We demonstrate that the collinear expansion fails in the case of gluon emission from a fast quark produced in $eA$ DIS. In this approximation the N=1 rescattering contribution to the gluon spectrum vanishes. We show that the higher-twist approach by Guo, Wang and Zhang [4,5] based on the collinear expansion is wrong. The nonzero gluon spectrum obtained in [4,5] is a consequence of unjustified neglecting some important terms in the collinear expansion.
We consider the multi-collinear limit of multi-gluon QCD amplitudes at tree level. We use the MHV rules for constructing colour ordered tree amplitudes and the general collinear factorisation formula to derive timelike splitting functions that are valid for specific numbers of negative helicity gluons and an arbitrary number of positive helicity gluons (or vice versa).
We show that the arguments against our recent paper on the failure of the collinear expansion in the calculation of the induced gluon emission raised by X.N. Wang are either incorrect or irrelevant.
At the non-perturbative stage of jet evolution, fluctuations of collinear gluons are less than those for coherent states that is indication of gluon squeezed states. We show that gluon entangled states which are closely related with two-mode squeezed states of gluon fields can appear by analogy with corresponding photon states in quantum optics.
We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next- to-next-to-next-to-leading order (N$^3$LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.
We analyze the structure of running coupling corrections to the gluon production cross section in the projectile-nucleus collisions calculated in the Color Glass Condensate (CGC) framework. We argue that for the gluon production cross section (and for gluon transverse momentum spectra and multiplicity) the inclusion of running coupling corrections brings in collinear singularities due to final state splittings completely unaffected by CGC resummations. Hence, despite the saturation/CGC dynamics, the gluon production cross section is not infrared-safe. As usual, regularizing the singularities r…
We clarify the confusion in a recent paper by Aurenche, Zakharov and Zaraket (AZZ) \cite{azz} over the procedure and region of validity of the collinear approximation in the twist expansion approach to induced gluon emission in deeply inelastic scattering (DIS) off a nucleus target. We point out that, in this approach to the semi-inclusive spectrum, the transverse momentum $\vec\ell_\perp$ of the induced gluon must be fixed in the collinear expansion in the transverse momentum $\vec k_\perp$ of the initial partons, therefore the result is valid for $<k_\perp^2> \ll \ell_\perp^2 \ll Q^2$. In th…
The factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators in an effective field theory. Exclusive hadronic factorization and inclusive partonic factorization follow as special cases. The leading order Lagrangian is derived using power counting and gauge invariance in the effective theory. Several species of gluons are required, and softer gluons appear as background fields to gluons with harder momenta. Two examples are given: the factorization of soft gluons in B->D pi, and the soft-collinear convolution for the B->Xs gamma spectrum.
I present analytical expressions for the collinear and soft gluon corrections to Higgs production via the process b bbar -> H as well as gg -> H through next-to-next-to-next-to-leading order (NNNLO). The soft corrections are complete while the collinear corrections include leading and some subleading logarithms. Numerical results at the Tevatron and the LHC are presented, primarily for b bbar -> H. It is shown that the collinear terms greatly improve the soft and virtual approximation at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO), especially when subleading terms are …
Glauber gluons in Drell-Yan processes are soft gluons with the transverse momenta much larger than their momentum components along the directions of initial hadrons. Their existence has been a serious challenge in proving the factorization of Drell-Yan processes. The recently proposed soft collinear effect theory of QCD can provide a transparent way to show factorizations for a class of processes, but it does not address the effect of glauber gluons. In this letter we first confirm the existence of glauber gluons through an example. We then add glauber gluons into the effective theory and stud…
We construct the Lagrangian for an effective theory of highly energetic quarks with energy Q, interacting with collinear and soft gluons. This theory has two low energy scales, the transverse momentum of the collinear particles, p_perp, and the scale p_perp^2/Q. The heavy to light currents are matched onto operators in the effective theory at one-loop and the renormalization group equations for the corresponding Wilson coefficients are solved. This running is used to sum Sudakov logarithms in inclusive B -> X_s gamma and B -> X_u \ell nu decays. We also show that the interactions with collinea…
We give a simple physical derivation of the photon emission rate from the weakly coupled quark-gluon plasma connected with the collinear processes $q\to γq$ and $q\bar{q}\to γ$. The analysis is based on the light-cone path integral approach to the induced radiation. Our results agree with that by Arnold, Moore and Yaffe obtained using the real-time thermal perturbation theory. It is demonstrated that the solution of the AMY integral equation is nothing but the time-integrated Green's function of the light-cone path integral approach written in the momentum representation.
