Soft Gluon Research Articles

Parton Energy Loss in the Generalized High-Twist Approach

We calculate the radiative parton energy loss in the deeply inelastic scattering (DIS) off a large nucleus within a generalized high-twist approach. The final gluon radiation spectrum is a convolution of the hard partonic part and the transverse momentum dependent (TMD) quark-gluon correlation function, without the twist expansion in the transverse momentum of the initial gluons used in original high twist approach. The TMD quark-gluon correlation function can be factorized approximately as the product of initial quark distribution and TMD gluon distribution which can be used to define the generalized or TMD jet transport coefficient. The radiation spectrum will recover the Gylassy-Levai-Vitev (GLV) result in the first order of the opacity expansion, under the static scattering center and soft gluon radiation approximation. We also investigate numerically the difference as a result of the soft gluon radiation approximation, under the static scattering center approximation.

Factorization and its breaking in dijet single transverse spin asymmetries in pp collisions

We study factorization in single transverse spin asymmetries for dijet production in proton-proton collisions, by considering soft gluon radiation at one-loop order. We show that the associated transverse momentum dependent (TMD) factorization is valid at the leading logarithmic level. At next-to-leading-logarithmic (NLL) accuracy, however, we find that soft gluon radiation generates terms in the single transverse spin dependent cross section that differ from those known for the unpolarized case. As a consequence, these terms cannot be organized in terms of a spin independent soft factor in the factorization formula. We present leading logarithmic predictions for the single transverse spin dijet asymmetry for $pp$ collisions at RHIC, based on quark Sivers functions constrained by semi-inclusive deep inelastic scattering data. We hope that our results will contribute to a better understanding of TMD factorization breaking effects at NLL accuracy and beyond.

Theory for quarkonium: from NRQCD factorization to soft gluon factorization * *Supported by the National Natural Science Foundation of China (11875071, 11975029) and the China Postdoctoral Science Foundation (2018M631234)

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 arrive at this conclusion. In the first method, we apply the two factorization theories to the physical process . Our explicit calculation shows that both SGF and NRQCD can correctly reproduce the low energy physics of full QCD, and the two factorizations are thus 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 relation as a byproduct. Compared with the NRQCD factorization, the advantage of SGF is that it resums the series of relativistic corrections originating from kinematic effects to all powers, yielding better convergence of the relativistic expansion.

Jet momentum broadening in the pre-equilibrium Glasma

Jets are important probes of heavy ion collisions as they can provide information on the interaction of a highly energetic parton with the medium it traverses. In the hydrodynamic stage of dense, strongly interacting matter, these interactions can be explained in terms of scattering processes, soft gluon emission and collinear parton splittings. However, jets originate even before the hydrodynamic stage. Here we report on the first numerical simulation of transverse momentum broadening of jets stemming from the interaction of partons with boost-invariantly expanding Glasma flux tubes. The Glasma stage is a pre-hydrodynamic stage based on the Color Glass Condensate framework. Our calculation shows strong time-dependence and an intrinsic anisotropy of momentum broadening in the directions transverse to the jet propagation direction.

Eikonal amplitudes for three-hard legs processes at finite-Nc

We extend our previous work on scattering amplitudes [1] to hadron collisions. We provide a general formalism for the computation of eikonal amplitudes squared for the radiation of soft and energy-ordered gluons off three massless hard-legs at finite-Nc to any order in the perturbative expansion. Examples of three-hard legs processes include vector/Higgs boson production in association with a single hard jet and dijet production in DIS. Explicit expressions for the radiation of up to four gluons are provided as an illustration of the formalism.

On Sugawara construction on celestial sphere

Conformally soft gluons are conserved currents of the Celestial Conformal Field Theory (CCFT) and generate a Kac-Moody algebra. We study celestial amplitudes of Yang-Mills theory, which are Mellin transforms of gluon amplitudes and take the double soft limit of a pair of gluons. In this manner we construct the Sugawara energy-momentum tensor of the CCFT. We verify that conformally soft gauge bosons are Virasoro primaries of the CCFT under the Sugawara energy-momentum tensor. The Sugawara tensor though does not generate the correct conformal transformations for hard states. In Einstein-Yang- Mills theory, we consider an alternative construction of the energy-momentum tensor, similar to the double copy construction which relates gauge theory amplitudes with gravity ones. This energy momentum tensor has the correct properties to generate conformal transformations for both soft and hard states. We extend this construction to supertranslations.

