We extend the auxiliary-mass-flow (AMF) method originally developed for Feynman loop integration to calculate integrals which also involve phase-space integration. The flow of the auxiliary mass from the boundary ( ) to the physical point ( ) is obtained by numerically solving differential equations with respective to the auxiliary mass. For problems with two or more kinematical invariants, the AMF method can be combined with the traditional differential-equation method, providing systematic boundary conditions and a highly nontrivial self-consistency check. The method is described in detail using a pedagogical example of at NNLO. We show that the AMF method can systematically and efficiently calculate integrals to high precision.

Fragmentation is the dominant mechanism for hadron production with high transverse momentum. For spin-triplet S-wave heavy quarkonium production, contribution of gluon fragmenting to color-singlet channel has been numerically calculated since 1993. However, there is still no analytic expression available up to now because of its complexity. In this paper, we calculate both polarization-summed and polarized fragmentation functions of gluon fragmenting to a heavy quarkantiquark pair with quantum number 3 S[1] 1 . Our calculations are performed in two different frameworks. One is the widely used nonrelativistic QCD factorization, and the other is the newly proposed soft gluon factorization. In either case, we calculate at both leading order and next-to-leading order in velocity expansion. All of our final results are presented in terms of compact analytic expressions.

We calculate the NLO corrections for the gluon fragmentation functions to a heavy quarkantiquark pair in 1 Sstate 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. ηc and J/ψ) production at the LHC. n are normalized long-distance matrix elements (LDMEs) 1 . 1 ŌH n can be related to the original definition of NRQCD LDME O H n [1] by the following rules. They are the same if n is color-octet, and ŌH n = O H n /(2Nc) if n is color-singlet.

The pole position of the σ meson is determined using the recently proposed new unitarization method combined with the two-loop SU(2) × SU(2) chiral perturbation theory results. The scattering length parameters and the effective range parameters predicted by our method are in good agreement with experimental results. It is found that

The next-to-leading order (NLO) ($$ \mathcal{O} $$ O ($$ {\alpha}_s^3 $$ α s 3 )) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in 3$$ {P}_J^{\left[1,8\right]} $$ P J 1 8 states are calculated within the NRQCD factorization. We use the integration-by-parts reduction and differential equations to semi-analytically calculate the 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 the matching procedure, infrared-safe short distance coefficients and $$ \mathcal{O} $$ O ($$ {\alpha}_s^2 $$ α s 2 ) perturbative NRQCD matrix elements ⟨$$ {\mathcal{O}}^3{P}_J^{\left[1,8\right]} $$ O 3 P J 1 8 (3$$ {S}_1^{\left[8\right]} $$ S 1 8 )⟩ are obtained simultaneously. The NLO short distance coefficients are found to have significant corrections comparing with the LO ones.

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