multi-Regge Limit Research Articles

Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function $U$, which is closely related to the parity-even part of the ratio function $V$, is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios $(u,v,w)$, as well as on a few two-dimensional planes. Finally, we present an empirical formula for $V$ in terms of elements of the coproduct of the six-gluon MHV remainder function $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).

The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at N3LLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.

Color-kinematics duality and the Regge limit of inelastic amplitudes

We investigate tree-level five-point amplitudes in scalar-QCD expressed in terms of Sudakov variables and find the equivalent "gravitational" counterparts using the color-kinematics duality proposed by Bern, Carrasco, and Johannson. Taking the multi-Regge limit in the gravitational amplitudes, we show that those pieces in the coupling of two reggeized gravitons to one on-shell graviton directly stemming from the double copy of the vertex for two reggeized gluons to one on-shell gluon are universal and properly reproduced by the duality.

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory

We consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2->4 and 3->3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2->4 and 3->3 NMHV amplitudes, as well as a particular sub-set of two loop 2 ->2 +n N^kMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.

The six-point remainder function to all loop orders in the multi-Regge limit

Abstract We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric $ \mathcal{N}=4 $ Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem — the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w * , which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w| → 0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log |w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log |w| is given in terms of modified Bessel functions.

Multi-Regge limit of the n-gluon bubble ansatz

Abstract We investigate n-gluon scattering amplitudes in the multi-Regge region of $ \mathcal{N}=4 $ supersymmetric Yang-Mills theory at strong coupling. Through a careful analysis of the thermodynamic bubble ansatz (TBA) for surfaces in AdS 5 with n-g(lu)on boundary conditions we demonstrate that the multi-Regge limit probes the large volume regime of the TBA. In reaching the multi-Regge regime we encounter wall-crossing in the TBA for all n > 6. Our results imply that, in this limit, amplitudes at strong coupling can be calculated through an auxiliary system of algebraic Bethe ansatz equations. Their solutions are expected to determine the strong coupling analogue of Regge cut contributions to the amplitude.

Single-valued harmonic polylogarithms and the multi-Regge limit

Abstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w ∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w ∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

Next-to-leading order jet vertex from Lipatov’s QCD effective action

Real and virtual contributions to the quark-initiated forward jet vertex in QCD are calculated at next-to-leading logarithmic accuracy using Lipatov's effective action, valid in the high-energy multi-Regge limit. A regularization of longitudinal divergencies is proposed which allows for the determination of jet observables. Agreement with previous results in the literature is found.

Graviton emission in Einstein-Hilbert gravity

The five-point amplitude for the scattering of two distinct scalars with the emission of one graviton in the final state is calculated in exact kinematics for Einstein-Hilbert gravity. The result, which satisfies the Steinmann relations, is expressed in Sudakov variables, finding that it corresponds to the sum of two gauge invariant contributions written in terms of a new two scalar - two graviton effective vertex. A similar calculation is carried out in Quantum Chromodynamics (QCD) for the scattering of two distinct quarks with one extra gluon in the final state. The effective vertices which appear in both cases are then evaluated in the multi-Regge limit reproducing the well-known result obtained by Lipatov where the Einstein-Hilbert graviton emission vertex can be written as the product of two QCD gluon emission vertices, up to corrections to preserve the Steinmann relations.

Bootstrapping the three-loop hexagon

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar $ \mathcal{N} = 4 $ super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol’s entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 → 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

AdS/CFT: SCATTERING AMPLITUDES IN THE REGGE LIMIT – FROM WEAK TO STRONG COUPLING

I briefly review recent work, within the AdS/CFT correspondence, on the determination of the remainder function of the six point scattering amplitude in the multi-Regge limit, both at weak and at strong coupling.

The Pomeranchuk singularity and vector boson reggeization in electroweak theory

We investigate the high energy behavior of vector boson scattering in the electroweak sector of the Standard Model. In analogy with the BFKL analysis in QCD we compute production amplitudes in the multi-Regge limit and derive, for the vacuum exchange channel, the integral equation for vector particle scattering. We also derive and solve the bootstrap equations for the isospin-1 exchange channel, both for the reggeizing charged and non-reggeizing neutral vector bosons.

Symmetry properties of the effective action for high-energy scattering in QCD.

