Modification of the $B$ meson mass in a magnetic field from QCD sum rules

In this paper, we extend the well-known QCD sum rules used in the calculation of the mass of heavy mesons to estimate the modification of the charged $B$-meson mass, $m_B$, in the presence of an external Abelian magnetic field, $eB$. Two simplifying limits were considered: the weak field limit, in which the external field satisfies $eB\ll m^2$ (with $m$ being any of the masses involved); and the strong field limit, in which the field strength is small in comparison to the bottom quark mass (or the $B$-meson mass) squared, but it is large compared to the mass of the light quarks, i.e., $m_{u,d}^2\ll eB\ll m_{b,B}^2$. We found that $m_B$ decreases with the magnetic field in both of these limits.

Figure 1

(a) Convergence of the OPE expansion and (b) pole dominance in the absence of a magnetic field.

Figure 2

Dependence of the mass (a) and coupling constant (b) of the $B$ meson, as functions of the Borel mass, in the absence of a magnetic field. The points indicate the Borel window.

Figure 3

(a) OPE convergence and (b) pole dominance for $eB=4\times 10^{-8}{\mathrm{GeV}}^2$ and $q_{\perp }^2=1{\mathrm{GeV}}^2$.

Figure 4

Mass of the $B$ meson, $m_B$, as a function of the Borel mass $\overline{M^2}$ for $eB=4\times 10^{-8}{\mathrm{GeV}}^2$ (the weak field limit) and $q_{\perp }^2=1{\mathrm{GeV}}^2$. The points indicate the Borel window.

Figure 5

(a) $B$-meson mass, $m_B$, as a function of the perpendicular momentum for a fixed magnetic field of $eB=2\times 10^{-6}{m}_{\pi }^2$ (the weak field limit) and (b) dependence of $m_B$ on the magnetic field for a fixed perpendicular momentum $q_{\perp }^2=1{\mathrm{GeV}}^2$.

Figure 6

(a) OPE convergence and (b) pole dominance for $eB=1.5{\mathrm{GeV}}^2$ (strong field limit) and $q_{\perp }^2=0.5{\mathrm{GeV}}^2$.

Figure 7

(a) Mass of the $B$ meson, $m_B$, as a function of the perpendicular momentum $q_{\perp }^2$ for fixed $eB=1.5{\mathrm{GeV}}^2$ (strong field limit), and (b) as a function of the magnetic field $eB$ for fixed $q_{\perp }^2=0.5{\mathrm{GeV}}^2$. The curves correspond to the central $\sqrt{s_0}\sim (m_H+600)\mathrm{MeV}$, lower $\sqrt{s_0}\sim (m_H+400)\mathrm{MeV}$, and upper $\sqrt{s_0}\sim (m_H+800)\mathrm{MeV}$ interpolated continuum threshold $s_0(eB)$ curves, as described in the main text.

Figure 8

Perturbative diagrams up to order $(eB{)}^2$.