We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in AdS/CFT and S-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in N=4 Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the external scalars and the spin J leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators of spin J.
The Operator Product Expansion is a useful tool to represent correlation functions. In this note we extend Conformal Regge theory to provide an exact OPE representation of Lorenzian four-point correlators in conformal field theory, valid even away from Regge limit. The representation extends convergence of the OPE by rewriting it as a double integral over continuous spins and dimensions, and features a novel "Regge block". We test the formula in the conformal fishnet theory, where exact results involving nontrivial Regge trajectories are available.
Continuing our investigations of the analytic structure of the 2->n scattering amplitudes in the planar limit of N=4 SYM in multi-Regge kinematics we compute, in all kinematic regions, the Regge cut contributions of the 2->6 process in leading order. Compared to previous studies of the 2->4 and the 2->5 processes we encounter two new features: the 3-reggon cut and the product of two 2-reggeon cut contributions.
We investigate the analytic structure of the $2\to5$ scattering amplitude in the planar limit of $\mathcal{N}=4$ SYM in multi-Regge kinematics in all physical regions. We demonstrate the close connection between Regge pole and Regge cut contributions: in a selected class of kinematic regions (Mandelstam regions) the usual factorizing Regge pole formula develops unphysical singularities which have to be absorbed and compensated by Regge cut contributions. This leads, in the corrections to the BDS formula, to conformal invariant 'renormalized' Regge pole expressions in the remainder function. We…
We study two kinematical limits, the Regge limit and the Lorentzian OPE limit, of the four-point function of the stress-tensor multiplet in Super Yang-Mills at weak coupling. We explain how both kinematical limits are controlled by the leading twist operators. We use the known expression of the four-point function up to three loops, to extract the pomeron residue at next-to-leading order. Using this data and the known form of pomeron spin up to next-to-leading order, we predict the behaviour of the four-point function in the Regge limit at higher loops. Specifically, we determine the leading l…
We define a Regge limit for off-shell Green functions in quantum field theory, and study it in the particular case of conformal field theories (CFT). Our limit differs from that defined in arXiv:0801.3002, the latter being only a particular corner of the Regge regime. By studying the limit for free CFTs, we are able to reproduce the Low-Nussinov, BFKL approach to the pomeron at weak coupling. The dominance of Feynman graphs where only two high momentum lines are exchanged in the t-channel, follows simply from the free field analysis. We can then define the BFKL kernel in terms of the two point…
In this second part of our investigation of the analytic structure of the $2\to5$ scattering amplitude in the planar limit of $\mathcal{N}=4$ SYM in multi-Regge kinematics we compute, in all kinematic regions, the Regge cut contributions in leading order. The results are infrared finite and conformally invariant.
The structure constants of twist-two operators with spin $j$ in the BFKL limit $g^2\rightarrow 0, j\rightarrow 1$ but ${g^2\over j-1}\sim 1$ are determined 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.
A dual conformal symmetry, analogous to the dual conformal symmetry observed for the scattering amplitudes of N=4 Super Yang-Mills theory, is identified in the Regge limit of QCD. Combined with the original two-dimensional conformal symmetry of the theory, this dual symmetry can potentially explain the integrability of the BFKL Hamiltonian. We also give evidence that the symmetry survives when a subset of unitarity corrections are taken into account by studying briefly the non-planar 2 to m reggeon transition vertices.
Traditional idea of Pomeron/Reggeon description for hadron scattering is now being given theoretical foundation in gravity dual descriptions, where Pomeron corresponds to exchange of spin-j\in 2Z states in the graviton trajectory. Deeply virtual compton scattering (DVCS) is essentially a 2 to 2 scattering process of a hadron and a photon, and hence one should be able to study non-perturbative aspects (GPD) of this process by the "Pomeron" process in gravity dual. We find, however, that even the most developed formulation of gravity dual Pomeron (Brower--Polchinski--Strassler--Tan (BPST) 2006) …
We study a holographic construction of conformal blocks in the Regge limit of four-point scalar correlation functions by using coordinates of the two-sided Rindler-AdS black hole. As a generalization of geodesic Witten diagrams, we construct diagrams with four external scalar fields in the Rindler-AdS black hole by integrating over two half-geodesics between the centers of Penrose diagrams and points at the AdS boundary. We demonstrate that late-time behaviors of the diagrams coincide with the Regge behaviors of conformal blocks. We also point out their relevance with the pole-skipping phenome…
The scattering amplitude of hadrons in high energy Regge limit can be rewritten in terms of reggeized gluons, i.e. Reggeons. We consider three-Reggeon states that possess either C=+1 or C=-1 parity. In this work using Janik-Wosiek method the spectrum of conformal charges is calculated for states with conformal Lorentz spin n_h=0,1,2,3,... . Moreover corrections to WKB approximation are computed.
