alpha}}(\alpha-1) \qquad \text{and}\qquad \eta\,
{\equiv \eta_{r,s}}=\frac{1}{2\sqrt{\alpha}}\left((r+1)\alpha-(s+1)\right) \,.\end{equation}
In what follows, $P_\La=P^{(\alpha)}_\La$ denotes the sJack with parameter $\alpha$ and indexed by the superpartition $\La$, while $v_\Lambda=v_\La^{(\alpha)}$ stands for some coefficient depending rationally on $\alpha$.
With the above parametrization, the positive-chirality R singular vector $\ket{\chi_{r,s}}$ can be represented as a superpolynomial
\begin{equation}
\label{defv}F_{r,s}= \sum_{\substack{\Lambda\\ m=0\,\text{mod}\,2\\|\La|=rs/2}}v_\Lambda P^{}_\La,
\end{equation}
if and only if
\begin{equation} {\mathcal{G}_1(F_{r,s})=0\qquad \text{and}\qquad \mathcal{L}_1(F_{r,s})=0 \, . }
\label{2sv}\end{equation}
The objective is to first characterize more precisely those $\La$ for which $v_{\La}\ne 0$ and then to find the explicit expression of $v_{\La}$. The formula to be presented below applies to the case $r$ odd. Recall that in the R sector, $r+s$ is odd. Therefore, in other to cover all situations, we need to be able to recover $F_{s,r}$ from $F_{r,s}$. This point is discussed in the following section.
\section{General formula for the coefficients of the sJacks-form of the Ramond singular vectors}
In this section, we present a conjecture for the explicit form of the coefficient $v_\Lambda$ in the representation of
\begin{align}
\ket{\chi_{r,s}}\longleftrightarrow \sum_{\Lambda \in \mathcal{A}_{r,s}}v_\Lambda P^{}_\Lambda.
\end{align}
$\mathcal{A}_{r,s}$ is the set of all allowed superpartitions of type $(r,s)$ defined below.
For convenience, we will work with singular vectors for which $r$ is odd and $s$ even. The case $r$ even and $s$ odd is recovered by a duality transformation that exchanges $r\leftrightarrow s$, as detailed in Section \ref{sectionduality}.
\subsection{Allowed superpartitions}
As previously observed in \cite{DLM_jhep}, those $\La$ that contribute to the R singular vector at level $rs/2$ belong to the set of $(r,s)$-self-complementary superpartitions of the R sector, which we denote by $\AR$ (cf. Section B.4 in \cite{DLM_jhep}). A superpartition $\Lambda$ belongs to $\AR$ if and only if the complement of the diagram
$\Lambda^*$, in the rectangle\footnote{To be clear, the leftmost upper corner of the rectangle is adjusted with the external boundaries of the $(1,1)$ box of $\La^*$.} with $r$ columns and $s$ rows, corresponds to
$\Lambda^*$ rotated by 180 degrees. Note that
\begin{equation} \Lambda\in\AR \quad \Longrightarrow \quad |\Lambda|= \frac{rs}{2} .
\end{equation}
Consider for instance the case with $(r,s)=(3,4)$. Suppose moreover that the fermionic degree is even. Then, there are 40 superpartitions of appropriate degree. However, as illustrated below, only 14 superpartitions amongst them are $(3,4)$-self-complementary: \footnote{This example was considered in Section B.4 of \cite{DLM_jhep} but the formulation of the rule for allowed superpartitions misses a condition that was tacitly assumed in the example (B.25) (namely, the third condition in eq. \eqref{selec} below). We thus rework this example properly.}
\begin{equation}
\tableau[scY]{&&\\&&\\\tf&\tf&\tf\\\tf&\tf&\tf\\}
\longleftrightarrow\;
\footnotesize{ \begin{matrix}(3,3)\\(3,0;3)\end{matrix}}\qquad
\normalsize{\tableau[scY]{&&\\&&\tf\\&\tf&\tf\\\tf&\tf&\tf\\}
\longleftrightarrow}\;
\footnotesize{ \begin{matrix}(3,2,1)\\(1,0;3,2)\\(2,0;3,1)\\(2,1;3)\\(3,0;2,1)\\(3,1;2)\\(3,2;1)\\(3,2,1,0;)\end{matrix} } \qquad \normalsize{
\tableau[scY]{&&\tf\\&&\tf\\&\tf &\tf\\&\tf&\tf\\}
\longleftrightarrow }\;
\footnotesize{ \begin{matrix}(2,2,1,1)\\(2,1;2,1)\\(2,0;2,1,1)\\ (1,0;2,2,1)\end{matrix}}\qquad
\end{equation}
where the boxes marked with thick frames correspond to the boxes of the rotated copy of $\La^*$.
The set of allowed superpartitions of type $(r,s)$ for $r$ odd, denoted $\mathcal{A}_{r,s}$, is the set of all superpartition $\Lambda$ satisfying the following selection rules:
\begin{subequations}\label{selec}
\begin{align}
&m=0\, \mathrm{mod} \, 2, \\
&\Lambda \in \AR, \\
&\ell(\Lambda) \leq s.
\end{align}
\end{subequations
In other words, $\La$ is an allowed superpartition of type $(r,s)$ if and only if $\La$ contains an even number of circles, the complement of $\La^*$ in the rectangle of width $r$ and height $s$ is a copy of $\La^*$, and the only cell of $\La$ that can be out of the latter region is a circle in the first row.\footnote{For $r$ even, the only cell of $\La$ that can be out of the $r\times s$ rectangle is a circle in the first column.}
This, as we said, defines the set $\mathcal{A}_{r,s}$:
\begin{equation}
\text{$r$ odd}:\quad \mathcal{A}_{r,s}=\{\La\,|\,\Lambda \in \AR,\, \text{$m$ even},\,\ell(\Lambda) \leq s\}.\end{equation}
Returning to the example for which $(r,s)=(3,4)$ we see that the superpartitions $(2,0;2,1,1)$ and $(1,0;2,2,1)$ are not allowed since they contradict the third selection rule; there are thus 12 allowed superpartitions of type $(3,4)$.
\subsection{The recursive structure of the allowed diagrams}
Since the allowed superpartitions must be self-complementary -- disregarding
the circles -- and fit within the $r\times s$ rectangle, we can view any diagram at a given level as being build up from a lower level one, by the addition of either a column or a row. For instance, $(r,s)$-type diagrams
are allowed to be wider than the $(r-2,s)$ ones. But the self-complementary requirement constrains their horizontal extension: every $(r-2,s)$ diagram can be transformed into a $(r,s)$ diagram by simply adding a column of length $s$ on the left side of the diagrams. However, not all the allowed $(r,s)$-level diagrams are generated by a column adjunction. Another generating source comes from the vertical extension of the $(r,s-2)$-level ones. The ``rectangle-fitting" rule and self-complementarity imply that this vertical extension must be done in a very specific way, namely by adding a row of length $r$ atop the diagram. \\
Let the core of $ \mathcal{A}_{r,s}$, denoted $ \mathcal{A}^*_{r,s}$,
be the set of $\La\in \mathcal{A}_{r,s}$ such that $\La=\La^*$ (i.e., those diagrams without circles).
Summarizing what has been said so far, the whole set $ \mathcal{A}^*_{r,s}$ can be obtained by adding a column of length $s$ to every elements of $ \mathcal{A}^*_{r-2,s}$ and adding a row of length $r$ to those of $ \mathcal{A}^*_{r,s-2}$. The full set
$ \mathcal{A}_{r,s}$ is recovered by adding pairs of circles to elements of $ \mathcal{A}^*_{r,s}$ in all the allowed way (including no circle addition).\\
A simple illustration of this recursive process is presented in
the following example:
\begin{equation}\label{ex123}
\tableau[scY]{\\ \\}\xrw{(1,4)\rw(3,4)} \tableau[scY]{\tf&\\\tf&\\\tf\\\tf}\qquad
\tableau[scY]{&&\\}\xrw{(3,2)\rw(3,4)}\tableau[scY]{\tf&\tf&\tf\\ &&}\qquad
\tableau[scY]{&\\ \\}\xrw{(3,2)\rw(3,4)} \tableau[scY]{\tf&\tf&\tf\\ &\\ \\}\qquad
\end{equation}
In this example, we obtain $ \mathcal{A}^*_{3,4}$ by adding one column of length $s=4$ to the unique element of $ \mathcal{A}^*_{1,4}$ displayed at the left in \eqref{ex123}, and adding a row of width $r=3$ atop the two elements of $ \mathcal{A}^*_{3,2}$. The result is $ \mathcal{A}^*_{3,4}=\lbrace (2^2,1^2),(3^2),(3,2,1) \rbrace$. In order to generate the full set $\mathcal{A}_{3,4}$, we decorate these core-diagrams with pairs of circles in all allowed ways, obtain thus
\begin{align}
\mathcal{A}_{3,4} :
\begin{array}{l}
\left\lbrace (3,3),(2,2,1,1),(3,2,1),(3,0;3),(2,1;2,1),(1,0;3,2),(2,0;3,1),\right. \\
\left. (3,0;2,1),(2,1;3),(3,1;2),(3,2;1),(3,2,1,0;) \right\rbrace.
\end{array}
\end{align}
\iffalse
\begin{figure}[h]
\psscalebox{.7}{
\begin{pspicture}(-2,-2)(5,3)
\def\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\psboxfill{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\rput(3,-2){\pspolygon[fillstyle=boxfill](0,0)(1,0)(1,2)(2,2)(2,4)(0,4)}
\psframe[linewidth=1.5pt](3,-2)(4,2)
\rput(0,-1){\pspolygon[fillstyle=boxfill](0,0)(1,0)(1,2)(0,2)}
\rput[b](.5,1){$r=1,s=4$}
\rput[b](4,2){$r=3,s=4$}
\psline{->}(1.5,0)(2.5,0)
\end{pspicture}
}
\caption{Going from $(1,4)$ to $(3,4)$}
\label{fig.r1s4tor3s4}\end{figure}
\begin{figure}[h]
\psscalebox{.7}{
\begin{pspicture}(-2,-.5)(5,2)
\def\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\psboxfill{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\rput(-2,0){\pspolygon[fillstyle=boxfill](0,0)(3,0)(3,1)(0,1)}
\rput(3,-.5){\pspolygon[fillstyle=boxfill](0,0)(3,0)(3,2)(0,2)}
\psframe[linewidth=1.5pt](3,.5)(6,1.5)
\rput[b](-.5,1){$r=3,s=2$}
\rput[b](4.5,1.5){$r=3,s=4$}
\psline{->}(1.5,.5)(2.5,.5)
\end{pspicture}
}
\caption{Going from $(3,2)$ to $(3,4)$}
\label{fig.r3s2tor3s4a}\end{figure}
\begin{figure}[h]
\psscalebox{.7}{
\begin{pspicture}(-2,-.5)(5,3)
\def\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\psboxfill{\pspicture(1,1)\psframe[dimen=middle](1,1)\endpspicture}
\psframe[linewidth=1.5pt](3,1.5)(6,2.5)
\rput(-1.5,0){\pspolygon[fillstyle=boxfill](0,0)(1,0)(1,1)(2,1)(2,2)(0,2)}
\rput(3,-.5){\pspolygon[fillstyle=boxfill](0,0)(1,0)(1,1)(2,1)(2,2)(3,2)(3,3)(0,3)}
\rput[b](-.5,2){$r=3,s=2$}
\rput[b](4.5,2.5){$r=3,s=4$}
\psline{->}(1.5,1)(2.5,1)
\end{pspicture}
}
\caption{Going from $(3,2)$ to $(3,4)$} \label{fig.r3s2tor3s4b}
\end{figure}
\fi
\begin{figure}[h]
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\caption{Illustration of the column-removal operation for self-complementary diagrams. Here, two removals of a column of length $6$ are displayed and after each operation, there results a self-complementary diagram.}\label{fig.r5s6tor1s6}
\end{figure}
\begin{figure}[h]
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\caption{Illustration similar to Figure \ref{fig.r5s6tor1s6} but here for row removal.}\label{fig.r5s6tor5s2}
\end{figure}
{
The completeness of this recursive construction of the $\mathcal{A}^*_{r,s}$ set is based on the simple observation that a self-complementary diagram necessarily has a column of length $s$ or a row of length $r$ (but not both). This, in turn, implies that any self-complementary diagram can be deconstructed uniquely down to a simple self-complementary diagram --a single row diagram corresponding to $(r,s)=(2k+1,2)$ for some $k\geq 0$ --, by successively deleting a column (of length $s$) or a row (of length $r$) -- with the understanding that one of the values of $r$ and $s$ is changing at every step.
