Publications

AbstractMellin-Barnes (MB) integrals appear in various branches of physics and mathematics and are, in particular, used as a standard tool for evaluating multi-loop, multi-scale Feynman integrals both analytically and numerically. Recent geometric approaches based on conic hulls and triangulations provide a systematic framework for computing multiple MB integrals in terms of multivariate series. These approaches have so far been limited to MB integrals whose integrands are ratios of products of Euler's gamma functions only. However, in Feynman integral calculus, MB integrals with polygamma functions naturally arise, for instance, after resolving singularities in the dimensional-regularisation parameter ε and expanding the MB integrand in powers of ε, as done by the public codes MB.m and MBresolve.m. In this paper, we extend the conic hull and triangulation methods to the computation of MB integrals having polygamma functions in their integrand. We show that the arguments of polygamma functions can be treated in a similar way to the arguments of gamma functions when applying the conic hull and triangulation techniques to identify poles that would contribute to different series solutions. However, since the singularity structure of the polygamma function is different from that of the gamma function, we propose two different ways to compute MB integrals involving polygamma functions, depending on whether the MB integral has straight or non-straight contours. We have implemented these algorithms in an updated version of the Mathematica package MBConicHulls.wl, which can be found at https://github.com/SumitBanikGit/MBConicHulls/, and we illustrate their use with a set of examples from Feynman integral calculus.

AbstractIn this paper, we present HyperPrecision, a Mathematica package for high-precision numerical evaluation of general Horn-type multivariate hypergeometric functions and their Laurent expansions in a small parameter ε. Such functions appear widely in physics and mathematics, with applications ranging from quantum field theory and string theory to number theory and statistics. Their high-precision numerical evaluation, however, remains challenging, since their defining series converge only in restricted domains and analytic continuation beyond these domains is, in general, non-trivial. HyperPrecision addresses this problem by automatically constructing the Pfaffian system of partial differential equations for a given hypergeometric function and restricting it to a one-dimensional contour in the space of variables connecting the starting to the target point. The resulting ordinary differential equation is then solved by the Frobenius method, with boundary conditions fixed analytically by the defining series. We illustrate the use of the package by evaluating commonly occurring multivariate hypergeometric functions, including the Appell F₁, F₂, F₃, and F₄ functions, the Horn G- and H-series, and the Lauricella F_A, F_B, F_C, and F_D functions, as well as by considering applications to angular integrals, Feynman integrals, and cosmological and holographic correlators.

AbstractSunset integrals are among the simplest of two-loop integrals that appear in perturbative quantum field theories and possess up to four distinct mass scales. By means of integration by parts identities, they can be written in terms of four distinct master integrals. In this article, we discuss the independent configurations of on-shell and off-shell sunset master integrals with one, two and three mass scales that arise in chiral perturbation theory. We derive Mellin-Barnes integral representations of these integrals and analytically solve them using various methods to obtain exact results in the form of single and double convergent series of the hypergeometric type, for the values of the mass parameters that allow us to do so. We then discuss how to analytically continue the results to other regions of the parameters and conclude by discussing a few applications in chiral perturbation theory.

AbstractWe compute the two-loop renormalization-group equations for the baryon-number-violating dimension-six operators in the SMEFT. This includes all three gauge interactions, the Yukawa, and Higgs self-interaction contributions. In addition, we present the one-loop matching of the S₁ scalar leptoquark on the SMEFT, which can generate the Wilson coefficients of all four gauge-invariant baryon-number-violating SMEFT operators. Using this example, we demonstrate the cancellation of scheme and matching-scale dependences. Together with the known two-loop renormalization-group evolution below the electroweak scale in the LEFT, as well as the one-loop matching of SMEFT onto LEFT, our results enable consistent next-to-leading-log analyses of nucleon decays, provided that the relevant matrix elements are known at next-to-leading-order accuracy.

