We calculate the light cone wave functions for the QCD Fock components in a transverse virtual photon necessary for applications at next-to-leading order in the QCD coupling, including quark masses. We present a detailed calculation of both the one loop wave function for the quark-antiquark Fock component and the tree level wave function for the quark-antiquark-gluon Fock component. The quark masses are renormalized in the pole mass scheme, satisfying constraints from the requirement of Lorentz invariance. In particular the quark Pauli form factor at NLO is recovered from the on-shell limit of the quark-antiquark Fock component. We use our result to calculate the next-to-leading order correction to the high energy deep inelastic scattering (DIS) transverse structure function on a dense target in the dipole factorization framework. Together with our earlier result for longitudinal photons, this completes the calculation of the total deep inelastic scattering cross section in the dipole picture with massive quarks at next-to-leading order, enabling a comparison with experimental data.
In this article we review recent advances in the area of QCD factorization. We begin with a brief outline of QCD factorization, as well as a description of some recent results regarding B-->VV decays. We examine what is necessary at NLO, look at recent advances towards that goal and consider what remains to be done.
Although the choice of a factorization scheme is as important as the choice of a factorization scale, the dependence of theoretical predictions (at finite order) on the choice of a factorization scheme has been little investigated. This is due to the fact that the freedom in the choice of a factorization scheme is enormous, even at NLO. One of the reason why to study factorization schemes is the possible exploitation of the freedom in their choice in the construction of NLO Monte Carlo event generators with NLO initial state parton showers. However, the ZERO factorization scheme, which should …
The NLO generalization of the dipole factorization formula for the structure functions F_2 and F_L at low x is calculated using light front perturbation theory. That result gives some interesting insight into the kinematics of initial state parton showers in mixed space.
In the hybrid kT-factorization formula, one initial-state parton momentum is space-like and carries non-vanishing transverse components, while the other is on-shell. We promote this factorization formula to next-to-leading order. Studying the partonic cross section, we identify all soft and collinear divergencies in the real and virtual contribution, and recognize that all non-cancelling ones can be attributed to PDF evolution, evolution kernel, and target impact factors. In result, we construct a framework that may be used to compute NLO impact factors in general. In particular, we recover kn…
We evaluate in perturbative QCD the semi-inclusive Drell-Yan cross-section for the production of a single hadron accompaining the lepton pair. We demonstrate to one loop level a collinear factorization formula within the fracture functions approach. We propose such a process as a factorization analyzer in hadronic collisions. Phenomenological implications at the hadron colliders are briefly discussed.
The aim of the present study is to show that: the redefinition of the factorization scale $Q_i\to z_i Q_i$ in the ladder can be traded exactly for the NLO correction to the LO evolution kernel, $P(z)\to P(z)+(2C_F α_S/π)Δ(z)$ The above observation was done/exploited in the literature, but the present study demonstrates how this phenomenon is realized within the Markovian Monte Carlo parton shower - hence it might be relevant in MC practice.
We discuss the inclusive production of jets in the central region of rapidity in the context of k_T-factorization at next-to-leading order (NLO). Calculations are performed in the Regge limit making use of the NLO BFKL results. We introduce a jet cone definition and carry out a proper phase--space separation into multi-Regge and quasi-multi-Regge kinematic regions. We discuss two situations: scattering of highly virtual photons, which requires a symmetric energy scale to separate impact factors from the gluon Green's function, and hadron-hadron collisions, where a non-symmetric scale choice is…
The inclusive production of jets in the central region of rapidity is studied in $k_T$-factorization at next-to-leading order (NLO) in QCD perturbation theory. Calculations are performed in the Regge limit making use of the NLO BFKL results. A jet cone definition is introduced and a proper phase--space separation into multi-Regge and quasi-multi-Regge kinematic regions is carried out. Two situations are discussed: scattering of highly virtual photons, which requires a symmetric energy scale to separate the impact factors from the gluon Green's function, and hadron-hadron collisions, where a no…
In this contribution we discuss the inclusive production of jets in central regions of rapidity in the context of $k_T$-factorization at next-to-leading order (NLO). We work in the Regge limit of QCD and use the NLO BFKL results. A jet cone definition is proposed together with a phase-space separation into multi-Regge and quasi-multi-Regge kinematics. We discuss scattering of highly virtual photons, with a symmetric energy scale to separate the impact factors from the gluon Green's function, and hadron-hadron collisions, with a non-symmetric scale choice.
