Pion distribution amplitudes within the instanton model of QCD vacuum

Pion transition form factor for the process γγ → π at space-like values of photon momenta is calculated within the effective quark-meson model with the interaction induced by instanton exchange. The leading and next-to-leading order power asymptotics of the form factor and the relation between the light-cone pion distribution amplitudes of twists 2 and 4 and the dynamically generated quark mass are found. The pion form factor Mπ0(q 2 1 , q 2 2) for the transition process γ (q1)γ (q2) → π (p), where q1 and q2 are photon momenta, is related to fundamental properties of QCD dynamics at low and high energies. At zero photon virtualities the observed value of the width for the two-photon decay of the π0−meson Γ(π → γγ) = emπ0 64π M π (0, 0) = 7.79(56) eV, (1) is consistent with the theoretical prediction due to the chiral anomaly for π0 Mπ0 (0, 0) = (4π fπ) , (2) where fπ = 92.4 MeV is the pion weak decay constant. The existing experimental data from CELLO [1] and CLEO [2] Collaborations on the form factor Mπ0 for one photon being almost real, q 2 ≈ 0, with the virtuality of the other photon scanned up to 8 GeV , can be fitted by a monopole form factor: Mπ0(q 2 1 = −Q , q 2 = 0) ∣∣ fit = gπγγ 1 +Q/Λπ , Λπ ≃ 0.77 GeV, (3) where gπγγ = 0.275 GeV −1 is the two-photon pion decay constant. The large Q behavior of the form factor (3) is in agreement with the lowest order perturbative QCD (pQCD) prediction [3] Mπ0(q 2 1 , q 2 2) ∣∣ Q→∞ = J (2) (ω) 1 Q2 + J (4) (ω) 1 Q4 +O( αs π ) +O( 1 Q6 ), (4) where the leading (LO) and next-to-leading (NLO) order asymptotic coefficients J (ω) are expressed in terms of the light-cone pion distribution amplitudes (DA), φπ(x): J (2) (ω) = 4 3 fπ ∫ 1 0 dx φ (2) π (x) 1− ω2(2x− 1)2 , J (4) (ω) = 4 3 fπ∆ 2 ∫ 1 0 dx 1 + ω(2x− 1)] [1− ω2(2x− 1)2]2 φ π (x). (5) In the above expressions Q = −(q 1 + q 2 2) ≥ 0 is the total virtuality of photons and ω = (q 2 1 − q 2 2)/(q 2 1 + q 2 2) is the asymmetry in their distribution. The distribution amplitudes are normalized as ∫ 1 0 dxφπ(x) = 1 and the parameter ∆ 2 characterizes the scale of the NLO power corrections. The first perturbative correction to the LO term in (4) has been found in [4] and the NLO power corrections have been discussed in [5] and more recently in [6] within the collinear operator product expansion. The leading momentum power dependence of the form factor (4) is dictated by the scaling property of the pion wave function. But the coefficients of the asymptotic expansion depend crucially on the internal pion dynamics, which is parameterized by the nonperturbative pion DAs, φπ(x), defined at some normalization scale μ, with x being the fraction of the pion momentum, p, carried by a quark. At asymptotically large normalization scale μ → ∞ the DAs are determined in pQCD: φ π,as(x) = 6x(1− x), φ (4) π,as(x) = 30x (1− x). (6) However, for the description of the experimentally observable hard exclusive processes one needs to know the DAs normalized at virtuality μ ∼ 1 GeV. The aim of this letter is to calculate the pion transition form factor in the kinematical region up to moderately large Q and extract from its asymptotics the pion DAs at normalization scale typical for hadrons. The calculations carried out within the effective model with nonlocal quark-quark interaction are consistent with the chiral anomaly and result in the relations between the DAs of twists 2 and 4 and the dynamically