AbstractIn this review, we present a comprehensive overview of some of our work carried out in numerous collaborations on important topics in the context of higher-order calculations in perturbative quantum field theories. The approach of this review is one where analytical methods are given prominence. Thus, we primarily concern ourselves with the study of multi-loop scalar Feynman integrals appearing in simplified and idealized models on the one hand, and on the other, to methods for obtaining analytic results for such integrals that are amenable to an implementation on Mathematica as the computer algebra software of choice. After a preliminary discussion of some of the commonly used parametric representations for Feynman integrals, we review (a) the construction of an algorithm and an automated program to find the ‘regions’ of Feynman integrals using Landau equations and power geometry, (b) the analysis of a non-trivial two-loop non-planar Feynman integral using Hopf algebras, (c) some basic aspects of multi-variable hypergeometric functions, namely, their regions of convergence and analytic continuations, (d) the interplay between the theory of multi-variable hypergeometric functions and Feynman integrals, including an algorithmic method for finding series representations for multi-fold Mellin-Barnes representations of Feynman integrals, the interpretation of Feynman integrals as GKZ hypergeometric functions and an automated program that uses this idea for obtaining series solutions, the ε-expansion for multi-variable hypergeometric functions arising from dimensionally regularized Feynman integrals, algebraic relations for products of propagators, and (e) the summation of large logarithms for renormalizable as well as non-renormalizable quantum field theories.

AbstractWe examine the correlations between new scalar boson decays to photons and electric dipole moments (EDMs) in the CP-violating flavor-aligned two-Higgs-doublet model (2HDM). It is convenient to work in the Higgs basis {H₁, H₂} where only the first Higgs doublet field H₁ acquires a vacuum expectation value. In light of the LHC Higgs data, which agree well with Standard Model (SM) predictions, it follows that the parameters of the 2HDM are consistent with the Higgs alignment limit. In this parameter regime, the observed SM-like Higgs boson resides almost entirely in H₁, and the other two physical neutral scalars, which reside almost entirely in H₂, are approximate eigenstates of CP (denoted by the CP-even H and the CP-odd A). In the Higgs basis, the scalar potential term Z₇H₁^†H₂H₂^†H₂ + h.c. governs the charged-Higgs loop contributions to the decay of H and A to photons. If Re Z₇ Im Z₇ ≠ 0, then CP-violating effects are present and allow for an H⁺H⁻A coupling, which can yield a sizable branching ratio for A → γγ. These CP-violating effects also generate non-zero EDMs for the electron, the neutron and the proton. We examine these correlations for the cases of m_A = 95 GeV and m_A = 152 GeV where interesting excesses in the diphoton spectrum have been observed at the LHC. These excesses can be explained via the decay of A while being consistent with the experimental bound for the electron EDM in regions of parameter space that can be tested with future neutron and proton EDM measurements. This allows for the interesting possibility where the 95 GeV diphoton excess can be identified with A, while m_H ≃ 98 GeV can account for the best fit to the LEP excess in e⁺e⁻ → ZH with H → bb.

AbstractDespite intensive searches at the LHC, no new fundamental particle has been discovered since the discovery of the 125 GeV Higgs boson. In general, a new physics discovery is challenging without a UV-complete model because different channels and observables cannot be combined directly and unambiguously. Moreover, without indirect hints for new particles, the parameter space to be searched is huge, resulting in diminished significance due to the look-elsewhere effect. Several LHC searches with multiple leptons in the final state point towards the existence of a new Higgs boson with a mass in the 140-160 GeV range, decaying mostly to a pair of W bosons. This dominant decay mode motivates a Higgs triplet with zero hypercharge, which also predicts a heavier-than-expected W-boson as indicated by the CDF-II measurement. Within this simple and predictive model, we simulate and combine channels of associated di-photon production. Considering the run-2 results of ATLAS, including those presented recently at the Moriond conference, a significance of 4.3σ is obtained for a mass of 152 GeV. This is the largest statistical evidence for a new narrow resonance observed at the LHC.

