Soft-collinear Effective Theory Approach Research Articles

Factorization at subleading power in deep inelastic scattering in the x→1 limit

We examine the endpoint region of inclusive deep inelastic scattering at next-to-leading power (NLP). Using a soft-collinear effective theory approach with no explicit soft or collinear modes, we discuss the factorization of the cross section at NLP and show that the overlap subtraction procedure introduced to eliminate double counting of degrees of freedom at leading power ensures that spurious endpoint divergences in the rate cancel at NLP at one loop. For this cancellation to occur at all renormalization scales a nontrivial relation between the anomalous dimensions of the leading and subleading operators is required, which is demonstrated to hold at one loop.

Charmless two-body B meson decays in the perturbative QCD factorization approach* *Supported by the National Science Foundation of China (11805060, 11975112, 12105028, 12070131001, 11775117), the Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (U1932110), the Natural Science Foundation of Jiangsu Province (BK20200980) and the National Key Research and Development Program of China (2020YFA0406400)

The perturbative quantum chromodynamics (PQCD) approach based on factorization has resulted in great achievements in the QCD calculation of hadronic B decays. By regulating the endpoint divergence by the transverse momentum of quarks in the propagators, one can perform the perturbation calculation for various diagrams, including annihilation type diagrams. In this paper, we review the current status of the PQCD factorization calculation of two-body charmless U decays up to next-to-leading order (NLO) QCD corrections. Two new power suppressed terms in the decaying amplitudes are also considered. Using universal input (non-perturbative) parameters, we collect the branching ratios and asymmetry parameters of all charmless two body B decays, which are calculated in the PQCD approach up to NLO. The results are compared with those of the QCD factorization approach, soft-collinear effective theory approach, and current experimental measurements. For most of the considered B meson decays, the PQCD results for branching ratios agree well with those of other approaches and experimental data. The PQCD predictions for the asymmetry parameters of many of the decay channels do not agree with those of other approaches but have better agreement with experimental data. The longstanding puzzle regarding the pattern of the direct CP asymmetries of penguin-dominated decays can be understood after the inclusion of NLO contributions in PQCD. The NLO corrections and power suppressed terms play an important role in color suppressed and pure annihilation type B decay modes. These rare decays are more sensitive to different types of corrections, providing an opportunity to examine the factorization approach with more precise experimental measurements.

The qT and \u2206\u03d5 spectra in W and Z production at the LHC at N3LL\u2032+N2LO

The production of weak gauge bosons, W± and Z, are at the core of the LHC precision measurement program. Their transverse momentum spectra as well as their pairwise ratios are key theoretical inputs to many high-precision analyses, ranging from the W mass measurement to the determination of parton distribution functions. Owing to the different properties of the W and Z boson and the different accessible fiducial regions for their measurement, a simple one-dimensional correlation is insufficient to capture the differing vector and axial-vector dynamics of the produced lepton pair. We propose to correlate them in two observables, the transverse momentum qT of the lepton pair and its azimuthal separation ∆ϕ. Both quantities are purely transverse and therefore accessible in all three processes, either directly or by utilising the missing transverse momentum of the event. We calculate all the single-differential qT and ∆ϕ as well as the double-differential (qT, ∆ϕ) spectra for all three processes at N3LL′+N2LO accuracy, resumming small transverse momentum logarithms in the soft-collinear effective theory approach and including all singlet and non-singlet contributions. Using the double-differential cross sections we build the pairwise ratios {mathrm{mathcal{R}}}_{W^{+}/Z} , {mathrm{mathcal{R}}}_{W^{-}/Z} , and {mathrm{mathcal{R}}}_{W^{+}/{W}^{-}} and determine their uncertainties assuming fully correlated, partially correlated, and uncorrelated uncertainties in the respective numerators and denominators. In the preferred partially correlated case we find uncertainties of less than 1% in most phase space regions and up to 3% in the lowest qT region.

A formalism for the resummation of non-factorizable observables in SCET

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 in SCET for several physical observables within a single framework.

SCET approach to regularization-scheme dependence of QCD amplitudes

We investigate the regularization-scheme dependence of scattering amplitudes in massless QCD and find that the four-dimensional helicity scheme (FDH) and dimensional reduction (DRED) are consistent at least up to NNLO in the perturbative expansion if renormalization is done appropriately. Scheme dependence is shown to be deeply linked to the structure of UV and IR singularities. We use jet and soft functions defined in soft-collinear effective theory (SCET) to efficiently extract the relevant anomalous dimensions in the different schemes. This result allows us to construct transition rules for scattering amplitudes between different schemes (CDR, HV, FDH, DRED) up to NNLO in massless QCD. We also show by explicit calculation that the hard, soft and jet functions in SCET are regularization-scheme independent.

