We present a new analysis of two-jet event shape distributions in soft collinear effective theory. Extending previous results, we observe that a large class of such distributions can be expressed in terms of vacuum matrix elements of operators in the effective theory. We match these matrix elements to the full theory in the two-jet limit without assuming factorization of the complete set of hadronic final states into independent sums over partonic collinear and soft states. We also briefly discuss the relationship of this approach to diagrammatic factorization in the full theory.
We analyze two consequences of the relationship between collinear factorization and $k_t$-factorization. First, we show that the $k_t$-factorization gives a fundamental justification for the choice of the hard scale $Q^2$ done in the collinear factorization. Second, we show that in the collinear factorization there is an uncertainty on this choice which will not be reduced by higher orders. This uncertainty is absent within the $k_t$-factorization formalism.
We examine some of the complications involved when combining (matching) TMD factorization with collinear factorization to allow accurate predictions over the whole range of measured transverse momentum in a process like Drell-Yan. Then we propose some improved methods for combining the two types of factorization. (This talk is based on work reported in arXiv:1605.00671.)
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.
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.
Hard scattering processes involving hadrons at small $x$ are described by a $k_T$-factorization formula driven by a BFKL gluon. We explore the equivalence of this description to a collinear-factorization approach in which the anomalous dimensions $γ_{gg}$ and $γ_{qg}/α_S$ are expressed as power series in $α_S \log (1/x)$, or to be precise $α_S/ω$ where $ω$ is the moment index. In particular we confront the collinear-factorization expansion with that extracted from the BFKL approach with running coupling included.
The soft-collinear effective theory has been recently applied to prove novel factorization theorems for many B decays. We describe here in some detail the factorization relation for color-supressed nonleptonic B -> D(*)0 pi0 decays and update the phenomenological analysis of these decays.
I consider the singular behaviour of tree-level QCD amplitudes when the momenta of three partons become simultaneously parallel and I discuss the universal factorization formula that controls the singularities of the multiparton matrix elements in this collinear limit.
We construct an improved implementation for combining transverse-momentum-dependent (TMD) factorization and collinear factorization. TMD factorization is suitable for low transverse momentum physics, while collinear factorization is suitable for high transverse momenta and for a cross section integrated over transverse momentum. The result is a modified version of the standard $W+Y$ prescription traditionally used in the Collins-Soper-Sterman (CSS) formalism and related approaches. We further argue that questions regarding the shape and $Q$-dependence of the cross sections at lower $Q$ are lar…
We define the collinear factorization scheme, which absorbs only the collinear physics into the parton distribution functions. In order to isolate the collinear physics, we introduce a procedure to combine real and virtual corrections, canceling infrared singularities prior to integration. In the collinear scheme, the factorization scale $μ$ has a simple physical interpretation as a collinear cutoff. We present a method for choosing the factorization scale and apply it to the Drell-Yan process; we find $μ\approx Q/2$, where $Q$ is the vector-boson invariant mass. We show that, for a wide varie…
The factorization of amplitudes into hard, soft and collinear parts is known to be violated in situations where incoming particles are collinear to outgoing ones. This result was first derived by studying limits where non-collinear particles become collinear. We show that through an effective field theory framework with Glauber operators, these factorization-violating effects can be reproduced from an amplitude that is factorized before the splitting occurs. We confirm results at one-loop, through single Glauber exchange, and at two-loops, through double Glauber exchange. To approach the calcu…
Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections. In this paper we extend our previous investigation of heavy-to-light currents in soft-collinear effective theory to final states with invariant mass Lambda^2 as is appropriate to exclusive B meson decays. The effective theory contains soft modes and two collinear modes with virtualities of order m_b*Lambda (`hard-collinear') and Lambda^2. I…
In this paper, we calculate the inclusive rate of $W^{+} W^{-}$ pair production through leptonic decay channels $W^+W^- \to l^+ν_l + l^{\prime -} ν_{l'}$ in the $k_t$-factorization framework. We also consider the exclusive $W^{+} W^{-}$ pair production through the one-loop induced $gg \to H \to W^+W^-$ channel that is important for the study of new physics beyond the Standard Model. The results are compared with predictions from the Herwig 7 event generator in the collinear factorization framework and with the experimental data from the ATLAS and the CMS collaborations. It will be shown that o…
