Two-loop Level Research Articles

Logarithmic EW corrections at one-loop

We present a fully automated implementation of next-to-leading order electroweak (NLO EW) corrections in the logarithmic approximation in OpenLoops. For energies above the electroweak scale NLO EW corrections are logarithmically enhanced and in tails of kinematic distributions of crucial LHC processes yield correction factors of several tens of percent. The implementation of the logarithmic Sudakov EW approximation in the amplitude generator OpenLoops is fully general, largely model independent, it supports the computation of EW corrections to resonant processes, and it is suitable for extensions to the two-loop NNLO EW level. The implementation is based on an efficient representation of the logarithmic approximation in terms of an effective vertex approach. Investigating a set of representative LHC processes we find excellent agreement between the logarithmic approximation and full one-loop results in observables where the assumptions of the EW Sudakov approximation are fulfilled.

Testing the equivalence between the planar Gross-Neveu and Thirring models at N=1

It is known that the Fierz identities predict that the Gross-Neveu and Thirring models should be equivalent when describing systems composed of a single fermionic flavor, N=1. Here, we consider the planar version of both models within the framework of the optimized perturbation theory at the two-loop level, in order to verify if the predicted equivalence emerges explicitly when different temperature and density regimes are considered. At vanishing densities, our results indicate that both models indeed describe exactly the same thermodynamics, provided that N=1. However, at finite chemical potentials we find that the N=1 Fierz equivalence no longer holds. After examining the relevant thermodynamic potentials, we have identified the contributions which lead to this puzzling discrepancy. Finally, we discuss different frameworks in which this (so far open) problem could be further understood and eventually circumvented. Published by the American Physical Society 2024

Improved thermal resummation for multi-field potentials

The resummation of large thermal corrections to the effective potential is mandatory for the accurate prediction of phase transitions. We discuss the accuracy of different prescriptions to perform this resummation at the one- and two-loop level and point out conceptual issues that appear when using a high-temperature expansion at the two-loop level. Moreover, we show how a particular prescription called partial dressing, which does not rely on a high-temperature expansion, consistently avoids these issues. We introduce a novel technique to apply this resummation method to the case of multiple mixing fields. Our approach significantly extends the range of applicability of the partial dressing prescription, making it suitable for phenomenological studies of beyond the Standard Model extensions of the Higgs sector.

Ladder top-quark condensation imprints in supercooled electroweak phase transition

The electroweak (EW) phase transition in the early Universe might be supercooled due to the presence of the classical scale invariance involving Beyond the Standard Model (BSM) sectors and the supercooling could persist down till a later epoch around which the QCD chiral phase transition is supposed to take place. Since this supercooling period keeps masslessness for all the six SM quarks, it has simply been argued that the QCD phase transition is the first order, and so is the EW one. However, not only the QCD coupling but also the top Yukawa and the Higgs quartic couplings get strong at around the QCD scale due to the renormalization group running, hence this scenario is potentially subject to a rigorous nonperturbative analysis. In this work, we employ the ladder Schwinger-Dyson (LSD) analysis based on the Cornwall-Jackiw-Tomboulis formalism at the two-loop level in such a gauge-Higgs-Yukawa system. We show that the chiral broken QCD vacuum emerges with the nonperturbative top condensate and the lightness of all six quarks is guaranteed due to the accidental U(1) axial symmetry presented in the top-Higgs sector. We employ a quark-meson model-like description in the mean field approximation to address the impact on the EW phase transition arising due to the top quark condensation at the QCD phase transition epoch. In the model, the LSD results are encoded to constrain the model parameter space. We then observe the cosmological phase transition of the first-order type and discuss the induced gravitational wave (GW) productions. We find that in addition to the conventional GW signals sourced from an expected BSM at around or over the TeV scale, the dynamical topponium-Higgs system can yield another power spectrum sensitive to the BBO, LISA, and DECIGO, etc.

Models of Radiative Linear Seesaw with Electrically Charged Mediators

Abstract We propose two versions of radiative linear seesaw models, where electrically charged scalars and vector-like leptons generate the Dirac neutrino mass submatrix at the one- and two-loop levels. In these models, the Standard Model charged lepton masses are generated from a one-loop-level radiative seesaw mechanism mediated by charged exotic vector-like leptons and electrically neutral scalars running in the loops. These models can successfully accommodate the current amount of dark matter and baryon asymmetries observed in the universe, as well as the muon anomalous magnetic moment.

