Lattice QCD at the physical point meets $SU(2)$ chiral perturbation theory

We perform a detailed, fully correlated study of the chiral behavior of the pion mass and decay constant, based on $2+1$ flavor lattice QCD simulations. These calculations are implemented using tree-level, $O(a)$-improved Wilson fermions, at four values of the lattice spacing down to 0.054 fm and all the way down to below the physical value of the pion mass. They allow a sharp comparison with the predictions of $SU(2)$ chiral perturbation theory ($\chi \mathrm{PT}$) and a determination of some of its low energy constants. In particular, we systematically explore the range of applicability of next-to-leading order (NLO) $SU(2)$ $\chi \mathrm{PT}$ in two different expansions: the first in quark mass ($x$ expansion), and the second in pion mass ($\xi$ expansion). We find that these expansions begin showing signs of failure for $M_{\pi }\gtrsim 300\mathrm{MeV}$, for the typical percent-level precision of our $N_f=2+1$ lattice results. We further determine the LO low energy constants (LECs), $F=88.0\pm 1.3\pm 0.2$ and $B^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=2.61(6)(1)\mathrm{GeV}$, and the related quark condensate, ${\mathrm{\Sigma }}^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=(272\pm 4\pm 1\mathrm{MeV}{)}^3$, as well as the NLO ones, ${\overline{\ell }}_3=2.6(5)(3)$ and ${\overline{\ell }}_4=3.7(4)(2)$, with fully controlled uncertainties. We also explore the next-to-next-to-leading order (NNLO) expansions and the values of NNLO LECs. In addition, we show that the lattice results favor the presence of chiral logarithms. We further demonstrate how the absence of lattice results with pion masses below 200 MeV can lead to misleading results and conclusions. Our calculations allow a fully controlled, ab initio determination of the pion decay constant with a total 1% error, which is in excellent agreement with experiment.

Figure 1

Example of a continuum extrapolation of the LO LEC $B$ (left panel) and of $F_{\pi }$ at physical $M_{\pi }$ (right panel). These plots are obtained from a typical $x$ expansion, NLO chiral fit of the type discussed in Sec. 3b, to all of our lattice simulation results up to $M_{\pi }=300\mathrm{MeV}$ (the fit considered is the same as in Fig. 2). This fit is then used to interpolate the lattice points, $\beta$ per $\beta$, to the relevant physical $m_{ud}$ and $m_s$ point in infinite volume, eliminating the dependence on all lattice parameters except for a possible $a$ dependence. This yields the blue crosses, corresponding to the results at the four lattice spacing considered here. We also show, as a green burst, the resulting continuum limit value for this example analysis. Since we are considering a particular analysis, error bars are statistical only. The black lines going through the points show the $a$ dependence of $B$ and $F_{\pi }$ given by the NLO fit. As the left panel shows, a linear dependence is visible in $B$. This dependence is inherited from the discretization corrections observed in $m_{ud}$. On the other hand, $F_{\pi }$ has no statistically significant $a$ dependence, as shown in the right panel.

Figure 2

Example of an NLO $SU(2)$ $\chi \mathrm{PT}$ fit (curves) of our lattice results (points with error bars) for $B_{\pi }=M_{\pi }^2/(2m_{ud})$ and $F_{\pi }$ as functions of $m_{ud}$, in the $x$ expansion. These are fully correlated fits to the NLO expressions of (14), which also account for discretization and strange quark mass corrections. Only points with $M_{\pi }\le M_{\pi }^{\max }=300\mathrm{MeV}$ (i.e. $m_{ud}^{\mathrm{RGI}}\lesssim 23\mathrm{MeV}$) are included in the fits, i.e. those left of the dashed vertical line. The more massive points are shown for illustration. The lattice results in the figure are corrected for discretization and strange mass contributions, using the fit parameters obtained. Thus, they are continuum limit results at the physical value of $m_s$ and their only residual dependence is on $m_{ud}$. Nevertheless, results obtained at different lattice spacings are plotted with different symbols. The fact that they lie on the same curve indicates that residual discretization errors are negligible. Note that the corrections made to the more massive points may not be optimal as these points are not included in the fit and, as we will see, the applicability of NLO $\chi \mathrm{PT}$ is questionable for these points. Error bars on all points are statistical only. Also shown, but not included in the fits, is the experimental value of $F_{\pi }$ [48]. Agreement with our results computed directly around the physical pion mass point is remarkable.