This is illustrated in Figures
\ref{fig.r5s6tor1s6} and \ref{fig.r5s6tor5s2}. The first of these figures shows the transformation of the core partition $(3^3,2^3)$ for $(r,s)=(5,6)$ down to the partition $(1^3)$ for $(r,s)=(1,6)$ by removing twice a column of length 6. Observe that at each step, the resulting core diagram is self-complementary. This is easily seen to be a consequence of the deconstruction mechanism. Indeed, the figure also displays the defining $r\times s$ rectangle in dashed lines. Now, in the column reduction process, two columns are removed from this rectangle, one at each extremity (at the exterior of the gray lines). But that shows clearly that the partition and its complement in the $r\times s$ rectangle are reduced in exactly the same way, ensuring the preservation of self-complementarity. That two columns are removed from the defining rectangle is the reason for which $r$ decreases by 2 at each step. Notice that the last resulting diagram can be further reduced by removing twice a row of length 1, ending up with the partition $(1)$ for $(r,s)=(1,2)$.
Figure \ref{fig.r5s6tor5s2} illustrates the similar process of removing rows of length $r$. Notice again that the last diagram can be further reduced to a single box by two column-removing steps.
It is clear that the choice of either a column or a row removal at a given step is determined by the partition under consideration and it is unambiguous. For instance, with a staircase diagram, these operations alternate.} \\
This recursive pattern allows us to identify every box of a diagram as belonging to a row or a column which has been inserted at a certain $(r,s)$ level. As we just argued, the increment in $r$ results from the addition of columns, while that of $s$ is due to the addition of rows. Therefore, within an allowed $(r,s)$ diagram, we write $({\tilde r}_j,{\tilde s}_i)$ in the box with coordinate $(i,j)$, where
\begin{align}
\tilde{r}_{j} &\equiv r-2(j-1)\, , \\
\tilde{s}_{i} &\equiv s-2(i-1) \, .
\end{align}
From these data, we now introduce two sets:
\begin{align}
\mathcal{S}_{\Lambda,s}&=\{(i,j)\in\La\,|\, l_{\Lambda^\ast}(i,j)+1=\tilde{s}_{i}\}\, ,\nonumber\\
\mathcal{R}_{\Lambda,r}&=\{(i,j)\in\La\,|\,a_{\Lambda^\ast}(i,j)+1=\tilde{r}_{j}\}\, .\end{align}
It will also be convenient to denote by $\mathcal{O}_\Lambda$ the set of indices $(i,j)$ of every circle in the diagram $\Lambda$, or equivalently, the boxes of $\La^{\circledast}$ that are not in $\La^*$:
\begin{align}
\mathcal{O}_{\Lambda}&=\{(i,j)\in\La^{\circledast}/\La^*\}.\end{align}
\begin{figure}[!h]
\centering
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\caption[Illustration of sets $\mathcal{S}_{\Lambda,s}$, $\mathcal{R}_{\Lambda,r}$ and $\mathcal{O}_{\Lambda}$]{Illustration of sets $\mathcal{S}_{\Lambda,s}$, $\mathcal{R}_{\Lambda,r}$ and $\mathcal{O}_{\Lambda}$}
\label{fig.IllusEnsembles}
\end{figure}
Figure \ref{fig.IllusEnsembles} illustrates these definitions. It displays a diagram in which we have identified boxes with their $(\ti r_j, \ti s_i)$ coordinates. For those boxes that belong to the set $\mathcal{R}_{\Lambda,r}$, the $\ti r$ coordinate is bold, while for those which are elements of the set $\mathcal{S}_{\Lambda,s}$, the $\ti s$ coordinate is bold. We then see that $\mathcal{R}_{(5,4,2,0;3,1),5}=\lbrace (1,1),(2,2),(3,3) \rbrace$, $\mathcal{S}_{(5,4,2,0;3,1),6}=\lbrace (2,1), (3,2) \rbrace$. Finally, the set $\mathcal{O}_{(5,4,2,0;3,1)}$ is clearly $\lbrace (1,6), (2,5), (4,3), (6,1) \rbrace$.
\subsection{The explicit formula for the singular-vector expansion coefficients}
We are now in position to present the explicit conjectural expression for the coefficients $v_\La$ in \eqref{defv}:
\begin{align}
v_{\Lambda}(r,s) &= \epsilon(\Lambda,r,s)B(\Lambda,r,s)C(\Lambda,r,s),
\end{align}
where $B(\Lambda,r,s)$ is a function that captures the part of the coefficient that depends solely on box structure of the diagram, while $C(\Lambda,r,s)$ depends upon the diagram's circle pattern.
Finally, the factor $\epsilon(\Lambda,r,s)$, which is explained latter in Section \ref{SectSign}, is a sign that also depends upon the circles arrangement.\\
In the following expressions, in order to shorten the notation, we denote a box by $t$ when the specification of its coordinates are not necessary. For $t=(i,j)$, ${\tilde r}_t={\tilde r}_j$ and ${\tilde s}_t={\tilde s}_i$.
The coefficient $B$ reads
\begin{align}
B(\Lambda,r,s)&= \prod_{t \in \mathcal{S}_{\Lambda,s}} k(t)f(\tilde{r}_t,\tilde{s}_t)\, , \end{align}
where
\begin{align}
&k(t) = \prod_{n=0}^{l_{\Lambda^\ast}(t)}
{h^{\downarrow\uparrow}_\Lambda(i+n,j)}\, , \\
&{h^{\downarrow\uparrow}_\Lambda(t)}=\frac{h^\downarrow_\Lambda(t)}{h^\uparrow_\Lambda(t)} \, , \\
\label{hdn}
&h^\downarrow_\La(t)=l_{\La^*}(t)+1+\a\,a_{\La^{\circledast}}(t) \, , \\
\label{hup}
&h^\uparrow_\La(t)=l_{\La^{\circledast}}(t)+\a(a_{\La^*}(t)+1)\, ,
\end{align}
and
\begin{align}
f(r,s)&=\frac{(r-1)\alpha}{(r-1)\alpha + s}\prod_{\substack{i=s-1 \\ i \text{ odd}}}^1 \frac{r\alpha + i}{(r-2)\alpha + i}\, .
\end{align}
The $C(\La,r,s)$ coefficient is
\begin{align}
C(\Lambda,r,s)& = \prod_{{t=}(i,j) \in \mathcal{R}_{\Lambda,r}} \left\lbrace
\prod_{\substack{(i^\prime,j^\prime) \in \mathcal{O}_\Lambda\\ 0 < j-j^\prime+ \tilde{r}_{j} \leq \tilde{r}_{j}}} \frac{||(i,j+\tilde{r}_{j}),(i^\prime-1,j^\prime)||^{(1-{\delta_{\ti l(t)+1,\tilde{s}_{i}})}}}
{||(i,j+\tilde{r}_{j}),(i^\prime,j^\prime)||^{(1-{\delta_{\ti a(t),\tilde{r}_{j}})}}} \right. \nonumber \\
&\quad\times \left.
\prod_{\substack{(i^\prime,j^\prime) \in \mathcal{O}_\Lambda \\ 0 \leq j^\prime - j < \tilde{r}_{j}}} \frac{||(i-1+\tilde{s}_{i},j),(i^\prime-1,j^\prime)||^{(1-{\delta_{\ti a(t),\tilde{r}_{j}})}}}
{||(i-1+\tilde{s}_{i},j),(i^\prime,j^\prime)||^{(1-{\delta_{\ti l(t)+1,\tilde{s}_{i}})}}} \right\rbrace,
\end{align}
where $\ti a=a_{\La^{\circledast}}$ and $\ti l=l_{\La^{\circledast}}$.
Here, $ \|t_1,t_2\|$ refers to the $\alpha$-distance between
$t_1$ and $t_2$: for $t_1=(i_1,j_1)$ and $t_2=(i_2,j_2)$,
this distance is defined as
\begin{equation} \|t_1,t_2\|=| i_1-i_2 + \alpha(j_2-j_1)|.
\end{equation}
\subsection{Illustration of the combinatorics underlying the general formula}
Since the explicit formula has a complicated looking form, we will illustrate it by considering separately its three components, the two functions $B(\Lambda,r,s)$ and $C(\Lambda,r,s)$, and finally, the sign $\epsilon(\La,r,s)$.
\subsubsection{The $B(\Lambda,r,s)$ function}
The $B(\La,r,s)$ function is built up from the elements of the set $\mathcal {S}(\La,s)$.
For each box $t=(i,j)$ such that $\tilde{s}_{i}-1$ is equal to the number of boxes below it, there is a contributing factor $f(\tilde{r}_{j},\tilde{s}_{i})$ multiplied by the function $k(\Lambda,i,j)$. The latter is given by the product of ${h^\downarrow}(t')/{h^\uparrow}(t')$ for $t'=t$ and every box below $t$:
\begin{equation} t'=\{(i',j)\, |\, i\leq i'\leq i+l_{\La^*}(t)\}\end{equation}
If the set $\mathcal {S}(\La,s)$ is empty, it is understood that $B(\Lambda,r,s)=1$.\\
\begin{figure}[!h]
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\caption[Illustration of the $B(\Lambda,r,s)$ coefficient]{}
\label{IllustrationB}
\end{figure}
Consider the superpartition $\La=(7,1;6,6,5,5,5,2,2,2,1)$ whose diagram is displayed in Figure \ref{IllustrationB}.
This is easily checked to be an element of $\mathcal{A}_{7,12}$.
We first label the boxes by their indices $(\tilde{r}_{j},\tilde{s}_{i})$.
A partial label-filling is illustrated in the figure. Those boxes that are elements of $\mathcal {S}_{\La,12}$ have their labels underlined. So here
$\mathcal {S}_{\La,12}=\lbrace (2,1),(4,2)\rbrace$. There are thus two contributing factors $f(\tilde{r}_{j},\tilde{s}_{i})$, namely $f(7,10)$ and $f(5,6)$. This
is multiplied by the ratios ${h^\downarrow(i,j)}/{h^\uparrow (i,j)}$ of those underlined-label boxes and each one underneath, which corresponds to the hatched boxes in the figure. Multiplying all these factors yields $B(\La,7,12)$:
\begin{align}
&B((7,1;6,6,5,5,5,2,2,2,1),7,12) \nonumber \\
&=f(7,10){h^{\downarrow\uparrow}(2,1) h^{\downarrow\uparrow}(3,1)\dots h^{\downarrow\uparrow}(11,1)}
\times f(5,6) {h^{\downarrow\uparrow}(4,2) h^{\downarrow\uparrow}(5,2) \dots h^{\downarrow\uparrow}(9,2)}
\nonumber \\
& = \frac { \left( 7\,\alpha+9 \right) \left( \alpha+2 \right) ^{2} \left( 4\,\alpha +7\right) \left( \alpha+5 \right) \left( \alpha+4 \right) }{ \left( 3\,\alpha+1 \right) \left( 3\,\alpha+5 \right) \left( 5\,\alpha +7\right) ^{2}} \times \frac { \left( 7\,\alpha+5 \right) \left( 7\,\alpha+3 \right) \left( 7\,\alpha+1 \right) \alpha}{ \left( 2\,\alpha+3 \right) ^{3} \left( 5\,\alpha+6 \right) \left( \alpha+1 \right) ^{5} \left( 4\,\alpha+5 \right) }. \label{eq.IllustrationBcoeff}
\end{align}
\subsubsection{The $C(\Lambda,r,s)$ function}
To calculate the part of the coefficient associated with $C(\Lambda,r,s)$, we first identify the boxes for which the number $\tilde{r}_{j}$ correspond to the arm of the box plus 1.