Highly energetic particles traveling in the background of an asymptotically AdS black hole experience a Shapiro time delay and an angle deflection. These quantities are related to the Regge limit of a heavy-heavy-light-light four-point function of scalar operators in the dual CFT. The Schwarzschild radius of the black hole in AdS units is proportional to the ratio of the conformal dimension of the heavy operator and the central charge. This ratio serves as a useful expansion parameter; its power counts the number of stress tensors in the multi-stress tensor operators which contribute to the fo…
We compute the masses of even spin glueball states $ J^{PC} $, with $ P=C=+1 $, using a twist two operator from an AdS/QCD model known as the hardwall model, using Dirichlet and Neumann boundary conditions. Within this approach, we found that the glueball masses are comparable with those in literature. From these masses, we obtained the Pomeron Regge trajectories for both boundary conditions in agreement with experimental data available and other holographic models.
We investigate the structure of conformal Regge trajectories for the maximally supersymmetric (2,0) theories in six dimensions. The different conformal multiplets in a single superconformal multiplet must all have similarly-shaped Regge trajectories. We show that these super-descendant trajectories interact in interesting ways, leading to new constraints on their shape. For the four-point function of the stress tensor multiplet supersymmetry also softens the Regge behavior in some channels, and consequently we observe that 'analyticity in spin' holds for all spins greater than -3. All the phys…
By restricting the functional integration to the Regge geometries, we give the discretized version of the well known path integral formulation of 2--dimensional quantum gravity in the conformal gauge. We analyze the role played by diffeomorphisms in the Regge framework and we give an exact expression for the Faddeev--Popov determinant related to a Regge surface; such an expression in the smooth limit goes over to the correct continuum result.
In this work, we consider a massive gauge boson field in AdS$_5$ dual to odd glueball states with twist-5 operator in 4D Minkowski spacetime. Introducing an IR cutoff we break the conformal symmetry of the boundary theory allowing us to calculate the glueball masses with odd spins using Dirichlet and Neumann boundary conditions. Then, from these masses we construct the corresponding Regge trajectories associated with the odderon. Our results are compatible with the ones in the literature.
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 $F^{(M)}(u,v)\sim {u}^{-δ}\,$, $δ>0$, in the li…
We show that a scale invariant approximation to the next-to-leading order BFKL kernel, constructed via transverse momentum diagrams, has a simple conformally invariant representation in impact parameter space i.e. K(r1,r2,r1',r2') = g^4 N^2 ln^4[(|r1-r1'||r2-r2'|)/(|r1-r2'||r2-r1'|)] That a conformally invariant representation exists is shown first by relating the kernel directly to Feynman diagrams contributing to two photon diffractive dissociation.
We study the pairwise interaction of reggeized gluons and quarks in the Regge limit of perturbative QCD. The interactions are represented as integral kernels in the transverse momentum space and as operators in the impact parameter space. We observe conformal symmetry and holomorphic factorization in all cases.
We review some of our recent work on the conformal geometry corresponding to the triangulated surfaces used in 2-dimensional simplicial quantum gravity. In particular, we discuss the regularized Liouville action associated with random Regge triangulations and its connection with Hodge-Deligne theory.
Simons Center for Geometry and Physics, Stony Brook University（石溪大学西蒙斯几何与物理中心）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学陈省身理论物理研究所）
We consider the structure of the operator product expansion (OPE) in conformal field theory by employing the OPE block formalism. The OPE block acted on the vacuum is promoted to an operator and its implications are examined on a non-vacuum state. We demonstrate that the OPE block is dominated by a light-ray operator in the Regge limit, which reproduces precisely the Regge behavior of conformal blocks when used inside scalar four-point functions. Motivated by this observation, we propose a new form of the OPE block, called the light-ray channel OPE block that has a well-behaved expansion domin…
We show how Boundary Conformal Field Theory deformation techniques allow for a complete characterisation of the coupling between the discrete geometry inherited uniformizing a random Regge triangulations and open string theory.
