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 su-persymmetric (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 [1], 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 yeyd=−12−2,yμys=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{y_e}{y_d}=-\frac{1}{2}{-}^2,\frac{y_{\mu }}{y_s} = 6 $$\end{document}, and yτyb=−32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{y_{\tau }}{y_b}=-\frac{3}{2} $$\end{document}, 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][3][4][5][6]) 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. [7][8][9][10]) and y µ = 3y s [11]. 1 Other promising quarklepton mass relations at the GUT scale have been discussed in [12,13], 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. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] for recent works.

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With the discovery of the Higgs boson at the LHC [32,33] 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 [34,35], 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. [36]), 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. [37][38][39][40][41], see e.g. [42] 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 JHEP07(2016)108 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 [11]. It was pointed out in [12] (see also [13]) 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 JHEP07(2016)108 can be consistent with the experimental data is provided by the -RG invariant and SUSY threshold correction invariant 3 -double ratio [36] y µ y s While the Georgi-Jarlskog relations [11] imply a double ratio of 9, disfavoured by more than 2σ, other combinations of CG factors [12,13], 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. [34,35,43,44]. A subset of these relation, y µ = 6y s , and y τ = − 3 2 y b , has been used in [45]. 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
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. The SUSY threshold corrections can be subdivided into three 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. Finally there are threshold corrections from exchange of heavy Higgs doublets and canonical normalization of external fields (i.e. from wavefunction renormalization diagrams), as shown in figures 4 and 5-7, respectively. Given the importance of the SUSY threshold corrections, we will discuss them and their implementation in SusyTC in detail in the next section.

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H 24 T FH 5 10 6 Table 1. Examples for dimension 5 effective operators (AB) R (CD)R and CG factors emerging from the supergraphs of figure 1 when R andR are integrated out [12].

SUSY threshold corrections & numerical procedure
We follow the notation of REAP [1] (see also [47]) and use a RL convention for the Yukawa matrices. The MSSM superpotential extended by a type-I seesaw mechanism [48][49][50][51][52][53][54] 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 [55]. 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 [47]), we have calculated the DR two-loop β-functions of the gaugino mass JHEP07(2016)108 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 [56]). 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 [57]. 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 by (in terms of DR quantities) where P d , P Q , and P h are the threshold corrections (3.12), (3.19), (3.20), and (3.22) due to canonical normalization.Ỹ d ij is given bỹ 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 threshold correction P h due to canonical normalization of external Higgs doublets is given by where h and H are the mass eigenstates doublets of the MSSM Higgs scalar fields and (3.14) with  15) JHEP07(2016)108 which correspond to the loops shown in figures 5 and 6. The loop functions B 2 andB 2 are given by The threshold corrections P f due to canonical normalization of the external fermion fields are given by For the threshold corrections of the down-type Yukawa matrix the K f are given by with corresponding to the loops in figure 7, and   Figure 6. Fermion loops contributing to threshold corrections from canonical normalization of external Higgs fields.

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with The diagrams for K d are similar to the ones for K Q , when Q is replaced with d, with the exception of the wino loop, which does not exist for external right-handed fermions. The loop-functionsB 1 and B 3 are given bỹ For the charged leptons, Y SM e is given bỹ where P e and P L are threshold corrections due to canonical normalization of external charged lepton fields and

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with the tan β-enhanced contributions The diagrams for the Y e SUSY threshold corrections are analogous to the ones in figures 2 and 3, with the exception that the loop diagrams shown in the top rows do not exist. The diagram of figure 4 also doesn't have an analogue for Y e . P e and P L are calculated from (3.19) with and

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Again, the loop diagrams for K L and K e can be easily obtained from the diagrams for K Q by suitable exchange of labels and indices and dropping non-existent gaugino loops. Turning to Y u , the types of diagrams which were tan β-enhanced for Y d and Y e are now tan β-suppressed. However, there also exist SUSY threshold corrections which are independent of tan β and enhanced by large trilinear couplings. These SUSY threshold corrections to Y u can have important effects. For example the SUSY threshold corrections to the top Yukawa coupling y t can be of significance in analyses of the Higgs mass and vacuum stability. The expression for the Y u SUSY threshold corrections can be readily obtained from the SUSY threshold corrections to Y d (3.4)-(3.7) and (3.19)-(3.23) by the replacement 5 with the exception of the bino-loops, whose contribution become After the SUSY threshold corrections are incorporated in the DR scheme, REAP converts the Yukawa and gauge couplings to the MS scheme following [58].
Finally SusyTC calculates the value of |µ| and m 3 from m 2 Hu , m 2 H d , tan β and M Z by requiring the existence of spontaneously broken EW vacuum, which is equivalent to vanishing one-loop corrected tad-pole equations of H u and H d 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 [59], but extended to include inter-generational 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

