Two-loop Renormalization Group Equations Research Articles

SARAH 4: A tool for (not only SUSY) model builders

We present the new version of the Mathematica package SARAH which provides the same features for a non-supersymmetric model as previous versions for supersymmetric models. This includes an easy and straightforward definition of the model, the calculation of all vertices, mass matrices, tadpole equations, and self-energies. Also the two-loop renormalization group equations for a general gauge theory are now included and have been validated with the independent Python code PyR@TE. Model files for FeynArts, CalcHep/CompHep, WHIZARD and in the UFO format can be written, and source code for SPheno for the calculation of the mass spectrum, a set of precision observables, and the decay widths and branching ratios of all states can be generated. Furthermore, the new version includes routines to output model files for Vevacious for both, supersymmetric and non-supersymmetric, models. Global symmetries are also supported with this version and by linking Susyno the handling of Lie groups has been improved and extended. Program summaryProgram title: SARAHCatalogue identifier: AEIB_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIB_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 271795No. of bytes in distributed program, including test data, etc.: 2612867Distribution format: tar.gzProgramming language: Mathematica.Computer: All for which Mathematica is available.Operating system: All for which Mathematica is available.Classification: 11.1, 11.6.Catalogue identifier of previous version: AEIB_v2_1Journal reference of previous version: Comput. Phys. Commun. 184 (2013) 2604Does the new version supersede the previous version?: Yes, the new version includes all known features of the previous versions but also provides the new features mentioned below.Nature of problem:A supersymmetric model is usually characterized by the particle content, the gauge sector and the superpotential. It is a time consuming process to obtain all necessary information for phenomenological studies from these basic ingredients.Solution method:Non-supersymmetric models are supported by the new possibility to define not only chiral superfields but also component fields. The renormalization group equations (RGEs) for a non-supersymmetric models are calculated by using the generic formulae for a general quantum field theory.Reasons for new version:New features in the definition of models and a full support of non-supersymmetric models. New output for Vevacious.Summary of revisions:Support of non-supersymmetric models; calculation of renormalization group equations for a general gauge theory; link to Susyno for handling of non-SU(N) gauge groups; support of global symmetries; output of model files for Vevacious; support of aligned VEVs; calculation of gauge dependent parts of RGEs for VEVs in running of supersymmetric and non-supersymmetric models.Restrictions:Only renormalizable terms in the Lagrangian are supported. No support of fields with spin 2 or 3/2.Unusual features:Calculation of non-supersymmetric RGEs includes effects of kinetic mixing as well as gauge dependence of running vacuum expectation values. SARAH is the first tool which can automatically create model files for Vevacious. Fully automatized derivation of all terms in the Lagrangian which are fixed by gauge invariance.Running time:Loading the Standard Model: 1.6 s; calculation of all vertices: 11.8 s; calculation of all RGEs: 130.2 s; output for Vevacious model files: 0.1 s; output of model files in UFO format: 0.8 s; output of model files for FeynArts: 0.1 s; output of model files for CalcHep: 0.8 s; output of model files for WHIZARD: 3.5 s; writing of source code for SPheno: 34.5 s. All times measured on Lenovo X220 with Intel(R) Core(TM) i7-2620M CPU @ 2.70 GHz.

Higgs ξ-inflation for the 125-126 GeV Higgs: a two-loop analysis

A non-minimal coupling xi of the Standard Model Higgs field to gravity can give rise to inflation, but large xi is required and thus leads to a violation of perturbative unitarity at Mp/xi, which is well below the inflationary scale Mp/sqrt(xi). We re-examine this claim for a Higgs mass in the range 125-126 GeV for which lambda_eff(mu) runs to very small values near the Planck scale and can significantly reduce the value of xi required for inflation. Using the two-loop renormalization group equations and effective potential for Higgs xi-inflation, we find that familiar inflationary solutions can have a non-minimal coupling as small as xi ~ 400 without the potential developing a second minimum. We also find a new observationally allowed region of Higgs xi-inflation with xi ~ 90 and distinct inflationary predictions, including an observable level of the tensor-to-scalar ratio r.

