We re-examine observables with rapidity divergences in the context of a formulation of Soft-Collinear Effective Theory in which infrared degrees of freedom are not explicitly separated into modes. We consider the Sudakov form factor with a massive vector boson and Drell-Yan production of lepton pairs at small transverse momentum as demonstrative examples. In this formalism, rapidity divergences introduce a scheme dependence into the effective theory and are associated with large logarithms appearing in the soft matching conditions. This scheme dependence may be used to derive the corresponding rapidity renormalization group equations, and rates naturally factorize into hard, soft and jet contributions without the introduction of explicit modes.
We study the (exclusive) kT jet algorithm using effective field theory techniques. Regularizing the virtualities and rapidities of graphs in the soft-collinear effective theory (SCET), we are able to write the next-to-leading-order dijet cross section as the product of separate hard, jet, and soft contributions. We show how to reproduce the Sudakov form factor to next-to-leading logarithmic accuracy previously calculated by the coherent branching formalism. Our result only depends on the renormalization group evolution of the hard function, rather than on that of the hard and jet functions as …
The artificial separation of a full-theory mode into distinct collinear and soft modes in SCET leads to divergent integrals over rapidity, which are not present in the full theory. Rapidity divergence introduces an additional scale into the problem, giving rise to its own renormalization group with respect to this new scale. Two contradicting claims exist in the literature concerning rapidity scale uncertainty. One camp has shown that the results of perturbative calculations depend on the precise choice of rapidity scale. The other has derived an all-order factorization formula with no depende…
We consider di-jet production in hadron collisions where a transverse veto is imposed on radiation for (pseudo-)rapidities in the central region only, where this central region is defined with rapidity cutoff. For the case where the transverse measurement (e.g., transverse energy or min $p_T$ for jet veto) is parametrically larger relative to the typical transverse momentum beyond the cutoff, the cross section is insensitive to the cutoff parameter and is factorized in terms of collinear and soft degrees of freedom. The virtuality for these degrees of freedom is set by the transverse measureme…
We summarize and extend previous results on the comparison of threshold resummation, performed, using soft-collinear effective theory (SCET), in the Becher-Neubert approach, to the standard perturbative QCD formalism based on factorization and resummation of Mellin moments of partonic cross sections. We show that the logarithmic accuracy of this SCET result can be extended by half a logarithmic order, thereby bringing it in full agreement with the standard QCD result if a suitable choice is made for the soft scale mu_s which characterizes the SCET result. We provide a master formula relating t…
A procedure is developed for using Soft Collinear Effective Theory (SCET) to generate fully exclusive events, which can then be compared to data from collider experiments. We show that SCET smoothly interpolates between QCD for hard emissions, and the parton shower for soft emissions, while resumming all large logarithms. In SCET, logarithms are resummed using the renormalization group, instead of classical Sudakov factors, so subleading logarithms can be resummed as well. In addition, all loop effects of QCD can be reproduced in SCET, which allows the effective theory to incorporate next-to-l…
In soft collinear effective theory (SCET) the interaction between high energy quarks moving in opposite directions involving momentum transfer much smaller than the center-of-mass energy is described by the Glauber interaction operator which has two-dimensional Coulomb-like behavior. Here, we determine this $n$-$\bar{n}$ collinear Glauber interaction operator and consider its renormalization properties at one loop. At this order a rapidity divergence appears which gives rise to an infrared divergent (IR) rapidity anomalous dimension commonly called the gluon Regge trajectory. We then go on to …
