multi-Regge Kinematics Research Articles

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Single and pair J/ψ production in the improved color evaporation model using the parton Reggeization approach

In the article, we study single and pair $J/\ensuremath{\psi}$ hadroproduction in the improved color evaporation model via the parton Reggeization approach. The last one is based on the ${k}_{T}$ factorization of hard processes in multi-Regge kinematics, the Kimber-Martin-Ryskin-Watt model for unintegrated parton distribution functions, and the effective field theory of Reggezied gluons and quarks, suggested by L.N. Lipatov. We compare contributions from the single and double parton scattering mechanisms in the pair $J/\ensuremath{\psi}$ production. The numerical calculations are realized using the Monte-Carlo event generator KaTie.

Heptagon functions and seven-gluon amplitudes in multi-Regge kinematics

We compute all 2 → 5 gluon scattering amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory in the multi-Regge limit that is sensitive to the non-trivial (“long”) Regge cut. We provide the amplitudes through four loops and to all logarithmic accuracy at leading power, in terms of single-valued multiple polylogarithms of two variables. To obtain these results, we leverage the function-level results for the amplitudes in the Steinmann cluster bootstrap. To high powers in the series expansion in the two variables, our results agree with the recently conjectured all-order central emission vertex used in the Fourier-Mellin representation of amplitudes in multi-Regge kinematics. Our results therefore provide a resummation of the Fourier-Mellin residues into single-valued polylogarithms, and constitute an important cross-check between the bootstrap approach and the all-orders multi-Regge proposal.

High energy scattering in Einstein\u2013Cartan gravity

We consider Riemann–Cartan gravity with minimal Palatini action, which is classically equivalent to Einstein gravity. Following the ideas of Lipatov (Nucl Phys B 365:614–632, 1991, Phys Part Nucl 44:391–413, 2013, Subnucl Ser 49:131, 2013, Subnucl Ser 50:213–225, 2014, Int J Mod Phys A 31(28/29):1645011, 2016, EPJ Web Conf 125:01010, 2016) and Bartels et al. (JHEP 07:056, 2014) we propose the effective action for this theory aimed at the description of the high-energy scattering of gravitating particles in the multi-Regge kinematics. We add to the Palatini action the new terms. These terms are responsible for the interaction of gravitational quanta with gravitational reggeons. The latter replace exchange by multiple gravitational excitations. We propose the heuristic explanation of its particular form based on an analogy to the reggeon field theory of QCD. We argue that Regge kinematics assumes the appearance of an effective two-dimensional model describing the high-energy scattering similar to that of QCD. Such a model may be formulated in a way leading to our final effective theory. It contains interaction between the ordinary quanta of spin connection and vielbein with the gravitational reggeons.

Towards stability of NLO corrections in high-energy factorization via modified multi-Regge kinematics approximation

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 DIS-like process, induced by the operator tr [GμνGμν ], which is sensitive to gluon PDF already in the LO, is studied. Moderate (O(20%)) NLO corrections to the inclusive structure function are found at small xB< 10−4, while for the pT -spectrum of a leading jet in the considered process, NLO corrections are small (< O(20%)) and LO of kT -factorization is a good approximation. The approach can be straightforwardly extended to the case of multi-scale hard processes in pp-collisions at high energies.

Two-Jet Correlations in Multijet Events in the Regge Limit of QCD

A new method for matching calculations in the leading approximation in the kT-factorization approach with the higher-order contributions of perturbation theory, which appear due to the emission of additional hard partons, is proposed. In the proposed method, azimuthal correlations between the leading and subleading jets in a multijet event are considered. The calculations are carried out within the parton Reggeization approach (PRA) that is based on kT-factorization of hard processes in multi-Regge kinematics, the Kimber–Martin–Ryskin model for nonintegrated parton distribution functions (nPDF), and the effective field theory of Reggeized gluons and quarks proposed by L.N. Lipatov. Theoretical predictions are compared with the CMS collaboration data collected at the Large Hadron Collider at energy $$\sqrt S = 13$$ TeV.

All-Order Amplitudes at any Multiplicity in the Multi-Regge Limit.