Figure 3

Example of an NLO $SU(2)$ $\chi \mathrm{PT}$ fit (curves) of our lattice results (points with error bars) for $1/B_{\pi }$ and $F_{\pi }$ as functions of $M_{\pi }^2$, in the $\xi$ expansion. Only points with $M_{\pi }\le M_{\pi }^{\max }=300\mathrm{MeV}$ are included in the fits, i.e. those left of the dashed vertical line. The description is the same as in Fig. 2, except that the functional forms used are those of (15).

Figure 4

Example of an NNLO $SU(2)$ $\chi \mathrm{PT}$ fit (curves) of our lattice results (points with error bars) for $B_{\pi }$ and $F_{\pi }$ as functions of $m_{ud}$, in the $x$ expansion. Only points with $M_{\pi }\le M_{\pi }^{\max }=500\mathrm{MeV}$ (i.e. $m_{ud}^{\mathrm{RGI}}\lesssim 65\mathrm{MeV}$) are included in the fits, i.e. those left of the dashed vertical line. The description is the same as in Fig. 2, except that the functional forms used are those of (14) with $B_{\pi }^{x-\mathrm{NLO}}$ and $F_{\pi }^{x-\mathrm{NLO}}$ replaced by $B_{\pi }^{x-\mathrm{NNLO}}$ and $F_{\pi }^{x-\mathrm{NNLO}}$, respectively.

Figure 5

Example of an NNLO $SU(2)$ $\chi \mathrm{PT}$ fit (curves) of our lattice results (points with error bars) for $1/B_{\pi }$ and $F_{\pi }$ as functions of $M_{\pi }^2$, in the $\xi$ expansion. Only points with $M_{\pi }\le M_{\pi }^{\max }=500\mathrm{MeV}$ are included in the fits, i.e. those left of the dashed vertical line. The description is the same as in Fig. 2, except that the function forms used are those of (15) with $B_{\pi }^{\xi -\mathrm{NLO}}$ and $F_{\pi }^{\xi -\mathrm{NLO}}$ replaced by $B_{\pi }^{\xi -\mathrm{NNLO}}$ and $F_{\pi }^{\xi -\mathrm{NNLO}}$, respectively.

Figure 6

Fit quality of the fully correlated $SU(2)$ $\chi \mathrm{PT}$ fits to our lattice results for $B_{\pi }$ and $F_{\pi }$ described in Sec. 3b. The fits to our lattice results for these quantities include points whose pion mass is in the range $[120\mathrm{MeV},M_{\pi }^{\max }]$. The $p$ values shown are those of NLO and NNLO fits in the $x$ and $\xi$ expansions. They are plotted as a function of $M_{\pi }^{\max }$. For $M_{\pi }^{\max }\le 350\mathrm{MeV}$, only the $p$ values of NLO fits are plotted as these ranges do not contain enough data to constrain NNLO chiral expressions. For $M_{\pi }^{\max }\in [400,450]\mathrm{MeV}$, the $p$ values of both NLO and NNLO fits are shown. For larger $M_{\pi }^{\max }$ only NNLO results are shown, as the $p$ values of NLO fits are negligibly small. The gray band corresponds to the $p$-value interval of 10% to 50% and the red one to that of 50% to 100%. Error bars on each point are the systematic uncertainties described in the text. Results obtained for the same $M_{\pi }^{\max }=250\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\max }$ value so that they can be visually distinguished.

