Using lattice chiral effective theory to study pi-pi scattering

Abstract

We use lattice field theory to study the finite-volume energy spectrum of the $ππ$ system in $SU(2)$ chiral effective field theory (ChEFT) at leading order in the chiral expansion. We compare our results with the finite volume spectrum obtained from lattice QCD (Blum et al., Phys. Rev. D 107(9):094512, 2023, arXiv:2301.09286 [hep-lat]). Our calculation and the lattice QCD calculation are both performed with the physical pion mass and the same physical volume. However, we find significant differences between the two calculations in the isospin $I=0$ channel. In particular, there is a nearly stable $σ$ resonance in our lattice ChEFT calculation, which is absent in the lattice QCD calculation. This likely indicates that ChEFT does not converge well with a naive lattice regularization.

Figures (6)

• Figure 1: The probability distribution of $\sum_i\phi_i^2(x)$ for various $\lambda$ on a $16^3\times 32$ lattice with $m^2/\lambda=-0.102$ and $\alpha=0.007$. These parameters are chosen so that we are near the physical point (defined as the point where $m_\pi/F_\pi=135/92\approx 1.47$) when $\lambda=10^4$ (see Table \ref{['ensemble_parameters']} for more information on the ensemble with $\lambda=10^4$). As can be seen from the figure, at large $\lambda$, the values of the fields are dynamically constrained to lie near the surface of a 3-sphere.
• Figure 2: The $\lambda$-dependence of the vacuum expectation value of $\phi_0$, the pion mass $m_\pi$, the effective sigma mass $m_\sigma^\text{eff}$, and the pion decay constant $F_\pi$ on a $16^3\times 32$ lattice. All values are in lattice units. We set $m^2/\lambda=-0.102$ and $\alpha=0.007$ so that we are near the physical point (defined as the point where $m_\pi/F_\pi=135/92\approx 1.47$) when $\lambda=10^4$ (see Table \ref{['ensemble_parameters']} for more information on the ensemble with $\lambda=10^4$). The effective sigma mass is based on the values of the $\sigma$-$\sigma$ correlation function at time-separations 0 and 1. As can be seen from the Figure, increasing $\lambda$ from $10^4$ to $10^5$ does not significantly affect $m_\sigma^\text{eff}$.
• Figure 3: The correlation function $\left\langle \left(\sum_{i=1}^3\phi_i(t)\right)\left(\sum_{j=1}^3\phi_j(t=0)\right)\right\rangle$ divided by a single-state fit $f(t) = \frac{A}{\cosh((N_t/2-1)m_\pi)}\cosh\left(\left(\frac{N_t}{2}-t\right)m_\pi\right)$, where $A$ and $m_\pi$ are the fit parameters, and $N_t$ is the time extent of the lattice. This correlation function was calculated on the $16^3\times 128$ ensemble with $m^2/\lambda=-0.102$, $\alpha=0.007$, and $\lambda=10^4$ (see Table \ref{['gevp_ensembles']} for more details on this ensemble). The fit gives $A=7238(102)$ and $m_\pi=0.2201(77)$. As can be seen from the figure, single-state fit works well even at small $t$.
• Figure 4: The pion mass $m_\pi$, the effective sigma mass $m_\sigma^\text{eff}$, and the pion decay constant $F_\pi$ (all measured in lattice units) versus $\frac{m^2}{\lambda}$ on a $8^3\times 16$ lattice (top) and a $16^3\times 32$ lattice (bottom). The value of $m^2/\lambda$ closest to the physical point (defined as the point where $m_\pi/F_\pi=135/92\approx 1.5$) is marked by a vertical line. $\lambda$ is held fixed at $10^4$, which we have determined is sufficiently large that further increases will not have a strong effect on the results. We set $\alpha=0.05$ for the $8^3\times 16$ lattice and $\alpha=0.007$ for the $16^3\times 32$ lattice. The resulting lattice spacing and spacial extent near the physical points are given in Table \ref{['ensemble_parameters']}. The effective sigma mass is based on the values of the $\sigma$-$\sigma$ correlation function at time-separations 0 and 1. Increasing the magnitude of $\frac{m^2}{\lambda}$ causes $F_\pi$ to increase while $m_\pi$ decreases, making it clear that the physical point is unique.
• Figure 5: The isospin-0 (left) and isospin-2 (right) energy spectrum as determined by solving the generalized eigenvalue problem using timeslices $t$ and $t+1$ on a $16^3\times 128$ lattice with $m^2/\lambda = -0.102$ and $\alpha=0.007$ (as usual, $\lambda=10^4$). These parameters are chosen so that we are near the physical point with $a^{-1}=0.6363(99)$ GeV and $L=4.961(77)\text{ fm}$ (see Table \ref{['gevp_ensembles']} for more details). The QCD spectrum, as calculated in RBC_pipi_scattering on a $24^3\times 64$ lattice with similar physical volume ($L=4.63(1)$ fm) and $a^{-1}=1.023(2)$ GeV, is shown to the right for each channel. The error bars on the QCD spectrum do not include the error in determining $m_\pi$. Non-interacting pion energies $2E_{\pi,p=0}$ and $2E_{\pi,p=2\pi/L}$ are shown on all plots for comparison.