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The Implementation of the Minimal Supersymmetric Standard Model in FeynArts and FormCalc

We describe the implementation of the MSSM in the diagram generator FeynArts and the calculational tool FormCalc. This extension allows to perform loop calculations of MSSM processes almost fully automatically. The actual implementation has two aspects: The MSSM Feynman rules are specified in a new model file for FeynArts. The computation of the parameters in the MSSM Lagrangian from the input parameters is realized as a Fortran subroutine in the framework of FormCalc. The model file does not depend on the latter, however, and can be used even if one does not want to continue the calculation with FormCalc. The Feynman rules have been entered in a very generic way to allow e.g. scenarios with complex parameters, and have been tested extensively by reproducing known results for several non-trivial scattering processes.


Introduction
One of the main problems of Feynman-diagrammatic computations is the enormous growth of the number of Feynman diagrams, not only with the loop order, but also with the number of particles in a model. While many precision calculations in the Standard Model (SM) could still be performed by hand exactly, the same is very difficult in models like the Minimal Supersymmetric Standard Model (MSSM). Yet it is highly desirable to perform unabridged calculations in the MSSM, too, since also the MSSM allows to make precise predictions in terms of a set of input parameters.
With the availability of powerful software packages, the basic problem of bookkeeping and calculation of the diagrams has been solved for many common cases. Still, it is not entirely trivial to code a model of the complexity of the MSSM in such a system, since this has to be done in a reasonably general way (i.e. not only for special cases of the parameters) and many checks have to be performed to test all sectors of the model.
The present paper documents the implementation of the MSSM in the FeynArts [1] and Form-Calc [2] packages. Other programs for which an MSSM model file exists are GRACE [3,4] and CompHEP [5,6]. Conceptually, the FeynArts-FormCalc system works in three stages, as sketched in the following diagram.

Diagram generation
FeynArts Mathematica

The MSSM model file
The model file is the source of all physics information in FeynArts. It declares the properties of the fields, their propagators, and their couplings. In the model file the parameters of the Lagrangian are used, not a restricted set of input parameters. There are two versions of the MSSM model file in FeynArts, both of which follow the conventions of [9,10,11]. The file MSSMQCD.mod defines the complete (electroweak and strong) MSSM, whereas MSSM.mod contains only the electroweak subset, defined as everything except the gluon, its ghost, and the gluino. The four-sfermion couplings appear in MSSM.mod although they have both electroweak and strong parts. Counter-terms are not yet entered in the model files. Table 1 gives the names for the fields and corresponding masses defined in MSSM.mod and MSSMQCD.mod. The symbols used for the MSSM parameters are specified in Table 2. The complete list of couplings is too long to be included here, but is contained in the FeynArts distribution as a PostScript file.
The MSSM model files further declare a number of restrictions with which certain groups of particles can be excluded in the diagram generation. These restrictions are listed in Table 3. 3 Algebraic simplification

Algorithms
The algebraic simplification has to address two problems that arise in one-loop calculations in supersymmetric theories in general and in the MSSM in particular. The first is the conceptual where the following indices are used:   problem of implementing a supersymmetry-preserving regularization scheme, the second is the technical problem of dealing with more diagrams and larger expressions. The default regularization scheme employed by FormCalc is dimensional regularization. Unfortunately, this scheme is known to break supersymmetry [12]. FormCalc has therefore been equipped with an alternative scheme suited for calculations in supersymmetric theories. It is the constrained differential renormalization scheme (CDR) [13] which is equivalent to regularization by dimensional reduction at the one-loop level [2]. The schemes are chosen with the Dimension option of the CalcFeynAmp function: Dimension -> D selects dimensional regularization (the default), Dimension -> 4 switches to CDR.
The second problem is a technical one. In the SM, both the number of diagrams and the comparatively simple coupling structure allow to perform calculations with "brute force." To wit, the diagrams are generated with all indices explicitly written out, and then calculated.
The situation is less favourable in the MSSM, however: not only is the number of diagrams considerably higher in most cases, but also the coupling structure is more involved. This is because particles like sfermions, or gauginos and higgsinos in general mix to form mass eigenstates and hence their couplings contain plenty of mixing matrices and can become rather lengthy. This means that the algebraic simplification has to be performed more carefully in order to maintain a decent performance and keep the size of the results as small as possible.
The way in which FormCalc proceeds is already suggested by the level structure of the Feyn-Arts amplitudes. The lowest (generic) level completely determines the kinematical structure of a diagram, so FormCalc first performs the time-consuming simplifications involving kinematical quantities, like the tensor reduction, on the far fewer generic diagrams only. For each diagram at the next higher (classes) level, it then substitutes the generic coupling constants and masses by their actual values. The diagrams are mostly specified by then, except that index summations, e.g. over fermion generations, are not yet carried out. These index summations are performed only in the Fortran code, which means that FormCalc has to keep track of all indices and write out the corresponding loop instructions in Fortran. For a model like the MSSM, the savings incurred with this method of simplification may easily amount to an order of magnitude in both CPU time and size of the Fortran code.

