22HDME : Two-Higgs-Doublet Model Evolver
Joel Oredsson
Department of Astronomy and Theoretical Physics, Lund University,S¨olvegatan 14A, SE 223-62 Lund, Sweden
Abstract
Two-Higgs-Doublet Model Evolver ( ) is a
C++ program that provides the functionality to perform fast renor-malization group equation running of the general, potentially CP -violating, 2 Higgs Doublet Model at 2-loop order.Simple tree-level calculations of masses; calculations of the oblique parameters S , T and U ; di ff erent parameteri-zations of the scalar potential; tests of perturbativity, unitarity and tree-level stability of the scalar potential are alsoimplemented. We briefly describe the ’s structure, provide a demonstration of how to use it and list some of themost useful functions. Keywords:
Higgs physics, 2HDM, RGE, 2-loop, C ++ PROGRAM SUMMARY
Program Title: 2HDMELicensing provisions: GNU GPLv3Programming language: C ++ Nature of problem: Renormalization group evolution of the general, potentially complex, 2 Higgs doublet model at 2-loop order.Also tests of perturbativity, unitarity and tree-level stability of the model.Solution method: Numerical solutions of systems of ordinary di ff erential equations.
1. Introduction
The discovery of a 125 GeV scalar particle at the LHC [1, 2] marks the beginning of an era of precision Higgsphysics. So far, it resembles the Higgs boson of the Standard Model (SM) [3]; however, further experimental investi-gation is required to decipher its true nature.The Two-Higgs-Doublet Model (2HDM) is a very popular extension of the standard model. By adding a secondHiggs doublet, it o ff ers a rich phenomenology with three neutral and one charged pair of Higgs bosons. Some of itsfeatures are the possibility of explicit and spontaneous CP -violation in the scalar sector; a dark matter candidate andlepton number violation. For a review of the 2HDM, we refer to ref. [4].A useful tool when investigating the 2HDM is to employ a renormalization group (RG) analysis. One can withsuch a method look for instabilities, fine-tuning and valid energy ranges in the 2HDM’s parameter space. The RGequations (RGEs) for any renormalizable quantum field theory at 2-loop order are well known [5, 6, 7, 8]. However,implementing them for a specific model to perform numerical calculations is a tedious and error-prone task.The purpose of 2 Higgs Doublet Model Evolver ( ) is to provide a fast C++ application programming interface(API) that can be used to evolve the 2HDM in renormalization scale by numerically solving its RGEs. workswith the general, potentially complex and CP -violating, 2HDM and both 1- and 2-loop RGEs are implemented.Furthermore, calculations of oblique parameters, tests of perturbativity, unitarity and tree-level stability of the scalarpotential are available.In this manual, we give instructions and showcase some of ’s functionalities. The source code can be foundat ref. [9] and we give some installation instructions in Appendix A. For a physics discussion, we refer to ref. [10];which employs to analyze Z breaking e ff ects in the evolution of the 2HDM. Email address: [email protected] (Joel Oredsson)
Preprint submitted to Computer Physics Communications August 21, 2019 a r X i v : . [ h e p - ph ] A ug e begin by briefly describing the structure of in section 2 and a demonstration of how to use the APIis given in section 3; installation instructions are given in Appendix A. Further details on the base classes and thefunctionality they provide are then given in section 4. The main class of is THDM which is described in section 5,where we also give a short review of the physics of the 2HDM. We give a short description of the algorithm used whenperforming RG evolution in section 6. Instructions of how to implement one’s own model or extend the
THDM classare given in section 7. Finally, we discuss other software that are capable of performing RG evolution in section 8 andconclude in section 9.
2. Structure of 2HDME is written in
C++11 and depends on GSL [11] for numerically solving the RGEs as well as on Eigen [12]for linear algebra operations, see Appendix A for installation instructions. All source code is fairly well documentedwith comments in the header files, which show the functionality of all the classes. originated from an extensionof [13] and hence share a similar structure.The purpose of is to provide an API that consists of methods to manipulate a 2HDM model; thus theidea is that the user should write their own executable code that uses the
THDM class of . A simple examplethat demonstrates how to use is provided in src / demos / DemoRGE.cpp ; which is explained in more detail insection 3.The main class of the is the
THDM object that describes a general, potentially complex, 2HDM; see section 5for a more detailed description of its functionality. For basic usage, one only needs to interact with the public methodsof
THDM (and the SM class to set boundary conditions). The framework to solve RGEs is contained in the RgeModel class, which
THDM inherits from. The
RgeModel , furthermore, inherits some basic functionality related to data andconsole output from the abstract class
BaseModel . To initialize a THDM object, one needs to specify three sectors:the Standard Model (SM) parameters, the scalar potential and the Yukawa sector.The SM parameters that need to be set are the Higgs vacuum expectation value, fermion masses, CKM matrix,gauge couplings and renormalization scale at which they are defined. This can be done manually with a memberfunction, where the user specifies all the input. Another method is to initialize the
THDM class using the SM class thatrepresents the SM. The SM is by default initialized at the top mass scale, ≈
173 GeV, see Appendix C for a detailedlist of all the values; however, these can all be easily changed using its member functions. Similar to the
THDM , the SM also inherits RGE functionality from the RgeModel class, with its own set of RGEs. Thus, if one wants the SMparameters at another renormalization scale, the SM can be evolved to an arbitrary energy scale. However, one shouldonly run models above the top mass scale, since the RGEs specified for the SM include all particles .After setting up all the SM parameters, the THDM needs a scalar potential and a Yukawa sector. The scalar sectorcan be specified with any of the 2HDM bases in section Appendix B; these bases are separate struct s that aredefined in
THDM bases.cpp . Note though that
THDM internally works in the generic basis.The Yukawa sector can be specified in three ways. One option is to impose a Z symmetry; type-I,-II,-III(Y) or-IV(X); these are defined in table 1. Another is to use a flavor alignment ansatz, where the Yukawa matrices for thedi ff erent Higgs fields are proportional to each other. Of course one can also set the Yukawa matrices manually as athird option.To evolve the THDM or SM , one simply uses their member function evolve() . The options for the RG evolutioncan be set with the RgeConfig struct . The 2-loop RG equations (RGEs) for massless parameters of any renormalizable quantum field theory in 4 di-mensions were derived in the seminal papers by Machacek and Vaughn [5, 6, 7]. That work has been supplementedwith the 2-loop RGEs of massive parameters in ref. [8], which is the source that we have used to derive the 1- and2-loop RGEs for a general 2HDM. Note though, that when working with quantum field theories with multiple indis-tinguishable scalar fields one must be careful when interpreting their formulas, since the formulas in ref. [5, 6, 7, 8]are written for the case of an irreducible representation of the scalar fields . In the case of a general 2HDM, one gets Running the SM downwards in energy below the top mass scale should incorporate some mechanism for integrating out particles at theircorresponding mass threshold. This is currently not implemented in . This is a subtlety that is also discussed in ref. [14, 15]. .The general 2-loop RGEs are very long and we will thus not write them down here, but are instead provided assupplementary material and are of course also available in C++ format in the source code of .