The resummation of radiative corrections to collider jet observables using soft collinear effective theory is encoded in differential renormalization group equations (RGEs), with anomalous dimensions depending on the observable under consideration. This observable dependence arises from the ultraviolet (UV) singular structure of real phase space integrals in the effective field theory. We show that the observable dependence of anomalous dimensions in SCETI problems can be disentangled by introducing a suitable UV regulator in real radiation integrals. Resummation in the presence of the new reg…
The hard-scattering contributions to heavy-to-light form factors at large recoil are studied systematically in soft-collinear effective theory (SCET). Large logarithms arising from multiple energy scales are resummed by matching QCD onto SCET in two stages via an intermediate effective theory. Anomalous dimensions in the intermediate theory are computed, and their form is shown to be constrained by conformal symmetry. Renormalization-group evolution equations are solved to give a complete leading-order analysis of the hard-scattering contributions, in which all single and double logarithms are…
Recently, Belle and BaBar Collaborations observed surprising suppression in the endpoint $J/ψ$ spectrum, which stimulates us to examine the endpoint behaviors of the $e^+ e^- \to J/ψgg$ production. We calculate the $J/ψ$ momentum and angular distributions for this process within the framework of the soft-collinear effective theory (SCET). The decreasing spectrum in the endpoint region is obtained by summing the Sudakov logarithms. We also find a large discrepancy between the NRQCD and SCET spectrum in the endpoint region even before the large logarithms are summed, which is probably due to the…
We argue that energetic muon-electron scattering in the backward direction can be viewed as a template case to study the resummation of large logarithms related to endpoint divergences appearing in the effective-theory formulation of hard-exclusive processes. While it is known since the mid sixties that the leading double logarithms from QED corrections resum to a modified Bessel function on the amplitude level, the modern formulation in Soft-Collinear Effective Theory (SCET) shows a surprisingly complicated and iterative pattern of endpoint-divergent convolution integrals. In contrast to the …
In the framework of soft-collinear effective theory (SCET), we show how to formulate the resummation for a broad family of final-state, global observables in high-energy collisions in a general way that is suitable for a numerical calculation. Contrary to the standard SCET approach, this results in a method that does not require an observable-specific factorization theorem. We present a complete formulation at next-to-next-to-leading logarithmic order for $e^+e^-$ observables, and show how to systematically extend the framework to higher orders. This work paves the way to automated resummation…
Using the framework of soft-collinear effective theory (SCET), we factorize and calculate $e^+e^-$ angularity distributions, including perturbative resummation and the incorporation of a universal model for the nonperturbative soft function. Angularities $τ_a$ are a class of event shapes varying in their sensitivity to the substructure of jets in the final state, controlled by a continuous parameter $a<2$. We calculate the jet and soft functions in factorized angularity distributions for all $a<1$ to first order in the strong coupling $α_s$ and resum large logarithms to next-to-leading logarit…
In this talk I give an overview of soft collinear effective theory (SCET), including a discussion of some recent advances. First, I briefly cover the foundation upon which SCET is built, namely QCD factorization, and review the theoretical framework of SCET. Next, I cover some recent calculations involving SCET. Since there are more interesting results than I can cover, I have picked four that that I find particularly exciting. The first two topics are dynamical threshold enhancements in Drell-Yan and resummation of higgs production at the LHC. Both topics center on the resummation of large Su…
We present a universal formalism for transverse momentum resummation in the view of soft-collinear effective theory (SCET), and establish the relation between our SCET formula and the well known Collins-Soper-Sterman's pQCD formula at the next-to-leading logarithmic order (NLLO). We also briefly discuss the reformulation of joint resummation in SCET.