The one-loop QCD corrections to the light-front wave-function for the quark-antiquark Fock state inside a transverse or longitudinal off-shell photon are explicitly calculated, both in full momentum space and in mixed space (a.k.a. dipole space). These results provide one of the main contributions to virtual NLO corrections to many DIS observables (inclusive or not) in the dipole factorization formalism at low Bjorken x. In a follow-up article, these one-loop corrections are combined with earlier results on the wave-function for the quark-antiquark-gluon Fock state, in order to get the full se…
In this paper we firstly demonstrate step by step that the factorization hypothesis is valid at the next-to-leading order (NLO) for the exclusive process $ργ^{\star} \to ρ$ by employing the collinear factorization approach, and then extend this proof to the case of the $k_T$ factorization by taking into account the transversal momentum of the light external quark (anti-quark) lines in the $ρ$ meson. At the NLO level, we then show that the soft divergences from different sub-diagrams will be canceled each other in the quark level, while the remaining collinear divergences can be absorbed into t…
In this paper, by employing the $k_{T}$ factorization theorem, we calculate firstly the next-to-leading-order (NLO) twist-3 contributions to the pion electromagnetic form factors in the $πγ^* \to π$ process. From the analytical and numerical calculations we find the following points: (a) For the leading order (LO) twist-2, twist-3 and the NLO twist-2 contributions, our results agree very well with those obtained in previous works; (b) We extract out two factors $F^{(1)}_{\rm T3}(x_i,t,Q^2)$ and $\ov{F}^{(1)}_{\rm T3}(x_i,t,Q^2)$, which describe directly the NLO twist-3 contributions to the pio…
We calculate the cross sections of inclusive J/psi production in photoproduction and two-photon scattering, involving both direct and resolved photons, and in e^+e^- annihilation at next-to-leading order (NLO) within the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD), including the full relativistic corrections due to the intermediate ^1S_0^{[8]}, ^3S_1^{[8]}, and ^3P_J^{[8]} color-octet (CO) states. Exploiting also our previous results on hadroproduction, we perform a combined fit of the respective CO long-distance matrix elements (LDMEs) to all available high-quali…
We discuss aspects of our recent derivation of the gluon Regge trajectory at two loop from Lipatov's high energy effective action. We show how the gluon Regge trajectory can be rigorously defined through renormalization of the high energy divergence of the reggeized gluon propagator. We furthermore provide some details on the determination of the two-loop reggeized gluon self-energy.
We study the factorization scheme dependence of the next-to-leading order inclusive one jet cross section d sigma/dE_T. The scheme is varied parametrically along the direction that transforms the MSbar scheme to the DIS scheme: we introduce a parameter lambda such that lambda = 0 is MSbar and lambda = 1 is DIS. The factorization scale mu is also varied. We observe a change of +/- 9% in the cross section for E_T = 50 GeV when mu and lambda are varied in the range -2 < lambda < 2, E_T/8 < mu < 2 E_T. This grows to +/- 32% for E_T = 400 GeV.
