Soft-Collinear Effective Theory (SCET) has been formulated since a decade now in covariant gauges. In this work we derive a modified SCET Lagrangian applicable in both classes of gauges: regular and singular ones. This extends the range of applicability of SCET. The new Lagrangian must be used to obtain factorization theorems in cases where the transverse momenta of the particles in the final states are not integrated over, such as semi-inclusive deep inelastic scattering, Drell-Yan and the Higgs production cross-section at low transverse momentum. By doing so all non-perturbative matrix elements appearing in the factorized cross-sections are gauge invariant.
Two topics in soft collinear effective theory (SCET) for gravitational interactions are explored. First, the collinear Wilson lines---necessary building blocks for maintaining multiple copies of diffeomorphism invariance in gravity SCET---are extended to all orders in the SCET expansion parameter $λ$, where it has only been known to $O(λ)$ in the literature. Second, implications of reparametrization invariance (RPI) for the structure of gravity SCET lagrangians are studied. The utility of RPI is illustrated by an explicit example in which $O(λ^2)$ hard interactions of a collinear graviton are …
Soft Collinear Effective Theory (SCET) is an effective field theory which describes the interactions of low invariant mass jets which are highly boosted with respect to one another. In the standard formulation of SCET, the effective Lagrangian for collinear fields is expanded in inverse powers of the energy. At leading order this leads to manifest decoupling of soft and collinear degrees of freedom; however, subleading terms in the effective Lagrangian violate this manifest decoupling. In this paper we point out that the collinear expansion in the SCET Lagrangian is unnecessary, and that the S…
We describe the construction of light-cone sum rules (LCSRs) for exclusive $B$-meson decays into light energetic hadrons from correlation functions within soft-collinear effective theory (SCET). As an example, we consider the SCET sum rule for the $B \to π$ transition form factor at large recoil, including radiative corrections from hard-collinear loop diagrams at first order in the strong coupling constant.
The accurate calculation of weak heavy-to-light form factors is crucial for the determination of CKM matrix elements from experimental data on B decays. In SCET, the soft-collinear effective theory, these form factors can, in the heavy quark limit, be split into a factorisable part that is calculable in perturbation theory, and a nonfactorisable part that observes certain spin-symmetries. I discuss the relation between the SCET factorisation formulas and the heavy quark limit of the corresponding QCD sum rules on the light-cone. I also analyse the numerical size of factorisable and nonfactoris…
Gauge invariance in soft-collinear effective theory (SCET) is discussed in regular (covariant) and singular (light-cone) gauges. It is argued that SCET, as it stands, is not capable to define in a gauge invariant way certain non-perturbative matrix elements that are an integral part of many factorization theorems. Those matrix elements involve two quark or gluon fields separated not only in light-cone direction but also in the transverse one. This observation limits the range of applicability of SCET. To remedy this we argue that one needs to introduce a new Wilson line as part of SCET formali…
We study the evolution of flavor-singlet, light-cone amplitudes in the soft-collinear effective theory (SCET), and reproduce results previously obtained by a different approach. We apply our calculation to the color-singlet contribution to the photon endpoint in radiative Upsilon decay. In a previous paper, we studied the color-singlet contributions to the endpoint, but neglected operator mixing, arguing that it should be a numerically small effect. Nevertheless the mixing needs to be included in a consistent calculation, and we do just that in this work. We find that the effects of mixing are…
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…
A complete basis for the next-to-next-to leading order heavy-to-light currents in the soft-collinear effective theory is constructed. Reparameterization invariance is imposed by deriving constraint equations. Their solutions give the set of allowed Dirac structures as well as relations between the Wilson coefficients of operators that appear at different orders in the power expansion. The completeness of reparameterization invariance constraints derived on a projected surface is investigated. We also discuss the universality of the ultrasoft Wilson line with boundary conditions.
