We perform a detailed investigation of multiple hard interactions in hadron-hadron collisions. We discuss the space-time, spin and color structure of multiple interactions, classify different contributions according to their power behavior and provide several elements required for establishing all-order factorization. This also allows us to analyze the structure of Sudakov logarithms in double hard scattering. We show how multiparton distributions can be constrained by lattice calculations, by connecting them with generalized parton distributions, and by calculating their behavior at large transverse parton momenta.
After a brief recapitulation of the general interest of parton densities, we discuss multiple hard interactions and multiparton distributions. We report on recent theoretical progress in their QCD description, on outstanding conceptual problems and on possibilities to use multiparton distributions as a laboratory to test and improve our understanding of hadron structure.
We investigate several ingredients for a theory of multiple hard scattering in hadron-hadron collisions. Issues discussed include the space-time structure of multiple interactions, their power behavior, spin and color correlations, interference terms, scale evolution and Sudakov logarithms. We discuss possibilities to constrain multiparton distributions by lattice calculations and by connecting them with generalized parton distributions. We show that the behavior of two-parton distributions at small interparton distances leads to problems with ultraviolet divergences and with double counting, …
We present the general expressions for the resummation, up to next-to-leading logarithmic accuracy, of Sudakov-type logarithms in processes with an arbirtrary number of hard-scattering partons. These results document the formulae used by the authors in several previous phenomenological studies. The resummation formulae presented here, which are valid for phase-space factorizable observables, determine the resummation correction in a process-independent fashion. All process dependence is encoded in the colour and flavour structure of the leading order and virtual one-loop amplitudes, and in Sud…
The multiparton density matrices of the QCD gluon cascade are investigated. The generating functional and master equation for momentum space multiparton density in the double logarithmic approximation (DLA) are proposed.
The connection between multiparton interaction, diffractive processes and saturation effects is discussed. The relation of the rise of the gluon density at small longitudinal momentum fractions x with the occurrence of saturation, diffraction and multiparton interaction is being studied both experimentally and theoretically. We illustrate key ideas underlying recent progress, and stress the role of different theoretical approaches to small-x QCD evolution in investigations of multiparton interactions.
In this talk we discuss an algorithm for the numerical calculation of one-loop QCD amplitudes and present results at next-to-leading order for jet observables in electron-positron annihilation calculated with the above-mentioned method. The algorithm consists of subtraction terms, approximating the soft, collinear and ultraviolet divergences of QCD one-loop amplitudes, as well as a method to deform the integration contour for the loop integration into the complex plane to match Feynman's i delta rule. The algorithm is formulated at the amplitude level and does not rely on Feynman graphs. There…
In this work, we present the extension of an alternative subtraction scheme for next-to-leading order QCD calculations to the case of an arbitrary number of massless final-state partons. The scheme is based on the splitting kernels of an improved parton shower and comes with a reduced number of final state momentum mappings. While a previous publication including the setup of the scheme has been restricted to cases with maximally two massless partons in the final state, we here provide the final state real emission and integrated subtraction terms for processes with any number of massless part…
The structure of the QCD gluonic cascade in configuration space is investigated. The explicit form of the inclusive single particle density in configuration space transverse coordinates is derived in the double logarithmic approximation (DLA) of QCD. The possible simplification of the multiparton density matrix formalism for DLA approach is found and discussed.
We present approximations to tree level multiparton scattering amplitudes which are appropriate when two partons are unresolved. These approximations are required for the analytic isolation of infrared singularities of n+2 parton scattering processes contributing to the next-to-next-to-leading order corrections to n jet cross sections. In each case the colour ordered matrix elements factorise and yield a function containing the singular factors multiplying the n parton amplitudes. When the unresolved particles are not colour connected, the approximations are simple products of the familar eiko…
Multiplicity fluctuations in rings around the jet axis and in off-axis cones have been measured by the DELPHI collaboration in $e^+e^-$ annihilations into hadrons at LEP energies. The measurements are compared with analytical perturbative QCD calculations for the corresponding multiparton system, using the concept of Local Parton Hadron Duality. Some qualitative features are confirmed by the data but substantial quantitative deviations are observed.