Corresponding to each of those boxes, we add two triangles (one up, one down), each equipped with two labels: $\vartriangle_{\tilde{r}_{j},\tilde{s}_{i}}$ in position $(i,j+\tilde{r}_{j})$ and $\triangledown_{\tilde{r}_{j},\tilde{s}_{i}}$ in position $(i-1+\tilde{s}_{i},j)$. The function $C(\La,r,s)$ is built from the various (shifted) $\a$-distance between these triangles and the circles in the diagram of $\La$, as we now detail. \\
{
Note that by construction, there is a triangle (of either type) at the end of each row (but not necessarily at the end of each column) and every circle becomes filled with a triangle.
Let us first argue that there is a triangle at the end of each row of a core diagram. Since any diagram of $\mathcal{A}_{r,s}$ is obtained by adding circles to elements of $\mathcal{A}^*_{r,s}$, the second statement (that circles are filled by triangles) will follow. The argument relies on the recursive
structure of self-complementary diagrams. The starting point is a row diagram with $2k+1$ boxes, corresponding to $(r,s)=(2k+1,2)$. We add a triangle at position $(1,2k+2)$ and a reversed one at position $(2,1)$. Keeping adding rows amounts to add a triangle at the end of each new row and pile reversed triangles at positions $(4,1),\, (6,1),\cdots (s',1)$. See for instance the first three steps in \eqref{extra} below, where the beginning of the recursive reconstruction of $\La^*=(7,6,6,5,5,5,2,2,2,1,1)\in\mathcal{A}^*_{7,12}$ is displayed. Adding a column of length $s'$ shift this pile of down triangles to the end of length-one rows, as in the last step in the following example:
\begin{equation}\label{extra}
{\tableau[scY]{&&&\bl\vartriangle{} \\ \bl\triangledown}}\quad\rw\quad
{\tableau[scY]{&&&\bl\vartriangle{} \\&&&\bl\vartriangle{} \\ \bl\triangledown\\ \bl\triangledown}}
\quad\rw\quad
{\tableau[scY]{&&&\bl\vartriangle{} \\&&&\bl\vartriangle{} \\&&&\bl\vartriangle{} \\ \bl\triangledown\\ \bl\triangledown\\ \bl\triangledown}}
\quad\rw\quad
{\tableau[scY]{&&&&\bl\vartriangle{} \\&&&&\bl\vartriangle{} \\&&&&\bl\vartriangle{} \\& \bl\triangledown\\& \bl\triangledown\\ &\bl\triangledown}}
\quad\rw\quad\cdots
\end{equation}
Adding more columns of length $s'$ just push these triangles further right. Since the addition of rows and columns are the two basic operations for building up self-complementary core diagrams, this shows that each row of such a core diagram ends with a triangle, up or down.}\\
The numerator of each product is the `upper-shifted' $\a$-distance between every triangle and
every circle. The upper-shifting means that the $\a$-distance is calculated not directly with a circle but rather with the box just above it (and the case where there is no box above the circle is treated below).
The first numerator captures the contributions of the distances between $\vartriangle_{\tilde{r}_{j},\tilde{s}_{i}}$ and the circles.
At first, the exponent $(1-\delta_{\ti l(t)+1,\tilde{s}_{i}})$ indicates that
if $\triangledown_{\tilde{r}_{j},\tilde {s}_{i}}$ is circled, this whole numerator factor reduces to 1.
Otherwise,
the constraint $0<j-j'+\ti r_j\leq \ti r_j$ means that only those circles that are southwest of $\vartriangle_{\tilde{r}_{j},\tilde{s}_{i}}$ and not horizontally further apart than $\ti r_j$ from $\vartriangle_{\tilde{r}_{j},\tilde{s}_{i}}$ do contribute.
The second numerator factor keeps track of the shifted $\a$-distances between the reversed triangles $\triangledown_{\tilde {r}_{j},\tilde {s}_{i}}$ and the circles. Again, if $\vartriangle_{\tilde {r}_{j},\tilde {s}_{i}}$ is circled, this whole factor reduces to one.
(This condition takes cares of the situation where there is no box above a circle, that is, the first row ends with a circle.) The condition $0\leq j'-j<\ti r_j$ states that only those circles above or northeast of $\triangledown_{\tilde {r}_{j},\tilde {s}_{i}}$ contribute, as long as their horizontal separation is strictly smaller than $\ti r_j$.
The denominator factors are similar except that the upper-shifted $\a$-distance is replaced by the unshifted $\a$-distance and if $\vartriangle$ or $\triangledown$ is circled, the corresponding factor is 1, i.e., the $\a$-distances from $\vartriangle$ or $\triangledown$ do not contribute).\\
We illustrate these rules by considering again the case $\La=(7,1;6,6,5,5,5,2,2,2,1)$ whose diagram, augmented with the $\vartriangle,\triangledown$-factors inserted, is presented in Figure \ref{IllusC}.
The solid lines link the pairs ($\vartriangle$, box-above-a-circle) or ($\triangledown$, box-above-a-circle) that contribute to the numerator and the dashed lines similarly related pairs ($\vartriangle$, circle) or ($\triangledown$, circle) that contribute to the denominator.
Consider first $\vartriangle_{7,12}$: there is one solid line linking it to the box atop the circle $(10,2)$; its contribution is $6\a+8$. There is no dashed line originating from $\vartriangle_{7,12}$ because it is circled. From $\vartriangle_{5,10}$ originates one line of each type for a factor $(5\a+7)/(5\a+8)$.
The circle in position $(10,2)$ is separated from $\vartriangle_{3,n}$ for $n=2,4,6$ by an horizontal distance 4; since this is larger than $\ti r_j=3$, these terms do not contribute. A similar conclusion holds for the three terms $\triangledown_{3,n}$. A solid line starts from $\triangledown_{5,8}$ and ends on the box just above, for a contribution of 1.
A line of each type originates
from $\triangledown_{5,10}$ for a factor $2/1$.
Finally, for $\triangledown_{7,12}$ there is no numerator term (no solid line) $\vartriangle_{7,12}$ is circled. The dashed line that starts from there only goes to the $(10,2)$ circle since the $(1,8)$ circle is at an horizontal distance of $\tilde {r}_{j}=7$. This yields $1/(\a+2)$.
Collecting these factors yields:
\begin{equation} C((7,1;6,6,5,5,5,2,2,2,1),7,12)=\frac{2(6\alpha+8)(5\alpha+7)}{(5\alpha+8)(5\alpha+7)(\alpha+2)}.
\end{equation}
\begin{figure}[!h]
\centering
\psscalebox{.7}{
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\rput(1,11){\underline{\textbf{5},10}}
\rput(1,10){\underline{\textbf{5},8}}
\rput(1,9){5,6}
\rput(2,9){\underline{\textbf{3},6}}
\rput(2,8){\underline{\textbf{3},4}}
\rput(2,7){\underline{\textbf{3},2}}
\rput(7,12){$\bigtriangleup_{7,12}$}\rput(0,1){$\bigtriangledown_{7,12}$} \pscurve[linestyle=dashed]{o->}(.5,1)(1.5,1.5)(1.5,3) \pscurve{o->}(6.75,12)(1.5,8)(1,4)
\rput(6,11){$\bigtriangleup_{5,10}$}\rput(1,2){$\bigtriangledown_{5,10}$} \pscurve[linestyle=dashed]{o->}(6,10.75)(7,8)(1.5,3) \pscurve[linestyle=dashed]{o->}(1,2.25)(1.5,2.75)(1.5,3) \pscurve{o->}(1,2.25)(.25,3)(1,4) \pscurve{o->}(6,10.75)(1.5,7)(1,4)
\rput(6,10){$\bigtriangleup_{5,8}$}\rput(1,3){$\bigtriangledown_{5,8}$} \pscurve[linestyle=dashed]{o->}(6,9.75)(6,7)(1.5,3) \pscurve[linestyle=solid]{o->}(1,3.25)(1.25,3.5)(1,4)
\rput(5,9){$\bigtriangleup_{3,6}$}\rput(2,4){$\bigtriangledown_{3,6}$}
\rput(5,8){$\bigtriangleup_{3,4}$}\rput(2,5){$\bigtriangledown_{3,4}$}
\rput(5,7){$\bigtriangleup_{3,2}$}\rput(2,6){$\bigtriangledown_{3,2}$}
\end{pspicture}
}
\caption[Illustration of the $C(\Lambda,r,s)$ coefficient]{Dressing of the diagram of Figure \ref{IllustrationB} with the $\vartriangle,\triangledown$-terms, together with the indication of the factors contributing to the numerator (solid lines) and the denominator (dashed lines) of the $C(\La,r,s)$ coefficient.}\label{IllusC}
\end{figure}
\begin{figure}[!h]
\centering
\psscalebox{.7}{
\begin{pspicture}(0,0)(5,-6)
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\rput(6,-1){$\bigtriangleup_{5,6}$}\rput(1,-6){$\bigtriangledown_{5,6}$}
\rput(5,-2){$\bigtriangleup_{3,4}$}\rput(2,-5){$\bigtriangledown_{3,4}$}
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\pscurve[linestyle=dashed]{o->}(4,-3.3)(3.7,-3.8)(3.5,-4)
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\pscurve[linestyle=solid]{o->}(4.65,-2)(3.75,-2.2)(3,-2.75)
\end{pspicture}
}
\caption[Illustration of the $C(\Lambda,r,s)$ coefficient for a 4 circle diagram]{Another illustration of the triangles insertion and the arrow-contribution indicators, this time for a diagram with four circles}
\label{fig.IllusC4cercles}
\end{figure}
It is clear from the above rules that for a given core diagram ($\La^*$), the more circles there are in $\La$, the least number of factors contribute to $C(\La,r,s)$. We illustrate this in Figure \ref{fig.IllusC4cercles} which displays a diagram with six rows but four of them ending with a circle. Let us detail the contributing terms. Both the $(5,6)$-indexed triangles are circled, so they do not contribute. $\vartriangle_{3,4}$ being circled cancels the solid lines of $\triangledown_{3,4}$ and its own dashed lines. Only the solid line linking it to the box in position $(3,3)$ contributes and this for a factor $(2 \alpha +1)$. Finally we only need to consider the two dashed lines ending on $\triangledown_{1,2}$; these contribute to $1/{(\alpha+1)^2}$. The corresponding $C(\La,r,s)$ factor for this diagram is thus:
\begin{align}
C((5,4,2,0;3,1),5,6)=\frac{(2 \alpha +1)}{(\alpha+1)^2}.
\end{align}
\subsubsection{The $\epsilon$ function} \label{SectSign}
As already indicated, each circle is filled by a triangle, either up or down. It turns out that the sign associated to a given diagram is fixed by the triangle-data in the circles. Let us associate a degree to a triangle, up or down, given by the product of its labels divided by 2 and indicate the degree as $|\!\vartriangle\!|$ and $|\triangledown|$. We now introduce two quantities:
\begin{align}
n_\vartriangle&=\text{number of circled $\vartriangle$},\nonumber\\
p_\diamond&=\text{number of circled pairs $(\vartriangle,\triangledown)$
such that $ | {\vartriangle}_{\ti r,\ti s} | < | {\triangledown}_{\ti r^\prime, \ti s^\prime} |$}.