AbstractIn this article, we examine the Standard Model extended by a Y = 0 real Higgs triplet, the ΔSM. It contains a CP-even neutral Higgs (Δ⁰) and two charged Higgs bosons (Δ^±), which are quasi-degenerate in mass. We first study the theoretical constraints from vacuum stability and perturbative unitarity and then calculate the Higgs decays, including the loop-induced modes such as di-photons (γγ) and Zγ. In the limit of a small mixing between the SM Higgs and Δ⁰, the latter decays dominantly to WW and can have a sizable branching ratio to di-photon. The model predicts a positive definite shift in the W mass, which agrees with the current global electroweak fit. At the Large Hadron Collider, it leads to a (i) stau-like signature from pp → Δ⁺Δ⁻ → τ⁺τ⁻νν, (ii) multi-lepton final states from pp → γ^* → Δ⁺Δ⁻ → W⁺W⁻ZZ and pp → W^* → Δ^±Δ⁰ → W^±ZW⁺W⁻ as well as (iii) associated di-photon production from pp → W^* → Δ^±(Δ⁰ → γγ). Concerning (i), the reinterpretation of the recent supersymmetric tau partner search by ATLAS and CMS excludes m_Δ^± < 110 GeV at 95% CL. From (ii), some of the signal regions of multi-lepton searches lead to bounds close to the predicted cross-section, but electroweak scale masses are still allowed. For (iii), the recast of the associated di-photon searches by ATLAS and a combined log-likelihood fit of signal and background to data find that out of the 25 signal regions, 10 provide relevant limits on Br(Δ⁰ → γγ) at the per cent level. Interestingly, 6 signal regions show excesses at around 152 GeV, leading to a preference for a non-zero di-photon branching ratio of about 0.7% with the corresponding significance amounting to about 4σ.

AbstractThe multi-lepton anomalies and searches for the associated production of a narrow resonance indicate the existence of a ≈151 GeV Higgs with a significance of > 5σ and > 3.9σ, respectively. On the one hand, these anomalies require a sizable branching fraction of the new scalar to WW, while on the other hand, no ZZ signal at this mass has been observed. This suggests that the new boson is the neutral component of an SU(2)_L triplet with zero hypercharge. This field leads to a positive definite shift in the W mass, as preferred by the current global fit, and is produced via the Drell-Yan process pp → W^* → Δ⁰Δ^±. We use the side-bands of the ATLAS analysis of the associated production of the Standard Model Higgs in the di-photon channel to search for this production mode of the triplet. Since the dominant decays of Δ^± depend only on its mass, the effect in the 22 signal categories considered by ATLAS is completely correlated. We find that the ones most sensitive to the Drell-Yan production of the triplet Higgs show consistent excesses at a mass of ≈151.5 GeV. Combining these channels in a likelihood ratio test, a non-zero Br[Δ⁰ → γγ] = 0.66% is preferred by ≈3σ, supporting our conjecture.

AbstractStatistically significant tensions between the Standard Model (SM) predictions and the measured lepton distributions in differential top cross-sections emerged in LHC Run 1 data and became even more pronounced in Run 2 analyses. Due to the level of sophistication of the SM predictions and the performance of the ATLAS and CMS detectors, this is very remarkable. Therefore, one should seriously consider the possibility that these measurements are contaminated by beyond-the-SM contributions. In this article, we use differential lepton distributions from the latest ATLAS tt analysis to study a new physics benchmark model motivated by existing indications for new Higgses: a new scalar H is produced via gluon fusion and decays to S^′ (95 GeV) and S (152 GeV), which subsequently decay to bb and WW, respectively. In this setup, the total χ² is reduced, compared to the SM, resulting in Δχ² = 34 to Δχ² = 158, depending on the SM simulation used. Notably, allowing m_S to vary, the combination of the distributions points towards m_S≈150 GeV which is consistent with the existing γγ and WW signals, rendering a mismodelling of the SM unlikely. Averaging the results of the different SM predictions, a non-vanishing cross-section for pp → H → SS^′ → bbWW of ≈13pb is preferred. If S^′ is SM-like, this cross-section, at the same time explains the 95 GeV γγ excess, while the dominance of S → WW suggests that S is the neutral component of the SU(2)_L triplet with hypercharge 0.