We investigate the breakdown of collinear factorization for non-inclusive observables in hadron-hadron collisions. For pure QCD processes, factorization is violated at the three-loop level and it has a structure identical to that encountered previously in the case of super-leading logarithms. In particular, it is driven by the non-commutation of Coulomb/Glauber gluon exchanges with other soft exchanges. Beyond QCD, factorization may be violated at the two-loop level provided that the hard subprocess contains matrix element contributions with phase differences between different colour topologie…
We provide a precise statement of hard-soft-collinear factorization of scattering amplitudes and prove it to all orders in perturbation theory. Factorization is formulated as the equality at leading power of scattering amplitudes in QCD with other amplitudes in QCD computed from a product of operator matrix elements. The equivalence is regulator independent and gauge independent. As the formulation relates amplitudes to the same amplitudes with additional soft or collinear particles, it includes as special cases the factorization of soft currents and collinear splitting functions from generic …
It is shown that the difference between the c-quark proton structure functions calculated in the k_T-factorization approach using different unintegrated gluon distribution functions is the same order as the difference between results obtained in the parton model and in the k_T-factorization approach.
Exclusive electroproduction of hard dijets can be described within the collinear factorization. This process has clear experimental signature and provides one with an interesting alternative venue to test QCD description of hard diffractive processes and extract information on generalized nucleon parton distributions. In this work we present detailed leading-order QCD calculations of the relevant cross sections, including longitudinal momentum fraction distribution of the dijets and their azimuthal angle dependence.
Collinear Factorization Violation and Reggeization
Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；SLAC National Accelerator Laboratory, Stanford University（斯坦福大学SLAC国家加速器实验室）；University of Cyprus（塞浦路斯大学）
We discuss the evaluation of the collinear single-logarithmic contributions to virtual electroweak corrections at high energies. More precisely, we proof the factorization of the mass singularities originating from loop diagrams involving collinear virtual gauge bosons coupled to external legs. We discuss, in particular, processes involving external longitudinal gauge bosons, which are treated using the Goldstone-boson equivalence theorem. The proof of factorization is performed within the 't Hooft--Feynman gauge at one-loop order and applies to arbitrary electroweak processes that are not mas…
The combination of collinear factorization with effective field theory originally developed for soft interactions of heavy quarks provides the foundations of the theory of exclusive and semi-inclusive B decays. In this article I summarize some of the later conceptual developments of the so-called QCD factorization approach that make use of soft-collinear effective theory. Then I discuss the status and results of the calculation of the hard-scattering functions at the next order, and review very briefly some of the phenomenology, covering aspects of charmless, electroweak penguin and radiative …
We consider the singular behaviour of tree-level QCD amplitudes when the momenta of three partons become simultaneously parallel. We discuss the universal factorization formula that controls the singularities of the multiparton matrix elements in this collinear limit and present the explicit expressions of the corresponding splitting functions. The results fully include spin (azimuthal) correlations up to ${\cal O}(\as^2)$. In the case of spin-averaged splitting functions we confirm similar results obtained by Campbell and Glover.
We construct a theoretical framework to match the formulas for forward inclusive hadron productions in pA collisions in the small-x saturation formalism and collinear factorization. The small-x calculation can be viewed as a power series in $Q_s^2/k_\perp^2$, in which the collinear factorization result corresponds to the leading term. At high transverse momentum, the subleading correction terms are insignficant, whereas at low $p_\perp$, the power corrections become important and the small-x resummation is essential to describe the differential cross section. We show that the familiar collinea…
We calculate the production and polarization of direct $J/ψ$ in the improved color evaporation model at $\mathcal{O}(α_s^3)$ in the collinear factorization approach. We present the first calculation of polarization parameters $λ_\vartheta$, $λ_ϕ$, and $λ_{\vartheta ϕ}$ in the helicity and the Collins-Soper frames, as well as the frame-invariant polarization parameter $\tildeλ$ as a function of transverse momentum. We find agreement with both $J/ψ$ cross sections and the invariant polarization parameters at small and moderate $p_T$.