We study the effective action describing high-energy scattering processes in the multi-Regge limit of QCD, which should provide the starting point for a new attempt to overcome the limitations of the leading logarithmic and the eikonal approximations. The action can be obtained via simple graphical rules or by integrating in the QCD functional integral over momentum modes of gluon and quark fields that do not appear explicitely as scattering or exchanged particles in the considered processes. The supersymmetry is used to obtain the terms in the action involving quarks fields from the pure gluonic ones. We observe a Weizs\"acker - Williams type relations between terms describing scattering and production of particles.

High-energy behaviour in a non-abelian gauge theory (II). First corrections to Tn→ m beyond the leading In s approximation

In this second part of our attempt to construct a unitary high-energy description of a spontaneously broken non-abelian gauge theory we calculate, for the n→ m amplitude in the multi-Regge limit, the first corrections beyond the leading logarithmic approximation. The resulting amplitudes come in the form of the reggeon calculus where the number of reggeons in each t-channel is restricted to one or two. We then study the limit where the mass of the vector particle is taken to zero: for the 2→2 amplitude we show that this limit exists, not only for the approximation of the present paper but also for higher-order corrections.

Multi-Regge factorization in inclusive two-particle production

We consider in detail the multi-Regge expansion for the eight-particle amplitude with one triple-Regge and two double-Regge couplings. We show that under trivial conditions and with a suitable choice of helicity contours, the double-Regge couplings in the multi-Regge (MR) limit are identical, to leading order, with couplings obtained in the mixed helicity-pole-Regge-pole (HPRP) limit appropriate to inclusive reactions. Since we do not need to introduce any dependences on any additional variables, our conditions are somewhat simpler than the results of others. As a plausibility argument for the identity of the MR and HPRP limit, we present an intuitive picture of how helicity poles may be visualized at the triple-Regge vertex. Assuming the mentioned identity, we calculate the diffractive contribution to ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}X$ and discuss the problem of isolating this mechanism experimentally. We propose experimental tests for the identity of the couplings and relate this to the problem of observing the ${A}_{1}$.

Multi-Regge limit of the dual Pomeranchuk amplitude and the absence of a sister for the pomeron

The linear triple-Regge limit of the one-loop non-planar orientable diagram is computed. It is proved that a pomeron sister trajectory does not appear in the pomeron sector, at lowest order, of the two-reggeon two-particle amplitude in the conventional dual model. As a consequence, non-planar diagrams, unlike planar ones, do not renormalize the sister Regge trajectory at lowest order. Both effects, appearance of the pomeron sister trajectory and renormalization of the sister Regge trajectory, might happen if higher-order corrections were considered, in agreement with the recent discovery of a pomeron sister in the Shapiro-Virasoro model by Barratt.

High-energy behaviour in a non-Abelian gauge theory: (I). Tn → m in the leading ln s approximation

This is the first part of an attempt to find a unitary high-energy description of a spontaneously broken non-Abelian gauge theory. In this paper we calculate the high-energy behaviour of n → m amplitudes in the leading ln s approximation. We start from tree approximations and then only use dispersion relations and unitarity equations for elastic and inelastic amplitudes in the multi-Regge limit. The resulting amplitudes have multi-Regge behaviour with simple pole exchange, and satisfy s-channel unitarity in all channels. There is, however, no vacuum quantum number exchange (pomeron) in this approximation. We briefly outline a scheme which may lead to a unitary description of the pomeron.

Possible phenomenological implications of the new trajectories on the pion trajectory

We discuss possible phenomenological consequences of the new Regge trajectories recently discovered in the conventional dual model (CDM) and the Neveu-Schwarz model (NSM). The contributions of the new trajectories to the scattering amplitude dominate at large enough momentum transfers in the multi-Regge limit and it is shown that the β π trajectory of the NSM should be the most easy to detect experimentally.

Multi-regge limit of the Veneziano model and the new trajectories (I)

We study multi-regge limits of the Veneziano model corresponding to the exchange of new trajectories. We investigate in detail the behaviour of the simplest six-, seven- and eight-point functions and exhibit contributions corresponding to the exchange of α, β, γ trajectories. We also derive the form of the two-β-trajectory—one-particle vertex function and compare it with the conventional trajectories vertex function. We also investigate the simplest higher point functions exhibiting contributions corresponding to the exchange of all trajectories from the family of new trajectories suggested by Hoyer. We discuss briefly the structure of regge vertices involving new trajectories.

Simple rules for the signature and the singularity structure of the multiparticle amplitudes

We show that the representation of the 2 → N amplitude in the multiregge limit which exhibits signature factors and singularity structure allowed by the Steinmann relations can be reproduced by means of a few simple rules. We illustrate the rules with several examples and point out their importance for the reggeon calculus.