Abstract
I give an overview about the features the Mathematica package SARAH provides to study new models. In general, SARAH can handle a wide range of models beyond the MSSM coming with additional chiral superfields, extra gauge groups, or distinctive features like Dirac gaugino masses. All of these models can be implemented in a compact form in SARAH and are easy to use: SARAH extracts all analytical properties of the given model like two-loop renormalization group equations, tadpole equations, mass matrices, and vertices. Also one- and two-loop corrections to tadpoles and self-energies can be obtained. For numerical calculations SARAH can be interfaced with other tools to get the mass spectrum, to check flavour or dark matter constraints, and to test the vacuum stability or to perform collider studies. In particular, the interface to SPheno allows a precise prediction of the Higgs mass in a given model comparable to MSSM precision by incorporating the important two-loop corrections. I show in great detail with the example of the B-L-SSM how SARAH together with SPheno, HiggsBounds/HiggsSignals, FlavorKit, Vevacious, CalcHep, MicrOmegas, WHIZARD, and MadGraph can be used to study all phenomenological aspects of a model.
1. Introduction
Supersymmetry (SUSY) has been the top candidate for beyond standard model (BSM) physics for many years [1–3]. This has many reasons. SUSY solves the hierarchy problem of the standard model (SM) [4, 5], provides a dark matter candidate [6–8], leads to gauge coupling unification [9–15], and gives an explanation for electroweak symmetry breaking (EWSB) [16, 17]. One might even consider the measured Higgs mass as a first hint for SUSY since it falls in the correct ballpark, while it could be much higher in the SM and other BSM scenarios. Before the LHC has been turned on, the main focus has been on the minimal supersymmetric extensions of the SM, the MSSM. The 105 additional parameters of this model, mainly located in the SUSY breaking sector, can be constrained by assuming a fundamental, grand unified theory (GUT) and a specific mechanism for SUSY breaking [18–27]. In these cases often four or five free parameters are left and the model becomes very predictive. However, the negative results from SUSY searches at the LHC1 as well as the measured Higgs mass of about 125 GeV [28, 29] put large pressure on the simplest scenarios. Wide regions of the parameter space, which had been considered as natural before LHC has started, have been ruled out. This has caused more interest in nonminimal SUSY models. Beyond-MSSM model can provide many advantages compared to the MSSM: they address not only the two issues mentioned so far. A more complete list of good reasons to take a look on extensions of the MSSM is as follows.
(i)Naturalness. The Higgs mass in SUSY is not a free parameter like in the SM. In the MSSM the tree-level mass is bounded from above by . Thus, about one-third of the Higgs mass has to be generated radiatively to explain the observation. Heavy SUSY masses are needed to gain sufficiently large loop corrections: that is, a soft version of the hierarchy problem appears again. The need for large loop corrections gets significantly softened if - or -terms are present which already give a push to the tree-level mass [30–34]. (ii)SUSY Searches. The negative results from all SUSY searches at the LHC have put impressive limits on the Sparticle masses. However, the different searches are based on certain assumptions like a stable, neutral, and colourless lightest SUSY particle (LSP), a sufficient mass splitting between the SUSY states, and so on. As soon as these conditions are no longer given like in models with broken -parity, the limits become much weaker [35–37]. Also in scenarios with compressed spectra where SUSY states are nearly degenerated, the strong limits do often not apply [38–49]. (iii)Neutrino Data. There is an overwhelming experimental evidence that neutrinos have masses and do mix among each other; see [50] and references therein. However, neutrino masses are not incorporated in the MSSM. To do that, either one of the different seesaw mechanisms can be utilised or -parity must be broken to allow a neutrino-neutralino mixing [51–62]. (iv)Strong CP-Problem. The strong CP-problem remains an open question not only in the SM but also in the MSSM. In principle, for both models the same solution exists to explain the smallness of the term in QCD: the presence of a broken Peccei-Quinn (PQ) symmetry [63]. In its supersymmetric version PQ models predict not only an axion but also an axino which could be another DM candidate [64–70]. In general, the phenomenological aspects of axion-axino models are often even richer, in particular if the DSFZ version is considered [71, 72]: the minimal, self-consistent supersymmetric DSFZ-axion model needs in total three additional superfields compared to the MSSM [73]. (v)-Problem. The superpotential of the MSSM involves one parameter with dimension mass: the -term. This term is not protected by any symmetry: that is, the natural values would be either exactly 0 or . However, both extreme values are ruled out by phenomenological considerations. The optimal size of this parameter would be comparable to the electroweak scale. This could be obtained if the -term is actually not a fundamental parameter but is generated dynamically. For instance, in singlet extensions an effective -term appears as consequence of SUSY breaking and is therefore naturally [30, 74]. (vi)Top-Down Approach. Starting with a GUT or String theory it is not necessarily clear that only the gauge sector and particle content of the MSSM are present at the low scale. Realistic UV completions come often with many additional matter close to the SUSY scale. In many cases also additional neutral and even charged gauge bosons are predicted [75–78]. (vii)-Symmetry. If one considers -symmetric models, Majorana gaugino masses are forbidden. To give masses to the gauginos in these models, a coupling to a chiral superfield in the adjoint representation is needed. This gives rise to Dirac masses for the gauginos which are in agreement with -symmetry [79–119]. Dirac gauginos are also attractive because they can weaken LHC search bounds [120–122] and flavour constraints [123–125].Despite the large variety and flexibility of SUSY, many dedicated public computer tools like Isajet [126–132], Suspect [133], SoftSUSY [134–136], SPheno [137, 138], or FeynHiggs [139, 140] are restricted to the simplest realization of SUSY, the MSSM, or small extensions of it. Therefore, more generic tools are needed to allow studying of nonminimal SUSY models with the same precision as the MSSM. This precision is needed to confront also these models with the strong limits from SUSY searches, flavour observables, dark matter observations, and Higgs measurements. The most powerful tool in this direction is the Mathematica package SARAH [141–145]. SARAH has been optimized for an easy, fast, and exhaustive study of nonminimal SUSY models. While the first version of SARAH has been focused on the derivation of tree-level properties of a SUSY model, that is, mass matrices and vertices, and interfacing this information with Monte-Carlo (MC) tools, with the second version of SARAH the calculation of one-loop self-energies as well as two-loop renormalization group equations (RGEs) has been automatized. With version 3, SARAH became the first “spectrum-generator-generator”: all analytical information derived by SARAH can be exported to Fortran code which provides a fully-fledged spectrum generator based on SPheno. This functionality has been later extended by the FlavorKit [146] interface which allows a modular implementation of new flavour observables based on the tools FeynArts/FormCalc–SARAH–SPheno. Also different methods to calculate the two-loop corrections to the Higgs states in a nonminimal model are available with SPheno modules generated by SARAH today: the radiative contributions to CP even scalar masses at the two-loop level can be obtained by using either the effective potential approach [147] based on generic results given in [16], or a fully diagrammatic calculation [148]. Both calculations provide Higgs masses with a precision which is otherwise just available for the MSSM. Beginning with SARAH 4, the package is no longer restricted to SUSY models but can handle also a general, renormalizable quantum field theory and provides nearly the same features as for SUSY models. Today, SARAH can be used for SUSY and non-SUSY models to write model files for CalcHep/CompHep [149, 150], FeynArts/FormCalc [151, 152], and WHIZARD/O’Mega [153, 154] as well as in the UFO format [155] which can be handled, for instance, by MadGraph 5 [156], GoSam [157], Herwig++ [158–160], and Sherpa [161–163]. The modules created by SARAH for SPheno calculate the full one-loop and partially two-loop-corrected mass spectrum, branching ratios and decays widths of all states, and many flavour and precision observables. Also an easy link to HiggsBounds [164, 165] and HiggsSignals [166] exists. Another possibility to get a tailor-made spectrum generator for a nonminimal SUSY model based on SARAH is the tool FlexibleSUSY [167]. Finally, SARAH can also produce model files for Vevacious [168]. The combination SARAH–SPheno–Vevacious provides the possibility to find the global minimum of the one-loop effective potential of a given model and parameter point.
