Light-like Limit Research Articles

Arising of trapped surfaces with nontrivial topology from colliding shock waves

The lightlike limit of boosted black hole solutions with one angular momentum is considered for $D \geq4$ dimensions. The boost is performed parallel to the angular momentum and the lightlike limit is done by means of perturbative expansions. We shown that for $D=4$ and $D> 5$ the lightlike limit cannot be extended inside the ring singularity. Then, for $D = 5$ we discuss the arising of trapped surfaces in the head-on collision. We find that, inside the validity of the perturbative analysis we do, a trapped surface with topology $\mathbb R \times \mathbb S_1 \times\mathbb S_1$ seems to appear over the past light cone of the collision below a critical value $a_c$ of the Kerr parameter $a$.

Ironing out the crease

The crease is a surface operator folded by a finite angle along an infinite line. Several realisations of it in the 6d mathcal{N} = (2, 0) theory are studied here. It plays a role similar to the generalised quark-antiquark potential, or the cusp anomalous dimension, in gauge theories. We identify a finite quantity that can be studied despite the conformal anomalies ubiquitous with surface operators and evaluate it in free field theory and in the holographic dual. We also find a subtle difference between the infinite crease and its conformal transform to a compact observable comprised of two glued hemispheres, reminiscent of the circular Wilson loop. We prove by a novel application of defect CFT techniques for the SO(1, 2) symmetry along the fold that the near-BPS behaviour of the crease is determined as the derivative of the compact observable with respect to its angle, as in the bremsstrahlung function. We also comment about the lightlike limit of the crease in Minkowski space.

Ten dimensional symmetry of mathcal{N} = 4 SYM correlators

We consider four-point correlation functions of protected single-trace scalar operators in planar mathcal{N} = 4 supersymmetric Yang-Mills (SYM). We conjecture that all loop corrections derive from an integrand which enjoys a ten-dimensional symmetry. This symmetry combines spacetime and R-charge transformations. By considering a 10D light-like limit, we extend the correlator/amplitude duality by equating large R-charge octagons with Coulomb branch scattering amplitudes. Using results from integrability, this predicts new finite amplitudes as well as some Feynman integrals.

Origin of the Six-Gluon Amplitude in Planar N=4 Supersymmetric Yang-Mills Theory.

We study the maximally-helicity-violating six-gluon scattering amplitude in planar N=4 super-Yang-Mills theory at finite coupling when all three cross ratios are small. It exhibits a double logarithmic scaling in the cross ratios, controlled by a handful of "anomalous dimensions" that are functions of the coupling constant alone. Inspired by known seven-loop results at weak coupling and the integrability-based pentagon operator product expansion, we present conjectures for the all-order resummation of these anomalous dimensions. At strong coupling, our predictions agree perfectly with the string theory analysis. Intriguingly, the simplest of these anomalous dimensions coincides with one describing the lightlike limit of the octagon, namely, the four-point function of large-charge Bogomol'nyi-Prasad-Sommerfield (BPS) operators.

The correlahedron

We introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar mathcal{N}=4 super Yang-Mills. Re-expressing the Grassmann dependence of correlation functions of n chiral stress-energy multiplets with Grassmann degree 4k in terms of 4(n + k)-linear bosonic variables, the resulting expressions have an interpretation as volume forms on a Gr(n+k, 4+n+k) Grassmannian, analogous to the expressions for planar amplitudes via the amplituhedron. The resulting volume forms are to be naturally associated with the correlahedron geometry. We construct such expressions in this bosonised space both directly, in general, from Feynman diagrams in twistor space, and then more invariantly from specific known correlator expressions in analytic superspace. We give a geometric interpretation of the action of the consecutive lightlike limit and show that under this the correlahedron reduces to the squared amplituhedron both as a geometric object as well as directly on the corresponding volume forms. We give an explicit easily implementable algorithm via cylindrical decompositions for extracting the squared amplituhedron volume form from the squared amplituhedron geometry with explicit examples and discuss the analogous procedure for the correlators.

Limit directions for Lorentzian Coxeter systems

Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space of a Coxeter group, limit directions arising from a point are accumulation points of the orbit of this point. In particular, limit directions arising from roots are called limit roots. Recent studies show that limit roots lie on the isotropic cone of the representation space. In this paper, we study limit directions of Coxeter groups arising from any point when the representation space is a Lorentz space. We prove that the limit roots are the only light-like limit directions, and characterize the limit roots using eigenvectors of infinite-order elements. Then we describe the structure of space-like limit directions in terms of the projective Coxeter arrangement. Some non-Lorentzian cases are also discussed.

