Towards a nonperturbative calculation of weak Hamiltonian Wilson coefficients

We propose a method to compute the Wilson coefficients of the weak effective Hamiltonian to all orders in the strong coupling constant using Lattice QCD simulations. We perform our calculations adopting an unphysically light weak boson mass of around 2 GeV. We demonstrate that systematic errors for the Wilson coefficients $C_1$ and $C_2$, related to the current-current four-quark operators, can be controlled and present a path towards precise determinations in subsequent works.

Figure 1

Connected diagram of the generic four-point function described in Eq. (6) and Fourier transformed with phases $e^{i{p}_k{y}_k}$. Disconnected contributions are forbidden by the flavor structure of the operators $O_i$. In our notation the four external quarks have all in-coming momenta.

Figure 2

Dependence of the bare lattice Wilson coefficients on the momentum of the external quark states for a fixed value of $m_{\mathrm{W}}\approx 1.8\mathrm{GeV}$. The two sets of data points correspond to the exceptional and nonexceptional kinematics described in Eq. (21) measured on the 24I ensemble with $am=0.04$. We perform combined polynomial fits where the constant term is constrained by the two data sets, which have been obtained from $\cancel{q}$ parity odd projectors.

Figure 3

Dependence of the lattice Wilson coefficient $C_2^{\mathrm{lat}}$ on the momentum of the external states for fixed $m_{\mathrm{W}}\approx 1.8\mathrm{GeV}$. The labels in the legend denote the choice of projectors ($\gamma$ or $\cancel{q}$) and momentum (injected along $xy$ spatial directions or time $t$). Both panels show points obtained from parity odd projectors. Left: results from the 16I ensemble; if the momentum is along the time direction, twice as big as the spatial ones, $C_2^{\mathrm{lat}}$ from $\cancel{q}$ projectors coincides with the measurements performed on the larger lattice. Right: results from the 24I ensemble where all combinations of momentum and projectors converge to the same point in the limit of small momenta.

Figure 4

Quark mass dependence of $C_2^{\mathrm{lat}}$ for fixed input $W$ mass at 1.8 GeV on the 24I ensemble. Crosses refer to parity even projectors and dots represent the opposite parity, all in the $\gamma$ scheme. No difference is observed if $\gamma$ projectors are replaced with the corresponding $\cancel{q}$ ones. The $\mathrm{VV}+\mathrm{AA}$ projectors produce results very sensitive to the quark mass, contaminated by infrared effects proportional to $1/p^2$. A small quark mass dependence is observed in $C_1^{\mathrm{lat}}$ for parity odd projectors as well.

Figure 5

Goldstone boson propagator and coupling to the quark current. In the latter we have already assumed degenerate quark masses and simplified the contribution proportional to the identity.

Figure 6

Contribution of the Goldstone boson exchange to the Wilson coefficients with parity even $\gamma$ projectors. The two lines are linear fits of $\log C_i^{\mathrm{lat}}$ in $p^2/m_{\mathrm{W}}^2$, which describe well the data. The suppression given by the prefactor in the vertices, $(2m/m_{\mathrm{W}}{)}^2=0.0064$, makes the two contributions negligible, even though they slightly reduce the discrepancy from the $C_i^{\mathrm{lat}}\mathrm{s}$ obtained with parity odd projectors.

Figure 7

Left: $\delta C_2$, as a function of $p_{\mathrm{cut}}$, decreases as more points at small momenta are excluded from the fit, suppressing nonperturbative effects. Right: measured difference of $C_2$ from odd and even projectors in the $\gamma$ scheme as function of the $W$ boson mass. The lines are linear fits forced to pass through the origin with excellent ${\chi }^2$. The nonperturbative contaminations which we quantify with $\delta C_i^{\mathrm{proj}}$ given in Eq. (24) seem to vanish only with one inverse power of $m_{\mathrm{W}}$. In the plot we present the data for the exceptional kinematics, but practically no difference is observed in the nonexceptional case.

Figure 8

Relative continuum extrapolations of $C_1^{\mathrm{RI}}$, renormalized at the value of the $W$ boson mass used to compute them. We normalize the $y$ axis with the measurement of $C_1^{\mathrm{RI}}$ from 32I. The slope of the extrapolations slightly changes with $m_{\mathrm{W}}$, but remains a subdominant effect. The points have been horizontally displaced for better visibility.

Figure 9

Results in the continuum limit for three-flavor QCD of the Wilson coefficients in the $\overline{\mathrm{MS}}$ scheme as a function of the $W$ boson mass. The shaded regions represent the perturbative one-loop result with error propagated from the $\mathrm{\Lambda }$-parameter. The plotted error bars include both the statistical and systematic uncertainties, described in previous sections. The latter dominate and gradually decrease with $m_{\mathrm{W}}$. The difference between the two intermediate RI schemes accounts for higher order effects in the $\mathrm{RI}\to \overline{\mathrm{MS}}$ conversion factors, which can be taken as an additional systematic error.