Building a consistent parton shower

Modern parton showers are built using one of two models: dipole showers or angular ordered showers. Both have distinct strengths and weaknesses. Dipole showers correctly account for wide-angle, soft gluon emissions and track the leading flows in QCD colour charge but they are known to mishandle partonic recoil. Angular ordered showers keep better track of partonic recoil and correctly include large amounts of wide-angle, soft physics but azimuthal averaging means they are known to mishandle some correlations. In this paper, we derive both approaches from the same starting point; linking our under- standing of the two showers. This insight allows us to construct a new dipole shower that has all the strengths of a standard dipole shower together with the collinear evolution of an angular-ordered shower. We show that this new approach corrects the next-to-leading- log errors previously observed in parton showers and improves their sub-leading-colour accuracy.

Infrared structure of SU(N) \xd7 U(1) gauge theory to three loops

We study the infrared (IR) structure of SU(N ) × U(1) (QCD × QED) gauge theory with nf quarks and nl leptons within the framework of perturbation theory. In particular, we unravel the IR structure of the form factors and inclusive real emission cross sections that contribute to inclusive production of color neutral states, such as a pair of leptons or single W/Z in Drell-Yan processes and a Higgs boson in bottom quark annihilation, in Large Hadron Collider (LHC) in the threshold limit. Explicit computation of the relevant form factors to third order and the use of Sudakov’s K + G equation in SU(N ) × U(1) gauge theory demonstrate the universality of the cusp anomalous dimensions (AI, I = q, b). The abelianization rules that relate AI of SU(N ) with those from U(1) and SU(N) × U(1) can be used to predict the soft distribution that results from the soft gluon emission subprocesses in the threshold limit. Using the latter and the third order form factors, we can obtain the collinear anomalous dimensions (BI) and the renormalisation constant Zb to third order in perturbation theory. The form factors, the process independent soft distribution functions can be used to predict fixed and resummed inclusive cross sections to third order in couplings and in leading logarithmic approximation respectively.

The transverse momentum spectrum of low mass Drell\u2013Yan production at next-to-leading order in the parton branching method

It has been observed in the literature that measurements of low-mass Drell–Yan (DY) transverse momentum spectra at low center-of-mass energies sqrt{s} are not well described by perturbative QCD calculations in collinear factorization in the region where transverse momenta are comparable with the DY mass. We examine this issue from the standpoint of the Parton Branching (PB) method, combining next-to-leading-order (NLO) calculations of the hard process with the evolution of transverse momentum dependent (TMD) parton distributions. We compare our predictions with experimental measurements at low DY mass, and find very good agreement. In addition we use the low mass DY measurements at low sqrt{s} to determine the width q_s of the intrinsic Gauss distribution of the PB-TMDs at low evolution scales. We find values close to what has earlier been used in applications of PB-TMDs to high-energy processes at the Large Hadron Collider (LHC) and HERA. We find that at low DY mass and low sqrt{s} even in the region of p_mathrm{T}/m_mathrm{DY}sim 1 the contribution of multiple soft gluon emissions (included in the PB-TMDs) is essential to describe the measurements, while at larger masses (m_mathrm{DY}sim m_{{mathrm{Z}}}) and LHC energies the contribution from soft gluons in the region of p_mathrm{T}/m_mathrm{DY}sim 1 is small.

Medium-induced cascade in expanding media

Detailed insight into the interplay between parton energy loss and the way deconfined medium created in heavy-ion collisions expands is of great importance for improving the understanding of the jet quenching phenomenon. In this paper we study the impact of the expansion of deconfined medium on the single-gluon emission spectrum, its resummation and the jet suppression factor (QAA) within the BDMPS-Z formalism. We calculate these quantities for three types of expansion scenarios, namely static, exponentially decaying and Bjorken expanding media. The distribution of medium-induced gluons is calculated using an evolution equation with splitting kernels derived from the gluon emission spectra. A universal behavior of splitting kernels is derived in the regime of soft gluon emissions when evaluated at a common effective evolution time τeff. Novel scaling features of the resulting gluon distribution and jet QAA are discussed. For realistic spectra valid beyond the soft-gluon emission limit, where the results are obtained by a numerical solution of the evolution equation, these features are partially replaced by a scaling expected from considering an averaged jet quenching parameter along the trajectory of propagation. Further we show that differences arising from different types of the medium expansion can be to a large extent scaled out by appropriate choice of the quenching parameter. Sizable differences among the values of the quenching parameter for different types of medium expansion point to the importance of the medium expansion for precise modeling of the jet quenching phenomenon.