3. Demonstration of usage
As an example of how to use the
API, we here go through the
DemoRGE in src / demos , where a 2HDM isinitialized at the top mass scale and then evolved up to the Planck scale. For instructions of how to install and run thedemo, see Appendix A.The first thing to note is that the is wrapped in a namespace , thus including using namespace THDME isconvenient.To initialize the THDM , we first need to specify the SM parameters. Rather then specifying them manually, we canuse the SM class to provide all the necessary inputs. It is constructed at the top mass scale and we print its parametersto the console with SM sm;sm.print all();
This SM is used to initialize the CKM matrix, fermion mass κ F matrices, gauge couplings, vacuum expectation value(VEV) v and the renormalization scale of the THDM . For instructions of how to obtain a SM at another renormalizationscale, see section 4.3. Note, that the SM parameters can also be set manually with member functions that are describedin section 4.3.3.To create a THDM and feed it a SM , use THDM thdm(sm);
Next up, we need to set the scalar potential. This can be done with any of the bases in section Appendix B. The basesare defined as struct objects which have member functions that can be used to convert one base to another. Here,we use the generic basis which we specify by Base generic gen;gen.beta = 1.46713;gen.M12 = std::complex
We set the potential with thdm.set param gen(gen);
Finally we set the Yukawa sector to be type I Z symmetric with thdm.set yukawa type(TYPE I); Now that we have fully initialized the
THDM and the SM , we can save them in SLHA-like text files with For more details about the RGEs of di ff erent renormalization schemes in theories with multiple indistinguishable scalar fields, we refer toref. [16]. They are collected in separate header files in src / RGEs . There is also the VEV phase ξ which is automatically initialized to zero; furthermore, it is fixed by the tadpole equations when actually settingthe THDM potential. m.write slha file();thdm.write slha file(); To evolve the
THDM , we need to configure the settings of the RG evolution. This is done by creating a
RgeConfigstruct and feeding it to the
THDM like
RgeConfig options;options.dataOutput = true;options.consoleOutput = true;options.evolutionName = "DemoRGE";options.twoloop = true;options.perturbativity = true;options.stability = false;options.unitarity = false;options.finalEnergyScale = 1e18;options.steps = 100;thdm.set rgeConfig(options);
The di ff erent options are explained in section 4.2. One can print the options to the console with options.print() .Now, we are ready to evolve the THDM with thdm.evolve(); which should only take a couple of seconds at 2-loop order. The parameters as functions of the renormalization scaleare listed in output / DemoRGE / data and basic plots are created in output / DemoRGE / plots , see figure 1 for an example.Afterwards we can print the parameters at the energy scale where the RG running stopped with thdm.print all(); .To retrieve the scalar potential, one can for example use thdm.get param gen(); Next up, we can evolve the
THDM to another energy. First we save the
THDM at the high scale with thdm.write slha file(sphenoLoopOrder, "DemoRGE evolvedThdm"); .It is always possible to evolve the thdm further. After the fist evolution, the
THDM is specified at the high scale and toevolve the it downwards to µ = finalEnergyScale of its RgeConfig beforeevolving, which is achieved with options.finalEnergyScale = 1e3;options.dataOutput = false;thdm.set rgeConfig( options);thdm.evolve(); where we also changed dataOutput to false ; thus preventing that the first plots are overwritten. A more compactmethod would be to use thdm.evolve to(1e3) , which sets the final scale to the given argument.
4. Classes
There are two main classes of : THDM and SM . These inherit the features from the base classes BaseModel and
RgeModel like
BaseModel ← RgeModel ← THDM , BaseModel ← RgeModel ← SM . We briefly describe the features of the
BaseModel , RgeModel and SM in the following subsections, while the THDM isdescribed in more detail in section 5. 4 Energy [GeV] λ λ λ λ λ λ λ Figure 1: The evolution of quartic couplings in the generic basis produced by
DemoRGE.cpp . BaseModel class
The
BaseModel is an abstract class that o ff ers some basic functionality such as input and output to data files aswell as to the console.The level of information printed to the console of the THDM and SM during computations can be set byFunction Description Returns set logLevel(LogLevel) Sets level of console output information void where
LogLevel can be either one of the following: • LOG INFO : Prints information of calculations performed and status updates. • LOG ERRORS : Only prints error messages. • LOG WARNINGS : Prints error messages and warning messages. • LOG DEBUG : Prints all the information as well as additional debugging information.
RgeModel class
The
RgeModel inherits the input / output functionality of the BaseModel and acts as a base class which o ff ers aframework to incorporate RG evolution in derived classes; both THDM and SM are derived classes of RgeModel .If one wants to create a new type of class that implements RGE functionality similar to
THDM and SM , it is easyto use RgeModel . For example one might want to extend the 2HDM with additional operators and consequently newparameters and RGEs. Instructions of how to do this is given in section 7. There is also the class
NewModel , thatserves as a minimalistic example of a new class that inherits from
RgeModel .See the header file
RgeModel.h for a list of all member functions. Some of the most important ones are:Function Description Returns set rgeConfig(RgeConfig)
Sets up the options for the RG evolution. voidget rgeConfig()
Retrieves the RG evolution options.
