arXiv subjects

Prashanth Raman

Publications and source records attributed to Prashanth Raman.

15 recordsLinked to original sources

arXiv2026

Analytic Continuation of Conformal Integrals in Momentum Space

Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide class of higher-point conformal correlators. These integrals belong to the class of generalised hypergeometric functions, and while their series representations are known in principle, none converges throughout the entire physical region of kinematic space selected by momentum conservation. In this work, we construct series representations adapted to exactly this physical domain. Using the method of brackets, which turns the evaluation of definite integrals into solving a linear system of algebraic equations, we derive various Lauricella-type series representations for multiple-$K$ integrals. For triple-$K$ integrals, conventionally expressed in terms of the Appell $F_4$ function whose series converges only outside the triangle-inequality region, we find instead a compact, two-branch series that converges throughout the entire physical region and for arbitrary scaling dimensions. We then extend this construction to general multiple-$K$ integrals: by introducing a new set of kinematic variables, we build an iterative series representation that converges for all physical kinematic configurations.

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arXiv2026

Approximating Feynman integrals using complete monotonicity and Stieltjes properties

We present two novel approaches for the numerical evaluation of Feynman integrals based on their universal analytic properties related to positivity, namely complete monotonicity (CM) and Stieltjes properties. Building on recent results, we exploit the fact that scalar Feynman integrals in the Euclidean region are completely monotonic functions, meaning that all their derivatives have a fixed sign. Building on this observation, the CM bootstrap allows one to reconstruct integrals from differential equations without explicit boundary data, yielding rigorous bounds. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, are not only CM but in fact Stieltjes functions. This enables the use of Pad\'e approximants with provable convergence properties in the cut complex plane, providing an efficient method for analytic continuation and fast numerical evaluation. We illustrate the method with simple examples such as the massive bubble integral and discuss applications to multi-loop integrals, including the 20-loop banana integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

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arXiv2026

Lecture Notes on Positivity Properties of Scattering Amplitudes

We review completely monotone (CM) and Stieltjes functions, which are classes of functions obeying an infinite hierarchy of positivity constraints. While these are classical concepts in analysis, such properties have recently been shown to arise in many fundamental building blocks and observables of quantum field theory (QFT), including scalar Feynman integrals in the Euclidean region and Coulomb branch amplitudes in $\mathcal{N}=4$ SYM. After reviewing their mathematical structure, we discuss the physical and geometric origins of these properties, ranging from unitarity and analyticity in scattering amplitudes to the structure of parametric representations of Feynman integrals. We then survey a number of applications, including constraints on the analytic S-matrix, implications for numerical bootstrap methods, and connections to positive geometry, where we present evidence for a close relation between these functions and geometric volume interpretations. These notes are based on an extended series of lectures delivered at the \emph{Positive Geometry in Scattering Amplitudes and Cosmological Correlators} workshop, held at the International Centre for Theoretical Sciences (ICTS), Bengaluru, in February 2025.

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arXiv2025

Approximating Feynman Integrals Using Complete Monotonicity and Stieltjes Properties

We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives have a fixed sign, in the Euclidean kinematic region. This imposes strong constraints on the function space. Simultaneously, these integrals obey systems of linear differential equations with respect to kinematic parameters. By imposing that the solutions to these differential equations satisfy complete monotonicity across the Euclidean region, we develop an efficient and highly constraining numerical bootstrap method. We provide a proof of principle of the power of our approach by applying it to a class of multi-loop Feynman integrals with internal masses. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, such as dimension and propagator exponents, are not only CM but in fact Stieltjes functions. The latter can be described efficiently by Pad\'e approximants that are known to converge in the cut complex plane. This means that these representations are valid also in analytically continued kinematics, such as physical scattering regions. These insights allow us to obtain rational approximations to Feynman integrals from minimal information, such as a Taylor expansion about a soft limit. We demonstrate the effectiveness of this method by applying it to a 20-loop banana-type Feynman integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

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arXiv2025

Canonical Forms as Dual Volumes

We study dual volume representations of canonical forms for positive geometries in projective spaces, expressing their rational canonical functions as Laplace transforms of measures supported on the convex dual of the semialgebraic set. When the measure is non-negative, we term the geometry completely monotone, reflecting the property of its canonical function. We identify a class of positive geometries whose canonical functions admit such dual volume representations, characterized by the algebraic boundary cut out by a hyperbolic polynomial, for which the geometry is a hyperbolicity region. In particular, simplex-like minimal spectrahedra are completely monotone, with representing measures related to the Wishart distribution, capturing volumes of spectrahedra or their boundaries. We explicitly compute these measures for positive geometries in the projective plane bounded by lines and conics or by a nodal cubic, revealing periods evaluating to transcendental functions. This dual volume perspective reinterprets positive geometries by replacing logarithmic differential forms with probability measures on the dual, forging new connections to partial differential equations, hyperbolicity, convexity, positivity, algebraic statistics, and convex optimization.

