Two-loop Renormalization Group Research Articles

Impacts of the Higgs mass on vacuum stability, running fermion masses, and two-body Higgs decays

The recent results of the ATLAS and CMS experiments indicate $116\text{ }\text{ }\mathrm{GeV}\ensuremath{\lesssim}{M}_{H}\ensuremath{\lesssim}131\text{ }\text{ }\mathrm{GeV}$ and $115\text{ }\text{ }\mathrm{GeV}\ensuremath{\lesssim}{M}_{H}\ensuremath{\lesssim}127\text{ }\text{ }\mathrm{GeV}$, respectively, for the mass of the Higgs boson in the standard model (SM) at the 95% confidence level. In particular, both experiments point to a preferred narrow mass range ${M}_{H}\ensuremath{\simeq}(124\ensuremath{\cdots}126)\text{ }\text{ }\mathrm{GeV}$. We examine the impact of this preliminary result of ${M}_{H}$ on the SM vacuum stability by using the two-loop renormalization-group equations, and arrive at the cutoff scale ${\ensuremath{\Lambda}}_{\mathrm{VS}}\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}{10}^{12}\text{ }\text{ }\mathrm{GeV}$ (for ${M}_{H}=125\text{ }\text{ }\mathrm{GeV}$, ${M}_{t}=172.9\text{ }\text{ }\mathrm{GeV}$, and ${\ensuremath{\alpha}}_{s}({M}_{Z})=0.1184$), where the absolute stability of the SM vacuum is lost and some kind of new physics might take effect. We update the values of running lepton and quark masses at some typical energy scales, including the ones characterized by ${M}_{H}$, 1 TeV and ${\ensuremath{\Lambda}}_{\mathrm{VS}}$, with the help of the two-loop renormalization-group equations. The branching ratios of some important two-body Higgs decay modes, such as $H\ensuremath{\rightarrow}b\overline{b}$, $H\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}}$, $H\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$, $H\ensuremath{\rightarrow}{W}^{+}{W}^{\ensuremath{-}}$, and $H\ensuremath{\rightarrow}ZZ$, are also recalculated by inputting the values of relevant particle masses at ${M}_{H}$.

LightestCP-even Higgs boson mass in the testable flippedSU(5)×U(1)Xmodels fromFtheory

We study the lightest CP-even Higgs boson mass in five kinds of testable flipped SU(5) x U(1)_X models from F-theory. Two kinds of models have vector-like particles around the TeV scale, while the other three kinds also have the vector-like particles at the intermediate scale that can be considered as messenger fields in gauge mediated supersymmetry breaking. We require that the Yukawa couplings for the TeV-scale vector-like particles and the third family of the Standard Model (SM) fermions are smaller than three from the electroweak scale to the SU(3)_C x SU(2)_L unification scale. With the two-loop renormalization group equation running for the gauge couplings and Yukawa couplings, we obtain the maximal Yukawa couplings between the TeV-scale vector-like particles and Higgs fields. To calculate the lightest CP-even Higgs boson mass upper bounds, we employ the renormalization group improved effective Higgs potential approach, and consider the two-loop leading contributions in the supersymmetric SM and one-loop contributions from the TeV-scale vector-like particles. We assume maximal mixings between the stops and between the TeV-scale vector-like scalars. The numerical results for these five kinds of models are roughly the same. In particular, we show that the lightest CP-even Higgs boson can have mass up to 146 GeV naturally, which is the current upper bound from the CMS and ATLAS collaborations.

Decoupling heavy sparticles in Effective SUSY scenarios: unification, Higgs masses and tachyon bounds

Using two-loop renormalization group equations implementing the decoupling of heavy scalars, Effective SUSY scenarios are studied in the limit in which there is a single low energy Higgs field. Gauge coupling unification is shown to hold with similar or better precision than in standard MSSM scenarios. b-tau unification is examined, and Higgs masses are computed using the effective potential, including two-loop contributions from scalars. A 125 GeV Higgs is compatible with stops/sbottoms at around 300 GeV with non-universal boundary conditions at the scale of the heavy sparticles if some of the trilinear couplings at this scale take values of the order of 1-2 TeV; if more constrained boundary conditions inspired by msugra or gauge mediation are set at a higher scale, heavier colored sparticles are required in general. Finally, since the decoupled RG flow for third-generation scalar masses departs very significantly from the MSSM DR-bar one, tachyon bounds for light scalars are revisited and shown to be relaxed by up to a TeV or more.