\end{align}
Then, the sign of a diagram with $m$ circles
reads
\begin{equation} \epsilon(\La,r,s)=(-1)^{\binom{m}{2}+\binom{n_\vartriangle}{2}+\, p_\diamond},\end{equation}
where it is understood that $\binom12=0$. For the example of Figure \ref{IllusC}, $m=2,\,n_\vartriangle=1 $, and because $|\triangledown_{5,8}|<|\!\!\vartriangle_{7,12}\!\!|$, $p_\diamond=0$, so that $\epsilon=1$. For the diagram in Figure \ref{fig.IllusC4cercles}, we have $m=4$, $n_\vartriangle=2 $ and $p_\diamond=1$, which yields $\epsilon=+1$.
\subsubsection{The formula for $\ket{\chi_{1,s}}$}
As a simple application of this general formula, we can readily obtain the superpolynomial representation of $\ket{\chi_{1,s}}$.
Given that $\mathcal{A}_{1,s}=\lbrace (1^{s/2}), (1,0;1^{s/2-1})\rbrace$, the set $\mathcal{S}_{\Lambda,s}=\emptyset$, so that $B(\La,1,s)=1$. Similarly, it is a simple execise to prove that the coefficient $C(\La,1,s)=1$. Hence, only the sign term does contribute for $(1,0;1^{s/2-1})$: since $n_\vartriangle=p_\diamond=0$, we get $\epsilon=-1$. The result is
\begin{align}
\ket{\chi_{1,s}} \longleftrightarrow P^\alpha_{(1^{s/2})} -P^\alpha_{(1,0;1^{s/2-1})}
\end{align}
which does correspond to the formula given in \cite[Eq. (B.22)]{DLM_jhep}.
\iffalse
We can assign a level to circles which corresponds to the level of the diagram where the circle was first allowed. For exemple, an antisymetric part of 0 can't be placed lower than $s$, so if the circle is at $s$, it has been added at the level $r*s/2$.\\
The method to calculate the sign is the following: For a 2 circle diagram, you write the $\vartriangle_{\ti r,\ti s}$ and $\triangledown_{\ti r,\ti s}$ as usual. Then, you calculate the level $\ti r \ti s / 2$ for every cicled triangle. If the smallest triangle level is a $\vartriangle$ then the coefficients picks up a minus sign, it does not otherwise. If the smallest level corresponds to a triangle of each type, the diagram does not pick up a sign.\\
If there are more than 2 circles, then the calculation must be done for every pair of circles on the diagram and the results must be multiplied together. \\
If we define the levels at which circles has been created by $|\vartriangle|$ and $|\triangledown|$ then the function is the following :
\begin{align}
\epsilon = \prod_{\substack{\text{every pair}\\ \text{of circles }t,t^\prime}} \eta(t,t^\prime)
\end{align}
where
\begin{align}
\eta(t,t^\prime)= \left\lbrace \begin{array}{l}
-1 \text{ if } | \vartriangle_{\ti r,\ti s} | < | \triangledown_{\ti r^\prime, \ti s^\prime} | \\
-1 \text{ if there are 2 } | \vartriangle | \\
\phantom{-}1 \text{ otherwise}
\end{array} \right.
\end{align}
\cb{
\begin{align}
\eta(t,t^\prime)= \left\lbrace \begin{array}{l}
-1 \text{ if } | \triangledown_{\ti r,\ti s} | \leq | \vartriangle_{\ti r^\prime, \ti s^\prime} | \\
-1 \text{ if there are 2 } | \triangledown | \\
\phantom{-}1 \text{ otherwise}
\end{array} \right.
\end{align}
}
Pour $(r,s)=(3,4)$
$${\tableau[scY]{&&&\bl\tcercle{}\\&&\\\bl\tcercle{} \\ }} \qquad | \triangledown_{3,2} |=3, \quad|\triangle_{3,4} |=6\qquad \epsilon=-1$$
$${\tableau[scY]{&&&\bl\tcercle{}\\&&\bl\tcercle{}\\ \\ }} \qquad | \triangle_{1,2} |=1, \quad|\triangle_{3,4} |=6\qquad \epsilon=-1\quad(\text{deux triangles up})$$
$${\tableau[scY]{&&&\bl\tcercle{}\\&\\ \\\bl\tcercle{} \\ }} \qquad | \triangledown_{3,4} |=6, \quad|\triangle_{3,4} |=6\qquad \epsilon=-1$$
\fi
\section{Duality transformation}\label{sectionduality}
Let $F_{r,s}(\alpha)$ be the superpolynomial representing the singular vector $|\chi_{r,s}\rangle$ in the R sector. Here we show that there is a simple
duality formula involving $F_{r,s}(\alpha)$ which allows to go from the superpolynomial representation of $|\chi_{r,s}\rangle$ to that of $|\chi_{s,r}\rangle$. This duality is slightly more complicated than in the NS sector \cite{DLM_jhep} and it will thus be worked out in detail.
The guiding observation in view of
establishing the formula implementing the interchange of the Kac labels is the following relation
\begin{equation}
\eta_{r,s}(\alpha)=-\eta_{s,r}(1/\alpha)
\end{equation}
that follows from the parametrization given in \eqref{parametsing}.
This indicates that the label swapping is accompanied by the interchange of $\a\lrw1/\a$.
The operation that implements this interchange at the level of the sJacks is
the following homomorphism naturally defined on power-sums (and whose action on sJacks is given below):
\begin{equation}\bar\omega_\alpha(p_n)=(-1)^{n-1}\alpha p_n,\qquad \bar\omega_\alpha(\tilde p_n)=(-1)^{n}\tilde p_n.
\end{equation}
The last equation, {combined with the preservation of the commutation relations $[\d_n,p_m]=n\delta_{n,m}$ and $\{ \ti\d_n,\ti p_m\}=\delta_{n,m}$,}
imply that
\begin{equation}
\bar \omega_\alpha(\partial_n)=(-1)^{n-1}\alpha^{-1}\partial_n,\qquad \bar \omega_\alpha(\tilde \partial_n)=(-1)^{n}\tilde \partial_n .
\end{equation}
Let us denote by
$\mathcal{G}_1(\alpha,r,s)$ the expression for $\mathcal{G}_1$ in \eqref{GLpol} evaluated with $\ga$ and $\eta$ replaced by their expressions in \eqref{parametsing}, and similarly for $\mathcal{L}_1$.
Consider the action of $\bar \omega_\alpha$ on the first equation of \eqref{2sv}:
\begin{equation}
\bar\omega_\alpha\circ \mathcal{G}_1(\alpha,r,s) \big(F_{r,s}(\a)\big) =\bar\omega_\alpha\circ \mathcal{G}_1(\alpha,r,s)\circ \bar\omega_\alpha^{-1} \bar\omega_\alpha \big(F_{r,s}(\a)\big)=0,
\end{equation}
and a similar equation with $\mathcal{G}$ replaced by $\mathcal{L}$.
We then determine by a direct calculation how the two fundamental differential operators $ \mathcal{G}_1$ and $ \mathcal{L}_1$ -- given in \eqref{GLpol} -- do transform under a similarity transformation with $ \bar\omega_\alpha$.
Using $\gamma(\alpha)=-\gamma(1/\alpha)$, we obtain
\begin{equation}
\bar\omega_\alpha\circ \mathcal{G}_1(\alpha,r,s) \circ \bar\omega_\alpha^{-1} = \mathcal{G}_1(1/\alpha,s,r) , \qquad \bar\omega_\alpha\circ \mathcal{L}_1(\alpha,r,s)\circ \bar\omega_\alpha^{-1}=-\mathcal{L}_1(1/\alpha,s,r).
\end{equation}
{Therefore, the transformation of $\mathcal {G}_1(\alpha,r,s)$ still reproduce $\mathcal{G}_1$ but with $\a$ changed into $1/\a$ and the Kac labels $r$ and $s$ interchanged. A similar result holds for $\mathcal{L}_1$, up to an overall sign which is irrelevant for the determination of singular vectors.}
The transformed singular-vector relations should read
\begin{equation}
\mathcal{G}_1(1/\alpha,s,r)\big(F_{s,r}(1/\a)\big)=0,\qquad \mathcal{L}_1(1/\alpha,s,r)\big(F_{s,r}(1/\a)\big)=0.
\end{equation}
Therefore, we have the identification
\begin{equation} \bar\omega_\alpha \big(F_{r,s}(\a)\big)=F_{s,r}(1/\a)
\end{equation}
or equivalently,
\begin{equation}
\bar{\omega}_{1/\alpha}\big(F_{r,s}(1/\alpha)\big)=F_{s,r}(\a)
\end{equation}
which is the desired result.
We have thus shown that, still assuming the parametrization given in \eqref{parametsing},
\begin{equation}
|\chi_{r,s}\rangle\quad \longleftrightarrow\quad F_{r,s}(\alpha)= \sum_{\Lambda\in \mathcal{A}_{r,s}}v_\Lambda^{(\alpha)}(r,s) P^{(\alpha)}_\La
\end{equation}
if and only if
\begin{equation}
|\chi_{s,r}\rangle\quad \longleftrightarrow\quad \bar{\omega}_{1/\alpha}\big(F_{r,s}(1/\alpha)\big)= \sum_{\Lambda\in \mathcal{A}_{r,s}}v_\Lambda^{(1/\alpha)}(r,s) \,\bar{\omega}_{1/\alpha}\big(P^{(1/\alpha)}_\La\big).
\end{equation}
This last expression can be made more explicit since
\begin{equation}
\tilde\omega_\alpha \big( P^{(\alpha)}_\La\big)=(-1)^{\binom{m}{2}}\alpha^
\, \bar\omega_\alpha \big( P^{(\alpha)}_\La \big),
\end{equation}
where $\tilde\omega_\alpha$ is the homomorphism such that (cf. \cite[Eq. (4.30)]{DLM_jhep})
\begin{equation}
\tilde\omega_\alpha \big( P^{(\alpha)}_\La \big)=j_\La(\alpha)\, P^{(1/\alpha)}_{\La'} .
\end{equation}
The coefficient $j(\La)$ is the norm (up to a sign) of $P_\La^{(\a)}$ given by
\begin{equation} j_\La(\La)=\a^m\prod_{t\in\La}\frac{h^{\uparrow}_\La(t)}{h^{\downarrow}_\La(t)},
\end{equation}
where $h^{\downarrow}_\La(t)$ and $h^{\uparrow}_\La(t)$ are defined in \eqref{hdn} and \eqref{hup}.
Therefore, we end up with the following expression representing $|\chi_{s,r}\rangle$:
\begin{equation}\label{swap}
|\chi_{s,r}\rangle\quad \longleftrightarrow\quad \sum_{\La\in \mathcal{A}_{r,s}}(-1)^{\binom{m}{2}} v_\Lambda^{(1/\alpha)}(r,s)\,\a^m
\, j_\La(1/\alpha) \, P^{(\alpha)}_{\La'}.
\end{equation}
Let us consider as an example, the construction of $\ket{\chi_{2,5}}$ out of $\ket{\chi_{5,2}}$, whose expression is
\begin{align}
\ket{\chi_{5,2}} \longleftrightarrow P^{(\alpha)}_{{(5)}}+2\,{\frac { \left( 2+3\,\alpha \right) \left( 5
\,\alpha+1 \right) P^{(\alpha)}_{{(4,1)}}}{ \left( 4\,\alpha+1 \right)
\left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) }}+2\,{\frac { \left( 5\,\alpha+1 \right)
\left( \alpha+2 \right) P^{(\alpha)}_{{(3,2)}}}{ \left( 3\,\alpha+1 \right) \left( 1+2\,
\alpha \right) ^{2}}}\nonumber \\
-4\,{
\frac { \left( \alpha+1 \right) \left( 5\,\alpha+1 \right) P^{(\alpha)}_{{(4,1;)}}}{ \left( 4\,\alpha+1 \right) \left( 3\,\alpha+1 \right) }}-2\,{\frac {
\left( 2+3\,\alpha \right) \left( 5\,\alpha+1 \right) P^{(\alpha)}_{{(3,2;)}}
}{ \left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) }}-P^{(\alpha)}_{{(
5,0;)}}.
