Regge Limit Research Articles

Computing one-loop corrections to effective vertices with two scales in the EFT for Multi-Regge processes in QCD

The computation of one-loop corrections to Reggeon-Particle-Particle effective vertices with two scales of virtuality is considered in the framework of gauge-invariant effective field theory for Multi-Regge processes in QCD. Rapidity divergences arising in loop integrals are regulated by “tilted Wilson lines” prescription. General analysis of rapidity divergences at one loop is given and necessary scalar integrals with one and two scales of virtuality are computed. Two examples of effective vertices at one loop are considered: the effective vertex of interaction of (space-like) virtual photon with one Reggeized and one Yang-Mills quark and the effective vertex of Reggeized gluon to Yang-Mills gluon transition with an insertion of the operator tr[GμνGμν] carrying the (space-like) off-shell momentum. All terms ∼r±ϵ in the rapidity-regulator variable r cancel between diagrams and only log⁡r-divergence is left. It is checked on several examples, that obtained results indeed allow one to reproduce Regge limit of one-loop QCD scattering amplitudes.

Black holes, heavy states, phase shift and anomalous dimensions

We compute the phase shift of a highly energetic particle traveling in the background of an asymptotically AdS black hole. In the dual CFT, the phase shift is related to a four point function in the Regge limit. The black hole mass is translated to the ratio between the conformal dimension of a heavy operator and the central charge. This ratio serves as a useful expansion parameter; its power measures the number of stress tensors appearing in the intermediate channel. We compute the leading term in the phase shift in a holographic CFT of arbitrary dimensionality using Conformal Regge Theory and observe complete agreement with the gravity result. In a two-dimensional CFT with a large central charge the heavy-heavy-light-light Virasoro vacuum block reproduces the gravity phase shift to all orders in the expansion parameter. We show that the leading order phase shift is related to the anomalous dimensions of certain double trace operators and verify this agreement using known results for the latter. We also perform a separate gravity calculation of these anomalous dimensions to second order in the expansion parameter and compare with the phase shift expansion.

Open-string singlet interaction as pomeron in elastic pp, pbar{p} scattering

We consider contributions of the open-string singlet (OSS) interaction to the proton–proton scattering. Modelling baryon and meson as instanton and open string in the effective QCD string model, the typical pomeron is identified with a massive spin-2 closed string. According to the open-closed string duality, the massive spin-2 closed string is accompanied by the open-string singlet interaction with universal coupling. The flat space 4-point open-string singlet amplitudes and cross sections are computed in the Regge limit. Including both the open and closed string pomerons, we fit the differential cross-section with experimental data of the proton–proton and proton–antiproton scattering from the CDF, E-710, D0, LHC-ATLAS and LHC-TOTEM collaborations. The fitting results show the universality of the string mass scale (or string tension) and consistency of the string model between the open and closed string parameters in describing the pp and pbar{p} scattering in the Regge limit. The fitting of the string model to experimental data gives the string scale M_{S}simeq 1.7–1.8 GeV and the corresponding glueball pole mass m_{g}simeq 1.6–1.7 GeV. This lies within experimental values of the observed 2^{++}, I=0 “meson” masses.

Measuring color memory in a color glass condensate at electron–ion colliders

The color memory effect is the non-abelian gauge theory analog of the gravitational memory effect, in which the passage of color radiation induces a net relative SU(3) color rotation of a pair of nearby quarks. It is proposed that this effect can be measured in the Regge limit of deeply inelastic scattering at electron–ion colliders.

A bound on massive higher spin particles

According to common lore, massive elementary higher spin particles lead to inconsistencies when coupled to gravity. However, this scenario was not completely ruled out by previous arguments. In this paper, we show that in a theory where the low energy dynamics of the gravitons are governed by the Einstein-Hilbert action, any finite number of massive elementary particles with spin more than two cannot interact with gravitons, even classically, in a way that preserves causality. This is achieved in flat spacetime by studying eikonal scattering of higher spin particles in more than three spacetime dimensions. Our argument is insensitive to the physics above the effective cut-off scale and closes certain loopholes in previous arguments. Furthermore, it applies to higher spin particles even if they do not contribute to tree-level graviton scattering as a consequence of being charged under a global symmetry such as ℤ2. We derive analogous bounds in anti-de Sitter space-time from analyticity properties of correlators of the dual CFT in the Regge limit. We also argue that an infinite tower of fine-tuned higher spin particles can still be consistent with causality. However, they necessarily affect the dynamics of gravitons at an energy scale comparable to the mass of the lightest higher spin particle. Finally, we apply the bound in de Sitter to impose restrictions on the structure of three-point functions in the squeezed limit of the scalar curvature perturbation produced during inflation.