We compute the amplitudes for non-leptonic annihilation decays of $B$ mesons into two particles within the pQCD collinear approach. The end point divergences are regulated with the help of an infrared finite gluon propagator characterized by a non-perturbative dynamical gluon mass consistent with recent lattice simulations. The divergences at twist-3 are regulated by a dynamical quark mass. Our results fit quite well the existent data of $B^0\to D_s^-K^+$ and $B^0\to D_s^{-*}\bar{K}^+$ for the expected range of dynamical gluon masses. We also make predictions for the rare decays $\bar{B}^0\to …
Effects of Glauber gluons, which cause the elastic scattering process between different jets, are studied in the frame of soft-collinear effective theory(SCET). Glauber modes are added into the Lagrangian before integrated out, which is helpful in studies on Glauber couplings of collinear and soft particles explicitly. It is proved that interactions after the hard collision cancel out in processes inclusive enough. So are interactions with the light cone coordinates Sx^{+}$ and $x^{-}$ greater than those of the hard collision. Eikonalization of Glauber couplings of active particles and absorpt…
We consider the multi-collinear limit of multi-gluon QCD amplitudes at tree level. We use the MHV rules for constructing colour ordered tree amplitudes and the general collinear factorization formula to derive timelike splitting functions that are valid for specific numbers of negative helicity gluons and an arbitrary number of positive helicity gluons (or vice versa). As an example we present new results describing the collinear limits of up to six gluons.
The full, explicit, scale dependence of the inclusive N3LO cross section for single Higgs hadroproduction is obtained by calculating the convolutions of collinear splitting kernels with lower-order partonic cross sections. We provide results for all convolutions of splitting kernels and lower-order partonic cross sections to the order in epsilon needed for the full N3LO computation, as well as their expansions around the soft limit. We also discuss the size of the total scale uncertainty at N3LO that can be anticipated with existing information.
Gluon interactions involving spectator partons in collisions at hadronic machines are investigated. We find a class of examples in which a mode, called Glauber gluons, must be introduced to the effective theory for consistency.
We demonstrate that the N=1 rescattering contribution to the gluon radiation from a fast massless quark in eA DIS vanishes in the collinear approximation. It is shown that the nonzero N=1 gluon spectrum obtained in the higher-twist approach by Guo, Wang and Zhang is a consequence of unjustified neglecting some important terms in the collinear expansion.
We study color entanglement effect for T-odd cases in collinear twist-3 factorization. As an example, we compute the transverse single spin asymmetry for direct photon production in pp collisions in pure collinear twist-3 approach. By analyzing the gauge link structure of the collinear gluon distribution on unpolarized target side, we demonstrate how color entanglement effect arises in the presence of the additional gluon attachment from polarized projectile. The result is consistent with that obtained from a hybrid approach calculation.
We study parton-branching solutions of QCD evolution equations and present a method to construct both collinear and transverse momentum dependent (TMD) parton densities from this approach. We work with next-to-leading-order (NLO) accuracy in the strong coupling. Using the unitarity picture in terms of resolvable and non-resolvable branchings, we analyze the role of the soft-gluon resolution scale in the evolution equations. For longitudinal momentum distributions, we find agreement of our numerical calculations with existing evolution programs at the level of better than 1 percent over a range…
We discuss how higher-point QCD amplitudes may be constructed from lower point ones by imposing the factorization constraints in the limits as external momenta become collinear. As a particular example, the all-$n$ gluon one-loop amplitude with maximal helicity violation is presented. We also discuss the necessary collinear behavior of the $n$-gluon amplitudes.
We outline a proof of factorization in exclusive processes, taking into account the presence of soft and collinear modes of arbitrarily low energy, which arise when the external lines of the process are taken on shell. Specifically, we examine the process of e^+e^- annihilation through a virtual photon into two light mesons. In an intermediate step, we establish a factorized form that contains a soft function that is free of collinear divergences. In contrast, in soft-collinear effective theory, the low-energy collinear modes factor most straightforwardly into the soft function. We point out t…
We present a conjecture for the $n$-gluon one-loop amplitudes with maximal helicity violation. The conjecture emerges from the powerful requirement that the amplitudes have the correct behavior in the collinear limits of external momenta. One implication is that the corresponding amplitudes where three or more gluon legs are replaced by photons vanish for $n>4$.