The next-to-leading order (NLO) evolution of the parton distribution functions (PDF's) in QCD is the "industry standard" in the lepton-hadron and hadron-hadron collider data analysis. The standard NLO DGLAP evolution is formulated for inclusive (integrated) PDFs and is done using inclusive NLO kernels. We report here on the ongoing project, called KRKMC, in which NLO DGLAP evolution is performed for the exclusive multiparton (fully unintegrated) distributions (ePDF's) with the help of the exclusive kernels. These kernels are calculated within the two-parton phase space for bremsstrahlung subse…
The structure of the QCD gluonic cascade in configuration space is investigated. The explicit form of the inclusive single particle density in configuration space transverse coordinates is derived in the double logarithmic approximation (DLA) of QCD. The possible simplification of the multiparton density matrix formalism for DLA approach is found and discussed.
This paper presents a new event generator, ALPGEN, dedicated to the study of multiparton hard processes in hadronic collisions. The code performs, at the leading order in QCD and EW interactions, the calculation of the exact matrix elements for a large set of parton-level processes of interest in the study of the Tevatron and LHC data. The current version of the code describes the following final states: (W -> ffbar') QQbar+ N jets (Q being a heavy quark, and f=l,q), with N<=4; (Z/gamma* -> f fbar)+QQbar+Njets (f=l,nu), with N<=4; (W -> ffbar') + charm + N jets (f=l,q), N<= 5; (W -> f fbar') +…
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,…
We unravel the structure of the spectrum of the anomalous dimensions of the quark-gluon twist-3 operators which are responsible for the multiparton correlations in hadrons and enter as a leading contribution to several physical cross sections. The method of analysis is bases on the recent finding of a non-trivial integral of motion for the corresponding Hamiltonian problem in multicolour limit which results into exact integrability of the three-particle system. Quasiclassical expansion is used for solving the problem. We address the chiral-odd sector as a case of study.
At the LHC all processes are QCD ones, whether "signal" or "background". In this review the frontiers of current QCD research are addressed, towards increased understanding, improved calculational precision, and role in potential future discoveries. Issues raised include the limits of perturbative QCD calculations and parton distribution usage, the nature of multiparton interactions, the impact of colour reconnection on physical observables, the need for progress on hadronization modelling, the improvements of parton showers and their combination with the matrix-element description, the use of…
Amplitudes of hard {\it exclusive} processes such as γ^*(Q^2) N \to γY, where Y=N (DVCS) or any other state with a limited mass (M_Y^2 << Q^2), factorize into a hard subprocess amplitude and a target (transition) GPD. The corresponding {\it inclusive} cross section, summed over all states Y of a given (limited) mass, is then given by the discontinuity of a forward multiparton distribution. An application to the Drell-Yan process π^+ N \to γ^*(x_F,Q^2)+Y allows to explain the observed longitudinal polarization of the virtual photon at high x_F.
The next-to-leading order (NLO) evolution of the parton distribution functions (PDFs) in QCD is a common tool in the lepton-hadron and hadron-hadron collider data analysis. The standard NLO DGLAP evolution is formulated for inclusive (integrated) PDFs and done using inclusive NLO kernels. We report here on the ongoing project, called KRKMC, in which NLO DGLAP evolution is performed for the exclusive multiparton (fully unintegrated) distributions (ePDFs) with the help of the exclusive kernels. These kernels are calculated within the two-parton phase space for the non-singlet evolution, using Cu…
We study production of two pairs of jets in %hard hadron--hadron collisions in view of extracting contribution of {\em double hard interactions} of three and four partons ($3\to4$, $4\to4$). Such interactions, in spite of being power suppressed at the level of the total cross section, become comparable with the standard hard collisions of two partons, $2\to4$, in the {\em back-to-back kinematics} when the transverse momentum imbalances of two pairing jets are relatively small. We express differential and total cross sections for two-dijet production in double parton collisions through the gene…
Instituto de Física Corpuscular, Universitat de València – Consejo Superior de Investigaciones Científicas（瓦伦西亚大学-西班牙国家科学研究委员会粒子物理研究所）；Dipartimento di Fisica e Astronomia, Università di Firenze and INFN Sezione di Firenze（佛罗伦萨大学物理与天文系及意大利国家核物理研究所佛罗伦萨分部）
We consider the radiation of a soft gluon ($g$) and a soft quark-antiquark ($q{\bar q}$) pair in QCD hard scattering. In the soft limit the scattering amplitude has a singular behaviour that is factorized and controlled by a soft current, which has a process-independent structure in colour space. We evaluate the soft $gq{\bar q}$ current at the tree level for an arbitrary multiparton scattering process. The irreducible correlation component of the current includes strictly non-abelian terms and also terms with an abelian character. Analogous abelian correlations appear for soft photon-lepton-a…
