$\bar{b}c$ susceptibilities from fully relativistic lattice QCD

We compute the $\bar{h}c$ (pseudo)scalar, (axial-)vector and (axial-)tensor susceptibilities as a function of $u=m_c/m_h$ between $u=m_c/m_b$ and $u=0.8$ using fully relativistic lattice QCD, employing nonperturbative current renormalisation and using the second generation 2+1+1 MILC HISQ gluon field configurations. We include ensembles with $a\approx 0.09\mathrm{fm}$, $0.06\mathrm{fm}$, $0.045\mathrm{fm}$ and $0.033\mathrm{fm}$ and we are able to reach the physical $b$-quark on the two finest ensembles. At the physical $m_h=m_b$ point we find $\overline{m}_b^2 \chi_{1^+}={0.720(34)\times 10^{-2}}$, $\overline{m}_b^2 \chi_{1^-}={1.161(54)\times 10^{-2}}$, $\chi_{0^-}={2.374(33)\times 10^{-2}}$, $\chi_{0^+}={0.609(14)\times 10^{-2}}$. Our results for the (pseudo)scalar, vector and axial-vector are compatible with the expected small size of nonperturbative effects at $u=m_c/m_b$. We also give the first nonperturbative determination of the tensor susceptibilities, finding $\overline{m}_b^2 \chi_{T}={0.891(44)\times 10^{-2}}$ and $\overline{m}_b^2 \chi_{AT}={0.441(33)\times 10^{-2}}$. Our value of $\overline{m}_b^2\chi_{AT}$ is in good agreement with the $\mathcal{O}(\alpha_s)$ perturbation theory, while our result for $\overline{m}_b^2\chi_{T}$ is in tension with the $\mathcal{O}(\alpha_s)$ perturbation theory at the level of $2\sigma$. These results will allow for dispersively bounded parameterisations to be employed using lattice inputs for the full set of $h\to c$ semileptonic form factors in future calculations, for heavy-quark masses in the range $1.25\times m_c \leq m_h \leq m_b$.