We present compact integral representations for the calculation of two-loop anomalous dimensions for a generic class of soft functions that are defined in terms of two light-like Wilson lines. Our results are relevant for the resummation of Sudakov logarithms for e+ e- event-shape variables and inclusive hadron-collider observables at next-to-next-to-leading logarithmic accuracy within Soft-Collinear Effective Theory (SCET). Our formalism applies to both SCET-1 and SCET-2 soft functions and we clarify the relation between the respective soft anomalous dimension and the collinear anomaly expone…
We consider the class of jet shapes known as angularities in dijet production at hadron colliders. These angularities are modified from the original definitions in e+e- collisions to be boost invariant along the beam axis. These shapes apply to the constituents of jets defined with respect to either k_T-type (anti-k_T, C/A, and k_T) algorithms and cone-type algorithms. We present an SCET factorization formula and calculate the ingredients needed to achieve next-to-leading-log (NLL) accuracy in kinematic regions where non-global logarithms are not large. The factorization formula involves previ…
In this article, we report how to use the "plus-type" subtraction scheme to deal with endpoint divergences in $h\rightarrowγγ$ or $gg\rightarrow h$ amplitudes induced by light quark loop, which can be formulated by next-to-leading power (NLP) SCET. This subtraction is ensured by two re-factorization conditions, which have been proven to all orders in $α_s$. Based on these conditions, cutoffs emerge naturally after some re-arrangements to handle endpoint divergences and renormalization is compatible with that. Our formalism can analytically reproduce three-loop amplitudes up to $\ln^3(-M_h^2/m_…
We reanalyze the factorization theorems for Drell-Yan process and for deep inelastic scattering near threshold, as constructed in the framework of the soft-collinear effective theory (SCET), from a new, consistent perspective. In order to formulate the factorization near threshold in SCET, we should include an additional degree of freedom with small energy, collinear to the beam direction. The corresponding collinear-soft mode is included to describe the parton distribution function (PDF) near threshold. The soft function is modified by subtracting the contribution of the collinear-soft modes …
We present a method to calculate the soft function in Soft-Collinear Effective Theory to NLO for N-jet events, defined with respect to arbitrarily complicated observables and algorithms, using a subtraction-based method. We show that at one loop the singularity structure of all observable/algorithm combinations can be classified as one of two types. Type I jets include jets defined with inclusive algorithms for which a jet shape is measured. Type II jets include jets found with exclusive algorithms, as well as jets for which only the direction and energy are measured. Cross sections that are i…
We discuss recent progress concerning the resummation of large logarithms at next-to-leading power (NLP) in scattering processes near threshold. We begin by briefly reviewing the diagrammatic and SCET approach, which are used to derive factorization theorems for physical observables in this kinematic limit. Then, we focus on the quark-gluon channel in deep inelastic scattering and Drell-Yan. We show that the use of consistency conditions for the cancellation of leading poles in the hadronic cross section can be used to achieve the resummation of large leading logarithms (LLs) at NLP, both with…
We study exclusive radiative decays of the $Υ$ using soft-collinear effective theory and non-relativistic QCD. In contrast to inclusive radiative decays at the endpoint we find that color-octet contributions are power suppressed in exclusive decays, and can safely be neglected, greatly simplifying the analysis. We determine the complete set of Lorentz structures that can appear in the SCET Wilson coefficients and match onto them using results from a previous calculation. We run these coefficients from the scale $\mups$ to the scale $Λ\sim 1 \textrm{GeV}$, thereby summing large logarithms. Fina…
The proton form factors at large momentum transfer are dominated by two contributions which are associated with the hard and soft rescattering respectively. Motivated by a very active experimental form factor program at intermediate values of momentum transfers, $Q^{2}\sim 5-15 \text{GeV}^{2}$, where an understanding in terms of only a hard rescattering mechanism cannot yet be expected, we investigate in this work the soft rescattering contribution using soft collinear effective theory (SCET). Within such description, the form factor is characterized, besides the hard scale $Q^2$, by a semi-ha…
In this article we show how the resummation of infrared and collinear logarithms within Soft-Collinear Effective Theory (SCET) can be formulated in a way that makes it suitable for a Monte-Carlo implementation. This is done by applying the techniques developed for automated resummation using the branching formalism, which have resulted in the general resummation approach CAESAR/ARES. This work builds a connection between the two resummation approaches, and paves the way to automated resummation in SCET. As a case study we consider the resummation of the thrust distribution in electron-positron…