RgeConfig
RgeConfig member variable rgeConfig . This RgeConfig has the following member variables: • bool twoloop : If true, uses 2-loop RGEs; otherwise uses 1-loop. • bool perturbativity : If true, the RG evolution stops when perturbativity is violated. • bool unitarity : If true, the RG evolution stops when unitarity is violated. • bool stability : If true, the RG evolution stops when tree-level stability is violated. • bool consoleOutput : If true, prints information to the console during and after RG evolution. • string evolutionName : Name of folder in output , where data and plots are stored. • bool dataOutput : If true, creates data files in output / ”evolutionName” / data folder. These files contain theparameters of the model at each step in the RG evolution. If GNUPLOT is enabled in the
Makefile , simple plotsare created in output / ”evolutionName” / plots . • double finalEnergyScale : The final energy scale, in GeV, for the RG running (from current renormaliza-tion scale). This can be both higher as well as lower than the current renormalization scale. • int steps : Number of steps for the RG evolution; which are logarithmically distributed. Perturbativity, uni-tarity and stability are being checked at each step.To evolve a class that inherits from RgeModel , one can use either of the following two functions:Function Description Returns evolve()
Evolves the model in renormalization scale. true/falseevolve to(double)
Evolves the model to given scale true/false
These evolve the model according to the configuration set by set rgeConfig(RgeConfig) . It returns false ifthe ODE solver runs into numerical problems, e.g. encounters a Landau pole. This does not usually happen if perturbativity is set to true in the
RgeConfig , since the RG running is stopped before the parameters becometoo large.The function evolve to first sets rgeConfig.finalEnergyScale to the given argument and then evolves themodel.The result of the RG evolution is collected in a
RgeResults struct . It can be retrieved with get rgeResults() or simply printed to the console with print rgeResults() . It stores any violation of perturbativity, unitarity orstability and at what energy scale it occurs.Since the evolve() function updates all the parameters, it can be useful to save the state of a model at a specificrenormalization scale usingFunction Description Returns save current state()
Saves the current state internally voidreset to saved state()
Resets to a previously saved state true/false
The model can afterwards be restored to this state with reset to saved state() . There is however currently noway of saving multiple states. If one wishes to do so, it might be easiest to simply copy
THDM objects instead.Some other useful functions are:Function Description Returns set final energy scale(double)
Sets finalEnergyScale for RG evolution voidget rgeResults(RgeResults)
Retrieves results of RG evolution
RgeResultsprint rgeResults(RgeResults)
Prints results of RG evolution to console voidset renormalization scale()
Sets µ voidget renormalization scale() Retrieves µ double All member variables of classes are denoted with an underline as a first character. .3. SM class A class that describes the SM. It inherits the RGE functionality from
RgeModel . The physics member variablesthat are evolved during RG evolution are • The three SU(3) c , SU(2) L and U(1) Y gauge couplings: g3 , g2 , g1 . • Quartic Higgs couplings λ : lambda ; normalized according to the scalar potential V = − m Φ † Φ + λ ( Φ † Φ ) . (1) • Higgs VEV v = ( √ G F ) − / ≈
246 GeV: v . • Complex 3 by 3 Yukawa matrices, yU , yD and yL .In total these sum up to 59 real parameters. The Yukawa matrices are in the fermion weak eigenbasis and initializedwith the CKM matrix and fermion masses: Y U = √ v M u , Y D = √ v V CKM M d , Y L = √ v M (cid:96) , (2)where M f are the diagonal fermion mass matrices. These parameters are at construction defined at the renormalizationscale µ = .
34 GeV. For the default numerical values and conventions for the CKM matrix, see Appendix C. Onecan modify all the default values with the member functions in section 4.3.3. There is no mechanism to generateneutrino masses implemented; hence the neutrinos are treated as being massless.
The SM can be saved to a SLHA-like text file withFunction Description Returns write slha file(string) Creates SLHA file. voidset from slha file(string)
Sets the SM from SLHA file. true/false Other useful functions areFunction Description Returns get v2()
Returns v doubleget gauge couplings() Returns { g , g , g } vector
Eigen::Matrix3cdget lambda()
Returns Higgs quartic coupling doubleprint all()
Prints parameters to console void
As previously mentioned, the SM is constructed at the top mass scale. It is however possible to obtain a SM definedat any other energy scale. One should use the functions evolve() or evolve to(double) of RgeModel to evolvethe SM . In the RG evolution, the mass matrices M f and CKM matrix are calculated at each step by diagonalizing the Y F matrices. Note though that the RGEs for the SM are the full ones, with 6 quarks for example, and no decouplingis performed which should be done when running at energy scales that are below the top mass scale. For example, toget the SM at 1 TeV, one can evolve a constructed SM with evolve to(1e3) .7 .3.3. Setting the parameters As previously mentioned the SM is created with some default numerical values for its parameters. Of course, thesecan also be set manually with the following functions:Function Description set params(mu,lambda,v2,g i,mup,mdn,ml,VCKM) Sets all parameters ( mu = renormalization scale). set v2(v2) Sets the squared VEV. set higgs(v2, lambda)
Sets the potential parameters. set gauge(g i)
Sets the gauge couplings from a vector
Eigen::Matrix3cd .Another method of setting the parameters manually is by loading a SLHA file. This can be done as follows: • Construct a SM and save the parameters to a SLHA file with write slha file("smSlhaFile") . • The SLHA file contain all the parameters of the SM , including the renormalization scale. It is readable and thusprovides an easy way to manually edit all the numerical values. • Then use the edited SLHA file to set a SM object with set from slha file("smSlhaFile") . THDM class
THDM is the main class of and describes a general, potentially complex, 2HDM. It inherits RGE function-ality from
RgeModel .Here, we give a short summary of the general 2HDM and the parameterization of it inside
THDM . For a thoroughreview of the 2HDM see ref. [4]. We use the notation of refs. [17, 18, 19] to describe the generic and Higgs bases ofthe 2HDM.
