The pion and kaon $\langle x^3 \rangle$ from lattice QCD and PDF reconstruction from Mellin moments

Abstract

We present a calculation of the pion and kaon Mellin moment x 3 extracted directly in lattice QCD using a three-derivative local operator. We use one ensemble of gauge configurations with two degenerate light, a strange and a charm quark (N f = 2 + 1 + 1) of maximally twisted mass fermions with clover improvement. The ensemble reproduces a pion mass ∼ 260 MeV, and a kaon mass ∼ 530 MeV. Excited-states contamination is evaluated using four values of the source-sink time separation within the range of 1.12 − 1.67 fm. We use an operator that is free of mixing, and apply a multiplicative renormalization function calculated non-perturbatively. Our results are converted to the MS scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are x 3 u + π = 0.024( 18)stat(2)syst, x 3 u + K = 0.035( 6)stat(3)syst, and x 3 s + K = 0.075( 5)stat(1)syst, where the systematic error is the uncertainty due to excited state contamination.We combine x 3 with the two lower moments, namely x and x 2 , to obtain the ratios x 3 / x and x 3 / x 2 , as well as x 3 u + K / x 3 u + π and x 3 u + K / x 3 u + π . In addition, we reconstruct the xdependence of the pion and kaon PDFs via 2-and 3-parameter fits to our results. We find that the reconstruction is feasible and that our lattice data favor a large x-dependence that falls as (1 − x) 2 for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with 4 ≤ n ≤ 6. Finally, we compare the pion and kaon PDFs, as well as the ratios of their moments, to address the effect of SU(3) flavor symmetry breaking.

The scalar, vector and tensor form factors for the pion and kaon from lattice QCD

“…Such effects have been observed in nature, for example, in the charge radii of π ± and K ± , as well as in π 0 and K 0 . We have previously investigated such effects in the Mellin moments of the pion and kaon PDFs and their reconstructed PDFs [57]. In this work, we draw conclusions on the SU(3) flavor symmetry breaking using the scalar, vector and tensor form factors.…”

“…We obtain a good signal up to Q 2 ∼ 2.5 GeV 2 , which corresponds to a ratio |Q max |/E 0 ∼ 2, similar to the ratio in the rest frame. As we have done in our previous work on the pion and kaon Mellin moments x , x 2 , and x 3 [56,57], we focus on four values of the source-sink time separation, that is t s /a = 12, 14, 16, 18. We have shown that at Q 2 = 0 a two-state fit on these separations can eliminate excited states.…”

“…We use momentum boost of the form p = 2πn /L with n = (±1, ±1, ±1), which results in an increase of the computational cost by a factor of eight. For Q 2 = 0 the eight combinations can be averaged, as we have done in the calculation for the Mellin moments x , x 2 , and x 3 [56,57]. However, this is not the case for the form factors because the various p do not correspond to the same value of Q 2 , as can be seen in Eq.…”

“…(11). The choice of the momentum boost is such that it can be used to extract matrix elements with up to three-covariant derivative operators [57] avoiding any mixing under renormalization. The statistics used for each frame and each value of the sourse-sink time separation, t s , is given in Tab.…”

Mellin moments of spin dependent and independent PDFs of the pion and rho meson

“…The study of mesonic structure using lattice QCD has, by now, a history of over three decades. While earlier simulations [18,[26][27][28][29][30][31] used quenched fermion representations, more recent simulations [32][33][34][35][36][37] use, for example, (clover-improved) dynamical Wilson fermions, where the fermion determinant, and thus the quark sea, is taken into account. However, what all these studies have in common is that they neglect disconnected contributions, because the latter are notoriously difficult to calculate and usually come with a large statistical error.…”