We calculate cross section for production of $χ_c$ pairs in proton-proton collisions. The cross section for the $g g \to χ_{c J_1} χ_{c J_2}$ is considerably smaller (especially for $χ_{c 1} χ_{c 1}$ final state) than that obtained recently in the $k_T$-factorization approach. We calculate therefore next-to-leading order contributions with $χ_c$ pair and one extra associated (mini-)jet. We find these contributions to be much larger than those for the $2 \to 2$ contribution. Especially the emission of a leading gluon (carrying a large momentum fraction of one of the incoming gluons) are importa…
We consider the production of $Z$ bosons associated with beauty quarks at the LHC using a combined $k_T$ + collinear QCD factorization approach, that interpolates between small $x$ and large $x$ physics. Our consideration is based on the off-shell gluon-gluon fusion subprocess $g^* g^* \to Z Q\bar Q$ at the leading order ${\cal O}(αα_s^2)$ (where the $Z$ boson further decays into a lepton pair), calculated in the $k_T$-factorization approach, and several subleading ${\cal O}(αα_s^2 )$ and ${\cal O}(αα_s^3 )$ subprocesses involving quark-antiquark and quark-gluon interactions, taken into accoun…
The structure of tree-level scattering amplitudes for collinear massless bosons is studied beyond their leading splitting function behavior. These near-collinear limits at sub-leading order are best studied using the Cachazo-He-Yuan (CHY) formulation of the S-matrix based on the scattering equations. We compute the collinear limits for gluons, gravitons and scalars. It is shown that the CHY integrand for an n-particle gluon scattering amplitude in the collinear limit at sub-leading order is expressed as a convolution of an (n-1)-particle gluon integrand and a collinear kernel integrand, which …
I present the resummation of collinear and soft gluon corrections to single top quark production in the t channel at next-to-next-to-leading logarithm (NNLL) accuracy using two-loop soft anomalous dimensions. The expansion of the resummed cross section yields approximate next-to-next-to-leading order (NNLO) cross sections. Numerical results for t-channel single top quark (or single antitop) production at the Tevatron and the LHC are presented, including the dependence of the cross sections on the top quark mass, and the uncertainties from scale variation and parton distributions. Combined resu…
We analyze soft and collinear gluon resummation effects at the N$^3$LL level for Standard Model Higgs boson production via gluon fusion $gg\to H$ and the neutral scalar and pseudoscalar Higgs bosons of the minimal supersymmetric extension at the N$^3$LL and NNLL level, respectively. We introduce refinements in the treatment of quark mass effects and subleading collinear gluon effects within the resummation. Soft and collinear gluon resummation effects amount to up to about 5% beyond the fixed-order results for scalar and pseudoscalar Higgs boson production.
I briefly review some general features and some recent developments concerning the resummation of long-distance singularities in QCD and in more general non-abelian gauge theories. I emphasize the field-theoretical tools of the trade, and focus mostly on the exponentiation of infrared and collinear divergences in amplitudes, which underlies the resummation of large logarithms in the corresponding cross sections. I then describe some recent results concerning the conformal limit, notably the case of N = 4 superymmetric Yang-Mills theory
In this paper we analyze the exclusive photoproduction of heavy quarkonia pairs in the collinear factorization framework. We evaluate the amplitude of the process for $J/ψ\,-η_{c}$ quarkonia pair in the leading order of the strong coupling $α_{s},$ and express it in terms of generalized parton distributions (GPDs) of gluons in the proton. We made numerical estimates in the kinematics of the Electron Ion Collider, and found that in the photoproduction regime, when the virtuality of the photon is much smaller than the quarkonia mass, the cross-section of the process is sufficiently large for exp…
Motivated by the recent work of Brzemiński, Motyka, Sadzikowski and Stebel in arXiv:1611.04449, where forward Drell--Yan production is studied in proton-proton collisions at the LHC, we improve their calculation by introducing an unintegrated gluon density obtained in arXiv:1209.1353 and arXiv:1301.5283 from a fit to combined HERA data at small values of Bjorken $x$. This gluon density was calculated within the BFKL formalism at next-to-leading order with collinear corrections. We show that it generates a good description of the forward Drell--Yan cross section dependence on the invariant mass…
