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Register a new userFragmentation, NRQCD and Factorization in Heavy Quarkonium Production
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Gouranga C. Nayak
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We discuss factorization in heavy quarkonium production in high energy collisions using NRQCD. Infrared divergences at NNLO are not matched by conventional NRQCD matrix elements. However, we show that gauge invariance and factorization require that conventional NRQCD production matrix elements be modified to include Wilson lines or non-abelian gauge links. With this modification NRQCD factorization for heavy quarkonium production is restored at NNLO.
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We discuss heavy quarkonium production through parton fragmentation, including a review of arguments for the factorization of high-p_T particles into fragmentation functions for hadronic initial states. We investigate the further factorization of fragmentation functions in the NRQCD formalism, and argue that this requires a modification of NRQCD octet production matrix elements to include nonabelian phases, which makes them gauge invariant. We describe the calculation of uncanceled infrared divergences in fragmentation functions that must be factorized at NNLO, and verify that they are absorbed into the new, gauge invariant matrix elements.
We study the transition of a heavy quark pair from octet to singlet color configurations at next-to-next-to-leading order (NNLO) in heavy quarkonium production. We show that the infrared singularities in this process are consistent with NRQCD factorization to all orders in the heavy quark relative velocity v. This factorization requires the gauge-completed matrix elements that we introduced previously to prove NNLO factorization to order v ^2.
We study the transverse-momentum spectrum of quarkonium production from single light-parton fragmentation mechanism. In the case of semi-inclusive deep inelastic scattering, we observe that there are two possible initiating processes, namely photon-gluon fusion and light-quark fragmentation. For the second case we derive the factorization theorem, which involves a new hadronic quantity: the quarkonium transverse-momentum-dependent fragmentation functions in NRQCD. We calculate their matching onto the non-perturbative long distance matrix elements at the lowest order in the strong-coupling constant (${mathcal O}(alpha_s^2)$). Focusing on the case of the electron-ion collider, we make a comparative phenomenological study of the two production mechanisms and find the regions of the phase space where one is dominant over the other.
The widely used nonrelativistic QCD (NRQCD) factorization theory now encounters some notable difficulties in describing quarkonium production. This may be due to the inadequate treatment of soft hadrons emitted in the hadronization process, which causes bad convergence of velocity expansion in NRQCD. In this paper, starting from QCD we propose a rigorously defined factorization approach, soft gluon factorization (SGF), to better deal with the effects of soft hadrons. After a careful velocity expansion, the SGF can be as simple as the NRQCD factorization in phenomenological studies, but has a much better convergence. The SGF may provide a new insight to understand the mechanisms of quarkonium production and decay.
In a recent paper (arXiv:1912.02253), Rothkopf claims that the Bryan method, which is widely used to obtain the solution in the maximum entropy method and makes use of the singular value decomposition of a matrix, limits the search space for the solution. He even presents a counterexample to the Bryan method. In this comment, we first recapitulate the mathematical basis of the Bryan method, and reconfirm that it makes use of no approximations and that it is therefore mathematically rigorous. In the second part, we explicitly show that Rothkopfs ``counterexample actually does not constitute a counterexample on the basis of the definition of singular value decomposition itself.
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