Numerical evaluation
In the FormCalc framework, the Mathematica expressions resulting from the algebraic simplification are translated to Fortran code for numerical evaluation. The generated Fortran code of course has to be provided with the proper numerical values for the parameters appearing in the model, i.e. the variables in Table 2. This is solved by a subroutine which is called at the beginning of the calculation to initialize all model parameters.
Technically, the input parameters are specified in common blocks defined in model.h. Some less commonly changed inputs, notably some breaking parameters in the sfermion sector, are realized as preprocessor variables which take default values if not defined by the user. Preprocessor variables are also used for switches, such as whether squark mixing should be turned on or off.

Parameters of the MSSM
Supersymmetry (SUSY) completely determines the supersymmetric part of the MSSM Lagrangian once the SM parameters are known. The complete MSSM, however, with softly broken SUSY in its general form, necessarily introduces a large number of masses, phases, and mixing angles to parametrize the SUSY breaking. At a final count, 105 of these degrees of freedom cannot be absorbed in some other quantity or rotated away [14]. In order to get a handle on so many parameters, several assumptions have been made in mssm_ini.F to arrive at a moderate number of input parameters while retaining reasonable generality.
The first assumption is that the SUSY-GUT relation holds, which relates the U(1), SU(2), and SU(3) gaugino mass parameters M 1 , M 2 , and M 3 according to where √ s is the CMS energy, c W = M W /M Z , and s W = 1 − c 2 W . The running couplings α and α s are MS values; we use the CERNlib function ALPHAS2 [15] for α s (s) and with 1/α(M 2 Z ) = 127.934 [16]. Secondly, we assume the SUSY-breaking parameters in the sfermion sector to be flavour-blind and distinguish only two A-parameters for different isospin values, hence the sfermion sector is governed by seven scalar parameters: where in addition the various MX get the default value M SUSY , a common SUSY-breaking mass, i.e.
The inputs to mssm_ini.F are summarized in Table 4. All other variables in the MSSM Lagrangian (see Table 2) are determined once this reduced set of MSSM input parameters and the SM inputs are specified and are calculated by mssm_ini.F. This is the reason why mssm_ini.F is much more involved than its companion file for the SM, sm_ini.F. The calculation of the parameters can be directed by several switches, which are also listed in Table 4. When the particle masses are calculated from these input parameters, mssm_ini.F checks whether they are consistent with the current exclusion limits and automatically omits alreadyexcluded points with a warning. The following bounds are used: mt 80 GeV [17] , mb 70 GeV [17] , mq =b,t 150 GeV [17] , ml 70 GeV [18] , M h 91 GeV for real input parameters [19] , 85 GeV for complex input parameters [20] , mχ 90 GeV [21] , mχ0 30 GeV [21] , mg 175 GeV [22] .
The detailed formulas built into mssm_ini.F for computing the various parameters are presented in the following sections, organized according to the various sectors of the MSSM.

The Higgs sector
The neutral Higgs sector is fixed by choosing a value for tan β = v 2 /v 1 (the ratio of the vacuum expectation values of the two Higgs doublets) and for the mass M A of the CP-odd neutral Higgs boson A 0 . For the CP-even Higgs masses, which receive sizable radiative corrections, we use the approximation formula of [23] which agrees with the full two-loop calculation [24]

COMPLEX_PARAMETERS
Uses a simpler (one-loop) approximation for the Higgs masses which is valid for all parameters. This switch must be set if complex input parameters are used because otherwise the more precise (two-loop) approximation for the Higgs masses is taken, which is valid only for real parameters.

