Predicting the Sparticle Spectrum from GUTs via SUSY Threshold Corrections with SusyTC

Grand Unified Theories (GUTs) can feature predictions for the ratios of quark and lepton Yukawa couplings at high energy, which can be tested with the increasingly precise results for the fermion masses, given at low energies. To perform such tests, the renormalization group (RG) running has to be performed with sufficient accuracy. In supersymmetric (SUSY) theories, the one-loop threshold corrections (TC) are of particular importance and, since they affect the quark-lepton mass relations, link a given GUT flavour model to the sparticle spectrum. To accurately study such predictions, we extend and generalize various formulas in the literature which are needed for a precision analysis of SUSY flavour GUT models. We introduce the new software tool SusyTC, a major extension to the Mathematica package REAP, where these formulas are implemented. SusyTC extends the functionality of REAP by a full inclusion of the (complex) MSSM SUSY sector and a careful calculation of the one-loop SUSY threshold corrections for the full down-type quark, up-type quark and charged lepton Yukawa coupling matrices in the electroweak-unbroken phase. Among other useful features, SusyTC calculates the one-loop corrected pole mass of the charged (or the CP-odd) Higgs boson as well as provides output in SLHA conventions, i.e. the necessary input for external software, e.g. for performing a two-loop Higgs mass calculation. We apply SusyTC to study the predictions for the parameters of the CMSSM (mSUGRA) SUSY scenario from the set of GUT scale Yukawa relations $y_e / y_d = - 1/2$, $y_\mu / y_s = 6$, and $y_\tau / y_b = - 3/2$, which has been proposed recently in the context of SUSY GUT flavour models.


Introduction
Supersymmetric (SUSY) Grand Unified Theory (GUT) models of flavour are promising candidates towards solving the open questions of the Standard Model (SM) of Particle Physics. They embrace the unification of the SM gauge couplings, a dark matter candidate, a solution to the gauge hierarchy problem and an explanation of the hierarchies of masses and mixing angles in the flavour sector. Whether a flavour GUT model can successfully explain the observations in the flavour sector, depends on the renormalization group (RG) evolution of the Yukawa matrices from the GUT scale to lower energies. Furthermore, it is known that tan β enhanced supersymmetric threshold corrections (see e.g. [2]) are essential in the investigation of mass (or Yukawa coupling) ratios predicted at the GUT scale. Interesting well-known GUT predictions in this context are b − τ and b − τ − t unification (for early work see e.g. [3]) and y µ = 3y s [4]. 1 Other promising quark-lepton mass relations at the GUT scale have been discussed in [5,6], e.g. y τ = ± 3 2 y b , y µ = 6y s or y µ = 9 2 y s . Various aspects regarding the impact of such GUT relations for phenomenology have been studied in the literature, see e.g. [7] for recent works.
With the discovery of the Higgs boson at the LHC [8] and the possible discovery of sparticles in the near future, the question whether a set of SUSY soft-breaking parameters can be in agreement with both, specific SUSY threshold corrections as required for realizing the flavour structure of a GUT model, and constraints from the Higgs boson mass and results on the sparticle spectrum, gains importance. To accurately study this question, we introduce the new software tool SusyTC, a major extension to the Mathematica package REAP [1].
REAP, which is designed to run the SM and neutrino parameters in seesaw scenarios with a proper treatment of the right-handed neutrino thresholds, is a convenient tool for top-down analyses of flavour (GUT) models, with the advantage of a user-friendly Wolfram Mathematica front-end. However, the SUSY sector is not included. In order to take supersymmetric threshold corrections into account in the analyses, for example of the A 4 flavour GUT models in [9,10], the following procedure was undertaken: First REAP was used to run the Yukawa matrices in the MSSM from the GUT-scale to a user-defined "SUSY"-scale. At this scale, the SUSY threshold corrections were incorporated as mere model parameters, in a simplified treatment assuming, e.g., degenerate first and second generation sparticle masses (cf. [11]), without specializing any details on the SUSY sector. Finally, the Yukawa matrices, corrected by these tan β-enhanced thresholds, were taken as input for a second run of REAP, evolving the parameters in the SM from the "SUSY"scale to the top-mass scale. 2 Although this procedure is quite SUSY model-independent, it only allows to study the constraints on the SUSY sector indirectly (i.e. via the introduced additional parameters and with simplifying assumptions), and it is unclear whether an explicit SUSY scenario with these assumptions and requirements can be realised.