AbstractStatistically significant excesses exist at around 152 GeV in associated di-photon production (γγ + X) in the sidebands of SM Higgs analyses of ATLAS (using the full run-2 dataset). They are most pronounced in the single-τ, missing-transverse-energy, four-jet and ⩾ 1ℓ+⩾ 1b-jet channels (≈ 3σ) and can be explained by the Drell-Yan production of new Higgs bosons, i.e. pp → W^* → H^±H⁰. We first examine the excesses in a simplified model approach, considering that H^± decays to τν, WZ or tb. Both the τν and tb decay modes individually lead to a significance of ⪅ 4σ while for WZ one can obtain at most 3.5σ. This is because the decays of WZ lead to multiple leptons contributing to the two-lepton channel which does not show an excess at 152 GeV. Next, we consider two-Higgs-doublet models where the charged Higgs does not decay to WZ at tree-level, finding a significance of ⪆ 4σ for a branching ratio of the new neutral Higgs to photons of ≈2%. Even though this branching fraction is quite sizable, it can be obtained in composite models or via the Lagrangian term λ₆H₁^†H₁H₂^†H₁ + h.c. breaking the commonly imposed Z₂ symmetry.

AbstractWe present a novel technique for the analytic evaluation of multifold Mellin-Barnes (MB) integrals, which commonly appear in physics, as for instance in the calculations of multi-loop multi-scale Feynman integrals. Our approach is based on triangulating a set of points which can be assigned to a given MB integral, and yields the final analytic results in terms of linear combinations of multiple series, each triangulation allowing the derivation of one of these combinations. When this technique is applied to the computation of Feynman integrals, the involved series are of the (multivariable) hypergeometric type. We implement our method in the Mathematica package MBConicHulls.wl, an already existing software dedicated to the analytic evaluation of multiple MB integrals, based on a recently developed computational approach using intersections of conic hulls. The triangulation method is remarkably faster than the conic hulls approach and can thus be used for the calculation of higher-fold MB integrals as we show here by computing triangulations for highly complicated objects such as the off-shell massless scalar one-loop 15-point Feynman integral whose MB representation has 104 folds. As other applications we show how this technique can provide new results for the off-shell massless conformal hexagon and double box Feynman integrals, as well as for the hard diagram of the two loop hexagon Wilson loop.

AbstractLeptoquarks are theoretically well-motivated and have received increasing attention in recent years as they can explain several hints for physics beyond the Standard Model. In this article, we calculate the renormalisation group evolution of models with scalar leptoquarks. We compute the anomalous dimensions for all couplings (gauge, Yukawa, Higgs and leptoquarks interactions) of the most general Lagrangian at the two-loop level and the corresponding threshold corrections at one-loop. The most relevant analytic results are presented in the Appendix, while the notebook containing the full expressions can be downloaded at https://github.com/SumitBanikGit/SLQ-RG. In our phenomenological analysis, we consider some exemplary cases with focus on gauge and Yukawa coupling unification.

AbstractWe explore the possibility that an SU(2)_L triplet scalar with hypercharge Y = 0 is the origin of the 95 GeV diphoton excess. For a small mixing angle with the Standard Model Higgs, its neutral component has naturally a sizable branching ratio to γγ such that its Drell-Yan production via pp → W^* → HH^± is sufficient to obtain the desired signal strength, where H^± is the charged Higgs component of the triplet. The predictions of this setup are: 1) The γγ signal has a p_T spectrum different from gluon fusion but similar to associated production. 2) Photons are produced in association with tau leptons and jets, but generally do not fall into the vector-boson fusion category. 3) The existence of a charged Higgs with m_H^± ≈(95 ± 5) GeV leading to σ(pp → ττνν) ≈ 0.4 pb, which is of the same level as the current limit and can be discovered with Run 3 data. 4) A positive definite shift in the W mass as suggested by the current global electroweak fit.