We consider unitary CFTs with continuous global symmetries in $d>2$. We consider a state created by the lightest operator of large charge $Q \gg 1$ and analyze the correlator of two light charged operators in this state. We assume that the correlator admits a well-defined large $Q$ expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists. We find that the crossing equations admit a consistent truncation, where only a finite number $N$ of Regge trajectories contribute to the correlator at leading nontrivial order. We classify all such truncated solutions to …
We analytically solve the conformal bootstrap equations in the Regge limit for large N conformal field theories. For theories with a parametrically large gap, the amplitude is dominated by spin-2 exchanges and we show how the crossing equations naturally lead to the construction of AdS exchange Witten diagrams. We also show how this is encoded in the anomalous dimensions of double-trace operators of large spin and large twist. We use the chaos bound to prove that the anomalous dimensions are negative. Extending these results to correlators containing two scalars and two conserved currents, we …
In this paper we continue our study of the dual SL(2,C) symmetry of the BFKL equation, analogous to the dual conformal symmetry of N=4 Super Yang Mills. We find that the ordinary and dual SL(2,C) symmetries do not generate a Yangian, in contrast to the ordinary and dual conformal symmetries in the four-dimensional gauge theory. The algebraic structure is still reminiscent of that of N=4 SYM, however, and one can extract a generator from the dual SL(2,C) close to the bi-local form associated with Yangian algebras. We also discuss the issue of whether the dual SL(2,C) symmetry, which in its orig…
Every conformal field theory (CFT) above two dimensions contains an infinite set of Regge trajectories of local operators which, at large spin, asymptote to "double-twist" composites with vanishing anomalous dimension. In two dimensions, due to the existence of local conformal symmetry, this and other central results of the conformal bootstrap do not apply. We incorporate exact stress tensor dynamics into the CFT$_2$ analytic bootstrap, and extract several implications for AdS$_3$ quantum gravity. Our main tool is the Virasoro fusion kernel, which we newly analyze and interpret in the bootstra…
We study the Regge trajectories of the Mellin amplitudes of the $0-,1-$ and $2-$ magnon correlators of the Fishnet theory. Since fishnet theory is both integrable and conformal, the correlation functions are known exactly. We find that while for $0$ and $1$ magnon correlators, the Regge poles can be exactly determined as a function of coupling, $2$-magnon correlators can only be dealt with perturbatively. We evaluate the resulting Mellin amplitudes at weak coupling, while for strong coupling we do an order of magnitude calculation.
We study the Regge limit of 4-point AdS$_3 \times S^3$ correlators in the tree-level supergravity approximation and provide various explicit checks of the relation between the eikonal phase derived in the bulk picture and the anomalous dimensions of certain double-trace operators. We consider both correlators involving all light operators and HHLL correlators with two light and two heavy multi-particle states. These heavy operators have a conformal dimension proportional to the central charge and are pure states of the theory, dual to asymptotically AdS$_3 \times S^3$ regular geometries. Devia…
We perform the OPE analysis of the contribution of colour-singlet compound states of reggeized gluons to a generic hard process in QCD and calculate the spectrum of the corresponding higher twist anomalous dimensions in multi-colour limit. These states govern high energy asymptotics of the structure functions and their energies define the intercept of the Regge singularities both in the Pomeron and the Odderon sectors. We argue that due to nontrivial analytical properties of the energy spectrum, the twist expansion does not hold for the gluonic states with the minimal energy generating the lea…
By regularizing the conical singularities by means of a segment of a sphere or pseudosphere and then taking the regulator to zero, we compute exactly the Faddeev--Popov determinant related to the conformal gauge fixing for a piece-wise flat surface with the topology of the sphere. The result is analytic in the opening angles of the conical singularities in the interval ($π$, $4π$) and in the smooth limit goes over to the continuum expression. The Riemann-Roch relation on the dimensions of ker$(L^†L)$ and ker$(LL^†)$ is satisfied.