The range of models covered by SARAH is very broad. SARAH and its different interfaces have been successfully used to study many different SUSY scenarios: singlet extensions with and without CP violation [169–179], triplet extensions [180, 181], models with -parity violation [182–188], different kinds of seesaw mechanisms [58, 60–62, 189–194], models with extended gauge sectors at intermediate scales [195–198] or the SUSY scale [34, 199–203], models with Dirac gauginos [109, 111, 204–206] or vector-like states [207], and even more exotic extensions [208–211]. In addition, SARAH can be also very useful to perform studies in the context of the MSSM which cannot be done with any other public tool out of the box. That is the case, for instance, if new SUSY breaking mechanisms should be considered [212–219] or if the presence of charge and colour breaking minima should be checked [220, 221]. For the NMSSM, despite the presence of specialized tools like NMSSMTools [222], SoftSUSY [223], or NMSSMCalc [224], the SPheno version created by SARAH is the only code providing two-loop corrections beyond not relying on MSSM approximations [225]. Also the full one-loop corrections to all SUSY states in the NMSSM have first been derived with SARAH [226].
This paper is organized as follows. in the next section an overview about the models supported by SARAH is given. In Section 3, I will discuss the possible analytical calculations which can be done with SARAH and list the possible output of the derived information for further evaluation. The main part of this paper is a detailed example of how SARAH can be used to study all phenomenological aspects of a model. That is done in Sections 4–8: in Section 4 the implementation of the B-L-SSM in SARAH is described, in Section 5 how the model can be understood at the analytical level in Mathematica is discussed, the SPheno output with all its features is presented in Section 6, in Section 7 I will show how other tools can be used together with SARAH and SPheno to study, for instance, the dark matter and collider phenomenology, and in Section 8, different possibilities to perform parameter scans are presented. I summarize in Section 9. Throughout the paper and in the given examples I will focus mainly on SUSY models, but many statements apply one-to-one also to non-SUSY models.
2. Models
2.1. Input Needed by SARAH to Define a Model
SARAH is optimized for the handling of a wide range of SUSY models. The basic idea of SARAH was to give the user the possibility to implement models in an easy, compact, and straightforward way. Most tasks to get the Lagrangian are fully automatized: it is sufficient to define just the fundamental properties of the model. That means that the necessary inputs to completely define the gauge eigenstates with all their interactions are(1)global symmetries,(2)gauge symmetries,(3)chiral superfields,(4)superpotential. That means that SARAH automatizes many steps to derive the Lagrangian from that input as follows: (1)All interactions of matter fermions and the -terms are derived from the superpotential.(2)All vector boson and gaugino interactions as well as -terms are derived from gauge invariance.(3)All gauge fixing terms are derived by demanding that scalar-vector mixing vanishes in the kinetic terms.(4)All ghost interactions are derived from the gauge fixing terms.(5)All soft-breaking masses for scalars and gauginos as well as the soft-breaking counterparts to the superpotential couplings are added automatically.Of course, the Lagrangian of the gauge eigenstates is not the final aim. Usually one is interested in the mass eigenstates after gauge symmetry breaking. To perform the necessary rotations to the new eigenstates, the user has to give some more information: (1)definition of the fields which get a vacuum expectation value (VEV) to break gauge symmetries,(2)definition of vector bosons, scalars, and fermions which mix among each other. Using this information, all necessary redefinitions and fields rotations are done by SARAH. Also the gauge fixing terms are derived for the new eigenstates and the ghost interactions are added. For all eigenstates plenty of information can be derived by SARAH as explained in Section 3. Before coming to that, I will give more details about what kind of models and what features are supported by SARAH.
2.2. Supported Models and Features
As we have seen in the introduction, there are many possibilities to go beyond the widely studied MSSM. Each approach modifies the on or the other sector of the model. In general, possible changes compared to the MSSM are (i) using other global symmetries to extent the set of allowed couplings, (ii) adding chiral superfields, (iii) extending the gauge sector, (iv) giving VEVs to other particles compared to only the Higgs doublets, (v) adding Dirac masses for gauginos, (vi) considering noncanonical terms like nonholomorphic soft-SUSY breaking interactions or Fayet-Iliopoulos -terms. All of these roads can in principle be gone by SARAH and I will briefly discuss what is possible in the different sectors and which steps are done by SARAH to get the Lagrangian. Of course, extending the gauge sector or adding Dirac masses to gauginos comes inevitable with an extended matter sector as well. Thus, often several new effects appear together and can be covered by SARAH.
2.2.1. Global Symmetries
SARAH can handle an arbitrary number of global symmetries which are either or symmetries. Also a continuous -symmetry is possible. Global symmetries are used in SARAH mainly for three different purposes. First, they help to constrain the allowed couplings in the superpotential. However, SARAH does not strictly forbid terms in the superpotential which violate a global symmetry. SARAH only prints a warning to point out the potential contradiction. The reason is that such a term might be included on purpose to explain its tininess. Global symmetries can also affect the soft-breaking terms written down by SARAH. SARAH always tries to generate the most general Lagrangian and includes also soft-masses of the form for two scalars , with identical charges. However, these terms are dropped if they are forbidden by a global symmetry. By the same consideration, Dirac gaugino mass terms are written down or not. Finally, global symmetries are crucial for the output of model files for MicrOmegas to calculate the relic density. For this output at least one unbroken discrete global symmetry must be present.
By modifying the global symmetries one can already go beyond the MSSM without changing the particle content: choosing a (Baryon triality) instead of -parity [227–231], lepton number violating terms would be allowed while the proton is still stable. SARAH comes not only with -parity violating models based on Baryon triality, but also with a variant for Baryon number violation but conserved Lepton number is included.
2.2.2. Gauge Sector
Gauge Groups. The gauge sector of a SUSY model in SARAH is fixed by defining a set of vector superfields. SARAH is not restricted to three vector superfields like in the MSSM, but many more gauge groups can be defined. To improve the power in dealing with gauge groups, SARAH has linked routines from the Mathematica package Susyno [232]. SARAH together with Susyno take care of all group-theoretical calculations: the Dynkin and Casimir invariants are calculated, and the needed representation matrices as well as Clebsch-Gordan coefficients are derived. This is done not only for and gauge groups, but also for and and expectational groups can be used. For all Abelian groups also a GUT normalization can be given. This factor comes usually from considerations about the embedding of a model in a greater symmetry group like or . If a GUT normalization is defined for a group, it will be used in the calculation of the RGEs. The soft-breaking terms for a gaugino of a gauge group are usually included as
Gauge Interactions. With the definition of the vector superfields already the self-interactions of vector bosons as well as the interactions between vector bosons and gauginos are fixed. Those are taken to be I am using here and in the following capital letters and to label the gauge groups and small letter , , and to label the generators, vector bosons, and gauginos of a particular gauge group. The field strength tensor is defined as and the covariant derivative is Here, is the structure constant of the gauge group . Plugging (3) in the first term of (2) leads to self-interactions of three and four gauge bosons. In general, the procedure to obtain the Lagrangian from the vector and chiral superfields is very similar to [233]. Interested readers might check this reference for more details.