The p_T-spectra for direct J/psi and psi' in hadroproduction at Tevatron energy have been calculated based on NRQCD formalism and fragmentation approximation in the collinear parton model and k_T-factorization approach. We have described the CDF data and obtained a good agreement between the predictions obtained in the parton model and k_T-factorization approach. We performed the calculations using the relevant leading order in alpha_s hard amplitudes and the equal values of the color-octet long-distance matrix elements for the both models.
We compare results of $k_t$-factorization approach and next-to-leading order collinear-factorization approach for dijet correlations in proton-proton collisions at RHIC energies. We discuss correlations in azimuthal angle as well as correlations in two-dimensional space of transverse momenta of two jets. Some $k_t$-factorization subprocesses are included for the first time in the literature. Different unintegrated gluon/parton distributions are used in the $k_t$-factorization approach. The results depend on UGDF/UPDF used. For collinear NLO case the situation depends significantly on whether w…
We examine the uncertainty of perturbative QCD factorization for hadron structure functions in deep inelastic scattering at a large value of the Bjorken variable xB. We analyze the target mass correction to the structure functions by using the collinear factorization approach in the momentum space. We express the long distance physics of structure functions and the leading target mass corrections in terms of parton distribution functions with the standard operator definition. We compare our result with existing work on the target mass correction. We also discuss the impact of a final-state jet…
We discuss the role of intrinsic charm (IC) in the nucleon for forward production of $c$-quark (or $\bar c$-antiquark) in proton-proton collisions for low and high energies. The calculations are performed in collinear-factorization approach with on-shell partons, $k_T$-factorization approach with off-shell partons as well as in a hybrid approach using collinear charm distributions and unintegrated (transverse momentum dependent) gluon distributions. For the collinear-factorization approach we use matrix elements for both massless and massive charm quarks/antiquarks. The distributions in rapidi…
We study the transverse single-spin asymmetry for single-hadron production in proton-proton collisions within the framework of collinear twist-3 factorization in Quantum Chromodynamics. By taking into account the contribution due to parton fragmentation we obtain a very good description of all high transverse-momentum data for neutral and charged pion production from the Relativistic Heavy Ion Collider. Our study may provide the crucial step towards a final solution to the longstanding problem of what causes transverse single-spin asymmetries in hadronic collisions within Quantum Chromodynamic…
We study the Sudakov form factor for a spontaneously broken gauge theory using a (new) Delta -regulator. To be well-defined, the effective theory requires zero-bin subtractions for the collinear sectors. The zero-bin subtractions depend on the gauge boson mass M and are not scaleless. They have both finite and 1/epsilon contributions, and are needed to give the correct anomalous dimension and low-scale matching contributions. We also demonstrate the necessity of zero-bin subtractions for soft-collinear factorization. We find that after zero-bin subtractions the form factor is the sum of the co…
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 derive a factorization theorem for the Higgs boson transverse momentum (p_T) and rapidity (Y) distributions at hadron colliders, using the Soft Collinear Effective Theory (SCET), for m_h>> p_T>> Λ_{QCD} where m_h denotes the Higgs mass. In addition to the factorization of the various scales involved, the perturbative physics at the p_T scale is further factorized into two collinear impact-parameter Beam Functions (iBFs) and an inverse Soft Function (iSF). These newly defined functions are of a universal nature for the study of differential distributions at hadron colliders. The additional f…
We discuss the soft-collinear effective theory (SCET) and kinematic expansions in B-decays, focusing on recent results for color suppressed B -> D^(*) X decays. In particular we discuss model independent predictions for B^0 -> D^0 pi^0 and B^0 -> D^*0 pi^0, and update the comparison using new experimental data. We show why HQET alone is insufficient to give these results. SCET predictions are also reviewed for other B and Lambda_b decay channels that are not yet tested by data.