The aim of this work is to make full use of the SUSY threshold corrections to gain information on the SUSY model parameters from GUTs. Towards this goal we extend and generalize various formulas in the literature which are needed for a precision analysis of SUSY flavour GUT models and implement them in SusyTC. For example, SusyTC includes the full (CP violating) MSSM SUSY sector, sparticle spectrum calculation, a careful calculation of the one-loop SUSY threshold corrections for the full down-type quark, up-type quark and charged lepton Yukawa coupling matrices in the electroweak-unbroken phase, and automatically performs the matching of the MSSM to the SM, including DR to MS conversion. Among other useful features, SusyTC calculates the one-loop corrected pole mass of the charged (or the CP-odd) Higgs boson as well as provides output in SLHA conventions, i.e. the necessary input for external software, e.g. for performing a two-loop Higgs mass calculation.
SusyTC is specifically developed to perform top-down analyses of SUSY flavour GUT models. This is a major difference to other well-known SUSY spectrum generators (e.g. [12,13,14,15], see e.g. [16] for a comparison), which run experimental constraints from low energies to high energies, apply GUT-scale boundary conditions, run back to low energies and repeat this procedure iteratively. SusyTC instead starts directly from the GUT-scale, allowing the user to define general (complex) Yukawa, trilinear, and soft-breaking matrices, as well as non-universal gaugino masses, as input. These parameters are then run to low energies, thereby enabling an investigation whether the GUT-scale Yukawa matrix structures of a given SUSY flavour GUT model are in agreement with experimental data.
We apply SusyTC to study the predictions for the parameters of the Constrained MSSM (mSUGRA) SUSY scenario from the GUT-scale Yukawa relations ye y d = − 1 2 , yµ ys = 6, and yτ y b = − 3 2 , which have been proposed recently in the context of SUSY GUT flavour models. With a Markov Chain Monte Carlo analysis we find a "best-fit" benchmark point as well as the 1σ ranges for the sparticle masses and the correlations between the SUSY parameters. Without applying any constraints from LHC SUSY searches or dark matter, we find that the considered GUT scenario predicts a CMSSM sparticle spectrum above past LHC sensitivities, but within reach of the current LHC run or a future high-luminosity upgrade. Furthermore, the scenario generically features a bino-like neutralino LSP and a stop NLSP with a mass that can be close to the present bounds. This paper is organized as follows: In section 2 we review GUT predictions for Yukawa coupling ratios. In section 3, we describe the numerical procedure in SusyTC and present the main used formulas. We give a short introduction to the new features SusyTC adds to REAP in section 4. In section 5 we study the predictions for the parameters of the CMSSM SUSY scenario from the above mentioned GUT-scale Yukawa relations with SusyTC. In the appendices we present other relevant formulas and a detailed documentation of SusyTC.

Predictions for Yukawa Coupling ratios from GUTs
GUTs not only contain a unification of the SM forces, they also unify fermions into joint representations. After the GUT gauge group is broken to the SM gauge group, this can lead to predictions for the ratios of down-type quark and charged lepton Yukawa couplings which result from group theoretical Clebsch Gordan (CG) factors. To confront such predictions of GUT models with the experimental data, the RG evolution of the Yukawa couplings from high to low energies has to be performed, including (SUSY) threshold corrections.
In SU (5) GUTs, for example, the right-handed down-type quarks and the lepton doublets are unified in five-dimensional representations of SU (5) and the quark doublets plus right-handed up-type quarks and right-handed charged leptons are unified in a tendimensional SU (5) representation. The Higgs doublets are supplemented by SU (3) c triplets and embedded into five-dimensional representations of SU (5). Using only these fields and a single renormalizable operator to generate the Yukawa couplings for the down-type quarks and charged leptons, so-called minimal SU (5) predicts Y e = Y T d for the Yukawa matrices at the GUT scale. To correct this experimentally challenged scenario, SU (5) GUT flavour models often introduce a 45-dimensional Higgs representation, which can lead to the Georgi-Jarlskog relations y µ = −3y s and y e = 1 3 y d [4]. It was pointed out in [5] (see also [6]) that other promising Yukawa coupling GUT ratios can emerge in SU (5), e.g. y τ = ± 3 2 y b , y µ = 6y s or y µ = 9 2 y s , and y e = − 1 2 y d from higher dimensional GUT operators containing for instance a GUT breaking 24-dimensional Higgs representation. A convenient test whether GUT predictions for the first two families can be consistent with the experimental data is provided by the -RG invariant and SUSY threshold correction invariant 3 -double ratio [11] y µ y s While the Georgi-Jarlskog relations [4] imply a double ratio of 9, disfavoured by more than 2σ, other combinations of CG factors [5,6], e.g. y µ = 6y s and y e = − 1 2 y d can be in better agreement (here with a double ratio of 12, within the 1σ experimental range).