AbstractThe two-Higgs-doublet model with a U(1)_H gauge symmetry (N2HDM-U(1)) has several advantages compared to the “standard”Z₂ version (N2HDM-Z₂): It is purely based on gauge symmetries, involves only spontaneous symmetry breaking, and is more predictive because it contains one parameter less in the Higgs potential, which further ensures CP conservation, i.e., avoiding the stringent bounds from electric dipole moments. After pointing out that a second, so far unknown version of the N2HDM-U(1) exists, we examine the phenomenological consequences for the Large Hadron Collider (LHC) of the differences in the scalar potentials. In particular, we find that while the N2HDM-Z₂ predicts suppressed branching ratios for decays into different Higgs bosons for the case of the small scalar mixing (as suggested by Higgs coupling measurements), both versions of the N2HDM-U(1) allow for sizable rates. This is particularly relevant in light of the CMS excess in resonant Higgs-pair production at around 650 GeV of a Standard Model Higgs boson subsequently decaying to photons and a new scalar with a mass of ≈ 90 GeV subsequently decaying to bottom quarks (i.e., compatible with the CMS and ATLAS γγ excesses at 95 GeV and ≈ 670 GeV). As we will show, this excess can be addressed within the N2HDM-U(1) in case of a nonminimal Yukawa sector, predicting an interesting and unavoidable Z + bb signal and motivating further asymmetric di-Higgs searches at the LHC.

AbstractWe show how the conic hull method, recently developed for the analytic and non-iterative evaluation of multifold Mellin-Barnes (MB) integrals, can be extended to the case where these integrals have straight contours of integration parallel to the imaginary axes in the complex planes of the integration variables. MB integrals of this class appear, for instance, when one computes the ε-expansion of dimensionally regularized Feynman integrals, as a result of the application of one of the two main strategies (called A and B in the literature) used to resolve the singularities in ε of MB representations. We upgrade the Mathematica package MBConicHulls.wl which can now be used to obtain multivariable series representations of multifold MB integrals with arbitrary straight contours, providing an efficient tool for the automatic computation of such integrals. This new feature of the package is presented, along with an example of application by calculating the ε-expansion of the dimensionally regularized massless one-loop pentagon integral in general kinematics and D = 4 − 2ε.

AbstractIn the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, and the packages AMBRE and Olsson.wl as dependencies. As applications of the package, we find series solutions to the GKZ systems of several one-loop and two-loop Feynman integrals. These are included in the file Examples.nb that can be downloaded along with the package from https://github.com/anant-group/FeynGKZ.

AbstractThe Method of Brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant process recently in the study of the latter due to the development of a new computational approach based on conic hulls (see Phys. Rev. Lett. 127, 151601 (2021)). The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added Rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.

AbstractMellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, until now there is no systematic computational technique of the multiple series representations of N-fold MB integrals for N > 2. Relying on a simple geometrical analysis based on conic hulls, we show here a solution to this important problem. Our method can be applied to resonant (i.e logarithmic) and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained the method also allows, in general, the determination of a single master series for each series representation, which considerably simplifies convergence studies and/or numerical checks. We provide, along with this paper, a Mathematica implementation of our technique with examples of applications. Among them, we present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers.

AbstractThe computational technique of N-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massive one-loop conformal n-point case. In particular, we prove two recent conjectures that give the massive one-loop conformal n-point integral (for generic propagator powers) in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in Hypergeometric Functions Theory. Finally, we also use our MB method to identify spurious contributions that can arise in the Yangian approach.

AbstractThe 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.

AbstractWe use the method of brackets to evaluate quadratic and quartic type integrals. We recall the operational rules of the method and give examples to illustrate its working. The method is then used to evaluate the quadratic type integrals which occur in entries 3.251.1,3,4 in the table of integrals by Gradshteyn and Ryzhik and obtain closed form expressions in terms of hypergeometric functions. The method is further used to evaluate the quartic integrals, entry 2.161.5 and 6 in the table. We also present generalization of both types of integrals with closed form expression in terms of hypergeometric functions.

AbstractThe conic hulls/triangulation method is extended to the analytic computation of multiple Mellin-Barnes having polygamma functions of arbitrary order in their integrand. This computational approach is automatized in the MBConicHulls.wl Mathematica package, which we give a brief description of the new corresponding functionalities, basing our discussion on the example of a simple two-loop sunset integral.