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…
A consistent factorization theorem is presented in the framework of effective field theories. Conventional factorization suffers from infrared divergences in the soft and collinear parts. We present a factorization theorem in which the infrared divergences appear only in the parton distribution functions by carefully reorganizing collinear and soft parts. The central idea is extracting the soft contributions from the collinear part to avoid double counting. Combining it with the original soft part, an infrared-finite kernel is obtained. This factorization procedure can be applied to various hi…
We consider heavy quark production in high energy pA collisions and investigate the contribution of interactions of valence quarks of proton with the nucleus. The often made assumption that valence quarks of proton can be factored out is justified only if the nucleus saturation momentum is much smaller than the heavy quark mass. This is not the case in phenomenologically relevant situations. Breakdown of factorization manifests itself in substantial decrease of the cross section at large total and small relative transverse momenta of the heavy quark -antiquark pair.
In this thesis we present an introduction to Soft-Collinear Effective Theory, which can be used to prove (or disprove) factorization theorems to all orders in the strong coupling constant for some B decays into light and energetic particles. Specifically, the factorizable amplitudes for inclusive B->X_u l nu and exclusive B->gamma l nu are calculated in renormalization-group improved perturbation theory to first non-trivial order. Form factors encoding the exclusive decay amplitudes for B->P l nu and B->V l nu (P=light pseudoscalar meson, V=light vector meson) are studied and proved to be domi…
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…
Proof of transverse-momentum-dependent(TMD) factorization for hadron-hadron collision is given in this paper. We focus on processes without detected soft final hadrons or detected final hadrons that are collinear to initial hadrons. This contradicts the widely accepted viewpoint that TMD- factorization does not hold in such processes even in the generalized sense. The key point is that singular points of the type l+ = 0 can be absorbed into Wilson lines of soft gluons, where l is collinear to the plus direction. Thus one should subtract such singular points from the collinear region of l. Afte…
We study the scattering of a single parton state with a multi-parton state to derive the complete results of perturbative coefficient functions at leading order, which appear in the collinear factorization for Single transverse-Spin Asymmetry(SSA) in Drell-Yan processes with a transversely polarized hadron in the initial state. We find that the factorization formula of SSA contains hard-pole-, soft-quark-pole- and soft-gluon-pole contributions. It is interesting to note that the leading order perturbative coefficient functions of soft-quark-pole- and soft-gluon-pole contributions are extracted…
Consistent factorization theorems in high-energy scattering near the threshold are presented in the framework of the soft-collinear effective theory. Traditional factorization theorem separates the soft and collinear parts successfully, but a final step should be supplemented if each part encounters infrared divergence. We present factorization theorems in which the infrared divergences appear only in the parton distribution functions and the infrared divergence is removed by carefully separating and reorganizing collinear and soft parts. The underlying physical idea is to isolate and remove t…
We propose the inclusive hadroproduction of a heavy-light dijet system, as a new channel for the investigation of high energy QCD. We build up a hybrid factorization that incorporates a partial next-to-leading BFKL resummation inside the standard collinear description of observables. We present a detailed analysis of different observables: cross-section summed over azimuthal angles and differential in rapidity and azimuthal distribution. The stability that these distributions show under higher-order corrections motivates our interest in future studies, doable at new generation colliding machin…
Recent works have discussed the violation of factorization and universality in hadronic hard scattering processes aimed at measurements of T-odd distributions. We use simple arguments to show that it is possible to restore an approximate factorization involving T-odd contributions if the factorization breaking interactions present a frequency spectrum dominated by by a narrow and regular peak whose maximum value corresponds to a respected factorization.