Two-loop contributions to the anomalous chromomagnetic dipole moment of the top quark in two-Higgs-doublet models

We present the results for the anomalous chromomagnetic moment of a quark at the one-loop and two-loop levels through model-independent parameterizations, considering the external gluon to be off-shell. Then, we consider different types of two-Higgs-doublet models with the momentum transfer of the external gluon, q2=±MZ2, and 0, for the numerical evaluations thereof for the top quark. We find that the contributions in the two-Higgs-doublet models are on the order of O(10−3) at the one-loop level and O(10−4) at the two-loop level, where the one-loop and two-loop contributions interfere destructively, resulting in the overall contributions being on the order of O(10−3) with a positive sign.

Leptoquark-mediated two-loop neutrino mass in unified theory

Scalar leptoquarks naturally arise within unified theories, offering a promising avenue for addressing one of the most significant challenges of the Standard Model–the existence of non-zero neutrino masses. In this work, we present a unified theory based on the SU(5) gauge group, where neutrino mass appears at the two-loop level via the propagation of scalar leptoquarks. Due to the unified framework, the charged fermion and neutrino masses and mixings are entangled and determined by a common set of Yukawa couplings. These exotic particles not only shed light on the neutrino mass generation mechanism but also help to achieve the unification of gauge couplings and are expected to lead to substantial lepton flavor violating rates, offering tangible opportunities for experimental verification. Reproducing the observed neutrino mass scale necessitates that a set of leptoquarks reside a few orders of magnitude below the unification scale–a specific feature of the proposed scenario. Moreover, maximizing the unification scale implies TeV scale new physics states, making them accessible at colliders. The diverse roles that leptoquarks play highlight the elegance and predictive ability of the proposed unified model.

Gravitational waves from a scotogenic two-loop neutrino mass model

We propose a framework to account for neutrino masses at the two-loop level. This mechanism introduces new scalars and Majorana fermions to the Standard Model. It assumes the existence of a global U(1)×Z2 symmetry which after partial breaking provides the stability of the dark matter candidates of the theory. The rich structure of the potential allows for the possibility of first-order phase transitions (FOPTs) in the early Universe which can lead to the generation of primordial gravitational waves. Taking into account relevant constraints from lepton flavor violation, neutrino physics, as well as the trilinear Higgs couplings at next-to-leading order accuracy, we have found a wide range of possible FOPTs which are strong enough to be probed at the proposed gravitational-wave interferometer experiments such as LISA. Published by the American Physical Society 2024

Probing Zee-Babu states at muon colliders

The Zee-Babu model is a minimal realization of radiative neutrino mass generation mechanism at the two-loop level. We study the phenomenology of this model at future multi-TeV muon colliders. After imposing all theoretical and low-energy experimental constraints on the model parameters, we find that the Zee-Babu states are expected not to reside below the TeV scale, making it challenging to probe them at the LHC. We first analyze the production rates for various channels, including multi singly charged and/or doubly charged scalars at muon colliders. For concreteness, we study several benchmark points that satisfy neutrino oscillation data and other constraints and find that most channels have large production rates. We then analyze the discovery reach of the model using two specific channels: the pair production of singly and doubly charged scalars. For the phenomenologically viable scenarios considered in this study, charged scalars with masses up to O(3–4) TeV can be probed for the center-of-mass energy of 10 TeV and total luminosity of 10 ab−1. Published by the American Physical Society 2024

Entanglement entropy in scalar quantum electrodynamics

We find the entanglement entropy of a subregion of the vacuum state in scalar quantum electrodynamics, working perturbatively to the two-loop level. Doing so leads us to derive the Maxwell-Proca propagator in conical Euclidean space. The area law of entanglement entropy is recovered in both the massive and massless limits of the theory, as is expected. These results yield the renormalization group flow of entanglement entropy, and we find that loop contributions suppress entanglement entropy. We highlight these results in the light of the renormalization group flow of couplings and correlators, which are increased in scalar quantum electrodynamics, so that the potential tension between the increase in correlations between two points of spacetime and the decrease in entanglement entropy between two regions of spacetime with energy is discussed. We indeed show that the vacuum of a subregion of spacetime purifies with energy in scalar quantum electrodynamics, which is related to the concept of screening. Published by the American Physical Society 2024

New symmetries of the two-Higgs-doublet model

The Two Higgs Doublet Model invariant under the gauge group SU(2)×U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SU(2)\ imes U(1)$$\\end{document} is known to have six additional global discrete or continuous symmetries of its scalar sector. We have discovered regions of parameter space of the model which are basis and renormalization group invariant to all orders of perturbation theory in the scalar and gauge sectors, but correspond to none of the hitherto considered symmetries. We therefore identify seven new symmetries of the model and discuss their phenomenology. Soft symmetry breaking is required for some of these models so that electroweak symmetry breaking can occur. We show that, at least at the two-loop level, it is possible to extend some of these symmetries to include fermions.

