First Physics Results at the Physical Pion Mass from $N_f = 2$ Wilson Twisted Mass Fermions at Maximal Twist

TL;DR

The study demonstrates first physics results at the physical pion mass using $N_f=2$ Wilson twisted mass fermions with a clover term, achieving automatic $O(a^2)$ improvement at maximal twist and suppressed isospin breaking. By computing meson and baryon observables, quark masses, and lepton hadronic vacuum polarization contributions at $a\approx 0.09$ fm, the work shows remarkable agreement with phenomenology despite using a single lattice spacing and volume. The results validate the viability of physical-point simulations with this action, laying groundwork for precise continuum extrapolations and future $N_f=2+1+1$ studies with controlled systematic uncertainties. This framework promises accurate inputs for heavy-flavor physics, hadron structure, and tests of the Standard Model, contingent on expanding to finer lattices and larger volumes.

Abstract

We present physics results from simulations of QCD using $N_f = 2$ dynamical Wilson twisted mass fermions at the physical value of the pion mass. These simulations were enabled by the addition of the clover term to the twisted mass quark action. We show evidence that compared to previous simulations without this term, the pion mass splitting due to isospin breaking is almost completely eliminated. Using this new action, we compute the masses and decay constants of pseudoscalar mesons involving the dynamical up and down as well as valence strange and charm quarks at one value of the lattice spacing, $a \approx 0.09$ fm. Further, we determine renormalized quark masses as well as their scale-independent ratios, in excellent agreement with other lattice determinations in the continuum limit. In the baryon sector, we show that the nucleon mass is compatible with its physical value and that the masses of the $Δ$ baryons do not show any sign of isospin breaking. Finally, we compute the electron, muon and tau lepton anomalous magnetic moments and show the results to be consistent with extrapolations of older ETMC data to the continuum and physical pion mass limits. We mostly find remarkably good agreement with phenomenology, even though we cannot take the continuum and thermodynamic limits.

Figures (17)

• Figure 1: Charged and neutral pion mass splittings (a): $a^2 (M^2_{\pi^\pm} - M^2_{\pi^{(0,c)}})$ and (b): $a^2 ( M^2_{\pi^\pm} - M^2_{\pi^0})$ as a function of the bare light quark mass $a \mu_\ell$.
• Figure 2: (a) Relative mass differences $\Delta_M$ for the $\Delta$ baryons as a function of $(a/r_0)^2$ for $N_f=2+1+1$ ensembles (open circles for $\beta=1.90$, squares for $\beta=1.95$ and diamonds for $\beta=2.10$) as well as for the ensemble at the physical value of the pion mass (filled red triangle). Red symbols represent the lightest pion mass, then blue, green and violet for increasing pion mass values for each lattice spacing. The data points have been slightly displaced horizontally for legibility. (b) Effective masses as a function of $t/a$ of the nucleon, kaon and pion for the physical pion mass ensemble cA2.09.48.
• Figure 3: (a)$r_0M_\pi^2/f_\pi$ as a function of $(r_0M_\pi)^2$ comparing $N_f=2$ results w/o clover term Baron:2009wt with the new results presented in this paper. The line is a NLO $\chi$PT fit to the data as explained in the text. (b) Ratio of the nucleon mass to the pion mass as a function of the pion mass squared in units of $r_0$. We show data for $N_f=2$ w/o clover term, $N_f=2+1+1$ and the new physical point result.
• Figure 4: Comparison of the chiral extrapolation of the light quark contributions to the three lepton anomalous magnetic moments obtained from $N_f=2+1+1$ simulations to the values obtained with the standard definition Eq. \ref{['eq:amudef']} at the physical value of the pion mass (black square). The dark green diamonds correspond to $a=0.086\mathrm{ fm}$ and $L=2.8\mathrm{ fm}$ and the circles to $a=0.078\mathrm{ fm}$, the violet one stands for $L=1.9\mathrm{ fm}$, the blue ones for $L=2.5\mathrm{ fm}$, and the pink for $L=3.7\mathrm{ fm}$. The orange triangle shows the value obtained for $a=0.061\mathrm{ fm}$ and $L=1.9\mathrm{ fm}$ and the light green triangle denotes $a=0.061\mathrm{ fm}$ and $L=2.9\mathrm{ fm}$.
• Figure 5: Ratios of lattice results and phenomenological values of the quantities in the legend with lattice decay constants computed via the continuum definition. For dimensional quantities, the inner error bar combines the statistical and systematic errors in quadrature while the outer error bar stems from the estimate of the lattice spacing from gluonic scales. The red bands show the phenomenological uncertainty on $Q_\mathrm{phys}$ separately (the respective experimental errors on $M_N$ and $M_{D_s}$ are too small to be visible). The dotted and dashed lines indicate per-mille and per-cent deviations from $1.0$ respectively.