We propose an all-loop expression for scattering amplitudes in planar N=4 super Yang-Mills theory in multi-Regge kinematics valid for all multiplicities, all helicity configurations, and arbitrary logarithmic accuracy. Our expression is arrived at from comparing explicit perturbative results with general expectations from the integrable structure of a closely related collinear limit. A crucial ingredient of the analysis is an all-order extension for the central emission vertex that we recently computed at next-to-leading logarithmic accuracy. As an application, we use our all-order formula to prove that all amplitudes in this theory in multi-Regge kinematics are single-valued multiple polylogarithms of uniform transcendental weight.

Small x phenomenology on gluon evolution through the BFKL equation in light of a constraint in multi-Regge kinematics

We investigate the impact of so-called kinematic constraint on gluon evolution at small x. Implanting the constraint on the real emission term of the gluon ladder diagram, we obtain an integro-differential form of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation. Later we solve the equation analytically using the method of characteristics. We sketch the Bjorken x and transverse momentum dependence of our solution of unintegrated gluon distributions in the kinematic constraint supplemented BFKL equation and contrasted the same with the original BFKL equation. Then we extract the integrated gluon density xg(x, Q2) from unintegrated gluon distributions and compared our theoretical prediction with that of global data fits, namely NNPDF3.1sx and CT14. Finally we illustrate the phenomenological implication of our solution for unintegrated gluon distribution towards exploring high precision HERA DIS data by the theoretical prediction of proton structure functions (F2 and FL).

Study of small x behaviour of unintegrated gluon distribution constraining gluon evolution in MD-BFKL equation

In this work we have proposed a modified form of MD-BFKL nonlinear evolution equation imposing a kinematic constraint on the gluon evolution in multi-Regge kinematics. Later we have suggested an analytical solution to our KC imposed MD-BFKL equation and studied nonlinear effect of the transverse momentum dependent (TMD) gluon distribution function at small x (10−6 ≤ x ≤ 10−2). Our results have been compared to various TMDlibuPDF datasets viz. PB-NLO-HERA I+II 2018, ccfm set A0.

The multi-Regge limit of the eight-particle amplitude beyond leading logarithmic accuracy

We present the computation of the eight-particle three-loop amplitude beyond leading logarithmic accuracy in the multi-Regge limit of planar mathcal{N} = 4 Super Yang-Mills theory. Starting from the all-loop dispersion integral form of the amplitude, we consider the eight-particle case and by analyzing said dispersion integral we associate it to a well-defined Fourier-Mellin transform. By using the properties of the Fourier-Mellin representation and its convolution product structure, we compute the three-loop eight-particle MHV amplitude at next-to-leading logarithmic accuracy. From this MHV result, we obtain the three-loop eight particle amplitude in multi-Regge kinematics for all helicity configurations, including next-to-next-to-MHV. Finally, we find that the result is described by combinations of single-valued multiple polylogarithms of uniform weight, the leading singularity structure of which corresponds to the classification shown at leading logarithmic accuracy.

Studies on gluon evolution and geometrical scaling in kinematic constrained unitarized BFKL equation: application to high-precision HERA DIS data

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution f(x,k_T^2) towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions (F_2 and F_L) as well as double differential reduced cross section (sigma _r). The validity of our theory in the low Q^2 transition region is established by studying virtual photon–proton cross section in light of HERA data.

Gravitational scattering in the high-energy limit

Any gravitational scattering amplitude takes a remarkably simple factorized form at tree level in multi-Regge kinematics (MRK), where the produced particles are strongly ordered in rapidity. Very recently, it was shown that the scattering equations also have a very simple structure in MRK. In this paper we study Einstein gravity amplitudes in MRK in the framework of the scattering equations. We present a new derivation of the multi-Regge factorization of tree-level amplitudes with any number of external gravitons and any helicity configuration.

All two-loop MHV remainder functions in multi-Regge kinematics

We introduce a method to extract the symbol of the coefficient of (2πi)2 of MHV remainder functions in planar mathcal{N} = 4 Super Yang-Mills in multi-Regge kinematics region directly from the symbol in full kinematics. At two loops this symbol can be uplifted to the full function in a unique way, without any beyond-the-symbol ambiguities. We can therefore determine all two-loop MHV amplitudes at function level in all kinematic regions with different energy signs in multi-Regge kinematics. We analyse our results and we observe that they are consistent with the hypothesis of a contribution from the exchange of a three-Reggeon composite state starting from two loops and eight points in certain kinematic regions.