Bootstrapping correlation functions in N = 4 $$ \mathcal{N}=4 $$ SYM

We describe a new approach to computing the chiral part of correlation functions of stress-tensor supermultiplets in N=4 SYM that relies on symmetries, analytic properties and the structure of the OPE only. We demonstrate that the correlation functions are given by a linear combination of chiral N=4 superconformal invariants accompanied by coefficient functions depending on the space-time coordinates only. We present the explicit construction of these invariants and show that the six-point correlation function is fixed in the Born approximation up to four constant coefficients by its symmetries. In addition, the known asymptotic structure of the correlation function in the light-like limit fixes unambiguously these coefficients up to an overall normalization. We demonstrate that the same approach can be applied to obtain a representation for the six-point NMHV amplitude that is free from any auxiliary gauge fixing parameters, does not involve spurious poles and manifests half of the dual superconformal symmetry.

Lagrangian insertion in the light-like limit and the super-correlators/super-amplitudes duality

In these notes we describe how to formulate the Lagrangian insertion technique in a way that mimics generalized unitarity. We introduce a notion of cuts in position space and show that the cuts of the correlators in the super-correlators/super-amplitudes duality correspond to generalized unitarity cuts of the equivalent amplitudes. The cuts consist of correlation functions of operators in the chiral part of the stress-tensor multiplet as well as other half-BPS operators. We will also discuss the application of the method to other correlators as well as non-planar contributions.

Simplifying instanton corrections to $ \mathcal{N} $ = 4 SYM correlators

We compute instanton corrections to non-minimal correlation functions of chiral primary operators in the N=4 super Yang-Mills super-current multiplet. Using a representation in terms of Mellin integrals, we find that these corrections can be written as conformal integrals in AdS_5 times certain "kinematic" prefactors that are independent of the instanton moduli. We then consider the consecutive, pairwise light-like limit x_{i, i+1}^2 -> 0 of such correlators, and prove that the ratio between the instanton contribution and the corresponding tree-level expression vanishes in this limit. We also speculate on the extension to the non-perturbative level of the proposed duality with polygonal Wilson loops and briefly discuss its potential implications for scattering amplitudes.

Systematics of the cusp anomalous dimension

We study the velocity-dependent cusp anomalous dimension in supersymmetric Yang-Mills theory. In a paper by Correa, Maldacena, Sever, and one of the present authors, a scaling limit was identified in which the ladder diagrams are dominant and are mapped onto a Schrodinger problem. We show how to solve the latter in perturbation theory and provide an algorithm to compute the solution at any loop order. The answer is written in terms of harmonic polylogarithms. Moreover, we give evidence for two curious properties of the result. Firstly, we observe that the result can be written using a subset of harmonic polylogarithms only, at least up to six loops. Secondly, we show that in a light-like limit, only single zeta values appear in the asymptotic expansion, again up to six loops. We then extend the analysis of the scaling limit to systematically include subleading terms. This leads to a Schrodinger-type equation, but with an inhomogeneous term. We show how its solution can be computed in perturbation theory, in a way similar to the leading order case. Finally, we analyze the strong coupling limit of these subleading contributions and compare them to the string theory answer. We find agreement between the two calculations.

Lightlike Wilson loops from AdS/CFT

We investigate the lightlike limit of stationary spacelike string configurations on a large class of five-dimensional asymptotically AdS backgrounds. Specific examples include gravity duals which incorporate finite 't Hooft coupling, curvature-squared corrections, and chemical potentials. A universal feature of these AdS/CFT models is that the string solution with minimum action yields a lightlike Wilson loop whose leading behavior is exponentially linear, rather than quadratic, in the width of the loop. Unless there is a compelling reason for discarding the leading saddlepoint contribution to the Wilson loop, following the proposal of Liu et. al. [2] leads to zero jet-quenching parameter for all of these models.

Spacelike strings and jet quenching from a Wilson loop

We investigate stationary string solutions with spacelike worldsheet in a five-dimensional AdS black hole background, and find that there are many branches of such solutions. Using a non-perturbative definition of the jet quenching parameter proposed by Liu et. al., hep-ph/0605178, we take the lightlike limit of these solutions to evaluate the jet quenching parameter in an N=4 super Yang-Mills thermal bath. We show that this proposed definition gives zero jet quenching parameter, independent of how the lightlike limit is taken. In particular, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear, rather than quadratic, in the quark separation.

Wave and particle scattering properties of high-speed Kerr black holes

The light-like limit of the Kerr gravitational field relative to a distant observer moving rectilinearly in an arbitrary direction is an impulsive plane gravitational wave with a singular point on its wave front. By colliding particles with this wave we show that they have the same focusing properties as high-speed particles scattered by the original black hole. By colliding photons with the gravitational wave we show that there is a circular disc, centred on the singular point on the wave front, having the property that photons colliding with the wave within this disc are reflected back and travel with the wave. This result is approximate in the sense that there are observers who can see a dim (as opposed to opaque) circular disc on their sky. By colliding plane electromagnetic waves with the gravitational wave we show that the reflected electromagnetic waves are the high-frequency waves.

Lightlike limit of the boosted Kerr black holes in higher-dimensional spacetimes

Deriving the gravitational field of the high-energy objects is an important theme because the black holes might be produced in particle collisions in the brane world scenario. In this paper, we develop a method for boosting the metric of the stationary objects and taking the lightlike limit in higher-dimensional spacetimes, using the Kerr black hole as one example. We analytically construct the metric of lightlike Kerr black holes and find that this method contains no ambiguity in contrast to the four-dimensional case. We discuss the possible implications of our results in the brane world context.