\end{align}
According to \eqref{swap}, the singular vector $\ket{\chi_{2,5}}$ is given by
\begin{align}
&\left[ j_{(5)}P^{(1/\alpha)}_{{(5)^\prime}}+j_{(4,1)}{\frac { 2\left( 2+3\,\alpha \right) \left( 5
\,\alpha+1 \right) P^{(1/\alpha)}_{{(4,1)^\prime}}}{ \left( 4\,\alpha+1 \right)
\left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) }}+j_{(3,2)}{\frac {2 \left( 5\,\alpha+1 \right)
\left( \alpha+2 \right) P^{(1/\alpha)}_{{(3,2)^\prime}}}{ \left( 3\,\alpha+1 \right) \left( 1+2\,
\alpha \right) ^{2}}} \right. +\frac{(-1)^{\binom{2}{2}}}{\alpha^2} \nonumber \\ &\left.
\times\left(-j_{(4,1;)}{
\frac {4 \left( \alpha+1 \right) \left( 5\,\alpha+1 \right) P^{(1/\alpha)}_{{(4,1;)^\prime}}}{ \left( 4\,\alpha+1 \right) \left( 3\,\alpha+1 \right) }}-j_{(3,2;)}{\frac {2
\left( 2+3\,\alpha \right) \left( 5\,\alpha+1 \right) P^{(1/\alpha)}_{{(3,2;)^\prime}}
}{ \left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) }}-j_{(5,0;)^\prime}P^{(1/\alpha)}_{{(
5,0;)^\prime}} \right) \right]_{\alpha \rightarrow \alpha^{-1}}.\label{exa}
\end{align}
The different $j_\Lambda$ are given by
\begin{align}
\begin{array}{l l l }
j_{(5)}=\frac {120{\alpha}^{5}}{ \left( 4\,\alpha+1 \right) \left( 3\,
\alpha+1 \right) \left( 1+2\,\alpha \right) \left( \alpha+1 \right)}
&j_{(4,1)}=\frac {6 \left( 4\,\alpha+1 \right) {\alpha}^{4}}{ \left( 2+3\,\alpha \right) \left( 1+2\,\alpha \right) \left( \alpha+1 \right) }
&j_{(3,2)}=\frac { \left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) {\alpha}^{3}}{ \left( \alpha+1 \right) ^{2} \left( \alpha+2 \right) }\\
j_{(5,0;)}={\frac {24{\alpha}^{6}}{ \left( 4\,\alpha+1 \right) \left( 3\,\alpha+1 \right) \left( 1+2\,\alpha \right) \left( \alpha+1 \right) }}
&j_{(4,1;)}={\frac {{\alpha}^{5} \left( 4\,\alpha+1 \right) }{ \left( 1+2\,\alpha \right) ^{2} \left( \alpha+1 \right) ^{2}}}
&j_{(3,2;)}={\frac {{\alpha}^{4} \left( 3\,\alpha+1 \right) }{ \left( \alpha+1 \right) ^{2} \left( 2+3\,\alpha \right) }}.
\end{array}
\end{align}
Substituting these expressions into \eqref{exa}, and removing an overall factor in order to set the coefficient of $P^{(\alpha)}_{{(2,2,1)}}$ equal to 1 yield
\begin{align}
P^{(\alpha)}_{{(2,2,1)}}+6\,{\frac { \left( \alpha+1 \right) P^{(\alpha)}_{{(2,1,1,1)}
}}{ \left( \alpha+3 \right) \left( \alpha+2 \right) }}+60\,{\frac {
\left( \alpha+1 \right) P^{(\alpha)}_{{(1,1,1,1,1)}}}{ \left( \alpha+5
\right) \left( \alpha+4 \right) \left( \alpha+3 \right) }}+\alpha\,
P^{(\alpha)}_{{(1,0;2,2)}}\nonumber\\
+2\,{\frac { \left( \alpha+1 \right) \alpha\,P^{(\alpha)}_{{(1,0;2,1,1)}}}{ \left( \alpha+3 \right) \left( \alpha+2 \right) }}+12\,
{\frac { \left( \alpha+1 \right) \alpha\,P^{(\alpha)}_{{(1,0;1,1,1,1)}}}{
\left( \alpha+5 \right) \left( \alpha+4 \right) \left( \alpha+3
\right) }}.
\end{align}
This agrees with the case $s=5$ of the general expression for the $|\chi_{2,s}{\rangle}$ singular vectors given in \cite[Eq. (B.23)]{DLM_jhep}.\\
More generally, we can recover the general expression for $|\chi_{2,s}{\rangle}$ (that is, \cite[Eq. (B.23)]{DLM_jhep}) by applying the duality transformation to the the closed-form expression for $|\chi_{s,2}{\rangle}$, but the analysis is somewhat technical and does not involve essential novelties.
\begin{appendix}
\section{Superpartitions, superpolynomials and sJacks: a brief review}
\label{SsJ}
\subsection{Superpartitions} \label{A1}
A superpartition $\La$ is
a pair of partitions
\begin{equation}\label{sppa}
\La=(\La^{a};\La^{s})=(\La_1,\ldots,\La_m;\La_{m+1},\ldots,\La_\ell),
\end{equation}
such that
\begin{equation}\label{sppb}
\La_1>\ldots>\La_m\geq0 \quad \text{ and}
\quad \La_{m+1}\geq \La_{m+2} \geq \cdots \geq
\La_\ell > 0 \, .\end{equation}
\noindent
The number $m$ is the fermionic degree of $\Lambda$ and $\ell$ is its length. The bosonic degree is $|\La|=\sum_i\La_i$.
By removing the semi-coma and reordering the parts, we obtain an ordinary partition that we denote $\La^*$. The diagram of $\La$ is that of $\La^*$ with circles added to the rows corresponding to the parts of $\La^a$
and ordered in length as if a circle was a half-box.
Finally, we will denote by $\La^{\circledast}$ the partition obtained from the diagram of $\La$ by replacing circles by boxes. Here is an example, for which $\La=(4,3,0;4)$:
\iffalse
\begin{figure}[h]\caption{Diagrammatic representation of superpartition }\label{fig3}
\begin{center}
{\small $$ \Lambda={\tableau[scY]{&&&&\bl\tcercle{}\\&&&\\&&&\bl\tcercle{}\\\bl\tcercle{} \\ }}
\quad \Longleftrightarrow \quad \Lambda^\circledast={\tableau[scY]{&&&&\\&&& \\&&&\\ \\ }} \quad \Lambda^*={\tableau[scY]{&&&\\&&&\\ &&\\ \bl\\ }} $$}
\end{center}
\end{figure}
\fi
\begin{equation} \Lambda={\tableau[scY]{&&&&\bl\tcercle{}\\&&&\\&&&\bl\tcercle{}\\\bl\tcercle{} \\ }}
\quad \Longleftrightarrow \quad \Lambda^\circledast={\tableau[scY]{&&&&\\&&& \\&&&\\ \\ }} \quad \Lambda^*={\tableau[scY]{&&&\\&&&\\ &&\\ \bl\\ }} .\end{equation}
Given a partition $\lambda$ (in our case, it is either $\La^*$ or $\La^{\circledast}$), its
conjugate $\lambda'$ is the diagram
obtained by reflecting $\lambda$ about the main diagonal.
In the main text, we make use of the following data: for a cell $t=(i,j)$ in $\lambda$, we define the arm and the leg of the box $t$ as
\begin{equation} \label{arml}
a_{\lambda}(t)=\lambda_i-j\, ,\qquad
l_{\lambda}(t)=\lambda_j'-i \, .
\end{equation}
Note that conjugation is defined for superpartitions in the same way as for partitions: rows of $\La$ becomes columns of $\La'$. For instance
\begin{equation}
\left(\;{\tableau[scY]{&&\bl\tcercle{}\\\bl\tcercle{} }}\right)'={\tableau[scY]{& \bl\tcercle{}\\ \\\bl\tcercle{} }}\end{equation}
\subsection{Superpolynomials}\label{A2}
Superpolynomials
polynomials in the usual commuting $N$ variables $x_1,\cdots ,x_N$ and the $N$ anticommuting variables $\ta_1,\cdots,\ta_N$.
Symmetric superpolynomials are invariant with respect to the interchange of $(x_i,\ta_i)\leftrightarrow (x_j,\ta_j)$ for any $i,j$
\cite{DLM1}. They are labelled by superpartitions.
The simplest example of a symmetric superpolynomial is the super-version of the monomial polynomials:\begin{equation}
m_\La(x,\theta)=\theta_{1}
\cdots\theta_{m} x_1^{\Lambda_1} \cdots x_N^{\Lambda_N}+\text{distinct permutations}
\end{equation}
{with the understanding that $\La_{\ell+1}=\cdots =\La_N=0$. }
An explicit example, for $N=4$ is
\begin{align}
m_{(1,0;1,1)}(x;\theta)=\;&\ta_1\ta_2(x_{1}-x_2)x_3x_4+\ta_1\ta_3(x_{1}-x_3)x_2x_4+\ta_1\ta_4(x_{1}-x_4)x_2x_3\nonumber\\+\,&\ta_2\ta_3(x_{2}-x_3)x_1x_4+\ta_2\ta_4(x_{2}-x_4)x_1x_3+\ta_3\ta_4(x_{3}-x_4)x_1x_2.
\end{align}
Another example is
the super-power-sums\begin{equation}\label{spower}
p_\La=\tilde{p}_{\La_1}\cdots\tilde{p}_{\La_m}p_{\La_{m+1}}\cdots p_{\La_\ell},\qquad\text{where}\quad \tilde{p}_n=\sum_i\theta_ix_i^n\quad\text{and}\quad p_n=
\sum_ix_i^n \, .
\end{equation}
Both $\{m_\La\}$ and $\{p_\La\}$ provide bases for the space of symmetric superpolynomials.
\subsection{Jack superpolynomials}\label{A3}
\label{charaphys}
The Jack superpolynomials (sJacks) $P_\La^{(\alpha)}$ can be characterized by the following
two conditions (e.g., see \cite{DLMeva}). The first is {\it triangularity in the monomial basis:}
\begin{equation} \label{Ptriangular}
P_\La^{(\a)} =m_\La+\sum_{\Om<\La}
c_{\La\Om}(\alpha)\,m_\Om \, ,
\end{equation}
where $<$ refers to the dominance order on
superpartitions \cite{DLMeva}:
\begin{equation} \label{eqorder1}
\Omega\leq\Lambda \quad \text{iff}
\quad \Omega^* \leq \Lambda^*\quad \text{and}\quad
\Omega^{\circledast} \leq \Lambda^{\circledast} .
\end{equation}
We recall that the order on partitions is the usual dominance ordering:
\begin{equation}
\lambda \geq \mu \quad \iff \quad \sum_{i} \lambda_i =\sum_i \mu_i
\quad \text{and}\quad \lambda_1+ \cdots +\lambda_k \geq
\mu_1+ \cdots +\mu_k\, \quad \forall \; k \,
\end{equation}
Pictorially, $\Om<\La$ if $\Om$ can be obtained from $\La$ successively by moving down a box or a circle, which can be viewed as a sort of super-squeezing rule.
