Novel quark smearing for hadrons with high momenta in lattice QCD

TL;DR

The paper presents momentum smearing, a phase-modulated generalization of quark smearing, to maintain strong ground-state overlaps for hadrons at high spatial momenta in lattice QCD. By centering the smeared quark wavefunctions around a finite momentum $\boldsymbol{k}$, implemented within Wuppertal smearing and extended to non-iterative approaches, the authors achieve robust plateaus for pion and nucleon energies up to $|\boldsymbol{p}| \approx 2.8$ GeV, in agreement with continuum and lattice dispersion relations. Boosted Lorentz-contracted smearing offered no clear advantage over momentum smearing. The method significantly enhances the feasibility of high-momentum calculations relevant to PDFs, TMDs, and quasi-distributions, using comparatively modest computational resources.

Abstract

Hadrons in lattice QCD are usually created employing smeared interpolators. We introduce a new quark smearing that allows us to maintain small statistical errors and good overlaps of hadronic wavefunctions with the respective ground states, also at high spatial momenta. The method is successfully tested for the pion and the nucleon at a pion mass $m_π\approx 295$ MeV and momenta as high as 2.8 GeV. We compare the results obtained to dispersion relations and suggest further optimizations.

Figures (11)

• Figure 1: Conventional smearing versus momentum smearing for the example of a Gaussian wave function in $d=1$ spatial dimensions. The momentum $k$ shifts the centre of the distribution in momentum space, resulting in an oscillatory behaviour in position space.
• Figure 2: Cross sections of the smearing density profile $\rho(\boldsymbol{x})$ Eq. \ref{['density']} in the $x_1$-$x_2$ plane. The colour encodes the phase Eq. \ref{['phase']} and the momentum smearing $\boldsymbol{k}$ parameter has the direction $(1,1,0)$. Top left: free field case. Top right: APE smeared gauge links. Bottom left: original gauge links. Bottom right: APE smeared links with an additional boost factor $\gamma=5.3$.
• Figure 3: Effective pion energies for $\boldsymbol{P}=(1,1,0)$, corresponding to $|\boldsymbol{p}|\approx 0.94\,\textmd{GeV}$ and different ratios $\zeta$, see Eq. \ref{['eq:zeta']}. The line is the expectation from the continuum dispersion relation. Symbols are shifted horizontally to enhance the legibility.
• Figure 4: Effective pion energies Eq. \ref{['effen']} for the lattice momentum $\boldsymbol{P}=(1,1,1)$, corresponding to $|\boldsymbol{p}|\approx 0.94\,\textmd{GeV}$. In the case of momentum smearing (squares), we set $\boldsymbol{K}=\zeta\boldsymbol{P}$ with $\zeta=0.8$. Solid symbols correspond to smeared-smeared, open symbols to smeared-point two-point functions. Some data points are shifted horizontally to enhance the legibility. The expectations from the continuum and lattice dispersion relations can be found in Eqs. \ref{['eq:disperse']} and \ref{['eq:dispersepi']}, respectively. Symbols are shifted horizontally for better legibility.
• Figure 5: The same as Fig. \ref{['fig:mes1']} for $\boldsymbol{P}=(1,1,1)$, corresponding to $|\boldsymbol{p}|\approx 1.88\,\textmd{GeV}$. The effective energies without momentum smearing cannot be determined, due to prohibitively large errors and non-monotonous behaviour of the central values of the respective two-point functions.