Gauge Interactions of Matter Fields. Vector superfields usually do not come alone but also matter fields are present. I am going to discuss the possibilities to define chiral superfields in Section 2.2.4. Here, I assume that a number of chiral superfields are present and I want to discuss the gauge interactions which are taken into account for those. First, the -terms stemming from the auxiliary component of the superfield are calculated. These terms cause four scalar interactions and read Here, the sum is over all scalars in the model, and are the generators of the gauge group for an irreducible representation . For Abelian groups simplify to the charges of the different fields. In addition, Abelian gauge groups can come also with another feature: a Fayet-Iliopoulos -term [234]: This term can optionally be included in SARAH for any .
The other gauge-matter interactions are those stemming from the kinetic terms: with covariant derivatives . The SUSY counterparts of these interactions are those between gauginos and matter fermions and scalars:
Gauge-Kinetic Mixing. The terms mentioned so far cover all gauge interactions which are possible in the MSSM. These are derived for any other SUSY model in exactly the same way. However, there is another subtlety which arises if more than one Abelian gauge group is present. In that case are allowed for field strength tensors of two different Abelian groups and [235]. is in general a matrix if Abelian groups are present. SARAH fully includes the effect of kinetic mixing independent of the number of Abelian groups. For this purpose SARAH is not working with field strength interactions like (9) but performs a rotation to bring the field strength in a diagonal form. That is done by a redefinition of the vector carrying all gauge fields : This rotation has an impact on the interactions of the gauge bosons with matter fields. In general, the interaction of a particle with all gauge fields can be expressed by where is a vector containing the charges of under all groups and is a diagonal matrix carrying the gauge couplings of the different groups. After the rotation according to (10) the interaction part can be expressed by with a general matrix which is no longer diagonal. In that way, the effect of gauge-kinetic mixing has been absorbed in “off-diagonal” gauge couplings. This means that the covariant derivative in SARAH reads where and are running over all groups and are the entries of the matrix . Gauge-kinetic mixing is included not only in the interactions with vector bosons, but also in the derivation of the -terms. Therefore, the -terms for the Abelian sector in SARAH read while the non-Abelian -terms keep the standard form equation (5). Finally, also “off-diagonal” gaugino masses are introduced. The soft-breaking part of the Lagrangian then reads SARAH takes the off-diagonal gaugino masses to be symmetric: .
2.2.3. Gauge Fixing Sector
All terms written down so far lead to a Lagrangian which is invariant under a general gauge transformation. To break this invariance one can add “gauge fixing” terms to the Lagrangian. The general form of these terms is Here, is usually a function involving partial derivatives of gauge bosons . SARAH uses gauge. This means that, for an unbroken gauge symmetry, the gauge fixing terms are For broken symmetries, the gauge fixing terms are chosen in a way where the mixing terms between vector bosons and scalars disappear from the Lagrangian. This generates usually terms of the form Here, is the Goldstone boson of the vector boson with mass . From the gauge fixing part, the interactions of ghost fields are derived by Here, assigns the operator for a BRST transformation. All steps to get the gauge fixing parts and the ghost interactions are completely done automatically by SARAH and adjusted to the gauge groups in the model.
2.2.4. Matter Sector
There can be up to 99 chiral superfields in a single SUSY model in SARAH. All superfields can come with an arbitrary number of generations and can transform as any irreducible representation with respect to the defined gauge groups. In the handling of nonfundamental fields under a symmetry, SARAH distinguishes if the corresponding symmetry gets broken or not: for unbroken symmetries it is convenient to work with fields which transform as vector under the symmetry with the appropriate length. For instance, a 6 under is taken to be That is, it carries one charge index. In contrast, nonfundamental fields under a broken gauge symmetry are represented by tensor products of the fundamental representation. For instance, a 3 under is taken to be Thus, the triplet can be given as usual as matrix.
For Abelian gauge groups not only one can define charges for superfields which are real numbers, but also variables can be used for that. All interactions are then expressed keeping these charges as free parameter.
For all chiral superfield SARAH adds the soft-breaking masses. For fields appearing in generations, these are treated as Hermitian matrices. As written above, also soft-terms mixing two scalars are included if allowed by all symmetries. Hence, the soft-breaking mass terms read, in general, Note that , label different scalar fields; generation indices are not shown. is 1, if fields and have exactly the same transformation properties under all local and global symmetries, and otherwise 0.
2.2.5. Models with Dirac Gauginos
Another feature which became popular in the last years is models with Dirac gauginos. In these models mass terms between gauginos and a fermionic component of the chiral superfield in the adjoint representation of the gauge group are present. In addition, also new -terms are introduced in these models [98]. Thus, the new terms in the Lagrangian are is the auxiliary component of the vector superfield of the group . To allow for Dirac mass terms, these models come always with an extended matter sector: to generate Dirac mass terms for all MSSM gauginos at least one singlet, one triplet under , and one octet under must be added. Furthermore, models with Dirac gauginos generate also new structures in the RGEs [236]. All of this is fully supported in SARAH.
If Dirac masses for gauginos are explicitly turned on in SARAH, it will check for all allowed combinations of vector and chiral superfields which can generate Dirac masses and which are consistent with all symmetries. For instance, in models with several gauge singlets, the bino might even get several Dirac mass terms.
2.2.6. Superpotential, Soft-Terms, and Noncanonical Interactions
The matter interactions in SUSY models are usually fixed by the superpotential and the soft-SUSY breaking terms. SARAH fully supports all renormalizable terms in the superpotential and generates the corresponding soft-breaking terms: , , and are real coefficients. All parameters are treated by default in the most general way by taking them as complex tensors of appropriate order and dimension. If identical fields are involved in the same coupling, SARAH derives also the symmetry properties for the parameter.
As discussed below, SARAH can also handle to some extent nonrenormalizable terms with four superfields in the superpotential: From the superpotential, all the -terms and interactions of matter fermions are derived. Here is the superpotential with all superfields replaced by their scalar component . is the fermionic component of that superfield.
Usually, the - and -terms and the soft-breaking terms for chiral and vector superfields fix the full scalar potential of the model. However, in some cases also noncanonical terms should be studied. These are, for instance, nonholomorphic soft-terms: Those can be added as well and they are taken into account in the calculation of the vertices and masses and as consequence also in all loop calculations. However, they are not included in the calculation of the RGEs because of the lack of generic results in the literature.
2.2.7. Symmetry Breaking and VEVs
All gauge symmetries can also be broken. This is in general done by decomposing a complex scalar into its real components and a VEV: Assigning a VEV to a scalar is not restricted to colourless and neutral particles. Also models with spontaneous colour or charge breaking (CCB) can be studied with SARAH. Also explicit CP violation in the Higgs sector is possible. There are two possibilities to define that. Either a complex phase is added or a VEV for the CP odd component is defined: Both options are possible in SARAH, even if the first one might often be preferred.
In the case of an extended gauge sector also additional gauge bosons are present. Depending on the quantum numbers of the states which get a VEV these gauge bosons might mix with the SM ones. Also this mixing is fully supported by SARAH. There is no restriction if the additional gauge bosons are ultralight (dark photons) or much heavier (, -bosons).
2.2.8. Mixing in Matter Sector
Mixing between gauge eigenstates to new mass eigenstate appears not only in the gauge but also in the matter sector. In general the mixing is induced via bilinear terms in the Lagrangian between gauge eigenstates. These bilinear terms either can be a consequence of gauge symmetry breaking or can correspond to bilinear superpotential or soft-terms. In general, four kinds of bilinear terms can show up in the matter part of the Lagrangian: Here, , , are vectors whose components are gauge eigenstates. are complex and are real scalars, , , and are Weyl spinors. The rotation of complex scalars to mass eigenstates happens via a unitary matrix which diagonalizes the matrix . For real scalars the rotation is done via a real matrix which diagonalizes : We have to distinguish for fermions if the bilinear terms are symmetric or not. In the symmetric case the gauge eigenstates are rotated to Majorana fermions. The mass matrix is then diagonalized by one unitary matrix. In the second case, two unitary matrices are needed to transform and differently. This results in Dirac fermions. Both matrices together diagonalize the mass matrix : There is no restriction in SARAH of how many states do mix. The most extreme case is the one with spontaneous charge, colour, and CP violation where all fermions, scalars, and vector bosons mix among each other. This results in a huge mass matrix which would be derived by SARAH. Phenomenological more relevant models can still have a neutralino sector mixing seven to ten states. That is done without any problem with SARAH. Information about the calculation of the mass matrices in SARAH is given in Section 3.3.