Soft-Collinear Effective Theory (SCET) is a framework for modeling the infrared structure of theories whose long distance behavior is dominated by soft and collinear divergences. This paper demonstrates that SCET can be made compatible with supersymmetry (SUSY). Explicitly, the effective Lagrangian for $\mathcal{N} = 1$ SUSY Yang-Mills is constructed and shown to be a complete description for the infrared of this model. For contrast, we also construct the effective Lagrangian for chiral SUSY theories with Yukawa couplings, specifically the single flavor Wess-Zumino model. Only a subset of the …
The jet function for the factorized cross section $e^+e^-$ into dijets is given as a function of the jet invariant mass s and with a generic jet algorithm at $\mathcal{O}(α_s)$. We demonstrate the results using the Sterman-Weinberg algorithm and show that the jet function is independent of the energy fraction $β$ of the soft radiation. The anomalous dimension has the same form with and without the cone half-angle $δ$. The dependence of the finite part of the jet function on the cone angle is given.
We analyze the transverse momentum broadening in the absence of radiation of an energetic parton propagating through quark-gluon plasma via Soft Collinear Effective Theory (SCET). We show that the probability for picking up transverse momentum k_\perp is given by the Fourier transform of the expectation value of two transversely separated light-like path-ordered Wilson lines. The subtleties about the ordering of operators do not change the \hat q value for the strongly coupled plasma of N=4 SYM theory.
Hidden self-energy contributions of collinear functions in SCET
Argonne National Laboratory（阿贡国家实验室）；Massachusetts Institute of Technology（麻省理工学院）；Fudan University（复旦大学）；Korea University（韩国大学）；Central China Normal University（华中师范大学）
Heavy-to-light weak form factor is calculated using the light-cone sum rule (LCSR)in the framework of soft-collinear effective theory (SCET). There are spin-symmetric and spin-nonsymmetric contributions. Leading order spin-symmetric contribution corresponds to the ''soft overlap'' where some of the partons carry very small momentum. The next-to-leading order spin-symmetric and spin-nonsymmetric parts are characterized by a collinear gluon exchange with the spectator quark. We reproduce the full theory LCSR results and give comments on recent LCSR in SCET.
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.
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 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 main application of Heavy Quark Effective Theory (HQET) and of Soft Collinear Effective Theory (SCET) is in establishing factorization theorems for exclusive and semi-inclusive decays of heavy mesons. However, the calculation of the soft factors from the HQET or SCET factorization relations is, as usual, impeded by the non-perturbative aspect of the strong interactions. In the hope of gaining some insights into some of these quantities we compute them in the 't Hooft model. We find that the B-meson shape function is exactly given by the square of the B-meson light-cone wave-function. The s…
We derive light-cone sum rules (LCSRs) for exclusive B-meson decays into light energetic hadrons from correlation functions within soft-collinear effective theory (SCET). In these sum rules the short-distance scale refers to ``hard-collinear'' interactions with virtualities of order (Lambda_{QCD}*m_b). Hard scales (related to virtualities of order m_b^2) are integrated out and enter via external coefficient functions in the sum rule. Soft dynamics is encoded in light-cone distribution amplitudes for the B-meson, which describe both the factorizable and non-factorizable contributions to exclusi…
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…
Calculations of collinear correlation functions in perturbative QCD and Soft-Collinear Effective Theory (SCET) require a prescription for subtracting soft or zero-bin contributions in order to avoid double counting the contributions from soft modes. At leading order in $λ$, where $λ$ is the SCET expansion parameter, the zero-bin subtractions have been argued to be equivalent to convolution with soft Wilson lines. We give a proof of the factorization of naive collinear Wilson lines that is crucial for the derivation of the equivalence. We then check the equivalence by computing the non-Abelian …
The form factors of $B\toπ(ρ)$ decays are analyzed using the light-cone sum rules in the framework of the soft-collinear effective theory (SCET). We establish the sum rules for the leading and the next-to-leading order (NLO) nonperturbative functions of the SCET. Explicit calculation shows that (leading)+(NLO) $\sim 1/E^2+1/E^3$, where $E$ is the large recoiling energy. The results are compatible with the literatures. Also the validity of this hybrid formalism is discussed.
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…
We study the dynamics of Wilson lines near the light cone in QCD. Lattice simulations of the near light cone Hamiltonian in SU(2) show that the correlation mass at strong coupling vanishes at intermediate coupling, which signals a continuum transition on the light cone. We point out the possible relevance of this mass for high energy scattering.
