Pseudoscalar mesons parton distribution amplitudes

We compute all kaon and pion parton distribution amplitudes (PDAs) to twistthree and find that all of the pion’s distributions are symmetric and all of the kaon’s distributions are skewed in favour of the heavier s-quark, which are clear signals of dynamical chiral symmetry breaking (DCSB). And we can also notice that only the pseudotensor PDA can reasonably be approximated by its conformal limit. At any realistic energy scale, the pseudovector and pseudoscalar PDAs differ markedly from the corresponding functions in QCD’s conformal limit. We can use these PDAs as the inputs to study hard exclusive processes and get trustable results.


Introduction
The PDAs of pion and kaon are very crucial to study hard exclusive processes [1,2].In the previous studies, people can only use the expressions of PDAs in the QCD's conformal limit [3,4].Now people can recalculate mesons' more realistic PDAs by using a new method.With this new method [5,6], we calculate six PDAs of pion and kaon to twist-three.

Distribution amplitudes and Bethe-Salpeter wave functions
A pseudoscalar meson, P ḡ f (q), has three two-particle distribution amplitudes on the light front, they can be expressed as [7] a. e-mail: chenchen.anl@gmail.comb. e-mail: shichao0820@gmail.com We can numerically solve the coupled integral equations [8][9][10][11], namely, QCD's gap and Bethe-Salpeter equations, to get the expressions of quark propagators and the mesons' Bethe-Salpeter amplitudes.Here we solve these equations in two symmetry-preserving truncations [12]: the rainbow-ladder (RL) truncation, which is the most widely used in the calculation; and the modern DCSB-improved (DB) kernels, which can give the most realistic predictions by now.

Pseudovector
We depict our pseudovector results in Fig. 1, and compare them with the asymptotic two-particle distribution which is the asymptotic distribution in QCD's conformal limit, and with which is a model result, it is practically indistinguishable from the DSE prediction.; and long-dashed curve (red) -QCD sum rules result for the kaon from Ref. [14].We do not plot the asymptotic form because it is effectively indistinguishable from our prediction for the pion.

Pseudoscalar
We depict our pseudoscalar results in Fig. 2 and 3, and compare them with the asymptotic distribution ω asy

Pseudotensor
We depict our pseudotensor results in Fig. 4, we do not plot the asymptotic form because it is effectively indistinguishable from our prediction for the pion.. put DCSB into the quark-gluon vertex, so the the DB-PDA is more realistic.In kaon's pseudovector and pseudotensor case, we can get similar observations.It can be shown that in all PDAs, the S U(3) flavour-symmetry breaking is a 13% effect [5,15].And we can also notice that only the pseudotensor PDA can reasonably be approximated by its conformal limit.At any realistic energy scale, the pseudovector and pseudoscalar PDAs differ markedly from the corresponding functions in QCD's conformal limit.Now we have calculated pion's and kaon's all the PDAs to twist-three, which are remarkably different from their expressions in the conformal limit except for the pseudotensor case.We can use these PDAs as the inputs to study hard exclusive processes and get trustable results.

Figure 4 .
Figure 4. Pseudotensor two-particle, twist-three PDAs, computed at ζ 2 : dot-dashed curve (dark blue) -pion, υ π (u); solid curve (black)kaon in DB truncation, υ DB K (u); dashed curve (dark green) -kaon in RL truncation υ RL K (u); and long-dashed curve (red) -QCD sum rules result for the kaon from Ref.[14].We do not plot the asymptotic form because it is effectively indistinguishable from our prediction for the pion.