---
abstract: 'The pion-photon Transition Distribution Amplitudes (TDAs) are studied, treating the pion as a bound state in the sense of Bethe-Salpeter, in the formalism of the NJL model. The results obtained explicitly verify support, sum rules and polynomiality conditions. The role of PCAC is highlighted.'
author:
- |
    A. Courtoy $^{1}$ and S. Noguera $^{1}$\
    $^{1} $ [Departamento de Fisica Teorica and Instituto de Física Corpuscular,]{}\
    [Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain.]{}
title: ' Pion-Photon TDAs in the NJL Model [^1] '
---

Hard reactions provide important information for unveiling the structure of hadrons. The large virtuality, $Q^{2}$, involved in the processes allows the factorization of the hard (perturbative) and soft (non-perturbative) contributions in their amplitudes. In recent years a large variety of processes governed by the Generalized Parton Distributions (GPDs), like the Deeply Virtual Compton Scattering, has been considered. A generalization of GPDs to non-diagonal transitions has been proposed in [@Pire:2004ie]. In particular, the easiest case to consider is the pion-photon TDA, governing processes like $\pi^{+}\pi^{-}\rightarrow\gamma^{\ast}\gamma$ or $\gamma
^{\ast}\pi^{+}\rightarrow\gamma\pi^{+}$ in the kinematical regime where the virtual photon is highly virtual but with small momentum transfer. At leading-twist, the vector and axial TDAs, respectively $V(x,\xi,t)$ and $A(x,\xi,t)$, are defined as [@Courtoy:2007vy] [$$\begin{aligned}
\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\left.  \left\langle
\gamma\left( p^{\prime}\right)  \right\vert \bar{q}\left(
-\frac{z}{2}\right)  \gamma ^{+}\hspace*{-0.05cm}\tau^{-}q\left(
\frac{z}{2}\right)  \left\vert \pi^+\left(  p\right)
\right\rangle \right\vert _{z^{+}=z^{\bot}=0}  &
=i\,e\,\varepsilon_{\nu}\,\epsilon^{+\nu\rho\sigma}\,P_{\rho}\,\Delta_{\sigma
}\,\frac{V^{\pi^{+}}\left(  x,\xi,t\right)  }{\sqrt{2}f_{\pi}}~,\label{vectcurr}\\
\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\left.  \left\langle
\gamma\left( p^{\prime}\right)  \right\vert \bar{q}\left(
-\frac{z}{2}\right)  \gamma
^{+}\hspace*{-0.05cm}\gamma_{5}\tau^{-}q\left(  \frac{z}{2}\right)
\left\vert \pi^+\left(  p\right)  \right\rangle \right\vert
_{z^{+}=z^{\bot}=0}  & =e\,\left(
\vec{\varepsilon}^{\bot}\cdot\vec{\Delta}^{\bot}\right)
\frac{A^{\pi^{+}}\left(  x,\xi,t\right)  }{\sqrt{2}f_{\pi}}\nonumber\\
&  +e\,\left(  \varepsilon\cdot\Delta\right)
\frac{2\sqrt{2}f_{\pi}}{m_{\pi }^{2}-t}~\epsilon\left(  \xi\right)
~\phi\left(  \frac{x+\xi}{2\xi}\right)
\quad, \label{axcurr}$$ ]{}where $t=\Delta^{2}=(p^{\prime}-p)^{2}$, $P=\left(
p+p^{\prime}\right)  /2$, $\xi=\left(  p-p^{\prime}\right)
^{+}/2P^{+}$, $\epsilon\left(  \xi\right) =1$ for $\xi>0$ and $-1$ for $\xi<0$ and where $f_{\pi}=93$ MeV. For any four-vector $v^{\mu},$ we have the light-cone coordinates $v^{\pm}=\left(
v^{0}\pm v^{3}\right)  /\sqrt{2}$ and the transverse components $\vec{v}^{\bot}=\left(  v^{1},v^{2}\right)  .$ Finally, $\phi\left(
x\right)  $ is the pion distribution amplitude (PDA).

