Two-loop Level Research Articles

γ5 in the four-dimensional helicity scheme

We investigate the regularization-scheme dependent treatment of $\gamma_{5}$ in the framework of dimensional regularization, mainly focusing on the four-dimensional helicity scheme (FDH). Evaluating distinctive examples, we find that for one-loop calculations, the recently proposed four-dimensional formulation (FDF) of the FDH scheme constitutes a viable and efficient alternative compared to more traditional approaches. In addition, we extend the considerations to the two-loop level and compute the pseudo-scalar form factors of quarks and gluons in FDH. We provide the necessary operator renormalization and discuss at a practical level how the complexity of intermediate calculational steps can be reduced in an efficient way.

Universality hypothesis breakdown at one-loop order

We probe the universality hypothesis by analytically computing, at least, the two-loop corrections to the critical exponents for $q$-deformed O($N$) self-interacting $\lambda\phi^{4}$ scalar field theories through six distinct and independent field-theoretic renormalization group methods and $\epsilon$-expansion techniques. We show that the effect of $q$-deformation on the one-loop corrections to the $q$-deformed critical exponents is null, so universality hypothesis is broken down at this loop order. Such effect emerges just, at least, at two-loop level and the validity of universality hypothesis is restored. The $q$-deformed critical exponents obtained through the six methods are the same and, furthermore, reduce to their non-deformed values in the appropriated limit.

Neutrino mass in flavor dependent gauged lepton model

We study a neutrino model introducing an additional nontrivial gauged lepton symmetry where the neutrino masses are induced at two-loop level while the first and second charged-leptons of the standard model are done at one-loop level. As a result of model structure, we can predict one massless active neutrino, and there is a dark matter candidate. Then we discuss neutrino mass matrix, muon anomalous magnetic moment, lepton flavor violations, oblique parameters, and relic density of dark matter taking into account the experimental constraints.

Model independent evaluation of the Wilson coefficient of the Weinberg operator in QCD

We derive a Wilson coefficient of a CP-violating purely gluonic dimension-6 operator called the Weinberg operator ($GG\tilde{G}$) generated by a scalar and two fermions at the two-loop level. We do not specify the representation of SU(3)$_c$ for the scalar and the fermions, and thus our result can be applied to a variety of models beyond the standard model. We estimate the nucleon EDMs induced by the Weinberg operator in some examples and discuss the importance of measuring EDMs. It is found that future measurements of the EDMs can probe physics at higher energy scale beyond the reach of collider experiments.

New form of the NSVZ relation at the two-loop level

Recently the exact NSVZ β-function was rewritten in the form of a relation between the β-function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. It was also suggested that this form of the NSVZ equation follows from an underlying equation relating two-point Green functions of the theory. Here we demonstrate that this relation is satisfied at the two-loop level for the non-Abelian N=1 supersymmetric gauge theories in the case of using the simplest (BRST non-invariant) version of the higher covariant derivative regularization. Consequently, the integrals giving the two-loop β-function can be reduced to the one-loop integrals giving the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields.

Implications of the complex singlet field for noncommutative geometry model

We consider a complex singlet scalar in the spectral action approach to the standard model. It is shown that there is a range of initial values at the unification scale which is able to produce Higgs and top quark masses at low energies. The stability of the vacuum and the deviation of gauge couplings from experimental values are discussed and compared at the two-loop level with a real scalar singlet and the pure standard model.

EFTofPNG: a package for high precision computation with the effective field theory of post-Newtonian gravity

We present a novel public package ‘EFTofPNG’ for high precision computation in the effective field theory of post-Newtonian (PN) gravity, including spins. We created this package in view of the timely need to publicly share automated computation tools, which integrate the various types of physics manifested in the expected increasing influx of gravitational wave (GW) data. Hence, we created a free and open source package, which is self-contained, modular, all-inclusive, and accessible to the classical gravity community. The ‘EFTofPNG’ Mathematica package also uses the power of the ‘xTensor’ package, suited for complicated tensor computation, where our coding also strategically approaches the generic generation of Feynman contractions, which is universal to all perturbation theories in physics, by efficiently treating n-point functions as tensors of rank n. The package currently contains four independent units, which serve as subsidiaries to the main one. Its final unit serves as a pipeline chain for the obtainment of the final GW templates, and provides the full computation of derivatives and physical observables of interest. The upcoming ‘EFTofPNG’ package version 1.0 should cover the point mass sector, and all the spin sectors, up to the fourth PN order, and the two-loop level. We expect and strongly encourage public development of the package to improve its efficiency, and to extend it to further PN sectors, and observables useful for the waveform modelling.