A tool box for implementing supersymmetric models

We present a framework for performing a comprehensive analysis of a large class of supersymmetric models, including spectrum calculation, dark matter studies and collider phenomenology. To this end, the respective model is defined in an easy and straightforward way using the Mathematica package SARAH. SARAH then generates model files for CalcHep which can be used with micrOMEGAs as well as model files for WHIZARD and O’Mega. In addition, Fortran source code for SPheno is created which facilitates the determination of the particle spectrum using two-loop renormalization group equations and one-loop corrections to the masses. As an additional feature, the generated SPheno code can write out input files suitable for use with HiggsBounds to apply bounds coming from the Higgs searches to the model. Combining all programs provides a closed chain from model building to phenomenology.Program summaryProgram title: SUSY Phenomenology toolbox.Catalog identifier: AEMN_v1_0.Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMN_v1_0.html.Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland.Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html.No. of lines in distributed program, including test data, etc.: 140206.No. of bytes in distributed program, including test data, etc.: 1319681.Distribution format: tar.gz.Programming language: Autoconf, Mathematica.Computer: PC running Linux, Mac.Operating system: Linux, Mac OS.Classification: 11.6.Nature of problem: Comprehensive studies of supersymmetric models beyond the MSSM is considerably complicated by the number of different tasks that have to be accomplished, including the calculation of the mass spectrum and the implementation of the model into tools for performing collider studies, calculating the dark matter density and checking the compatibility with existing collider bounds (in particular, from the Higgs searches).Solution method: The presented scripts create a framework for the study of SUSY models using the public codes SARAH, SPheno, WHIZARD, CalcHep, MicrOmegas, HiggsBounds and SSP. This includes the download, the configuration and the installation of different tools as well as a script to automatically implement new models into these codes.Unusual features: The study of extended SUSY models is considerably simplified, with the precision of the spectrum calculation matching the precision usually achieved for the MSSM.Running time: Tested on Lenovo X220: The download, configuration and building the tools take 7 min and 40 s. The time needed for the implementation of new models depends on the model under consideration: It takes 42 min for the MSSM including the compilation of all parts.

Transport properties of a multichannel Kondo dot in a magnetic field

We study the nonequilibrium transport through a multichannel Kondo quantum dot in the presence of a magnetic field. We use the exact solution of the two-loop renormalization group equation to derive analytical results for the g factor, the spin relaxation rates, the magnetization, and the differential conductance. We show that the finite magnetization leads to a coupling between the conduction channels which manifests itself in additional features in the differential conductance.

Renormalization group analysis of a spin-1 Kondo dot: Nonequilibrium transport and relaxation dynamics

We study non-equilibrium transport through a spin-1 Kondo dot in a local magnetic field. To this end we perform a two-loop renormalization group analysis in the weak-coupling regime yielding analytic results for (i) the renormalized magnetic field and the g-factor, (ii) the time evolution of observables and the relevant decay rates, (iii) the magnetization and anisotropy as well as (iv) the current and differential conductance in the stationary state. In particular, we find that compared to a spin-1/2 Kondo dot there exist three additional decay rates resulting in an enhanced broadening of the logarithmic features observed in stationary quantities. Additionally, we study the effect of anisotropic couplings between reservoir and impurity spin.

Probing High-Scale and Split Supersymmetry with Higgs mass measurements

We study the range of Higgs masses predicted by High-Scale Supersymmetry and by Split Supersymmetry, using the matching condition for the Higgs quartic coupling determined by the minimal field content. In the case of Split Supersymmetry, we compute for the first time the complete next-to-leading order corrections, including two-loop renormalization group equations and one loop threshold effects. These corrections reduce the predicted Higgs mass by a few GeV. We investigate the impact of the recent LHC Higgs searches on the scale of supersymmetry breaking. In particular, we show that an upper bound of 127 GeV on the Higgs mass implies an upper bound on the scale of Split Supersymmetry of about 10 8 GeV , while no firm conclusion can yet be drawn for High-Scale Supersymmetry.

Running soft parameters in SUSY models with multiple [formula omitted] gauge factors

We generalize the two-loop renormalization group equations for the parameters of the softly broken SUSY gauge theories given in the literature to the most general case when the gauge group contains more than a single Abelian gauge factor. The complete method is illustrated at two-loop within a specific example and compared to some of the previously proposed partial treatments.

The renormalization group and the effective action

The renormalization group is used to sum the leading-log (LL) contributions to the effective action for a large constant external gauge field in terms of the one-loop renormalization group (RG) function β, the next-to-leading-log (NLL) contributions in terms of the two-loop RG function, etc. The log-independent pieces are not determined by the RG equation, but can be fixed by considering the anomaly in the trace of the energy-momentum tensor. Similar considerations can be applied to the effective potential V for a scalar field [Formula: see text]; here the log-independent pieces are fixed by the condition [Formula: see text].