The 2HDM contains two hypercharge + Φ and Φ . First of all, since the scalarfields have identical quantum numbers, one can always perform a field redefinition of the scalar fields, i.e. a non-singular complex transformation Φ a → B a ¯ b Φ b . The Lagrangian of the 2HDM exhibits a U(2) Higgs flavor symmetry, Φ a → U a ¯ b Φ b ; since the Lagrangian keeps the same form after such a transformation. We will denote 2HDMs relatedby such Higgs flavor transformations as di ff erent bases of the 2HDM. All the di ff erent bases that are implemented in are listed and described in Appendix B. The most general 2HDM gauge invariant renormalizable scalar potential can be written −L V = m Φ † Φ + m Φ † Φ − ( m Φ † Φ + h.c.) + λ (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + (cid:34) λ (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + λ (cid:16) Φ † Φ (cid:17) (cid:16) Φ † Φ (cid:17) + h.c. (cid:35) , (3)where m , λ , λ and λ are potentially complex while all the other parameters are real; resulting in a total of 14degrees of freedom. However, three of these are fixed by the tadpole equations and one can be removed by a re-phasing of the second Higgs doublet. The bases in eq. (3) will be referred to as the generic basis; which is the internalbasis used in the THDM class. 8fter electroweak symmetry breaking, SU(2) × U(1) Y → U(1) em , both of the scalar fields acquire VEVs, which canbe expressed in terms of a unit vector in the Higgs flavor space (cid:104) Φ a (cid:105) = v √ (cid:32) v a (cid:33) , ˆ v a ≡ (cid:32) c β s β e i ξ (cid:33) , (4)where the unit vector is normalized to ˆ v ∗ ¯ a ˆ v a =
1. By convention, we take 0 ≤ β ≤ π/ ≤ ξ ≤ π . Here, wehave used up all our gauge freedom, when setting the VEV in the lower component of the doublets with a SU(2)transformation and removing any phase in the Φ VEV with a U(1) Y transformation. We also define ˆ w b ≡ ˆ v ∗ ¯ a (cid:15) ab ,where (cid:15) = − (cid:15) = (cid:15) = (cid:15) =
0. The angle β can also be expressed as the ratio of the Higgs fields’ vacuumexpectation values, tan β = |(cid:104) Φ (cid:105)| / |(cid:104) Φ (cid:105)| . (5)The Yukawa interactions in the generic basis are −L Y = ¯ Q L ˜ Φ ¯ a η U , a U R + ¯ Q L Φ a η D , a D R + ¯ L L Φ a η L , a E R + h.c. , (6)where the left-handed fermion fields in the weak eigenbasis are Q L ≡ (cid:32) U L D L (cid:33) , L L ≡ (cid:32) ν L E L (cid:33) (7)and ˜ Φ ≡ i σ Φ ∗ .The 129 parameters of the 2HDM are stored as member variables in THDM : • The SU(3) c × SU(2) W × U(1) Y gauge couplings: g3 , g2 and g1 respectively. • The Higgs VEV v = v + v : v2 . This is initialized when feeding a SM to the THDM . • The potential parameters in the generic basis: base generic ; which also includes the angles β and ξ . SeeAppendix B for a detailed description. • The 6 Yukawa matrices in the fermion weak eigenbasis: eta1U , eta2U , eta1D , eta2D , eta1L , eta2L .These are the variables that have their RGEs defined in RGE.cpp . Note though that the angles β and ξ are calculatedfrom the VEVs of the Higgs fields, v a = v ˆ v a ; which run according to the anomalous dimensions of the fields.In addition to the member variables above, the THDM also stores the Yukawa sector in the fermion mass eigenbasis.To go to the fermion mass eigenbasis, the
THDM first calculates the Yukawa matrices in the Higgs basis; which has theLagrangian −L Y = ¯ Q L ˜ H κ U , U R + ¯ Q L H κ D , D R + ¯ L L H κ L , E R + ¯ Q L ˜ H ρ U , U R + ¯ Q L H ρ D , D R + ¯ L L H ρ L , E R + h.c. , (8)where only H acquires a VEV. The κ F , and ρ F , matrices are given by κ U , = ˆ v ∗ a η U , a , ρ U , = ˆ w ∗ a η U , a ,κ D , = ˆ v a η D , a , ρ D , = ˆ w a η D , a ,κ L , = ˆ v a η L , a , ρ L , = ˆ w a η L , a . (9)Note that ˆ v a and ˆ w are defined in terms of the VEVs in the generic basis, which run during RG evolution; thus thetransformation to the Higgs basis is µ -dependent.After going to the Higgs basis, THDM performs biunitary transformations to diagonalize the κ F , matrices. Thefermions in the mass eigenbasis are defined as F L ≡ V FL F L , F R ≡ V FR F R , (10)9here F ∈ { U , D , E } is each fermion species. The diagonal Yukawa matrices are κ U = V UL κ U , V U † R = √ v diag( m u , m c , m t ) ,κ D = V DL κ D , V D † R = √ v diag( m d , m s , m b ) ,κ L = V LL κ L , V L † R = √ v diag( m e , m µ , m τ ) , (11)while ρ F = V FL ρ F , V F † R are potentially non-diagonal; which would mean that tree-level FCNCs are present. The CKMmatrix is composed out of the left-handed transformation matrices, V CKM ≡ V UL V D † L .To summarize, in addition to the parameters in the generic basis, the THDM stores • The diagonal κ F matrices: kU , kD and kL . The fermion masses are also stored as mup[3] , mdn[3] and ml[3] . • The non-diagonal ρ F matrices: rU , rD and rL . • The CKM matrix:
VCKM
THDM
To fully initialize the
THDM , one needs to do three things in order: • Construct a
THDM object and either feed it a SM object or set the SM parameters manually. • Set the scalar potential with any of the available bases. • Fix the Yukawa sector. This can be done with a Z symmetry or a flavor ansatz, which produces a Yukawasector without FCNCs. However, the Yukawa matrices can also be set manually.. The
THDM needs the gauge couplings, CKM matrix, fermion masses and the square of the VEV before setting upthe scalar potential and Yukawa sector. This can be done manually by specifying all of these parameters, or throughtree-level matching to the SM.The SM can be given at construction or with set sm(SM) . This will set the VEV, gauge couplings, renormalizationscale, κ F matrices and CKM matrix. A more sophisticated matching procedure than tree-level is beyond the scopeof this program. There are however physics scenarios where higher order matching, of at least the most importantparameters, can have a very large e ff ect [20]. In such a scenario the user should derive all the THDM ’s parameters inwhatever matching scheme of their choosing. Afterwards, these can be fed into the
THDM with the following commands(all returning void ):Function Description set sm(mu,v2,g i,mup,mdn,ml,VCKM)
Sets all the necessary SM parameters ( mu = renormalization scale). set v2(v2) Sets the squared VEV. set gauge couplings(g i)
Sets the gauge couplings ( g i=vector
Sets the CKM matrix from an
Eigen::Matrix3cd .10 .5. Setting the scalar potential
The scalar potential can be set with any of the bases described in Appendix B. Internally though, the
THDM usesthe generic basis. The functions are Returns set param gen(Base generic,bool=true) true/falseset param higgs(Base higgs,bool=true) true/falseset param invariant(Base invariant,bool=true) true/falseset param hybrid(Base hybrid,bool=true) true/false
All of the functions set the parameters in the generic basis after transforming the basis that is given. The optional bool argument refers to if the tree-level tadpole equations should be enforced; which they are by default. If tan β (cid:44) m , m and ξ . Otherwise the Higgs basis tadpole equations areused, which fix m = − v λ / m = − v λ /
2. These functions will return false if the tree-level Higgs massesare imaginary or if the tadpole equations could not be set.