The NLO corrections to the DIS structure functions $F_2$ and $F_L$ (or equivalently the photon-target cross sections $σ^{γ^*}_{T}$ and $σ^{γ^*}_{L}$) at low $x_{Bj}$ are obtained, as a generalization of the dipole factorization formula. For the first time, the contributions of both the $q\bar{q}$ and the $q\bar{q}g$ Fock states in the photon are directly calculated, using earlier results for the $q\bar{q}$ light-front wave-functions at one loop inside a dressed virtual photon. Both the $q\bar{q}$ and the $q\bar{q}g$ contributions have UV divergences, which are shown to cancel each other, using…
The perturbatively-stable scheme of Next-to-Leading order (NLO) calculations of cross-sections for multi-scale hard-processes in DIS-like kinematics is developed in the framework of High-Energy Factorization. The evolution equation for unintegrated PDF, which resums $\log 1/z$-corrections to the coefficient function in the Leading Logarithmic approximation together with a certain subset of Next-to-Leading Logarithmic and Next-to-Leading Power corrections, necessary for the perturbative stability of the formalism, is formulated and solved in the Doubly-Logarithmic approximation. An example of D…
In this paper, we calculate the next-to-leading-order (NLO) twist-3 contribution to the form factors of $B \to π$ transitions by employing the $k_{T}$ factorization theorem. All the infrared divergences regulated by the logarithms $\ln(k_{iT}^{2})$ cancel between those from the quark diagrams and from the effective diagrams for the initial $B$ meson wave function and the final pion meson wave function. An infrared finite NLO hard kernel is therefore obtained, which confirms the application of the $k_{T}$ factorization theorem to $B$ meson semileptonic decays at twist-3 level. From our analytic…
Next-to-leading order correction to the one-particle inclusive cross section in the framework of high energy factorization is calculated. Numerical results for midrapidity region are compared with predictions of conventional calculations based on collinear factorization.
In this paper we calculate the next-to-leading-order (NLO) corrections to $ρ$-meson electromagnetic form factors by employing the $k_T$ factorization approach. We find that the NLO correction to $F_i (Q^2)(i=LT,TL)$ is around $30\%$ of the leading-order (LO) contributionin the region $Q^2>2GeV^2$. The NLO correction to $F_{LL}(Q^2)$ is close to $20\%$ of the LO one in the region $Q^2>3GeV^2$. The NLO radiative corrections to the electric, magnetic, and quadruple form factors $F_j(Q^2) (j=1,2,3)$ are sizeable in magnitude and agree with those from other approaches.
Single inclusive forward hadron production in high-energy hadron collisions can provide an important test of the Color Glass Condensate picture at small $x$. Recent studies of this process at next-to-leading order have led to problematic results, with cross sections becoming negative at large transverse momenta. We study a new formulation of this quantity proposed recently by Iancu et al. We show that it leads to physical results up to large transverse momenta at fixed coupling. Taking into account running coupling effects in a way that is consistent with existing DIS calculations still poses …
We investigate spin-dependent parton distributions in the polarized virtual photon target in perturbative QCD up to the next-to-leading order (NLO). In the case $Λ^2 \ll P^2 \ll Q^2$, where $-Q^2$ ($-P^2$) is the mass squared of the probe (target) photon, parton distributions can be predicted completely up to NLO, but they are factorization-scheme-dependent.We analyze parton distributions in six different factorization schemes and discuss their scheme dependence. We study, in particular, the QCD and QED axial anomaly effects on the first moments of parton distributions to see the interplay bet…
The possibility of an improvement of current NLO Monte Carlo event generators by means of choosing a suitable factorization scheme is studied. The optimal factorization scheme for combining initial state parton showers and NLO hard scattering cross-sections is the ZERO factorization scheme, in which all NLO splitting functions vanish. However, it has turned out that the ZERO factorization scheme has a limited range of practical applicability. Hence, this paper is focused on searching for a factorization scheme which is applicable at the NLO in the full range of x relevant for QCD phenomenology…
We investigate the differences between the NLO evolution kernels in the Curci-Furmanski-Petronzio (CFP) and Monte Carlo (MC) factorization schemes for the non-singlet case. We show the origin of these differences and present them explicitly. We examine the influence of the choice of the factorization scale in the MC scheme (given by the upper phase space limit) on the evolution kernels in this scheme.
The photon impact factor for the BFKL pomeron is calculated in the next-to-leading order (NLO) approximation using the operator expansion in Wilson lines. The result is represented as a NLO $k_T$-factorization formula for the structure functions of small-$x$ deep inelastic scattering.
