Lattice computation of the electromagnetic contributions to kaon and pion masses

We present a lattice calculation of the electromagnetic (EM) effects on the masses of light pseudoscalar mesons. The simulations employ $2+1$ dynamical flavors of asqtad QCD quarks and quenched photons. Lattice spacings vary from $\approx 0.12\mathrm{fm}$ to $\approx 0.045\mathrm{fm}$. We compute the quantity $\epsilon$, which parametrizes the corrections to Dashen’s theorem for the $K^{+}$–$K^0$ EM mass splitting, as well as ${\epsilon }_{K^0}$, which parametrizes the EM contribution to the mass of the $K^0$ itself. An extension of the nonperturbative EM renormalization scheme introduced by the BMW group is used in separating EM effects from isospin-violating quark mass effects. We correct for leading finite-volume effects in our realization of lattice electrodynamics in chiral perturbation theory, and remaining finite-volume errors are relatively small. While electroquenched effects are under control for $\epsilon$, they are estimated only qualitatively for ${\epsilon }_{K^0}$ and constitute one of the largest sources of uncertainty for that quantity. We find $\epsilon =0.78(1{)}_{\text{stat}}(\genfrac{}{}{0ex}{}{+8}{-11}{)}_{\text{syst}}$ and ${\epsilon }_{K^0}=0.035(3{)}_{\text{stat}}(20{)}_{\text{syst}}$. We then use these results on $2+1+1$ flavor pure QCD highly improved staggered quark (HISQ) ensembles and find $m_u/m_d=0.4529(48{)}_{\text{stat}}(\genfrac{}{}{0ex}{}{+150}{-67}{)}_{\text{syst}}$.

Figure 1

The Goldstone pion propagator as a function of Euclidean time on the $a\approx 0.045\mathrm{fm}$ ensemble with $a{m}_l^{\prime }=0.0028$ and $a{m}_s^{\prime }=0.014$. The grid size is $64^3\times 192$. The meson propagators are periodic in time and have been folded over, so the maximum time is 96. Four charge combinations are plotted: (0,0), $(2/3,2/3)$, $(1/3,-2/3)$, and $(2/3,-2/3)$, with total charges $q_{xy}=0$, 0, 1 and $4/3$, respectively. These charges are all in units of $e$. The valence quark and antiquark masses are both 0.0014 in lattice units.

Figure 2

Fits of the four pion propagators shown in Fig. 1. Fits are from $D_{min}$ to the center of the lattice. The symbol sizes are proportional to the $p$ value of the fit. Black crosses denote single-particle fits and red squares denote two-particle fits.

Figure 3

The Goldstone pion mass as a function of the square of the meson charge on the $a\approx 0.045\mathrm{fm}$ ensemble with $a{m}_l^{\prime }=0.0028$ and $a{m}_s^{\prime }=0.014$. Four charge combinations are plotted: (0,0), $(2/3,2/3)$, $(1/3,-2/3)$, and $(2/3,-2/3)$, with total charges $Q=0$, 0, 1 and $4/3$, respectively. These charges are all in units of $e$. The valence quark and antiquark masses are both 0.0014, in lattice units.

Figure 4

Fits of the ratio of the propagators shown in Fig. 1 for mesons with charged quarks, divided by the propagator for the meson with neutral quarks. The vertical axis gives the mass difference between the two propagators in the ratio. Fits are from $D_{\min }$ to the center of the lattice. The symbol sizes are proportional to the $p$ value of the fit. The horizontal, red (color online), solid and dashed lines show the central value and error for the mass difference given by subtracting the masses in Table 4 and propagating the errors using the covariance matrix determined by jackknife, as described in the text.

Figure 5

Feynman diagrams that contribute to the meson-mass at $O(p^4,e^2{p}^2)$. Straight lines are the pseudoscalar meson propagator and wiggly lines are the photon. A filled dot represents a vertex from the $O(p^2,e^2)$ Lagrangian, ${\mathcal{L}}^{(\mathrm{LO})}$, while an open square represents an insertion of the $O(p^4,e^2{p}^2)$ Lagrangian, ${\mathcal{L}}^{(\mathrm{NLO})}$. (a) photon tadpole; (b) photon sunset; (c) meson tadpole; (d) $O(p^4,e^2{p}^2)$ tree-level insertion.

Figure 6

The FV effect from photon diagrams, ${\delta }_{\mathrm{FV}}^{\gamma }$ for ${\mathrm{QED}}_L$ and ${\mathrm{QED}}_{TL}$, as a function of $mL$. In the ${\mathrm{QED}}_L$ case, the dark green line shows the result when $T=\infty$ from Hayakawa and Uno [73], while the dark red diamonds and purple squares show our evaluation at $T/L=2.29$ and $T/L=5.33$, respectively. For ${\mathrm{QED}}_{TL}$, the lines give our results for six values of $T/L$ ranging between 2.25 and 3.43, which are the values relevant to the bulk of our data. The numerical errors in the points and lines are too small to be seen on this scale.