The scale of soft resummation in SCET vs perturbative QCD

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 the 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 μs which characterizes the SCET result. We provide a master formula relating the two approaches for other scale choices. We then show that with the Becher-Neubert scale choice the Landau pole, which in the perturbative QCD approach is usually removed through power- or exponentially suppressed terms, in the SCET approach is removed by logarithmically subleading terms which break factorization. Such terms may become leading for generic choices of parton distributions, and are always leading when resummation is used far enough from the hadronic threshold.

Two-photon exchange corrections to elastic electron-proton scattering at large momentum transfer within the SCET approach

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 formulas for the TPE amplitudes at the leading power and leading logarithmic accuracy. The momentum region where both photons are hard is described by only one new nonperturbative SCET form factor. It turns out that the same form factor also arises for wide-angle Compton scattering which is also described in the framework of the SCET approach. This allows us to estimate the soft-spectator contribution associated with the hard photons in a model independent way. The main unknown in our description of the TPE contribution is related with the configuration where one photon is soft. The nonperturbative dynamics in this case is described by two unknown SCET amplitudes. We use a simple model in order to estimate their contribution. The formalism is then applied to a phenomenological analysis of existing data for the reduced cross section as well as for the transverse and longitudinal polarization observables.

Threshold resummation in SCET vs. perturbative QCD: An analytic comparison

We compare threshold resummation in QCD, as 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 consider various forms of the SCET result, which correspond to different choices of the soft scale μs that characterizes this approach. We derive a master formula that relates the SCET resummation to the QCD result for any choice of μs. We then use it first, to show that if SCET resummation is performed in N-Mellin moment space by suitable choice of μs it is equivalent to the standard perturbative approach. Next, we show that if SCET resummation is performed by choosing for μs a partonic momentum variable, the perturbative result for partonic resummed cross-sections is again reproduced, but, like its standard perturbative counterpart, it is beset by divergent behaviour at the endpoint. Finally, using the master formula we show that when μs is chosen as a hadronic momentum variable the SCET and standard approach are related through a multiplicative (convolutive) factor, which contains the dependence on the Landau pole and associated divergence. This factor depends on the luminosity in a non-universal way; it lowers by one power of log the accuracy of the resummed result, but it is otherwise subleading if one assumes the luminosity not to contain logarithmically-enhanced terms. Therefore, the SCET approach can be turned into a prescription to remove the Landau pole from the perturbative result, but the price to pay for this is the reduction by one logarithmic power of the accuracy at each order and the need to make assumptions on the parton luminosity.

Soft spectator scattering in the nucleon form factors at largeQ2within the soft collinear effective theory approach

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-hard scale $Q \Lambda$, which arises due to presence of soft spectators, with virtuality $\Lambda^2$ ($\Lambda \sim 0.5$ GeV), such that $Q^{2}\gg Q\Lambda\gg \Lambda^{2}$. We show that in this case a two-step factorization can be successfully carried out using the SCET approach. In a first step (SCET$_I$), we perform the leading order matching of the QCD electromagnetic current onto the relevant SCET$_I$ operators and perform a resummation of large logarithms using renormalization group equations. We then discuss the further matching onto a SCET$_{II}$ framework, and propose the complete factorization formula for the Dirac form factor, accounting for both hard and soft contributions. We also present a qualitative discussion of the phenomenological consequences of this new framework.

Effective predictions of event shapes: factorized, resummed and gapped angularity distributions

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 τa 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 αs and resum large logarithms in these distributions to next-to-leading logarithmic (NLL) accuracy. Our expressions for the next-to-leading order (NLO) (αs) partonic jet and soft functions in the factorization theorem for angularity distributions are given for the first time. We employ a model for the nonperturbative soft function with a gap parameter which cancels the renormalon ambiguity in the partonic soft function. We explore the relation between the SCET approach to resummation and past approaches in QCD, and discuss the advantages of the effective theory approach. In addition, we draw from the NLO calculations of the jet and soft functions an intuitive lesson about how factorization breaks down in the effective theory as a → 1.

Resummation and NLO matching of event shapes with effective field theory

The resummed differential thrust rate in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation is calculated using soft-collinear effective theory (SCET). The resulting distribution in the two-jet region ($T\ensuremath{\sim}1)$ is found to agree with the corresponding expression derived by the standard approach. A matching procedure to account for finite corrections at $Tl1$ is then described. There are two important advantages of the SCET approach. First, SCET manifests a dynamical seesaw scale $q={p}^{2}/Q$ in addition to the center-of-mass energy $Q$ and the jet mass scale $p\ensuremath{\sim}Q\sqrt{(1\ensuremath{-}T)}$. Thus, the resummation of logs of $p/q$ can be cleanly distinguished from the resummation of logs of $Q/p$. Second, finite parts of loop amplitudes appear in specific places in the perturbative distribution: in the matching to the hard function, at the scale $Q$, in matching to the jet function, at the scale $p$, and in matching to the soft function, at the scale $q$. This allows for a consistent merger of fixed order corrections and resummation. In particular, the total NLO ${e}^{+}{e}^{\ensuremath{-}}$ cross section is reproduced from these finite parts without having to perform additional infrared regulation.