Spin-dependent parton distributions in the polarized virtual photon are investigated in QCD up to the next-to-leading order (NLO). In the case $Λ^2 \ll P^2 \ll Q^2$, where $-Q^2$ ($-P^2$) is the mass squared of the probe (target) photon, parton distributions can be predicted completely up to the NLO, but they are factorization-scheme-dependent. Parton distributions are analyzed in four different factorization schemes and their scheme dependence are discussed. Particular attentions are paid to the axial anomaly effect on the first moments of quark parton distributions, and also to the large-$x$…
We present a calculation of the B0 - anti-B0 mixing matrix element in the framework of QCD sum rules for three-point functions. We compute alpha_s corrections to a three-point function at the three-loop level in QCD perturbation theory, which allows one to extract the matrix element with next-to-leading order (NLO) accuracy. This calculation is imperative for a consistent evaluation of experimentally-measured mixing parameters since the coefficient functions of the effective Hamiltonian for B0 - anti-B0 mixing are known at NLO. We find that radiative corrections violate factorization at NLO; t…
Within the Color Glass Condensate effective theory, we reconsider the next-to-leading order (NLO) calculation of the single inclusive particle production at forward rapidities in proton-nucleus collisions at high energy. Focusing on quark production for definiteness, we establish a new factorization scheme, perturbatively correct through NLO, in which there is no `rapidity subtraction'. That is, the NLO correction to the impact factor is not explicitly separated from the high-energy evolution. Our construction exploits the skeleton structure of the (NLO) Balitsky-Kovchegov equation, in which t…
We calculate the time-like pion electromagnetic form factor in the $k_T$ factorization formalism with the inclusion of the next-to-leading-order(NLO) corrections to the leading-twist and sub-leading-twist contributions. It's found that the total NLO correction can enhance (reduce) the magnitude (strong phase) of the leading order form factor by $20\% - 30\%$ ( $< 15^o$) in the considered invariant mass squared $q^2 > 5$ GeV$^2$, and the NLO twist-3 correction play the key role to narrow the gap between the pQCD predictions and the measured values for the time-like pion electromagnetic form fac…
In this paper, we investigate the factorization hypothesis step by step for the exclusive process $B \to ρ$ at next-to-leading order (NLO) with the collinear factorization approach, and then we extend our results to the $k_T$ factorization frame. We show that the soft divergence from the specific NLO diagrams will cancel each other at the quark level, while the remaining collinear divergence can be absorbed into the NLO wave functions. The full NLO amplitudes can be factorized into two parts: the NLO $B$ and $ρ$ meson wave functions containing the collinear divergence and the leading order (LO…
We discuss the exclusive radiative B-meson decays B -> K* gamma and B -> rho gamma in a model-independent manner. The analysis is based on the heavy-quark limit of QCD. This allows a factorization of perturbatively calculable contributions to the B -> V gamma matrix elements from non-perturbative form factors and universal light-cone distribution amplitudes. These results allow us to compute exclusive b -> s(d) gamma decays systematically beyond the leading logarithmic approximation. We present results for these decays complete to next-to-leading order in QCD and to leading order in the heavy-…
The factorisation scheme (FS) abbreviated as KRK FS including a new definition of the PDFs for initial hadrons was formulated while developing KrkNLO scheme of matching QCD NLO corrections for the hard process with the parton shower heavy boson production in hadron-hadron collision and for deep inelastic lepton-hadron scattering. KRK FS (originally called Monte Carlo FS) can be regarded as a variant of the $\overline{MS}$ system. It is therefore trivially universal, that is process independent. The question of its universality is formulated differently: As the basic role of KRK FS is to simpli…
We present results on the NLO (alpha_s^2) spectator-scattering corrections to the topological penguin amplitudes for charmless hadronic two-body B-decays in QCD factorization. The corrections can be sizable for the colour-suppressed electroweak penguin amplitudes alpha_{4,EW}^p but otherwise are numerically small. Our results explicitly demonstrate factorization at this order. To assess the phenomenological viability of the framework, we consider penguin-to-tree ratios in the penguin-dominated pi K system and find agreement to the expected precision (i.e., a power correction).
