Abstract
We calculate in the non-relativistic QCD (NRQCD) factorization framework all leading relativistic corrections to the exclusive production of $\chi_{cJ}+\gamma$ in $e^+ e^-$ annihilation. In particular, we compute for the first time contributions induced by octet operators with a chromoelectric field. The matching coefficients multiplying production long distance matrix elements (LDMEs) are determined through perturbative matching between QCD and NRQCD at the amplitude level. Technical challenges encountered in the non-relativistic expansion of the QCD amplitudes are discussed in detail. The main source of uncertainty comes from the not so well known LDMEs. Accounting for it, we provide the following estimates for the production cross sections at $\sqrt{s} = 10.6\textrm{ GeV}$: $\sigma (e^+ e^- \to \chi_{ c0} + \gamma) = (1.3 \pm 0.4) \textrm{ fb}$, $\sigma (e^+ e^- \to \chi_{ c1} + \gamma) = (15.4 \pm 6.7) \textrm{ fb}$, and $\sigma (e^+ e^- \to \chi_{ c2} + \gamma) = (4.7 \pm 2.6) \textrm{ fb}$.
Highlights
More than four decades have passed since the experimental groups of Samuel Ting and Burton Richter [1,2] discovered the J=ψ, the first observed bound state formed by a heavy quark and a heavy antiquark
The Effective field theories (EFTs) resulting from integrating out modes associated with the energy scale of the heavy-quark mass or larger is known as nonrelativistic quantum chromodynamics (QCD) (NRQCD) [7]
We believe that the present work is the first study, in which matching coefficients multiplying long distance matrix elements (LDMEs) with chromoelectric fields are explicitly computed for a heavy quarkonium production process and cannot be traded with other operators
Summary
More than four decades have passed since the experimental groups of Samuel Ting and Burton Richter [1,2] discovered the J=ψ, the first observed bound state formed by a heavy (charm) quark and a heavy antiquark. The EFT resulting from integrating out modes associated with the energy scale of the heavy-quark mass or larger is known as nonrelativistic QCD (NRQCD) [7] It conjectures a factorization theorem allowing one to write the quarkonium production cross section as an expansion where each term is the product of a short-distance coefficient and a longdistance matrix element (LDME). The former is computed from matching to perturbative QCD, it is a series in the strong coupling αs, and it incorporates effects from the hard scale m (heavy quark mass) and above. Appendix D provides a derivation of the generalized Gremm-Kapustin relations for 3PJ quarkonia
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