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

Abstract

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 $1.25\times m_c\le m_h\le m_b$.

Comments: 13 pages, 10 figures