Figure 7

LO LECs as a function of $M_{\pi }^{\max }$ obtained from the $SU(2)$ $\chi \mathrm{PT}$ fits to our lattice results for $B_{\pi }$ and $F_{\pi }$ in the pion-mass range $[120\mathrm{MeV},M_{\pi }^{\max }]$, as described in Sec. 3b. Results are shown for NLO and NNLO fits in the $x$ and $\xi$ expansions (see the caption of Fig. 6 for additional details). In the top panel of each of the two figures, it is the LEC in physical units which is shown. The horizontal gray band denotes our final result for the corresponding LEC, given in Table 4, and obtained as described in Sec. 4. In the lower panel of each figure it is the difference of the LEC obtained from a fit with $M_{\pi }\in [120\mathrm{MeV},M_{\pi }^{\max }]$ to that obtained from the NLO fit in the range [120, 300] MeV, in the corresponding expansion. As argued in the text, this reference domain is in the range of applicability of NLO $\chi \mathrm{PT}$ at our level of accuracy. Error bars on each point are the statistical and the quadratically combined statistical-plus-systematic uncertainties. Results obtained for a same $M_{\pi }^{\max }=250\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\max }$ value so that they can be visually distinguished.

Figure 8

NLO LECs as a function of $M_{\pi }^{\max }$ obtained from the $SU(2)$ $\chi \mathrm{PT}$ fits to our lattice results for $B_{\pi }$ and $F_{\pi }$ in the pion mass range $[120\mathrm{MeV},M_{\pi }^{\max }]$, as described in Sec. 3b. This figure is the same as Fig. 7, but for NLO instead of LO LECs.

Figure 9

LECs which appear at NNLO in the $SU(2)$ $\chi \mathrm{PT}$ expansions of $B_{\pi }$ and $F_{\pi }$ given in Eqs. (2)–(9). The top figure shows the results obtained for $k_M$ and $k_F$ from NNLO fits in the $x$ expansion to our lattice results with $M_{\pi }\in [120\mathrm{MeV},M_{\pi }^{\max }]$. The results are plotted as functions of $M_{\pi }^{\max }$. In the middle figure are plotted the NNLO LECs $c_M$ and $c_F$, which appear in the $\xi$ expansion. The NLO LEC combination ${\overline{\ell }}_{12}=(7{\overline{\ell }}_1+8{\overline{\ell }}_2)/15$ appears in the $x$ and $\xi$ expansions of $B_{\pi }$ and $F_{\pi }$ at NNLO. The results that we obtain for this LEC in each of the expansions are plotted in the bottom panel. The horizontal gray band denotes our final result for ${\overline{\ell }}_{12}$, obtained as described in Sec. 4. Error bars on each point are the statistical and the quadratically combined statistical-plus-systematic uncertainties. Results obtained for the same $M_{\pi }^{\max }=250\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\max }$ value so that they can be visually distinguished.

Figure 10

Ratios of the NLO contributions to $B_{\pi }$ and $F_{\pi }$ with respect to the LO ones, as a function of $m_{ud}$ in the $SU(2)$ chiral $x$ expansion (left panel) and of $M_{\pi }^2$ in the $\xi$ expansion (right panel). The values of the LECs used are those given in Table 3 for the respective expansions.

Figure 11

Typical ratios of the NLO and NNLO contributions to $B_{\pi }$ (left panel) and $F_{\pi }$ (right panel) with respect to the LO ones, as a function of $m_{ud}$ in the $SU(2)$ chiral $x$ expansion. The values of the LECs used are those obtained from the fit shown in Fig. 4.

Figure 12

Typical ratios of the NLO and NNLO contributions to $1/B_{\pi }$ (left panel) and $F_{\pi }$ (right panel) with respect to the LO ones, as a function of $M_{\pi }^2$ in the $SU(2)$ chiral $\xi$ expansion. The values of the LECs used are those obtained from the fit shown in Fig. 5.

Figure 13

Systematic error distributions for the LO LECs. These are obtained by varying the analysis procedure, as described in the text. The total distribution is delineated by the solid black line. It is the sum of the distributions corresponding to the analyses performed in the $x$ and $\xi$ expansions. These are shown as a red dotted line and a blue dashed line, respectively. Where only the $x$ or $\xi$ expansion distributions contribute, they partially hide the line corresponding to the total distribution. In the plots, the central, vertical, dotted line is the mean of the total distribution, i.e. our final central value. The central, vertical green band denotes the systematic error, the larger pink one, the statistical error and the largest gray one, the sum in quadrature of these two errors.