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arXiv2025

Mastering Cosmological Amplitudes Using Generalized Ramanujan's Theorem

We present a systematic method for computing cosmological amplitudes, including in-in correlators and wavefunction coefficients, in FRW spacetime. Specializing to cases with conformally-coupled external scalars and massive scalar exchanges, we introduce a decomposition into massive family trees, which capture the nested time structure common to these observables. We then evaluate these building blocks using the Method of Brackets (MoB), a multivariate extension of Ramanujan's master theorem that operates directly on the integrand, translating integrals into discrete summations via a compact set of algebraic rules. This yields infinite series representations valid across the full space of external momenta and internal energies. We also develop Feynman-like diagrammatic rules that map interaction graphs to summand structures, enabling efficient and scalable computation. The resulting expressions make time evolution manifest, smoothly interpolate to the conformal limit, and are well suited for both numerical evaluation and analytic analysis of massive field effects in cosmology.

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arXiv2024

Positivity properties of scattering amplitudes

We investigate positivity properties in quantum field theory (QFT). We provide evidence,and in some case proofs, that many building blocks of scattering amplitudes, and in some cases the full amplitudes, satisfy an infinite number of positivity conditions: the functions, as well as all their signed derivatives, are non-negative in a specified kinematic region. Such functions are known as completely monotonic(CM) in the mathematics literature. A powerful way to certify complete monotonicity is via integral representations. We thus show that it applies to planar and non-planar Feynman integrals possessing a Euclidean region,as well as to certain Euler integrals relevant to cosmological correlators and stringy integrals. This implies that in particular that many basic building blocks of perturbation theory, such as master integrals, can be chosen to be completely monotone. We also discuss two pathways for showing complete monotonicity for full amplitudes. One is related to properties of the analytic S-matrix. The other one is a close connection between the CM property and Positive Geometry. Motivated by this, we investigate positivity properties in planar maximally supersymmetric Yang-Mills theory. We present evidence, based on known analytic multi-loop results, that the CM property extends to several physical quantities in this theory. This includes the (suitably normalized) finite remainder function of the six-particle maximally-helicity-violating (MHV) amplitude, four-point scattering amplitudes on the Coulomb branch,four-point correlation functions, as well as the angle-dependent cusp anomalous dimension. Our findings are however not limited to supersymmetric theories. It is shown that the CM property holds for the QCD and QED cusp anomalous dimensions, to three and four loops, respectively. We comment on open questions, and on possible numerical applications of complete monotonicity.

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arXiv2022

QFT, EFT and GFT

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.

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arXiv2022

Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.

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arXiv2022

Celestial insights into the S-matrix bootstrap

We consider 2-2 scattering in four spacetime dimensions in Celestial variables. Using the crossing symmetric dispersion relation (CSDR), we recast the Celestial amplitudes in terms of crossing symmetric partial waves. These partial waves have spurious singularities in the complex Celestial variable, which need to be removed in local theories. The locality constraints (null constraints) admit closed form expressions, which lead to novel bounds on partial wave moments. These bounds allow us to quantify the degree of low spin dominance(LSD) for scalar theories. We study a new kind of positivity that seems to be present in a wide class of theories. We prove that this positivity arises only in theories with a spin-0 dominance. The crossing symmetric partial waves with spurious singularities removed, dubbed as Feynman blocks, have remarkable properties in the Celestial variable, namely typically realness, in the sense of Geometric Function Theory (GFT). Using GFT techniques we derive non-projective bounds on Wilson coefficients in terms of partial wave moments.

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arXiv2020

Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

Stringy canonical forms are a class of integrals that provide $α'$-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebra, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type $A_n$ and $B_n$ generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite $α'$, and the configuration space is binary although the $u$ equations take a more general form than those "perfect" ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type $A_n$ and $B_n$ integrals, which have perfect $u$ equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

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arXiv2019

Stokes Polytopes : The positive geometry for $ϕ^{4}$ interactions

In a remarkable recent work [arXiv : 1711.09102] by Arkani-Hamed et al, the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless $ϕ^{3}$ theory (and its close cousin, bi-adjoint $ϕ^{3}$ theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude is nothing but residue of the canonical form associated to the associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless $ϕ^{4}$ theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that residues of the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude. Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of $ϕ^{3}$ theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.

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arXiv2019

The positive geometry for $ϕ^{p}$ interactions

Starting with the seminal work of Arkani-Hamed et al arXiv:1711.09102, in arXiv:1811.05904, the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless $ϕ^{4}$ theory. In this paper we show that the program can be further extended to include $ϕ^{p}$ ($p>4$) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.

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arXiv2018

Scalar Blocks as Gravitational Wilson Networks

In this paper we continue to develop further our prescription [arXiv:1602.02962] to holographically compute the conformal partial waves of CFT correlation functions using the gravitational open Wilson network operators in the bulk. In particular, we demonstrate how to implement it to compute four-point scalar partial waves in general dimension. In the process we introduce the concept of OPE modules, that helps us simplify the computations. Our result for scalar partial waves is naturally given in terms of the Gegenbauer polynomials. We also provide a simpler proof of a previously known recursion relation for the even dimensional CFT partial waves, which naturally leads us to an odd dimensional counterpart.

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arXiv2016

Holographic Conformal Partial Waves as Gravitational Open Wilson Networks

We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.

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