Abstract
We report on the calculation [1] of the relativistic O ( α s 0 ν 2 ) corrections to the quarkonium production process e + e − → χ cJ + γ in non-relativistic QCD (NRQCD). In our work we incorporate effects from operators that contribute through the sub-leading Fock state | Q Qg 〉, that were not taken into account by previous studies. We determine the corresponding matching coeffcients that should be included into theoretical predictions for the electromagnetic production cross-section of χ cJ . This process could be, in principle, measured by the Belle II experiment.
Highlights
Heavy quarkonia provide a privileged window to study the interplay of perturbative and nonperturbative effects in QCD
The non-relativistic nature of the system and the presence of a hierarchy of well-separated scales makes the application of the the effective field theory (EFT) approach advantageous, where non-relativistic QCD (NRQCD) [2] allows to factorize cross-sections and decay rates of heavy quarkonia into the short-distance matching coefficients and long-distance matrix elements (LDME)
By comparing the production cross-sections obtained from the NRQCD amplitudes Eqs. (5)-(7) to Eqs. (2)-(4) we can read off the values of the matching coefficients
Summary
Sub-leading Fock states that contain gluons are usually suppressed by power of v as compared to the dominant state |QQ They must be taken into account when higher order relativistic or loop corrections are computed. At O(α0sv2) operators that contribute through the sub-leading Fock state |QQg must be taken into account This prompts us to consider relativistic corrections generated by these operators. To this purpose, in the perturbative matching between QCD and NRQCD we include diagrams that contain an ultrasoft (|kg| ∼ mv2) final state gluon and treat this gluon as a part of the quarkonium system. We determine new relativistic corrections induced by operators that contain one chromoelectric field and give no contribution through the dominant Fock state.
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