Figure 14

Systematic error distributions for the NLO LECs. The different components of the graphs have the same meaning as in Fig. 13.

Figure 15

Comparison of the $p$ values obtained in fits of NLO $SU(2)$ expansion, including and omitting the chiral logarithms. These fully correlated fits to our lattice results for $B_{\pi }$ and $F_{\pi }$ include points whose pion mass is in the range $[120\mathrm{MeV},M_{\pi }^{\max }]$. Results are shown for the $x$ and $\xi$ expansions. In the top panel the individual $p$ values are shown. Those of fits including the logarithms are the same as the ones given in Fig. 6. In the lower panel it is the difference of the $p$ value obtained omitting logarithms minus the $\chi \mathrm{PT}$ one, normalized by the latter. Error bars on each point are the systematic uncertainties discussed in Sec. 3c. Results obtained for the same $M_{\pi }^{\max }=200\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\max }$ value so that they can be visually distinguished.

Figure 16

$F_{\pi }$ as a function of $M_{\pi }^{\max }$, obtained from NLO fits with and without logarithms, in the $x$ and $\xi$ expansions (upper panel). In the lower panel it is the difference (no logarithm minus logarithm) of these highly correlated results that are shown. Error bars on each point are the statistical and the quadratically combined statistical-plus-systematic uncertainties. Results obtained for a same $M_{\pi }^{\max }=200\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\max }$ value so that they can be visually distinguished.

Figure 17

The $p$ value as a function of $M_{\pi }^{\min }$. The $p$ values are obtained by performing fully correlated NLO, $SU(2)$ $\chi \mathrm{PT}$ fits to lattice results for $B_{\pi }$ and $F_{\pi }$ with pion masses in the range $[M_{\pi }^{\min },450\mathrm{MeV}]$. Both the $x$ and $\xi$ expansions are considered. The points with $M_{\pi }^{\min }=120\mathrm{MeV}$ are the same as those with $M_{\pi }^{\max }=450\mathrm{MeV}$ in Fig. 6. These and the points at $M_{\pi }^{\min }=150\mathrm{MeV}$ have significantly smaller $p$ values because NLO, $SU(2)$ $\chi \mathrm{PT}$ does not adequately describe the behavior of $B_{\pi }$ and $F_{\pi }$ for pion masses ranging from around its physical value up to $M_{\pi }^{\max }=450\mathrm{MeV}$, as discussed at length in Sec. 3. The horizontal bands have the same meaning as in Fig. 6. Error bars on each point are the systematic uncertainties discussed in Sec. 3c. Results obtained for a same $M_{\pi }^{\min }=120,150\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\min }$ value so that they can be visually distinguished.

Figure 18

LO LECs as a function of $M_{\pi }^{\min }$ (upper panel of each plot). The LECs are obtained from the fits described in Fig. 17. The horizontal gray band denotes our final result for the corresponding LEC, given in Table 4, and obtained as described in Sec. 4. In the lower panel corresponding to each LEC, it is the difference of this LEC with the one obtained from fits in our canonical range, $M_{\pi }\in [120,300]\mathrm{MeV}$. Error bars on each point are the statistical and the quadratically combined statistical-plus-systematic uncertainties. Results obtained for the same $M_{\pi }^{\min }=150\mathrm{MeV},\dots$, are displaced horizontally by a small amount around that $M_{\pi }^{\min }$ value so that they can be visually distinguished.

Figure 19

Same as Fig. 17, but for NLO LECs.

Figure 20

Illustration of the NLO and NNLO lower bounds on ${\overline{\ell }}_4$ coming from the requirement that there is a physical solution for $F_{\pi }$ assuming that the NLO or the NNLO $\xi$ expansion expressions of (7) hold exactly. ${\overline{\ell }}_4$ must lie above the given curve for each order in the expansion. To plot these curves, we use our final result for $F$ given in Table 4. The dashed curves delimit the $1\sigma$ error band on each bound arising from the total uncertainty on $F$.