aa r X i v : . [ h e p - ph ] M a r Pion-Photon TDAs in the NJL Model ∗ A. Courtoy and S. Noguera Departamento de Fisica Teorica and Instituto de F´ısica Corpuscular,Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain.
November 15, 2018
Abstract
The pion-photon Transition Distribution Amplitudes (TDAs) are studied, treating the pion asa bound state in the sense of Bethe-Salpeter, in the formalism of the NJL model. The resultsobtained explicitly verify support, sum rules and polynomiality conditions. The role of PCAC ishighlighted.
Hard reactions provide important information for unveiling the structure of hadrons. The largevirtuality, Q , 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 governedby the Generalized Parton Distributions (GPDs), like the Deeply Virtual Compton Scattering, hasbeen considered. A generalization of GPDs to non-diagonal transitions has been proposed in [1]. Inparticular, the easiest case to consider is the pion-photon TDA, governing processes like π + π − → γ ∗ γ or γ ∗ π + → γπ + in the kinematical regime where the virtual photon is highly virtual but with smallmomentum transfer. At leading-twist, the vector and axial TDAs, respectively V ( x, ξ, t ) and A ( x, ξ, t ),are defined as [2] Z dz − π e ixP + z − (cid:10) γ (cid:0) p ′ (cid:1)(cid:12)(cid:12) ¯ q (cid:16) − z (cid:17) γ + τ − q (cid:16) z (cid:17) (cid:12)(cid:12) π + ( p ) (cid:11)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = i e ε ν ǫ + νρσ P ρ ∆ σ V π + ( x, ξ, t ) √ f π , (1) Z dz − π e ixP + z − (cid:10) γ (cid:0) p ′ (cid:1)(cid:12)(cid:12) ¯ q (cid:16) − z (cid:17) γ + γ τ − q (cid:16) z (cid:17) (cid:12)(cid:12) π + ( p ) (cid:11)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = e (cid:16) ~ε ⊥ · ~ ∆ ⊥ (cid:17) A π + ( x, ξ, t ) √ f π + e ( ε · ∆) 2 √ f π m π − t ǫ ( ξ ) φ (cid:18) x + ξ ξ (cid:19) , (2) where t = ∆ = ( p ′ − p ) , P = ( p + p ′ ) / ξ = ( p − p ′ ) + / P + , ǫ ( ξ ) = 1 for ξ > − ξ < f π = 93 MeV. For any four-vector v µ , we have the light-cone coordinates v ± = ( v ± v ) / √ ~v ⊥ = ( v , v ) . Finally, φ ( x ) is the pion distribution amplitude (PDA).Apart from the axial TDA A ( x, ξ, t ), the axial current Eq.(2) contains a pion pole contribution,which can be understood as a consequence of PCAC because the axial current must be coupled to thepion. This second term has been isolated in a model independent way. Therefore, all the structure ofthe incoming pion remains in A ( x, ξ, t ). The pion pole term is not a peculiarity of the pion-photonTDAs: a similar contribution would be present in the Lorentz decomposition, in terms of distributionamplitudes, of the axial current for any pair of external particles. This term is only non-vanishing inthe ERBL region, i.e. the x ∈ [ − ξ, ξ ] region, whose kinematics allow the emission or absorption of apion from the initial state, which is described through the PDA. ∗ 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) V ( x , ξ , t ) x m π =140 MeV and t=-0.5 GeV ξ =0.25 ξ =0.5 ξ =0.75 -1,0 -0,5 0,0 0,5 1,0-0,010,000,010,020,030,04 A ( x ,, t ) x =0.75 =0.25 =-0.25 =-0.75 m =140 MeV and t = - 0.5 GeV The π - γ TDAs are related to the vector and axial transition form factors through the sum rules Z − dx V π + ( x, ξ, t ) = √ f π m π F V ( t ) , Z − dx A π + ( x, ξ, t ) = √ f π m π F A ( t ) . (3)As usual, we consider that the currents present in Eqs. (1) and (2) are dominated by the handbagdiagram. The method of calculation developed in [3] 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 theproblem are then preserved. As a consequence, the correct support is obtained, i.e. x ∈ [ − , , vectorand axial TDAs obey the sum rules, Eq.(3), and the polynomiality expansion is recovered in both cases.Moreover, for the DGLAP region, we have obtained the isospin relations V ( − x, ξ, t ) = − V ( x, ξ, t ) , A ( − x, ξ, t ) = 2 A ( x, ξ, t ) , | ξ | < x < . (4)In the figures are depicted both the vector and axial TDAs, which explicit expression are given in[2], for m π = 140 MeV, t = − . and different values of ξ ranging between t/ (2 m π − t ) < ξ < ξ and we have depicted only positive values of ξ. For the axial TDA, two quite differentbehaviours are observed according to the sign of ξ . The value we numerically obtain for F V (0) is inagreement with [4], while the one we obtain for F A (0) is twice the expected value [4].Previous studies of the pion-photon TDAs have been released [5]. Since both these studies parametrizeTDAs by means of double distributions, Ref. [2] is the first study of the polynomiality property of TDAs.Moreover, in Ref. [2], 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 exclusivemeson pair production in γγ ∗ scattering [1]. In particular, a deeper study of the pion pole contributionshould allow us to give a cross-section estimate for the ππ pair case. References [1] B. Pire and L. Szymanowski, Phys. Rev. D (2005) 111501. J. P. Lansberg et al. , Phys. Rev. D (2006) 074014.[2] A. Courtoy and S. Noguera, Phys. Rev. D (2007) 094026 [arXiv:0707.3366 [hep-ph]].[3] S. Noguera, L. Theussl and V. Vento, Eur. Phys. J. A (2004) 483.[4] W. M. Yao et al. [Particle Data Group], J. Phys. G (2006) 1.[5] B. C. Tiburzi, Phys. Rev. D (2005) 094001. W. Broniowski and E. R. Arriola, Phys. Lett. B649