2.2.9. Superheavy Particles
Extensions of the MSSM can not only be present at the SUSY scale but also appear at much higher scales. These superheavy states have then only indirect effects on the SUSY phenomenology compared to the MSSM: they alter the RGE evolution and give a different prediction for the SUSY parameters. In addition, they can also induce higher-dimensional operators which are important. SARAH provides features to explore models with superheavy states: it is possible to change stepwise the set of RGEs which is used to run the parameters numerically with SPheno. In addition, the most important thresholds are included at the scale at which the fields of mass are integrated out. These are the corrections to the gauge couplings and gaugino masses [237]: is the Dynkin index of a superfield transforming as representation with respect to the gauge group . When evaluating the RGEs from the low scale to the high scale the contribution is positive; when running down, it is negative. Equations (36) assume that the mass splitting between the components of the chiral superfield integrated out is negligible. That is often a good approximation for very heavy states. Nevertheless, SARAH can also take into account the mass splitting among the components if necessary.
Also higher-dimensional operators can be initialized which give rise to terms like (26). However, those are only partially supported in SARAH. This means that only the RGEs are calculated for these terms and the resulting interactions between two fermions and two scalars are included in the Lagrangian. The six scalar interactions are not taken into account. This approach is, for instance, sufficient to work with the Weinberg operator necessary for neutrino masses [238, 239].
2.3. Checks of Implemented Models
After the initialization of a model SARAH provides functions to check the (self-) consistency of this model. The function CheckModel performs the following checks.
What Causes the Particle Content Gauge Anomalies? Gauge anomalies are caused by triangle diagrams with three external gauge bosons and internal fermions [240]. The corresponding conditions for all groups to be anomaly free are Again, are the generators for a fermion transforming as irreducible representation under the gauge group . The sum is taken over all chiral superfields. In the Abelian sector several conditions have to be fulfilled depending on the number of gauge groups: The mixed condition involving Abelian and non-Abelian groups is Finally, conditions involving gravity are If one if these conditions is not fulfilled a warning is printed by SARAH. If some charges were defined as variable, the conditions on these variables for anomaly cancellation are printed.
What Leads the Particle Content to the Witten Anomaly? SARAH checks that there is an even number of doublets. This is necessary for a model in order to be free of the Witten anomaly [241].
Are All Terms in the (Super)Potential in Agreement with Global and Local Symmetries? As mentioned above, SARAH does not forbid including terms in the superpotential which violate global or gauge symmetries. However, it prints a warning if this happens.
Are There Other Terms Allowed in the (Super)Potential by Global and Local Symmetries? SARAH will print a list of renormalizable terms which are allowed by all symmetries but which have not been included in the model file.
Are All Unbroken Gauge Groups Respected? SARAH checks what gauge symmetries remain unbroken and if the definitions of all rotations in the matter sector and of the Dirac spinors are consistent with that.
Are There Terms in the Lagrangian of the Mass Eigenstates Which Can Cause Additional Mixing between Fields? If in the final Lagrangian bilinear terms between different matter eigenstates are present this means that not the entire mixing of states has been taken into account. SARAH checks if those terms are present and returns a warning showing the involved fields and the nonvanishing coefficients.
Are All Mass Matrices Irreducible? If mass matrices are block diagonal, a mixing has been assumed which is actually not there. In that case SARAH will point this out.
Are the Properties of All Particles and Parameters Defined Correctly? These are formal checks about the implementation of a model. It is checked, for instance, if the number of PDGs fits the number of generations for each particle class, if LaTeX names are defined for all particles and parameters, if the positions in a Les Houches spectrum file are defined for all parameters, and so forth. Not all of these warnings have to be addressed by the user, especially if he/she is not interested in the output which would fail because of missing definitions.
3. Calculations and Output
SARAH can perform in its natural Mathematica environment many calculations for a model on the analytical level. For an exhaustive numerical analysis usually one of the dedicated interfaces to other tools is the best approach. I give in this section an overview about what SARAH calculates itself and how that information is linked to other codes.
3.1. Renormalization Group Equations
SARAH calculates the SUSY RGEs at the one- and two-loop level. In general, the -function of a parameter is parametrized by and are the coefficients at one- and two-loop level. For the gauge couplings the generic one-loop expression is rather simple and reads is the Dynkin index for the gauge group summed over all chiral superfields charged under that group and is the Casimir of the adjoint representation of the group. The two-loop expressions are more complicated and are skipped here. They are, for instance, given in [242].
The starting points for the calculation of the RGEs for the superpotential terms in SARAH are the anomalous dimensions for all superfields. These can be also parametrized by I want to stress again that are not generation indices but label the different fields. Generic formulas for the one- and two-loop coefficients and are given in [242] as well. SARAH includes the case of an anomalous dimension matrix with off-diagonal entries: that is, . That is, for instance, necessary in models with vector-like quarks where the superpotential reads is not vanishing but receives already at one-loop contributions .
From the anomalous dimensions it is straightforward to get the -functions of the superpotential terms: for a generic superpotential of the form equations (24) and (26) the coefficients are given by up to constant coefficients. In the soft-breaking sector SARAH includes also all standard terms of the form The generic expressions for ’s, ’s, ’s, and ’s up to two-loop are given again in [242] which are used by SARAH. The -function for the linear soft-term is calculated using [243]. For the quartic soft-term the approach of [244] is adopted. In this approach is defined by The -functions for can then be expressed by and : In principle, the same approach can also be used for and terms as long as no gauge singlet exists in the model. Because of this restriction, SARAH uses the more general expressions of [242].
The running of the Fayet-Iliopoulos -term receives two contributions: The first part is already fixed by the running of the gauge coupling of the Abelian group, the second part, , is known even to three loops [245, 246]. SARAH has implemented the one- and two-loop results which are rather simple: and are traces which are also used to express the -functions of the soft-scalar masses at one- and two-loop; see, for instance, [242].
Finally, the -functions for the gaugino mass parameters are where the expressions for are also given in [242, 243, 247]. has actually a rather simple form similar to the one of the gauge couplings. One finds Therefore, the running of the gaugino masses is strongly correlated with the one of the gauge couplings. Thus, for a GUT model the hierarchy of the running gaugino masses is the same as the one for the gauge couplings.
The expressions presented in the early works of [242, 243, 247] did actually not cover all possibilities and are not sufficient for any possible SUSY models which can be implemented in SARAH. Therefore, SARAH has implemented also some more results from the literature which became available in the last few years. In the case of several ’s, gauge-kinetic mixing can arise if the groups are not orthogonal. Substitution rules to translate the results of [242] to those including gauge-kinetic mixing were presented in [248] and have been implemented in SARAH2. For instance, to include gauge-kinetic mixing in the running of the gauge couplings and gaugino masses (42) and (52) can be used together with the substitutions: Here, and are matrices carrying the gauge couplings and gaugino masses of all groups; see also Section 2.2, and I introduced . The sums are running over all chiral superfields . Also for all other terms involving gauge couplings and gaugino masses appearing in the -functions similar rules are presented in [248] which are used by SARAH.
