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

We compute the $\overline{h}c$ (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 renormalization and using the second generation $2+1+1$ MILC HISQ gluon field configurations. We include ensembles with $a\approx 0.09$, 0.06, 0.045, and 0.033 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 ${\overline{m}}_b^2{\chi }_{1^{+}}=0.720\left(34\right)\times {10}^{-2}$, ${\overline{m}}_b^2{\chi }_{1^{-}}=1.161\left(54\right)\times {10}^{-2}$, ${\chi }_{0^{-}}=2.374\left(33\right)\times {10}^{-2}$, and ${\chi }_{0^{+}}=0.609\left(14\right)\times {10}^{-2}$. 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 ${\overline{m}}_b^2{\chi }_T=0.891\left(44\right)\times {10}^{-2}$ and ${\overline{m}}_b^2{\chi }_{AT}=0.441\left(33\right)\times {10}^{-2}$. Our value of ${\overline{m}}_b^2{\chi }_{AT}$ is in good agreement with the $O\left({\alpha }_s\right)$ perturbation theory, while our result for ${\overline{m}}_b^2{\chi }_T$ is in tension with the $O\left({\alpha }_s\right)$ perturbation theory at the level of $2\sigma$. These results will allow for dispersively bounded parametrizations 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 $1.25\times m_c\le m_h\le m_b$.