Hadronic quarkonium decays at order $v^7$

We compute the complete imaginary part of the nonrelativistic QCD (NRQCD) Lagrangian at order $1/M^4$ in the heavy-quark mass expansion, which includes center of mass operators, and at order ${\alpha }_s^2$ in the matching coefficients. We also compute the imaginary part of the NRQCD Lagrangian at order $1/M^6$ and at order ${\alpha }_s^2$ that contributes to the S-wave and P-wave inclusive decay widths of heavy quarkonium into light hadrons at order $v^7$ in the heavy-quark velocity expansion. If we count ${\alpha }_s(M)\sim v^2$, the calculation provides the complete next-to-leading order corrections to the P-wave hadronic widths, and in the original NRQCD power counting, the complete next-to-leading order corrections to the vector S-wave widths, and part of the next-to-next-to leading order corrections to the pseudoscalar S-wave widths. In the S-wave case, we confirm previous findings and add new terms in a more conservative power counting. In the P-wave case, our results are in disagreement with previous ones. Constraints induced by Poincaré invariance on the NRQCD four-fermion sector are studied for the first time and provide an additional check of the calculation. Perspectives for phenomenological applications are discussed.

Figure 1

QCD Feynman diagrams describing the amplitude $Q\overline{Q}\to Q\overline{Q}$ at order ${\alpha }_s^2$.

Figure 2

Generic NRQCD four-fermion Feynman diagram. The empty box stands for one of the four-fermion vertices induced by the operators listed in Appendix , Eqs. (a16, a17, a18, a19, a20, a21, a26, a27, a28, a29, a30, a31).

Figure 3

Box diagrams: the gluon in the initial state interacts with a fermion leg. The other six diagrams, in which the two gluon propagators cross, have not been displayed.

Figure 4

Box diagrams: the gluon in the initial state interacts with a gluon propagator. The other four diagrams, in which the two gluon propagators cross, have not been displayed.

Figure 5

Vertex corrections: the gluon in the initial state couples to a fermion leg or to a three-gluon vertex.

Figure 6

Vertex corrections: the gluon in the initial state couples to a fermion leg or to a three-gluon vertex.

Figure 7

Vertex corrections: the gluon in the initial state interacts with a gluon propagator.

Figure 8

Vacuum polarization: the gluon in the initial state interacts with a fermion leg.

Figure 9

Generic NRQCD four-fermion Feynman diagrams involving an ingoing $Q\overline{Q}$ pair and a gluon and an outgoing $Q\overline{Q}$ pair. The black box with a gluon attached to it and the empty box stand, respectively, for one of the four-fermion–one-gluon vertices and for one of the four-fermion vertices induced by the operators listed in Appendix , Eqs. (a16, a17, a18, a19, a20, a21, a22, a23, a24, a25, a26, a27, a28, a29, a30, a31). The black dot with a gluon attached to it stands for one of the quark-gluon vertices induced by the bilinear part of the NRQCD Lagrangian given in Eq. (a1).