Figure 7

A comparison of the full FV effect for ${\mathrm{QED}}_{TL}$, coming from one-loop photon diagrams with a pointlike meson, and the corresponding asymptotic forms determined in Ref. [72], for various values of $T/L$. The squares show our calculations of the full effect, while the lines are the asymptotic forms. The numerical errors in the points are small and are just barely visible in some of the points at the left.

Figure 8

Effective mass plots for a “$K^{+}$ meson” in pure QCD (left) and in $\mathrm{QCD}+\mathrm{quenched}{\mathrm{QED}}_{TL}$ (right). The data is from the ensemble listed first in Table 1, with $a\approx 0.12\mathrm{fm}$, $L/a=12$ and $T/a=64$. The valence masses are 0.01 and 0.04.

Figure 9

Finite volume effects at $a\approx 0.12\mathrm{fm}$ and $a{m}_l^{\prime }=0.01$, $a{m}_s^{\prime }=0.05$ as a function of spatial lattice length $L$ for two different meson masses: a unitary ‘pion’ (blue) with degenerate valence masses $m_x=m_y=m_l^{\prime }$, and a ‘kaon’ (red) with valence masses $m_x=m_l^{\prime }$ and $a{m}_y=0.04$, close to the physical strange quark mass. The fit lines are to our FV form for ${\mathrm{QED}}_{TL}$ (omitting the negligible meson tadpole term), and have one free parameter each, the infinite volume value (shown by horizontal solid lines with dotted lines for errors).

Figure 10

Products $B_0{m}_l$ (left) and $B_0{m}_s$ (right) vs $a^2$. The black squares show our data at nonzero lattice spacing, while the blue octagons show our continuum values. Two simple linear extrapolations are shown for comparison. The red line in each plot is a fit to all four lattice spacings, and the red cross is its extrapolation. Similarly, the green line and fancy cross in each plot comes from a fit that omits the coarsest lattice spacing.

Figure 11

Finite-volume corrections to $\mathrm{\Delta }{M}_{xy}^2$ for a small subset of our data vs $M_{xy}^2$ itself, where both quantities are expressed in $r_1$ units. For each pair of points with the same color and symbol, the lower point shows the raw datum, while the upper shows the result after correction for FV effects, i.e., in infinite volume. Colors and symbols identify the lattice spacing and light sea-quark mass, and (in one case) the spatial lattice size, as shown in the legend. Points in the left-hand cluster are pionlike and unitary ($m_x=m_y=m_l^{\prime }$), while those in the right-hand cluster are kaonlike and almost always unitary ($m_x=m_l^{\prime }$, $m_y=m_s^{\prime }$). The exceptions are kaonlike points for $a\approx 0.12\mathrm{fm}$, which have $m_y=0.8m_s^{\prime }$, which is closer to the physical strange mass than $m_s^{\prime }$ itself. The locations of the physical pion and kaon masses are indicated by the vertical dot-dashed black lines. All points shown are for mesons with charge $\pm e$.

Figure 12

Central fit to the EM splitting $\mathrm{\Delta }{M}_{xy}^2$ vs the sum of the valence-quark masses, after correction for FV effects. The same small subset of the data as in Fig. 11 is shown. The blue, light green, and dark green lines show the fit to the $a\approx 0.09$, 0.06, and 0.045 fm data, respectively. The largest lattice spacing (red and magenta points, $a\approx 0.12\mathrm{fm}$) is not included in the fit; the dashed red lines show how the fit does in “predicting” these points. The horizontal dotted line shows the experimental value of the ${\pi }^{+}$–${\pi }^0$ splitting; the vertical dashed-dotted lines show the quark mass values for physical $\pi$ and $K$ mesons. The black and brown curves are extrapolations of $\mathrm{\Delta }{M}_{xy}^2$ to the continuum, with and without the NLO effects of sea quark charges, respectively. (The brown curve is barely visible under the right hand black curve; the curves are identical for the pions at left, and only the black curve is visible.) The solid purple curves are obtained from the black ones by subtracting $\mathrm{\Delta }{M}_{xy}^2$ for the corresponding neutral mesons, $K^0$ and “${\pi }^0$.” The dashed-dotted line and the dashed purple curves show the LO, and $\mathrm{LO}+\mathrm{NLO}$ contributions to the total solid purple lines, respectively.