SM_ONLY
Calculates M h as usual, but then reverse-engineers the mixing in the Higgs sector (α and β) such that the MSSM Higgs sector looks like a SM Higgs sector (see [11], p. 356), with the light CP-even Higgs boson h figuring as the SM Higgs boson.

NO_EXCLUSION_LIMITS
Ignores the experimental bounds, i.e. does not exclude points in parameter space if the bounds in Eq. (6) are violated. Table 4: The inputs and switches for mssm_ini.F. Note: A preprocessor variable is defined with "#define var value" and a switch is set with "#define switch". The #define must stand at the beginning of the line.

The neutral CP-even Higgs bosons h and H
In this section we discuss the case of real MSSM parameters, for which a more sophisticated computation of the Higgs masses is implemented. The case of complex parameters is treated in Sect. 4.2.2. The gauge eigenstates φ 1 and φ 2 of the neutral CP-even Higgs bosons mix via the mass matrix Diagonalizing this matrix yields the Higgs masses The mixing angle α of the CP-even Higgs doublet is hence The presence of renormalized self-energiesΣ indicates that the mass matrix contains significant radiative corrections which must be taken into account to make quantitatively correct predictions. The self-energies of course receive contributions from all sectors of the MSSM, but not all are numerically of equal relevance. We take into account only the following terms: •Σ (1,rest) : the one-loop leading-log contributions from all other sectors [27].
In the following, we give the explicit expressions for these contributions as implemented in mssm_ini.F.
• The one-loop t/t-contributions at zero momentum transfer up to m 4 t : with • The dominant two-loop t/t-contributions of O(αα s ) at zero momentum transfer and the leading two-loop Yukawa correction of O(α 2 ): wherẽ and U t is the stop mixing matrix defined in Eq. (35). We use the MS top mass instead of the pole mass to include also the leading t/t-contributions beyond O(αα s ).
• The one-loop leading-log contributions from all other sectors:

The case of complex parameters
The sophisticated calculation of the Higgs masses as in the preceding section is valid only for real parameters. When complex parameters are used, the preprocessor variable COMPLEX_PARAMETERS must be defined (see Table 4), which makes mssm_ini.F use a simple (one-loop) approximation for the Higgs masses that is valid also for complex parameters. For the Higgs masses the dominant one-loop contributions are the m 4 t terms [25]. Neglecting the splitting of thet masses, the φ 2 self-energy of Eq. (12) becomeŝ with the running top mass m t as defined in Eq. (19). All other radiative corrections in the Higgs mass matrix are set to zero.

The charged Higgs bosons
There are only small radiative corrections for the charged Higgs masses. We use the following relation, valid for M A ∼ O(M Z ) [27], up to M A -values of 250 GeV: with N c = N g = 3 and the running top mass m t as defined in Eq. (19). Above 250 GeV, this formula reduces to M 2

The chargino sector
The charged gauginos and higgsinos mix via the mass matrix [9, 10]

Supersymmetric contributions to ∆ρ
In addition to the experimental bounds on the squark masses, mssm_ini.F also checks whether the MSSM contributions to the ρ-parameter are consistent with the current exclusion limits (see Eq. (6)). This bound becomes relevant mainly when parameters are chosen to achieve a large mass splitting betweent 1 andt 2 orb 1 andb 2 .
We have implemented the calculation of the MSSM contributions to ∆ρ according to [29]. The sfermion contributions are significant only when the masses of the isospin partners are very different, and since this is governed by the quark masses (see Eq. (36)), only corrections from t/b-loops have been included. Contributions from gluino exchange, which are very lengthy and vanish for large mg, have similarly been neglected.
These contributions to ∆ρ are given by where the U f are the sfermion mixing matrices defined in Eq. (35) and the function includes the one-and two-loop contributions The two-loop contribution is of the order of 10 to 15% of the one-loop result.

Tests
The model files MSSM.mod and MSSMQCD.mod have been checked against results known from the literature for a variety of scattering processes. In cases where the Fortran programs of the original authors were available, we found perfect agreement for all differential cross-sections. In the other cases we could reproduce qualitatively the figures. The list of processes we checked together with the agreement we achieved is given in Table 5.