[Paper Review] Detecting the flavor content of the vacuum using the Dirac operator spectrum

Jian Liang, Andrei Alexandru+4 more|arXiv (Cornell University)|Feb 10, 2021

Experimental and Theoretical Physics Studies1 references4 citations

TL;DR

This paper uses the overlap Dirac operator spectrum on 2+1 flavor domain wall fermion ensembles to precisely determine the chiral condensate and quark masses in QCD. It resolves the flavor content of the vacuum by fitting the spectral density to NLO chiral perturbation theory, yielding Σ = (260.3(0.7)(1.3)(0.7)(0.8) MeV)³ and Σ₀ = (232.6(0.9)(1.2)(0.7)(0.8) MeV)³ at 2 GeV in the $ar{ extrm{MS}}$ scheme, with 20% uncertainties on light and strange quark masses.

ABSTRACT

We compute the overlap Dirac spectrum on three gauge ensembles generated using $2+1$-flavor domain wall fermions. The three ensembles have different lattice spacings and two of them have quark masses tuned to the physical point. The spectral density is determined up to $λ\sim$100 MeV with subpercentage statistical uncertainty. We find that the density is close to a constant below $λ\sim$ 20 MeV as predicted by chiral perturbative theory ($χ$PT), and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order $χ$PT form and using the non-perturbative RI/MOM renormalization, the $ m SU(2)$ (keeping the strange quark mass at the physical point) and $ m SU(3)$ chiral condensates at $\overline{ extrm{MS}}$ 2 GeV are determined to be $Σ=(265.4(0.5)(4.2)\ extrm{MeV})^3$ and $Σ_0=(234.3(0.5)(25.8)\ extrm{MeV})^3$, respectively. The pion decay constants are also determined to be $F=84.1(1.9)(8.0)$ and $F_0=58.6(0.5)(10.0)$ MeV. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant $L_6$. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with {$\sim10\%-20\%$} statistical uncertainties using the Dirac spectral density.

Motivation & Objective

• To determine the chiral condensate and quark masses in the $N_f=2$ and $N_f=3$ chiral limits using the Dirac spectrum.
• To resolve the flavor content of the QCD vacuum by analyzing the spectral density of the overlap Dirac operator.
• To extract light and strange quark masses with ~20% uncertainties by fitting the spectral density to NLO chiral perturbation theory.
• To achieve sub-percentage statistical uncertainty in the spectral density up to λ ~ 100 MeV on physical-mass ensembles.
• To compare results from different renormalization schemes and assess systematic uncertainties from higher-order corrections.

Proposed method

• Computes the overlap Dirac spectrum on three 2+1 flavor domain wall fermion ensembles with different lattice spacings and physical quark masses.
• Uses non-perturbative RI/MOM scheme for renormalization of the chiral condensate to $ar{ extrm{MS}}$ at 2 GeV.
• Employs NLO partially quenched chiral perturbation theory to model the spectral density $\rho(\lambda)$ as a function of quark masses.
• Fits the measured spectral density to the NLO $χ$PT form to extract $\Sigma$, $\Sigma_0$, $F_0$, $m_l$, and $m_s$.
• Performs continuum extrapolation and accounts for statistical, renormalization, and lattice spacing uncertainties.
• Uses stochastic eigensolver techniques to compute low-lying eigenvalues of the Dirac operator with high precision.

Experimental results

Research questions

• RQ1 Can the spectral density of the overlap Dirac operator resolve the flavor content of the QCD vacuum in 2+1 flavor QCD?
• RQ2 To what extent does the strange quark mass influence the spectral density $\rho(\lambda)$ in the range $\lambda \in (20, 100)$ MeV?
• RQ3 Can the chiral condensate in the $N_f=2$ and $N_f=3$ chiral limits be extracted with sub-percentage statistical uncertainty?
• RQ4 How accurately can the light and strange quark masses be determined from the Dirac spectrum using NLO $χ$PT?
• RQ5 What is the impact of higher-order corrections (NNLO) on the determination of the light quark mass from $\rho(\lambda)$?

Key findings

• The spectral density $\rho(\lambda)$ is nearly constant below $\lambda \leq 20$ MeV, with a 10% enhancement over the 2-flavor chiral limit, consistent with NLO $χ$PT.
• The spectral density increases linearly above $\lambda \sim 20$ MeV due to the finite strange quark mass, confirming its influence on the vacuum structure.
• The chiral condensate in the $N_f=2$ chiral limit is determined as $\Sigma = (260.3(0.7)(1.3)(0.7)(0.8)\ \textrm{MeV})^3$ at $\overline{\textrm{MS}}$ 2 GeV, with total uncertainty ~1.0%.
• The chiral condensate in the $N_f=3$ chiral limit is $\Sigma_0 = (232.6(0.9)(1.2)(0.7)(0.8)\ \textrm{MeV})^3$, with similar total uncertainty.
• The ratio $\Sigma / \Sigma_0 = 1.40(2)(2)$ is significantly larger than 1, reflecting the effect of the strange quark mass in the $N_f=3$ limit.
• The light quark mass is extracted as $m_l = 2.95(1)(4)(5)(1)$ MeV, which is 12% lower than the FLAG average, likely due to missing NNLO effects in the fit.