We 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 S_1 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.

In 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 point to the target point. The resulting ordinary differential equation is then solved by the Frobenius method, with the boundary conditions analytically determined 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 FA, FB, FC, and FD functions, as well as by considering applications to angular integrals, Feynman integrals, and cosmological and holographic correlators.PROGRAM SUMMARYProgram Title: HyperPrecision.wl, version 1.2CPC Library link to program files: https://doi.org/10.17632/xp9y5hsm9f.1Developer’s repository link: github.com/HyperPrecision/HyperPrecisionLicensing provisions: GNU General Public License v3Programming language: Wolfram Mathematica, version 12.3 or higherExternal routines/libraries used: FiniteFlow, DESolverNature of problem: Numerical evaluation of multivariate hypergeometric functions is challenging, since their defining series converge only in limited regions of argument space, and analytic continuation beyond these regions is generally not available in closed form for arbitrary Horn-type functions.Solution method: Given a Horn-type hypergeometric function, the package automatically constructs the associated Pfaffian system of partial differential equations from the series definition. This system is then restricted to a one-dimensional path, reducing the problem to an ordinary differential equation. The latter is then solved numerically using the Frobenius generalised power-series method, with boundary conditions determined directly from the original hypergeometric series at the origin.Restrictions: The current version of the package is restricted to complete Horn-type hypergeometric functions with real arguments, and its performance depends on the rank of the underlying holonomic system, the complexity of the associated Pfaffian system and on the requested numerical precision.References:∘Wolfram Mathematica; proprietary software.∘FiniteFlow, open-source software.∘DESolver, open-source software.

Mellin-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.

After 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 tt̅Z 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.

Sunset 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.

We 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 {711; 712} where only the first Higgs doublet field 711 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 711, and the other two physical neutral scalars, which reside almost entirely in 712, are approximate eigenstates of CP (denoted by the CP-even H and the CP-odd A). In the Higgs basis, the scalar potential term Z771 dagger 171271 dagger 2712 + H.c. governs the charged-Higgs loop contributions to the decay of H and A to photons. If ReZ7 Im Z7 not equal 0, then CP-violating effects are present and allow for an H+H-A coupling, which can yield a sizable branching ratio for A -> 77. These CP-violating effects also generate nonzero EDMs for the electron, the neutron and the proton. We examine these correlations for the cases of mA = 95 GeV and mA = 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 mH similar or equal to 98 GeV can account for the best fit to the LEP excess in e+e- -> ZH with H -> bb.

Statistically 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 $t\bar t$ 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 $S^\prime$ ($95\,$GeV) and $S$ ($152\,$GeV), which subsequently decay to $b\bar b$ and $WW$, respectively. In this setup, the total $\chi^2$ is reduced, compared to the SM, resulting in $\Delta\chi^2=34$ to $\Delta\chi^2=158$, depending on the SM simulation used. Notably, allowing $m_S$ to vary, the combination of the distributions points towards $m_S\!\approx\!150\,$GeV which is consistent with the existing $\gamma \gamma$ and $WW$ signals, rendering a mismodelling of the SM unlikely. Averaging the results of the different SM predictions, a non-vanishing cross-section for $pp\to H\to SS^\prime\to b\bar b WW$ of $\approx\!13$pb is preferred. If $S^\prime$ is SM-like, this cross-section, at the same time explains the $95\,$GeV $\gamma\gamma$ excess, while the dominance of $S\to WW$ suggests that $S$ is the neutral component of the $SU(2)_L$ triplet with hypercharge~0.