We calculate the anomalous dimensions of the next-to-leading order dijet operators in soft-collinear effective theory (SCET). We use a formulation of SCET where the Lagrangian is multiple copies of QCD and the interactions between sectors occur through light-like Wilson lines in external currents. We introduce a small gluon mass to regulate the infrared divergences of the individual loop diagrams in order to properly extract the ultraviolet divergences. We discuss this choice of infrared regulator and contrast it with the $δ$-regulator. Our results can be used to increase the theoretical preci…
Vacuum expectation values for one Wegner-Wilson loop representing a moving quark-antiquark pair are calculated in four-dimensional Euclidean and Minkowski space-time. The calculation uses gluon field strength correlators with perturbative gluon exchange and non-perturbative correlations from the stochastic vacuum model. The expectation value of a Wegner-Wilson loop forming a hyperbolic angle in Minkowski space-time is connected by an analytical continuation to the expectation value of the Wegner-Wilson loop in Euclidean four-space. The obtained result shows how confinement enters into the ligh…
The transverse broadening of an energetic jet passing through a non-Abelian plasma is believed to be described by the thermal expectation value of a light-cone Wilson loop. In this exploratory study, we measure the light-cone Wilson loop with classical lattice gauge theory simulations. We observe, as suggested by previous studies, that there are strong interactions already at short transverse distances, which may lead to more efficient jet quenching than in leading-order perturbation theory. We also verify that the asymptotics of the Wilson loop do not change qualitatively when crossing the li…
We consider the problems of gauge invariance, path-dependence and treatment of overlapping UV/rapidity divergences peculiar to the transverse-momentum dependent parton distribution functions (TMDs). For different formulations of the TMDs available in the literature, we check the consistency of the TMD matrix elements with the collinear parton distribution functions possessing the well-known operator structure. Comparative on- and off-light-cone layout of the Wilson lines which secure the gauge-invariance of the TMDs is presented and briefly discussed.
I review the resummation formalism for organizing large logarithms in perturbative expansion of collinear subprocesses through the variation of Wilson lines off the light cone. A master equation is derived, which involves the evolution kernel resulting from this variation. It is then demonstrated that all the known single- and double-logarithm summations for a parton distribution function or a transverse-momentum-dependentparton distribution can be reproduced from the master equation by applying appropriate soft-gluon approximations to the evolution kernel. Moreover, jet substructures, informa…
Complete two-loop calculation of a dimensionally regularized Wilson loop with light-like segments is performed in the light-like axial gauge with the Mandelstam-Leibbrandt prescription for the gluon propagator. We find an expression which {\it exactly} coincides with the one previously obtained for the same Wilson loop in covariant Feynman gauge. The renormalization of Wilson loop is performed in the $\MS-$scheme using a general procedure tailored to the light-like axial gauge. We find that the renormalized Wilson loop obeys a renormalization group equation with the same anomalous dimensions a…
The study of amplitudes and cross sections in the soft and collinear limits allows for an understanding of their all orders behavior, and the identification of universal structures. At leading power soft emissions are eikonal, and described by Wilson lines. Beyond leading power the eikonal approximation breaks down, soft fermions must be added, and soft radiation resolves the nature of the energetic partons from which they were emitted. For both subleading power soft gluon and quark emissions, we use the soft collinear effective theory (SCET) to derive an all orders gauge invariant bare factor…
We exhibit the geometric structure of the convex cone in the linear space of the Wilson coefficients for the dimension-8 operators involving the left-handed lepton doublet $L$ and the Higgs doublet $H$ in the Standard Model effective field theory (SMEFT). The boundary of the convex cone gives rise to the positivity bounds on the Wilson coefficients, while the extremal ray corresponds to the unique particle state in the theory of ultra-violet completion. Among three types of canonical seesaw models for neutrino masses, we discover that only right-handed neutrinos in the type-I seesaw model show…
Applying the vacuum-to-$B$-meson correlation functions with an interpolating current for the light vector meson we construct the light-cone sum rules (LCSR) for the "effective" form factors $ξ_{\parallel}(n \cdot p)$, $ξ_{\perp}(n \cdot p)$, $Ξ_{\parallel}(τ, n \cdot p)$ and $Ξ_{\perp}(τ, n \cdot p)$, defined by the corresponding hadronic matrix elements in soft-collinear effective theory (SCET), entering the leading-power factorization formulae for QCD form factors responsible for $B \to V \ell \bar ν_{\ell}$ and $B \to V \ell \bar \ell$ decays at large hadronic recoil at next-to-leading-orde…