Note that tan β must be set before fixing the Yukawa sector. After that has been done, it can initialized withDescription set yukawa type(Z2symmetry) Fixes all Yukawa matrices from a Z symmetry. set yukawa aligned(double,double,double) Sets the Yukawa matrices from a flavor ansatz. set yukawa manual(Matrix3cd, ...)
Sets the ρ F matrices manually. set yukawa eta(Matrix3cd, ...) Sets all η F , manually.The type of Z symmetry is specified by a enum , Z2symmetry , which can be set to either
NO SYMMETRY , TYPE I , TYPE II , TYPE III or TYPE IV . Imposing a Z symmetry makes the Yukawa matrices proportional to each other, ρ F = a F κ F , (12)where the coe ffi cients a F are fixed by β as in table 1. These a F coe ffi cients can also be set manually with set yukawa aligned(aU,aD,aL) .Type U R D R L R a U a D a L I + + + cot β cot β cot β II + − − cot β − tan β − tan β Y / III + − + cot β − tan β cot β X / IV + + − cot β cot β − tan β Table 1: Di ff erent Z symmetries that can be imposed on the 2HDM. Φ is odd( −
1) and Φ is even( + Z symmetry, the ρ F matrices become proportional to the diagonal mass matrices, ρ F = a F κ F . Some of the checks that are implemented are: Returns is perturbative() true / false is unitary() true / false is stable() true / false is cp conserved() true / falseThese are simple tree-level constraints. It should be straightforward, however, to manually edit these functions andextend them with one’s own algorithms if that is needed. Using set yukawa type(NO SYMMETRY) does nothing in terms of fixing the ρ F matrices. erturbativity The perturbativity limit is reached when any of the λ i parameters is larger than a specific limit. By default, thislimit is set to 4 π ; however, it can be changed with set perturbativity limit(double) . The default value issomewhat arbitrary and should not be interpreted as a strict limit for when perturbation theory breaks down. In fact,4 π is a very large when dealing with scalar quartic couplings. When some λ i ∼ O (1) there are usually large quantumcorrections to scalar masses for example and many tree-level quantities cannot be trusted. Since the quartic couplingsrun very fast for λ i > π , the perturbativity-violation scale can be interpreted as an approximation for the scale wherea Landau pole is encountered. Turning o ff the perturbativity limit is not recommended; since running a THDM abovethis limit produces numerical problems for the ODE solver and consequently slows down any evolution.
Perturbative unitarity
Tree-level perturbative unitarity of the scattering matrix of scalars at large √ s produces constraints on the quarticcouplings. These constraints are laid out in Appendix D. Note, that they are only valid for large scattering energiesand there are scenarios where one needs to apply more careful checks [21]. Tree-level stability of the scalar potential
The stability conditions at tree-level are given in Appendix E.Similar to the above checks there are more sophisticated methods that can be used that includes quantum correc-tions. In ref. [22] it is shown that by analyzing the 1-loop e ff ective potential using Vevacious [23], one can in fact savemany parameter points that are deemed unstable at tree-level. There are a number of functions that return useful quantities from the
THDM :Function Returns get param gen() Base genericget param higgs() Base higgsget param invariant() Base invariantget param hybrid() Base hybridget yukawa type() Z2symmetryget aF() {| a U | , | a D | , | a L |} get v2() v get gauge couplings() { g , g , g } get mup() { m u , m c , m t } get ml() { m e , m µ , m τ } get vCkm() V CKM get yukawa eta() all { η Fi } get vevs() { v cos β, v sin β e i ξ } get higgs treeLvl masses() { m h i , m H ± } get largest diagonal rF() max( ρ Fii ) get largest nonDiagonal rF() max( ρ Fi (cid:44) j ) get largest lambda() max( λ i ) get largest nonDiagonal lamF() max( λ Fi (cid:44) j ) get lamF element(FermionSector,i,j) λ Fi j get lamF(FermionSector) λ F get oblique() { S , T , U } Note that the aligned parameters a F are only meaningful when the Yukawa matrices are diagonal, which may changeduring RG evolution.The get largest nonDiagonal lamF() and get lamF element(FermionSector,i,j) functions returns the ρ F Yukawa matrices in terms of the Cheng-Sher ansatz [24] defined by λ Fi j ≡ v (cid:114) ρ Fij m i m j , where FermionSector iseither UP , DOWN or LEPTON . 12he oblique parameters S , T and U are calculated with the formulas in ref. [19].There are also a number of functions that print information to the console: • print higgs masses() : Prints the tree-level Higgs boson masses. • print fermion masses() : Prints the tree-level fermion masses. • print potential() : Displays a table showing the scalar potential parameters. Both the generic and the Higgsbasis are shown. • print yukawa() : Prints all the η F , , κ F and ρ F Yukawa matrices. • print CKM() : Prints the CKM matrix. • print param gen() : Prints the scalar potential in the generic basis as well as the η F , Yukawa matrices. • print param compact() : Prints the scalar potential in the compact basis. • print oblique() : Prints the oblique parameters S , T and U . • print param higgs() : Prints the scalar potential in the Higgs basis as well as the κ F and ρ F Yukawa matrices. • print features() : Prints checks showing whether the model is CP conserving, perturbative, unitary andstable, where either is true or false. It also prints the results of any RGE running that might have been performed.If one wants to calculate the results of performing a RG evolution of the 2HDM without updating its parameters,one can use Function Description Returns calc rgeResults() Calculates RG evolution void
The results can be printed to the console with the
RgeModel ’s function print rgeResults .One can create SLHA-like files and setting
THDM objects withFunction Description Returns write slha file(string file)
Creates SLHA file voidset from slha file(string file)
Sets
THDM from SLHA file true/false
These files contain all the information of the
THDM .