Structure constants of twist-two light-ray operators in the triple Regge limit

The structure constants of twist-two operators with spin j in the BFKL limit g2 → 0, j → 1 and frac{g^2}{j-1} ∼ 1 are found from the calculation of the three-point correlator of twist-two light-ray operators in the triple Regge limit. It is well known that the anomalous dimensions of twist-two operators in this limit are determined by the BFKL intercept. Similarly, the obtained structure constants are determined by an analytic function of three BFKL intercepts.

Shockwaves from the operator product expansion

We clarify and further explore the CFT dual of shockwave geometries in Anti-de Sitter. The shockwave is dual to a CFT state produced by a heavy local operator inserted at a complex point. It can also be created by light operators, smeared over complex positions. We describe the dictionary in both cases, and compare to various calculations, old and new. In CFT, we analyze the operator product expansion in the Regge limit, and find that the leading contribution is exactly the shockwave operator, ∫ duhuu, localized on a bulk geodesic. For heavy sources this is a simple consequence of conformal invariance, but for light operators it involves a smearing procedure that projects out certain double-trace contributions to the OPE. We revisit causality constraints in large-N CFT from this perspective, and show that the chaos bound in CFT coincides with a bulk condition proposed by Engelhardt and Fischetti. In particular states, this reproduces known constraints on CFT 3-point couplings, and confirms some assumptions about double-trace operators made in previous work.

Particle Production at High Energy: DGLAP, BFKL and Beyond

Particle production in high energy hadronic/nuclear collisions in the Bjorken limit Q 2 , s → ∞ can be described in the collinear factorization framework of perturbative Quantum ChromoDynamics (QCD). On the other hand in the Regge limit, at fixed and not too high Q 2 with s → ∞ , a k ⊥ factorization approach (or a generalization of it) is the appropriate framework. A new effective action approach to QCD in the Regge limit, known as the Color Glass Condensate (CGC) formalism, has been developed which allows one to investigate particle production in high energy collisions in the kinematics where collinear factorization breaks down. Here we give a brief overview of particle production in CGC framework and the evolution equation which governs energy dependence of the observables in this formalism. We show that the new evolution equation reduces to previously known evolution equations in the appropriate limits.

The analytic functional bootstrap. Part II. Natural bases for the crossing equation

We clarify the relationships between different approaches to the conformal bootstrap. A central role is played by the so-called extremal functionals. They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap. We explain in detail that the extremal functionals probe the Regge limit. We construct two complete sets of extremal functionals for the crossing equation specialized to z=overline{z} , associated to the generalized free boson and fermion theories. These functionals lead to non-perturbative sum rules on the CFT data which automatically incorporate Regge boundedness of physical correlators. The sum rules imply universal properties of the OPE at large Δ in every unitary solution of SL(2) crossing. In particular, we prove an upper and lower bound on a weighted sum of OPE coefficients present between consecutive generalized free field dimensions. The lower bound implies the ϕ × ϕ OPE must contain at least one primary in the interval [2Δϕ + 2n, 2Δϕ + 2n + 4] for all sufficiently large integer n. The functionals directly compute the OPE decomposition of crossing-symmetrized Witten exchange diagrams in AdS2. Therefore, they provide a derivation of the Polyakov bootstrap for SL(2), in particular fixing the so-called contact-term ambiguity. We also use the resulting sum rules to bootstrap several Witten diagrams in AdS2 up to two loops.