For example,
\begin{equation}
{\tableau[scY]{&&&&\bl\tcercle{}}}
\quad > \quad
{\tableau[scY]{&&\bl\tcercle{}\\ &}}\quad > \quad
{\tableau[scY]{&\\&\\ \bl\tcercle{}}} \;. \end{equation}
The second condition is {\it orthogonality in the power-sum basis:}
\begin{equation} \label{scap}
\ensuremath{\langle\!\langle} \,
{P_\La} \, | \, {P_\Om }\, \ensuremath{\rangle\!\rangle}_\alpha=0\quad \text{if} \quad \La\ne \Om\, .
\end{equation}
The scalar product is defined as follows:
\begin{equation} \label{scap} \ensuremath{\langle\!\langle} \,
{p_\La} \, | \, {p_\Om }\, \ensuremath{\rangle\!\rangle}_\alpha=(-1)^{\binom{m}2}\, \alpha^{{\ell}(\La)}\, z_{\La^s}
\delta_{\La,\Om}\,, \end{equation}
where $z_{\La^s} $ is given by
\begin{equation} \label{zlam}
z_{\La^s}=\prod_{i \geq 1} i^{n_{\La^s}(i)} {n_{\La^s}(i)!}\, ,
\end{equation}
with $n_{\La^s}(i)$ the number of parts in $\La^s$ equal to $i$.
\end{appendix}
|
{
"timestamp": "2013-10-01T02:15:55",
"yymm": "1309",
"arxiv_id": "1309.7965",
"language": "en",
"url": "https://arxiv.org/abs/1309.7965"
}
|
\section{Introduction}
The production of B-hadrons at the Large Hadron Collider (LHC) provides particular challenges and opportunities for insight into Quantum Chromodynamics (QCD), especially as a probe of the properties of the fundamental constituents of matter and their interactions in a new energy regime. Mesons and baryons containing a b quark can be seen as the hydrogen and helium atoms of QCD, therefore the measurement of their spectra plays an especially important role in understanding the strong interactions. The characteristices of b-hadrons at the LHC can also give input for tuning models and refining event generators in a new energy regime.\\
The most important elements of the ATLAS detector for B physics measurements are the Inner Detector (ID) tracker, the Electromagnetic Calorimeter, and the Muon Spectrometer (MS); details can be found in \cite{detector}. Dedicated B physics triggers are based on both single muons and di-muons with different thresholds and mass ranges.
\section{Measurement of the $B^+$ production cross-section}
\begin{figure}[!htb]
\centering
\includegraphics[height=2in]{bplusfit.png}
\caption{ The observed invariant mass distribution of $B^\pm$ candidates, $m_{J/\psi K^\pm}$, with transverse momentum and rapidity in the range 20 GeV $ <p_T < $ 25 GeV, 0.5 $ < |y| < $ 1 (dots), compared to the binned maximum likelihood fit (solid line). The error bars represent the statistical uncertainty. Also shown are the components of the fit as described in the legend.}
\label{fig:bplusfit}
\end{figure}
Several B-hadron production cross sections have been measured \cite{bplus1,bplus2,bplus3,bplus4,bplus5,bplus6,bplus7,bplus8,bplus9,bplus10,bplus11,bplus12}, but the theoretical uncertainty remains up to 40\% due to uncertainty in the factorization and renormalization scales of the b-quark production, and the b-quark mass $m_b$ \cite{mb}. The LHC provides an opportunity to perform more precise measurements of B-hadron production cross-sections. Also the B-hadron production cross-section measurements in pp collisions at the LHC can provide tests of QCD calculations for heavy-quark production at high center-of-mass energies and in wide transverse momentum ($p_T$) and rapidity ($y$) ranges.\\
\begin{figure}[!htb]
\centering
\subfigure[]{\label{a}\includegraphics[height=2in]{bpluscs.png}}
\subfigure[]{\label{b}\includegraphics[height=2in]{bpluscsNLO.png}}
\subfigure[]{\label{c}\includegraphics[height=2in]{bpluscsNLO2.png}}
\caption{ (a) The doubly-differential cross-section for $B^+$ production as a function of $p_T$ and $y$, averaged over each ($p_T$, $y$) interval and quoted at its center. The data points are compared to NLO predictions from POWHEG and MC@NLO. The shaded areas around the theoretical predictions reflect the uncertainty from renormalization and factorization scales and the b-quark mass. The ratio of the measured cross-section to the theoretical predictions ($\sigma/\sigma_{NLO}$) of POWHEG (b) and MC@NLO (c) in eight $p_T$ intervals in four rapidity ranges is shown. The points with error bars correspond to data with their associated uncertainties, which is the combination of the statistical and systematic uncertainty. The shaded areas around the theoretical predictions reflect the uncertainty from renormalization and factorization scales and the b-quark mass.}
\label{fig:bpluscs}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[height=2in]{bpluscsFONLL.png}
\caption{ The differential cross-section for $B^+$ production versus $p_T$, integrated over rapidity. The solid points with error bars correspond to the differential cross-section measurement by ATLAS with total uncertainty (statistical and systematic) in the rapidity range $|y|<$ 2.25, averaged over each $p_T$ interval and quoted at its center. For comparison, data points from CMS are also shown, for a measurement covering $p_T<$ 30 GeV and $|y|<$ 2.4 \cite{bplusCMS}. Predictions of the FONLL calculation for b-quark production are also compared with the data, assuming a hadronization fraction $f_{b\rightarrow B^+}$ of $(40.1\pm 0.8)\%$ \cite{bplusf} to fix the overall scale. Also shown is the ratio of the measured cross-section to the predictions by the FONLL calculation ($\sigma/\sigma_{FONLL}$). The upper and lower uncertainty limits on the prediction were obtained by considering scale and b-quark mass variations.}
\label{fig:bpluscsFONLL}
\end{figure}
ATLAS has measured the $B^+$ production cross-section using the decay channel $B^+\rightarrow J/\psi K^+$ in pp collisions at $\sqrt{s}$ = 7 TeV, as a function of $B^+$ transverse momentum and rapidity \cite{bplus}. The measurement uses data of integrated luminosity of 2.4 $fb^{-1}$ recorded in early 2011 data $|y| < $ 2.25 and $p_T$ up to 100 GeV. Events were selected using a di-muon trigger where both muons are required to have $p_T > $ 4 GeV and pass a loose selection requirement compatible with $J/\psi$ meson decay into a muon pair. The $B^+$ candidates are reconstructed from a $J/\psi$ candidate combined with a hadron track with $p_T>$ 1 GeV and required to have $p_{T} >$ 9 GeV, $|y| <$ 2.3, and $\chi^2/N_{d.o.f}<$ 6. The muon tracks and hadron tracks are required to have sufficient numbers of hits in the Pixel, Semiconductor Tracker (SCT) and Transition Radiation Tracker (TRT) detectors to ensure accurate ID measurements.\\
The differential cross-section for $B^+$ production is given by\\
$$\frac{\mathrm{d}^2\sigma(pp\rightarrow B^+X)}{\mathrm{d}p_T\mathrm{d}y}\cdot{\cal{B}} = \frac{N^{B^{+}}}{{{\cal L}}\cdot\Delta p_{T} \cdot \Delta y},$$\\
where $N^{B^{+}}$ is derived from the average yield of the two reconstructed charged states $B^+$ and $B^-$ in each ($p_T$, $y$) interval after correcting for detector effects and acceptance. The total number of $B^\pm$ events is extracted using a binned maximum likelihood fit in each ($p_T$, $y$) bin ({\it e.g.}~Figure~\ref{fig:bplusfit}). The differential cross-section is calculated in 8 $p_T$ bins and 4 $y$ bins in the full kinematic range 9 GeV $<p_T<$ 120 GeV and $|y|<$ 2.25 as shown in Figure~\ref{fig:bpluscs}(a). The data points have been compared to next-to-leading order (NLO) \cite{mb,NLO} predictions from POWHEG \cite{NLOP,NLOP2} and MC@NLO \cite{NLOM,NLOM2}. As shown in Figure~\ref{fig:bpluscs}(b)(c), the POWHEG prediction shows good agreement with data in both cross section and shape while MC@NLO predicts a lower cross section and softer $p_T$ spectrum at low $y$ and a harder $p_T$ spectrum in high $y$. The results have been compared to fixed order plus next-to-leading logarithms (FONLL) \cite{FONLL,FONLL2} as well. As shown in Figure~\ref{fig:bpluscsFONLL}, the FONLL prediction shows good agreement especially with $p_T<$ 30 GeV, and is still compatible in the high $p_T$ range even up to 100 GeV.
\section{$B_c$ meson observation}
The $B_{c}^\pm$ meson is a bound state of the two heave quarks able to form a stable state. Weak decays of the $B_{c}^\pm$ meson provide a unique probe of heavy quark dynamics that is inaccessible to $b\bar{b}$ or $c\bar{c}$ bound states. \\
\begin{figure}[!htb]
\centering
\includegraphics[height=2in]{bc.png}
\caption{The invariant mass distribution of reconstructed $B_c^{\pm} \to J/\psi \pi^{\pm}$ candidates. The points with error bars are the data. The solid line is the projection of the results of the unbinned maximum likelihood fit to all candidates in the mass range 5770-6820 MeV. The dashed line is the projection for the background component on the same fit.}
\label{fig:bc}
\end{figure}
The $B_{c}^\pm$ meson is reconstructed using ATLAS pp collision data collected in the year 2011 at $\sqrt{s}$ = 7 TeV using the decay mode $B_c\rightarrow J\psi\pi$ \cite{bc}. Single and di-muon triggers with various transverse momentum have been used. Each muon candidate must have a track reconstructed in the MS combined with a track reconstructed in the ID. The di-muon selection requires a pair of oppositely charged muons with $p_T>$ 4 GeV for both in the first half year and $p_T>$ (6, 4) GeV in the second half year. Additionally they are required to pass a loose selection requirement compatible with $J/\psi$ meson decay into a muon pair. The $B_c$ candidate is reconstructed from the $J/\psi$ candidate combined with a hadron track with $p_T>$ 4 GeV. Unlike $B^+$, the short $B_c$ meson lifetime means that lifetime cuts are not efficient in separating the $B_c$ signal from direct $J/\psi$ combinations. The transverse impact parameter significance has been proven to be more efficient, and it is required to be larger than 5. The reconstructed $B_c$ candidates are required to have $p_{T} >$ 15 GeV and $\chi^2/N_{d.o.f}<$ 2.\\
With the full 2011 integrated luminosity of 4.3 $fb^{-1}$ of data, $82\pm17$ $B_c$ ground state mesons have been extracted using an unbinned maximum likelihood fit (Figure~\ref{fig:bc}). The $B_c$ mass returned by the fit is $6282\pm7 $ MeV, which is consistent with the PDG mass\cite{bcPDG} of $6277\pm7$ MeV.
