Mellin-Barnes Representation Research Articles

Expansion by regions meets angular integrals

We study the small-mass asymptotic behavior of so-called angular integrals, appearing in phase-space calculations in perturbative quantum field theory. For this purpose we utilize the strategy of expansion by regions, which is a universal method both for multiloop Feynman integrals and various parametric integrals. To apply the technique to angular integrals, we convert them into suitable parametric integral representations, which are accessible to existing automation tools. We use the code asy.m to reveal regions contributing to the asymptotic expansion of angular integrals. To evaluate the contributions of these regions in an epsilon expansion we apply the method of Mellin-Barnes representation. Our approach is checked against existing results on angular integrals revealing a connection between contributing regions and angular integrals constructed from an algebraic decomposition. We explicitly calculate the previously unknown asymptotics for angular integrals with three and four denominators and formulate a conjecture for the leading asymptotics and the pole part for a general number of denominators and masses.

Multiple Mellin-Barnes integrals and triangulations of point configurations

Mellin-Barnes (MB) integrals are a well-known type of integrals appearing in diverse areas of mathematics and physics, such as in the theory of hypergeometric functions, asymptotics, quantum field theory, solid-state physics, etc. Although MB integrals have been studied for more than a century, it is only recently that, due to a remarkable connection found with conic hulls, N-fold MB integrals can be computed analytically for N>2 in a systematic way. In this article, we present an alternative novel technique by unveiling a new connection between triangulations of point configurations and MB integrals, to compute the latter. To make it ready to use, we have implemented our new method in the package oniculls.wl, an already existing software dedicated to the analytic evaluation of MB integrals using conic hulls. The triangulation method is remarkably faster than the conic hull approach and can thus be used for the calculation of higher-fold MB integrals, as we show here by testing our code on the case of the off-shell massless scalar one-loop N-point Feynman integral up to N=15, for which the MB representation has 104 folds. Among other examples of applications, we present new simpler solutions for the off-shell one-loop massless conformal hexagon and two-loop double-box Feynman integrals, as well as for some complicated 8-fold MB integrals contributing to the hard diagram of the two-loop hexagon Wilson loop in general kinematics. Published by the American Physical Society 2024

Complex and rational hypergeometric functions on root systems

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems An and Cn to the hyperbolic hypergeometric integrals, we apply the limit ω1→−ω2 for their quasiperiods (corresponding to b→i in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the Cn-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ω1→ω2 (or b→1) and obtain new hypergeometric identities for sums of integrals of rational functions.

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.

Analytical calculations of the tenth order QED radiative corrections to lepton anomalies within the Mellin–Barnes representation

We investigate the radiative QED corrections to the lepton (L = e, μ and τ) anomalous magnetic moment due to the contributions of diagrams with insertions of the photon vacuum polarisation operator consisting solely of four closed lepton (l = e, μ and τ) loops. Moreover, we focus on specific operators with two loops formed by leptons L of the same type as the external one, the other two formed by leptons ℓ different from L. The approach is essentially based on the employment of the Mellin–Barnes representation of the x-parametrization of the corresponding Feynman diagrams. This allows one to obtain, for the first time, exact analytical expressions for the radiative corrections of the tenth order w.r.t. the electromagnetic coupling constant e. Analytically, the radiative corrections are expressed in terms of the ratio r = m ℓ /m L of the internal to external lepton masses. The dependence on r is investigated numerically in the whole interval of r, 0 < r < ∞ . To make comparisons with earlier published results possible, our exact analytical expressions are expanded as r → 0 and r → ∞ and compared with the corresponding asymptotic expansions known in the literature.

Lerch $\Phi$ Asymptotics

We use a Mellin-Barnes integral representation for the Lerch transcendent $\Phi(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For non-integer $s$ the asymptotic approximations consists of the sum of two series. The first one is in powers of $(\ln z)^{-1}$ and the second one is in powers of $z^{-1}$. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible.