Strong Interactions in the Regge Limit and Infrared Region

In this talk, 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. 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 by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (Pomeron states) contributing to the high-energy scattering

Light cone bootstrap in general 2D CFTs and entanglement from light cone singularity

The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the general CFT 2 with c > 1. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit z → 1, which was unknown until now. In this study, we computed it in general by studying the pole structure of the fusion matrix (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value frac{c-1}{12} if the total Liouville momentum exceeds beyond the BTZ threshold. This might be interpreted as a black hole formation in AdS3.As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench.We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general n-th Renyi entropy after a local quench and out of time ordered correlators.

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.

Light-ray operators in conformal field theory

We argue that every CFT contains light-ray operators labeled by a continuous spin J. When J is a positive integer, light-ray operators become integrals of local operators over a null line. However for non-integer J , light-ray operators are genuinely nonlocal and give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in our construction is played by a novel set of intrinsically Lorentzian integral transforms that generalize the shadow transform. Matrix elements of light-ray operators can be computed via the integral of a double-commutator against a conformal block. This gives a simple derivation of Caron-Huot’s Lorentzian OPE inversion formula and lets us generalize it to arbitrary four-point functions. Furthermore, we show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-point functions. The average null energy operator is an important example of a light-ray operator. Using our construction, we find a new proof of the average null energy condition (ANEC), and furthermore generalize the ANEC to continuous spin.

Minkowski conformal blocks and the Regge limit for Sachdev-Ye-Kitaev-like models

We discuss scattering in a CFT via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an OPE with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular $F^{(M)}(u,v)\sim {u}^{-\delta}\,$, $\delta>0$, in the limit $u \rightarrow 0$ and $v\rightarrow 1$. Here, $\delta=(\ell_{eff}-1)/2$, with $\ell_{eff}$ serving as an effective spin and it can be determined through an OPE. In particular, it is bounded from above, $\ell_{eff} \leq 2$, for all CFTs with a gravity dual, and it can be associated with string modes interpolating the graviton in AdS. This singularity is historically referred to as the Pomeron. This bound is nearly saturated by SYK-like effective $d=1$ CFT, and its stringy and thermal corrections have piqued current interests. Our analysis has been facilitated by dealing with Wightman functions. We provide a direct treatment in diagonalizing dynamical equations via harmonic analysis over physical scattering regions. As an example these methods are applied to the SYK model.

A conformal collider for holographic CFTs

We develop a formalism to study the implications of causality on OPE coefficients in conformal field theories with large central charge and a sparse spectrum of higher spin operators. The formalism has the interpretation of a new conformal collider-type experiment for these class of CFTs and hence it has the advantage of requiring knowledge only about CFT three-point functions. This is accomplished by considering the holographic null energy operator which was introduced in [1] as a generalization of the averaged null energy operator. Analyticity properties of correlators in the Regge limit imply that the holographic null energy operator is a positive operator in a subspace of the total CFT Hilbert space. Utilizing this positivity condition, we derive bounds on three-point functions 〈TO1O2〉 of the stress tensor with various operators for CFTs with large central charge and a sparse spectrum. After imposing these constraints, we also find that the operator product expansions of all primary operators in the Regge limit have certain universal properties. All of these results are consistent with the expectation that CFTs in this class, irrespective of their microscopic details, admit universal gravity-like holographic dual descriptions. Furthermore, this connection enables us to constrain various inflationary observables such as the amplitude of chiral gravity waves, non-gaussanity of gravity waves and tensor-to-scalar ratio.

Biscalar Integrable Conformal Field Theories in Any Dimension.

We propose a D-dimensional generalization of 4D biscalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a particular strong-twist limit of γ-deformed N=4 supersymmetric Yang-Mills theory. Similar to the 4D case, the planar correlators of this D-dimensional theory are conformal and dominated by "fishnet" Feynman graphs. The dynamics of these graphs is described by the integrable conformal SO(1,D+1) spin chain. In 2D, it is the analogue of Lipatov's SL(2,C) spin chain for the Regge limit of QCD but with the spins s=1/4 instead of s=0. Generalizing recent 4D results of Grabner, Gromov, Korchemsky, and one of the authors to any D, we compute exactly at any coupling a four-point correlation function dominated by the simplest fishnet graphs of cylindric topology and extract from it exact dimensions of operators with chiral charge 2 and any spin together with some of their operator product expansion structure constants.