We calculate the two-photon exchange (TPE) corrections in the region where the kinematical variables describing the elastic $ep$ scattering are moderately large momentum scales relative to the soft hadronic scale. For such kinematics we use the QCD factorization approach formulated in the framework of the soft-collinear effective theory (SCET). Such technique allows us to develop a description for the soft-spectator scattering contribution which is found to be important in the region of moderately large scales. Together with the hard-spectator contribution we present the complete factorization…
We considered the power corrections to the amplitude describing hard exclusive pion pair production in gamma gamma collisions at large energy and momentum transfer. In order to derive a factorization formula for the subleading power corrections one has to take into account the so-called soft-overlap (or soft) contributions. These terms can be naturally introduced within the SCET framework as a matrix elements of the SCET-I operators. The so-called end-point singularities must cancel in the sum of the hard and soft contributions after matching onto SCET-II. Using the so-called physical subtract…
We derive an all-order factorization theorem for the narrow jet broadening event shape, a measure of the transverse momentum in jet events. This is a non-global observable which receives logarithmically enhanced contributions associated with the large rapidity difference between soft and collinear radiation and which is also sensitive to soft recoil effects. Our work is the first factorization analysis of an observable of this type and we show that with regard to the non-global nature, the rapidity logarithms do not constitute an essential complication since they can be tied to the jet functio…
These lectures aim to provide a pedagogical introduction to the philosophical underpinnings and technical features of Effective Field Theory (EFT). Improving control of $S$-matrix elements in the presence of a large hierarchy of physical scales $m \ll M$ is emphasized. Utilizing $λ\sim m/M$ as a power counting expansion parameter, we show how matching an ultraviolet (UV) model onto an EFT makes manifest the notion of separating scales. Renormalization Group (RG) techniques are used to run the EFT couplings from the UV to the infrared (IR), thereby summing large logarithms that would otherwise …
Soft-collinear effective theory (SCET) is used to sum Sudakov double-logarithms in the x ->1 endpoint region for the deep inelastic scattering structure function. The calculations are done in both the target rest frame and the Breit frame. The separation of scales in the effective theory implies that the anomalous dimension of the SCET current is linear in ln mu, and the anomalous dimension for the Nth moment of the structure function is linear in ln N, to all orders in perturbation theory. The SCET formulation is shown to be free of Landau pole singularities. Some important differences betwee…
Perturbative corrections beyond leading-log accuracy to BFKL and BK equations, describing the rapidity evolution of QCD scattering amplitudes at high energy, exhibit strong convergence problems due to radiative corrections enhanced by large single and double transverse logs. We identify explicitly the physical origin of double transverse logs and resum them directly in coordinate space as appropriate for BK equation, in terms of an improved local-in-rapidity evolution kernel. Numerical results show the crucial role of double-logarithmic resummation for BK evolution, which is stabilized and slo…
Soft and Jet functions for SCET at four loops in QCD
The Institute of Mathematical Sciences（马哈拉施特拉数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国家研究所）；II. Institute for Theoretical Physics, Hamburg University（汉堡大学第二理论物理研究所）
In soft-collinear effective theory, we analyze the structure of rapidity divergence due to the collinear and soft modes residing in disparate phase spaces. The idea of an effective theory is applied to a system of collinear modes with large rapidity and soft modes with small rapidity. The large-rapidity (collinear) modes are integrated out to obtain the effective theory for the small-rapidity (soft) modes. The full SCET with the collinear and soft modes should be matched onto the soft theory at the rapidity boundary, and the matching procedure becomes exactly the zero-bin subtraction. The larg…
We present angularity differential cross-section for the deep inelastic scattering process (DIS) in the framework of soft-collinear effective theory (SCET). Using SCET, the cross-section is factorized in terms of hard, jet, beam and soft functions. Our result includes resummation of the large logarithms up to next-to-next leading logarithmic (NNLL) accuracy. The numerical results presented here for DIS angularity cross-section can be explored by the future EIC.