6. RG evolution summary
Here, we briefly describe the procedure that is used when evolving a 2HDM. As an example, we initialize the2HDM at the top mass scale and evolve upwards in energy .First one must initialize the SM parameters, e.g. by feeding THDM with a SM . Then one must set the scalar potentialand Yukawa sector . The options for the RG evolution are specified by a RgeConfig , which is given to the
THDM with set rgeConfig . After that, the
THDM can be evolved in energy with evolve() . During the RG evolution, thefollowing is happening: • At each intermediate step as specified in the
RgeConfig , perturbativity, unitarity and stability are checked. • The parameters are evolved in the generic basis, but the other bases are calculated with the µ -dependent tan β and ξ at each step. • If dataOutput=true , the THDM stores the parameters as a function of µ in text files in output / ”evolutionName” .If GNUPLOT is enabled, it also creates simple plots of the running parameters afterwards. • When the RG evolution stops depends on the settings in
RgeConfig . By default, it stops when perturbativity isviolated; which with the default settings is very close to a Landau pole. Note that evolving a
THDM downwards in energy is also possible. One must then, however, fix the high scale boundary condition first in someway. Alternatively, one can set the
THDM from a SLHA file instead of performing the first few steps. . Extending 2HDME To use to evolve another QFT model or some 2HDM with additional degrees of freedom, one can eithercreate a new model that inherits RGE functionality from
RgeModel or extend the
THDM class.As a pedagogical example, there is a minimalistic class that simply describes the evolution of the gauge couplingin quantum electrodynamics called
NewModel . A demonstration of it is provided in src / demos / DemoNewModel . Tocreate one’s own model, the procedure is the following: • Store all the models parameters as member variable of a class that inherits from
RgeModel . All of the virtual functions must be overwritten in the derived class, see
NewModel for an example. • The member variables that should be evolved with RGEs should be transformed into an array with the set y function and the inverted transformation should be performed by set model from y . • The variables in the array are evolved according to the RGEs that are contained in two functions: rgeFuncNewModel 1loop and rgeFuncNewModel 2loop . • The function rge update should update the class’s member variables and, as previously mentioned, the mostminimalistic version of such a function is provided in
NewModel .The same procedure is applicable if one wants to add additional parameters to the
THDM class.Implementing higher order RGEs can also be done fairly easy. The RGEs are divided into di ff erent functionsin order to speed up evolutions of scenarios where one only needs the 1-loop order. The 2-loop order functions in RGE.h do however compute all the loop contributions. So if one wants to add additional loop orders, e.g. the 3-loopcontributions to the scalar potential in ref. [14], simply modify rgeFuncThdm 2loop to also compute these additionalterms in the RGEs.
8. Comparison to other software
Currently there is a number of di ff erent software programs that can be used to calculate RGEs of QFT models.To generate RGEs up to 2-loop order, one can use PyR@TE [25] and
SARAH [26, 27]; they provide RGEs in L A TEXorpython / Mathematica code . The results of these programs do agree with the RGEs that we have derived from ref. [8]in the Z symmetric 2HDM case. However, we have not been able to generate consistent 2-loop RGEs in the general2HDM with complex parameters with these programs; making a complete comparison di ffi cult. serves a function as an API that provides fast numerical evolution of the general 2HDM in C++ . The userdoes not have to go through the trouble of creating a 2HDM model from scratch. In addition, is easily modifiedand self-contained and any numerical calculation should be completely transparent in an investigation of the sourcecode.
9. Conclusion
We have described the
C++ program , which provides an API for evolving a general, potentially CP -violating, 2HDM in renormalization scale by numerically solving its 2-loop RGEs. Its main feature is the class THDM that represents a 2HDM object which is easily manipulated; with several parameterizations of the scalar poten-tial available. Furthermore, tree-level constraints of perturbativity, unitarity and stability are implemented; as well ascalculations of the oblique parameters S , T and U . SARAH also provides the option to export model files that can be used with
SPheno [28, 29] that can perform numerical evolution of the models. Except for linear algebra libraries such as gsl and
Eigen . cknowledgments originated from an extension of [13] in collaboration with Johan Rathsman, who also provideduseful feedback on this manuscript.This work is supported in part by the Swedish Research Council grants contract numbers 621-2013-4287 and2016-05996 and by the European Research Council (ERC) under the European Unions Horizon 2020 research andinnovation programme (grant agreement No 668679). 15 ppendix A. Installation instructions Source code
The source code is available at https://github.com/jojelen/2HDME . Dependencies
The requires the following to be installed: • A C++11 compiler such as gcc . • += -I / ... / eigen3 in the Makefile. • To solve the RGEs, 2HDME uses the GNU GSL library [11], which is usually included in GNU / Linux distri-butions. See https: // / software / gsl / for details. Additional dependencies
These dependencies are optional and can be enabled / disabled by commenting the relevant lines in the Makefile: • The can automatically create simple plots of the RG running of the parameters with the help of
GNUPLOT ,see http: // / for details. Compilation
First, make sure that all the requirements are properly installed. One might need to configure
Makefile to linkall the libraries if they are not installed in the usual location. After that, one can proceed to compile: In terminal,type make in the main directory that contains the
Makefile . Please note that the RGEs in
RGE.cpp are not written in an optimalform. However, the compiler with optimization level -03 will take care of this. This takes some time and compiling
RGE.cpp takes roughly 10 min on a laptop.