(3.41)
As in the previous formulas, the self energies Π H + H − and Π AA are based on [59], 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-MPT [1] (available at http://reapmpt.hepforge.org) needs to be installed on your system. SusyTC consists out of the REAP model file RGEMSSMsoftbroken.m, which is based on the model file RGEMSSM.m of REAP 1.11.2 and additionally contains, among other things, the RGEs of the MSSM soft-breaking parameters and the matching to the SM, and the file SusyTC.m, which includes the formulas for the sparticle spectrum and SUSY threshold correction calculations. Both files can be downloaded from http://particlesandcosmology.unibas.ch/pages/SusyTC.htm and have to be copied into the REAP directory.
To begin a calculation with SusyTC, one first needs to import RGEMSSMsoftbroken.m: The model MSSMsoftbroken is then defined by RGEAdd, including additional options such as RGEtanβ: RGEAdd["MSSMsoftbroken",RGEtanβ → 30];

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In MSSMsoftbroken all REAP options of the model MSSM are available. The options additionally available in SusyTC are given in appendix C. The input is given by RGESetInitial.
Let us illustrate some features of SusyTC: to test for example the GUT scale prediction for the Yukawa coupling ratio yµ ys = 6, considering a given example parameter point in the Constrained MSSM, one can type: Of course, any general matrices can be used as input for the Yukawa, trilinear, and softbreaking matrices, as given by the specific SUSY flavour GUT model under consideration. Also, non-universal gaugino masses can be specified. Repeating this calculation with all SU(5) CG factors listed in table 2 of [12], one obtains the results shown in figure 8. As described in appendix C, SusyTC can also read and write "Les Houches" files [55,58] 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 0 = 125.09 ± 0.21 ± 0.11 GeV [68] 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 [36], 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.

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9/2 6 9 0 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. [12]): 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. [34,35,43,44]). For the soft-breaking parameters we restrict our analysis to the Constrained MSSM parameters m 0 , m 1/2 , A 0 and tan β, with µ determined from requiring the breaking of electroweak symmetry as in (3.36) and set sgn(µ) = +1. We note that in specific models for the GUT Higgs potential, for instance in [43], µ can be realized as an effective parameter of the superpotential with a fixed phase, including the case that µ is real. 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 JHEP07(2016)108 sector is decoupled from the main analysis. Such small values of the neutrino Yukawa couplings are e.g. expected in the models [34,35,44], where they arise as effective operators.
Using one-loop RGEs, REAP 1.11.3 and SusyTC 1.1 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 2.11.3 [60][61][62][63][64][65][66][67] 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 [36].
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 = 0.9: input GUT scale parameters Looking at our results for the low-energy Yukawa coupling ratios, ye y d = 0.16, yµ ys = 1.92, 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 9. Additionally, as shown in figure 10, SUSY threshold corrections also affect the CKM mixing angles.
The SUSY spectrum obtained by SusyTC is shown in figure 11. The lightest supersymmetric particle (LSP) is a bino-like neutralino of about 827 GeV. The SUSY scale is obtained as Q = 3014 GeV. The µ parameter obtained from requiring spontaneous electroweak symmetry breaking is given by µ = 2634 GeV. Note that the only experimental constraints we used were the results for quark and charged lepton masses as well as m h 0 . 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). 6

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Due to the large (absolute) values of the trilinear couplings, we find using the constraints from [69], 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 [70,71] (following [72]) 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 7 from a Markov Chain Monte Carlo sample of 1.2 million points, using a Metropolis algorithm. We note that we did not compute the lifetime of the vacuum for each point in the MCMC analysis, which would take far too much computation time. This means that the obtained confidence intervals should be regarded as conservative, in the sense that including the lifetime constraints the upper bounds on the masses could become smaller. For some example points within the 1σ HPD regions we have checked that the lifetime constraints are satisfied. We applied the following priors to the MCMC analysis: m 0 ∈ [0, 4] TeV, A 0 ∈ [−10, 0] TeV, m 1/2 ∈ [0, 6] TeV and tan β ∈ [2,40]. As shown in figure 12 our results for the 1σ HPD intervals for the Constrained MSSM softbreaking parameters are well within these intervals. The 1σ HPD results of the sparticle masses are shown in figure 13. Furthermore we find that for about 70% of the data points of the MCMC analysis, the lightest MSSM sparticle is a neutralino, while for the others it is the lightest charged slepton. 8 The HPD interval for the SUSY scale is obtained as Q HPD = [2048, 5108] GeV.

