Light Spectrum and Decay Constants in Full QCD with Wilson Fermions

TL;DR

This paper analyzes the light spectrum and decay constants in full QCD using Wilson fermions, providing insights into hadronic properties at a specific lattice setup.

Contribution

It offers new lattice QCD results for light spectrum and decay constants with Wilson fermions at beta=5.6 on a 16^3x32 lattice.

Findings

Results for light spectrum consistent with previous studies.

Decay constants f_pi and f_V^{-1} calculated with Wilson fermions.

Provides data relevant for understanding QCD in the non-perturbative regime.

Abstract

We present results from an analysis of the light spectrum and the decay constants f_{\pi} and f_V^{-1} in Full QCD with n_f=2 Wilson fermions at a coupling of beta=5.6 on a 16^3x32 lattice.

Tables

Table 1. Table 1: Extrapolation results with fixed sea-quark mass. Errors quoted are purely statistical

${\kappa }_{\mathrm{sea}}$	${\kappa }_C$	$a{M}_{\rho }$	$a_{\rho }^{-1}$ [GeV]	$M_N$ [GeV]	$M_{\Delta }$ [GeV]	$f_{\pi }$ [MeV]	$f_V^{-1}$
0.1560	0.16065(8)	0.359(8)	2.14(5)	1.09(7)	1.30(9)	125(9)	0.33(3)
0.1570	0.15987(6)	0.341(8)	2.23(7)	1.14(6)	1.40(10)	101(8)	0.35(2)
0.1575	0.15963(11)	0.316(10)	2.44(8)	1.09(10)	1.22(8)	118(15)	0.30(3)

Table 2. Table 2: Results from extrapolations in κ sea subscript 𝜅 sea \kappa_{\rm{sea}} .

$M_N/M_{\rho }$	1.6(2)
$M_{\Delta }/M_{\rho }$	1.8(2)
$f_{\pi }/M_{\rho }$	0.14(3)
$f_V^{-1}$	0.32(6)

Equations

$$M_{PS}^2=b{m}_q+c{m}_q^2$$

$$M_V$$

$$<0|A|\pi >Z_k{Z}_A$$

$$<0|V_i|\rho >Z_k{Z}_V$$

Full text

1 INTRODUCTION

This analysis was performed as part of the SESAM project [1] to investigate Sea quark Effects on Spectrum And Matrix-elements. Configurations were generated using the Hybrid-Monte-Carlo algorithm with the standard Wilson action at a coupling of $\beta=5.6$ on a $16^{3}\times 32$ lattice. We work at three different values of the dynamical quark mass: $\kappa_{\rm{sea}}=0.1560$, $0.1570$ and $0.1575$ corresponding to $m_{\pi}/m_{\rho}$-ratios of $0.83(1)$, $0.76(1)$ and $0.71(2)$ [2]. Our lightest quark-mass is approximately equal to the strange quark-mass. Up to now we generated about 8000 trajectories of unit-length with a time-step of $\Delta t=0.01$. After measuring the integrated autocorrelation time for the plaquette and the pion-correlator at fixed timeslice [2, 3] we decided to use configurations separated by 25 trajectories for the present spectrum analysis. Our sample consists of $100$, $160$ and $100$ configurations for our 3 sea-quark values. Since we aim at a final sample of $3\times 200$ configurations this talk should be considered as a half-time status report.

Quark propagators for the set of valence $\kappa$-values { 0.1555, 0.1560, 0.1565, 0.1570, 0.1575} were computed using the standard overrelaxed Minimal Residual Algorithm with smeared sources and local as well as smeared sinks. We used the Wuppertal gauge-invariant gaussian smearing method with $N=50$ iterations and $\alpha=4$, fitting the hadron-correlators to single-exponential functions. Throughout this analyis we neglect correlations in time and in $\kappa$. We hope to present a stable correlated analysis on our final sample.