Multi-Regge kinematics and the scattering equations

We study the solutions to the scattering equations in various quasi-multi-Regge regimes where the produced particles are ordered in rapidity. We observe that in all cases the solutions to the scattering equations admit the same hierarchy as the rapidity ordering, and we conjecture that this behaviour holds independently of the number of external particles. In multi-Regge limit, where the produced particles are strongly ordered in rapidity, we determine exactly all solutions to the scattering equations that contribute to the Cachazo-He-Yuan (CHY) formula for gluon scattering in this limit. When the CHY formula is localised on these solutions, it reproduces the expected factorisation of tree-level amplitudes in terms of impact factors and Lipatov vertices. We also investigate amplitudes in various quasi-MRK. While in these cases we cannot determine the solutions to the scattering equations exactly, we show that again our conjecture combined with the CHY formula implies the factorisation of the amplitude into universal buildings blocks for which we obtain a CHY-type representation.

A functional RG approach for the BFKL Pomeron

In this paper we encode the perturbative BFKL leading logarithmic resummation, relevant for the Regge limit behavior of QCD scattering amplitudes, in the IR-regulated effective action which satisfies exact functional renormalization group equations. This is obtained using a truncation with a specific infinite set of non local vertices describing the multi-Regge kinematics (MRK). The goal is to use this framework to study, in the high energy limit and at larger transverse distances the transition to a much simpler effective local reggeon field theory, whose critical properties were recently investigated in the same framework. We perform a numerical analysis of the spectrum of the BFKL Pomeron deformed by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (states) contributing to the high energy scattering.

The dimensionally reduced description of the high energy scattering and the effective action for the reggeized gluons

We discuss the high energy scattering of hadrons with the multi-Regge kinematics. The effective 2D model for the interaction between the produced hard gluons appears as a result of the dimensional reduction, which is similar to the dimensional reduction 3+1 D rightarrow 3 D of the high temperature gauge theory. It is demonstrated that being supplemented by the exchange by virtual 3+1 D gluons this 2D model gives rise to the hermitian version of the effective action of Lipatov for the interaction between the ordinary and the reggeized gluons.

The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

We present an all-loop dispersion integral, well-defined to arbitrary logarithmic accuracy, describing the multi-Regge limit of the 2 → 5 amplitude in planar mathcal{N}=4 super Yang-Mills theory. It follows from factorization, dual conformal symmetry and consistency with soft limits, and specifically holds in the region where the energies of all produced particles have been analytically continued. After promoting the known symbol of the 2-loop N -particle MHV amplitude in this region to a function, we specialize to N = 7, and extract from it the next-to-leading order (NLO) correction to the BFKL central emission vertex, namely the building block of the dispersion integral that had not yet appeared in the well-studied six-gluon case. As an application of our results, we explicitly compute the seven-gluon amplitude at next-to-leading logarithmic accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the two independent NMHV helicity configurations, respectively.

From LO to NLO in the parton Reggeization approach

We present recent developments of the parton Reggeization approach (PRA), which is based on high-energy factorization of hard processes in the multi-Regge kinematics and Lipatov’s effective theory of Reggeized gluons and Reggeized quarks. The scheme of calculations in the leading order (LO) of the PRA is discussed. We present important examples of LO PRA applications for cross section calculations of multi-scale hard processes, such as pair production ofBB¯-mesons, pair production of photons, and pair production of jets. Also the problem of matching of NLO calculations in PRA with NLO results in the Collinear Parton Model is discussed.

Hard processes at high energies in the Reggeized-parton approach

Dominant contributions to the cross sections of hard processes at high energies come from the processes with multi-Regge kinematics which reflect the Reggeization of partonic amplitudes as a fundamental property of quantum-field gauge theories. The report briefly describes the Reggeized-parton approach based on the k T factorization at high energies and on the Lipatov’s effective field theory for Reggeized gluons and quarks.

Inclusive four-jet production at 7 and 13 TeV: Azimuthal profile in multi-Regge kinematics

Recently, new observables in LHC inclusive events with three tagged jets were proposed. Here, we extend that proposal to events with four tagged jets. The events are characterized by one jet in the forward direction, one in the backward direction with a large rapidity distance Y from the first one and two more jets tagged in more central regions of the detector. In our setup, non-tagged associated mini-jet multiplicity is present and needs to be accounted for by the inclusion of BFKL gluon Green functions. The projection of the cross section on azimuthal-angle components opens up the opportunity for defining new ratios of correlation functions of the azimuthal-angle differences among the tagged jets that can be used as probes of the BFKL dynamics.