AbstractAfter the Higgs discovery, the question of whether particles beyond those of the Standard Model exist is more pressing than ever. In this context, the scalar sector is particularly promising, since it lies at the core of the internal problems of the Standard Model, while extensions of it allow us to resolve them and can provide explanations for Dark matter, non-zero neutrino masses, inflation etc. In these proceedings, we review the indications for new Higgs bosons at the electroweak scale with masses of ≈95 GeV and ≈152 GeV. These excesses are most significant in the di-photon channel but are supported by weaker-than-expected limits in other decay modes. While for the 95 GeV candidate the production mechanism is mostly unknown, the (hypothetical) 152 GeV Higgs is dominantly produced in association with leptons, (b) jets and missing energy, pointing towards the Drell-Yan production of an SU(2)_L triplet with Y = 0. Interestingly, this model predicts t → H^±b with H^± → WZ, which resembles the signature of ttZ production in the Standard Model and is in fact preferred by current data. Finally, we investigate the possibility that the significant tensions between the Standard Model predictions and the measurements in differential top-quark distributions are due to contamination from new physics involving both the 152 GeV and the 95 GeV scalar.

AbstractIndications for new Higgs bosons at 95 GeV and 152 GeV with significance of > 3σ and > 4σ, respectively, have been found. While the former mainly originates from the inclusive γγ channel, the latter is obtained by combining several modes of associated di-photon production, i.e. γγ + X with X = ℓ, ℓb, MET, τ, etc. In particular, the Drell-Yan production pp → W^* → H⁰H^±, either in the ΔSM (the Standard Model extended by a real triplet) or a 2HDM can explain the 152 GeV excess. The former option offers a connection to tt differential distributions, which show strong tensions with the SM predictions, since the experimental signature is WWbb: A simplified model with a new scalar H, produced via gluon fusion and decaying to two neutral Higgses S and S^′ with subsequent (dominant) decays to WW and bb, respectively, leads to the same final state. Since WW is the dominant decay of a triplet Higgs with a mass around 152 GeV, this motivates that the anomalies might be connected. Furthermore, assuming that S^′ is an SU(2)_L singlet, the resulting di-photon signal strength is compatible with the 95 GeV γγ measurements. A possible UV completion of this simplified model is the Δ2HDMS, i.e. a 2HDM supplemented with a singlet and a real triplet, which can successfully accommodate electroweak scale Baryogenesis.

AbstractWe summarize two geometrical approaches to analytically evaluate higher-fold Mellin-Barnes (MB) integrals in terms of hypergeometric functions. The first method is based on intersections of conic hulls, while the second one, which is more recent, relies on triangulations of a set of points. We demonstrate that, once automatized, the triangulation approach is computationally more efficient than the conic hull approach. As an application of this triangulation approach, we describe how one can derive simpler hypergeometric solutions of the conformal off-shell massless two-loop double box and one-loop hexagon Feynman integrals than those previously obtained from the conic hull approach. Lastly, by applying the above techniques on the MB representation of multiple polylogarithms, we show how to obtain new convergent series representations for these functions. These new analytic expressions were numerically cross-checked with GINAC.

AbstractFeynman integrals at any order of perturbation, in the Lee-Pomeransky representation, could be realised as a subset of Euler-Mellin integrals. Such integrals satisfy the Gelfand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. In an ongoing collaboration, we automate the derivation of the associated GKZ system for a given Feynman diagram from either its Lee-Pomeransky representation or its Mellin-Barnes representation. We also present the automation of two mathematically equivalent techniques, namely the Gröbner deformation method and the method of triangulations of point configurations to solve this system. We have implemented these in the Mathematica package FeynGKZ [1], which is the first public proof-of-concept software that evaluates Feynman integrals in the GKZ framework.