We discuss conformal symmetry on the two dimensional noncommutative plane equipped with Moyal product in the twist deformed context. We show that the consistent use of the twist procedure leads to results which are free from ambiguities. This lends support to the importance of the use of twist symmetries in noncommutative geometry.
A Lagrangian of the topological field theory is found in the twisted osp(2|2)\oplus osp(2|2)conformal super algebra. The reduction on a moduli space is then elaborated through the vanishing Noether current.
From the perspective of AdS/CFT the Pomeron is identified with a Reggeized Graviton, while the Odderons correspond to Reggeized anti-symmetric $AdS_5$ Kalb-Ramond tensor-fields. In this paper, we consider the strong coupling expansion of the dimension of the leading twist operators dual to these Regge trajectories, $Δ(j)$, to determine its analytic continuation in $j$ beyond the diffusion limit. In particular, we compute the strong coupling expansion of the intercept to order $λ^{-3}$, where $λ$ is the t'Hooft coupling, for both the Pomeron, which is $C=+1$ crossing-even, and the "Odderons", w…
Operator nucleon vertices are constructed in composite superconformal string model. Splitting of baryon Regge trajectories with the same quantum numbers but with opposite parity is provided by inclusion of simple additional components to superconformal generators.
We study an inequality between a scaling exponent $A$ in the Regge limit of tree-level flat space S-matrices with external massless scalars and another scaling exponent $A'$ in the Regge limit of the corresponding four-point scalar conformal correlators by using scaling exponents of Mellin amplitudes. We derive $A'\ge A$, which leads to the Regge growth bound of tree-level flat space S-matrices from the chaos bound in the flat space limit of the AdS/CFT correspondence, from polynomial boundedness of the Mellin amplitudes for local bulk descriptions. We also show $A'=A$ from the conformal block…
We analyze conformal field theory 4-point functions of the form A ~ O_1(x_1) O_2(x_2) O_1(x_3) O_2(x_4), where the operators O_i are scalar primaries. We show that, in the Lorentzian regime, the limit x_1 -> x_3 is dominated by the exchange of conformal partial waves of highest spin. When partial waves of arbitrary spin contribute to A, the behavior of the Lorentzian amplitude for x_1 -> x_3 must be analyzed using complex-spin techniques, leading to a generalized Regge theory for CFT's. Whenever the CFT is dual to a string theory, the string tree-level contribution A_tree to the amplitude A pr…
In this paper we use the analyticity properties of the scattering amplitude in the context of the conformal mapping techniques. The Schwarz-Christoffel and Riemann-Schwarz functions are used to map the upper half -plane onto a triangle. We use the known asymptotic and threshold behaviors of the scattering amplitude to establish a connection between the values of the Regge trajectory functions and the angles of the triangle. This geometrical interpretation allows a link between values of the Regge trajectory functions and the generators of the invariance group of Moebius transformations associa…
We consider the Regge limit of a CFT correlation function of two vector and two scalar operators, as appropriate to study small-x deep inelastic scattering in N=4 SYM or in QCD assuming approximate conformal symmetry. After clarifying the nature of the Regge limit for a CFT correlator, we use its conformal partial wave expansion to obtain an impact parameter representation encoding the exchange of a spin j Reggeon for any value of the coupling constant. The CFT impact parameter space is the three-dimensional hyperbolic space H3, which is the impact parameter space for high energy scattering in…
The relation between the spin and the mass of an infinite number of particles in a $q$-deformed dual string theory is studied. For the deformation parameter $q$ a root of unity, in addition to the relation of such values of $q$ with the rational conformal field theory, the Fock space of each oscillator mode in the Fubini-Veneziano operator formulation becomes truncated. Thus, based on general physical grounds, the resulting spin-(mass)$^2$ relation is expected to be below the usual linear trajectory. For such specific values of $q$, we find that the linear Regge trajectory turns into a square-…
The conformal weights of spherical twist operators at non--zero (Euclidean) chemical potential are computed for free Dirac fields in arbitrary dimensions. An image technique, equivalent to replicas, is again used to obtain the $n$--fold cover quantities. The proof of a conjecture made by Bueno, Myers and Witczac--Krempa regarding the relation between the conformal weights and a corner coefficient in the Rényi entropy, given before for scalar fields, is extended to the fermion case. The variation of the weights with chemical potential indicates phase changes.