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 an all-order generalized factorization formula for QCD scattering amplitudes in kinematical configurations where two or more momenta of the external partons become collinear. The singular behaviour of the scattering amplitudes in the collinear limit is encoded by collinear splitting matrices. In the space-like region and beyond the tree level, the collinear splitting matrices depend also on the momenta and quantum numbers of the non-collinear partons, thus breaking strict collinear factorization. Although the factorization breaking contribution partly cancels for squared amplitudes,…
Universality in QCD factorization of parton densities, fragmentation functions, and soft factors is endangered by the process dependence of the directions of Wilson lines in their definitions. We find a choice of directions that is consistent with factorization and that gives universality between e^+e^- annihilation, semi-inclusive deep-inelastic scattering, and the Drell-Yan process. Universality is only modified by a time-reversal transformation of the soft function and parton densities between Drell-Yan and the other processes, whose only effect is the known reversal of sign for T-odd parto…
We propose the study of the inclusive hadroproduction of a heavy-flavored jet in association with a light jet, as a probe channel of strong interactions at high energies. We build up a hybrid factorization that encodes genuine high-energy effects, provided by a partial next-to-leading BFKL resummation, inside the standard collinear structure of the cross section. We present a detailed analysis of different distributions, shaped on kinematic ranges typical of experimental analyses at the Large Hadron Collider, and differential in rapidity, azimuthal angle and transverse momentum. The fair stabi…
We show that the amplitudes for B decays into two light mesons at leading order in soft-collinear effective theory are factorized to all orders in alpha_s. We construct gauge-invariant four-quark operators by employing the effective theories SCET_I for sqrt{m_b Λ} <μ<m_b, and SCET_II for μ<sqrt{m_b Λ} with Lambda ~0.5 GeV. These operators do not allow gluon exchanges between different sectors of the current and the matrix elements are reduced to the products of current matrix elements. The spectator interactions are also factorized at leading order in SCET and to all orders in alpha_s.
We analyze transverse thrust in the framework of Soft Collinear Effective Theory and obtain a factorized expression for the cross section that permits resummation of terms enhanced in the dijet limit to arbitrary accuracy. The factorization theorem for this hadron-collider event-shape variable involves collinear emissions at different virtualities and suffers from a collinear anomaly. We compute all its ingredients at the one-loop order, and show that the two-loop input for next-to-next-to-leading logarithmic accuracy can be extracted numerically, from existing fixed-order codes.
Soft threshold factorization has been used extensively to study hadronic collisions. It is derived in the limit where the momentum fractions $x_{a,b}$ of both incoming partons approach $x_{a,b}\to 1$. We present a generalized threshold factorization theorem for color-singlet processes, which holds in the weaker limit of only $x_a \to 1$ for generic $x_b$ (or vice versa), corresponding to the limit of large rapidity but generic invariant mass of the produced color singlet. It encodes the complete soft and/or collinear singular structure in the partonic momentum fractions to all orders in pertur…
In this work we consider how a parton distribution function, with an explicit transverse momentum dependence can be properly defined in a regularization-scheme independent manner. We argue that by considering a factorized form of the transverse momentum dependent spectrum for the production of a heavy lepton pair in Drell-Yan reaction, one should first split the relevant soft function into two boost invariant contributions. When those soft contributions are added to the pure collinear contributions, well-defined hadronic matrix elements emerge, i.e., the transverse momentum dependent distribut…
In perturbative QCD, the masses of the hadrons involved in high energy reactions can usually be neglected. However, in the case of Kaon production in electron-proton collisions at low (and not so low) beam energies this may not be a good approximation. In particular, a recent proposal to include hadron masses in theoretical calculations shows how these Hadron Mass Corrections (HMCs) can explain a large discrepancy observed in measurements performed at the HERMES and COMPASS experiments. In this talk, we present preliminary results of a spectator model calculation designed to test the range of …
We consider the singular behaviour of QCD scattering amplitudes in kinematical configurations where two or more momenta of the external partons become collinear. At the tree level, this behaviour is known to be controlled by factorization formulae in which the singular collinear factor is universal (process independent). We show that this strict (process-independent) factorization is not valid at one-loop and higher-loop orders in the case of the collinear limit in space-like regions (e.g., collinear radiation from initial-state partons). We introduce a generalized version of all-order colline…
The hadroproduction of direct J/Psi in the framework of the k_\perp-factorization approach is studied. The color-singlet contribution is essentially larger than in the collinear approach but is still an order of magnitude below the data. The deficit may be well described by the color octet contribution with the value of the matrix element <0|O^{J/Ψ}_8(^3S_1)|0> substantially decreased in comparison with the fits in the collinear factorization. This should lead to a reduction of the large transverse polarization, predicted in the collinear approach.
We reformulate the soft-collinear effective theory which includes the collinear quark and soft gluons. The quark form factor is used to prove that SCET$_{\rm II}$ reproduces the IR physics of the full theory. We give a factorization proof in deep inelastic lepton-hadron scattering by use of the position space formulation.
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 …