Multicomponent Dark Matter in Radiative Seesaw Models

We discuss radiative seesaw models, in which an exact $Z_2\times Z_2'$ symmetry is imposed. Due to the exact $Z_2\times Z_2'$ symmetry, neutrino masses are generated at a two-loop level and at least two extra stable electrically neutral particles are predicted. We consider two models: one has a multi-component dark matter system and the other one has a dark radiation in addition to a dark matter. In the multi-component dark matter system, non-standard dark matter annihilation processes exist. We find that they play important roles in determining the relic abundance and also responsible for the monochromatic neutrino lines resulting from the dark matter annihilation process. In the model with the dark radiation, the structure of the Yukawa coupling is considerably constrained and gives an interesting relationship among cosmology, lepton flavor violating decay of the charged leptons and the decay of the inert Higgs bosons.

Loop suppressed light fermion masses with U(1)R gauge symmetry

We propose a model with two Higgs doublet where quark and charged-lepton masses in the first and second families are induced at one-loop level, and neutrino masses are induced at the two-loop level. In our model we introduce an extra $U(1)_R$ gauge symmetry that plays a crucial role in achieving desired terms in no conflict with anomaly cancellation. We show the mechanism to generate fermion masses, the resultant mass matrices and Yukawa interactions in mass eigenstates, and discuss several interesting phenomenologies such as muon anomalous magnetic dipole moment and dark matter candidate that are arisen from this model.

Higgs-boson masses and mixing matrices in the NMSSM: analysis of on-shell calculations

We analyze the Higgs-boson masses and mixing matrices in the NMSSM based on an on-shell (OS) renormalization of the gauge-boson and Higgs-boson masses and the parameters of the top/scalar top sector. We compare the implementation of the OS calculations in the codes NMSSMCALC and NMSSM-FeynHiggs up to {mathcal {O}}(alpha _talpha _s). We identify the sources of discrepancies at the one- and at the two-loop level. Finally we compare the OS and overline{mathrm {DR}} evaluation as implemented in NMSSMCALC. The results are important ingredients for an estimate of the theoretical precision of Higgs-boson mass calculations in the NMSSM.

Gauge coupling unification with hidden photon, and minicharged dark matter

We show that gauge coupling unification is realized with a greater accuracy in the presence of a massless hidden photon which has a large kinetic mixing with hypercharge. We solve the renormalization group equations at two-loop level and find that the GUT unification scale is around 1016.5GeV which sufficiently suppresses the proton decay rate, and that the unification is essentially determined by the kinetic mixing only, and it is rather insensitive to the hidden gauge coupling or the presence of vector-like matter fields charged under U(1)H and/or SU(5). Matter fields charged under the unbroken hidden U(1)H are stable and they contribute to dark matter. Interestingly, they become minicharged dark matter which carries a small but non-zero electric charge, if the hidden gauge coupling is tiny. The minicharged dark matter is a natural outcome of the gauge coupling unification with a hidden photon.

Radiative neutrino mass in alternative left\u2013right model

We propose a radiative seesaw model in an alternative left–right model without any bidoublet scalar fields, in which all the fermion masses in the standard model are generated through a canonical seesaw mechanism at the tree level. On the other hand the observed neutrino masses are generated at two-loop level. In this paper we focus on the neutrino sector and show how to induce the active neutrino masses. Then we discuss the observed neutrino oscillation, constraints from lepton flavor violations, new sources of the muon anomalous magnetic moment, a long-lived dark matter candidate with keV scale mass, and collider physics.