Furthermore, also the changes in the RGEs in the presence of Dirac gaugino mass terms are known today at the two-loop level; see [236]. SARAH makes use of [236] to obtain the -functions for the new mass parameters as well as to include new contribution to the RGEs of tadpole terms in presence of Dirac gauginos. The -functions of a Dirac mass terms are related to the anomalous dimension of the involved chiral superfield , whose fermionic component is , and to the running of the corresponding gauge coupling: The tadpole term receives two new contributions from Fayet-Iliopoulos terms discussed above and terms mimicking insertions: Thus, the only missing piece is which is also calculated by SARAH up to two-loop based on [236].
Finally, the set of SUSY RGEs is completed by using the results of [249, 250] to get the gauge dependence in the running of the VEVs. As a consequence, the -functions for the VEVs consist of two parts which are calculated independently by SARAH: is the anomalous dimension of the scalar which receives the VEV . The gauge dependent parts which vanish in Landau gauge are absorbed in . All details about this calculation and the generic results for are given in [249, 250].
I want to mention that SARAH provides the same accuracy also for the RGEs for a nonSUSY model by making use of the generic results of [251–254]. These results are completed by [255] to cover gauge-kinetic mixing and again by [249, 250] to include the gauge dependence of the running of VEVs also in the non-SUSY case.
Output. The RGEs calculated by SARAH are outputted in different formats: (i) they are written in the internal SARAH format in the output directory, (ii) they are included in the LaTeX output in a much more readable format, (iii) they are exported into a format which can be used together with NDSolve of Mathematica to solve the RGEs numerically within Mathematica, and (iv) they are exported into Fortran code which is used by SPheno.
3.2. Tadpole Equations
During the evaluation of a model, SARAH calculates “on the fly” all minimum conditions of the tree-level potential, the so-called tadpole equations. In the case of no CP violation, in which complex scalars are decomposed as the expressions are calculated. These are equivalent to . For models with CP violation in the Higgs sector, that is, either where complex phases appear between the real scalars or where the VEVs have an imaginary part, SARAH calculates the minimum conditions with respect to the CP even and CP odd components: The set of all tadpole equations is in this case .
Output. The tadpole equations are exported into LaTeX format as well as in Fortran code used by SPheno. This ensures that all parameter points evaluated by SPheno are at least sitting at a local minimum of the scalar potential. Moreover, the tadpole equations are included in the model files for Vevacious which is used to find all possible solutions of them with respect to the different VEVs.
3.3. Masses and Mass Matrices
SARAH uses the definition of the rotations defined in the model file to calculate the mass matrices for particles which mix. The mass matrices for scalars are calculated by where can be either real or complex: that is, the resulting corresponds to or of (33). In the mass matrices of states which include Goldstone bosons also the dependent terms are included.
The mass matrices for fermions are calculated as with for Majorana fermions, and and for Dirac fermions.
SARAH calculates for all states which are rotated to mass eigenstates the mass matrices during the evaluation of a model. In addition, it checks if there are also particles where gauge and mass eigenstates are identical. In that case, it calculates also the expressions for the masses of these states.
Output. The tree-level masses and mass matrices are also exported to LaTeX files as well as to Fortran code for SPheno. In addition, they are used in the Vevacious output to enable the calculation of the one-loop effective potential. The mass matrices can also be exported to the CalcHep model files if the user wants to calculate the masses internally with CalcHep instead of using them as input.
3.4. Vertices
Vertices are not automatically calculated during the initialization of a model like it is done for mass matrices and tadpole equations. However, the calculation can be started very easily. In general, SARAH is optimized for the extraction of three- and four-point interactions with renormalizable operators. This means that usually only the following generic interactions are taken into account in the calculations: interactions of two fermions or two ghosts with one scalar or vector bosons (FFS, FFV, GGS, and GGV), interactions of three or four scalars or vector bosons (SSS, SSSS, VVV, and VVVV), and interactions of two scalars with one or two vector bosons (SSV and SSVV) or two vector bosons with one scalar (SVV).
In this context, vertices not involving fermions are calculated by Here, are either scalars, vector bosons, or ghosts. The results are expressed by a coefficient which is a Lorentz invariant and a Lorentz factor which involves , , or . Vertices for Dirac fermions are first expressed in terms of Weyl fermions. The two vertices are then calculated separately. Taking two Dirac fermions and and distinguishing the two cases for fermion-vector and fermion-scalar couplings, the vertices are calculated and expressed by Here, the polarization operators are used.
The user can either calculate specific vertices for a particular set of external states or call functions where SARAH derives all existing interactions from the Lagrangian. The first option might be useful to check the exact structure of single vertices, while the second one is needed to get all vertices to write model files for other tools.
Output. The vertices are exported into many different formats. They are saved in the SARAH internal format and they can be written to LaTeX files. The main purpose is the export into formats which can be used with other tools. SARAH writes model files for FeynArts, WHIZARD/OMEGA, and CalcHep/CompHep as well as in the UFO format. The UFO format is supported by MadGraph, Herwigg+, and Sherpa. Thus, by the output of the vertices into these different formats, SARAH provides an implementation of a given model in a wide range of HEP tools. In addition, SARAH generates also Fortran code to implement all vertices in SPheno.
3.5. One- and Two-Loop Corrections to Tadpoles and Self-Energies
3.5.1. One-Loop Corrections
SARAH calculates the analytical expressions for the one-loop corrections to the tadpoles and the one-loop self-energies for all particles. For states which are a mixture of several gauge eigenstates, the self-energy matrices are calculated. For doing that, SARAH is working with gauge eigenstates as external particles but uses mass eigenstates in the loop. The calculations are performed in -scheme using ’t Hooft gauge. In the case of non-SUSY models SARAH switches to -scheme. This approach is a generalization of the procedure applied in [256] to the MSSM. In this context, the following results are obtained: (i)the self-energies of scalars and scalar mass matrices, (ii)the self-energies , , and for fermions and fermion mass matrices, (iii)the transversal self-energy of massive vector bosons. The approach to calculate the loop corrections is as follows: all possible generic diagrams at the one-loop level shown in Figure 1 are included in SARAH. Each generic amplitude is parametrized by Here “Symmetry” and “Colour” are real factors. The loop functions are expressed by standard Passarino-Veltman integrals and and some related functions: , , , , , and as defined in [256].
As first step to get the loop corrections, SARAH generates all possible Feynman diagrams with all field combinations possible in the considered model. The second step is to match these diagrams to the generic expressions. All calculations are done without any assumption and always the most general case is taken. For instance, the generic expression for a purely scalar contribution to the scalar self-energy reads In the case of an external charged Higgs together with down- and up-squarks in the loop the correction to the charged Higgs mass matrix becomes is the charged Higgs-sdown-sup vertex where the rotation matrix of the charged Higgs is replaced by the identity matrix to get the projection on the gauge eigenstates. One can see that all possible combinations of internal generations are included: that is, also effects like flavour mixing are completely covered. Also the entire dependence is kept.
Output. The one-loop expressions are saved in the SARAH internal Mathematica format and can be included in the LaTeX output. In addition, all self-energies and one-loop tadpoles are exported into Fortran code for SPheno. This enables SPheno to calculate the loop-corrected masses for all particles as discussed below.
3.5.2. Two-Loop Corrections
It is even possible to go beyond one-loop with SARAH and to calculate two-loop contributions to the self-energies of real scalars. There are two equivalent approaches implemented in the SPheno interface of SARAH to perform these calculations: an effective potential approach and a diagrammatic approach with vanishing external momenta. Because of the very complicated form of the results there is no output of the corresponding expressions in the Mathematica or LaTeX format but the results are just included in the Fortran code for numerical evaluation. I will discuss both calculations a bit more.