The combination of GUT-scale Yukawa relations y e = − 1 2 y d , y µ = 6y s , and y τ = − 3 2 y b (as direct result of CG factors, cf. Table 1 and Figure 1, or as approximate relation after diagonalization of the GUT-scale Yukawa matrices Y d and Y e ) has been used to construct SU (5) SUSY GUT flavour models in Refs. [9,10,17,18]. A subset of these relation, y µ = 6y s , and y τ = − 3 2 y b , has been used in [19]. In addition to providing viable quark and charged lepton masses, the GUT CG factors (Y e ) ji /(Y d ) ij = − 1 2 and 6 can also be applied to realize the promising relation between the lepton mixing angle θ PMNS 13 and the Cabibbo angle, θ PMNS 13 θ C sin θ PMNS 23 , in flavour models, as discussed in [20]. As mentioned above, in supersymmetric GUT models the SUSY threshold corrections can have an important influence on the Yukawa coupling ratios. When the MSSM is matched to the SM, integrating out the sparticles at loop-level leads to the emergence of effective operators, which can contribute sizeably to the Yukawa couplings, depending on the values of the sparticle masses, tan β, and the soft-breaking trilinear couplings. Thereby, via the SUSY threshold corrections, a given set of GUT predictions for the ratios yτ y b , yµ ys and ye y d imposes important constraints on the SUSY spectrum.
H 24 T FH 5 10 6 Table 1: Dimension 5 effective operators (AB) R (CD)R and CG factors emerging from the supergraphs of Figure 1 when R andR are integrated out [5].  Table 1 generating effectively Yukawa couplings when the pair of messengers fields R andR is integrated out.
The SUSY threshold corrections can be subdivided into two classes: While at treelevel the down-type quarks only couple to the Higgs field H d , via exchange of sparticles at one-loop level they can also couple to H u , as shown in Figure 2 in section 3. When the sparticles are integrated out the emerging effective operator is enhanced for large tan β (i.e. "tan β-enhanced"). Analogously, there are also tan β-enhanced threshold corrections to the charged lepton Yukawa couplings. For Y u , however, the threshold effects emerging from effective couplings to H d are tan β-suppressed. The second class of threshold corrections emerges from the supersymmetric loops shown in Figure 3. While some of them are strongly suppressed, others lead to the emergence of effective operators proportional to the soft SUSY-breaking trilinear couplings. For large trilinear couplings, they too can become important. Given the importance of the SUSY threshold corrections, we will discuss them and their implementation in SusyTC in detail in the next section.

SUSY threshold corrections & numerical procedure
We follow the notation of REAP [1] (see also [21]) and use a RL convention for the Yukawa matrices. The MSSM superpotential extended by a type-I seesaw mechanism [22] is thus given by where the left-chiral superfields Φ c contain the charge conjugated fields ψ † andφ * R . We use the totally antisymmetric SU (2) tensor 12 = − 21 = 1 for the product Φ · Ψ ≡ ab Φ a Ψ b . The soft-breaking Lagrangian is given by Note that these conventions differ from SUSY Les Houches Accord 2 [23]. They can easily be translated by Since REAP includes the RG running in the type-I seesaw extension of the MSSM (with the DR two-loop β-functions for the MSSM parameters and the neutrino mass operator given in [21]), we have calculated the DR two-loop β-functions of the gaugino mass parameters M a , the trilinear couplings T f , the sfermion squared mass matrices m 2 f , and soft-breaking Higgs mass parameters m 2 Hu and m 2 H d in the presence of Y ν , M n and m 2 ν (using the general formulas of [24]). We list these β-functions in appendix A. The Yukawa matrices and soft-breaking parameters are evolved to the SUSY scale where the stop masses are defined by the up-type squark mass eigenstatesũ i with the largest mixing tot 1 andt 2 . 4 REAP automatically integrates out the right-handed neutrinos, as described in [1]. We assume that M n is much larger than the SUSY scale Q. REAP also features the possibility to add one-loop right-handed neutrino thresholds for the SM parameters, following [25]. At the SUSY scale Q the tree-level sparticle masses and mixings are calculated. Considering heavy sparticles and large Q TeV the SUSY threshold corrections are calculated in the electroweak (EW) unbroken phase. In the EW unbroken phase there are in total twelve types of loop diagrams contributing to the SUSY threshold corrections for Y d (cf. Figures 2 and 3). The SUSY threshold corrections to Y d are calculated in the basis of diagonal squark masses and are given bỹ correspond