Endpoint resummation in squark decays

We study soft and collinear gluon emission in squark decays to quark--neutralino pair, at next-to-next-to-leading logarithmic (NNLL) accuracy in the end-point region, using Soft Collinear Effective Theory (SCET), and at next-to-leading (NLO) fixed order in the rest of the phase space. As a phenomenological case study we discuss the impact of radiative corrections on the simultaneous measurements of squark and neutralino masses at a linear $e^{+}e^{-}$ collider based on $\sqrt{s} = 3$ TeV Compact Linear Collider (CLIC). Since the majority of mass measurement techniques are based on edges in kinematic distributions, and these change appreciably when there is additional QCD radiation in the final state, the knowledge of higher-order QCD effects is required for precise mass determinations.

TMD soft function from large-momentum effective theory

We study Euclidean formulations of the transverse-momentum-dependent (TMD) soft function, which is a cross section for soft gluon radiations involving color charges moving in two conjugate lightcone directions in quantum chromodynamics. We show it is related to a special form factor of a pair of color sources traveling with nearly-lightlike velocities, which can be matched to TMD physical observables in semi-inclusive deep-inelastic scattering and Drell-Yan process in the framework of large momentum effective theory. It can also be extracted by combining a large-momentum form factor of light meson and its leading TMD wave function. These formulations are useful for initiating nonperturbative calculations of this useful quantity.

Production of f0(980) meson at the LHC: Color evaporation versus color-singlet gluon-gluon fusion

The production of the f0(980) meson at high energies is not well understood. We investigate two different potential mechanisms for inclusive scalar meson production in the kt-factorization approach: color-singlet gluon-gluon fusion and color evaporation model. The γ⁎γ⁎→f0(980) form factor(s) can be constraint from the f0(980) radiative decay width. The g⁎g⁎→f0(980) form factors are obtained by a replacement of αem electromagnetic coupling constant by αs strong coupling constant and appropriate color factors. The form factors for the two couplings are parametrized with a function motivated by recent results for scalar quarkonia. The differential cross sections are calculated in the kt-factorization approach with modern unintegrated gluon distributions. Unlike for quarkonia it seems rather difficult to describe a preliminary ALICE data for inclusive production of f0(980) exclusively by the color singlet gluon-gluon fusion mechanism. Two different scenarios for flavor structure of f0(980) are considered in this context. We consider also mechanism of fusion of quark-antiquark associated with soft gluon emission in a phenomenological color evaporation model (CEM) used sometimes for quarkonium production. Here we use kt-factorization version of CEM to include higher-order contributions. In addition, for comparison we consider also NLO collinear approach with qq¯q and qq¯g color octet partonic final states. Both approaches lead to a similar result. However, very large probabilities are required to describe the preliminary ALICE data. The pomeron-pomeron fusion mechanism is also discussed and results are quantified.

Soft gluon emission at two loops in full color

The soft emission factor is a central ingredient in the factorization of generic n-particle gauge theory amplitudes with one soft gluon in the external state. We present the complete two-loop soft factor, capturing the leading power behavior in the soft-gluon momentum. At two loops, the color structure and the kinematic dependence of the soft factor become nontrivial as the soft gluon can couple to three hard partons for the first time (tripole terms). The nontrivial kinematic dependence of the tripole terms is of uniform, maximal transcendental weight, and can be expressed (in a “Euclidean” region) in terms of single-valued harmonic polylogarithms. Our results are consistent with the behavior of the recently computed symbol of the two-loop five-particle amplitude in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory. In the limit where the outgoing soft gluon is also collinear with an incoming hard parton, potentially dangerous factorization-violating terms can arise.

JIMWLK evolution, Lindblad equation and quantum-classical correspondence

In the Color Glass Condensate (CGC) effective theory, the physics of valence gluons with large longitudinal momentum is reflected in the distribution of color charges in the transverse plane. Averaging over the valence degrees of freedom is effected by integrating over classical color charges with some quasi probability weight functional W [j] whose evolution with rapidity is governed by the JIMWLK equation. In this paper, we reformulate this setup in terms of effective quantum field theory on valence Hilbert space governed by the reduced density matrix hat{rho} for hard gluons, which is obtained after properly integrating out the soft gluon “environment”. We show that the evolution of this density matrix with rapidity in the dense and dilute limits has the form of Lindblad equation. The quasi probability distribution (weight) functional W is directly related to the reduced density matrix hat{rho} through the generalization of the Wigner-Weyl quantum-classical correspondence, which reformulates quantum dynamics on Hilbert space in terms of classical dynamics on the phase space. In the present case the phase space is non Abelian and is spanned by the components of transverse color charge density j. The same correspondence maps the Lindblad equation for hat{rho} into the JIMWLK evolution equation for W .