Figure 13

Same fit as Fig. 12, but showing the neutral mesons made out of quarks with charges $\pm e/3$. The meaning of the blue, light green, dark green, and dashed red curves is the same as in Fig. 12; note the difference in vertical scale between the two plots. The solid black curves are extrapolations of $\mathrm{\Delta }{M}_{xy}^2$ to the continuum. The dashed black lines show the NLO contribution to the solid black curves; there are no LO contributions. The discretization errors and sea-mass dependence for neutral mesons are very small, as are the nonlinear contributions to the valence-mass dependence.

Figure 14

Two alternative chiral-discretization fits. The data included in the fits, as well as the meaning of symbols and curves, is the same as for the central fit, Fig. 12. Fit (a) sets parameters ${\rho }_6$ and ${\rho }_7$ to zero, as well as ${\kappa }_2$. The differences with Fig. 12 are small: all curves are slightly higher in the chiral limit, and the predictions for the $a\approx 0.12\mathrm{fm}$ points (dashed red curves) are noticeably higher. Fit (b) does not fix the parameter ${\kappa }_2$ (or ${\rho }_6$ and ${\rho }_7$) to zero. The main difference from Fig. 12 that is apparent in fit (b) is the relative size of the contributions to the continuum result of the LO contribution (horizontal, purple dashed-dotted line), and the $\mathrm{LO}+\mathrm{NLO}$ contribution (dashed purple curves). The full continuum-extrapolated results (solid purple curves) are however very close to those in Fig. 12.

Figure 15

An alternative chiral-discretization fit to the same data as for the central fit. The meaning of symbols and curves is the same as for the Fig. 12. This fit puts a strict prior on ${\mathrm{\Delta }}_{\mathrm{EM}}$ to force it to be very near to the value that would give the physical pion splitting at LO. Note that the higher chiral orders reduce the pion chiral limit significantly below the LO contribution.

Figure 16

Two examples of chiral-discretization fit and extrapolation like the central fit in Fig. 12, but including the points from the $a\approx 0.12\mathrm{fm}$ ensembles. Fit (a) is most similar to the central fit, in that ${\kappa }_2$ is fixed to zero and ${\mathrm{\Delta }}_{\mathrm{EM}}$ is unconstrained. Fit (b) allows ${\kappa }_2$ to vary (with a prior $0\pm 40$), and imposes a prior of $0.003\pm 0.001$ on $r_1^2{e}^2{\mathrm{\Delta }}_{\mathrm{EM}}$. Nevertheless the fit value of ${\mathrm{\Delta }}_{\mathrm{EM}}$ is negative.

Figure 17

$m_u/m_d$ on the physical quark mass HISQ ensembles, and the continuum extrapolation. The red line is the fit used for our central value, and the blue lines three of the alternative fits used for estimating systematic error from the continuum extrapolation. These alternate fits are a quadratic fit including all the data points, a linear fit omitting the 0.15 fm. data, and a linear fit omitting both the 0.15 and 0.12 fm data.

Figure 18

Comparison of $\epsilon$ in Eq. (74) (magenta burst) with previous unquenched lattice-QCD calculations. The open symbols with dashed error bars represent early work, with only statistical errors quoted. The references are RM123 17 [19], MILC 16 [15] (a preliminary result), BMW 16 [17], QCDSF 15 [16], Blum et al. 10 [10], RM123 13 [18], and RBC 07 [9].

Figure 19

Comparison of $m_u/m_d$ in Eq. (76) (magenta burst) with previous unquenched lattice-QCD calculations that include a lattice evaluation of the EM effects. For comparison, we also show, with an open triangle, the MILC 09 result, which uses a phenomenological estimate of the EM effects. An early result, RBC 07, just quotes statistical errors and is shown with dashed error bars and an open symbol. The current (MILC 18) result should be considered an update of the Fermilab/MILC 17 result [see discussion following Eq. (76)]. The references are Fermilab/MILC 17 [85], RM123 17 [19], ETM 14 [87], BMW 16 [17], QCDSF 15 [16], Blum et al. 10 [10], MILC 09 [4], RM123 13 [18], and RBC 07 [9].

Figure 20

Location of the poles of the momentum space propagator ${\tilde{G}}_{\text{cont}}(p_0)$ for routing B, for three different values of $mT$ and $mL$. The quantities $(y/m{)}^2$ (black lines) and $1+\mathrm{\Delta }\sigma (iy)/m^2$ (red lines) are shown as a function of $y/m$, where $y$ is the imaginary part of the Euclidean energy $p_0$. The $y/m$ values of the poles are given by the locations of the crossings of the two curves.

Figure 21

Relative size of the routing dependence of the self-energy contribution at $p_0=im$ of the sunset graph, for the same data at 0.12 fm that was presented in Fig. 9. The red squares are for “kaon” points; the blue diamonds, for “pion” points. For the three leftmost diamonds and the three leftmost squares, where no deviation from 0 is visible, the actual deviation varies between $\approx 5\times 10^{-5}$ and $\approx 1\times 10^{-4}$.