to the loops in Figure 3, respectively, where the contributions ζ G ij and ζ B ij can become important in cases of small tan β and large trilinear couplings. The loop functions H 2 and C 00 are defined as C 00 (q, x, y) := 1 4 Y ,T are the Yukawa-and trilinear coupling matrices rotated into the basis where the squark mass matrices are diagonal, using the transformations and analogously for down-type (s)quarks and charged (s)leptons. The SUSY threshold corrections to Y e are given bỹ In the real (CP conserving) MSSM the phase φ µ is restricted to 0 and π. The expressions for the one-loop tadpoles t u , t d and the transverse Z-boson self energy Π T ZZ are based on [26], but extended to include intergenerational mixing, and are presented in appendix B. Because µ enters the one-loop formulas for the threshold corrections, treating t u , t d and Π T ZZ as functions of tree-level parameters is sufficiently accurate. The one-loop expression of the soft-breaking mass m 3 is calculated as If desired, SusyTC allows to outsource a two-loop Higgs mass calculation to external software, e.g. FeynHiggs [27,28,29], by calculating the pole mass m H + (m A ) as input for the complex (real) MSSM whereM W is the DR W-boson mass given aŝ with M Z and M W pole masses and the DR vacuum expectation valuev(Q) given bŷ As in the previous formulas, the self energies Π H + H − and Π AA are based on [26], but extended to include inter-generational mixing, and are understood as functions of treelevel parameters. They are given in appendix B.

The REAP extension SusyTC
In this section we provide a "Getting Started" calculation for SusyTC. A full documentation of all features is included in appendix C. Since SusyTC is an extension to REAP, an up-todate version of REAP Repeating this calculation with all SU (5) CG factors listed in Table 2 of [5], one obtains the results shown in Figure 4.   Table 2 of [5], i.e. the GUT predictions yµ ys = CG, for a given example Constrained MSSM parameter point with tan β = 30, m 1/2 = 2000 GeV, A 0 = 1000 GeV, and m 0 = 2500 GeV. The area between the dashed gray lines corresponds to the experimental 1σ range [11].
As described in appendix C, SusyTC can also read and write "Les Houches" files [23,30] as input and output.

The Sparticle Spectrum predicted from CG factors
In this section we apply SusyTC to investigate the constraints on the sparticle spectrum which arise from a set of GUT scale predictions for the quark-lepton Yukawa coupling ratios ye y d , yµ ys , and yτ y b . As GUT scale boundary conditions for the SUSY-breaking terms we take the Constrained MSSM. The experimental constraints are given by the Higgs boson mass m H = 125.7 ± 0.4 GeV [31] as well as the charged fermion masses (and the quark mixing matrix). We use the experimental constraints for the running MS Yukawa couplings at the Z-boson mass scale calculated in [11], where we set the uncertainty of the charged lepton Yukawa couplings to one percent to account for the estimated theoretical uncertainty (which here exceeds the experimental uncertainty). When applying the measured Higgs mass as constraint, we use a 1σ interval of ±3 GeV, including the estimated theoretical uncertainty.
For our study, we consider GUT scale Yukawa coupling matrices which feature the GUT-scale Yukawa relations ye y d = − 1 2 , yµ ys = 6, and yτ y b = − 3 2 (cf. [5]): These GUT relations can emerge as direct result of CG factors in SU(5) GUTs or as approximate relation after diagonalization of the GUT-scale Yukawa matrices Y d and Y e (cf. [9,10,17,18]). For the soft-breaking parameters we restrict our analysis to the Constrained MSSM parameters m 0 , m 1/2 , and A 0 , with µ determined from requiring the breaking of electroweak symmetry as in (16) and set sgn(µ) = +1. We note that in specific models for the GUT Higgs potential, for instance in [17], µ can be realized as an effective parameter of the superpotential with a fixed phase, including the case that µ is real. The value of tan β is fixed at the SUSY scale Q to tan β = 30. 5 We note that we have also added a neutrino sector, i.e. a neutrino Yukawa matrix Y ν and and a mass matrix M n of the right-handed neutrinos, but we have set the entries of Y ν to very small values below O(10 −3 ), such that their effects on the RG evolution can be safely neglected, and the masses of the right-handed neutrinos to values many orders of magnitude higher than the expected SUSY scale. With these parameters, the neutrino sector is decoupled from the main analysis. Such small values of the neutrino Yukawa couplings are e.g. expected in the models [9,10,18], where they arise as effective operators.