Run Demo
To see that everything works, the demo in section 3 is included. The demo evolves a Z symmetric CP conserved2HDM from the top mass scale to the Planck scale. The source code is located in src / demos / DemoRGE.cpp . To run it,type bin/DemoRGE in the terminal. If the
GNUPLOT functionality hasn’t been disabled (by commenting out lines in the Makefile), plots ofthe parameters should have been created in output / DemoRGE / plots . Appendix B. Bases of 2HDM’s potential
There are four bases for the most general scalar potential of the 2HDM implemented in as well as onebasis that describes the CP conserved 2HDM with a softly broken Z symmetry. The base struct for these bases is ThdmBasis , which has the member variables that defines the VEV in eq. (4), i.e. β and ξ .More details about these bases and their relations to each other can be found in refs. [17, 18]. Note that THDM worksin the generic basis internally. 16 ase generic
The generic basis of 2HDM is described in section 5.2. It consists of the additional parameters:Parameter
Base generic m M112 m M222 m M12 λ Lambda1 λ Lambda2 λ Lambda3 λ Lambda4 λ Lambda5 λ Lambda6 λ Lambda7
A generic basis can be converted to other bases with the functions: Returns convert to compact() Base compactconvert to higgs() Base higgsconvert to invariant(double v2) Base invariant
Base compact
The compact basis is defined by −L V = Y a ¯ b Φ † ¯ a Φ b + Z a ¯ bc ¯ d ( Φ † ¯ a Φ b )( Φ † ¯ c Φ d ) . (B.1)All these parameters are stored as complex numbers in Base compact :Parameter
Base compact Y a ¯ b Y[2][2] Z a ¯ bc ¯ d Z[2][2][2][2]
Base higgs
The Higgs basis is defined by the basis where only one scalar doublet acquires a VEV, (cid:104) H (cid:105) = v √ (cid:32) (cid:33) , (cid:104) H (cid:105) = . (B.2)The Lagrangian takes the form −L V = Y H † H + Y H † H + (cid:16) Y H † H + h . c . (cid:17) + Z ( H † H ) + Z ( H † H ) + Z ( H † H )( H † H ) + Z ( H † H )( H † H ) + (cid:40) Z ( H † H ) + (cid:104) Z ( H † H ) + Z ( H † H ) (cid:105) H † H + h.c. (cid:41) . (B.3)The Y , Y , Z , , , are invariants under a Higgs flavor U(2) transformation, while Y , Z , , are pseudoinvariants thattransform as { Y , Z , Z } → (det U ) − { Y , Z , Z } , (B.4) Z → (det U ) − Z . (B.5)This e ff ectively means that the Higgs basis is unique up to a rephasing of the H field. The parameters of Base higgs are: 17arameter
Base higgs Y Y1 Y Y2 Y Y3 Z Z1 Z Z2 Z Z3 Z Z4 Z Z5 Z Z6 Z Z7 And it can be transformed to other bases with the functions: Returns convert to generic() Base genericconvert to compact() Base compactconvert to invariant(double v2) Base invariant
Base invariant
The invariant basis describes the general 2HDM with only Higgs flavor U(2) invariant quantities.Four invariant quantities are the tree-level masses of the Higgs bosons. The charged Higgs boson mass is given by m H ± = Y + Z v (B.6)and the three neutral Higgs bosons’ masses are given by the mass matrix in the Higgs basis, M ≡ v Z Re ( Z ) − Im ( Z )Re ( Z ) [ Z + Z + Re ( Z )] + Y / v − Im ( Z ) − Im ( Z ) − Im ( Z ) [ Z + Z − Re ( Z )] + Y / v . (B.7)This mass matrix can be diagonalized with the rotation matrix R ≡ c c − c s − c s s − c c s + s s c s c c − s s s − c s s − c s s c s c c , (B.8)to produce M D = R M R T = diag( m h , m h , m h ). We will, without loss of generality, assume ordered masses, m h < m h < m h . The eigenvalues of the mass matrix are invariant under Higgs flavor transformations, even though thematrix elements are not. Consequently, the rotation matrix is not invariant either. While the angles θ and θ areinvariant, θ changes so that e i θ → (det U ) e i θ is a pseudo-invariant quantity [18]. In ref. [18], they define a U(2)invariant mass matrix, ˜ M ≡ v Z Re (cid:16) Z e − i θ (cid:17) − Im (cid:16) Z e − i θ (cid:17) Re (cid:16) Z e − i θ (cid:17) Re (cid:16) Z e − i θ (cid:17) + A / v − Im (cid:16) Z e − i θ (cid:17) − Im (cid:16) Z e − i θ (cid:17) − Im (cid:16) Z e − i θ (cid:17) A / v , (B.9)where A ≡ Y + (cid:104) Z + Z − Re (cid:16) Z e − i θ (cid:17)(cid:105) v . This matrix is diagonalized with the rotation matrix˜ R ≡ c c − s − c s c s c − s s s c , (B.10)such that M D = ˜ R ˜ M ˜ R T . The angles lie in the range − π/ ≤ θ , θ < π/ { m h i } , m H ± , θ , θ , Z , Z and the complexparameter Z e − i θ : 18arameter Description Base invariant { m h , m h , m h } Ordered neutral Higgs boson masses mh[3] m H ± Charged Higgs boson mass mHc s ∈ [ − ,
1] Mixing angle of neutral Higgs mass matrix s12 c ∈ [0 ,
1] Mixing angle of neutral Higgs mass matrix c13 { Z , Z } Real quartic couplings
Z2,Z3 Z e − i θ Complex quartic coupling
Z7inv
The invariant basis can be converted to other bases with Returns convert to generic(double v2) Base genericconvert to compact(double v2) Base compactconvert to higgs(double v2) Base higgs
Base hybrid
The hybrid basis presented in ref. [30] is describing a CP conserved 2HDM with a softly broken Z symmetry. Itconsists of a combination of tree-level masses and quartic couplings:Parameter Description Base invariant m h Lightest CP even Higgs boson mh m H Heaviest CP even Higgs boson mH cos( β − α ) Mixing angle of CP even Higgs mass matrix cba tan β Ratio of Higgs VEVs in generic basis tanb { Z , Z , Z } Real quartic couplings
Z4,Z5,Z7 and can be converted to the general bases with Returns convert to generic(double v2) Base genericconvert to higgs(double v2) Base higgsconvert to invariant(double v2) Base invariant
Appendix C. SM input
The SM is defined at the top quark mass scale, M t = .