Closer look at the matching condition for radiative QCD θ parameter

In this paper, we scrutinize a radiatively generated QCD θ parameter at the two-loop level based on both full analytical loop functions with the Fock-Schwinger gauge method and the effective field theory approach, using simplified models. We observe that the radiatively generated θ parameters at the low energy scale precisely match between them. It provides validity to perturbative loop calculations of the QCD θ parameter with the Fock-Schwinger gauge method. Furthermore, it is also shown that the ordinary Fujikawa method for the radiative θ parameter by using θ¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ heta} $$\\end{document} = −arg det Mqloop\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_q^{\ extrm{loop}} $$\\end{document} does not cover all contributions in the simplified models. But, we also find that when there is a scale hierarchy in CP-violating sector, evaluation of the Fujikawa method is numerically sufficient. As an application, we calculate the radiative θ parameter at the two-loop level in a slightly extended Nelson-Barr model, where the spontaneous CP violation occurs to solve the strong CP problem. It is found a part of the radiative θ parameters cannot be described by the Fujikawa method.

Minimal Inert Doublet benchmark for dark matter and the baryon asymmetry

In this article we discuss a minimal extension of the Inert Doublet Model (IDM) with an effective CP-violating D=6 operator, involving the inert Higgs and weak gauge bosons, that can lift it to a fully realistic setup for creating the baryon asymmetry of the Universe (BAU). Avoiding the need to stick to an explicit completion, we investigate the potential of such an operator to give rise to the measured BAU during a multi-step electroweak phase transition (EWPhT) while sustaining a viable DM candidate in agreement with the measured relic abundance. We find that the explored extension of the IDM can account quantitatively for both DM and for baryogenesis and has quite unique virtues, as we will argue. It can thus serve as a benchmark for a minimal realistic extension of the SM that solves some of its shortcomings and could represent the low energy limit of a larger set of viable completions.After discussing the impact of a further class of operators that open the possibility for a larger mass splitting (enhancing the EWPhT) while generating the full relic abundance also for heavy inert-Higgs DM, we ultimately provide a quantitative evaluation of the induced lepton electric dipole moments in the minimal benchmark for the BAU. These arise here at the two-loop level and are therefore less problematic compared to the ones that emerge when inducing CP violation via an operator involving the SM-like Higgs.

Analytic results for loop-level momentum space Witten diagrams

This paper presents an evaluation of the wave function coefficients for conformally coupled scalars at both one and two-loop levels at leading order in the coupling constant, in momentum space. We take cues from time-dependent interactions in flat spacetime and under suitable approximations, these can also be used to study the wave function coefficients for general cosmologies. We make use of recursion relations developed in [20] to regularize certain bulk-point integrals and express the wave function coefficients in a form that simplifies the loop integrals. We utilize hard-cutoff regularization to regularize the loop integrals and further provide a discussion on their renormalization. Our results can also be analytically continued to obtain answers for transition amplitudes in AdS.

Renormalization group evolution with scalar leptoquarks

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.

Integrating by parts at finite density

Both nonzero temperature and chemical potentials break the Lorentz symmetry present in vacuum quantum field theory by singling out the rest frame of the heat bath. This leads to complications in the application of thermal perturbation theory, including the appearance of novel infrared divergences in loop integrals and an apparent absence of four-dimensional integration-by-parts (IBP) identities, vital for high-order computations. Here, we propose a new strategy that enables the use of IBP techniques in the evaluation of Feynman integrals, in particular vacuum or bubble diagrams, in the limit of vanishing temperature T but nonzero chemical potentials μ. The central elements of the new setup include a contour representation for the temporal momentum integral, the use of a small but nonzero T as an IR regulator, and the systematic application of both temporal and spatial differential operators in the generation of linear relations among the loop integrals of interest. The relations we derive contain novel inhomogeneous terms featuring differentiated Fermi-Dirac distribution functions, which severely complicate calculations at nonzero temperature, but are shown to reduce to solvable lower-dimensional objects as T tends to zero. Pedagogical example computations are kept at the one- and two-loop levels, but the application of the new method to higher-order calculations is discussed in some detail.

