We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element M(P z3, z 2 3 ) as a function of the Ioffe time ν = P z3 and the distance z3. The next step is to divide M(P z3, z 2 3 ) by the rest-frame density M(0, z 2 3 ). Our lattice calculation shows a linear exponential z3-dependence in the rest-frame function, expected from the Z(z 2 3 ) factor generated by the gauge link. Still, we observe that the ratio M(P z3, z 2 3 )/M(0, z 2 3 ) has a Gaussian-type behavior with respect to z3 for 6 values of P used in the calculation. This means that Z(z 2 3 ) factor was canceled in the ratio. When plotted as a function of ν and z3, the data are very close to z3-independent functions. This phenomenon corresponds to factorization of the x-and k ⊥ -dependence for the TMD F(x, k 2 ⊥ ). For small z3 ≤ 4a, the residual z3-dependence is explained by perturbative evolution, with αs/π = 0.1.

We report on new results on the infrared behavior of the three-gluon vertex in quenched Quantum Chromodynamics, obtained from large-volume lattice simulations. The main focus of our study is the appearance of the characteristic infrared feature known as ‘zero crossing’, the origin of which is intimately connected with the nonperturbative masslessness of the Faddeev–Popov ghost. The appearance of this effect is clearly visible in one of the two kinematic configurations analyzed, and its theoretical origin is discussed in the framework of Schwinger–Dyson equations. The effective coupling in the momentum subtraction scheme that corresponds to the three-gluon vertex is constructed, revealing the vanishing of the effective interaction at the exact location of the zero crossing.The research of J.P. and J.R-Q is supported by the Spanish MINECO under grant FPA2014-53631-C2-1-P and FPA2014-53631-C2-2-P and SEV-2014-0398, and Generalitat Valenciana under grant Prometeo II/2014/066. S. Z. acknowledges support by the Alexander von Humboldt Foundation. We thank K. Cichy, M. Creutz, O. Pene, O. Philipsen, M. Teper, J. Verbaarschot for fruitful discussions. Numerical computations have used resources of CINES and GENCI-IDRIS as well as resources at the IN2P3 computing facility in Lyon

Using lattice configurations for quantum ​​​​​chromodynamics (QCD) generated with three domain-wall fermions at a physical pion mass, we obtain a parameter-free prediction of QCD’s renormalisation-group-invariant process-independent effective charge, . Owing to the dynamical breaking of scale invariance, evident in the emergence of a gluon mass-scale, GeV, this coupling saturates at infrared momenta: . Amongst other things: is almost identical to the process-dependent (PD) effective charge defined via the Bjorken sum rule; and also that PD charge which, employed in the one-loop evolution equations, delivers agreement between pion parton distribution functions computed at the hadronic scale and experiment. The diversity of unifying roles played by suggests that it is a strong candidate for that object which represents the interaction strength in QCD at any given momentum scale; and its properties support a conclusion that QCD is a mathematically well-defined quantum field theory in four dimensions.