\section{Measurement of the $\Lambda_b$ lifetime and mass}
$\Lambda_{b}^0$ is the lightest b-baryon. Although it has been measured by many experiments \cite{lambdab2, lambdab3,lambdab4}, its lifetime still has large experimental uncertainty (2-3\%), and the discrepancy between the CDF and D0 measurements is high (1.8$\sigma$).\\
\begin{figure}[!htb]
\centering
\subfigure[]{\label{a}\includegraphics[height=2in]{lambdaba.png}}
\subfigure[]{\label{b}\includegraphics[height=2in]{lambdabb.png}}
\caption{ Projection of the fitted probability density function onto the mass (a) and the proper decay time (b) axis for $\Lambda_{b}$ candidates. The displayed errors are statistical only. The $\chi^2/N_{d.o.f}$ value is calculated from the dataset binned in mass and decay time with the number of degrees of freedom $N_{d.o.f}$ = 61.}
\label{fig:lambdab}
\end{figure}
This analysis is based on 4.9 $fb^{-1}$ data collected by ATLAS in 2011 using single muon and di-muon $J/\psi$ triggers and reconstructed through the decay channel of $\Lambda_b^0\rightarrow J/\psi(\mu^+\mu^-)\Lambda^0,~ \Lambda^0\rightarrow p^+\pi^-$ \cite{lambdab}. Muon tracks are reconstructed in the ID and identified in the MS, with $p_T$ above 400 MeV and pseudorapidity $|\eta| < $ 2.5 required. The reconstructed $J/\psi$ candidates are selected within a mass window of 2.8 GeV $<m(\mu^+\mu^-)<$ 3.4 GeV with a vertex fit constrained to the $J/\psi$ mass of 3069.92 MeV and muon $p_T>$ 4 GeV. Reconstructed $\Lambda^0$ candidates are selected within a mass window of 1.08 GeV $<m(p\pi)<$ 1.15 GeV with a vertex fit constrained to the $\Lambda^0$ mass of 1115.68 MeV; additionally the $\Lambda^0$ vertex is required to point to the $J/\psi$ vertex. Finally the $\Lambda_{b}^0$ candidates are reconstructed using a cascade vertex fit applied to the four tracks $\mu^+,~\mu^-,~p,~\pi$ simultaneously with all the $J/\psi$ and $\Lambda^0$ constraints. For comparison, $B_d$ candidates with similar topology are reconstructed as well. The reconstructed $\Lambda_{b}^0$ candidates are required to have $p_T>$ 3.5 GeV, transverse decay length larger than 10 mm, global $\chi^2/N_{d.o.f}<$ 3, and the difference between the cumulative $\chi^2$ probabilities of the two fits larger than 0.05.\\
The mass and the proper decay time, defined as:
$$\tau=\frac{L_{xy}m^{PDG}}{p_T}$$
of the $\Lambda_{b}^0$ are extracted using a simultaneous unbinned maximum likelihood fit with per event error (Figure~\ref{fig:lambdab}). Including systematic uncertainties on event selection and reconstruction bias, the fit model, $B_d^0$ contamination, and the $p_T$ scale, the mass and proper decay time measured are:
$$m_{\Lambda_b^0} = 5619.7 \pm 0.7(stat) \pm 1.1(syst)~\rm{MeV}$$
$$\tau_{\Lambda_b^0} = 1.449 \pm0.036(stat)\pm0.017(syst)~\rm{ps}.$$
They are consistent with the PDG value \cite{bplusf} and the LHCb result \cite{lambdabLHCb}.\\
The ratio of the $\Lambda_{b}^0$ and $B_{d}^0$ lifetimes has been measured:
$$R = \tau_{\Lambda_b^0}/\tau_{B_d^0} = 0.960 \pm 0.025(stat) \pm 0.016(syst).$$
This ratio is intermediate to the recent determination by D0, $R^{D0} = 0.864 \pm 0.052(stat)\\ \pm 0.033(syst)$ \cite{lambdabD0}, and the measurement by CDF, $R^{CDF} = 1.020 \pm 0.030(stat) \pm
0.008(syst)$ \cite{lambdabCDF}. It agrees with heavy quark expansion calculations which predict the value of the ratio to be between 0.88 and 0.97 \cite{hqe} and is compatible with the next-to-leading order QCD predictions with central values ranging between 0.86 and 0.88 (uncertainty of $\pm~0.05$) \cite{NLOQCD}.
\section{Observation of a new $\chi_{b}$ state in radiative transitions to $\Upsilon(1S)$ and $\Upsilon(2S)$}
\begin{figure}[!htb]
\centering
\includegraphics[height=3in]{chibpredict.png}
\caption{ Masses and radiative transitions of the $\chi_{b}$ states as observed in ATLAS. The first column shows the world-average masses of the Υ states. The second column displays the ATLAS mass barycenter determination for each of the $\chi_{b}(1P,2P,3P)$ triplets, with the bands of vertical stripes indicating the measurements from the analysis using unconverted photons and the solid band representing the measurement from the converted photon analysis ($\chi_{b}(3P)$ only). Dashed lines between the states in the second and first columns indicate the radiative transitions observed by ATLAS. In the third column, the world averages (or Potential Model predictions in the case of the 3P) are shown for the three multiplets. Progressively paler shades of blue indicate the J = 0, 1, 2 states.}
\label{fig:chibpredict}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[height=2in]{chibmass.png}
\caption{ The invariant mass of selected di-muon candidates. The shaded regions A and B show the selections for $\Upsilon(1S)$ and $\Upsilon(2S)$ candidates respectively.}
\label{fig:chibmass}
\end{figure}
The $\chi_{b}(1P)$ and $\chi_{b}(2P)$ states have been observed in previous experiments, but the $\chi_{b}(3P)$ state has not. The $\chi_{b}(3P)$ state is the highest P state predicted below the $B-\bar{B}$ threshold as shown in Figure~\ref{fig:chibpredict} with mass about 10.52 GeV and hyperfine splitting about 10-20 MeV. The $\chi_{b}(nP)$ states are sought using a data sample corresponding to an integrated luminosity of 4.9 $fb^{-1}$ of ATLAS 2011 data \cite{chib}, through decay modes of $\chi_{b}(nP)\rightarrow\Upsilon(1S)\gamma$ and $\chi_{b}(nP)\rightarrow\Upsilon(2S)\gamma$. \\
\begin{figure}[htb]
\centering
\subfigure[]{\label{a}\includegraphics[height=2in]{chib.png}}
\subfigure[]{\label{b}\includegraphics[height=2in]{chib1.png}}
\caption{ (a) The mass distribution of $\chi_{b}\rightarrow\Upsilon(1S)\gamma$ candidates for unconverted photons reconstructed from energy deposits in the electromagnetic calorimeter ($\chi^2_{fit}/N_{d.o.f.}$ = 0.85). (b) The mass distributions of $\chi_{b}\rightarrow\Upsilon(kS)\gamma$ (k = 1, 2) candidates formed using photons which have converted and been reconstructed in the ID ($\chi^2_{fit}/N_{d.o.f.}$ = 1.3). Data are shown before the correction for the energy loss from the photon conversion electrons due to bremsstrahlung and other processes. The data for decays of $\chi_{b}\rightarrow\Upsilon(1S)\gamma$ and $\chi_{b}\rightarrow\Upsilon(2S)\gamma$ are plotted using circles and triangles respectively. Solid lines represent the total fit result for each mass window. The dashed lines represent the background components only.}
\label{fig:upsilon}
\end{figure}
In this measurement, a set of muon triggers designed to select events containing muon pairs or single high transverse momentum muons was used to collect the data sample. Each muon candidate must have a track reconstructed in the MS combined with a track reconstructed in the ID with $p_{T} >$ 4 GeV and pseudorapidity $|\eta| <$ 2.3. The di-muon selection requires a pair of oppositely charged muons which are fitted to a common vertex. The di-muon candidate is also required to have $p_{T} >$ 12 GeV and $|\eta| <$ 2.0. $\Upsilon(1S)\rightarrow\mu\mu$ candidates with masses in the range $9.25 < m_{\mu\mu} < 9.65$ GeV and $\Upsilon(2S)\rightarrow\mu\mu$ candidates with masses in the range $9.80 < m_{\mu\mu} < 10.10$ GeV are selected (Figure~\ref{fig:chibmass}). This asymmetric mass window for $\Upsilon(2S)$ candidates is chosen in order to reduce contamination from the $\Upsilon(3S)$ peak and continuum background contributions. A photon is combined with each $\Upsilon$ candidate. Converted photons reconstructed by ID tracks from $e^+e^-$ pairs with a conversion vertex and unconverted photons reconstructed by electromagnetic calorimeter energy deposit are used. The converted photon candidates are required to be within $|\eta| <$ 2.30 while the unconverted photon candidates are required to be within $|\eta| <$ 2.37. Unconverted photons must also be outside the transition region between the barrel and endcap calorimeters, 1.37 $< |\eta| <$ 1.52. Requirements of $p_{T}(\mu^+\mu^-) >$ 20 GeV and $p_{T}(\mu^+\mu^-) >$ 12 GeV are applied to $\Upsilon$ candidates with unconverted and converted photon candidates respectively. These thresholds are chosen in order to optimize signal significance in the $\chi_{b}(1P,2P)$ peaks.\\
As shown in the mass difference $m(\mu^+\mu^-\gamma)-m(\mu^+\mu^-)+m_{PDG}(\Upsilon)$ distributions (Figure~\ref{fig:upsilon}), in addition to the mass peaks corresponding to the decay modes of $\chi_{b}(1P,2P)\rightarrow\Upsilon(1S,2S)$, a new structure centered at mass $10.530 \pm 0.005 (stat.) \pm 0.009 (syst.)$ GeV is observed with a significance of more than 6$\sigma$ in both of those two independent samples. This is interpreted as the $\chi_{b}(3P)$ state.\\
|
{
"timestamp": "2013-10-01T02:15:48",
"yymm": "1309",
"arxiv_id": "1309.7962",
"language": "en",
"url": "https://arxiv.org/abs/1309.7962"
}
|
\section{Introduction}
One of the challenging question in the modern stellar physics is to know if the Sun has a peculiar abundance of Li regarding to other stars of the same type. The present day solar Li abundance (A(Li) = 1.05) is much lower than the meteoritic Li abundance (A(Li) = 3.26; Asplund {\em et al.\/} \cite{asplund2009}). Standard models fail to reproduce solar Li destruction during the main sequence evolution.
The most recent observations show solar twins exhibiting lithium abundances comparable to the solar value: HIP 56948: A(Li) $= 1.28 \pm 0.05$ (Mel\'endez {\em et al.\/} \cite{melendez2012}); HIP 73815: A(Li) $< 0.90 \pm 0.20$ (Mel\'endez \& Ram\'irez \cite{melendez&ramirez2007}); YPB637, YPB1194, and YPB1787 in the open cluster M67: A(Li) $= 1.5$, $<1.3$, and $1.6$, respectively (Castro {\em et al.\/} \cite{castro2011}); CoRoT ID102684698: A(Li) $= 0.85 \pm 0.35$ (do Nascimento {\em et al.\/} \cite{donascimento2013}).
Those observations and the improvements of evolution models tend to show that solar Li abundance is not peculiar but a product of depletion due to non-standard mixing which affects both the Sun and the solar twins. Many authors have studied different mixing processes that could cause Li depletion, such as internal gravity waves (Montalban \& Schatzmann \cite{montalban&schatzman2000}; Charbonnel \& Talon \cite{charbonnel&talon2005}), overshooting mixing (Xiong \& Deng \cite{xiong&deng2009}), and meridional circulation (do Nascimento {\em et al.\/} \cite{donascimento2009}). All these studies suggest that Li depletion is a function of several parameters such as mass, age, metallicity, and angular momentum history (Charbonnel \& Talon \cite{charbonnel&talon2005}; do Nascimento {\em et al.\/} \cite{donascimento2009}; Meléndez {\em et al.\/} \cite{melendez2010}; Baumann {\em et al.\/} \cite{baumann2010}; Castro {\em et al.\/} \cite{castro2011}).
We studied the evolution of the Li abundance in three open clusters of different ages (Hyades, NGC752, and M67). We collected all published data to the best of our knowledge, re-analysed them and compared with TGEC evolution models calibrated on the Sun. In Sec. \ref{sec_models}, we present the physics in our models. In Secs. \ref{sec_M67}, \ref{sec_NGC752}, and \ref{sec_Hyades}, we compare our models to the observations for the open clusters M67, NGC752, and Hyades, respectively. In Sec. \ref{sec_Conclusions}, we announce some conclusions.
\section{TGEC models}
\label{sec_models}
The Toulouse Geneva Evolution Code (TGEC) compute 1D evolution models from ZAMS to the top of the RGB. Details about the input physics and the mixing processes in the models can be found in Pace {\em et al.\/} \cite{pace2012}.
With the assumption that the Sun is a common star among solar-type stars, we chose to calibrate our models on the Sun. We used the method described in Richard {\em et al.\/} \cite{richard2004}. We obtained at the solar age, a Li abundance $A(\mathrm{Li}) = 1.04$.