Using one-loop RGEs, REAP 1.11.2 and SusyTC we determine the soft-breaking parameters and µ at the SUSY scale, as well as the pole mass m H + . This output is then passed to FeynHiggs 2.11.2 [27,28,29] in order to calculate the two-loop corrected pole masses of the Higgs bosons in the complex MSSM. The MSSM is automatically matched to the SM and we compare the results for the Yukawa couplings at the Z-boson mass scale with the experimental values reported in [11].
When fitting the GUT-scale parameters to the experimental data, we found that our results for the up-type quark Yukawa couplings and CKM angles and CP-phase could be fitted to agree with observations to at least 10 −3 relative precision, by adjusting the parameters of Y u . The remaining six parameters are used to fit the Yukawa couplings of down-type quarks and charged leptons, as well as the mass of the SM-like Higgs boson. We find a benchmark point with a χ 2 = 3.75: input GUT-scale parameters Looking at our results for the low-energy Yukawa coupling ratios, ye y d = 0.17, yµ ys = 2.10, and yτ y b = 0.61, the importance of SUSY threshold corrections in evaluating the GUT-scale Yukawa ratios becomes evident. This can also be seen in Figure 5. Additionally, as shown in Figure 6, SUSY threshold corrections also affect the CKM mixing angles. Finally, in Figure 7 we show the results for the RG running of the soft-breaking mass parameters.
The SUSY spectrum obtained by SusyTC is shown in Figure 8. The lightest supersymmetric particle (LSP) is a bino-like neutralino of about 445 GeV. The next-to-lightest supersymmetric particle (NLSP) is a stop of about 656 GeV. The SUSY scale is obtained as Q = 1128 GeV. The µ parameter obtained from requiring spontaneous electroweak symmetry breaking is given by µ = 2630 GeV. Note that the only experimental constraints we used were the results for quark and charged lepton masses as well as m H . In particular, no bounds on the sparticle masses were applied as well as no restrictions from the neutralino relic density (which would require further assumptions on the cosmological evolution).
Due to the large (absolute) values of the trilinear couplings, we find using the constraints from [32], that the vacuum of our benchmark point is meta-stable. The scalar potential possesses charge and colour breaking (CCB) vacua, as well as one "unbounded from below" (UFB) field direction in parameter space. However, estimating the stability of the vacuum via the Euclidean action of the "bounce" solution [33,34] (following [35]) shows that the lifetime of the vacuum is many orders larger than the age of the universe.
Confidence intervals for the sparticle masses are obtained as Bayesian "highest posterior density" (HPD) intervals 6 from a Markov Chain Monte Carlo sample of two million points, using a Metropolis algorithm. As additional constraint we restricted |A 0 | < 7.5 TeV to avoid possibly dangerous vacuum decay rates. The 1σ HPD intervals for the Constrained MSSM soft-breaking parameters are shown in Figure 9. The 1σ HPD results of the sparticle masses are shown in Figure 10. As for the benchmark point, for all other parameter points the LSP and NLSP are a neutralino and stop, respectively. The HPD interval for the SUSY scale is obtained as Q HPD = [760, 3412] GeV. In Figure 11, we show two-dimensional HPD regions for the correlations between Constrained MSSM soft-breaking parameters and the mass of the Higgs bosons. Finally, in Figure 12 we show the two-dimensional HPD regions for the correlations between the masses of the lightest stop, the neutralino LSP, and the gluino.

Summary and Conclusion
In this work we discussed how predictions for the sparticle spectrum can arise from GUTs, which feature predictions for the ratios of quark and lepton Yukawa couplings at high energy. To test them by comparing with the experimental data, the RG running between high and low energy has to be performed with sufficient accuracy, including threshold corrections. In SUSY theories, the one-loop threshold corrections when matching the SUSY model to the SM are of particular importance, since they can be enhanced by tan β or large trilinear couplings, and thus have the potential to strongly affect the quark-lepton mass relations. Since the SUSY threshold corrections depend on the SUSY parameters, they link a given GUT flavour model to the SUSY model. In other words, via the SUSY threshold corrections, GUT models can predict properties of the sparticle spectrum from the pattern of quark-lepton mass ratios at the GUT scale.
