Electric dipole moment of the $\rho$ meson

At a hadronic scale the effect of CP-violating interactions that typically appear in extensions of the Standard Model may be described by an effective Lagrangian, in which the operators are expressed in terms of lepton and partonic gluon and quark fields and ordered by their mass dimension, $k⩾4$. Using a global-symmetry-preserving truncation of QCD Dyson-Schwinger equations, we compute the $\rho$-meson's electric dipole moment (EDM), $d_{\rho }$, as generated by the leading dimension-four and dimension-five CP-violating operators and an example of a dimension-six four-quark operator. The two dimension-five operators, i.e., the quark EDM and quark chromo-EDM, produce contributions to $d_{\rho }$ whose coefficients are of the same sign and within a factor of 2 in magnitude. Moreover, should a suppression mechanism be verified for the $\theta$ term in any beyond-Standard-Model theory, the contribution from a four-quark operator can match the quark EDM and quark chromo-EDM in importance. This study serves as a prototype for the more challenging task of computing the neutron's EDM.

Figure 1

Impulse approximation to the $\rho$-$\gamma$-$\rho$ vertex, Eq. (20). Solid lines denote the dressed-quark propagators and shaded circles are clockwise from top the Bethe-Salpeter vertex for quark-photon coupling and Bethe-Salpeter amplitudes for the ${\rho }^{+}$ meson.

Figure 2

Evolution of $\rho$-meson magnetic moment with current-quark mass. $m=170$ MeV corresponds to the mass of the $s$ quark in our treatment of the contact interaction [22], so the difference between ${\mathcal{M}}_{\rho }\left(0\right)$ and ${\mathcal{M}}_{\varphi }\left(0\right)$ is just 1$\%$.

Figure 3

(a) Correction to the quark-photon vertex generated by the four-fermion operator in Eq. (30). The unmodified quark-photon vertex is the left dot, whereas the right dot locates the insertion of ${\mathcal{L}}_6$. If the internal line represents a circulating $d$ quark then, owing to the ${\mathcal{L}}_6$ insertion, the external lines are $u$ quarks, and vice versa. (b) Analogous correction to the $\rho$ meson Bethe-Salpeter amplitude. The unmodified amplitude is the left dot, whereas the right dot locates the insertion of ${\mathcal{L}}_6$. The lower internal line is an incoming $d$ quark and the upper external line is an outgoing $u$ quark. (c) ${\mathcal{L}}_6$ correction to the dressed-quark propagator, with the dot locating the operator insertion. If the outer line is a $u$ quark, then the internal line is a $d$ quark, and vice versa.

Figure 4

Correction to the quark-photon vertex generated by the quark chromo-EDM operator in Eq. (2): the incoming and outgoing quark lines have the same flavor, $f$. The dot in the left two diagrams locates the insertion of ${\mathcal{L}}_{CEDM}$, while that in the rightmost diagram indicates the second term in Eq. (23), i.e., the explicit quark EDM.

Figure 5

Correction to the $\rho$ meson Bethe-Salpeter amplitude generated by the quark chromo-EDM operator in Eq. (2): the incoming line is a $d$ quark and the outgoing line is a $u$ quark. In each case the dot locates the insertion of ${\mathcal{L}}_{CEDM}$.

Figure 6

Correction to the dressed-quark propagator generated by the quark chromo-EDM operator in Eq. (2). In each image the dot locates the insertion of ${\mathcal{L}}_{CEDM}$.

Figure 7

Evolution of the quark EDM component of the $\rho$ meson's EDM with current-quark mass, under the assumption that $d_{-}$ is independent of $m$. $m=170$MeV corresponds to the mass of the $s$ quark in our treatment of the contact interaction [22], so the difference between $d_{\rho }^{\gamma }$ and $d_{\varphi }^{\gamma }$ is 10$\%$.