The production of hard photons in hadronic collisions is studied using Soft-Collinear Effective Theory (SCET). This is the first application of SCET to a physical, observable cross section involving energetic partons in more than two directions. A factorization formula is derived which involves a non-trivial interplay of the angular dependence in the hard and soft functions, both quark and gluon jet functions, and multiple partonic channels. The relevant hard, jet and soft functions are computed to one loop and their anomalous dimensions are determined to three loops. The final resummed inclus…
Recent concerns about the very large next-to-leading logarithmic (NLL) corrections to the BFKL equation are addressed by the introduction of a physical rapidity-separation parameter $Δ$. At the leading logarithm (LL) this parameter enforces the constraint that successive emitted gluons have a minimum separation in rapidity, $y_{i+1}-y_i>Δ$. The most significant effect is to reduce the BFKL Pomeron intercept from the standard result as $Δ$ is increased from 0 (standard BFKL). At NLL this $Δ$-dependence is compensated by a modification of the BFKL kernel, such that the total dependence on $Δ$ is…
A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum ($q_T$) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-$x$ logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consisten…
We consider Higgs boson production by gluon fusion in hadron collisions. We study the doubly-differential transverse-momentum (q_T) and rapidity (y) distribution of the Higgs boson in perturbative QCD. In the region of small q_T (q_T << M_H, M_H being the mass of the Higgs boson), we include the effect of logarithmically-enhanced contributions due to multiparton radiation to all perturbative orders. We use the impact parameter and double Mellin moments to implement and factorize the multiparton kinematics constraint of transverse- and longitudinal-momentum conservation. The logarithmic terms a…
Soft Collinear Effective Theory (SCET) is an effective field theory of Quantum Chromodynamics (QCD) for processes where there are energetic, nearly lightlike degrees of freedom interacting with one another via soft radiation. SCET has found many applications in high-energy and nuclear physics, especially in recent years the physics of hadronic jets in $e^+e^-$, lepton-hadron, hadron-hadron, and heavy-ion collisions. SCET can be used to factorize multi-scale cross sections in these processes into single-scale hard, collinear, and soft functions, and to evolve these through the renormalization g…
The Wilson Coefficients for all 4-parton operators which arise in matching QCD to Soft-Collinear Effective Theory (SCET) are computed at 1-loop. Any dijet observable calculated in SCET beyond leading order will require these results. The Wilson coefficients are separated by spin and color, although most applications will involve only the spin-averaged hard functions. The anomalous dimensions for the Wilson coefficients are given to 2-loop order, and the renormalization group equations are solved explicitly. This will allow for analytical resummation of dijet observables to next-to-next-to-lead…
We derive and solve renormalization group equations that allow for the resummation of subleading power rapidity logarithms. Our equations involve operator mixing into a new class of operators, which we term the "rapidity identity operators", that will generically appear at subleading power in problems involving both rapidity and virtuality scales. To illustrate our formalism, we analytically solve these equations to resum the power suppressed logarithms appearing in the back-to-back (double light cone) limit of the Energy-Energy Correlator (EEC) in $\mathcal{N}$=4 super-Yang-Mills. These logar…
Using soft-collinear effective theory (SCET), which provides a unified framework for factorization, resummation of logarithms, and incorporation of universal nonperturbative functions in hard-scattering QCD cross-sections, we present a new prediction of angularity distributions in e+e- annihilation. Angularities tau_a are an infinite class of event shapes which vary in their sensitivity to the substructure of jets in the final state, controlled by a continuous parameter a<2. We calculate angularity distributions for all a<1 to first order in the strong coupling alpha_s and resum large logarith…
We study the effects of threshold resummation on the rapidity dependence of single-particle-inclusive cross sections, using the prompt photon cross section as an example. We make use of the full resummation formula at next-to-leading logarithmic accuracy and develop a new technique for treating rapidity in resummation. We compare our phenomenological results with those of previous studies and discuss differences and similarities of the two existing resummation formalisms.