Comments and discussion
We would like to emphasize that the results described above have been obtained under specific assumptions for the input parameters at the GUT scale. In the following, we discuss these assumptions, how they may be obtained and/or generalized in fully worked out models, and also some limitations and uncertainties of our analysis.
To start with, we have chosen the specific GUT scale predictions yτ y b = − 3 2 , yµ ys = 6 and ye y d = − 1 2 . This is indeed only one of the possible predictions that can arise from GUTs. Other possibilities can be found, e.g., in [11][12][13]. We have chosen the above set of GUT predictions since they are among the ones recently used successfully in GUT model building (see e.g. [34,35,43,44]). In the future, it will of course be interesting to also test other combinations of promising Clebsch factors, and compare the predictions/constraints on the SUSY spectra. Of course, one can also construct GUT models which do not predict the quark-lepton Yukawa ratios. For such GUT models the constraints discussed here would not apply.
Furthermore, for our study we have assumed CMSSM boundary conditions for the soft breaking parameters, which is quite a strong assumption that will probably often be relaxed in realistic models. On the other hand, universal boundary conditions may also be a result of a specific SUSY breaking mechanism. Apart from the SUSY breaking 7 An 1σ HPD interval is the interval [θL,θH ] 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 [73]. 8 Note that since in the latter case the LSP may be the gravitino, we do not exclude those points.

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mechanism, GUTs themselves "unify" the soft breaking parameters since they unify the different types of SM fermions in GUT representations. In SU(5) GUTs, for example, one is left with only two soft breaking mass matrices at the GUT scale per family, one for the sfermions in the five-dimensional matter representation and one for the sfermions in the ten-dimensional representation. In SO(10) GUTs, there is only one unified sfermion mass matrix. In addition, the symmetries of GUT flavour models like [34,35,43,44] include various (non-Abelian) "family symmetries", which lead to hierarchical Yukawa matrices and impose (partially) universal soft breaking mass matrices among different generations. The combination of these effects can indeed lead to GUT scale boundary conditions close to the CMSSM. Finally, we like to note that the absence of deviations from the SM in flavour physics processes leads to constraints on flavour non-universalities in the SUSY spectrum (if the sparticles are not too heavy) and can be seen as an experimental hint that, if SUSY exists at a comparably low scale, it should indeed be close to flavour-universal. In any case, it will be interesting to see how the constraints on the SUSY spectrum change when the assumption of an exact CMSSM at the GUT scale is relaxed.
On the other hand, although we have analyzed here a specific example only, some of the key effects that lead to a predicted sparticle spectrum seem rather general, as long as the quark-lepton Yukawa ratios are predicted at the GUT scale together with (close-to) universal soft-breaking parameters (which we assume for the remainder of this discussion): • The main reason for the predictions/constraints on the SUSY spectrum is the fact that (to our knowledge) all the possible sets of GUT predictions for the quark-lepton Yukawa ratios require a certain amount of SUSY threshold corrections for each generation. 9 In general, to obtain the required size of the threshold corrections, one cannot have a sparticle spectrum which is too "split" (as e.g. in [74,75]), since otherwise the loop functions (cf. section 3) get too suppressed. More specifically, the required threshold corrections constrain the ratios of trilinear couplings, gaugino masses, µ and sfermion masses. In a CMSSM-like scenario, this means the ratios between m 0 , m 1/2 and A 0 are constrained. Furthermore, since the most relevant threshold corrections are the ones which are tan β-enhanced, tan β cannot bee too small.
• Finally, with the ratios between m 0 , m 1/2 and A 0 constrained and a moderate to large value of tan β, the measured value of the mass m h of the SM-like Higgs boson allows to constrain the SUSY scale. We note that this is an important ingredient, since the threshold corrections themselves depend only on the ratios of trilinear couplings, gaugino masses, µ and sfermion masses, and do not constrain the overall scale of the soft breaking parameters. The combination of the two effects leads to the result of a predicted sparticle spectrum from the assumed GUT boundary conditions.
Since m h plays an important role, we would like to comment that it would be highly desirable to have a more precise computation of the Higgs mass available for the "large stopmixing" regime. As discussed above, we use a theoretical uncertainty of ±3 GeV, which JHEP07(2016)108 is dominating the 1σ interval for m h entering our fit. This theoretical uncertainty should, strictly speaking, not be treated on the same footing as a pure statistical uncertainty (but the same of course also holds true for systematic experimental uncertainties). Furthermore, there are indications that the theoretical uncertainty in the m h calculation in the most relevant regions of parameter space of our analysis may be larger, as recently discussed e.g. in [76], however there is no full agreement on this aspect. For our analysis we have used the external software FeynHiggs 2.11.3, the current version when our numerical analysis was performed, and the most commonly assumed estimate ±3 GeV for the theoretical uncertainty.
One may also ask why we did not emphasize the aspect of gauge coupling unification. From a bottom-up perspective, one could indeed conclude that in order to make the gauge couplings meet at high energy, the SUSY scale cannot be too high. However, in realistic GUTs, such statements are strongly affected by GUT threshold corrections. They can be implemented easily with REAP once the specific GUT model is known, however they are very model-dependent. In particular, for more general GUT-Higgs potentials it is known that gauge coupling unification is not resulting in a relevant constraint on the SUSY scale. For our analysis, we have therefore simply set the GUT scale to 2 × 10 16 GeV. In an explicit model, where the GUT threshold corrections can be computed and in particular if they significantly shift the GUT scale, we expect that the predictions for the SUSY spectrum will also be modified correspondingly, e.g. due to the increased amount of RG evolution.
Let us also comment on the fact that SusyTC is matching the MSSM to the SM at one common "SUSY scale". This is certainly a limitation of the current version of SusyTC. On the other hand, as discussed above, in the scenarios where SUSY threshold corrections play an important role, the sparticle spectrum cannot be too "split". This means that for the main applications of SusyTC, like the example presented in this section, one-step matching is sufficiently accurate.