Beyond a = c : gravitational couplings to matter and the stress tensor OPE

We derive constraints on the operator product expansion of two stress tensors in conformal field theories (CFTs), both generic and holographic. We point out that in large N CFTs with a large gap to single-trace higher spin operators, the stress tensor sector is not only universal, but isolated: that is, leftlangle TTmathcal{O}rightrangle =0 , where mathcal{O}ne T is a single-trace primary. We show that this follows from a suppression of leftlangle TTmathcal{O}rightrangle by powers of the higher spin gap, Δgap, dual to the bulk mass scale of higher spin particles, and explain why leftlangle TTmathcal{O}rightrangle is a more sensitive probe of Δgap than a − c in 4d CFTs. This result implies that, on the level of cubic couplings, the existence of a consistent truncation to Einstein gravity is a direct consequence of the absence of higher spins. By proving similar behavior for other couplings leftlangle T{mathcal{O}}_1{mathcal{O}}_2rightrangle where {mathcal{O}}_i have spin si ≤ 2, we are led to propose that 1/Δgap is the CFT “dual” of an AdS derivative in a classical action. These results are derived by imposing unitarity on mixed systems of spinning four-point functions in the Regge limit. Using the same method, but without imposing a large gap, we derive new inequalities on these three-point couplings that are valid in any CFT. These are generalizations of the Hofman-Maldacena conformal collider bounds. By combining the collider bound on TT couplings to spin-2 operators with analyticity properties of CFT data, we argue that all three tensor structures of 〈TTT〉 in the free-field basis are nonzero in interacting CFTs.

Subleading Regge limit from a soft anomalous dimension

Wilson lines capture important features of scattering amplitudes, for example soft effects relevant for infrared divergences, and the Regge limit. Beyond the leading power approximation, corrections to the eikonal picture have to be taken into account. In this paper, we study such corrections in a model of massive scattering amplitudes in mathcal{N}=4 super Yang-Mills, in the planar limit, where the mass is generated through a Higgs mechanism. Using known three-loop analytic expressions for the scattering amplitude, we find that the first power suppressed term has a very simple form, equal to a single power law. We propose that its exponent is governed by the anomalous dimension of a Wilson loop with a scalar inserted at the cusp, and we provide perturbative evidence for this proposal. We also analyze other limits of the amplitude and conjecture an exact formula for a total cross-section at high energies.

Soft collinear effective theory for gravity

We present how to construct a Soft Collinear Effective Theory (SCET) for gravity at the leading and next-to-leading powers from the ground up. The soft graviton theorem and decoupling of collinear gravitons at the leading power are manifest from the outset in the effective symmetries of the theory. At the next-to-leading power, certain simple structures of amplitudes, which are completely obscure in Feynman diagrams of the full theory, are also revealed, which greatly simplifies calculations. The effective lagrangian is highly constrained by effectively multiple copies of diffeomorphism invariance that are inevitably present in gravity SCET due to mode separation, an essential ingredient of any SCET. Further explorations of effective theories of gravity with mode separation may shed light on lagrangian-level understandings of some of the surprising properties of gravitational scattering amplitudes. A gravity SCET with an appropriate inclusion of Glauber modes may serve as a powerful tool for studying gravitational scattering in the Regge limit.

Infrared singularities of QCD scattering amplitudes in the Regge limit to all orders

Scattering amplitudes of partons in QCD contain infrared divergences which can be resummed to all orders in terms of an anomalous dimension. Independently, in the limit of high-energy forward scattering, large logarithms of the energy can be resummed using Balitsky-Fadin-Kuraev-Lipatov theory. We use the latter to analyse the infraredsingular part of amplitudes to all orders in perturbation theory and to next-to-leading-logarithm accuracy in the high-energy limit, resumming the two-Reggeon contribution. Remarkably, we find a closed form for the infrared-singular part, predicting the Regge limit of the soft anomalous dimension to any loop order.