Twist-two matrix elements at finite and infinite volume

TL;DR

This work delivers a comprehensive one-loop HBχPT treatment of twist-two matrix elements (unpolarised, helicity, and transversity) in partially quenched and quenched QCD, across finite and infinite volumes. By matching QCD twist-two operators onto hadronic operators in SU(4|2) PQχPT (and SU(6|3)/SU(2|2) variants) and accounting for the Δ resonance, it derives explicit finite-volume corrections that are relevant for lattice QCD extrapolations, revealing typical FV shifts of 5–10% for current simulations and potentially larger effects in quenched calculations. The analysis includes forward and off-forward (GPD-related) matrix elements, highlighting how FV effects depend on whether momentum is injected in the baryon or meson lines and on the nature of the momentum transfer (space-like vs time-like). These results guide lattice practitioners in correcting for finite-volume effects and in interpreting moments of parton distributions, with appendices providing full PQχPT and quenched results for various flavor sectors. The work thus tightly integrates effective field theory with lattice QCD needs, enabling more reliable extractions of twist-two observables from finite-volume simulations.

Abstract

We present one-loop results for the forward twist-two matrix elements relevant to the unpolarised, helicity and transversity baryon structure functions, in partially-quenched (N_f=2 and N_f=2+1) heavy baryon chiral perturbation theory. The full-QCD limit can be straightforwardly obtained from these results and we also consider SU(2|2) quenched QCD. Our calculations are performed in finite volume as well as in infinite volume. We discuss features of lattice simulations and investigate finite volume effects in detail. We find that volume effects are not negligible, typically around 5--10% in current partially-quenched and full QCD calculations, and are possibly larger in quenched QCD. Extensions to the off-forward matrix elements and potential difficulties that occur there are also discussed.

Figures (7)

• Figure 1: Diagrams contributing to nucleon matrix elements of the twist-two operators. The black square corresponds to an interaction from the strong Lagrangian and the gray circle represents an insertion of the twist-two operators in Eqs. (\ref{['eq:hadron_op']})--(\ref{['eq:hadron_sigmaop']}). The thin, thick and dashed lines are 70--plet baryons, 44--plet baryons and mesons respectively. The first two diagrams represent the wave-function renormalisation and the remainder are operator renormalisations. Diagrams (e) and (f) are absent for the unpolarised, and isoscalar operator matrix elements, and diagrams in which the twist-two operator is inserted on a meson line are only present in the unpolarised case.
• Figure 2: Dependence of finite volume effects on the mass-splitting $\Delta$ in individual integrals/sums corresponding to diagrams (a)---(f) in Fig. \ref{['fig:nucleondiagrams']}. The point $m=0.3$ GeV corresponds to $m L=3.8$.
• Figure 3: Dependence of finite volume effects in double-pole contributions on the mass-splitting $\Delta$. Note that the scale here is ten times that in Fig. \ref{['fig:FV_inloopsums']}. The point $m=0.3$ GeV corresponds to $m L=3.8$.
• Figure 4: Indicative finite volume effects in SU(4$|$2) matrix elements. The results in the first row are for the isovector unpolarised (left) and helicity (right) operators and those in the second row are similarly for the isoscalar unpolarised (left) and helicity (right) operators. The third row corresponds to the transversity "isovector" $y_i=-1$ (left) and "isoscalar" $y_i=1$ (right) matrix elements. In each plot, the solid curve shows the total result, whilst the short-, medium- and long-dashed curves correspond to the individual FV effects arising from diagrams (c)--(f), diagrams (a) and (b), and diagram (i) in Fig. \ref{['fig:nucleondiagrams']}. In all of these results, we have considered a (2.5 fm)$^3$ box and set $g_A=g_1=1.3$ and $|g_{\Delta N}|=1.5$. $M_\pi=0.25$ GeV corresponds to $M_\pi L=3.2$.
• Figure 5: As in Fig. \ref{['fig:FV_generic_g1gA']} except with $g_1=-g_A$. $M_\pi=0.25$ GeV corresponds to $M_\pi L=3.2$.