Novel quark smearing for hadrons with high momenta in lattice QCD

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 wave functions 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 $m_{\pi }\approx 295\mathrm{MeV}$ and momenta as high as 2.8 GeV. We compare the results obtained to dispersion relations and suggest further optimizations.

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 center of the distribution in momentum space, resulting in an oscillatory behavior in position space.

Figure 2

Cross sections of the smearing density profile $\rho (\mathit{x})$ Eq. (45) in the $x_1-x_2$ plane. The color encodes the phase Eq. (46) and the momentum smearing $\mathit{k}$ 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 $\mathit{P}=(1,1,0)$, corresponding to $|\mathit{p}|\approx 0.94\mathrm{GeV}$ and different ratios $\zeta$, see Eq. (47). The line is the expectation from the continuum dispersion relation. Symbols are shifted horizontally to enhance the legibility.

Figure 4

Effective pion energies Eq. (48) for the lattice momentum $\mathit{P}=(1,1,1)$, corresponding to $|\mathit{p}|\approx 0.94\mathrm{GeV}$. In the case of momentum smearing (squares), we set $\mathit{K}=\zeta \mathit{P}$ 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. (39) and (40), respectively. Symbols are shifted horizontally for better legibility.

Figure 5

The same as Fig. 4 for $\mathit{P}=(1,1,1)$, corresponding to $|\mathit{p}|\approx 1.88\mathrm{GeV}$. The effective energies without momentum smearing cannot be determined, due to prohibitively large errors and nonmonotonous behavior of the central values of the respective two-point functions.

Figure 6

The same as Fig. 4 for the nucleon and $\zeta =0.5$. The horizontal lines correspond to the expectations from the continuum and lattice dispersion relations Eqs. (39) and (41), respectively.

Figure 7

The same as Fig. 6 for $\mathit{P}=(2,2,2)$ ($\zeta =0.4$) and $\mathit{P}=(3,3,3)$ ($\zeta =0.44$), corresponding to $|\mathit{p}|\approx 1.88\mathrm{GeV}$ and $|\mathit{p}|\approx 2.82\mathrm{GeV}$, respectively. Again, the conventional Wuppertal smearing data are too noisy to allow for the extraction of effective energies.

Figure 8

The nucleon smeared-smeared effective energy with one and the same momentum smearing $\mathit{K}=(0.8,0.8,0.8)$, optimized for $\mathit{P}=(2,2,2)$, but for momenta pointing into different directions. Symbols are shifted horizontally for better legibility.

Figure 9

Effective smeared-smeared pion energies for $\mathit{P}=(3,0,0)$, corresponding to $|\mathit{p}|\approx 1.63\mathrm{GeV}$: momentum smearing with the optimized parameter $\zeta =0.8$ and boosted momentum smearing Eq. (a6) for $\gamma \approx 5.6$ with $\zeta =0.5$, 0.8, 0. Symbols are shifted horizontally for better legibility.

Figure 10

Pion energies for different lattice momenta. in comparison to the continuum (solid curve) and lattice (points connected by dotted lines) dispersion relations Eqs. (39) and (40).

Figure 11

Nucleon energies for different lattice momenta. in comparison to the continuum (solid curve) and lattice (points connected by dotted lines) dispersion relations Eqs. (39) and (41).