The soft quark Sudakov

There has been recent interest in understanding the all loop structure of the subleading power soft and collinear limits, with the goal of achieving a systematic resummation of subleading power infrared logarithms. Most of this work has focused on subleading power corrections to soft gluon emission, whose form is strongly constrained by symmetries. In this paper we initiate a study of the all loop structure of soft fermion emission. In mathcal{N} = 1 QCD we perform an operator based factorization and resummation of the associated infrared logarithms using the formalism introduced in [1], and prove that they exponentiate into a Sudakov due to their relation to soft gluon emission. We verify this result through explicit calculation to mathcal{O}left({alpha}_s^3right) . We show that in QCD, this simple Sudakov exponentiation is violated by endpoint contributions proportional to (CA−CF)n which contribute at leading logarithmic order. Combining our mathcal{N} = 1 result and our calculation of the endpoint contributions to mathcal{O}left({alpha}_s^3right) , we conjecture a result for the soft quark Sudakov in QCD, a new all orders function first appearing at subleading power, and give evidence for its universality. Our result, which is expressed in terms of combinations of cusp anomalous dimensions in different color representations, takes an intriguingly simple form and also exhibits interesting similarities to results for large-x logarithms in the off diagonal splitting functions.

Exclusive quarkonium production or decay in soft gluon factorization

In this paper, we study the application of the recently proposed soft gluon factorization (SGF) to exclusive quarkonium production or decay. We find that in the nonrelativistic QCD factorization framework there are too many nonperturbative parameters. Thanks to the factorization of kinematical physics from dynamical physics, the SGF significantly reduces the number of nonperturbative parameters. Therefore, the SGF can improve our predictive power of exclusive quarkonium production or decay. By applying to ηc + γ production at B-factories, our result is the closest one to data among all theoretical calculations.

Drell\u2013Yan production with the CCFM-K evolution

We discuss the Drell–Yan dilepton production using the transverse momentum dependent parton distributions evolved with the Catani–Ciafaloni–Fiorani–Marchesini–Kwieciński (CCFM-K) equations in the single loop approximation. Such equations are obtained assuming angular ordering of emitted partons (coherence) for xsim 1 and transverse momentum ordering for x ll 1. This evolution scheme also contains the Collins–Soper–Sterman (CSS) soft gluon resummation. We make a comparison with a broad class of data on transverse momentum spectra of low mass Drell–Yan dileptons.

Extracting many-body correlators of saturated gluons with precision from inclusive photon+dijet final states in deeply inelastic scattering

We highlight the principal results of a computation in the Color Glass Condensate effective field theory (CGC EFT) of the next-to-leading order (NLO) impact factor for inclusive photon+dijet production at Bjorken $x_{\rm Bj} \ll 1$ in deeply inelastic electron-nucleus (e+A DIS) collisions. When combined with extant results for next-to-leading log $x_{\rm Bj}$ JIMWLK renormalization group (RG) evolution of gauge invariant two-point ("dipole") and four-point ("quadrupole") correlators of light-like Wilson lines, the inclusive photon+dijet e+A DIS cross-section can be determined to $\sim 10$\% accuracy. Our computation simultaneously provides the ingredients to compute fully inclusive DIS, inclusive photon, inclusive dijet and inclusive photon+jet channels to the same accuracy. This makes feasible quantitative extraction of many-body correlators of saturated gluons and precise determination of the saturation scale $Q_{S,A}(x_{\rm Bj})$ at a future Electron-Ion Collider. An interesting feature of our NLO result is the structure of the violation of the soft gluon theorem in the Regge limit. Another is the appearance in gluon emission of time-like non-global logs which also satisfy JIMWLK RG evolution.

An effective field theory approach to quarkonium at small transverse momentum

In this work we apply effective field theory (EFT) to observables in quarkonium production and decay that are sensitive to soft gluon radiation, in particular measurements that are sensitive to small transverse momentum. Within the EFT framework we study χQ decay to light quarks followed by the fragmentation of those quarks to light hadrons. We derive a factorization theorem that involves transverse momentum distribution (TMD) fragmentation functions and new quarkonium TMD shape functions. We derive renormalization group equations, both in rapidity and virtuality, which are used to evolve the different terms in the factorization theorem to resum large logarithms. This theoretical framework will provide a systematic treatment of quarkonium production and decay processes in TMD sensitive measurements.