Renormalization group fixed point with a fourth generation: Higgs-induced bound states and condensates

In the Standard Model with four generations, the two-loop renormalization group equations for the Higgs quartic and Yukawa couplings show a quasi fixed point structure which does not appear at the one-loop level. This quasi fixed point behavior indicates a possible restoration of scale symmetry above some physical cut-off scale Λ FP . We conjecture that there exists a true fixed point which is reached at a similar energy scale. If the masses of the fourth family are sufficiently large, this cut-off scale, Λ FP , is situated in the range of a few TeV to the order of 10 2 TeV , above which the Higgs quartic and Yukawa couplings become practically constant. We found that around Λ FP the strong Yukawa couplings make it possible for the fourth generation to form bound states, including composite extra Higgs doublets. In this scenario the fourth generation condensates are obtained without introducing Technicolor or other unknown interactions.

Variety of Fixed Points of Two-Level Kondo Model with Spin Degrees of Freedom

It is shown that the two-level Kondo model with real spin degrees of freedom has a fixed point of the Ising sd-model on the basis of the two-loop renormalization group method in contrast to the case of spinless model. The phase diagram of fixed point Hamiltonian is shown to be also qualitatively different from that expected from the spinless two-level Kondo model.

Two-loop RG functions of the massive $\phi^4$ field theory in general dimensions

The dependence of critical exponents of statistical mechanical systems on the physical space dimension is one of the most fundamental features of critical phenomena [1]. Continuous change of the space dimension d has been the basic principle for constructing the famous Wilson-Fisher epsilon expansion [2] with e = 4−d. However, the Renormalization Group (RG) [3], which provides the fundamental theoretical basis for such calculations, is nonperturbative in its nature. It does not require that the critical exponents or any other universal quantities should be expanded in some space deviation, coupling constant, inverse number of order-parameter components, etc. While investigating the critical phenomena, the RG allows, in principle, to work in any setting of interest provided that we are smart enough to apply the appropriate analytical or numerical tools. The e-expansion has turned out to be a very useful tool in studying the qualitative features of systems in critical state even in its lowest-order approximations (see, e.g., [4]). Being pushed to higher orders, it allowed to produce very accurate numerical extrapolations for critical exponents of three-dimensional N -vector models [5]. An alternative calculational scheme used with comparative success is the so-called g-expansion within the massive field theory in fixed dimension put forward by Parisi [6]. This approach has been mainly applied directly in three dimensions to systems of different complexity [7–14] (for a recent review see [15]). It was used in calculating the exponents of φ models in two dimensions in [8, 9], albeit with somewhat less accuracy. In [16], the massive theory approach was applied to disordered spin systems in two and three dimensions, while Yu. Holovatch and the present author analyzed the critical behavior of pure and disordered Ising systems in general dimensions 2 < d < 4 [17]. A natural access to non-integer dimensions is also provided by the well-known large-N expansion [18–21]. In its general scope, it gives critical exponents as functions of d. These are valid in the whole range between the lower and upper critical dimensionalities, and can be handled analytically in lower-order approximations. The large-N expansion is capable of yielding information on dimensional dependencies that are hardly accessible by other means, for example, from the epsilon expansion. Unfortunately, it is very hard to obtain such results in higher orders in 1/N , while short series expansions in 1/N usually fail to give very accurate numerical estimates for relatively small values of N , say N = 3. The convergence of truncated 1/N expansions has been analyzed on the basis of the field-theoretical approach at d = 3 in [22].

Scaling and the continuum limit of the finite temperature deconfinement transition inSU(Nc)pure gauge theory

We investigated the finite temperature (T) phase transition for SU(Nc) gauge theory with Nc=4, 6, 8 and 10 at lattice spacing, a, of 1/(6T) or less. We found that these theories have first order transitions at such small a. In many cases we were able to find the critical couplings with precision as good as a few parts in 10^4. We also investigated the use of two-loop renormalization group equations in extrapolating the lattice results to the continuum, thus fixing the temperature scale in units of the phase transition temperature, Tc. We found that when a \le 1/(8Tc) the two-loop extrapolation was accurate to about 1--2%. However, we found that trading Tc for the QCD scale, Lambda_MSbar, increases uncertainties significantly, to the level of about 5--10%.

Mechanism for the Singlet to Triplet Superconductivity Crossover in Quasi-One-Dimensional Organic Conductors

Superconductivity of quasi-one-dimensional organic conductors with a quarter-filled band is investigated using the two-loop renormalization group approach to the extended Hubbard model for which both the single electron hopping t_{\perp} and the repulsive interaction V_{\perp} perpendicular to the chains are included. For a four-patches Fermi surface with deviations to perfect nesting, we calculate the response functions for the dominant fluctuations and possible superconducting states. By increasing V_{\perp}, it is shown that a d-wave (singlet) to f-wave (triplet) superconducting state crossover occurs, and is followed by a vanishing spin gap. Furthermore, we study the influence of a magnetic field through the Zeeman coupling, from which a triplet superconducting state is found to emerge.

