Understanding the Symmetric Mass Generation in Lattice-QCD

Abstract

Signatures of symmetric mass generation (SMG) have recently been reported in lattice QCD calculations employing staggered fermions. We discuss the general criteria for SMG, and demonstrate that these conditions are indeed met by the staggered fermion action, in particular by the continuum action derived by Lee and Sharpe. We propose possible RG flow around the SMG transition, guided by the numerical results. We also point out that the Goldstone tetraquark meson states provide a phenomenological signature of the "type-II" SMG phase.

Figures (7)

• Figure 1: Mass of the pseudoscalar state $\bar{\psi}\gamma_5 \xi_5 \psi$. In the weak-coupling phase with $\beta > \beta_c$, the mass scales as $1/L$, consistent with a massless phase. In the strong-coupling regime $\beta < \beta_c$, the pseudoscalar is gapped. The critical coupling $\beta_c$ drifts with the system size; we have shifted the curves corresponding to different lattice sizes so that they all share a common transition point $\beta_c$ corresponding to the $L \to \infty$ transition.
• Figure 2: The mass ratio $R = M_{\rm PS}/M_{i5}$ as a function of $\beta$. Here $M_{\rm PS}$ denotes the mass of the pseudoscalar meson with taste $\xi_5$, and $M_{i5}$ denotes the mass of the pseudoscalar meson with taste $\xi_i\xi_5$. As before, the curves have been shifted such that they share a common critical coupling $\beta_c$.
• Figure 3: Two possible RG diagrams for the type-I SMG are shown, motivated by numerical simulations of $N_f = 4$ flavor $\mathrm{SU}(2)$ QCD. The product $N_f N_c$ is constrained to be a multiple of 8. In the plots, $g^2$ denotes the gauge coupling, and $u$ parametrizes operators that explicitly break the chiral symmetry $\mathrm{SU}(N_f)_L \times \mathrm{SU}(N_f)_R$, such as the term in Eq. \ref{['deltaS']} and other dimension-6 operators discussed in Ref. leesharpe. We assume that the $(N_f, N_c)$) system lies within the conformal window, so that there exists an infrared fixed point (IRFP) on the $u = 0$ axis. In diagram $(a)$, there are two fixed points: an IRFP at $u = 0$ characterized by an emergent $\mathrm{SU}(N_f)_L \times \mathrm{SU}(N_f)_R$ symmetry in the infrared, and a second ultraviolet fixed point at a finite value $u^\ast$, which is identified with the SMG critical fixed point. By appropriately tuning $N_f$ and $N_c$, these two fixed points merge into one, resulting in the single-fixed-point structure depicted in diagram $(b)$. The solid red line denotes the phase boundary separating the weak-coupling conformal phase from the strong-coupling SMG phase, while the dashed green line represents the trajectory corresponding to the lattice action of staggered fermion simulations.
• Figure 4: The predicted $R = M_{{\rm PS}}/M_{i5}$ based on RG diagram Fig. \ref{['RG']}$(a)$ and Fig. \ref{['RG']}$(b)$. In RG diagram Fig. \ref{['RG']}$(a)$, since the UVFP (the SMG transition) has lower symmetry than the IRFP, the mass ratio should differ from $1$ right at the transition $\beta = \beta_c$. The functions $R(\beta)_L$ with different $L$ are expected to cross at the same point $R_c$. In RG diagram Fig. \ref{['RG']}$(b)$, there is no extra UV fixed point for the SMG transition, therefore functions $R(\beta)_L$ do not need to meet at the same point $R_c$.
• Figure 5: The attempted data collapse for $L M_{\rm PS}$ for $N_f = 4$ SU(2) QCD. For the strongly coupled phase $\beta < \beta_c$, we rescaled the horizontal axis as $|\beta - \beta_c| L^{1/\nu}$.