We calculate and analyze Z and W production in association with quark-antiquark pair in k_T-factorization. Numerical calculations are performed using the Monte Carlo generator CASCADE for proton proton collisions at LHC energy. We compare total and differential cross sections calculated in k_T-factorization approach with total differential cross sections obtained in LO and NLO calculations in collinear factorization approach. We provide strong evidence that some of the effects of the NLO and even higher order collinear calculation are already included in the LO k_T-factorization calculation.
We calculate large mass diphoton exclusive photoproduction in the framework of collinear QCD factorization at next to leading order in αs and at leading twist. Collinear divergences of the coefficient function are absorbed by the evolution of the generalized parton distributions (GPDs). This result enlarges the existing factorization proofs to 2 -> 3 processes, opening new reactions to a trustable extraction of GPDs.
In this paper, we firstly varify that the factorization hypothesis is valid for the exclusive process $ργ^{\star} \to π$ at the next-to-leading order (NLO) with the collinear factorization approach, and then extend this proof to the case of the $k_T$ factorization approach. We particularly show that at the NLO level, the soft divergences in the full quark level calculation could be canceled completely as for the $πγ^{\star} \to π$ process where only the pseudoscalar $π$ meson involved, and the remaining collinear divergences can be absorbed into the NLO hadron wave functions. The full amplitud…
We compute the next-leading-order cross-sections for diffractive electro- or photoproduction of a pair of hadrons with large $p_T$, out of a nucleus or a nucleon. A hybrid factorization is used, mixing collinear and small-$x$ factorizations, more precisely shockwave formalism. We demonstrate the cancellation of divergences and extract the finite parts of the differential cross-section in general kinematics.
We have calculated the diffractive dijet cross section in low-Q^2 ep scattering in the HERA regime. The results of the calculation in LO and NLO are compared to recent experimental data of the H1 collaboration. We find that in LO the calculated cross sections are in reasonable agreement with the experimental results. In NLO, however, some of the cross sections disagree, showing that factorization breaking occurs in that order. By suppressing the resolved contribution by a factor of approximately three, good agreement with all the data is found. The size of the factorization breaking effects in…
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…
The next-to-leading order (NLO) ($\mathcal{O}(α_s^3)$) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in ${^{3}\hspace{-0.6mm}P_{J}^{[1,8]}}$ states are calculated within the NRQCD factorization. We use integration-by-parts reduction and differential equations to semi-analytically calculate fragmentation functions in full-QCD, and find that infrared divergences can be absorbed by the NRQCD long distance matrix elements. Thus, the NRQCD factorization conjecture is verified at two-loop level via a physical process, which is free of artificial ultraviolet divergence…
In nonrelativistic QCD (NRQCD), we study $e^+ e^- \to J/ψ(ψ(2S)) +χ_{c0}$ at $B$ factories, where the P-wave state $χ_{c0}$ is associated with an S-wave state $J/ψ$ or $ψ(2S)$. In contrast to the failure of factorization in most cases involving P-wave states, e.g. in B decays, we find that factorization holds in this process at next to leading order (NLO) in $α_s$ and leading order (LO) in $v$, where the associated S-wave state plays a crucial rule in canceling the infrared (IR) divergences. We also give some general analyses for factorization in various double charmonium production. The NLO c…
We investigate parton distributions in the virtual photon target, both polarized and unpolarized, up to the next-leading order (NLO) in QCD. Parton distributions can be predicted completely up to NLO, but they are factorization-scheme-dependent. We analyze parton distributions in several factorization schemes and discuss their scheme dependence. Particular attentions are paid to the axial anomaly effect on the first moments of the polarized quark parton distributions, and also to the large-$x$ behaviors of polarized and unpolarized parton distributions.
In this work, we will present the first complete calculation of the one-loop longitudinal photon-to-quark-antiquark light cone wave function, with massive quarks. The quark masses are renormalized in the pole mass scheme. The result is used to calculate the next-to-leading order correction to the high energy Deep Inelastic Scattering longitudinal structure function on a dense target in the dipole factorization framework. For massless quarks the next-to-leading order correction was already known to be sizeable, and our result makes it possible to evaluate it also for massive quarks.
We discuss the order α_s^2 gluon initiated QCD corrections to one particle inclusive deep inelastic processes. We focus in the NLO evolution kernels relevant for the non homogeneous QCD scale dependence of these cross sections and factorization.