We develop the technology for Polyakov-Mellin (PM) bootstrap in one-dimensional conformal field theories (CFT$_1$). By adding appropriate contact terms, we bootstrap various effective field theories in AdS$_2$ and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for $O(N)$ theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for $N=1$). We perform cross-checks which include con…
This paper has two parts. We first compute the leading contribution to the strong-coupling mixing between the Konishi operator and a double-trace operator composed of chiral primaries by using flat-space vertex operators for the string-duals of the operators. We then compute the three-point functions for protected or unprotected scalar operators with higher spin operators on the leading Regge trajectory. Here we see that the nontrivial spatial structures required by conformal invariance arise naturally from the form of the polarization tensors in the vertex operators. We find agreement with re…
We reinterpret the Scherk-Schwarz (SS) boundary condition for SU(2)_R in a compactified five-dimensional (5D) Poincare supergravity in terms of the twisted SU(2)_U gauge fixing in 5D conformal supergravity. In such translation, only the compensator hypermultiplet is relevant to the SS twist, and various properties of the SS mechanism can be easily understood. Especially we show the equivalence between the SS twist and boundary constant superpotentials at the full supergravity level.
We investigate the relationship between conformal and spin structure on lorentzian manifolds and see how their compatibility influences the formulation of rigid supersymmetric field theories. In dimensions three, four, six and ten, we show that if the Dirac current associated with a generic spinor defines a null conformal Killing vector then the spinor must obey a twistor equation with respect to a certain connection with torsion. Of the theories we consider, those with classical superconformal symmetry in Minkowski space can be reformulated as rigid supersymmetric theories on any lorentzian m…
We apply the framework of Rychkov-Tan arXiv:1505.00963 to the codimension two twist defect at the Wilson-Fisher fixed point in $4-ε$ dimensions. We obtain the scaling dimensions of the operators on the defect up to the lowest nontrivial order in the $ε$-expansion without using Feynman diagram computation. Our results agree with the known results.
We explore the structures of light cone and Regge limit singularities of $n$-point Virasoro conformal blocks in $c>1$ two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full $1/c$ corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the $n$-th ($n>2$) Renyi entropy afte…
Four-dimensional conformal fishnet theory is an integrable scalar theory which arises as a double scaling limit of $γ$-deformed maximally supersymmetric Yang-Mills. We give a perturbative reformulation of $γ$-deformed super-Yang-Mills theory in twistor space, and implement the double scaling limit to obtain a twistor description of conformal fishnet theory. The conformal fishnet theory retains an abelian gauge symmetry on twistor space which is absent in space-time, allowing us to obtain cohomological formulae for scattering amplitudes that manifest conformal invariance. We study various class…
An approach to special relativistic dynamics using the language of spinors and twistors is presented. Exploiting the natural conformally invariant symplectic structure of the twistor space, a model is constructed which describes a relativistic massive, spinning and charged particle, minimally coupled to an external electro-magnetic field. On the two-twistor phase space the relativistic Hamiltonian dynamics is generated by a Poincare scalar function obtained from the classical limit (appropriately defined by us) of the second order, to an external electro-magnetic field minimally coupled, Dirac…
C. N. Yang Institute for Theoretical Physics, Stony Brook University（杨振宁理论物理研究所，石溪大学）；Simons Center for Geometry and Physics, Stony Brook University（西蒙斯几何与物理中心，石溪大学）
本文在 pp 波背景上研究共形场论中大自旋算符的渐近行为，通过热单点函数预测高温与低温区域的渐近行为，并在自由标量理论及大 N O(N) 模型中验证。研究揭示了 Regge 极限与热场论及解析自举的联系。
We apply Kadanoff's theory of marginal deformations of conformal field theories to twistfield deformations of Z_2 orbifold models in K3 moduli space. These deformations lead away from the Z_2 orbifold sub-moduli-space and hence help to explore conformal field theories which have not yet been understood. In particular, we calculate the deformation of the conformal dimensions of vertex operators for p^2<1 in second order perturbation theory.
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twis…