PyR@TE: Renormalization group equations for general gauge theories

Although the two-loop renormalization group equations for a general gauge field theory have been known for quite some time, deriving them for specific models has often been difficult in practice. This is mainly due to the fact that, albeit straightforward, the involved calculations are quite long, tedious and prone to error. The present work is an attempt to facilitate the practical use of the renormalization group equations in model building. To that end, we have developed two completely independent sets of programs written in Python and Mathematica, respectively. The Mathematica scripts will be part of an upcoming release of SARAH 4. The present article describes the collection of Python routines that we dubbed PyR@TE which is an acronym for “Python Renormalization group equations At Two-loop for Everyone”. In PyR@TE, once the user specifies the gauge group and the particle content of the model, the routines automatically generate the full two-loop renormalization group equations for all (dimensionless and dimensionful) parameters. The results can optionally be exported to LaTeX and Mathematica, or stored in a Python data structure for further processing by other programs. For ease of use, we have implemented an interactive mode for PyR@TE in form of an IPython Notebook. As a first application, we have generated with PyR@TE the renormalization group equations for several non-supersymmetric extensions of the Standard Model and found some discrepancies with the existing literature. Program summaryProgram title: PyR@TECatalogue identifier: AERV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 924959No. of bytes in distributed program, including test data, etc.: 495197Distribution format: tar.gzProgramming language: Python.Computer: Personal computer.Operating system: Tested on Fedora 15, MacOS 10 and 11, Ubuntu 12.Classification: 11.1.External routines: SymPy, PyYAML, NumPy, IPython, SciPyNature of problem:Deriving the renormalization group equations for a general quantum field theory.Solution method:Group theory, tensor algebraRunning time:Tens of seconds per model (one-loop), tens of minutes (two-loop)

Renormalization group equations and matching in a general quantum field theory with kinetic mixing

We work out a set of simple rules for adopting the two-loop renormalization group equations of a generic gauge field theory given in the seminal works of Machacek and Vaughn to the most general case with an arbitrary number of Abelian gauge factors and comment on the extra subtleties possibly encountered upon matching a set of effective gauge theories in such a framework.

Running soft parameters in SUSY models with multiple U(1) gauge factors

The two-loop renormalization group equations for the parameters of softly broken supersymmetric gauge theories given in the literature are generalized to the case when the gauge group contains more than a single abelian factor. This contribution to the proceedings of the DISCRETE 2012 symposium is based on reference [1].

Bare Higgs mass at Planck scale

We compute one- and two-loop quadratic divergent contributions to the bare Higgs mass in terms of the bare couplings in the Standard Model. We approximate the bare couplings, defined at the ultraviolet cutoff scale, by the MS-bar ones at the same scale, which are evaluated by the two-loop renormalization group equations for the Higgs mass around 126GeV in the Standard Model. We obtain the cutoff scale dependence of the bare Higgs mass, and examine where it becomes zero. We find that when we take the current central value for the top quark pole mass, 173GeV, the bare Higgs mass vanishes if the cutoff is about 10^{23}GeV. With a 1.3 sigma smaller mass, 170GeV, the scale can be of the order of the Planck scale.

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.

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.

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.

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%.

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.

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.

Implications of canonical gauge coupling unification in high-scale supersymmetry breaking

We systematically construct two kinds of models with canonical gauge coupling unification and universal high-scale supersymmetry breaking. In the first we introduce standard vector-like particles while in the second we also include non-standard vector-like particles. We require that the gauge coupling unification scale is from 5 × 10 15 GeV to the Planck scale, that the universal supersymmetry breaking scale is from 10 TeV to the unification scale, and that the masses of the vector-like particles ( M V ) are universal and in the range from 200 GeV to 1 TeV. Using two-loop renormalization group equation (RGE) running for the gauge couplings and one-loop RGE running for Yukawa couplings and the Higgs quartic coupling, we calculate the supersymmetry breaking scales, the gauge coupling unification scales, and the corresponding Higgs mass ranges. When the vector-like particle masses are less than 1 TeV, these models can be tested at the LHC.

Two-loop renormalization group theory for the quasi-one-dimensional Hubbard model at half filling

We derive two-loop renormalization-group equations for the half-filled one-dimensional Hubbard chains coupled by the interchain hopping. Our renormalization-group scheme for the quasi-one-dimensional electron system is a natural extension of that for the purely one-dimensional systems in the sense that transverse-momentum dependences are introduced in the g-ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half-filled Hubbard model and obtain constraints on the g-ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization-group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical bosonization and fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have SO(3) x SO(3) x U(1) symmetry when the interchain hopping exceeds the magnitude of the charge gap.