We discuss the possible relation between certain geometrical properties of the loop space and energy evolution of the cusped Wilson exponentials defined on the light-cone. Analysis of the area differential equations for this special class of the Wilson loops calls for careful treatment of the ultraviolet and rapidity divergences which make those loops non-multiplicatively-renormalizable. We propose to consider the renormalization properties of the light-cone cusped Wilson loops from the point of view of the universal quantum dynamical approach introduced by Schwinger. We conjecture and discuss…
We present a systematic algorithm for the perturbative computation of soft functions that are defined in terms of two light-like Wilson lines. Our method is based on a universal parametrisation of the phase-space integrals, which we use to isolate the singularities in Laplace space. The observable-dependent integrations can then be performed numerically, and they are implemented in the new, publicly available package SoftSERVE that we use to derive all of our numerical results. Our algorithm applies to both SCET-1 and SCET-2 soft functions, and in the current version it can be used to compute …
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…
Transverse Momentum Dependent (TMD) parton distribution functions (PDFs) also take into account the transverse momentum ($p_T$) of the partons. The $p_T$-integrated analogues can be linked directly to quark and gluon matrix elements using the operator product expansion in QCD, involving operators of definite twist. TMDs also involve operators of higher twist, which are not suppressed by powers of the hard scale, however. Taking into account gauge links that no longer are along the light-cone, one finds that new distribution functions arise. They appear at leading order in the description of az…
We discuss the relation between partonic distributions near the phase space boundary and Wilson loop expectation values calculated along paths partially lying on the light-cone. Due to additional light-cone singularities, multiplicative renormalizability for these expectation values is lost. Nevertheless we establish the renormalization group equation for the light like Wilson loops and show that it is equivalent to the evolution equation for the physical distributions. By performing a two-loop calculation we verify these properties and show that the universal form of the splitting function fo…
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 …
We discuss the most general form of the leading power suppressed collinear operators in the soft-collinear effective theory. Such operators appear in the description of power corrections to exclusive heavy flavor decays into energetic light hadrons. Reparametrization invariance in the SCET provides powerful constraints on the Wilson coefficients of the subleading collinear operators. We present explicit results for the matrix elements of these operators on pseudoscalar and vector mesons, which are expressed in terms of twist-2 and twist-3 light-cone wave functions. We consistently include the …
Soft-collinear effective theory (SCET) is used to demonstrate factorization for event shape distributions in the two-jet region. The leading nonperturbative power corrections to these distributions can be characterized as shape functions defined in terms of Wilson lines of soft gluons. The relation of these results to the well-known predictions for shifts in the distributions induced by the leading power corrections is discussed.
Two effects, jet broadening and gluon bremsstrahlung induced by the propagation of a highly energetic quark in dense QCD matter, are reconsidered from effective theory point of view. We modify the standard Soft Collinear Effective Theory (SCET) Lagrangian to include Glauber modes, which are needed to implement the interactions between the medium and the collinear fields. We derive the Feynman rules for this Lagrangian and show that it is invariant under soft and collinear gauge transformations. We find that the newly constructed theory SCET$_{\rm G}$ recovers exactly the general result for the…
We present the first analytic ${\cal O}(α_s)$-computation at twist-$1$,$2$ of the $\bar B_{u,d,s} \to γ$ form factors within the framework of sum rules on the light-cone. These form factors describe the charged decay $\bar B_u \to γ\ell^- \barν$, contribute to the flavour changing neutral currents $\bar B_{d,s} \to γ\ell^+ \ell^-$ and serve as inputs to more complicated processes. We provide a fit in terms of a $z$-expansion with correlation matrix and extrapolate the form factors to the kinematic endpoint by using the $g_{BB^*γ}$ couplings as a constraint. Analytic results are available in te…
