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
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.
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 $n>4$.
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 …
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 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 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 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.
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 $\bar{B}^0\to …
This chapter reviews the construction of ``soft-collinear gravity'', the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion, focusing on the essential concepts. Among them are an emergent soft background gauge symmetry, which lives on the light-like trajectories of energetic particles and allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative and the soft Riemann tensor, and a systematic treatment of collinear interact…
Effects of Glauber gluons, which cause the elastic scattering process between different jets, are studied in the frame of soft-collinear effective theory(SCET). Glauber modes are added into the Lagrangian before integrated out, which is helpful in studies on Glauber couplings of collinear and soft particles explicitly. It is proved that interactions after the hard collision cancel out in processes inclusive enough. So are interactions with the light cone coordinates Sx^{+}$ and $x^{-}$ greater than those of the hard collision. Eikonalization of Glauber couplings of active particles and absorpt…
We study collinear and soft singularities in perturbative quantum gravity by constructing an effective field theory similar to soft-collinear effective theory for QCD (SCET). We find that the soft sector exhibits factorization properties similar to those of SCET. The collinear sector is, however, quite different. While the leading-power collinear effective Lagrangian is trivial, the presence of the metric field $h_{++}$ with negative scaling dimension allows for collinear divergences in loop diagrams with couplings to non-collinear sources. We provide a compact proof of the well-known fact tha…
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 formulate N=4 supersymmetric Yang-Mills theory in terms of soft-collinear effective theory. The effective Lagrangian in soft-collinear effective theory is developed according to the power counting by a small parameter η\sim p_{\perp}/Q. All the particles in this theory are in the adjoint representation of the SU(N) gauge group, and we derive the collinear gauge-invariant Lagrangian in the adjoint and fundamental representations respectively. We consider collinear and ultrasoft Wilson lines in this theory, and show the ultrasoft factorization of the collinear Lagrangian by redefining the col…
We provide a systematic BRST formalism for the soft-collinear effective theory describing interactions of soft and collinear degrees of freedom in the presence of a hard interaction. In particular, we develop full BRST symmetry transformation for SCET theory. We further extend the BRST formulation by making the transformation field dependent. This establishes a mapping between several SCET actions consistently when defined in different gauge conditions. In fact, a definite structure of gauge-fixed actions corresponding to any particular gauge condition can be generated for SCET theory using ou…
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…
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.
In asymptotically free theories with collinear divergences it is sometimes claimed that these divergences cancel if one sums over initial and final state degenerate cross-sections and uses an off-shell renormalisation scheme. We show for scalar $ϕ^3$ theory in six dimensions that there are further classes of soft collinear divergences and that they do not cancel. Furthermore, they yield a non-convergent series of terms at a fixed order of perturbation theory. Similar effects in gauge theories are also summarised.
Factorization theorems in soft-collinear effective theory at subleading order in power counting involve ``radiative jet functions'', defined in terms of matrix elements of collinear fields with a soft momentum emitted from inside the jet. Of particular importance are the radiative quark jet functions with an external photon or gluon, which arise e.g. in the factorization theorems for the Higgs-boson amplitudes $h\toγγ$, $h\to gg$ and $gg\to h$ induced by light-quark loops. While the photon case has been studied extensively in previous work, we present here a detailed study of the radiative jet…
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 construct "soft-collinear gravity", the effective field theory which describes the interaction of collinear and soft gravitons with matter (and themselves), to all orders in the soft-collinear power expansion. Despite the absence of collinear divergences in gravity at leading power, the construction exhibits remarkable similarities with soft-collinear effective theory of QCD (gauge fields). It reveals an emergent soft background gauge symmetry, which allows for a manifestly gauge-invariant representation of the interactions in terms of a soft covariant derivative, the soft Riemann tensor, a…
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…
It is argued that soft-collinear effective theory for processes involving both soft and collinear partons, such as exclusive B-meson decays, should include a new mode in addition to soft and collinear fields. These "soft-collinear messengers" can interact with both soft and collinear particles without taking them far off-shell. They thus can communicate between the soft and collinear sectors of the theory. The relevance of the new mode is demonstrated with an explicit example, and the formalism incorporating the corresponding quark and gluon fields into the effective Lagrangian is developed.