Dark matter in CP-violating Supersymmetry

Supersymmetry provides ideal candidates for explaining the dark matter mystery of our universe. Compelling cold dark matter (CDM) candidates are the neutralino, gravitino and the superpartner of the elusive axion, the axino. The LSP neutralino is perhaps the best motivated candidate for CDM and the popular SUSY models can be in agreement with accelerator and WMAP data. Supersymmetric CP-violating phases, although phenomenologically and theoretically interesting, especially for baryogenesis scenarios, are tightly constrained by electric dipole moment (EDM) data. Two-loop renormalization group equation (RGE) running from the unification to the electroweak scale renormalizes the gaugino mass phases with important consequences for EDMs. In minimal CP-violating extensions of mSUGRA, with non-universal boundary conditions at the unification scale, EDMs and WMAP data can be sumiltaneously satisfied for large values of the phases in regions where neutralinos annihilate through a rapid Higgs resonance. These regions of the parameter space are accessible to LHC.

Two-loop renormalization group calculation of response functions for a two-dimensional flat Fermi surface

We present the formalism for a two-loop renormalization group (RG) calculation of some order-parameter susceptibilities associated with a two-dimensional (2D) flat Fermi-surface model. In this order of perturbation theory, one must take into account the self-energy effects directly in all RG flow equations. In one-loop order, our calculation reproduces the well-known results obtained previously by other RG schemes. That is, for repulsive interactions all susceptibilities diverge in the low-energy limit and the antiferromagnetic (AF) spin-density-wave correlations produce indeed the leading instability in the system. In contrast, in two-loop order, only the AF susceptibility diverges for this model. However, even this divergence takes place at a much slower rate than in the one-loop RG approach. The purpose of this paper is to show in a very simple setting how to assess the importance of two-loop quantum fluctuations in 2D interacting fermionic models. With some modifications, the present formalism can also be extended to discuss more realistic models such as the paradigmatic 2D Hubbard model.

Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Systems described by an O(n) symmetrical varphi;{4} Hamiltonian are considered in a d-dimensional film geometry at their bulk critical points. The critical Casimir forces between the film's boundary planes B_{j}, j=1,2, are investigated as functions of film thickness L for generic symmetry-preserving boundary conditions partial differential_{n}phi=c[over composite function]_{j}phi. The L-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form f_{res} approximately D(c_{1}L;{Phi/nu},c_{2}L;{Phi/nu})/L;{d-1} when d<4, where c_{i} are scaling fields associated with the variables c[over composite function]_{i} and Phi is a surface crossover exponent. Explicit two-loop renormalization group results for the function D(c_{1},c_{2}) at d=4- dimensions are presented. These show that (i) the Casimir force can have either sign, depending on c_{1} and c_{2}, and (ii) for appropriate choices of the enhancements c[over composite function]_{j}, crossovers from attraction to repulsion and vice versa occur as L increases.

Erratum: Two-loop renormalization group equations for soft supersymmetry-breaking couplings [Phys. Rev. D 50, 2282 (1994)

We compute the two-loop renormalization group equations for all soft supersymmetry-breaking couplings in a general softly broken N=1 supersymmetric model. We also specialize these results to the Minimal Supersymmetric Standard Model.

Refining the predictions of supersymmetricCP-violating models: A top-down approach

We explore in detail the consequences of the CP-violating phases residing in the supersymmetric and soft SUSY breaking parameters in the approximation that family flavour mixings are ignored. We allow for non-universal boundary conditions and in such a consideration the model is described by twelve independent CP-violating phases and one angle which misaligns the vacuum expectation values (VEVs) of the Higgs scalars. We run two-loop renormalization group equations (RGEs), for all parameters involved, including phases, and we properly treat the minimization conditions using the one-loop effective potential with CP-violating phases included. We show that the two-loop running of phases may induce sizable effects for the electric dipole moments (EDMs) that are absent in the one-loop RGE analysis. Also important corrections to the EDMs are induced by the Higgs VEVs misalignment angle which are sizable in the large tan β region. Scanning the available parameter space we seek regions compatible with accelerator and cosmological data with emphasis on rapid neutralino annihilations through a Higgs resonance. It is shown that large CP-violating phases, as required in Baryogenesis scenarios, can be tuned to obtain agreement with WMAP3 cold dark matter constraints, EDMs and all available accelerator data, in extended regions of the parameter space which may be accessible to LHC.

Towards realistic standard model from D-brane configurations

Effective low energy models arising in the context of D-brane configurations with standard model (SM) gauge symmetry extended by several gauged Abelian factors are discussed. The models are classified according to their hypercharge embeddings consistent with the SM spectrum hypercharge assignment. Particular cases are analyzed according to their perspectives and viability as low energy effective field theory candidates. The resulting string scale is determined by means of a two-loop renormalization group calculation. Their implications in Yukawa couplings, neutrinos and flavor changing processes are also presented.