Matching lattice and continuum four-fermion operators with nonrelativistic QCD and highly improved staggered quarks

We match continuum and lattice heavy-light four-fermion operators at one loop in perturbation theory. For the heavy quarks we use nonrelativistic QCD and for the massless light quarks the highly improved staggered quark action. We include the full set of $\mathrm{\Delta }B=2$ operators relevant to neutral $B$ mixing both within and beyond the standard model and match through order ${\alpha }_s$, ${\mathrm{\Lambda }}_{\mathrm{QCD}}/M_b$, and ${\alpha }_s/(a{M}_b)$.

Figure 1

Tree-level diagrams representing the matrix elements of operators $Q1$, $Q2$, $Q4$, and $Q6$. The incoming state is a heavy antiquark and a light quark and the outgoing state is a heavy quark and a light antiquark. The letters $A$, $B$, $C$, and $D$ are color indices and correspond to the conventions of Eq. (10).

Figure 2

One loop diagrams representing the corrections to matrix elements of the operators $Qi$. The external states are those of Fig. 1 and Eq. (10).

Figure 3

Sample one loop diagrams representing the corrections to matrix elements of the $1/M$ operators $\widehat{Q}j1$. The black dot represents a derivative acting on the heavy (anti)quark propagator. The external states are those of Fig. 1. We show the corrections associated with diagrams (a), (b), (${\mathrm{a}}^{\prime }$), and (${\mathrm{b}}^{\prime }$) of Fig. 2. Analogous diagrams exist for diagrams (c) to (${\mathrm{f}}^{\prime }$). In general diagrams such as a.ii and b.ii vanish, because the derivative acts on an external heavy (anti)quark with zero momentum.

Figure 4

Sample one loop diagrams representing the one loop corrections to matrix elements of the $1/M$ operators $\widehat{Q}j1$. We show the four corrections associated with diagram (a) of Fig. 2. Analogous diagrams exist for diagrams (b) to (${\mathrm{f}}^{\prime }$). For more details, see the caption of Fig. 2. We do not include these contributions in our matching procedure.

Figure 5

Mass dependence of the lattice coefficients $c_{ij}^{\text{latt}}$, for $i=j$. Statistical uncertainties from numerical integration are $\pm 0.002$ and smaller than the data points on this scale.

Figure 6

Mass dependence of the lattice coefficients $c_{ij}^{\text{latt}}$, for $i\ne j$. Statistical uncertainties from numerical integration are $\pm 0.002$ and smaller than the data points on this scale.

Figure 7

Mass dependence of the leading order matching coefficients ${\rho }_{ij}$, for $i=j$. Statistical uncertainties from numerical integration are $\pm 0.002$ and smaller than the data points on this scale.

Figure 8

Mass dependence of the matching coefficients ${\rho }_{ij}$, for $i\ne j$.

Figure 9

Mass dependence of the next-to-leading order matching coefficients ${\zeta }_{ij}$, for $i=j$. Statistical uncertainties from numerical integration are $\pm 0.002$ and smaller than the data points on this scale.

Figure 10

Mass dependence of the next-to-leading order matching coefficients ${\zeta }_{ij}$, for $i\ne j$. Statistical uncertainties from numerical integration are $\pm 0.002$ and smaller than the data points on this scale.