We demonstrate that in the back-to-back kinematics the production of four jets in the collision of two partons is suppressed in the leading log approximation of pQCD, compared to the hard processes involving the collision of four partons. We derive the basic equation for four-jet production in QCD in terms of the convolution of generalized two-parton distributions of colliding hadrons in the momentum space representation. Our derivation leads to geometrical approach in the impact parameter space close to that suggested within the parton model and used before to describe the four-jet production…
This writeup summarizes the main experimental studies of the strong interaction, theoretically described by quantum chromodynamics (QCD), that were presented during the ICHEP-2020 conference. The latest results, measured mostly in p-p collisions at the LHC, are categorized in seven broad topics: (i) Extractions of the strong coupling constant $α_s(m_{\rm Z})$; (ii) Comparison of data to fixed-order (N$^{\rm n}$LO) perturbative QCD calculations; (iii) Determinations of parton distribution functions (PDFs); (iv) Comparison of data to resummed (N$^{\rm n}$LL) pQCD calculations; (v) Parton showeri…
We summarize recent progress in understanding the effects of nonperturbative short-range interactions in QCD on the nucleon's partonic structure at a low scale: (a) Sea quarks have intrinsic transverse momenta up to the chiral symmetry-breaking scale rho^{-1} ~ 0.6 GeV, much larger than those of valence quarks. (b) Sea quarks in the nucleon light-cone wave function exist partly in correlated in pairs of transverse size rho with sigma and pi-like quantum numbers and a distinctive spin structure (L = 1 components). The effects are demonstrated in an effective model of the low-energy dynamics res…
We discuss several examples of how the transverse spatial distribution of partons in the nucleon, as well as multiparton correlations, can be probed by observing hard processes (dijets) in high-energy pp (pbar p) and pA (dA) collisions. Such studies can complement the information gained from measurements of hard exclusive processes in ep scattering. The transverse spatial distribution of partons determines the distribution over pp impact parameters of events with hard dijet production. Correlations in the transverse positions of partons can be studied in multiple dijet production. We find that…
We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two…
We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two…
We present a phenomenological analysis of events with two high transverse momentum ($p_T$) jets separated by a large (pseudo-)rapidity interval void of particle activity, also known as jet-gap-jet events. In the limit where the collision energy $\sqrt{s}$ is much larger than any other momentum scale, the jet-gap-jet process is described in terms of perturbative pomeron exchange between partons within the Balitsky--Fadin--Kuraev--Lipatov (BFKL) limit of perturbative quantum chromodynamics (QCD). The BFKL pomeron exchange amplitudes, with resummation at the next-to-leading logarithmic approximat…
The Froissart (\F) asymptotic behavior of high energy cross-sections, if considered in a parton picture, is usually represented as a kind of behavior that occurs in the process of a collision of two almost black disks, filled with partons, when radiuses of these \F - disks grow proportional to log's of there energies. In this article we briefly summarize the main asymptotic properties of \F - disks that one can expect in QCD. ~~Then we consider if it is possible to guarantee the boost-invariance of transparency $T(s,b) = 1 - σ_{in}(s,b)$, where $σ_{in}(s,b)$ is the total inelastic cross-sectio…
We consider QCD factorization between hard and soft subprocesses in inclusive reactions where the momentum fraction x of one parton approaches unity as the hard scale Q^2 -> \infty, such that Q^2(1-x) is fixed. In this "BB limit" the entire (multi-parton) Fock state containing the high x parton is coherent with the hard subprocess. The soft contribution is given by a forward multiparton matrix element. The BB limit corresponds to a fixed (large or small) missing mass and is thus closely connected to exclusive production. We analyze the Drell-Yan process h + N -> γ^* + X in detail, explaining w…
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…
We use QCD sum rules to determine the twist-two non-singlet operator matrix elements and fixed x structure functions paying particular regard to the estimate of the errors. Particularly for the matrix element determination, we find large uncertainties due to radiative and higher dimension contributions. We consider the origin of these large corrections and comment on their consequences for other operator matrix element determinations.
Recent developements in the calculation of the NNLO QCD corrections to the charm quark mass dependent matrix elements in B -> X_s gamma are reported. Special emphasis is put on the new results of the virtual O(alpha_s^2) fermionic contribution to these matrix elements [1].