With the discovery of a Higgs boson with a mass of 125 gigaelectronvolts(GeV)at the Large Hadron Collider(LHC)at CERN in 2012,the Standard Model(SM)is complete,and despite intensive searches,no new fun-damental particle has been observed since then.In fact,a discovery can be challenging without a predictive new physics model because different channels and observables cannot be combined directly and unambiguously.Further-more,without supporting indirect hints,the signal space to be searched is huge,resulting in diluted significances ow-ing to the look-elsewhere effect.Several LHC processes with multiple leptons in the final state point towards the ex-istence of a new Higgs boson with a mass between 140 GeV to 160 GeV decaying mostly to W bosons.While the former strongly reduces the look-elsewhere effect,the latter indicates that it could be a Higgs triplet with zero hyper-charge.Within this simple and predictive extension of the SM,we simulate and combine different channels of di-photon production in association with leptons,missing energy,jets,etc...Using the full run-2 results by ATLAS,in-cluding those presented recently at the Moriond conference,an increased significance of 4 standard deviations is ob-tained for a ≈ 152 GeV Higgs.Due to the previously predicted mass range,the look-elsewhere effect is negligible,and this constitutes the highest statistical evidence for a new narrow resonance obtained at the LHC.Furthermore,the model predicts a heavier-than-expected W boson,as indicated by the global electroweak fit.If further substanti-ated,the discovery of a new Higgs would overthrow the SM,provide a compelling case for the construction of fu-ture particle colliders,and pave the way to a novel understanding of the known shortcomings of the SM.In particu-lar,the triplet Higgs field can lead to a strong first-order phase transition and could thus be related to the matter anti-matter asymmetry in our Universe.

In 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.

In this article, we examine the Standard Model extended by a Y = 0 real Higgs triplet, the ∆SM. It contains a CP-even neutral Higgs (∆0) and two charged Higgs bosons (∆±), 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 ∆0, 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 is compatible with the current global electroweak fit. At the Large Hadron Collider, it leads to a (i) stau-like signature from pp → ∆+∆− → τ^+τ^-νν , (ii) multi-lepton final states from pp → γ* → ∆+∆− → W+W−ZZ and pp → W* → ∆±∆0 → W±ZW+W− as well as (iii) associated di-photon production from pp → W* → ∆±(∆0 → γγ). Concerning (i), the reinterpretation of the recent supersymmetric tau partner search by ATLAS and CMS excludes m_Δ ^± < 110 GeV at 95

The multi-lepton anomalies suggest the existence of a new scalar with a mass between 145 GeV and 155 GeV, which is compatible with indications of the associated production of a narrow resonance with a mass of ≈151 GeV. These anomalies require a sizable branching fraction of the new scalar to WW. However, because no ZZ signal at this mass has been observed, this implies that the new boson could be the neutral component of an SU(2)L triplet with zero hypercharge. This field leads to a positive definite shift in the W mass and is produced via the Drell-Yan process pp→W⁎→Δ0Δ±. We use the side-bands of the ATLAS analysis [1] 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. Combining all channels in a likelihood ratio test, we find that a non-zero Br[Δ0→γγ], with a best-fit value of 0.66%, is preferred by ≈3σ for a mass of ≈151.5 GeV. While this mass is consistent with the multi-lepton anomalies, there is also an excess at ≈127 GeV with a significance of 3.6σ. However, the latter is inconsistent with the mass predicted by the multi-lepton anomalies and is in close proximity to the SM Higgs mass, suggesting the possibility of enhanced associated production of the SM Higgs.

We 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.

Two recently developed techniques of analytic evaluation of multifold Mellin-Barnes (MB) integrals are presented. Both approaches rest on the definition of geometrical objets 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.

Abstract Statistically 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 $${\lambda }_{6}{H}_{1}^{†}{H}_{1}{H}_{2}^{†}{H}_{1}$$ + h.c. breaking the commonly imposed Z2 symmetry.

Despite 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 4.3$\sigma$ is obtained for a mass of 152 GeV. This is the largest statistical evidence for a new narrow resonance observed at the LHC.

We 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.

Feynman 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.

Leptoquarks 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.

We 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 $\gamma\gamma$ such that its Drell-Yan production via $pp\to W^*\to H H^\pm$ is sufficient to obtain the desired signal strength, where $H^\pm$ is the charged Higgs component of the triplet. The predictions of this setup are: 1) The $\gamma\gamma$ 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 $m_{H^\pm}\approx\!(95\pm5)\,$GeV leading to $\sigma(pp\to \tau\tau\nu\nu)\approx0.4\,$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.