\section{Li in the open cluster M67}
\label{sec_M67}
Our data base is composed of 103 stars from a compilation of literature sources of Li abundance measurements, namely Canto Martins {\em et al.\/} \cite{cantomartins2011}, Castro {\em et al.\/} \cite{castro2011}, Pasquini {\em et al.\/} \cite{pasquini2011}, Randich {\em et al.\/} \cite{randich2007}, Jones {\em et al.\/} \cite{jones99}, and Balachandran \cite{balachandran95}.
In the left panel of Figure \ref{fig_M67}, we show the color-magnitude diagram of the open cluster M67 (Pace {\em et al.\/}, 2012), obtained with a reddening $E(B - V) = 0.02$ mag and a distance modulus $(m - M)_0 = 9.68$ mag. The isochrone of age 3.87 Gyr was calculated with models of metallicity [Fe/H] = 0.01 and masses from 0.90 to 1.31 $\mathrm{M_{\odot}}$ \ with a step of 0.01 $\mathrm{M_{\odot}}$.
\begin{figure}
\centering
\includegraphics[width=5.5cm]{castro_m_fig1.eps}
\qquad
\includegraphics[width=5.8cm,height=5.4cm]{castro_m_fig2.eps}
\caption{\textit{Left}: Color-magnitude diagram of the open cluster M67. \textit{Right}: Li abundance as a function of stellar mass for the open cluster M67. From Pace \etal \ \cite{pace2012}.}
\label{fig_M67}
\end{figure}
The right panel of Figure \ref{fig_M67} shows the Li abundance as a function of mass. The masses of the observational points were determined by interpolation in the CMD with the isochrone.
For stars with mass lower than 1.10 $\mathrm{M_{\odot}}$ (until the point I in Fig. \ref{fig_M67}), the Li depletion progressively increases toward lower masses. A dispersion is observed in this mass range and confirmed by several authors (Pasquini {\em et al.\/} \cite{pasquini97}; Jones {\em et al.\/} \cite{jones99}; Randich {\em et al.\/} \cite{randich2002},\cite{randich2007}; Pasquini {\em et al.\/} \cite{pasquini2008}), probably due to the history of angular momentum (Charbonnel \& Talon \cite{charbonnel&talon2005}).
Stars with masses between 1.10 $\mathrm{M_{\odot}}$ \ and 1.28 $\mathrm{M_{\odot}}$ \ (from I to II) are close to the turn off (or about to leave the MS). The surface convection zone and the mixed layers below the convection zone are too thin to reach temperatures high enough to cause the nuclear destruction of Li, whose abundance remains large. It exists an offset of 0.45 dex between our models and observations, showing an insufficient destruction of Li by rotationally driven mixing mechanisms.
Stars with mass larger than 1.28 $\mathrm{M_{\odot}}$ \ (from II to III) are sub-giants following an evolutionary path from the turn-off to the RGB. This range of masses corresponds to the Li-dip of M67 (Balachandran \cite{balachandran95}).
\section{Li in the open cluster NGC752}
\label{sec_NGC752}
For the open cluster NGC752, we adopted the Li equivalent width measurements from the 2.1 m and 4 m telescopes of Kitt Peak National Observatory (Hobbs \& Pilachowsky \cite{hobbs&pilachowsky86}; Pilachowsky \& Hobbs \cite{pilachowsky&hobbs88}; Pilachowsky \etal \ \cite{pilachowsky88}) and the 3.58 m SARG@TNG (Sestito {\em et al.\/} \cite{sestito2004}). We used the photometry from Daniel {\em et al.\/} (\cite{daniel94}). We found two different metallicities in the literature: [Fe/H] $= -0.15 \pm 0.05$ dex obtained from Daniel {\em et al.\/} (\cite{daniel94}), and [Fe/H] $\sim +0.10$ dex inferred by high-resolution spectroscopy of 4 giants by Carrera \& Pancino (\cite{carrera&pancino2011}).
A color-magnitude diagram of the open cluster NGC752 was obtained with a reddening $E(B - V) = 0.035$ and a distance modulus $(m - M)_{0} = 8.25$ mag (Daniel {\em et al.\/} \cite{daniel94}) for each metallicity (Fig. \ref{fig_cmd_NGC752}). In the left panel of Fig. \ref{fig_cmd_NGC752}, the isochrone of age 1.90 Gyr were constructed with models of metallicity [Fe/H] = -0.15 and masses from 0.80 to 1.59 $\mathrm{M_{\odot}}$ \ with a step of 0.01 $\mathrm{M_{\odot}}$, whereas in the right panel, the isochrone of age 1.49 Gyr was constituted by models of metallicity [Fe/H] = +0.10 and masses from 0.80 to 1.81 $\mathrm{M_{\odot}}$ \ with a step of 0.01 $\mathrm{M_{\odot}}$.
\begin{figure}
\centering
\includegraphics[width=5.5cm,height=5.5cm]{castro_m_fig3.eps}
\qquad
\includegraphics[width=5.7cm,height=5.5cm]{castro_m_fig4.eps}
\caption{Color-magnitude diagram of the open cluster NGC752 for metallicities [Fe/H] = -0.15 (\textit{left}) and [Fe/H] = +0.10 (\textit{right}).}
\label{fig_cmd_NGC752}
\end{figure}
In Figure \ref{fig_Li_NGC752}, we show the evolution of Li abundance as a function of stellar mass for each metallicity. In both cases, we have a good agreement for solar-type stars ($M < 1.10$ $\mathrm{M_{\odot}}$), and no significant dispersion at this age. However, the Li-dip is not reproduced by our models, which shows a lack of an extra-mixing mechanism, probably the internal gravity waves (Charbonnel \& Talon \cite{charbonnel&talon2005}).
\begin{figure}
\centering
\includegraphics[width=5.5cm,height=5.5cm]{castro_m_fig5.eps}
\qquad
\includegraphics[width=5.7cm,height=5.5cm]{castro_m_fig6.eps}
\caption{Li abundance as a function of stellar mass for the open cluster NGC752 for metallicities [Fe/H] = -0.15 (\textit{left}) and [Fe/H] = +0.10 (\textit{right}).}
\label{fig_Li_NGC752}
\end{figure}
Comparing the right panel of Fig. \ref{fig_M67} and the left panel of Fig. \ref{fig_Li_NGC752}, one notices that the Li-dip in M67 occurs at higher masses as compared to NGC752 if the metallicity given by Daniel {\em et al.\/} (\cite{daniel94}) is adopted for the latter, thus making Li-dip stars in NGC752 more Li-depleted than M67 members of the same mass. However, if we assume, as reasonable, the metallicity of [Fe/H]=0.10 for NGC752, which was found by Carrera \& Pancino through a high-resolution spectroscopic study, stellar evolution models make stars in this cluster more massive, and the masses of the Li-dip match for the three clusters. This suggests that - while our adopted value for metallicity affects the mass evaluation of any star, therefore also that of Li-dip stars - in reality it is difficult to infer a causal relation between the cluster metallicity and the mass at which the Li-dip occurs.
\section{Li in the open cluster Hyades}
\label{sec_Hyades}
Our sample is composed of 233 stars with photometry from Tycho-2 and parallaxes from Hipparcos (de Bruijne {\em et al.\/} \cite{debruijne2001}). The metallicity [Fe/H] = +0.13 comes from Paulson {\em et al.\/} \cite{paulson2003}. For 67 stars of this sample, we have the Li abundances determined by spectral synthesis by Takeda {\em et al.\/} \cite{takeda2013}.
The color-magnitude diagram of the open cluster Hyades (left panel of Fig. \ref{fig_Hyades}) is constructed with a reddening $E(B - V) = 0.010$ mag and a distance modulus $(m - M)_0 =$ 3.30 mag. Models of metallicity [Fe/H] = +0.13 and masses from 0.80 to 2.25 $\mathrm{M_{\odot}}$ \ with a step of 0.01 $\mathrm{M_{\odot}}$ \ allowed to construct the isochrone of age 0.79 Gyr.
\begin{figure}
\centering
\includegraphics[width=5.5cm, height=5.5cm]{castro_m_fig7.eps}
\qquad
\includegraphics[width=5.7cm, height=5.5cm]{castro_m_fig8.eps}
\caption{\textit{Left}: Color-magnitude diagram of the open cluster Hyades. The red points represent the 67 stars of our sample for which we have Li abundance determination. \textit{Right}: Li abundance as a function of stellar mass for the Hyades.}
\label{fig_Hyades}
\end{figure}
Right panel of Fig. \ref{fig_Hyades} shows the Li abundance evolution as a function of stellar mass for the Hyades. We can see that large destruction of Li occurs very soon in lower mass stars, even if our models destroy too much Li. The masses of stars in the Li-dip of the Hyades match with the masses of stars in the Li-dip of M67 and of NGC752 with the larger metallicity.
\section{Conclusions}
\label{sec_Conclusions}
We have a good qualitative general agreement between observations of Li abundances and models isochrones in the three clusters. However, the Li-dip is not reproduced by the models. We think that the inclusion of internal gravity waves could resolve the problem, even if other extra mixing mechanism cannot be excluded. If we adopt for NGC 752 the metallicity measured through high resolution spectroscopy ([Fe/H]=0.11, Carrera \& Pancino \cite{carrera&pancino2011}), the position in mass of the Li-dip matches for the three clusters, thus suggesting that NGC752 metallicity is underestimated by photometric measurements.
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{
"timestamp": "2013-10-01T02:15:57",
"yymm": "1309",
"arxiv_id": "1309.7969",
"language": "en",
"url": "https://arxiv.org/abs/1309.7969"
}
|
"\\section*{Acknowledgements}\n\n\\vspace{0.3in}\n\\noindent\\textbf{Acknowledgements:} \nWe would l(...TRUNCATED)
| {"timestamp":"2013-10-01T02:15:47","yymm":"1309","arxiv_id":"1309.7960","language":"en","url":"https(...TRUNCATED)
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"\\section{Introduction}\\label{s:intro}\n\nRelativistic fluids are expected in accretion discs arou(...TRUNCATED)
| {"timestamp":"2013-10-01T02:15:57","yymm":"1309","arxiv_id":"1309.7968","language":"en","url":"https(...TRUNCATED)
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"\\section{Introduction and Overview}\n\\label{sec:intro}\n\n\nNeutrino oscillations, for which conv(...TRUNCATED)
| {"timestamp":"2013-11-06T02:12:56","yymm":"1309","arxiv_id":"1309.7987","language":"en","url":"https(...TRUNCATED)
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"\\section{Appendix}\\setcounter{subsection}{0}\\renewcommand{\\thesubsection}{\\Alph{subsection}}}\(...TRUNCATED)
| {"timestamp":"2013-10-01T02:16:20","yymm":"1309","arxiv_id":"1309.7993","language":"en","url":"https(...TRUNCATED)
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"\\section{Introduction}\nIn theory, barycentric interpolation at the Chebyshev points of the second(...TRUNCATED)
| {"timestamp":"2013-10-01T02:15:58","yymm":"1309","arxiv_id":"1309.7970","language":"en","url":"https(...TRUNCATED)
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"\\section*{Introduction}\n\\label{intro}\n\\setcounter{equation}{0}\n\\setcounter{footnote}{0}\n\nT(...TRUNCATED)
| {"timestamp":"2013-10-01T02:16:18","yymm":"1309","arxiv_id":"1309.7988","language":"en","url":"https(...TRUNCATED)
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"\\section{Introduction}\nSince the discovery of a non-zero value for $\\theta_{13}$~\\cite{DYB1}, D(...TRUNCATED)
| {"timestamp":"2013-10-01T02:15:47","yymm":"1309","arxiv_id":"1309.7961","language":"en","url":"https(...TRUNCATED)
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