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Register a new userGluon fragmentation into ${^{3}hspace{-0.6mm}P_{J}^{[1,8]}}$ quark pair and test of NRQCD factorization at two-loop level
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The next-to-leading order (NLO) ($mathcal{O}(alpha_s^3)$) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in ${^{3}hspace{-0.6mm}P_{J}^{[1,8]}}$ states are calculated within the NRQCD factorization. We use integration-by-parts reduction and differential equations to semi-analytically calculate fragmentation functions in full-QCD, and find that infrared divergences can be absorbed by the NRQCD long distance matrix elements. Thus, the NRQCD factorization conjecture is verified at two-loop level via a physical process, which is free of artificial ultraviolet divergences. Through matching procedure, infrared-safe short distance coefficients and $mathcal{O}(alpha_s^2)$ perturbative NRQCD matrix elements $langle{mathcal O}^{{^{3}hspace{-0.6mm}P_{J}^{[1,8]}}}({^{3}hspace{-0.6mm}S_{1}^{[8]}})rangle_{mathrm{NLO}}$ are obtained simultaneously. The NLO short distance coefficients are found to have significant corrections comparing with the LO ones.
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We calculate the NLO corrections for the gluon fragmentation functions to a heavy quark-antiquark pair in ${^{1}hspace{-0.6mm}S_{0}^{[1]}}$ or ${^{1}hspace{-0.6mm}S_{0}^{[8]}}$ state within NRQCD factorization. We use integration-by-parts reduction to reduce the original expression to simpler master integrals (MIs), and then set up differential equations for these MIs. After calculating the boundary conditions, MIs can be obtained by solving the differential equations numerically. Our results are expressed in terms of asymptotic expansions at singular points of $z$ (light-cone momentum fraction carried by the quark-antiquark pair), which can not only give FFs results with very high precision at any value of $z$, but also provide fully analytical structure at these singularities. We find that the NLO corrections are significant, with K-factors larger than 2 in most regions. The NLO corrections may have important impact on heavy quarkonia (e.g. $eta_c$ and $J/psi$) production at the LHC.
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.
We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy quarkonium production or decay, which means that for any given process these two factorization theories are either both valid or both violated. We use two methods to achieve this conclusion. In the first method, we apply the two factorization theories to the physical process $J/psi to e^+e^-$. Our explicit calculation shows that both SGF and NRQCD can correctly reproduce low energy physics of full QCD, and thus the two factorizations are equivalent. In the second method, by using equations of motion we successfully deduce SGF from NRQCD effective field theory. By identifying SGF with NRQCD factorization, we establish relations between the two factorization theories and prove the generalized Gremm-Kapustin relations as a by product. Comparing with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originated from kinematic effects to all powers, which gives rise to a better convergence in relativistic expansion.
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 present the first calculation at next-to-leading order (NLO) in $alpha_s$ of a fragmentation function into quarkonium whose form at leading order is a nontrivial function of $z$, namely the fragmentation function for a gluon into a spin-singlet S-wave state at leading order in the relative velocity. To calculate the real NLO corrections, we introduce a new subtraction scheme that allows the phase-space integrals to be evaluated in 4 dimensions. We extract all ultraviolet and infrared divergences in the real NLO corrections analytically by calculating the phase-space integrals of the subtraction terms in $4-2epsilon$ dimensions. We also extract the divergences in the virtual NLO corrections analytically, and detail the cancellation of all divergences after renormalization. The NLO corrections have a dramatic effect on the shape of the fragmentation function, and they significantly increase the fragmentation probability.
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