We extract the matrix elements of four quark operators O_{L,S} relevant to the B_s and \bar B_s life time difference from QCD sum rules. We find the vacuum saturation approximation works reasonably well, i.e., within 10%. We discuss the implications of our results and compare them with the recent lattice QCD determination.
A new method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states is outlined. The aim is to provide at least a leading-order description of all hard multi-jet configurations together with jet fragmentation to next-to-leading logarithmic accuracy, while avoiding the most serious problems of double counting.
We present a simple way of separating the overlap between the soft and collinear factorization formulae of QCD squared matrix elements. We check its validity explicitly for single and double unresolved emissions of tree-level processes. The new method makes possible the definition of helicity-dependent subtraction terms for regularizing the real contributions in computing radiative corrections to QCD jet cross sections. This implies application of Monte Carlo helicity summation in computing higher order corrections.
A new method to construct event-generators based on next-to-leading order QCD matrix-elements and leading-logarithmic parton showers is proposed. Matrix elements of loop diagrams as well as those of a tree level can be generated using an automatic system. A soft/collinear singularity is treated using a leading-log subtraction method. Higher order re-summation of the soft/collinear correction by the parton shower method is combined with the NLO matrix-element without any double-counting in this method.
We describe how to disentangle the singly- and doubly-unresolved (soft and/or collinear) limits of tree-level QCD squared matrix elements. Using the factorization formulae presented in this paper, we outline a viable general subtraction scheme for computing next-to-next-to-leading order corrections for electron-positron annihilation into jets.
We use QCD sum rules to determine directly the leading-twist non-singlet operator matrix elements based on calculations of three-point correlator functions in configuration space. We find a different result from that obtained by integrating the structure functions' expressions obtained by Belyaev and Ioffe based on calculations of four-point correlators in momentum space. The origin of this discrepancy remains unclear.
We use QCD sum rules to determine the difference between moments of the non-singlet structure functions. This combination decouples from the singular behaviour of the structure functions near x=1 as calculated in the quark-gluon basis and thus should lead to improved sum rule predictions. However, we find there are still very large errors due to higher order corrections. In order to refine the error analysis, we study the effect of renormalon ambiguities on our QCD sum rules results.
We present a consistent way of combining associated weak boson radiation in hard dijet events with hard QCD radiation in Drell-Yan-like scatterings. This integrates multiple tree-level calculations with vastly different cross sections, QCD- and electroweak parton shower resummation into a single framework. The new merging strategy is implemented in the PYTHIA event generator and predictions are confronted with LHC data. Improvements over the previous strategy are observed. Results of the new electroweak-improved merging at a future 100 TeV proton collider are also investigated.
The correlation function of a $V-A$ current with a $V+A$ current is discussed within the framework of QCD in the limit of a large number of colours $N_c$. Applications to the evaluation of chiral condensates of dimension six and higher, as well as to the matrix elements of the $Q_7$ and $Q_8$ electroweak penguin operators are discussed. A critical comparison with previous determinations of the same parameters has also been made.
We describe the calculation of the $B\rightarrow π$ hadronic matrix element from operator product expansion near the light-cone in QCD. In the resulting sum rules, the form factors $f^+$ and $F_0$ determining this matrix element are expressed in terms of pion light-cone wave functions. We present predictions for $f^+$ and $F_0$ in the region of momentum transfer $0 \leq p^2 < m_b^2 - {\cal O}(\mbox{1 GeV}^2)$ taking into account the wave functions up to twist 4. We also discuss the extrapolation of the form factors to higher $p^2$ and their asymptotic dependence on the heavy quark mass.
The possibility of treating colour in one-loop amplitude calculations alike the other quantum numbers is briefly discussed for semi-numerical algorithms based on generalized unitarity and parametric integration techniques. Numerical results are presented for the calculation of virtual corrections in multi-gluon scattering.