The distributions of two-jet event shapes contain information on hadronization in QCD. Near the two-jet limit, these distributions can be described by convolutions of nonperturbative event shape functions with the same distributions calculated in resummed perturbation theory. The shape functions, in turn, are determined by correlations of momentum flow operators with each other and with light-like Wilson lines, which describe the coupling of soft, wide-angle radiation to jets. We observe that leading power corrections to the mean values of event shapes are determined by the correlation of a si…
We construct a class of pure quark and gluon observables by using the collinear drop grooming technique. The construction is based on linear combinations of multiple cumulative distributions of the jet mass in collinear drop, whose specific weights are fully predicted perturbatively. This yields observables which obtain their values purely from quarks (or purely from gluons) in a wide region of phase space. We demonstrate this by showing that these observables are effective in two phase space regions, one dominated by perturbative resummation and one dominated by nonperturbative effects. The n…
We study the role of running coupling and the effect of variation of the thermal quark mass on contribution of the collinear bremsstrahlung and annihilation to photon emission in $AA$ collisions in a scheme similar to that used in our previous jet quenching analyses. We find that for a scenario with the thermal quark mass $m_q\sim 50-100$ MeV contribution of the higher order collinear processes summed with the $2\to 2$ processes can explain a considerable part ($\sim 50$\%) of the experimental photon spectrum at $k_T\sim 2-3$ GeV for Au+Au collisions at $\sqrt{s}=0.2$ TeV. But for $m_q=300$ Me…
The effects of the soft gluon emission in hard scattering processes at the phase boundary are resummed in the soft-collinear effective theory (SCET). In SCET, the soft gluon emission is decoupled from the energetic collinear part, and is obtained by the vacuum expectation value of the soft Wilson-line operator. The form of the soft Wilson lines is universal in deep inelastic scattering, in the Drell-Yan process, in the jet production from e+e- collisions, and in the gamma* gamma* -> pi0 process, but its analytic structure is slightly different in each process. The anomalous dimensions of the s…
The thrust distribution in e+e- annihilation is calculated exploiting its exponentiation property in the two-jet region t = 1-T << 1. We present a general method (DGE) to calculate a large class of logarithmically enhanced terms, using the dispersive approach in renormalon calculus. Dressed Gluon Exponentiation is based on the fact that the exponentiation kernel is associated primarily with a single gluon emission, and therefore the exponent is naturally represented as an integral over the running coupling. Fixing the definition of Lambda is enough to guarantee consistency with the exact expon…
A new type of parametrization for parton distribution functions in a proton, based on their $Q^2$-evolution at large and small $x$ values, is constructed. In our analysis, the valence and nonsinglet parts obey the Gross-Llewellyn-Smith and Gottfried sum rules, respectively. For the singlet quark and gluon densities momentum conservation is taken into account. Then, using the Kimber-Martin-Ryskin prescription, we extend the consideration to Transverse Momentum Dependent (TMD, or unintegrated) gluon and quark distributions in a proton, which currently plays an important role in a number of pheno…
We discuss the relation between the treatments of the higher twist nuclear effects in eA DIS based on the pQCD collinear approximation and the light-cone path integral formalism. We show that in the collinear approximation the N=1 rescattering contribution to the gluon emission vanishes. It is demonstrated that the nonzero gluon spectrum obtained by Guo, Wang and Zhang is a consequence of unjustified neglect of some terms in the collinear expansion.
The energy loss of hard jets produced in the Deep-Inelastic scattering (DIS) off a large nucleus is considered in the collinear limit. In particular, the single gluon emission cross section due to multiple scattering in the medium is calculated. Calculations are carried out in the higher-twist scheme, which is extended to include contributions from multiple transverse scatterings on both the produced quark and the radiated gluon. The leading length enhanced parts of these power suppressed contributions are resummed. Various interferences between such diagrams lead to the Landau-Pomeranchuk-Mig…
We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.