Electroweak vacuum stability in classically conformal B-L extension of the standard model

We consider the minimal U(1)_{B-L} extension of the standard model (SM) with the classically conformal invariance, where an anomaly-free U(1)_{B-L} gauge symmetry is introduced along with three generations of right-handed neutrinos and a U(1)_{B-L} Higgs field. Because of the classically conformal symmetry, all dimensional parameters are forbidden. The B-L gauge symmetry is radiatively broken through the Coleman–Weinberg mechanism, generating the mass for the U(1)_{B-L} gauge boson (Z^prime boson) and the right-handed neutrinos. Through a small negative coupling between the SM Higgs doublet and the B-L Higgs field, the negative mass term for the SM Higgs doublet is generated and the electroweak symmetry is broken. In this model context, we investigate the electroweak vacuum instability problem in the SM. It is well known that in the classically conformal U(1)_{B-L} extension of the SM, the electroweak vacuum remains unstable in the renormalization group analysis at the one-loop level. In this paper, we extend the analysis to the two-loop level, and perform parameter scans. We identify a parameter region which not only solve the vacuum instability problem, but also satisfy the recent ATLAS and CMS bounds from search for Z^prime boson resonance at the LHC Run-2. Considering self-energy corrections to the SM Higgs doublet through the right-handed neutrinos and the Z^prime boson, we derive the naturalness bound on the model parameters to realize the electroweak scale without fine-tunings.

Asymptotic Behaviour of the QED Perturbation Series

I will summarize the present state of a long-term effort to obtain information on the large-order asymptotic behaviour of the QED perturbation series through the effective action. Starting with the constant-field case, I will discuss the Euler-Heisenberg Lagrangian in various dimensions and up to the three-loop level. This Lagrangian holds the information on the N-photon amplitudes in the low-energy limit, and combining it with Spinor helicity methods explicit all-N results can be obtained at the one-loop and, for the “all +” amplitudes, also at the two-loop level. For the imaginary part of the Euler-Heisenberg Lagrangian, an all-loop formula has been conjectured independently by Affleck, Alvarez, and Manton for Scalar QED and by Lebedev and Ritus for Spinor QED. This formula can be related through a Borel dispersion relation to the leading large-N behaviour of the N-photon amplitudes. It is analytic in the fine structure constant, which is puzzling and suggests a diagrammatic investigation of the large-N limit in perturbation theory. Preliminary results of such a study for the 1+1 dimensional case throw doubt on the validity of the conjecture.

Precise predictions for the Higgs-boson masses in the NMSSM

The particle discovered in the Higgs-boson searches at the LHC with a mass of about 125 , mathrm{GeV} can be identified with one of the neutral Higgs bosons of the Next-to-Minimal Supersymmetric Standard Model (NMSSM). We calculate predictions for the Higgs-boson masses in the NMSSM using the Feynman-diagrammatic approach. The predictions are based on the full NMSSM one-loop corrections supplemented with the dominant and sub-dominant two-loop corrections within the Minimal Supersymmetric Standard Model (MSSM). These include contributions at mathcal {O}{(alpha _t alpha _s, alpha _b alpha _s, alpha _t^2,alpha _talpha _b)}, as well as a resummation of leading and subleading logarithms from the top/scalar top sector. Taking these corrections into account in the prediction for the mass of the Higgs boson in the NMSSM that is identified with the observed signal is crucial in order to reach a precision at a similar level as in the MSSM. The quality of the approximation made at the two-loop level is analysed on the basis of the full one-loop result, with a particular focus on the prediction for the Standard Model-like Higgs boson that is associated with the observed signal. The obtained results will be used as a basis for the extension of the code FeynHiggs to the NMSSM.

The Higgs mass in the CP violating MSSM, NMSSM and beyond

We discuss the automatised calculation of the Higgs mass in renormalisable supersymmetric models with complex parameters at the two-loop level. Our setup is based on the public codes SARAH and SPheno, which can now compute the two-loop corrections to masses of all neutral scalars in such theories. The generic ansatz for these calculations and the handling of the ‘Goldstone Boson catastrophe’ is described. It is shown that we find perfect agreement with other existing two-loop calculations performed in the overline{mathrm {DR}} scheme. We also use the functionality to derive results for the MSSM and NMSSM not available before: the Higgs mass in the constrained version of the complex MSSM and the impact of CP phases in the two-loop corrections beyond O(alpha _s alpha _t) for the scale-invariant NMSSM are briefly analysed.