To accurately study such predictions, we extend and generalize various formulas in the literature which are needed for a precision analysis of SUSY flavour GUT models: For example, we extend the RGEs for the MSSM soft breaking parameters at two-loop by the additional terms in the seesaw type-I extension (cf. appendix A). We generalize the one-loop calculation of µ and pole mass calculation of m A and m H + to include inter-generational mixing in the self energies (cf. appendix B). Furthermore, we calculate the full one-loop SUSY threshold corrections for the down-type quark, up-type quark and charged lepton Yukawa coupling matrices in the electroweak unbroken phase (cf. section 3).
We introduce the new software tool SusyTC, a major extension to the Mathematica package REAP, where these formulas are implemented. In addition, SusyTC calculates the DR sparticle spectrum and the SUSY scale Q, and can provide output in SLHA "Les 6 An 1σ HPD interval is the interval [θ L ,θ H ] such that θ H θ L p(θ)dθ = 0.6826 . . . and the posterior probability density p(θ) inside the interval is higher than for any θ outside of the interval [31].
Houches" files which are the necessary input for external software, e.g. for performing a two-loop Higgs mass calculation. REAP extended by SusyTC accepts general GUT scale Yukawa, trilinear and soft breaking mass matrices as well as non-universal gaugino masses as input, performs the RG evolution (integrating out the right-handed neutrinos at their mass thresholds in the type I seesaw extension of the MSSM) and automatically matches the MSSM to the SM, making it a convenient tool for top-down analyses of SUSY flavour GUT models.
We applied SusyTC to study the predictions for the parameters of the Constrained MSSM SUSY scenario from the set of GUT-scale Yukawa relations ye y d = − 1 2 , yµ ys = 6, and yτ y b = − 3 2 , which has been proposed recently in the context of GUT flavour models. With a Markov Chain Monte Carlo analysis we find a "best-fit" benchmark point where the LSP is a bino-like neutralino with a mass of about 450 GeV and the NLSP a stop with a mass of 656 GeV. We also find the 1σ Bayesian confidence intervals for the sparticle masses and the correlations between the SUSY parameters. Without applying any constraints from LHC SUSY searches or dark matter, we find that the considered GUT scenario predicts a sparticle spectrum above past LHC sensitivities, but within reach of the current LHC run or a future high-luminosity upgrade.    A The β-functions in the seesaw type-I extension of the MSSM In this appendix we list the β-functions of the SUSY soft-breaking parameters in the MSSM extended by the additional terms in the seesaw type-I extension (obtained using the general formulas of [24]). Our conventions for W and L soft are given in (2) and (3).

B Self-energies and one-loop tadpoles including intergenerational mixing
Here we present the used formulas for the self-energies Π T ZZ , Π H + H − , Π AA , and the oneloop tadpoles t u , t d , which are based on [26] but generalized to include inter-generational mixing. In this appendix we employ SLHA 2 conventions [23] in the Super-CKM and Super-PMNS basis, to agree with the convention of [26]. The soft-breaking mass matrices in the Super-CKM/Super-PMNS basis are obtained from our flavour basis conventions (2) and (3) by Let us briefly review our generalization to the sfermion mass matrices of [26]: We define the sfermion mixing matrices by 7 with the sfermion mass matrices in the Super-CKM/Super-PMNS basis The D-terms are given by where I 3 denotes the SU (2) L isospin and Q e the electric charge of the flavour f , and θ W denotes the weak mixing angle. Note that our convention for µ differs by a sign from the convention in [26].
For the sake of completeness we also list the conventions for neutralino and chargino mass matrices and mixing matrices: The neutralino mixing matrix is defined by with (63) The chargino mixing matrix is defined by with We now present the generalization of Π T ZZ , Π H + H − , Π AA , t u , and t d of [26] to include inter-generational mixing. For all we have checked that our equations reduce to the corresponding equations in [26] when where i = 1 . . . 3.
We keep the abbreviations of [26]: The conventions for the one-loop scalar functions A 0 , B 22 ,B 22 , H, G, and F [36] are adopted from appendix B of [26]. Summations f are over all fermions, whereas summations fu , f d are restricted to up-type and down-type fermions, respectively. Summations Q , Q are over SU (2) (s)quark doublets, and analogously for (s)leptons. In summations over sfermions the indices i, j, s, and t run from 1 to 6 forũ,d, andẽ and from 1 to 3 forν. In summations of neutralinos (charginos) the indices i, j run from 1 to 4 (2). The summations h 0 runs over all neutral Higgs-and Goldstone bosons, the summation h + over the charged ones.