When comparing recent experimental data from the H1 and ZEUS Collaborations at HERA for diffractive dijet production in deep-inelastic scattering (DIS) and photoproduction with next-to-leading order (NLO) QCD predictions using diffractive parton densities, good agreement is found for DIS. However, the dijet photoproduction data are overestimated by the NLO theory, showing that factorization breaking occurs at this order. While this is expected theoretically for resolved photoproduction, the fact that the data are better described by a global suppression of direct and resolved contribution by a…
We have calculated the diffractive dijet cross section in low-Q^2 ep scattering in the HERA regime. The results of the calculation in LO and NLO are compared to recent experimental data of the H1 collaboration. We find that in LO the calculated cross sections are in reasonable agreement with the experimental results. In NLO, however, some of the cross sections disagree, showing that factorization breaking occurs in that order. By suppressing the resolved contribution by a factor of approximately three, good agreement with all the data is found. The size of the factorization breaking effects in…
In this paper, we calculated the branching ratios and direct CP violation of the four $B\to Kπ$ decays with the inclusion of all currently known next-to-leading order (NLO) contributions by employing the perturbative QCD (pQCD) factorization approach. We found that (a) Besides the 10% enhancement from the NLO vertex corrections, the quark-loops and magnetic penguins, the NLO contributions to the form factors can provide an additional $\sim 15%$ enhancement to the branching ratios, and lead to a very good agreement with the data; (b) The NLO pQCD predictions are $\acp^{dir}(B^0\to K^+π^-)=(-6.5…
We compute the 1-loop (alpha_s^2) correction to hard spectator scattering in non-leptonic B decay tree amplitudes. This forms part of the NNLO contribution to the QCD factorization formula for hadronic B decays, and introduces a new rescattering phase that corrects the leading-order result for direct CP asymmetries. Among the technical issues, we discuss the cancellation of infrared divergences, and the treatment of evanescent four-quark operators. The infrared finiteness of our result establishes factorization of spectator scattering at the 1-loop order. Depending on the values of hadronic in…
In this paper we study the factorization and resummation of s-channel single top quark production in the Standard Model at both the Tevatron and the LHC. We show that the production cross section in the threshold limit can be factorized into a convolution of hard function, soft function and jet function via soft-collinear-effective-theory (SCET), and resummation can be performed using renormalization group equation in the momentum space resummation formalism. We find that in general, the resummation effects enhance the Next-to-Leading-Order (NLO) cross sections by about $3%-5%$ at both the Tev…
The perturbative QCD (PQCD) approach based on $k_T$ factorization has made a great achievement for the QCD calculation of the hadronic B decays. Regulating the endpoint divergence by the transverse momentum of quarks in the propagators, one can do the perturbation calculation for kinds of diagrams including the annihilation type diagrams. In this paper, we review the current status of PQCD factorization calculation of two-body charmless $B\to PP, PV, VV$ decays up to the next-to-leading order (NLO) QCD corrections. two new power suppressed terms in decaying amplitudes are also taken into accou…
We use a factorization scheme analogous to one proposed for single inclusive forward hadron production to factorize the soft gluon divergence present in the deep inelastic scattering cross sections in the dipole picture at next-to-leading order (NLO). We show numerically that in this carefully constructed scheme it is possible to obtain meaningful results for the DIS cross sections at NLO, and so we are able to quantitatively study the recently derived NLO corrections to the DIS cross sections. We find that the NLO corrections can be significant and sensitive to the details of the factorizatio…
Methodology of including QCD NLO corrections in the quark--gluon Monte Carlo shower is outlined. The work concentrates on two issues: (i) constructing leading order (LO) parton shower Monte Carlo from scratch, such that it rigorously extends collinear factorization into the exclusive (fully unintegrated) one which we call the Monte Carlo factorization scheme; (ii) introducing next-to-leading-order (NLO) corrections to the hard process in this new environment. The presented solution is designed to be extended to the full NLO level Monte Carlo, including NLO corrections not only in the hard proc…