Bounds on the Higgs mass in the standard model for one-loop and two-loop effective potentials and their perturbative validity

The purpose of this paper is to obtain the bounds on the Higgs mass in the Standard Model for one and two–loops Higgs potential, which includes the contributions of scalar, vector and fermion loops at one –loop and two-loop levels for a bound on the coupling constant λ given in the work of P. Kielanowski and S.R.Juarez W as 0.369 ≤ λ ≤ 0.613. On the basis of these studies, we obtain the upper and lower bounds on the Higgs mass as 143 GeV ≤ mH ≤ 200 GeV. These results will be very useful to LHC experiments for the detection of the Higgs particle and are matching with the results obtained by a few contemporary workers. We also find that the condition for perturbative validity given by Miller is satisfied, when the Higgs mass is calculated perturbatively from the effective potential given by the mass less φ 4 theory at one and two-loop level.

Charged lepton flavor violating processes and scalar leptoquark decay branching ratios in the colored Zee-Babu model

We consider a neutrino mass generating model which employs a scalar leptoquark, $\Delta$, and a scalar diquark, $S$. The new scalars $\Delta$ and $S$ carry the standard model $SU(3)_c\times SU(2)_L\times U(1)_Y$ quantum numbers $(3,1,-1/3)$ and $(6,1,-2/3)$, respectively. The neutrino masses are generated at the two-loop level, as in the Zee-Babu model\cite{Zee-Babu}, and $\Delta/S$ plays the role of the doubly/singly charged scalar in the Zee-Babu model. With a moderate working assumption that the magnitudes of the six Yukawa couplings between $S$ and the down-type quarks are of the same order, strong connections are found between the neutrino masses and the charged lepton flavor violating processes. In particular, we study $Z\rightarrow \overline{l} l'$, and $l\rightarrow l' \gamma$ and find that some portions of the parameter space of this model are within the reach of the planned charged lepton flavor violating experiments. Interesting lower bounds are predicted that $B(Z\rightarrow \overline{l} l')\gtrsim 10^{-16}-10^{-14}(10^{-14})\times(1\mbox{TeV}\cdot m_S/7 m_\Delta^2)^2$ and $B(l\rightarrow l' \gamma)\gtrsim 10^{-17}-10^{-16}(10^{-18}-10^{-16})\times(1 \mbox{TeV}\cdot m_S/7 m_\Delta^2)^2$ for neutrino masses being the normal(inverted) hierarchical pattern. The type of neutrino mass hierarchy could also be determined by measuring the charged lepton flavor violating double ratios. Moreover, definite leptoquark decay branching ratios are predicted when there is no Yukawa interaction between the right-handed fermions and $\Delta$ ( the branching fraction of $\Delta$ to a charged lepton and a quark is 50\%), which could help refine the collider search limit on the scalar leptoquark mass.

Brief description of the flavor-changing neutral scalar interactions at two-loop level

In this letter we show a general description about flavor-changing neutral currents (FCNC) mediated by scalars. The analysis is extended at two-loop level for the Two-Higgs Doublet Model type-III because others models have strong constraints on its parameters, even at high orders of the perturbation. For this letter we focus on the standard model, calculating the amplitude for the h→γγ process and discussing the results briefly.

Unification of gauge couplings in radiative neutrino mass models

We investigate the possibility of gauge coupling unification in various radiative neutrino mass models, which generate neutrino masses at one- and/or two-loop level. Renormalization group running of gauge couplings is performed analytically and numerically at one- and two-loop order, respectively. We study three different classes of neutrino mass models: (I) minimal ultraviolet completions of the dimension-7 $\Delta L=2$ operators which generate neutrino masses at one- and/or two-loop level without and with dark matter candidates, (II) models with dark matter which lead to neutrino masses at one-loop level and (III) models with particles in the adjoint representation of $\mathrm{SU}(3)$. In class (I), gauge couplings unify in a few models and adding dark matter amplifies the chances for unification. In class (II), about a quarter of the models admit gauge coupling unification. In class (III), none of the models leads to gauge coupling unification. Regarding the scale of unification, we find values between $10^{14}$ GeV and $10^{16}$ GeV for models belonging to class (I) without dark matter, whereas models in class (I) with dark matter as well as models of class (II) prefer values in the range $5 \cdot 10^{10}-5 \cdot 10^{14}$ GeV.