2 CHIRAL EXTRAPOLATIONS WITH FIXED SEA-QUARK MASS

To determine the critical hopping-parameter $\kappa_{c}$ at a given value of $\kappa_{\rm{sea}}$ we fit the pseudoscalar masses according to

[TABLE]

with $m_{q}=1/2(1/\kappa-1/\kappa_{c})$ being proportional to the quark mass. We find $c$ to be significantly different from zero: a linear fit leads to a decrease of $\kappa_{c}$ by four standard deviations with respect to the quadratic fit. The resulting values of $\kappa_{c}$ are shown in table 1 together with their statistical errors. To estimate systematic errors on $\kappa_{c}$, we exclude the largest quark-mass from our data and find a change in $\kappa_{c}$ of the order of one standard deviation.

For the chiral extrapolation of vector masses we use two different parametrizations

[TABLE]

which are motivated by chiral perturbation theory. In contrast to the $\Pi$ extrapolation we find that the values obtained from 2- and 3-parameter fits agree within errors; moreover eqs. 2 and 3 yield identical results. In table 1 we quote the values for the $\rho$-masses and the corresponding lattice-spacings as obtained from the 2-parameter-fits. The masses are calculated at $\kappa_{c}$ because we find the mass-shifts induced by $\Delta\kappa=\kappa_{c}-\kappa_{\rm{light}}$ to be negligible compared to the statistical errors111In fact $\kappa_{c}$ and $\kappa_{\rm{light}}$ still agree within errors. We will discuss the light quark masses estimated from $\Delta\kappa$ in detail in a fourthcoming publication [2]. Figure 1 illustrates the variation of $m_{\rho}$ with $m_{q}$ and $\kappa_{\rm{sea}}$.

The nucleon data (see figure 2) cannot be fitted with $c=0$, but ansätze eqs. 2 and 3 do equally well. The quadratic extrapolation to $\kappa_{c}$ leads to the values $M_{N}$ quoted in table 1 together with the values for $M_{\Delta}$. With the present statistics we do not see any sea-quark dependence of the nucleon mass in physical units.

We now consider the decay-constants, $f_{\pi}$ and $f_{V}^{-1}$, defined as

[TABLE]

$A$ and $V_{i}$ are the local axial and vector currents on the lattice and $Z_{k}$, $Z_{A}$ and $Z_{V}$ the renormalization constants which connect the expectation values of equation 4 and 5 to the continuum. We determine these renormalization constants using tadpole improved perturbation theory [2, 4].

The $m_{q}$-dependence of the decay-constants is presented in figure 3. Obviously the statistical accuracy is not yet high enough to resolve a $\kappa_{\rm{sea}}$-dependence of $f_{\pi}$ and $f_{V}^{-1}$ at the chiral point (see also table 1).

3 EXTRAPOLATIONS IN $\kappa_{\rm{sea}}$

A consistent method to extrapolate to zero sea-quark mass is to use only the data-points with $\kappa_{\rm{sea}}=\kappa_{\rm{val}}$ for the extrapolation. In the case of dimensionful observables one might argue about the impact of the varying scale along the trajectory. For this reason we consider only mass-ratios as shown in figure 4. We use linear fits only. The extrapolated values, quoted in table 2, agree (within large errorbars) with those obtained from extrapolations at fixed $\kappa_{\rm{sea}}$.

4 CONCLUSIONS AND OUTLOOK

On our present statistics, uncorrelated data analyses of $t$- and $\kappa_{val}$ distributions allow for stable chiral extrapolations in valence quark mass, $m_{q}$. We find the $\rho$-meson mass to be be consistent with a linear chiral ansatz, while the remaining quantities considered exhibit nonlinear behaviour in $m_{q}$. $m_{\rho}$ varies on the level of 15 % across our range of sea quark masses. Baryon masses and decay constants so far indicate a trend but produce no compelling evidence for such variation.

Chiral extrapolations of our present data in $\kappa_{\rm{sea}}$ itself still carry too large uncertainties to be reliable. Better statistics and deeper penetration into the chiral regime are needed to improve on this situation.