The couplings f 0 Z , f + Z , g 0 Z , and g + Z are given in Eqs. (A.7) and (D.5) of [26]. of [26]. The couplings to sfermions in the case of inter-generational mixing are given by The couplings f 0 A , g 0 A , f + A , and g + A are given in Eq. (D.70) and Eqs. (D.34-D.38) of [26]. The couplings λ Ah 0 h 0 , λ AAh 0 h 0 , and λ AAh + h + are defined in Eqs. (D.63-D.65) and Eq. (D.67) of [26]. The couplings to sfermions in the case of inter-generational mixing are given by C SusyTC documentation Here we present a documentation of the REAP extension SusyTC. To get started, please follow first the steps described in Section 4. We now describe that additional features of SusyTC: In addition to the features of REAP package RGEMSSMsoftbroken.m (described in the REAP documentation), SusyTC adds the following options to the command RGEAdd: • STCsignµ is the general factor e iφµ in front of µ in (16). • STCSearchSMTransition is a switch to enable or disable the matching to the SM and the calculation of supersymmetric threshold corrections and sparticle spectrum.
(default: True) • STCCCBConstraints is a switch to enable or disable a warning message for potentially dangerous charge and colour breaking vacua, if large trilinear couplings violate the constraints of [37] at the SUSY scale Q where m L , m R and m H f denote the soft-breaking mass parameters of the scalar fields associated with the trilinear coupling T in the basis of diagonal Yukawa matrices.
(default: True) • STCUFBConstraints is a switch to enable or disable a warning message for possibly dangerous "unbounded from below" directions in the scalar potential, if the constraints of [32] are violated at the SUSY scale Q In addition to the parameters known from the MSSM REAP model, the following softbreaking parameters are available for RGEGetSolution at all energy scales higher than the SUSY scale Q: • RGETu, RGETd, RGETe, and RGETν are used to get the soft-breaking trilinear coupling matrices.
• RawTν is used to get the raw (internal representation) of the soft-breaking trilinear matrix for sneutrinos.
• RGEM1, RGEM2, and RGEM3 are used to get the soft-breaking gaugino mass parameters.
• RGEm2Q, RGEm2L, RGEm2u, RGEm2d, RGEm2e, RGEm2ν are used to get the soft-breaking squared mass matrices m 2 f for the sfermions. • RGEm2Hd and RGEm2Hu are used to get the soft-breaking squared masses for H d and H u , respectively.
To obtain the running DR gluino mass at a scale of two TeV for example, one uses

RGEGetSolution[2000,RGEM3];
With SusyTC the DR sparticle spectrum is automatically calculated. The following functions are included in SusyTC: • STCGetSUSYScale[] returns the SUSY scale Q.
• STCGetSUSYSpectrum[] returns a list of replacement rules for the SUSY scale Q, the DR tree-level values of µ and m 3 , and the DR sparticle masses and (tree-level) mixing matrices at the SUSY scale. In detail it contains -"Q" the SUSY scale Q.
-"mχp" a list of the two chargino masses.
-"msν" a list of the three light sneutrino masses.
-"tanα" the mixing angle of the CP-even Higgs bosons.
-"N" the mixing matrix of neutralinos.
-"Wude" a list of the three sparticle mixing matrices for up-type squarks, downtype squarks and charged sleptons.
-"Wν" the mixing matrix of the three light sneutrinos.
To obtain for example the SUSY scale and the tree-level masses of the charginos call "Q"/.STCGetSUSYSpectrum[]; "mχp"/.STCGetSUSYSpectrum[]; The squark masses and charged slepton masses are contained in a joint list as {mũ,md,mẽ}, and analogously for the sfermion mixing matrices. To obtain for example the up-type squark masses, the charged slepton mixing matrix, and the sneutrino masses type • STCGetOneLoopValues[] returns a list of replacement rules containing -"µ","m 3 " the one-loop corrected DR µ-parameter and m 3 as in (16) and (17) at the SUSY scale Q.
-"vev" the one-loop DR vevv as in (21) at the SUSY scale Q.
The value of µ can for example be obtained from "µ"/.STCGetOneLoopValues[]; • STCGetSCKMValues[] returns a list of replacement rules with the soft-breaking mass squared and trilinear coupling matrices in the SCKM basis, where sparticles are rotated analogously with their corresponding superpartners 9 . Since they are used for the self-energies calculation as described in the previous appendix, they are returned in SLHA2 convention [23]! In detail, there are -"VCKM" the CKM mixing matrix.
