Model-independent study of electric dipole transitions in quarkonium

The paper contains a systematic, model-independent treatment of electric dipole (E1) transitions in heavy quarkonium. Within the effective field theory framework of potential nonrelativistic QCD, we derive the complete set of relativistic corrections of relative order $v^2$ both for weakly and strongly coupled quarkonia. The result supports and complements former results from potential model calculations.

Figure 1

Leading-order diagram with a $c$-quark loop coupled to an external photon in NRQCD. Dashed lines stand for longitudinal gluons.

Figure 2

Matching of the amplitude (vi): before the arrow are the NRQCD diagrams, after the arrow the pNRQCD ones. The continuous line in the pNRQCD diagrams stands for the quark-antiquark singlet propagator.

Figure 3

Examples of diagrams at order $1/m$, where the electromagnetic coupling is contained in a covariant derivative.

Figure 4

Examples of diagrams at order $1/m$ with an external longitudinal photon.

Figure 5

Some diagrams that contribute to spin-dependent operators of order $1/m^2$ containing an electric field.

Figure 6

Some diagrams contributing to the matching of spin-independent operators of order $1/m^2$.

Figure 7

Kinematics of the transition $H\to H^{\prime }\gamma$ in the center of mass frame.

Figure 8

Color-octet contributions to the E1 transition $n^3{P}_J\to n^{\prime 3}{S}_1\gamma$ for weakly coupled states. The double line stands for an intermediate octet state.

Figure 9

Some diagrams contributing to the matching at order $1/m^3$.