pi NN and pseudoscalar form factors from lattice QCD.

The πNN form factor gπNN (q ) is obtained from a quenched lattice QCD calculation of the pseudoscalar form factor gP (q ) of the proton with pion pole dominance. We find that gπNN (q ) fitted with the monopole form agrees well with the Goldberger-Treiman relation and is much preferred over the dipole form. The monopole mass is determined to be 0.75 ± 0.14GeV which shows that gπNN (q ) is rather soft. The extrapolated πN coupling constant gπNN = 12.7±2.4 is quite consistent with the phenomenological values. We also compare gπNN (q ) with the axial form factor gA(q ) to check the pion dominance in the induced pseudoscalar form factor hA(q ) vis à vis chiral Ward identity. PACS numbers: 12.38.Gc, 14.20.Dh, 13.75.Gx, 13.60.-r The πNN form factor gπNN(q ) is a fundamental quantity in low-energy pionnucleon and nucleon-nucleon dynamics. Many dynamical issues like πN elastic and inelastic scattering, NN potential, three-body force (triton and He binding energies), pion photoproduction and electroproduction all depend on it. Similarly, the pseudoscalar form factor is important in testing low-energy theorems, the chiral Ward identity and the understanding of the explicit breaking of the chiral symmetry. Yet, compared with the electromagnetic form factors and the isovector axial form factor of the nucleon, the pseudoscalar form factor gP (q ) and the πNN form factor gπNN(q ) are poorly known either experimentally or theoretically. Notwithstanding decades of interest and numerous work, the shape and slope of gπNN(q ) remain illusive and unsettled. Upon parametrizing gπNN(q ) in the monopole form gπNN(q ) = gπNN Λ2πNN −m 2 π Λ2πNN − q 2 (1) with gπNN ≡ gπNN(m 2 π), the uncertainty in the parametrized monopole mass ΛπNN can be as large as a factor 2 or 3. For the sake of having a sufficiently strong tensor force to reproduce the asymptotic Dto Swave ratio and the quadrupole moment in the deuteron, ΛπNN is shown to be greater than 1GeV [1]. Consequently, ΛπNN in the realistic NN potentials are typically fitted with large ΛπNN (e.g ΛπNN ranges from 1.3GeV [2] to 2.3 – 2.5GeV [3]). On the other hand, arguments based on resolving the discrepancy of the Goldberger-Treiman relation [4] and the discrepancy between the ppπ and pnπ couplings [5] suggest a much softer gπNN(q ) with ΛπNN around 0.8GeV. Furthermore, hadronic models of baryons with meson clouds like the skyrmion typically have a rather soft form factor (i.e. ΛπNN∼ 0.6GeV) [6] due to its large pion cloud and such a small ΛπNN is needed for the high energy elastic pp scattering [7]. In view of the large uncertainty in gπNN(q ), it is high time to study it with a lattice QCD calculation. Since our recent calculations of the nucleon axial and electromagnetic form factors are within 10% of the experimental results [8, 9], a prediction of gπNN(q ) with a similar accuracy should be enough to adjudicate on the controversy over the πNN form factor. In this letter, we extend our lattice calculation to the proton pseudoscalar form factor for a range of light quark masses. gπNN(q ) is obtained by considering the pion pole dominance in gP (q ) when the latter is extrapolated to the quark mass which corresponds to the physical pion mass. In analogy with the study of the electromagnetic and axial form factors [8, 9] of the nucleon, we calculate the following twoand three-point functions for the proton G pp (t, ~ p) = ∑ ~x e 〈0|T (χ(x)χ̄(0)|0〉 (2) G pPp(tf , ~p, t, ~q) = ∑ ~xf ,~x e p·~xf+i~ 〈0|T (χ(xf )P (x)χ̄ (0))|0〉, (3)