Prashanth Raman

A bstract: 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 Padé 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.

A bstract: We present a systematic method for computing cosmological amplitudes, including in-in correlators and wavefunction coefficients. Specialising 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.

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.

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.

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 in particular that many basic building blocks appearing in 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.

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.

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.

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.

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.

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.