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{\large \bf $\eta$ Transition Form Factors from Rational Approximants
\\ }
\vspace{5mm} P.~Masjuan

\vspace{5mm}
Institut f\"ur Kernphysik, Johannes Gutenberg Universt\"at Mainz, Germany\\
\end{center}
\vspace{5mm}

The pseudoscalar transition form factor (TFF) describes the effect of the strong interaction on the $\gamma^*\gamma^* - P$ transition, where $P=\pi^0, \eta, \eta^\prime \dots$, and is represented by a function $F_{P\gamma^*\gamma^*}(q_1^2,q_2^2)$ of the photon virtualities $q_1^2$, and $q_2^2$.

From the experimental point of view, one can study such TFF from both space-like and time-like energy regimes. The time-like TFF can be accessed from a single Dalitz decay $P \to l^+l^- \gamma$ process which contains an off-shell photon with the momentum transfer $q_1^2$ and defines a $F_{P\gamma^*\gamma}(q_1^2,0)$ covering the $4m_l^2<q^2<m_P^2$ region.  The space-like TFF can be accessed in $\ee$ colliders by the two-photon-fusion reaction $e^+e^-\to e^+e^-P$. The common practice is to extract the TFF when one of the outgoing leptons is tagged and the other is not, that is, the single-tag method.
The tagged lepton emits a highly off-shell photon with the momentum transfer $q_1^2\equiv -Q^2$ and is detected,
while the other, untagged, is scattered at a small angle and its momentum transfer $q_2^2$ is near zero, i.e., 
$F_{P\gamma^*\gamma}(Q^2)\equiv F_{P\gamma^*\gamma^*}(-Q^2,0)$.

Theoretically, the limits $Q^2=0$ and $Q^2\rightarrow\infty$ are well known in terms of the axial anomaly in the chiral limit of QCD \cite{Adler:1969gk} and pQCD \cite{Lepage:1980fj}, respectively. The TFF is then calculated as a convolution of a perturbative hard-scattering amplitude and a gauge-invariant meson distribution amplitude (DA)~\cite{Mueller:1994cn} which incorporates the nonperturbative dynamics of the QCD bound-state~\cite{Lepage:1980fj}. Some model needs to be used either for the DA or the TFF itself. The discrepancy among different approaches reflects the model-dependency of that procedure. A different procedure might be, then, desirable.

We propose~\cite{EscribanoMasjuan}  to use a sequence of rational approximants called Pad\'e approximants (PA)~\cite{Baker} constructed from the Taylor expansion of the $F_{\eta\gamma^*\gamma}(Q^2)$ to fit the Space- and time-like experimental data, Refs.~\cite{SL} and~\cite{TL} resp., and obtain, in such a way, the derivatives of the $F_{\eta\gamma^*\gamma}(Q^2)$ at the origin of energies in a simple, systematic and model-independent way~\cite{Masjuan:2012wy}. Including the decays of the $\eta^{(')}\rightarrow \gamma \gamma$ in our set of data, we can systematically predict the slope and the curvature of both $\eta^{(')}$-TFFs. The low-energy parameters obtain with this method can be used to constrain the hadronic models used to account for the light-by-light scattering contribution part of the anomalous magnetic moment of the muon~\cite{g2}, rare $\eta$ decays and continuum cross section determinations in the charmonium region~\cite{inprep}.


The physical $\eta$ and $\eta'$ mesons are an admixture of the $SU(3)$ Lagrangian eignestates~\cite{Leutwyler:1997yr}. Deriving the parameters governing the mixing is a challenging task. Usually, these are determined through the use of $\eta(')\rightarrow \gamma\gamma$ decays as well as vector radiative decays into $\eta(')$ together with $\Gamma(J/\Psi \to \eta^\prime \gamma)/\Gamma(J/\Psi \to \eta \gamma)$~\cite{Leutwyler:1997yr}. However, since pQCD predicts 
that the asymptotic limit of the TFF for the $\eta(')$ is essentially given in terms of these mixing 
parameters, we use our TFF parametrization to estimate the asymptotic limit and further constrain the mixing parameters with compatible results compared to standard (but more sophisticated) determinations.




\begin{thebibliography}{99}
\setlength{\itemsep}{0cm}

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  %%CITATION = ZEPYA,C49,401;%%
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\bibitem{TL}
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  %%CITATION = ARXIV:0902.2547;%%
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  %``Determination of the eta-transition form factor in the gamma p ---> p eta ---> p gamma e+ e- reaction,''
  Phys.\ Lett.\ B {\bf 701} (2011) 562;
  %%CITATION = PHLTA,B701,562;%%
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  %%CITATION = ARXIV:1309.5648;%%

  
  %\cite{Masjuan:2012wy}
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  %%CITATION = ARXIV:1206.2549;%%
  P.~Masjuan, S.~Peris and J.~J.~Sanz-Cillero, 
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\bibitem{g2}    
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  %%CITATION = ARXIV:0902.3360;%%
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  %``Ballpark prediction for the hadronic light-by-light contribution to the muon (g-2)_{\mu},''
  arXiv:1212.0357 [hep-ph].
  %%CITATION = ARXIV:1212.0357;%%

  
  \bibitem{inprep}
P. Masjuan and P. Sanchez-Puertas, in preparation.  
  
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\bibitem{Masjuan:2013jha}
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