Renormalization of the tensor current in lattice QCD and the $J/\psi$ tensor decay constant

Lattice QCD calculations of form factors for rare Standard Model processes such as $B\to K{\ell }^{+}{\ell }^{-}$ use tensor currents that require renormalization. These renormalization factors, $Z_T$, have typically been calculated within perturbation theory and the estimated uncertainties from missing higher order terms are significant. Here we study tensor current renormalization using lattice implementations of momentum-subtraction schemes. Such schemes are potentially more accurate but have systematic errors from nonperturbative artifacts. To determine and remove these condensate contributions we calculate the ground-state charmonium tensor decay constant, $f_{J/\psi }^T$, which is also of interest in beyond the Standard Model studies. We obtain $f_{J/\psi }^T(\overline{\mathrm{MS}},2\mathrm{GeV})=0.3927(27)\mathrm{GeV}$, with ratio to the vector decay constant of 0.9569(52), significantly below 1. We also give $Z_T$ factors, converted to the $\overline{\mathrm{MS}}$ scheme, corrected for condensate contamination. This contamination reaches 1.5% at a renormalization scale of 2 GeV (in the preferred Regularisation Invariant Symmetric Momentum subtraction scheme) and so must be removed for accurate results.

Figure 1

Valence mass dependence of tensor current renormalization factors in the RI-SMOM (upper) and ${\mathrm{RI}}^{\prime }$-MOM (lower) schemes. The values shown are for $\mu =2\mathrm{GeV}$ on set 7. Both show reasonably mild dependence on the valence mass but the dependence is smaller for RI-SMOM.

Figure 2

The taste splitting of the $J/\psi$ masses of tastes ${\xi }_{\mu }{\xi }_t$ and ${\xi }_{\mu }$ as a function of $(a{m}_c{)}^2$. Clearly this difference decreases with the lattice spacing. The fit line shown has the form of Eq. (14).

Figure 3

Continuum extrapolation of the $J/\psi$ tensor decay constant in the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV using lattice tensor current renormalization in the RI-SMOM scheme at multiple $\mu$ values. Three different values of the renormalization scale $\mu$ are used in the lattice calculation of $Z_T^{\mathrm{SMOM}}$ to allow nonperturbative $\mu$ dependence to be fitted. These three different $\mu$ values are shown as different colored lines. The blue line is 2 GeV, the orange, 3 GeV and the purple, 4 GeV. The black hexagon is the physical result for $f_{J/\psi }^T(2\mathrm{GeV})$ obtained from the fit of Eq. (16) (with the condensate pieces removed).

Figure 4

The correction, $C^{\mathrm{SMOM}}(\mu )$, to the tensor current renormalization factor, $Z_T^{\overline{\mathrm{MS}}}(2\mathrm{GeV})$, required to account for nonperturbative effects arising from condensate contributions to the lattice calculation of $Z_T^{\mathrm{SMOM}}(\mu )$ and missing ${\alpha }_s^4$ terms in the matching to $\overline{\mathrm{MS}}$. The correction is defined in terms of a subset of the fit posteriors of the fit shown in Fig. 3 (see text).

Figure 5

Continuum extrapolation of the $J/\psi$ tensor decay constant in the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV using an intermediate nonperturbative renormalization of the tensor current in the ${\mathrm{RI}}^{\prime }$-MOM scheme on the lattice. Multiple values of the renormalization scale $\mu$ have been used so that $\mu$-dependent nonperturbative effects can be removed in the fit. The blue points and line are for $\mu =2\mathrm{GeV}$, orange for 3 GeV and purple for 4 GeV. The value obtained in the continuum limit with the condensate terms removed is shown as a black hexagon. The result is in agreement with that using RI-SMOM renormalization (Fig. 3) which is shown as the green square.

Figure 6

The same as Fig. 4 but for a correction term, $C^{\mathrm{MOM}}(\mu )$, needed for the tensor current renormalization factor when using the intermediate ${\mathrm{RI}}^{\prime }$-MOM scheme.

Figure 7

Continuum extrapolation of the ratio of the tensor and vector $J/\psi$ decay constants using intermediate lattice renormalization factors in the RI-SMOM scheme. Blue points and lines show $\mu =2\mathrm{GeV}$ results and fit lines, orange are 3 GeV and purple 4 GeV. The bold dashed lines are continuum extrapolations at each $\mu$ value with the condensate and ${\alpha }_s^4$ terms left in. The black hexagon is the continuum extrapolation with condensates and ${\alpha }_s^4$ errors removed.

Figure 8

Correction $C^{\mathrm{SMOM}}(\mu )$ for the tensor current renormalization factor, $Z_T$, determined from a fit to the ratio of the $J/\psi$ tensor and vector decay constants using intermediate renormalization factors in the RI-SMOM scheme. This agrees with the results shown in Fig. 4, as expected because the lattice RI-SMOM vector current renormalization factor has no condensate contamination [35].

Figure 9

A comparison plot of results for the tensor $J/\psi$ decay constant in the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV. The top two results are from this work [using both RI-SMOM (Eq. (21)) and ${\mathrm{RI}}^{\prime }$-MOM (Eq. (19)) intermediate schemes] and then we include results from [12, 37].

Figure 10

A comparison plot of results for the ratio of tensor and vector $J/\psi$ decay constants. The upper result is from this work [Eq. (20)] using the RI-SMOM intermediate scheme and gives a value significantly below 1 (marked with the black dashed line) for this ratio. The lower two results are from [12].