The effective operator tr(G^3) is the only dimension-6 gluonic operator that cannot be related to four-quark operators. A peculiar property of this operator is that it does not contribute to two-jet production at hadron colliders, at the level of one operator insertion and leading-order in alpha_s; therefore we study its effects on three jet events. To calculate the helicity amplitudes induced by this operator we make extensive use of collinear factorization. We propose several ways of detecting the tr(G^3) signal, one of which exploits its non-trivial behavior under azimuthal rotations of two…
For the case of $n$-jet production at next-to-next-to-leading order in the QCD coupling, in the infrared divergent corners of phase space where particles are collinear or soft, one must evaluate $(n+1)$-parton final-state one-loop amplitudes through $\Ord(\eps^2)$, where $\eps$ is the dimensional regularization parameter. For the case of gluons, we present to all orders in $\eps$ the required universal functions which describe the behavior of one-loop amplitudes in the soft and collinear regions of phase space. An explicit example is discussed for three-parton production in multi-Regge kinemat…
We calculate the transverse momentum dependent gluon-to-gluon splitting function within $k_T$-factorization, generalizing the framework employed in the calculation of the quark splitting functions in [1-3] and demonstrate at the same time the consistency of the extended formalism with previous results. While existing versions of $k_T$ factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel or the Ciafaloni-Catani-Fiore-Marchesini (CCFM) kernel, the obtained splitting function has the important propert…
We investigate the effect of the resummation of collinear double logarithms in the BFKL gluon Green function using the Monte Carlo event generator BFKLex. The resummed collinear terms in transverse momentum space were calculated in Ref. [1] and correspond to the addition to the NLO BFKL kernel of a Bessel function of the first kind whose argument contains the strong coupling and a double logarithm of the ratio of the squared transverse momenta of the reggeized gluons. We discuss how these additional terms improve the collinear convergence of the whole approach and reduce the asymptotic growth …
Early local equilibration of a hot medium created in high-energy heavy ion collisions has been one of the long standing issues in hadron physics. The glasma model predicts that the medium initially has a large amount of high-momentum gluons but few quarks. We develop a phenomenological model based on collinear splitting and recombination of quarks and gluons for a simplified description of thermalization and chemical equilibration. We find that both could be achieved in a short time but chemical equilibration is slower than thermalization.
We present the first application of a recently-developed effective theory of jet propagation in matter SCETG to inclusive hadron suppression in nucleus-nucleus collisions at the LHC. SCETG-based splitting kernels allow us to go beyond the traditional energy loss approximation and unify the treatment of vacuum and medium-induced parton showers. In the soft gluon emission limit, we establish a simple analytic relation between the QCD evolution and energy loss approaches to jet quenching. We quantify the uncertainties associated with the implementation of the in-medim modification of hadron produ…
We derive the splitting kernels for partons produced in large $Q^2$ scattering processes that subsequently traverse a region of strongly-interacting matter using a recently-developed effective theory \SCETG. We include all corrections beyond the small-$x$ approximation, consistent with the power counting of \SCETG. We demonstrate how medium recoil, geometry and expansion scenarios, and phase space cuts can be implemented numerically for phenomenological applications. For the simplified case of infinite transverse momentum kinematics and a uniform medium, we provide closed-form analytic results…
We present an extension to next-to-leading order in the strong coupling constant $g$ of the AMY effective kinetic approach to the energy loss of high momentum particles in the quark-gluon plasma. At leading order, the transport of jet-like particles is determined by elastic scattering with the thermal constituents, and by inelastic collinear splittings induced by the medium. We reorganize this description into collinear splittings, high-momentum-transfer scatterings, drag and diffusion, and particle conversions (momentum-preserving identity-changing processes). We show that this reorganized de…
In position space the interaction terms of soft-collinear effective theory must be multipole-expanded to obtain interaction terms with homogeneous scaling behaviour. In this note we provide a manifestly gauge-invariant formulation of the theory after this expansion in the presence of non-abelian gauge fields, extending our previous result. We give the effective Lagrangian (including the Yang-Mills Lagrangian for collinear and ultrasoft gluons) and heavy-to-light transition currents to second order in the power expansion, paying particular attention to the field redefinitions that lead to the g…
In this contribution, the one-gluon radiation spectrum off a massive quark-antiquark ($q {\bar q}$) antenna traversing a QCD medium is calculated in the eikonal approximation. The interference between emissions from the quark and the antiquark is considered. The gluon spectrum computed at first order in the opacity expansion is collinear finite but infrared divergent, which is in contrast with the result obtained from an independent emitter which is both infrared and collinear finite. Phenomenological consequences on the broadening of the emitted gluon are investigated. In the soft gluon emiss…
I present calculations of soft and collinear corrections to the cross sections for single top quark production, and for Higgs production via b bbar -> H at the Tevatron and the LHC. I show that the corrections provide significant enhancements to the cross sections in both cases. For single top production the soft gluon corrections dominate the cross section, particularly in the s channel and in tW production. For Higgs production it is shown that purely collinear terms have to be included as well to provide an accurate calculation.
The medium-induced one-gluon radiation spectrum off a massive quark-antiquark ($q {\bar q}$) antenna traversing a colored QCD medium is calculated in this contribution. The gluon spectrum off the antenna computed at first order in the opacity expansion is collinear finite but infrared divergent, which is different from the result obtained from an independent emitter which is both infrared and collinear finite. The interference between emitters dominates the soft gluon radiation when the antenna opening angle is small and the emitted gluon is soft, whereas the antenna behaves like a superpositi…
