Mellin Representation Research Articles

Secondary massive quarks with the Mellin-Barnes expansion

Processes involving only massless or massive quarks at tree-level get corrections from massive (lighter, heavier, or equal-mass) secondary quarks starting at two-loop order, generated by a virtual gluon splitting into a massive quark anti-quark pair. One convenient approach to compute such two-loop corrections is starting with the one-loop diagram considering the virtual gluon massive. Carrying out a dispersive integral with a suitable kernel over the gluon mass yields the desired two-loop result. On the other hand, the Mellin-Barnes representation can be used to compute the expansion of Feynman integrals in powers of a small parameter. In this article we show how to combine these two ideas to obtain the corresponding expansions for large and small secondary quark masses to arbitrarily high orders in a straightforward manner. Furthermore, the convergence radius of both expansions can be shown to overlap, being each series rapidly convergent. The advantage of our method is that the Mellin representation is obtained directly for the full matrix element from the same one-loop computation one needs in large-β0 computations, therefore many existing results can be recycled. With minimal modifications, the strategy can be applied to compute the expansion of the one-loop correction coming from a massive gauge boson. We apply this method to a plethora of examples, in particular those relevant for factorized cross sections involving massless and massive jets, recovering known results and obtaining new ones. Another bonus of our approach is that, postponing the Mellin inversion, one can obtain the small- and large-mas expansions for the RG-evolved jet functions. In many cases, the series can be summed up yielding closed expressions.

Twist decomposition of non-linear effects in Balitsky–Kovchegov evolution of proton structure functions

Effects of non-linear small-x evolution of the gluon distribution given by the Balitsky–Kovchegov equation are analyzed within the collinear approximation framework. We perform a twist decomposition of the proton structure functions F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_2$$\\end{document} and FL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\ extrm{L}}$$\\end{document} obtained from the Balitsky–Kovchegov equation using the Mellin representation of the scattering cross-sections at high energies. In both the structure functions we find strong corrections coming from the non-linear effects in the gluon evolution at twist 2, and strongly suppressed higher twist effects. This implies that unitarization effects of high energy scattering amplitudes are mostly the leading twist effect. Furthermore we consider the double logarithmic limit of the Balitsky–Kovchegov equation for the collinear gluon distribution, and compare the result to the Gribov–Levin–Ryskin equation. We find that these two equations differ by two powers of the hard scale logarithm for the large scales.

A note on defect Mellin amplitudes

We generalize the Mellin representation for a generic co-dimension flat defect CFT. We study the analytic structure of the Mellin amplitudes. We also compute Witten diagrams for a generic co-dimension flat defect CFT.

Computing Mellin Representations and Asymptotics of Nested Binomial Sums in a Symbolic Way: The RICA Package

Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums.

Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time.

Maximal U(1)Y-violating n-point correlators in mathcal{N} = 4 super-Yang-Mills theory

This paper concerns a special class of n-point correlation functions of operators in the stress tensor supermultiplet of mathcal{N} = 4 supersymmetric SU(N) Yang-Mills theory. These are “maximal U(1)Y-violating” correlators that violate the bonus U(1)Y charge by a maximum of 2(n − 4) units. We will demonstrate that such correlators satisfy SL(2, ℤ)-covariant recursion relations that relate n-point correlators to (n − 1)-point correlators in a manner analogous to the soft dilaton relations that relate the corresponding amplitudes in flat-space type IIB superstring theory. These recursion relations are used to determine terms in the large-N expansion of n-point maximal U(1)Y-violating correlators in the chiral sector, including correlators with four superconformal stress tensor primaries and (n − 4) chiral Lagrangian operators, starting from known properties of the n = 4 case. We concentrate on the first three orders in 1/N beyond the supergravity limit. The Mellin representations of the correlators are polynomials in Mellin variables, which correspond to higher derivative contact terms in the low-energy expansion of type IIB superstring theory in AdS5× S5 at the same orders as R4, d4R4 and d6R4. The coupling constant dependence of these terms is found to be described by non-holomorphic modular forms with holomorphic and anti-holomorphic weights (n − 4, 4 − n) that are SL(2, ℤ)-covariant derivatives of Eisenstein series and certain generalisations. This determines a number of non-leading contributions to U(1)Y-violating n-particle interactions (n > 4) in the low-energy expansion of type IIB superstring amplitudes in AdS5× S5.

Conformal correlators as simplex integrals in momentum space

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Lorentzian inversion and anomalous dimensions in Mellin space

In this note, we derive a Mellin space version of the Lorentzian inversion formula for CFTs by explicitly integrating over the cross-ratios in d = 2 and d = 4 spacetime dimensions. We use the simplicity of the Mellin representation of Witten diagrams and the double discontinuity to find the OPE coefficients and anomalous dimensions of double- trace primaries in large N CFTs to order frac{1}{N^4} . We find that our results match analytically at order frac{1}{N^2} , and numerically at order frac{1}{N^4} with existing literature.

Nonperturbative Mellin amplitudes: existence, properties, applications

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.

On conformal blocks, crossing kernels and multi-variable hypergeometric functions

In this note, we present an alternative representation of the conformal block with external scalars in general spacetime dimensions in terms of a finite summation over Appell fourth hypergeometric function F4. We also construct its generalization to the non-local primary exchange operator with continuous spin and its corresponding Mellin representation which are relevant for Lorentzian spacetime. Using these results we apply the Lorentzian inversion formula to compute the so-called crossing kernel in general spacetime dimensions, the resultant expression can be written as a double infinite summation over certain Kampé de Fériet hypergeometric functions with the correct double trace operator singularity structures. We also include some complementary computations in AdS space, demonstrating the orthogonality of conformal blocks and performing the decompositions.