A new method to construct event-generators based on next-to-leading order QCD matrix-elements and leading-logarithmic parton showers is proposed. Matrix elements of loop diagram as well as those of a tree level can be generated using an automatic system. A soft/collinear singularity is treated using a leading-log subtraction method. Higher order re-summation of the soft/collinear correction by the parton shower method is combined with the NLO matrix-element without any double-counting in this method. An example of the event generator for Drell-Yan process is given for demonstrating a validity …
We propose a method for combining QCD matrix elements and parton showers in Monte Carlo simulations of hadronic final states in $e^+e^-$ annihilation. The matrix element and parton shower domains are separated at some value $y_{ini}$ of the jet resolution, defined according to the $k_T$-clustering algorithm. The matrix elements are modified by Sudakov form factors and the parton showers are subjected to a veto procedure to cancel dependence on $y_{ini}$ to next-to-leading logarithmic accuracy. The method provides a leading-order description of hard multi-jet configurations together with jet fr…
We present a GUT-model-independent calculation of hadron matrix elements for all dimension-six operators associated with baryon number violating processes using lattice QCD. Our results cover all the matrix elements required to estimate the partial lifetimes of (proton,neutron)$\to$($π,K,η$) +(${\bar ν},e^+,μ^+$) decay modes. We point out the necessity of disentangling two form factors that contribute to the matrix elements; previous calculations did not make the separation, which led to an underestimate of the physical matrix elements. With a correct separation, we find that the matrix elemen…
We present a new computation of S-wave color-singlet nonrelativistic QCD matrix elements for the J/psi and the eta_c. We compute the matrix elements of leading order in the heavy-quark velocity v and the matrix elements of relative order v^2. Our computation is based on the electromagnetic decay rates of the J/psi and the eta_c and on a potential model that employs the Cornell potential. We include relativistic corrections to the electromagnetic decay rates, resumming a class of corrections to all orders in v, and find that they significantly increase the values of the matrix elements of leadi…
We have implemented a systematic procedure for combining parton shower algorithms with next-to-leading order QCD calculations for the case of jet production in deep-inelastic electron-proton scattering. Using this method we have computed inclusive jet cross sections and jet shapes for the case of single-jet production and compared them to data from the ZEUS collaboration at HERA. We found good agreement between the data and our calculations, both for the jet shapes and the inclusive spectra.
We compute the ${\cal O}(α_s^3)$ virtual QCD corrections to the $γ^*\to q\bar q g$ matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the $\bar{MS}$ scheme. The infrared pole structure of the matrix elements…
We investigate the off-forward matrix element of the light cone vector operator for a dressed quark state in light-front Hamiltonian perturbation theory. We obtain the corresponding splitting functions in a straightforward way. We show that the end point singularity is canceled by the contribution from the normalization of state. Considering mixing with the gluon operator, we verify the helicity sum rule in perturbation theory. We show that the quark mass effects are suppressed in the plus component of the matrix element but in the transverse component, they are not suppressed. We emphasize th…
We derive an improved prescription for the merging of matrix elements with parton showers, extending the CKKW approach. A flavour-dependent phase space separation criterion is proposed. We show that this new method preserves the logarithmic accuracy of the shower, and that the original proposal can be derived from it. One of the main requirements for the method is a truncated shower algorithm. We outline the corresponding Monte Carlo procedures and apply the new prescription to QCD jet production in e+e- collisions and Drell-Yan lepton pair production. Explicit colour information from matrix e…
We present a method to combine next-to-leading order (NLO) matrix elements in QCD with leading logarithmic parton showers by applying a suitably modified version of the phase-space-slicing method. The method consists of subsuming the NLO corrections into a scale-dependent phase-space-slicing parameter, which is then automatically adjusted to cut out the leading order, virtual, soft and collinear contributions in the matrix element calculation. In this way a positive NLO weight is obtained, which can be redistributed by a parton shower algortihm. As an example, we display the method for single-…
We present a subtraction scheme for eliminating the ultraviolet, soft, and collinear divergences in the numerical calculation of an arbitrary one-loop QCD amplitude with an arbitrary number of external legs. The subtractions consist of local counter terms in the space of the four-dimensional loop momentum. The ultraviolet subtraction terms reproduce MSbar renormalization. The key point in the method for the soft and collinear subtractions is that, although the subtraction terms are defined graph-by-graph and the matrix element is also calculated graph-by-graph, the sum over graphs of the integ…
Models of Weakly Interacting Massive Particles (WIMPs) specified at the electroweak scale are systematically matched to effective theories at hadronic scales where WIMP-nucleus scattering observables are evaluated. Anomalous dimensions and heavy quark threshold matching conditions are computed for the complete basis of lowest-dimension effective operators involving quarks and gluons. The resulting QCD renormalization group evolution equations are solved. The status of relevant hadronic matrix elements is reviewed and phenomenological illustrations are given, including details for the computati…