Perturbative expansion in DLCQ of field and string theories

We explore the perturbative aspects in discrete light-cone quantization (DLCQ) of field and string theories. In field theory, we develop a systematic DLCQ perturbation and establish the Feynman rules. From the Feynman rules, we show that DLCQ S-matrix does not have a covariant continuum limit for processes with p+ = 0 exchange. Furthermore, we study the relationship between Feynman rules in (conventional) DLCQ and those in DLCQ as a light-like limit (L3) and show that the difference between DLCQ and L3 is only the existence of the divergences by the zero-mode loop diagrams in L3, which are absent in DLCQ. In string theory, we study the scattering amplitudes in L3 and discuss that at any order in perturbation, the zero-mode loop divergences are absent even in L3. This result suggests that the scattering amplitudes in L3 string theory are well-defined and coincides with those in DLCQ string theory.

On the DLCQ as a light-like limit in string theory

We study the issue of defining the discrete light-cone quantization (DLCQ) in perturbative string theory as a light-like limit. While this limit is unproblematic at the classical level, it is non-trivial at the quantum level due to the divergences by the zero-mode loops. We reconsider this problem in bosonic string theory. We construct the multi-loop scattering amplitudes in both open and closed string theories by using the method of Kikkawa, Sakita and Virasoro (KSV), and then we show that these scattering amplitudes are perfectly well-defined in this limit. We also discuss the vacuum amplitudes of the string theory. They are, however, ill-defined in the light-like limit due to the zero-mode loop divergences, and hence we want supersymmetry to cure those pathological divergences even in string theory.

Compactification in the lightlike limit

We study field theories in the limit that a compactified dimension becomes lightlike. In almost all cases the amplitudes at each order of perturbation theory diverge in the limit, due to strong interactions among the longitudinal zero modes. The lightlike limit generally exists nonperturbatively, but is more complicated than might have been assumed. Some implications for the matrix theory conjecture are discussed.

$\mathrm{AdS}_7$/$\mathrm{CFT}_6$ correspondence and matrix models of M5-branes

We study the large N limit of matrix models of M5-branes, or (2,0) sixdimensional superconformal field theories, by making use of the Bulk/Boundary correspondence. Our emphasis is on the relation between the near-horizon limit of branes and the light-like limit of M-theory. In particular we discuss a conformal symmetry in the D0 + D4 system, and interpret it as a conformal symmetry in the discrete light-cone formulation of M5-branes. We also compute two-point functions of scalars by applying the conjecture for the AdS/CFT correspondence to the near-horizon geometry of boosted M5branes. We find an expected result up to a point subtle, but irrelevant to the IR behavior of the theory. Our analysis matches with the Seiberg and Sen’s argument of a justification for the matrix model of M-theory. E-mail address: awata@yukawa.kyoto-u.ac.jp E-mail address: hirano@yukawa.kyoto-u.ac.jp

DLCQ of M-theory as the light-like limit

We investigate whether DLCQ of M-theory can be defined as a limit of M-theory compactified on an almost light-like circle. This is of particular interest since the proofs of the matrix description of M-theory by Seiberg and Sen rely on this assumption. By the standard relation between M-theory on S 1 and IIA string theory, we translate this question into the corresponding one about the existence of the light-like limit of IIA superstring theory for any string coupling g s . We argue that perturbative string loop amplitudes should have a finite and well-defined light-like limit provided the external momenta are chosen to correspond to a well-defined DLCQ set-up. On the non-perturbative side we consider states and amplitudes. We show that an appropriate class of non-perturbative states (wrapped D-branes) precisely have the right light-like limit. We give some indications that non-perturbative corrections to string amplitudes, too, may behave as required in the light-like limit. Having perturbative and non-perturbative evidence, this suggests that type IIA superstring theory as a whole has a well-defined light-like limit (for any string coupling g s ) and hence that the same is true for M-theory.

A comment on compactification of M-theory on an (almost) light-like circle

In perturbative quantum field theory the limit of compactification on an almost light-like circle has recently been shown to be plagued by divergences. We argue that the light-like limit for M-theory probably is free of such divergences due to, among others, the existence of the wrapping modes of the membranes. To illustrate this, we consider superstring theory compactified on an almost light-like circle. Specifically, we compute a one-loop four-point amplitude in type II theory. As is well known, if the external states have vanishing momenta in the compact dimension, the divergence in the light-like limit is even stronger than in field theory. However, in the case of present interest, where these external momenta are non-vanishing, there is a subtle compensation and the resulting amplitude has a well defined and finite light-like limit. The net effect of taking the light-like limit is to replace the integration over one of the moduli of the four-punctured torus by a sum over a discrete modulus taking values in a finite lattice on the torus. The same result can also be obtained from a suitably “Wick rotated” amplitude computed directly with a compact light-like circle.