Inflation correlators with multiple massive exchanges

The most general tree-level boundary correlation functions of quantum fields in inflationary spacetime involve multiple exchanges of massive states in the bulk, which are technically difficult to compute due to the multi-layer nested time integrals in the Schwinger-Keldysh formalism. On the other hand, correlators with multiple massive exchanges are well motivated in cosmological collider physics, with the original quasi-single-field inflation model as a notable example. In this work, with the partial Mellin-Barnes representation, we derive a simple rule, called family-tree decomposition, for directly writing down analytical answers for arbitrary nested time integrals in terms of multi-variable hypergeometric series. We present the derivation of this rule together with many explicit examples. This result allows us to obtain analytical expressions for general tree-level inflation correlators with multiple massive exchanges. As an example, we present the full analytical results for a range of tree correlators with two massive exchanges.

Inflation correlators at the one-loop order: nonanalyticity, factorization, cutting rule, and OPE

Inflation correlators with one-loop massive exchange encode rich information about the dynamics of the massive loop particles. Their nonanalytic behavior in certain soft limits leads to characteristic oscillatory pattern, which is the leading signal of many particle models of cosmological collider physics. In this work, we investigate systematically such nonanalyticity for arbitrary one-particle-irreducible (1PI) one-loop correlators in various soft limits. With the partial Mellin-Barnes representation, we present and prove a factorization theorem and a cutting rule for arbitrary 1PI one-loop inflation correlators, which is reminiscent of the on-shell cutting rule for flat-space scattering amplitudes. We also show how to understand this factorization theorem from the viewpoint of operator product expansion on the future boundary. As an application of the one-loop factorization theorem, we derive new analytic and exact formulae for nonlocal cosmological collider signals for massive one-loop four-point inflation correlators of all possible 1PI topologies, including the bubble, the triangle, and the box graphs. Finally, we show how to push the computation of nonlocal signals to higher orders in the momentum ratio.

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller’s transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

GKZ-system of the 2-loop self energy with 4 propagators

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

Helical inflation correlators: partial Mellin-Barnes and bootstrap equations

Massive spinning particles acquire helicity-dependent chemical potentials during the inflation from axion-type couplings. Such spinning fields can mediate sizable inflaton correlators which we call the helical inflation correlators. Helical inflaton correlators are approximately scale invariant, dS boost breaking, parity violating, and are promising observables of cosmological collider physics. In this work, we present complete and analytical results for 4-point helical inflation correlators with tree-level exchanges of massive spinning particles, including both the smooth background and the oscillatory signals. We compute the bulk Schwinger-Keldysh integrals in two independent ways, including the partial Mellin-Barnes representation and solving bootstrap equations. We also present new closed-form analytical results for 3-point functions with massive scalar or helical spinning exchanges. The analytical results allow us to concretely and efficiently explore the phenomenological consequences of helicity-dependent chemical potentials. In particular, we show that the chemical potential can exponentially enhance oscillatory signals of both local and nonlocal types, but only affects the background in a rather mild way. Our results extend the de Sitter bootstrap program to include nonperturbative breaking of de Sitter boosts. Our results also explicitly verify the recently proposed cutting rule for cosmological collider signals.

Lepton anomaly from QED diagrams with vacuum polarization insertions within the Mellin–Barnes representation

Lepton anomaly from QED diagrams with vacuum polarization insertions within the Mellin–Barnes representation

MB tools reloaded

We address the problem of evaluation of multiloop Feynman integrals by means of their Mellin-Barnes representation. After a brief overview of available capabilities through open source toolkits and their application in various circumstances, we introduce a new code MBcreate which allows one to automatically deduce a concise Mellin-Barnes representation for a given parametric integral. A thorough discussion of its implementation and use is provided.