We present a computation of the global soft function for single-inclusive hadronic jet production at next-to-next-to-leading order (NNLO) in the strong coupling constant involving four light-like Wilson lines. Our soft function belongs to SCET$_{\text I}$ observables obeying the non-Abelian exponentiation theorem. We provide the calculation of the soft function involving four collinear light-cone directions for all $2\rightarrow2$ processes extending the previous result. We perform the threshold resummation for these processes at the next-to-next-to-leading logarithmic accuracy and present the…
We develop a new classical action that in addition to $\mathrm{MHV}$ vertices contains also $\mathrm{N^kMHV}$ vertices, where $1\leq k \leq n-4$ with $n$ the number of external legs. The lowest order vertex is the four-point MHV vertex -- there is no three point vertex and thus the amplitude calculation involves fewer vertices than in the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action. The action is obtained by a canonical transformation of the Yang-Mills action in the light-cone gauge, where the field transformations are based on the Wilson line functiona…
We consider the effects of a light quark mass in the soft-collinear effective theory (SCET) and we apply them to B -> X_s gamma in the endpoint region. We find that the reparameterization invariance can be extended by including the collinear quark mass in the SCET Lagrangian. This symmetry constrains the theory with the quark mass terms, and we present explicit results at one loop. It also relates the Wilson coefficients of some mass operators to those of the leading operators, which are useful in organizing the subleading effects due to the quark mass in B -> X_s gamma. We present strange qua…
$B_{(s)}$ semi-leptonic decays to the light scalar meson, $B_{(s)}\to S l\barν_l, S l \bar{l}\,\,(l=e,μ,τ)$, are investigated in the QCD light-cone sum rules (LCSR) with chiral current correlator. Having little knowledge of ingredients of the scalar mesons, we confine ourself to the two quark picture for them and work with the two possible Scenarios. The resulting sum rules for the form factors receive no contributions from the twist-3 distribution amplitudes (DA's), in comparison with the calculation of the conventional LCSR approach where the twist-3 parts play usually an important role. We …
We investigate the appearance of straight infinite Wilson lines lying on the self-dual plane in the context of the Self Dual sector of the Yang Mills theory and in a connection to the Lagrangian implementing the MHV vertices (MHV Lagrangian) according to the Cachazo-Svrcek-Witten method. It was already recognized in the past by two of the authors, that such Wilson line functional provides the field transformation of positive helicity fields between the Yang-Mills theory on the light-cone and the MHV Lagrangian. Here we discuss in detail the connection to the Self Dual sector and we provide a n…
The one-loop dilatation operator in Yang-Mills theory possesses a hidden integrability symmetry in the sector of maximal helicity Wilson operators. We calculate two-loop corrections to the dilatation operator and demonstrate that while integrability is broken for matter in the fundamental representation of the SU(3) gauge group, for the adjoint SU(N_c) matter it survives the conformal symmetry breaking and persists in supersymmetric N=1, N=2 and N=4 Yang-Mills theories.
We construct a quark target model (QTM) to incorporate intrinsic glue into effective low-energy models of QCD, which often contain only quark degrees of freedom. This method guarantees the gauge invariance of observables order-by-order in the strong coupling. The quark and gluon PDFs for the dressed quarks are obtained in the QTM at leading order. We demonstrate gauge invariance of the results by comparing both covariant and light cone gauges, with the former including an explicit Wilson line contribution. A key finding is that in covariant gauges the Wilson line can carry a significant amount…
Equations of motion for the light-like QCD Wilson loops are studied in the generalized loop space (GLS) setting. To this end, the classically conformal-invariant non-local variations of the cusped Wilson exponentials lying (partially) on the light-cone are formulated in terms of the Fréchet derivative. The rapidity and renormalization-group behaviour of the gauge-invariant quantum correlation functions (in particular, the three-dimensional parton densities) are demonstrated to be connected to certain geometrical properties of the Wilson loops defined in the GLS.
Recent developments showed that hadron light-cone parton distributions could be directly extracted from spacelike correlators, known as quasi parton distributions, in the large hadron momentum limit. Unlike the normal light-cone parton distribution, a quasi parton distribution contains ultraviolet (UV) power divergence associated with the Wilson line self energy. We show that to all orders in the coupling expansion, the power divergence can be removed by a "mass" counterterm in the auxiliary $z$-field formalism, in the same way as the renormalization of power divergence for an open Wilson line…