34 GeV [31]. See section 4.3 for instructions of how tocreate a SM object at another renormalization scale. At construction, we use the following input to fix its parameters: • The SM Higgs VEV is taken to be v = ( √ G F ) − / = . • The fermion masses are used to fix the Yukawa matrix elements in the fermion mass eigenbasis. We use theones from ref. [33]: m u = .
22 MeV , m c = .
590 GeV , m t = . , m d = .
76 MeV , m s =
52 MeV , m b = .
75 GeV , m e = . , m µ = . , m τ = . . (C.1) • Gauge couplings from ref. [32]: g = . g = . g = . Y , SU(2) W and SU(3) c respectively. 19 For the CKM matrix, we use the standard parametrization V CKM = c c s c s e − i δ − s c − c s s e i δ c c − s s s e i δ s c s s − c c s e i δ − c s − s c s e i δ c c , (C.3)where the angles in terms of the Wolfenstein parameters are s = λ, s = A λ , s e i δ = A λ ( ¯ ρ + i ¯ η ) √ − A λ √ − λ (cid:2) − A λ ( ¯ ρ + i ¯ η ) (cid:3) . (C.4)The numerical values λ = . , A = . , ¯ ρ = . , ¯ η = . , (C.5)are extracted from the PDG [34]. Appendix D. Tree-level unitarity conditions
The tree-level unitarity conditions for a general 2HDM have been worked out in ref. [35]. There, they work outthe following scattering matrices: Λ ≡ λ λ √ λ λ ∗ λ √ λ ∗ √ λ ∗ √ λ λ + λ , (D.1) Λ ≡ λ − λ , (D.2) Λ ≡ λ λ λ λ ∗ λ λ λ λ ∗ λ ∗ λ ∗ λ λ ∗ λ λ λ λ , (D.3) Λ ≡ λ λ + λ λ λ ∗ λ + λ λ λ λ ∗ λ ∗ λ ∗ λ + λ λ ∗ λ λ λ λ + λ . (D.4)In the end, the unitarity constraint put upper limits on the absolute value of the eigenvalues, Λ i , of these matrices, | Λ i | < π. (D.5) Appendix E. Tree-level stability of the scalar potential
Here, we give the conditions for the scalar potential to be bounded from below, as worked out in ref. [36, 37].20hen working out these conditions, ref. [36, 37] constructed a Minkowskian formalism of the 2HDM that usesgauge-invariant field bilinears, r ≡ Φ † Φ + Φ † Φ , (E.1) r ≡ (cid:16) Φ † Φ (cid:17) , (E.2) r ≡ (cid:16) Φ † Φ (cid:17) , (E.3) r ≡ Φ † Φ − Φ † Φ . (E.4)These can be used to create a four-vector r µ = ( r ,(cid:126) r ); where one can raise and lower the indices as usual with the flatMinkowski metric η µν = diag(1 , − , − , − V = − M µ r µ + r µ Λ νµ r ν , (E.5)where M µ = (cid:32) −
12 ( Y + Y ) , Re ( Y ) , − Im ( Y ) , −
12 ( Y − Y ) (cid:33) (E.6)and Λ νµ = ( Z + Z ) + Z − Re ( Z + Z ) Im ( Z + Z ) − ( Z − Z )Re ( Z + Z ) − Z − Re ( Z ) Im ( Z ) − Re ( Z − Z ) − Im ( Z + Z ) Im ( Z ) − Z + Re ( Z ) Im ( Z − Z ) ( Z − Z ) − Re ( Z − Z ) Im ( Z − Z ) − ( Z + Z ) + Z . (E.7)The scalar potential is bounded from below if and only if all of the below requirements are fulfilled: • All the eigenvalues of Λ νµ are real. • There exists a largest eigenvalue that is positive, Λ > { Λ , Λ , Λ } and Λ > • There exist four linearly independent eigenvectors; one V ( a ) for each eigenvalue Λ a . • The eigenvector to the largest eigenvalue is timelike, while the others are spacelike, V (0) · V (0) = (cid:16) V (0)0 (cid:17) − (cid:88) i = (cid:16) (cid:126) V (0) i (cid:17) > , (E.8) V ( i ) · V ( i ) = (cid:16) V ( i )0 (cid:17) − (cid:88) j = (cid:16) (cid:126) V ( i ) j (cid:17) < . (E.9) Appendix F. Example of output
The
DemoRGE program evolves a 2HDM from the top mass scale to the Planck scale, as explained in section 3.It produces a SLHA text file that contains di ff erent blocks with the numerical values of parameters and test results.Here, we list some of the blocks for the 2HDM after RG evolution.The first two blocks contain the renormalization scale where all parameters are defined, as well as potential viola-tions of tree-level tests and the results from any RG evolution that may have been performed of the model: Block THDME The scalar potential parameters are stored in the following blocks:
Block MIJ2
Also the VEVs and gauge couplings have blocks:
Block HMIXIN
All six η F , a Yukawa matrices are stored in two blocks each, one for the real and one for the imaginary part. Forexample, one of these look like:
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