Non-supersymmetric SO(10) models with Gauge and Yukawa coupling unification

We study a non-supersymmetric SO(10) Grand Unification Theory with a very high energy intermediate symmetry breaking scale in which not only gauge but also Yukawa coupling unification are enforced via suitable threshold corrections and matching conditions. For gauge unification, we focus on a few symmetry breaking patterns with the intermediate gauge groups text {SU}(4)_{textrm{C}} times text {SU}(2)_{textrm{L}} times text {SU}(2)_{textrm{R}} (Pati–Salam) and text {SU}(3)_{textrm{C}} times text {SU}(2)_{textrm{L}} times text {SU}(2)_{textrm{R}}times text {U}(1)_{{textrm{B}}-{textrm{L}}} (minimal left-right symmetry) assuming an additional global U(1) Peccei–Quinn symmetry, and having the Standard Model supplemented by a second Higgs doublet field at the electroweak scale. We derive the conditions as well as the approximate analytical solutions for the unification of the gauge coupling constants at the two-loop level and discuss the constraints from proton decay on the resulting high scale. Specializing to the case of the Pati–Salam intermediate breaking pattern, we then impose also the unification of the Yukawa couplings of third generation fermions at the high scale, again at the two-loop level. In the considered context, Yukawa unification implies a relation between the fermion couplings to the 10- and 126-dimensional scalar representations of the SO(10) group. We consider one such possible relation which is obtainable in an text {E}_6 model where the previous two scalar fields are part of a single multiplet. Taking into account some phenomenological features such as the absence of flavor changing neutral currents at tree-level, we derive constraints on the parameters of the low energy model, in particular on the ratio of the two Higgs doublets vacuum expectation values tan beta .

Novel loop-diagrammatic approach to QCD θ parameter and application to the left-right model

When the QCD axion is absent in full theory, the strong CP problem has to be explained by an additional mechanism, e.g., the left-right symmetry. Even though tree-level QCD overline{theta} parameter is restricted by the mechanism, radiative corrections to overline{theta} are mostly generated, which leads to a dangerous neutron electric dipole moment (EDM). The ordinary method for calculating the radiative overline{theta} utilizes an equation overline{theta}=-arg det {m}_q^{textrm{loop}} based on the chiral rotations of complex quark masses. In this paper, we point out that when full theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark contributions and does not contain its full radiative corrections. We formulate a novel method to calculate the radiative corrections to overline{theta} through a direct loop-diagrammatic approach, which should be more robust than the ordinary one. As an application, we investigate the radiative overline{theta} in the minimal left-right symmetric model. We first confirm a seminal result that two-loop level radiative overline{theta} completely vanishes (corresponding to one-loop corrections to the quark mass matrices). Furthermore, we estimate the size of a non-vanishing radiative overline{theta} at three-loop level. It is found that the resultant induced neutron EDM is comparable to the current experimental bound, and the expected size is restricted by the perturbative unitarity bound in the minimal left-right model.

Probing new physics in the vector-like lepton model by lepton electric dipole moments

We examine the lepton dipole moments in an extension of the Standard Model (SM), which contains vector-like leptons that couple only to the second-generation SM leptons. The model naturally leads to sizable contributions to the muon g − 2 and the muon electric dipole moment (EDM). One feature of this model is that a sizable electron EDM is also induced at the two-loop level due to the existence of new vector-like leptons in the loops. We find parameter regions that can explain the muon g − 2 anomaly and are also consistent with the experimental constraints coming from the electron EDM and the Higgs decay h → μ+μ−. The generated EDMs can be as large as mathcal{O} (10−22)e∙cm for the muon and mathcal{O} (10−30)e∙cm for the electron, respectively, which can be probed in future experiments for the EDM measurements.

A nice two-loop next-to-next-to-MHV amplitude in mathcal{N} = 4 super-Yang-Mills

We study a scalar component of the 8-point next-to-next-to-maximally-helicity-violating (N2MHV) amplitude at two-loop level in mathcal{N} = 4 super-Yang-Mills theory; it has a leading singularity proportional to the inverse of the four-mass-box square root and receives contributions from only two types of non-trivial integrals with one-loop infrared (IR) divergences. We compute such two-loop 8-point integrals by taking (double-)collinear limits of certain finite, dual-conformal-invariant integrals, and they nicely give the IR-safe ratio function after subtracting divergences. As the first genuine two-loop N2MHV amplitude computed explicitly, we find remarkable structures in its symbol and alphabet: similar to the next-to-MHV (NMHV) case, there are still 9 algebraic letters associated with the square root, and the latter also becomes a letter for the first time; unlike the NMHV case, such algebraic letters appear at either one or all of the second, third and last entry, and the part with three odd letters is particularly simple.