Resummation at finite conformal spin

We generalize the computation of anomalous dimension and correction to OPE coefficients at finite conformal spin considered recently in [1, 2] to arbitrary space-time dimensions. By using the inversion formula of Caron-Huot and the integral (Mellin) representation of conformal blocks, we show that the contribution from individual exchanges to anomalous dimensions and corrections to the OPE coefficients for “double-twist” operators {left[{mathcal{O}}_1{mathcal{O}}_2right]}_{Delta, J} in s-channel can be written at finite conformal spin in terms of generalized Wilson polynomials. This approach is democratic with respect to space-time dimensions, thus generalizing the earlier findings to cases where closed form expressions of the conformal blocks are not available.

Anomalous dimensions at finite conformal spin from OPE inversion

We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of [1], both from the position space representation as well as from the integral viz. Mellin representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the t-channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin β = Δ + J in the s-channel through the inversion formula. Our results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.

Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

AbstractWe study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)‐conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed‐form expressions for crossing kernels of spinning CPWs in terms of the hypergeometric function 4F3. We consider numerous examples involving both arbitrary internal and external spins, and for both leading and sub‐leading twist operators. As an application, working in general d we extract new results for anomalous dimensions of double‐trace operators induced by double‐trace deformations constructed from single‐trace operators of generic twist and integer spin. In particular, we extract the anomalous dimensions of double‐trace operators with a single‐trace operator of integer spin J.

Towards spinning Mellin amplitudes

We construct the Mellin representation of four point conformal correlation function with external primary operators with arbitrary integer spacetime spins, and obtain a natural proposal for spinning Mellin amplitudes. By restricting to the exchange of symmetric traceless primaries, we generalize the Mellin transform for scalar case to introduce discrete Mellin variables for incorporating spin degrees of freedom. Based on the structures about spinning three and four point Witten diagrams, we also obtain a generalization of the Mack polynomial which can be regarded as a natural kinematical polynomial basis for computing spinning Mellin amplitudes using different choices of interaction vertices.

A Note on (Non)-Locality in Holographic Higher Spin Theories

It was argued recently that the holographic higher spin theory features non-local interactions. We further elaborate on these results using the Mellin representation. The main difficulty previously encountered in this method is that the Mellin amplitude for the free theory correlator is ill-defined. To resolve this problem, instead of literally applying the standard definition, we propose to define this amplitude by linearity using decompositions, where each term has the associated Mellin amplitude well-defined. Up to a sign, the resulting amplitude is equal to the Mellin amplitude for the singular part of the quartic vertex in the bulk theory and, hence, can be used to analyze bulk locality. From this analysis we find that the scalar quartic self-interaction vertex in the holographic higher spin theory has a singularity of a special form, which can be distinguished from generic bulk exchanges. We briefly discuss the physical interpretation of such singularities and their relation to the Noether procedure.

The Mellin formalism for boundary CFTd

We extend the Mellin representation of conformal field theory (CFT) to allow for conformal boundaries and interfaces. We consider the simplest holographic setup dual to an interface CFT — a brane filling an AdSd subspace of AdSd+1 — and perform a systematic study of Witten diagrams in this setup. As a byproduct of our analysis, we show that geodesic Witten diagrams in this geometry reproduce interface CFTd conformal blocks, generalizing the analogous statement for CFTs with no defects.

Unitarity and positivity constraints for CFT at large central charge

We consider the four-point correlator of the stress tensor multiplet in mathcal{N}=4 SYM in the limit of large central charge c ∼ N2. For finite values of g2N single-trace intermediate operators arise at order 1/c and this leads to specific poles in the Mellin representation of the correlator. The sign of the residue at these poles is fixed by unitarity. We consider solutions consistent with crossing symmetry and this pole structure. We show that in a certain regime all solutions result in a negative contribution to the anomalous dimension of twist four operators. The reason behind this is a positivity property of Mack polynomials that leads to a positivity condition for the Mellin amplitude. This positivity condition can also be proven by assuming the correct Regge behaviour for the Mellin amplitude. For large g2N we recover a tower of solutions in one to one correspondence with local interactions in a effective field theory in the AdS bulk, with the appropriate suppression factors, and with definite overall signs. These signs agree with the signs that would follow from causality constraints on the effective field theory. The positivity constraints arising from CFT for the Mellin amplitude take a very similar form to the causality constraint for the forward limit of the S-matrix.

A Mellin space approach to the conformal bootstrap

We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFTd four point functions and expands them in terms of crossing symmetric combinations of AdSd+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O(ϵ3) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at ϵ = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

Conformal Bootstrap in Mellin Space.

We propose a new approach towards analytically solving for the dynamical content of conformal field theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built-in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the ε expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in ε than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement between certain observables in the 3D Ising model and the precise numerical values that have been recently obtained.

Exploring perturbative conformal field theory in Mellin space

We explore the Mellin representation of correlation functions in conformal field theories in the weak coupling regime. We provide a complete proof for a set of Feynman rules to write the Mellin amplitude for a general tree level Feynman diagram involving only scalar operators. We find a factorised form involving beta functions associated to the propagators, similar to tree level Feynman rules in momentum space for ordinary QFTs. We also briefly consider the case where a generic scalar perturbation of the free CFT breaks conformal invariance. Mellin space still has some utility and one can consider non-conformal Mellin representations. In this context, we find that the beta function corresponding to conformal propagator uplifts to a hypergeometric function.