Effective Potential Calculation. The first calculation of the two-loop self-energies is based on the effective potential approach. The starting points of the calculation are the generic results for the two-loop effective potential given in [16]. These have been translated to four component notations and were implemented in SARAH. When SARAH creates the SPheno output it writes down the amplitude for all two-loop diagrams which do not vanish in the gaugeless limit. This limit means that contributions from broken gauge groups are ignored. The remaining generic diagrams which are included are shown in Figure 2. Using these diagrams includes, for instance, all two-loop contributions which are also taken into account in the MSSM. To get the values for the two-loop self-energies and two-loop tadpoles, the derivatives of the potential with respect to the VEVs are taken numerically as proposed in [257]. There are two possibilities for this derivation implemented in SARAH/SPheno: (i) a fully numerical procedure which takes the derivative of the full effective potential with respect to the VEVs and (ii) a semianalytical derivation which takes analytically the derivative of the loop functions with respect to involved masses but derives the masses and coupling numerically with respect to the VEVs. More details about both methods and the numerical differences are given in [147].
Diagrammatic Calculation. A fully diagrammatic calculation for two-loop contributions to scalar self-energies with SARAH–SPheno became available with [148]. In this setup a set of generic expressions first derived in [148] is used. All two-loop diagrams shown in Figure 3 are included in the limit . These are again the diagrams which do not vanish in general in the gaugeless limit. The results of [148] have the advantage that the expressions which are derived from the effective potential are much simpler than taking the limit in other two-loop functions available in the literature [258]. The diagrammatic method gives completely equivalent results to the effective potential calculation but is usually numerically more robust.
The Need for Both Calculations. Since both calculations are based on a completely independent implementation and use a different approach they are very useful to perform cross-checks. For the MSSM and NMSSM both calculations reproduce exactly the results obtained by widely used routines based on [259–264]. However, for nonminimal SUSY models there are no references available to compare with. Thus, the only possibility to cross check the results is within SPheno and comparing of the two different methods.
Output. The two-loop expressions for the effective potential, the tadpoles, and the self-energies are just exported to SPheno at the moment to calculate the loop-corrected mass spectrum.
3.6. Loop-Corrected Mass Spectrum
The information about the one- and two-loop corrections to the one- and two-point functions introduced in Section 3.5 can be used to calculate the loop-corrected mass spectrum. Sticking to approach of [256], the renormalized mass matrices (or masses) are related to the tree-level mass matrices (or masses) and the self-energies as follows.
3.6.1. Loop-Corrected Masses
Real Scalars. For a real scalar , the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop-corrected mass matrix squared is related to the tree-level mass matrix squared and the self-energies via The one-shell condition for the eigenvalue of the loop-corrected mass matrix reads A stable solution of (68) for each eigenvalue is usually just found via an iterative procedure. In this approach one has to be careful how is defined: this is the tree-level mass matrix where the parameters are taken at the minimum of the effective potential evaluated at the same loop level at which the self-energies are known. The physical masses are associated with the eigenvalues . In general, for each eigenvalue the rotation matrix is slightly different because of the dependence of the self-energies. The convention by SARAH and SPheno is that the rotation matrix of the lightest eigenvalue is used in all further calculations and the output.
Complex Scalars. For a complex scalar the one-loop-corrected mass matrix squared is related to the tree-level mass and the one-loop self-energy via The same on-shell condition (68) as for real scalars is used.
Vector Bosons. For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosons are given by
Majorana Fermions. The one-loop mass matrix of a Majorana fermion is related to the tree-level mass matrix and the different parts of the self-energies by Note that is used to assign tree-level values while denotes a transposition. Equation (68) can also be used for fermions by taking the eigenvalues of .
Dirac Fermions. For a Dirac fermion one obtains the one-loop-corrected mass matrix via Here, the eigenvalues of are used in (68) to get the pole masses.
3.6.2. Renormalization Procedure
I have explained so far how SPheno does calculate the one- and two-loop self-energies and how these are related to the loop-corrected masses. Now, it is time to put this in a more global picture by describing step-by-step the entire renormalization procedure that SPheno uses.(1)Everything starts with calculating the running parameters at the renormalization scale from the given input parameters. Either the parameters can be given directly at as input or they are fixed by some GUT conditions and a RGE running is performed. itself either can be a fixed value or can be dynamically chosen. It is common to choose the geometric mean of the stop masses because this usually minimizes the scale dependence of the Higgs mass prediction.(2)Not all parameters are fixed by the input but some parameters are kept free. These parameters are arranged in a way that all further calculations are done at the minimum of the potential. For this purpose the tadpole equations are solved at tree-level with respect to these free parameters.(3)As soon as all running parameters are known at the SUSY scale, they are used to calculate the tree-level mass spectrum.(4)The tree-level masses are used to calculate the self-energies of the -boson, , where is the pole mass.(5), are used to get the tree-level value of the electroweak VEV . and the running value of are used to get tree-level VEVs , . Note that in this step it is assumed that always two Higgs doublets are present in SUSY models which give mass to up- and down-quark as well as leptons and gauge bosons.(6)Now, all tree-level parameters are known and the tree-level masses and rotation matrices are recalculated using the obtained values.(7)Tree-level masses, rotation matrices, and parameter are used to get all vertices at tree-level. The vertices and masses are then plugged in the expressions for the one- and two-loop corrections to the tadpoles . The conditions to work at the minimum of the effective potential are These equations are again solved for the same set of parameters as at tree-level.(8)The self-energies for all particles are calculated at the highest available loop level as explained above. Note, these calculations involve purely tree-level parameters but not the ones obtained from (73).(9)Equations (67)–(72) are used to get the loop-corrected mass matrices for all particle. Now, the parameters coming from loop-corrected tadpoles are used to express the tree-level mass matrices. All calculations are iterated until the on-shell condition is satisfied for all masses.
3.6.3. Thresholds
So far, I have not mentioned another subtlety: in general, the running SM parameters depend on the SUSY masses. The reason is the thresholds used to match the running parameters to the measured ones. These thresholds change, when the mass spectrum changes. Therefore, the above procedure is iterated until the entire loop-corrected mass spectrum has converged. The calculation of the thresholds is also dynamically adjusted by SARAH to include all new physics contributions. The general procedure to obtain the running gauge and Yukawa at is as follows.
(1) The first step is the calculation of , via Here, and are taken as input and receive corrections from the top loops as well as form new physics (NP): The sum runs over all particles which are not present in the SM and which are either charged or coloured. The coefficient depends on the charge, respectively, colour representation, the generic type of the particle (scalar, fermion, and vector), and the degrees of freedom of the particle (real/complex boson, respectively, Majorana/Dirac fermion).
(2) The next step is the calculation of the running Weinberg angle and electroweak VEV . For that the one-loop corrections and to the -mass and -mass are needed. And an iterative procedure is applied with in the first iteration together with Here, is the Fermi constant and is defined by where are the corrections to the muon decay which are calculated at one-loop as well. The parameter is calculated also at full one-loop and the known two-loop SM corrections are added.
(3) With the obtained information, the running gauge couplings at are given by
(4) The running Yukawa couplings are also calculated in an iterative way. The starting points are the running fermion masses in obtained from the pole masses given as input: with The two-loop parts are taken from [265, 266]. The masses are matched to the eigenvalues of the loop-corrected fermion mass matrices calculated as Here, the pure QCD and QED corrections are dropped in the self-energies . Inverting this relation to get the running tree-level mass matrix leads to , , and . Since the self-energies depend also on the Yukawa matrices, this calculation has to be iterated until a stable point is reached. Optionally, also the constraint that the CKM matrix is reproduced can be included in the matching.
Output. The calculation of the loop-corrected mass spectrum and the thresholds is included in the SPheno output.
3.7. Decays and Branching Ratios
The calculation of decays widths and branching ratios can be done by using the interface between SARAH and SPheno. SPheno modules created by SARAH calculate all two-body decays for SUSY and Higgs states as well as for additional gauge bosons. In addition, the three-body decays of a fermion into three other fermions and of a scalar into another scalar and two fermions are included.