Apart from the axial TDA $A(x,\xi,t)$, the axial current Eq.(\[axcurr\]) contains a pion pole contribution, which can be understood as a consequence of PCAC because the axial current must be coupled to the pion. This second term has been isolated in a model independent way. Therefore, all the structure of the incoming pion remains in $A\left(  x,\xi,t\right)  $. The pion pole term is not a peculiarity of the pion-photon TDAs: a similar contribution would be present in the Lorentz decomposition, in terms of distribution amplitudes, of the axial current for any pair of external particles. This term is only non-vanishing in the ERBL region, i.e. the $x\in\left[  -\xi,\xi\right]  $ region, whose kinematics allow the emission or absorption of a pion from the initial state, which is described through the PDA.

The $\pi$-$\gamma$ TDAs are related to the vector and axial transition form factors through the sum rules$$\int_{-1}^{1}dx~V^{\pi^{+}}\left(  x,\xi,t\right)  =\frac{\sqrt{2}f_{\pi}}{m_{\pi}}F_{V}\left(  t\right)
~,~~~~~\int_{-1}^{1}dx~A^{\pi^{+}}\left( x,\xi,t\right)
=\frac{\sqrt{2}f_{\pi}}{m_{\pi}}F_{A}\left(  t\right)  \quad.
\label{sumerule}$$

As usual, we consider that the currents present in Eqs. (\[vectcurr\]) and (\[axcurr\]) are dominated by the handbag diagram. The method of calculation developed in [@Theussl:2002xp] is here applied. The pion is treated as a bound-state in a fully covariant manner using the Bethe-Salpeter equation and solving it in the NJL model. Gauge invariance is ensured by using the Pauli-Villars regularization scheme. All the invariances of the problem are then preserved. As a consequence, the correct support is obtained, i.e. $x\in\lbrack-1,1],$ vector and axial TDAs obey the sum rules, Eq.(\[sumerule\]), and the polynomiality expansion is recovered in both cases. Moreover, for the DGLAP region, we have obtained the isospin relations $$V\left(  -x,\xi,t\right)  =-2V\left(  x,\xi,t\right)  ,~~~~A\left(
-x,\xi,t\right)  =2A\left(  x,\xi,t\right)  ,~~~~~~~~~~~\left\vert
\xi\right\vert <x<1\quad.\label{isospin}$$

In the figures are depicted both the vector and axial TDAs, which explicit expression are given in [@Courtoy:2007vy], for $m_{\pi}=140$ MeV, $t=-0.5$ GeV$^{2}$ and different values of $\xi$ ranging between $t/(2m_{\pi}^{2}-t)<\xi<1$: the process here does not constrain the skewness variable to be positive. The vector TDA is mainly a function of $\xi^{2}$ and we have depicted only positive values of $\xi.$ For the axial TDA, two quite different behaviours are observed according to the sign of $\xi$. The value we numerically obtain for $F_{V}\left(  0\right)  $ is in agreement with [@Yao:2006px], while the one we obtain for $F_{A}\left(
0\right)  $ is twice the expected value [@Yao:2006px].

![image](t5mpi140b){width="6.5cm"}

![image](Fig2b.eps){width="6.5cm"}

Previous studies of the pion-photon TDAs have been released [@Tiburzi:2005nj]. Since both these studies parametrize TDAs by means of double distributions, Ref. [@Courtoy:2007vy] is the first study of the polynomiality property of TDAs. Moreover, in Ref. [@Courtoy:2007vy], the support, sum rules and polynomiality expansion are results (and not inputs) of the calculation. The study of TDAs should lead to interesting estimates of cross-sections for exclusive meson pair production in $\gamma\gamma^{\ast}$ scattering [@Pire:2004ie]. In particular, a deeper study of the pion pole contribution should allow us to give a cross-section estimate for the $\pi\pi$ pair case.

[9]{} -2pt

B. Pire and L. Szymanowski, Phys. Rev. D **71** (2005) 111501. J. P. Lansberg *et al.*, Phys. Rev. D **73** (2006) 074014.

A. Courtoy and S. Noguera, Phys. Rev.  D [**76**]{} (2007) 094026 \[arXiv:0707.3366 \[hep-ph\]\]. S. Noguera, L. Theussl and V. Vento, Eur. Phys. J. A **20** (2004) 483.

W. M. Yao *et al.* \[Particle Data Group\], J. Phys. G **33** (2006) 1.

B. C. Tiburzi, Phys. Rev. D **72** (2005) 094001. W. Broniowski and E. R. Arriola, Phys. Lett. B **649** (2007) 49.

[^1]: This work has been supported by the 6th Framework Program of the European Commission No. 506078 and MEC (FPA 2007-65748-C02-01 and AP2005-5331)
