Pion-Nucleon Sigma Term from Lattice QCD

We present an analysis of the pion-nucleon $\sigma$-term ${\sigma }_{\pi N}$ using six ensembles with $2+1+1$-flavor highly improved staggered quark action generated by the MILC Collaboration. The most serious systematic effect in lattice calculations of nucleon correlation functions is the contribution of excited states. We estimate these using chiral perturbation theory ($\chi \mathrm{PT}$) and show that the leading contribution to the isoscalar scalar charge comes from $N\pi$ and $N\pi \pi$ states. Therefore, we carry out two analyses of lattice data to remove excited-state contamination, the standard one and a new one including $N\pi$ and $N\pi \pi$ states. We find that the standard analysis gives ${\sigma }_{\pi N}=41.9(4.9)\mathrm{MeV}$, consistent with previous lattice calculations, while our preferred $\chi \mathrm{PT}$-motivated analysis gives ${\sigma }_{\pi N}=59.6(7.4)\mathrm{MeV}$, which is consistent with phenomenological values obtained using $\pi N$ scattering data. Our data on one physical pion mass ensemble were crucial for exposing this difference, therefore, calculations on additional physical mass ensembles are needed to confirm our result and resolve the tension between lattice QCD and phenomenology.

Figure 1

The upper connected (left) and disconnected (right) diagrams contribute to the three-point function that determines the matrix element of flavor-diagonal scalar operators (shown by symbol $\otimes$ at time slice $t$) within the nucleon state. The black and gray blobs denote nucleon source and sink, separated by Euclidean time $\tau$. The bottom diagram illustrates that the disconnected diagrams include an $N\pi$ intermediate-state configuration that can give an enhanced contribution.

Figure 2

Results of the $\{4,3^{*}\}$ (top row) and $\{4^{N\pi },3^{*}\}$ (bottom row) fits to the sum of the connected and disconnected data plotted versus $(t-\tau /2)/a$ for ensembles $a09m130$ and $a06m220$. Result of the fit is shown by lines of the same color as the data for various $\tau /a$ listed in the label, and the $\tau =\infty$ value is given by the gray band.

Figure 3

Data for the $\sigma$-term, ${\sigma }_{\pi N}=m_{ud}{g}_S^{u+d}$, from the two ESC strategies $\{4,3^{*}\}$ (gray) and $\{4^{N\pi },3^{*}\}$ (color) are shown as a function of $a$ and $M_{\pi }^2$. The two left panels show the chiral-continuum (CC) fit $\{2,2a,3,4\}$ and the two right the CC fit $\{2,2a,3^{\chi },4,4L\}$ described in the text. The result at $M_{\pi }=135\mathrm{MeV}$ and [${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$] of the two fits are given in the legend.

Figure 4

Results for ${\sigma }_{\pi N}=m_{ud}{g}_S^{u+d}$ from $2+1$- and $2+1+1$-flavor lattice calculations. The BMW 20 result from $1+1+1+1$-flavor lattices is listed along with the other $2+1+1$-flavor calculations for brevity. Following the FLAG conventions, determinations via the direct approach are indicated by squares and the FH method by triangles. Also, the symbols used for lattice estimates that satisfy the FLAG criteria for inclusion in averages are filled green, and those not included are open red. The references from which lattice results have been taken are as follows: JLQCD 18 [68], $\chi \mathrm{QCD}$ 15A [65], BMW 15 [64], ETM 14A [71], ETM 19 [69], and BMW 20 [70]. Phenomenological estimates using $\pi N$ scattering data (blue filled circles) are from Gasser et al. 91 [20], Pavan et al. 02 [26], Alarcón et al. 11 [28], Hoferichter et al. 15 [39], and Ruiz de Elvira et al. 17 [45].