In the Higgs sector, possible decays into two SUSY particles, leptons, and massive gauge bosons are calculated at tree-level. For two quarks in the final state the dominant QCD corrections due to gluons are included [267]. The loop induced decays into two photons and gluons are fully calculated at leading-order (LO) with the dominant next-to-leading-order corrections known from the MSSM. For the LO contributions all charged and coloured states in the given model are included in the loop: that is, new contributions rising in a model beyond the MSSM are fully covered at one-loop. In addition, in the Higgs decays also final states with off-shell gauge bosons (, ) are included. The only missing piece is the channel. The corresponding loops are not yet derived by SARAH and the partial width is set to zero.
In contrast to other spectrum generators, SPheno modules by SARAH perform a RGE running to the mass scale of the decaying particle. This should give a more accurate prediction for the decay width and branching ratios. However, the user can also turn off this running and use always the parameters as calculated at in all decays as this is done by other codes.
Output. All necessary routines to calculate the two- and three-body decays are included by default in the Fortran output for SPheno.
3.8. Higgs Coupling Ratios
With the discovery of the Higgs boson at the LHC and the precise measurements of its mass and couplings to other particles, a new era of high energy physics has started. Today, many SUSY models not only have been confronted with the exclusion limits from direct searches, but also have to reproduce the Higgs properties correctly. The agreement with respect to the mass can be easily read off a spectrum file. For the rates this is usually not so easy. One can parametrize how “SM-like” the couplings of a particular scalar are by considering the ratio Here, is the calculated coupling between a scalar and two SM particles and for a particular parameter point in a particular model. This coupling is normalized to the SM expectation for the same interaction. Nowadays, all are constrained to be rather close to 1 if should be associated with the SM Higgs. SARAH uses the information which is already available from the calculation of the decays to obtain also values for . Of course, also here the channel is missing and is therefore put always to 0.
Output. All necessary routines to calculate the Higgs coupling ratios are included by default in the Fortran output for SPheno.
3.9. Flavour and Precision Observables
Constraints for new physics scenarios come not only from direct searches and the Higgs mass observation but also from the measurement of processes which happen only very rarely in the SM and/or which are known to have a very high accuracy. These are in particular flavour violation observables. When using the SPheno output of SARAH, routines for the calculation of many quark and lepton flavour (QFV and LFV) observables are generated.
Lepton Flavour Violation. The radiative decays of a lepton into another lepton and a photon and the purely leptonic three-body decays of leptons are included . Also flavour violating decays of the -boson are tested by SARAH/SPheno. Moreover, there are also semihadronic observables in the output: conversion in nuclei , where the considered nuclei are ( = Al, Ti, Sr, Sb, Au, and Pb), as well as decays of ’s into pseudoscalars, with .
Quark Flavour Violation. The radiative -decay and a set of -decays stemming from four-fermion operators are calculated: , , , and . Also Koan decays are considered , , and as well as CP observables (, , and ). Finally, some decays which take place already at tree-level are included: namely, and .
The approach in SARAH to generate the routines to calculate all these observables is similar to the approach used for loop calculations needed for radiative corrections to the masses: generic formulas for all possible Feynman diagrams which contribute to the Wilson coefficients3 of widely used dimension 5 and 6 operators are implemented in SARAH. I show in Figures 4 and 5 only the topologies which are considered because these are already many. For each topology the amplitudes with all possible generic insertions are included. Today, these implementations are mainly based on the FlavorKit functionality discussed below. SARAH generates all possible one-loop diagrams and uses the generic expression to get their amplitudes. In this context, not only all possible particles in the loop are included but also all different propagators for penguin diagrams are considered. Thus, not only photonic penguins which are often considered to be dominant in many processes are taken into account, but also all Higgs and—if existing— penguins are generated. After the calculation of the Wilson coefficients, these are then combined to calculate the observables. This can easily be done by using expressions from the literature which are usually model independent.
With the development of the FlavorKit [146] interface all information to calculate flavour observables is no longer hard-coded in SARAH but is provided by external files. This makes it possible for the user to extent the list of flavour observables when necessary. The FlavorKit is an automatization of the procedure presented in [268] to implement in SARAH and SPheno. Users interested in the internal calculation might take a look at these two references.
Also some other observables are calculated by the combination SARAH–SPheno which are measured with high precision: electric dipole moments (EDMs) and anomalous magnetic moments of leptons () and
Output. When generating SPheno code with SARAH, the above-listed flavour and precision observables are included in the Fortran code. In addition, SARAH writes also LaTeX files with all contribution to the Wilson coefficients from any possible diagram.
3.10. Fine-Tuning
A measure for the electroweak fine-tuning was introduced in [269, 270] is a set of independent parameters. gives an estimate of the accuracy to which the parameter must be tuned to get the correct electroweak breaking scale [271]. Using this definition the fine-tuning of a given model depends on the choice of what parameters are considered as fundamental and at which scale they are defined. The approach by SARAH is that it takes by default the scale at which the SUSY breaking parameters are set. This corresponds to models where SUSY is broken by gravity to the scale of grand unification (GUT scale), while for models with gauge mediated SUSY breaking (GMSB) the messenger scale would be used. For simplicity, I call both . The choice of the set of parameters is made by user. Usually, one uses in scenarios motivated by supergravity the universal scalar and gaugino masses (, ) as well as the parameters relating to the superpotential terms and the corresponding soft-breaking terms (, ) to calculate the fine-tuning. However, since also these parameters are related in specific models for SUSY breaking, it might be necessary to consider even more fundamental parameters like the gravitino mass . In addition, also the fine-tuning with respect to the superpotential parameters themselves as well as to the strong coupling might be included because they can even supersede the fine-tuning in the soft-SUSY breaking sector.
To calculate the fine-tuning in practice, an iteration of the RGEs between and happens using the full two-loop RGEs. In each iteration one of the fundamental parameters is slightly varied and the running parameters at are calculated. These parameters are used to solve the tadpole equations numerically with respect to all VEVs and to recalculate the -boson mass. To give an even more accurate estimate, also one-loop corrections to the mass stemming from can be included.
Output. A fine-tuning calculation is optionally included in the Fortran output for SPheno.
3.11. Summary
SARAH derives a lot of information about a given model. This information can be used in different interfaces to study a model in all detail. In general, one can get (i) LaTeX files, (ii) a spectrum generator based on SPheno, and (iii) model files for different HEP tools.
LaTeX. All analytical information derived about a model can be exported to LaTeX files. These files provide in a human readable format the following information: (i) list of all superfields as well as component fields for all eigenstates; (ii) the superpotential and important parts of the Lagrangian like soft-breaking and gauge fixing terms added by SARAH; (iii) all mass matrices and tadpole equations; (iv) the full two-loop RGEs; (v) analytical expressions for the one-loop self-energies and tadpoles; (vi) all interactions and the corresponding Feynman diagrams; and (vii) details about the implementation in SARAH. Separated files are also generated for the flavour observables showing all contributing diagrams with their amplitudes.