Bootstrapping Energy-Energy Correlation in Planar \U0001d4dd = 4 Supersymmetric Yang-Mills

The two-point energy flow correlation, alternatively dubbed the energy-energy correlation (EEC), is a class of conformal field theory observable in the maximally supersymmetric Yang-Mills theory (\U0001d4dd = 4) related to the event shapes in scattering experiments. It has been calculated up to the next-to-leading order (NLO) recently, showcasing the simplicity of the correlation function. This paper calculates the EEC using an approach based on its polylogarithmic functions. Using the amplitude bootstrap method, two ansatzes are made for the energy flow operators, namely the polylogarithm ansatz crafted using the Symbols method, and the polynomial ansatz based on the results from the NLO correction. The computation is carried out in the leading order (LO) and NLO order. After the computation is made, physical constraints are discovered and accordingly applied to the ansatz, namely the symmetry and end-point kinematics constraints. The resulting computation retrieved the energy flow correlation calculated previously using the Mellin-Barnes representation. The non-trivial nature of the result implies a simpler way to calculate the energy flow correlation without the conformal field theory-based approach.

Mellin–Barnes Representation for SL(2,ℂ)-Magnet

Mellin–Barnes Representation for SL(2,ℂ)-Magnet

From dS to AdS and back

We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing

We present a simple general relation between tree-level exchanges in AdS and dS. This relation allows to directly import techniques and results for AdS Witten diagrams, both in position and momentum space, to boundary correlation functions in dS. In this work we apply this relation to define Mellin amplitudes and a spectral representation for exchanges in dS. We also derive the conformal block decomposition of a dS exchange, both in the direct and crossed channels, from their AdS counterparts. The relation between AdS and dS exchanges itself is derived using a recently introduced Mellin-Barnes representation for boundary correlators in momentum space, where (A)dS exchanges are straightforwardly fixed by a combination of factorisation, conformal symmetry and boundary conditions.

On the consistency of (partially-)massless matter couplings in de Sitter space

We study the consistency of the cubic couplings of a (partially-)massless spinning field to two scalars in (d + 1)-dimensional de Sitter space. Gauge invariance of observables with external (partially)-massless spinning fields translates into Ward-Takahashi identities on the boundary. Using the Mellin-Barnes representation for boundary correlators in momentum space, we give a systematic study of Ward-Takahashi identities for tree-level 3- and 4-point processes involving a single external (partially-)massless field of arbitrary integer spin-J. 3-point Ward-Takahashi identities constrain the mass of the scalar fields to which a (partially-)massless spin-J field can couple. 4-point Ward-Takahashi identities then constrain the corresponding cubic couplings. For massless spinning fields, we show that Weinberg’s flat space results carry over to (d+1)-dimensional de Sitter space: for spins J = 1, 2 gauge-invariance implies charge-conservation and the equivalence principle while, assuming locality, higher-spins J > 2 cannot couple consistently to scalar matter. This result also applies to anti-de Sitter space. For partially-massless fields, restricting for simplicity to those of depth-2, we show that there is no consistent coupling to scalar matter in local theories. Along the way we also give a detailed account of how contact amplitudes with and without derivatives are represented in the Mellin-Barnes representation. Various new explicit expressions for 3- and 4-point functions involving (partially-)massless fields and conformally coupled scalars in dS4 are given.

On generalized Bessel\u2013Maitland function

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by mathcal{J}_{nu , lambda }^{mu }, where mu >0 and lambda ,nu in mathbb{C } get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of mathcal{J}_{nu ,lambda }^{mu } by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of mathcal{J}_{nu ,lambda }^{mu } is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions mathcal{I}_{nu ,lambda }^{mu } defined by mathcal{I}_{nu ,lambda }^{mu }(z)=i^{-2lambda -nu }mathcal{J}_{ nu ,lambda }^{mu }(iz) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for mathcal{I}_{nu ,lambda }^{mu }(z) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

Double box and hexagon conformal Feynman integrals

The off-shell massless six-point double box and hexagon conformal Feynman integrals with generic propagator powers are expressed in terms of linear combinations of multiple hypergeometric series of the generalized Horn type. These results are derived from 9-fold Mellin-Barnes representations obtained from their dual conformal Feynman parameter representations. The individual terms in the presented expressions satisfy the differential equation that relates the double box in $D$ dimensions to the hexagon in $D+2$ dimensions.