-SCKMBasis["T"] a list of the three trilinear coupling matrices for up-type squarks, down-type squarks and charged sleptons in the SCKM basis with SLHA2 conventions.
To obtain the down-type trilinear coupling matrix and the mass squared matrix of the left-handed up-type squarks in the SLHA basis for example, type As additional feature, SusyTC optionally supports input and output as SLHA "Les Houches" files. These files follow SLHA conventions [23,30]: • STCSLHA2Input["Path"] loads an "Les Houches" input file stored in "Path" and executes REAP and SusyTC. If no path is given, the default path is assumed as "SusyTC.in" in the Mathematica Notebook Directory. An important difference to other spectrum calculators is the pure "top-down" approach by SusyTC, i.e. there is no attempt of fitting SM inputs at a low scale or calculating a GUT-scale from gauge couplings unification. Instead, all input is given at a user-defined high energy scale, which is then evolved to lower scales. The input should be given in the flavour basis in SLHA 2 convention [23], with analogous convention for Y ν and convention for M n as in (2). The relation between the SusyTC conventions and the SLHA 2 conventions is given in Section 3. In the following, we list all SLHA 2 input blocks, which are available in SusyTC:  [23,30]. Note however, that the input value of tan β is interpreted to be given at the SUSY scale. Hu (M input ) Imaginary components for the gaugino masses can be given in Block IMEXTPAR.
-Block IMEXTPAR: as defined in [23]. -Block IMYU, Block IMYD, Block IMYE, Block IMYN: The imaginary parts of the Yukawa matrices Y u , Y d , Y e , and Y ν in the flavour basis [23]. They are given in the same format as the real parts.
-Block MN: The real part of the symmetric Majorana mass matrix M n of the right-handed neutrinos in the flavour basis (2). Only the "upper-triangle" entries should be given, the input format is as for the Yukawa matrices.
-Block IMMN: The imaginary part of the symmetric Majorana mass matrix M n of the right-handed neutrinos in the flavour basis (2). Only the "upper-triangle" entries should be given, the input format is as for the Yukawa matrices.
The remaining blocks can be given optionally to overwrite Constrained MSSM input boundary conditions: -Block TU, Block TD, Block TE, Block TN: The real parts of the trilinear soft-breaking matrices T u , T d , T e , and T ν in the flavour basis [23]. They should be given in the same format as the Yukawa matrices.
-Block IMTU, Block IMTD, Block IMTE, Block IMTN: The imaginary parts of the trilinear soft-breaking matrices T u , T d , T e , and T ν in the flavour basis [23]. They should be given in the same format as the Yukawa matrices.
-Block MSQ2, Block MSU2, Block MSD2, Block MSL2, Block MSE2, Block MSN2: The real parts of the soft-breaking mass squared matrices m 2Q , m 2 u , m 2 d , m 2 L , m 2 e , and m 2 ν in the flavour basis [23]. Only the "upper-triangle" entries should be given, the input format is as for the Yukawa matrices.
-Block IMMSQ2, Block IMMSU2, Block IMMSD2,Block IMMSL2, Block IMMSE2, Block IMMSN2: The imaginary parts of the soft-breaking mass squared matrices m 2Q , m 2 u , m 2 d , m 2 L , m 2 e , and m 2 ν in the flavour basis [23]. Only the "upper-triangle" entries should be given, the input format is as for the Yukawa matrices.
• STCWriteSLHA2Output["Path"] writes an "Les Houches" [23,30] output file to "Path". If no path is given, the output is saved in the Mathematica Notebook directory as "SusyTC.out". The output follows SLHA conventions, with the following exceptions: -Block MASS: The mass spectrum is given as DR masses at the SUSY scale. The only exception is the pole mass M H + (M A ) for CP violation turned on (off).
-Block ALPHA: the tree-level Higgs mixing angle α tree .
-Block HMIX: Instead of M A we give 101 : m 3 The other blocks follow the SLHA2 output conventions, e.g. DR values at the SUSY scale in the Super-CKM/Super-PMNS basis. To avoid confusion, the blocks Block DSQMIX, Block USQMIX, Block SELMIX, Block SNUMIX and the corresponding blocks for the imaginary entries, return the sfermion mixing matrices Rf in SLHA 2 convention.