AbstractTwo recently developed techniques of analytic evaluation of multifold Mellin-Barnes (MB) integrals are presented. Both approaches rest on the definition of geometrical objects conveniently associated with the MB integrands, which can then be used along with multivariate residues analysis to derive series representations of the MB integrals. The first method is based on introducing conic hulls and considering specific intersections of the latter, while the second one rests on point configurations and their regular triangulations. After a brief description of both methods, which have been automatized in the MBConicHulls.wl Mathematica package, we review some of their applications. In particular, we show how the conic hulls method was used to obtain the first analytic calculation of complicated Feynman integrals, such as the massless off-shell conformal hexagon and double-box. We then show that the triangulation method is even more efficient, as it allows one to compute these nontrivial objects and harder ones in a much faster way.

AbstractATLAS found that none of their Standard Model simulations can describe the measured differential lepton distributions in their tt analysis reasonably well. Therefore, we study the possibility that this measurement has a new physics contamination. We consider a benchmark model motivated by the indications for di-photon resonances: A heavy scalar decays into two lighter Higgs bosons with masses of 152 GeV and 95 GeV, with subsequent decay to WW and bb, respectively. In this setup, the description of data is improved by at least 5.6σ.

AbstractExperimental measurements of physical observables in particle colliders rely on high-precision theoretical calculations to validate the predictions of any theoretical model and detect signals of new physics. These theoretical calculations, in the perturbative framework of quantum field theory, require the computation of complicated integrals, commonly known as Feynman integrals. Therefore, the evaluation of Feynman integrals is the backbone of theoretical precision calculations. However, this is a highly non-trivial task as we go to higher order in the perturbative series expansion, where the number of scales and loops of the associated Feynman diagrams become large, and the number of Feynman diagrams grows factorially. Nevertheless, such computations are necessary to match the extraordinary precision level of modern and future colliders, and thus the study of Feynman integrals has become a very active field of research in recent years.In this thesis, we focus on studying the mathematical aspects of Feynman integrals and devise techniques to compute them. This field of research is at the cutting edge of theoretical physics, mathematics and computer programming, with new efficient tools proposed every year. Some of the well-known techniques to compute Feynman integrals include the method of differential equations, Mellin-Barnes (MB) representation, hypergeometric functions,Gelfand-Kapranov-Zelevinsky (GKZ) system of equations etc. Our primary focus in this thesis is the study of the Mellin-Barnes (MB) representation approach to solving Feynman integrals in terms of hypergeometric functions. In one of the projects reported in this thesis, we present the first systematic computational technique to solve N-fold MB integrals commonly appearing in the context of Feynman integrals. This technique is based on a surprising geometrical connection between MB integrals and the theory of conic hulls, and therefore we call this the conic hull method. This method yields the final solution in terms of hypergeometric series, which are useful for further analytic study as well as numerical computation. Along with this project, we also present a computer implementation of the conic hull method in the form of the package MBConicHulls.wl based on Mathematica.In another project in this thesis, we apply the conic hull method to solve the previously unsolved dual-conformal hexagon and double box Feynman integrals. These integrals remained unsolved as they involve solving a nine-fold MB integral. However, the conic hull method is successful in solving both the above integrals as it is applicable to N-fold MB integrals. The solutions we obtain for the hexagon and double box are cross-checked numerically against direct integration of Feynman parametrization and analytically using the differential equation satisfied by the double box and hexagon integrals.In another project, we show yet another application of the conic hull method by computing one-loop N-point massive conformal Feynman integrals. Here, we prove two conjectures, first proposed from the Yangian bootstrap approach, stating that any one-loop N-point massive conformal Feynman integral can be written as a single multi-fold hypergeometric series. To prove this, we show that there is always a conic hull, associated with the MB integral of the Feynman integrals, which does not intersect with any other conic hulls, and therefore yields a series solution with a single hypergeometric series.Finally, in the last project of this thesis, we use the conic hull method to illustrate the limitations of the method of brackets. The latter is a computational technique based on the Ramanujan Master Theorem, and was devised originally to evaluate Feynman integrals but is also useful to evaluate certain definite integrals. However, the method is plagued with divergences whose origin is not from the ultraviolet divergence of the Feynman integral, but from the breakdown of one of the rules of the method of brackets. Studying this method in parallel with the conic hull method helps us show some of the fundamental issues of the method of brackets and point out the domain of validity of the method.