Limit on the pion distribution amplitude

Abstract

The pion distribution amplitude (DA) can be related to the fundamental QCD Greenʼs functions as a function of the quark self-energy and the quark-pion vertex, which in turn are associated with the pion wave function through the Bethe-Salpeter equation. Considering the extreme hard asymptotic behavior in momentum space allowed for a pseudoscalar wave function, which is limited by its normalization condition, we compute the pion DA and its second moment. From the resulting amplitude, representing the field theoretical upper limit on the DA behavior, we calculate the photon-pion transition form factor πγγ F Q ( ) * 2 . The resulting upper limit on the pion transition form factor is compared with existing data published by CLEO, BaBar and Belle Collaborations.Keywords: non-perturbative QCD, pion distribution amplitude, hadron physics (Some figures may appear in colour only in the online journal)A few years ago new data were published [1,2] for the γ γ π → * 0 process, where one of the photons is far off mass shell (large Q 2 ) and the other one is near mass shell ( ≈ Q 0 2 ). These measurements of the photon-pion transition form factor πγγ F Q ( ) * 2 , taken in the single-taggede e e e 0 reaction, were performed in a wide range of momentum transfer squared (4-40) GeV 2 ). At sufficiently high Q 2 it is expected that the standard factorization approach can be applied [3][4][5] (for a review, see [6]). The amplitude for this process at high virtuality has the form As a consequence of these experiments there were many theoretical papers speculating why the data should (or not) obey the BL limit [9][10][11][12][13][14][15][16][17][18][19]. The first attempt to explain the BaBar result can be found in [20]. Among these there were proposals claiming that the pion distribution amplitude should be modified [9][10][11]16], leading to a broader or flatter distribution in the place of the asymptotic form φas [21]. A flat DA would be consistent with the BaBar data, although a field theoretical support for such possibility is still missing. Some papers claim that other transition form factors of heavier mesons are compatible among themselves and with the saturation required by factorization theorems obtained from pQCD [13,22]. However, for heavier mesons than the pion the DA may be more peaked away from the end points [12]. A common statement in all papers is the need for more data to settle this problem.Meanwhile, the pion transition form factor is the most sensitive physical quantity to observe a non-perturbative contribution to the DA. Other quantities, for instance, like the pion form factor, may already contain a hard scattering amplitude at leading order with a soft behavior, due to the effect of extra coupling constants or gluon propagators [23]. This means that they do not lead to such a simple integral over a DA as the one shown by equation (1). As claimed in [24], we may assume that at present there is no definite conclusion on which is the asymptotic form of the pion DA, and it is possible that in the futur...

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