Low energy constants of $SU(2)$ partially quenched chiral perturbation theory from $N_f=2+1$ domain wall QCD

We have performed fits of the pseudoscalar masses and decay constants, from a variety of the RBC-UKQCD Collaboration’s domain wall fermion ensembles, to $SU(2)$ partially quenched chiral perturbation theory at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). We report values for nine NLO and eight linearly independent combinations of NNLO partially quenched low-energy constants, which we compare to other lattice and phenomenological determinations. We discuss the size of successive terms in the chiral expansion and use our large set of low-energy constants to make predictions for mass splittings due to QCD isospin-breaking effects and the S-wave $\pi \pi$ scattering lengths. We conclude that, for the range of pseudoscalar masses explored in this work, $115\mathrm{MeV}\lesssim m_{\mathrm{PS}}\lesssim 430\mathrm{MeV}$, the NNLO $SU(2)$ expansion is quite robust and can fit lattice data with percent-scale accuracy.

Figure 1

Stacked histograms of the signed deviation of the data from the fit in units of the standard deviation.

Figure 2

Chiral extrapolation of unitary $m_{\pi }^2$ data. The fit has been used to correct each data point from the simulated strange quark mass to the physical strange quark mass, as well as to take the infinite volume limit. Filled symbols correspond to sub-ensembles which were included in the fit, and open symbols correspond to sub-ensembles which were excluded from the fit based on the pion mass cut. The dashed vertical line corresponds to the heaviest unitary point included in the fit. “Physical point” is the prediction for the physical pion mass obtained by interpolating the fit to $m_l^{\mathrm{phys}}$.

Figure 3

Chiral extrapolation of unitary $f_{\pi }$ data. The fit has been used to correct each data point from the simulated strange quark mass to the physical strange quark mass, as well as to take the infinite volume and continuum limits. Filled symbols correspond to sub-ensembles which were included in the fit, and open symbols correspond to sub-ensembles which were excluded from the fit based on the pion mass cut. The dashed vertical line corresponds to the heaviest unitary point included in the fit. “Physical point” is the prediction for the physical pion decay constant obtained by interpolating the fit to $m_l^{\mathrm{phys}}$.

Figure 4

Decomposition of the $SU(2)$ chiral expansion into LO, NLO, and NNLO terms, normalized by LO. The pion mass (top) and pion decay constant (bottom) are plotted as a function of the light quark mass, using the LECs obtained from a fit with a pion mass cut of 370 MeV (left) and 450 MeV (right). The vertical dashed line corresponds to the heaviest unitary point included in the fit, and the horizontal dotted line marks zero.

Figure 5

Relative sizes of the LO, NLO, and NNLO terms in the $SU(2)$ chiral expansion for $m_{\pi }^2$ (left) and $f_{\pi }$ (right) using the LECs obtained from a fit with a pion mass cut of 450 MeV. The vertical dashed line corresponds to the heaviest unitary point included in the fit.

Figure 6

Leading-order $SU(2)$ ChPT LECs compared to the 2013 FLAG lattice averages.

Figure 7

Next-to-leading-order $SU(2)$ ChPT LECs compared to the 2013 FLAG lattice averages and two phenomenological determinations.

Figure 8

Percent deviation between fits and data. We plot stacked histograms of the quantity $\mathrm{\Delta }\equiv 200\times (Y-Y^{\text{fit}})/(Y+Y^{\text{fit}})$.

Figure 9

Molecular dynamics evolution of the plaquette, chiral and pseudoscalar condensates, pion propagator at $t/a=20$, square of the topological charge, and clover discretized action density computed at the Wilson flow times $t_0$ and $w_0^2$ as a function of MD time on the 32ID-M1 ensemble. The first three quantities were computed every MD time step as part of the evolution. The topological charge and Wilson flow scales were computed every 10 and 20 MD time steps, respectively, after the ensemble was thermalized. The dashed vertical lines mark the range of MD times used to perform calculations of the spectrum.

