We study the γ γ → M + M − (M = π, K ) processes with the contributions from the two-particle twist-2 and twist-3 distribution amplitudes of pion and kaon mesons on BHL prescription in the standard hard-scattering approach. The results show that the contributions from twist-3 parts are actually not power suppressed compared with the leading-twist contributions in the low energy region. The cross sections with twist-3 corrections agree well with the experimental data in the two-photon center-of-mass energy W > 2.8 GeV and we also predict the cross section ratio, which is compatible with the experimental data from TPC and Belle.

Using the chromomagnetic interaction Hamiltonian with proper account for the SU (3) flavor symmetry breaking, we have performed a schematic study on the masses of S−wave heavy tetraquarks as bq bq (q denotes u, d, s quark). It is found that the numeral results for bu b d or bd bū with 1 + quantum number are 10612 MeV and 10683 MeV respectively, which are well compatible with the recent detected charged bottomonium-like Z b (10610) and Z b (10650). Theoretically, we also investigate the possible tetraquark states of 1 ++ and 2 + due to the charge conjugation as the potential candidates for the updating experiments.

We apply the background field method to calculate the moments of the pion two-particles twist-3 distribution amplitude (DA) φ p (ξ) in QCD sum rules. In this paper,we do not use the equation of motion for the quarks inside the pion since they are not on shell and introduce a new parameter m p 0 to be determined. We get the parameter m p 0 ≈ 1.30GeV in this approach. If assuming the expansion of φ p (ξ) in the series in Gegenbauer polynomials C 1/2 n (ξ), one can obtain its approximate expression which can be determined by its first few moments.

The decays of χ cJ → π + π − , K + K − (J = 0, 2) are discussed within the standard and modified hard scattering approach when including the contributions from twist-3 distribution amplitudes and wave functions of the light pseudoscalar meson. A model for twist-2 and twist-3 distribution amplitudes and wave functions of the pion and kaon with BHL prescription are proposed as the solution to the end-point singularities. The results show that the contributions from twist-3 parts are actually not power suppressed comparing with the leading-twist contribution. After including the effects from the transverse momentum of light meson valence-quark state and Sudakov factors, the decay widths of the χ cJ into pions or kaons are comparable with the their experimental data.The factorization form in the framework of the hard-scattering picture [1] is usually used in hadronic processes with large momentum transfer. In this picture, the full amplitude is factorized as a convolution of process independent distribution amplitudes of hadrons and process-dependent hard-scattering amplitude. However, the applicability of this approach at experimentally accessible momentum transfers, typically a few GeV, is questionable[2]. One of the reasons is the large contributions from the soft end-point regions. A modified perturbative approach, so-call modified hard-scattering approach(mHSA), has been proposed by Li and Sterman [3], where quark transverse momenta and Sudakov suppressions are taken into account. The advantage of this modified perturbative approach is the strong suppression of the soft end-point regions where the pQCD can not be applied.The exclusive charmonium decays have attracted interest for decades as they are an excellent laboratory for studying quark-gluon dynamics at relatively low energies. In the decay of P-wave charmonium χ c0,2 to a pair of pseudoscalars, one finds that the lowest Fock state, the color-singlet contribution, alone is not sufficient to accommodate the experimental data. This discrepancy provides an important arena in which to test our understanding of the boundary domain between perturbative and non-perturbative QCD. From a naive point of view, QCD radiative correction is suppressed by the factor α s (Q 2 ) and the contribution from higher Fock states, higher twist distribution amplitudes are suppressed by the factor 1/Q 2 . But this is not always the real case. In Ref. [4], the author showed that in the decay of χ c0,2 → P P (where P represents a light pseudoscalar meson.) the color-octet contribution from the higher Fock state contributes the same level as the color singlet state.To reveal the decay mechanism more clear, the systemic reanalysis on the contributions from higher twist distribution amplitudes will be significative and interested. When the amplitude of a physical process with large momentum transfer Q 2 is only related to one hadron wave function, there is a suppression for the contribution from higher Fock states and higher twist distribution amplitudes. For example, the dominating con...

Twist-3 distribution amplitudes of the pion and kaon are studied in this paper. We calculate the fist several moments for the twist-3 distribution amplitudes (φ π p,σ and 15 GeV after including the α s -corrections to the perturbative part. These moments will be helpful for constructing the twist-3 wave functions of the pion and kaon.