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Abstract We study the mass spectra of hidden-charm tetraquark systems with quantum numbers (I G )J P = (1+)1+ (and their I = 1 2 $$ \frac{1}{2} $$ partners) using QCD sum rules. The analysis incorporates the complete next-to-leading order (NLO) contribution to the perturbative QCD part of the operator product expansions, with particular attention to operator mixing effects due to renormalization group evolution. We find that both the parametric dependence and the perturbative convergence are significantly improved for the two mixed operators J 1 , 5 Mixed $$ {J}_{1,5}^{\textrm{Mixed}} $$ and J 2 , 6 Mixed $$ {J}_{2,6}^{\textrm{Mixed}} $$ , compared with those for the unmixed meson-meson or diquark-antidiquark type ones. For the d ¯ c c ¯ u $$ \overline{d}c\overline{c}u $$ system, the masses of J 1 , 5 Mixed $$ {J}_{1,5}^{\textrm{Mixed}} $$ and J 2 , 6 Mixed $$ {J}_{2,6}^{\textrm{Mixed}} $$ are determined to be 3.89 − 0.12 + 0.18 $$ {3.89}_{-0.12}^{+0.18} $$ GeV and 4.03 − 0.07 + 0.06 $$ {4.03}_{-0.07}^{+0.06} $$ GeV, respectively, closely matching those of Z c (3900) and Z c (4020). Similarly, for the s ¯ c c ¯ u $$ \overline{s}c\overline{c}u $$ states, the masses of J 1 , 5 Mixed $$ {J}_{1,5}^{\textrm{Mixed}} $$ and J 2 , 6 Mixed $$ {J}_{2,6}^{\textrm{Mixed}} $$ are found to be 4.02 − 0.09 + 0.17 $$ {4.02}_{-0.09}^{+0.17} $$ GeV and 4.21 − 0.07 + 0.08 $$ {4.21}_{-0.07}^{+0.08} $$ GeV, respectively, in close proximity to Z cs (3985)/Z cs (4000) and Z cs (4220), consistent with the expectation that they are the partners of Z c (3900) and Z c (4020). Our results highlight the crucial role of operator mixing, an inevitable effect in a complete NLO calculation, in achieving a robust phenomenological description for the tetraquark system.

Abstract We present an analytic calculation of the one-loop correction to the double-real emission contribution to the zero-jettiness soft function at N3LO in QCD, accounting for both gluon-gluon and quark-antiquark soft final-state partons. We explain all the relevant steps of the computation including the reduction of phase-space integrals to master integrals in the presence of Heaviside functions, and the methods we employed to compute them.

Abstract We study the mass spectra of Q ¯ Q Q ¯ Q $$ \overline{Q}Q\overline{Q}Q $$ (Q = c, b) systems in QCD sum rules with the complete next-to-leading order (NLO) contribution to the perturbative QCD part of the correlation functions. Instead of meson-meson or diquark-antidiquark currents, we use diagonalized currents under operator renormalization. We find that differing from conventional mesons q ¯ q $$ \overline{q}q $$ and baryons qqq, a unique feature of the multiquark systems like Q ¯ Q Q ¯ Q $$ \overline{Q}Q\overline{Q}Q $$ is the operator mixing or color configuration mixing induced by NLO corrections, which is crucial to understand the color structure of the states. Our numerical results show that the NLO corrections are very important for the Q ¯ Q Q ¯ Q $$ \overline{Q}Q\overline{Q}Q $$ system, because they not only give significant contributions but also reduce the scheme and scale dependence and make Borel platform more distinct, especially for the b ¯ b b ¯ b $$ \overline{b}b\overline{b}b $$ in the MS ¯ $$ \overline{\textrm{MS}} $$ scheme. We use currents that have good perturbation convergence in our phenomenological analysis. With the MS ¯ $$ \overline{\textrm{MS}} $$ scheme, we get three J PC = 0++ states, with masses 6.35 − 0.17 + 0.20 $$ {6.35}_{-0.17}^{+0.20} $$ GeV, 6.56 − 0.20 + 0.18 $$ {6.56}_{-0.20}^{+0.18} $$ GeV and 6.95 − 0.35 + 0.21 $$ {6.95}_{-0.35}^{+0.21} $$ GeV, respectively. The first two seem to agree with the broad structure around 6.2 ~ 6.8 GeV measured by the LHCb collaboration in the J/ψJ/ψ spectrum, and the third seems to agree with the narrow resonance X(6900). For the 2++ states we find one with mass 7.03 − 0.26 + 0.22 $$ {7.03}_{-0.26}^{+0.22} $$ GeV, which is also close to that of X(6900), and another one around 7.25 − 0.35 + 0.21 $$ {7.25}_{-0.35}^{+0.21} $$ GeV, which has good scale dependence but slightly large scheme dependence.

Abstract We compute the non-factorisable O $$ \mathcal{O} $$ ( α s 2 $$ {\alpha}_s^2 $$ ) corrections to t-channel single-top quark production at the LHC. These peculiar corrections arise because of interactions between the heavy- and the light-quark lines and appear for the very first time at next-to-next-to-leading order in perturbative QCD. We find that the non-factorisable corrections change the single-top production cross section and the relevant kinematic distributions in this process by about half a percent.

Abstract We discuss peculiarities that arise in the computation of real-emission contributions to observables that contain Heaviside functions. A prominent example of such a case is the zero-jettiness soft function in SCET, whose calculation at next-to-next-to-next-to-leading order in perturbative QCD is an interesting problem. Since the zero-jettiness soft function distinguishes between emissions into different hemispheres, its definition involves θ-functions of light-cone components of emitted soft partons. This prevents a direct use of multi-loop methods, based on reverse unitarity, for computing the zero-jettiness soft function in high orders of perturbation theory. We propose a way to bypass this problem and illustrate its effectiveness by computing various non-trivial contributions to the zero-jettiness soft function at NNLO and N3LO in perturbative QCD.

Abstract We compute the non-factorisable contribution to the two-loop helicity amplitude for t-channel single-top production, the last missing piece of the two-loop virtual corrections to this process. Our calculation employs analytic reduction to master integrals and the auxiliary mass flow method for their fast numerical evaluation. We study the impact of these corrections on basic observables that are measured experimentally in the single-top production process.

Abstract We compute the top quark contribution to the two-loop amplitude for on-shell Z boson pair production in gluon fusion, gg → ZZ. Exact dependence on the top quark mass is retained. For each phase space point the integral reduction is performed numerically and the master integrals are evaluated using the auxiliary mass flow method, allowing fast computation of the amplitude with very high precision.

Abstract We compute the contribution of third generation quarks (t, b) to the two-loop amplitude for on-shell W boson pair production in gluon fusion gg → WW. We present plots for the amplitude across partonic phase space as well as reference values for two kinematic points. The master integrals are efficiently evaluated by numerically solving a system of ordinary differential equations.

Abstract We calculate the NLO corrections for the gluon fragmentation functions to a heavy quark-antiquark pair in 1 S 0 [1] or 1 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. η c and J/ψ) production at the LHC.