Figure 10

Molecular dynamics evolution of the plaquette, chiral and pseudoscalar condensates, pion propagator at $t/a=20$, square of the topological charge, and clover discretized action density computed at the Wilson flow times $t_0$ and $w_0^2$ as a function of MD time on the 32ID-M2 ensemble. The first three quantities were computed every MD time step as part of the evolution. The topological charge and Wilson flow scales were computed every 2 and 40 MD time steps, respectively, after the ensemble was thermalized. The dashed vertical lines mark the range of MD times used to perform calculations of the spectrum.

Figure 11

Integrated autocorrelation times for the observables plotted in Figs. 9 and 10.

Figure 12

The residual mass, from Eq. (c3), on the 32ID-M1 (left) and 32ID-M2 (right) ensembles.

Figure 13

Light-light pseudoscalar mass on the 32ID-M1 (left) and 32ID-M2 (right) ensembles. We simultaneously fit a common mass $m_{ll}$ to the three correlators $⟨P{P}^{LW}⟩$, $⟨P{P}^{WW}⟩$, and $⟨A{P}^{LW}⟩$ on each ensemble.

Figure 14

Heavy-light pseudoscalar mass on the 32ID-M1 (left) and 32ID-M2 (right) ensembles. We simultaneously fit a common mass $m_{lh}$ to the three correlators $⟨P{P}^{LW}⟩$, $⟨P{P}^{WW}⟩$, and $⟨A{P}^{LW}⟩$ on each ensemble.

Figure 15

The vector current renormalization coefficient on the 32ID-M1 (left) and 32ID-M2 (right) ensembles. In the upper plot, we show the dependence of the ratio (c6) on the source-sink separation: the point plotted for each separation is evaluated at the midpoint $t=|t_{\mathrm{src}}-t_{\mathrm{snk}}|/2a$. Points which were included in the fit are marked in red. In the lower plot, we show an example of the fit to $Z_V$ overlaying the ratio (c6) for one of the source-sink separations included in the fit.

Figure 16

Light-light effective amplitudes ${\mathcal{N}}_{{\mathcal{O}}_1{\mathcal{O}}_2}^{\mathrm{eff}}(t)\equiv ⟨{\mathcal{O}}_1(t){\mathcal{O}}_2(0)⟩/(e^{-m_{\mathrm{eff}}t}\pm e^{-m_{\mathrm{eff}}(T-t)})$ on the 32ID-M1 (left) and 32ID-M2 (right) ensembles. The sign is $+(-)$ for the PP(AP) correlator. These are related to the light-light pseudoscalar decay constant according to Eq. (c7).

Figure 17

Heavy-light effective amplitudes on the 32ID-M1 (left) and 32ID-M2 (right) ensembles.

Figure 18

The axial current renormalization coefficient, from Eq. (c5), on the 32ID-M1 (left) and 32ID-M2 (right) ensembles.

Figure 19

The $\mathrm{\Omega }$ baryon mass on the 32ID-M1 (left) and 32ID-M2 (right) ensembles. The wall source and $Z_3$ box source correlators are simultaneously fit to double exponential Ansätze with common mass terms [Eq. (c11)]. Here we overlay the data with the effective mass curves obtained from the fit.

Figure 20

The Wilson flow scales $t_0^{1/2}$ (top) and $w_0$ (bottom) on the 32ID-M1 (left) and 32ID-M2 (right) ensembles.

Figure 21

Left: the correlation matrix ${\rho }^{ij}$ corresponding to fits with a 370 MeV cut. The dashed lines mark the division into sub-blocks by ensemble. From left to right these are: 32I ($m_l=0.004$), 32I ($m_l=0.006$), 24I ($m_l=0.005$), 48I, 64I, 32I-fine, 32ID ($m_l=0.001$), 32ID ($m_l=0.0042$), and 32ID-M1. Right: the eigenvalue spectrum of ${\rho }^{ij}$.

Figure 22

Sub-blocks of the correlation matrix corresponding to the 32I ensembles. Panel (c) shows the cross-correlations between the $m_l=0.004$ and $m_l=0.006$ ensembles induced by the use of $Z_V$ extrapolated to the chiral limit to normalize the decay constants.