We study the semi-inclusive limit of the deep inelastic scattering and Drell-Yan (DY) processes in soft collinear effective theory. In this regime so-called threshold logarithms must be resummed to render perturbation theory well behaved. Part of this resummation occurs via the Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP) equation, which at threshold contains a large logarithm that calls into question the convergence of the anomalous dimension. We demonstrate here that the problematic logarithm is related to rapidity divergences, and by introducing a rapidity regulator can be tamed. …
We compute the transverse momentum dependent (TMD) soft function for the production of a color-neutral final state at the LHC within the rapidity renormalization group (RRG) framework to next-to-next-to-leading order (NNLO). We use this result to extract the universal renormalized TMD beam functions (aka TMDPDFs) in the same scheme and at the same order from known results in another scheme. We derive recurrence relations for the logarithmic structure of the soft and beam functions, which we use to cross check our calculation. We also explicitly confirm the non-Abelian exponentiation of the TMD…
We construct an evolution equation for the $B$ meson wave functions in the $k_T$ factorization theorem, whose solutions sum the double logarithms associated with the light-cone singularities, namely, the rapidity logarithms. The derivation is subtler than that of the Sudakov resummation for an energetic light hadron, due to the involvement of the effective heavy-quark field. The renormalization-group evolution in the factorization scale needs to be included in order to derive an ultraviolet-finite and scale-invariant kernel for resumming the rapidity logarithms. It is observed that this kernel…
We propose a new method to evaluate jet substructure observables in inclusive jet measurements, based upon semi-inclusive jet functions in the framework of Soft Collinear Effective Theory (SCET). As a first example, we consider the jet fragmentation function, where a hadron $h$ is identified inside a fully reconstructed jet. We introduce a new semi-inclusive fragmenting jet function ${\mathcal G}^h_i(z= ω_J/ω,z_h=ω_h/ω_J,ω_J, R,μ)$, which depends on the jet radius $R$ and the large light-cone momenta of the parton `$i$' initiating the jet ($ω$), the jet ($ω_J$), and the hadron $h$ ($ω_h$). The…
To sum high energy leading logarithms in a consistent way, one has to impose the strong ordering in both projectile rapidity and dense target rapidity simultaneously, which results in a kinematically improved Balitsky-Kovchegov(BK) equation. We find that beyond this strong ordering region, the important sub-leading double logarithms arise at high order due to the incomplete cancellation between real corrections and virtual corrections in a t-channel calculation. Based on this observation, we further argue that these double logarithms are the Sudakov type ones, and thus can be resummed into an …
We discuss rapidity-gap events between two jets produced at high momentum transfer in $p$ $\bar{p}$ scattering, from the point of view of the soft energy flow into the interjet region. We define a gap cross section and, in perturbative QCD (pQCD), resum all the leading logarithms in the soft intermediate energy. We show that the numerical result from our cross section reproduces the shape of the D0 and CDF \cite{D0,CDF,D0fig} experimental data.
Jets are an important probe to identify the hard interaction of interest at the LHC. They are routinely used in Standard Model precision measurements as well as in searches for new heavy particles, including jet substructure methods. In processes with several jets, one typically encounters hierarchies in the jet transverse momenta and/or dijet invariant masses. Large logarithms of the ratios of these kinematic jet scales in the cross section are at present primarily described by parton showers. We present a general factorization framework called SCET$_+$, which is an extension of Soft-Collinea…
We study the production of Mueller-Navelet jets at 13 TeV LHC, within collinear factorization and including the BFKL resummation of energy logarithms in the next-to-leading approximation. We calculate several azimuthal correlations for different values of the rapidity separation $Y$ between the two jets and evaluate the effect of excluding those events where, for a given $Y$, one of the two jets is produced in the central region.
We investigate the soft spectator scattering contribution for the FF $F_{1}$. We focus our attention on factorization of the hard-collinear scale $\sim QΛ$ corresponding to transition from SCET-I to SCET-II. We compute the leading order jet functions and find that the convolution integrals over the soft fractions are logarithmically divergent. This divergency is the consequence of the boost invariance and does not depend on the model of the soft correlation function describing the soft spectator quarks. Using as example a two-loop diagram we demonstrated that such a divergency corresponds to t…
We propose the addition of a new "soft-collinear" mode to soft collinear effective theory (SCET) below the usual soft scale to factorize and resum logarithms of jet radii $R$ in jet cross sections. We consider exclusive 2-jet cross sections in $e^+e^-$ collisions with an energy veto $Λ$ on additional jets. The key observation is that there are actually two pairs of energy scales whose ratio is $R$: the transverse momentum $QR$ of the energetic particles inside jets and their total energy $Q$, and the transverse momentum $ΛR$ of soft particles that are cut out of the jet cones and their energy …
Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the rele- vant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group". That is, a Collin-Soper like equation is rea…