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 JHEP07(2016)108 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 Houches" files which are the necessary input for external software, e.g. for performing a twoloop 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 as well as the 1σ Bayesian confidence intervals for the sparticle masses. 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 partly within reach of (a high-luminosity upgrade of) the LHC, and possibly fully testable at a future O(100 TeV) pp collider like the FCC-hh [80] or the SPPC [81].
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 [56]). Our conventions for W and L soft are given in (3.1) and (3.2).

B Self-energies and one-loop tadpoles including inter-generational mixing
Here we present the used formulas for the self-energies Π T ZZ , Π H + H − , Π AA , and the one-loop tadpoles t u , t d , which are based on [59] but generalized to include inter-generational mixing. In this appendix we employ SLHA 2 conventions [55] in the Super-CKM and Super-PMNS basis, to agree with the convention of [59]. The soft-breaking mass matrices in the Super-CKM/Super-PMNS basis are obtained from our flavour basis conventions (3.1) and (3.2) by Let us briefly review our generalization to the sfermion mass matrices of [59]: we define the sfermion mixing matrices by 10

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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 [59].
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 (B.6) The chargino mixing matrix is defined by We now present the generalization of Π T ZZ , Π H + H − , Π AA , t u , and t d of [59] to include inter-generational mixing. For all we have checked that our equations reduce to the corresponding equations in [59] when We keep the abbreviations of [59]: e ≡ g 2 sin θ W , (B.14) The conventions for the one-loop scalar functions A 0 , B 22 ,B 22 , H, G, and F [77,78] are adopted from appendix B of [59]. 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. .

(B.22)
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 [59]. 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 [59]. The couplings to sfermions in the case of inter-generational mixing are given by (B.28) The couplings to sfermion in the case of inter-generational mixing are given by JHEP07(2016)108 The couplings to sfermion in the case of inter-generational mixing are given by (B.34)

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: • 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.
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 (3.36) and (3.37) at the SUSY scale Q.
-"vev" the one-loop DR vevv as in (3.41) 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. 12 Since they are used for the self-energies calculation as described in the previous appendix, they are returned in SLHA2 convention [55]! 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 [55,58]: • 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 [55], with analogous convention for Y ν and convention for M n as in (3.1). 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:  [55,58]. 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 [55]. -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 [55]. 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 righthanded neutrinos in the flavour basis (3.1). 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 (3.1). 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 [55]. They should be given in the same format as the Yukawa matrices.

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-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 [55]. 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 [55]. 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 [55]. Only the "upper-triangle" entries should be given, the input format is as for the Yukawa matrices.
• STCWriteSLHA2Output["Path"] writes an "Les Houches" [55,58] 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.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.