Starting with QCD, we derive an effective field theory description for forward scattering and factorization violation as part of the soft-collinear effective field theory (SCET) for high energy scattering. These phenomena are mediated by long distance Glauber gluon exchanges, which are static in time, localized in the longitudinal distance, where $|t| \ll s$. In hard scattering, Glauber gluons can induce corrections which invalidate factorization. With SCET, Glauber exchange graphs can be calculated explicitly, and are distinct from graphs with soft, collinear, or ultrasoft gluons. We derive a…
This letter provides a superfield based approach to constructing a collinear slice of $\mathcal{N}$ = 1 superspace. The strategy is analogous to integrating out anti-collinear fermionic degrees-of-freedom as was developed in the context of soft-collinear effective theory. The resulting Lagrangian can be understood as an integral over collinear superspace, where half the supercoordinates have been integrated out. The application to $\mathcal{N}$ = 1 super Yang-Mills is presented. Collinear superspace provides the foundation for future explorations of supersymmetric soft-collinear effective theo…
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…
Collinear matching for leading power gluon transverse momentum distributions
Università degli Studi di Pavia（帕维亚大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Università degli Studi di Cagliari（卡利亚里大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari（意大利国家核物理研究所卡利亚里分部）
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 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…
It has recently been argued that soft-collinear effective theory for processes involving both soft and collinear partons contains a new soft-collinear mode, which can communicate between the soft and collinear sectors of the theory. The formalism incorporating the corresponding fields into the effective Lagrangian is extended to include external current and four-quark operators relevant to weak interactions. An explicit calculation of the anomalous dimensions of these operators reveals that soft-collinear modes are needed for correctly describing the ultraviolet behavior of the effective theor…
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…
Starting from the first renormalized factorization theorem for a process described at subleading power in soft-collinear effective theory, we discuss the resummation of Sudakov logarithms for such processes in renormalization-group improved perturbation theory. Endpoint divergences in convolution integrals, which arise generically beyond leading power, are regularized and removed by systematically rearranging the factorization formula. We study in detail the example of the $b$-quark induced $h\toγγ$ decay of the Higgs boson, for which we resum large logarithms of the ratio $M_h/m_b$ at next-to…
We construct the soft-collinear effective Lagrangian which is manifestly gauge invariant order by order. Field redefinitions of collinear gauge fields and a proper decomposition of quark fields are necessary to make the Lagrangian gauge invariant. We can obtain the effective Lagrangian order by order in SCET_I and SCET_II by adopting appropriate power counting methods. Various types of current operators can be investigated using this formalism, and we choose the soft-collinear current as a specific example to present explicit radiative corrections. The hard-collinear, collinear, and ultrasoft …
We obtain analytic results for integrated triple-collinear splitting functions that emerge as collinear counter-terms in the context of the nested soft-collinear subtraction scheme arXiv:1702.01352 . With these results, all integrated subtraction terms required for NNLO QCD computations within this scheme are known analytically. In addition to improving efficiency and numerical stability of practical computations, the availability of these results will contribute towards establishing a general NNLO QCD subtraction formula for generic hard scattering processes in hadron collisions, similar to C…
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.
Collinear fields in soft collinear effective theory (SCET) can be made invariant under collinear gauge transformations by multiplying them with collinear Wilson lines. We discuss how we can quantize SCET directly in terms of these gauge invariant fields, allowing to directly calculate S matrix elements using the gauge invariant collinear fields. We also show how for each collinear direction SCET can be written in terms of fields whose interactions are given by the usual QCD Lagrangian, and how external operators coupling these different directions can be constructed.