Spectrum Generator. SARAH 3 has been the first “spectrum-generator-generator”: using the derived information about the mass matrices, tadpole equations, vertices, loop corrections, and RGEs for the given model SARAH writes Fortran source code for SPheno. Based on this code the user gets a fully functional spectrum generator for a new model. The features of a spectrum generator created in this way are as follows.(1)RGE Running. The full two-loop RGEs are included.(2)Precise Mass Spectrum Calculation. SPheno modules created by SARAH include the one-loop corrections to all SUSY particles. For Higgs states the full one-loop and in addition dominant two-loop corrections are included.(3)Calculation of Decays. SPheno calculates all two-body decays for SUSY and Higgs states. In addition, the three-body decays of a fermion into three other fermions and the three-body decays of scalar into another scalar and a pair of fermions are included.(4)FlavorKit Interface. SPheno modules calculate out of the box many flavour observables for a given model.(5)Calculation of Precision Observables. SPheno does also calculate , electromagnetic dipole moments, and anomalous magnetic moments of leptons.(6)Output for HiggsBounds and HiggsSignals. SPheno generates all necessary files with the Higgs properties (masses, widths, and couplings to SM states) which are needed to run HiggsBounds and HiggsSignals.(7)Estimate of the Fine-Tuning. SPheno modules can calculate the electroweak fine-tuning with respect to a set of defined parameters.
Model Files. In particular the vertex lists can be exported into several formats to create model files for FeynArts/FormCalc, CalcHep/CompHep, and WHIZARD/O’Mega as well as for MadGraph, Herwig++, or Sherpa based on the UFO format. Also model files for Vevacious can be generated which include the tree-level potential as well as the mass matrices to generate the one-loop effective potential.
4. Example—Part I: The B-L-SSM and Its Implementation in SARAH
I will discuss in this section and the subsequent ones the implementation of the B-L-SSM in SARAH and how all phenomenological aspects of this model can be studied. The B-L-SSM is for sure not the simplest extension of the MSSM, but it provides many interesting features. There are some subtleties in the implementation which will not show up in singlet extensions, for instance. I hope that the examples presented in the following sections show how even such a complicated model can be studied with a very high precision without too much effort. Applying the same methodology to other SUSY or non-SUSY models should be straightforward. However, I cannot discuss here all topics which might be interesting and useful for some models. In particular, I will not show how models with threshold scales are implemented. SARAH supports thresholds where heavy particles get integrated out and where the gauge symmetries do either change or not. Users interested in that topic might take a look at the manual as well as the implementations of seesaw models4 or left-right symmetric models5. A brief summary of the general approach to include thresholds is also given in Appendix A.5.
In the first part of this section I will give a short introduction to the B-L-SSM, before I come to the tools which are going to be used. The implementation of the B-L-SSM in SARAH is discussed in Section 4.4 and how it is evaluated is shown in Section 4.5. The next section explains what can be done with the model using just Mathematica. It is shown how the LaTeX output is generated, how mass matrices and tadpole equations can be handled with Mathematica, and how RGEs are calculated and solved. Section 6 is about the interface between SARAH and SPheno. The mass spectrum calculation is explained, and what else can be obtained with SPheno is discussed: decays and branching ratios, flavour and precision observables, and the fine-tuning. Afterwards, we include more tools in our study in Section 7: HiggsBounds/HiggsSignals to test Higgs constraints, Vevacious to check the vacuum stability, MicrOmegas to calculate the dark matter relic density and direct detections rates, WHIZARD/O’Mega to produce monojet events, and MadGraph to make a simple dilepton analysis. Section 8 is about parameter scans and how the different tools can be interfaced.
4.1. The Model
Supersymmetric models with an additional gauge symmetry at the TeV scale have recently received considerable attention: they can explain neutrino data, they might help to understand the origin of -parity and its possible spontaneous violation [272–276], and they have a rich phenomenology [277]. In the -parity conserving case, these models come with a new Higgs which mixes with the MSSM one [199], they provide new dark matter candidates [278], and they can have an impact on the Higgs decays [279]. In both cases of broken and unbroken -parity these models have interesting consequences for LHC searches [280–283].
4.1.1. Particle Content and Superpotential
We study the minimal supersymmetric model where the SM gauge sector is extended by a and where -parity is not broken by sneutrino VEVs. This model is called the B-L-SSM. In this model the matter sector of the MSSM is extended by three generations of right-handed neutrino superfields and two fields which are responsible for breaking, and . These fields carry lepton number 2 and are called “bileptons.” The chiral superfields and their quantum numbers are summarized in Table 1. The superpotential of the B-L-SSM is given byHere, and are generation indices and we suppressed all colour and isospin indices. The first line is identical to the MSSM, and all new terms are collected in the second line of (86). After breaking a Majorana mass term for the right-handed neutrinos is generated. This term causes a mass splitting between the CP even and odd parts of the complex sneutrinos.
The additional soft-breaking terms compared to the MSSM are The electroweak and symmetry are broken to by the following VEVs of Higgs states and bileptons: In analogy to the MSSM definition , we call the ratio of the two bilepton VEVs . As mentioned above, the Majorana mass term causes a mass splitting between the CP even and odd parts of the right sneutrinos. This makes it necessary to defineHere, is the generation index of the sneutrinos.
4.1.2. Gauge-Kinetic Mixing
The particle content of the B-L-SSM gives rise to gauge-kinetic mixing because the matrix is not diagonal. The indices and run over all groups and over all superfields. For the particle content of Table 1 we find contains the GUT normalization of the two Abelian gauge groups. We will take for and for . From (91) we see that even if gauge-kinetic mixing vanishes at one scale, it is induced again by RGE running and must be included therefore.
However, as long as the two Abelian gauge groups are unbroken, we can make a change of basis. This freedom is used to go to a basis where electroweak precision data is respected in a simple way: by choosing a triangle form of the gauge coupling matrix, the bilepton contributions to the mass vanish: The gauge couplings are related by [284]:
4.1.3. Mass Eigenstates
After electroweak and breaking and also because of gauge-kinetic mixing there is a mixing between the neutral SUSY particles from the MSSM and the new sector.
Neutral Gauge Bosons. In the gauge sector, the neutral gauge bosons , , and mix. This gives three mass eigenstates , , and which are related by a rotation matrix which can be expressed by two angles and :The entire mixing between the and the SM gauge boson comes from which can be approximated by [285] with and .
Neutral Higgs Bosons. In the Higgs sector, the CP even and odd parts of the doublets mix with the corresponding CP eigenstates of the bileptons. This leads to four physical scalar Higgs particles: and four pseudoscalars. Two pseudoscalars are physical and the other two are Goldstone bosons of the and :
Neutralinos. There are seven neutralinos in the model which are an admixture of the three gauginos, the two Higgsinos, and the two bileptinos:
Neutrinos and Sneutrinos. There are six Majorana neutrinos which are a superposition of the left and right neutrino gauge eigenstates: Similarly, the sneutrinos are admixtures of left and right sneutrino gauge eigenstates. Because of the mass splitting due to the Majorana mass term the CP even and odd parts mix separately, and there are 12 real mass eigenstates:
4.2. Constrained Model
The B-L-SSM has 55 additional parameters, including all phases, compared to the general MSSM. Therefore, an organizing principle to relate these parameters to reduce the number of free parameters is often helpful. We choose a CMSSM-like setup inspired by minimal supergravity. We set the boundary conditions at the GUT scale which is taken to be We assume that at the GUT gauge-kinetic mixing is absent but gets induced below that scale. This gives as additional conditions: Thus, at the GUT scale the relations and hold.
In minimal supergravity all scalar and gaugino soft-masses unify at the GUT scale: that is, there are only two free parameters to fix all soft-masses: and . The boundary conditions at the GUT scale are with the identity matrix 1. We assume also that no gauge-kinetic mixing is present in the gaugino sector at this scale: that is, For the trilinear soft-breaking terms we use in analogy to the CMSSM the relation with a free parameter . The remaining soft-terms and are not taken to be input, but the tadpole equations are, if not stated otherwise, solved with respect to , and . The advantage of this choice is that these parameters do not enter the RGE evaluation of all other terms and they can be considered independently.
To fix the gauge part of the sector, we need two more input parameters. These are the mass of the and the ratio of the bilepton VEVs. Finally, there are two more Yukawa-like matrices, and . Thus, the full set of input parameter is