Automatised matching between two scalar sectors at the oneloop level
Abstract
Nowadays, one needs to consider seriously the possibility that a large separation between the scale of new physics and the electroweak scale exists. Nevertheless, there are still observables in this scenario, in particular the Higgs mass, which are sensitive to the properties of the UV theory. In order to obtain reliable predictions for a model which involves very heavy degrees of freedom, the precise matching to an effective theory is necessary. While this has been so far only studied for a few selected examples, we present an extension of the Mathematica package SARAH to perform automatically the matching between two scalar sectors at the full oneloop level for general models. We show that we can reproduce all important results for commonly studied models like split or highscale supersymmetry. One can now easily go beyond that and study new ideas involving very heavy states, where the effective model can either be just the standard model or an extension of it. Also scenarios with several matching scales can be easily considered. We provide model files for the MSSM with seven different mass hierarchies as well as two highscale versions of the NMSSM. Moreover, it is explained how new models are implemented.
1 Introduction
The Standard Model (SM) of particle physics is a very successful theory which has been completed with the discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2]. On the other side, there are observations like dark matter for which no viable candidate exists within the SM. While it has been expected that solutions to the open problems of the SM, like e.g. supersymmetry (SUSY), exist close to the electroweak scale, the LHC has not found any direct signal for new physics so far. Therefore, the possibility of a large gap between the electroweak (EW) and the scale of new physics has been studied more intensively in the recent years. The most prominent idea in this direction is ‘split supersymmetry’ (splitSUSY) in which the SUSY scalars are much heavier than the SM particles and the SUSY fermions [3, 4, 5]. In this setup, most of the appealing properties of SUSY like gauge coupling unification and a dark matter candidate are kept, but the coloured particles are too heavy to be produced at the LHC. Mechanisms have been proposed how splitSUSY could arise from string theory [6, 7], and also the question of naturalness has been discussed [8]. Moreover, the ansatz of highscale SUSY, i.e. that all SUSY particles are much heavier than the EW scale, is taken seriously nowadays [9, 10]. While it is widely believed that these models suffer from a large finetuning, it has pointed out that large SUSY scales can be combined with the relaxion mechanism to solve the big and the small hierarchy problem simultaneously [11]. The idea of SUSY with very large mass scales is not restricted to the Minimal Supersymmetric extension of the SM (MSSM), but has also been applied to other SUSY models like the NexttoMSSM (NMSSM) [12] or models with Dirac gauginos [13, 14, 15, 16, 17].
Even if states beyond the SM (BSM) are too heavy to be produced at current colliders, they often still have an inprint in experimental results, see e.g. Refs. [18, 19]. The precise measurement of the Higgs boson mass of \(m_h = 125.09\) GeV [20] at the LHC has added another very important constraint in this direction. Consequently, large efforts were put in a precise Higgs boson mass calculation in split or highscale SUSY [9, 10, 21, 22, 23]. The reason for this endeavour is that the commonly used fixed order calculations of the Higgs boson mass in SUSY models should only be applied in the case of a small separation between the EW scale and the SUSY scale. Otherwise, the presence of large logarithms introduces a large uncertainty in the prediction of the numerical value of \(m_h\) [23, 24, 25, 26]. This can be resolved either by the standard ansatz of an effective field theory (EFT) in which the heavy states are integrated out [27, 28, 29, 30, 31, 32, 33, 34, 35, 36], or by a hybrid method in which the fixedorder calculation is combined with the higherorder leading logarithms extracted from an EFT [37, 38, 39, 40]. In both cases, one needs to know how the couplings among the light states depend on the full theory. In terms of the EFT ansatz this means that the full model involving heavy and light states must be matched to an effective theory at the scale at which the heavy degrees of freedom are integrated out. The matching at leading order is straightforward and the relations often can be read off from the treelevel Lagrangians of both models. However, treelevel relations are usually not sufficient to obtain the necessary precision in the Higgs boson mass prediction. Therefore, higherorder corrections are needed. Of course, the matching procedure at the full oneloop level is already much more timeconsuming. Depending on the details of the full and effective model also several subtleties like infrared divergences can occur as discussed in Ref. [41].
In order to facilitate these studies, we have developed an automatised process to perform the matching between the scalar sectors of two renormalisable theories. This feature has been implemented in the Mathematica package SARAH [42, 43, 44, 45, 46] and provides the functionality to obtain analytical expressions for the matching conditions at the oneloop level. Also the interface between SARAH and SPheno [47, 48] has been extended to include the matching between an EFT and a UVcomplete theory. In that way, one can obtain very quickly numerical predictions for the Higgs boson mass but also for all kind of other observables that concern the Higgs boson. It is worth to stress that this functionality is not restricted to split or highscale versions of the MSSM. A large variety of SUSY, but also nonSUSY models, with large BSM scales can be studied with the presented toolchain. Also the considered EFT need not to be the SM, but could be a singlet extension, a TwoHiggsDoubletModel (THDM), or an even more complicated model. Concerning the nature of the heavy states, we restrict our attention to heavy fermions and scalars. The implementation of integrating out heavy vector bosons at the oneloop level is reserved for future work. However, the lowenergy EFT can still contain an extended gauge sector which is also matched at the oneloop level. Nevertheless, we will mainly concentrate in the given examples on the established MSSM scenarios because they offer the possibility to compare our generic approach with results available in the literature.
2 Generic matching between two scalar sectors
2.1 General ansatz
2.2 Renormalisation scheme
A simple renormalisation scheme which is applicable to a wide range of models is the \(\overline{\text {MS}}\)/\(\overline{\text {DR}}\) scheme. Therefore, we are going to stick mainly to this scheme. The only exception is the treatment of the offdiagonal wavefunction renormalisation (WFR) of the scalar fields. It has been proposed in Ref. [10] that these contributions can be dropped by assuming finite counterterms for some input parameters. For instance, in the highscale MSSM one could assume a counterterm for \(\tan \beta \) which exactly cancels the offdiagonal WFR contributions. This approach is a more economic calculation and can lead to performance improvements in the runtime. However, it depends on the considered model and the chosen input parameters if such a scheme is possible. Therefore, we provide the possibility to include or exclude the oneloop contributions from offdiagonal WFR constants during the calculation.
For an appropriate choice of the WFR treatment it is worth to mention the equivalence between excluding the offdiagonal WFR constants and the extraction of effective quartic couplings from a polemass matching [24, 41]. Thus, for the comparison with tools that use a polemass matching, the inclusion of offdiagonal WFR constants should be disabled in the calculation.
2.3 Parametrisation of the results at the matching scale
 1.Thresholds from heavy fields:where \(g_i\) is the gauge coupling with respect to the gauge group i, \(I_2^i(x)\) is the Dynkin index of the field x with respect to the gauge group i, and \(C^i(x)\) is a multiplicity factor taking into account the charges under nonAbelian gauge groups others than i, i.e. in the case of the SM gauge group, this counts the colour and isospin multiplicity in the loop.$$\begin{aligned} \delta g_i&= \sum _{\phi ^H} \frac{1}{16 \pi ^2} \frac{C^i(\phi ^H)}{12} I_2^i(\phi ^H) g_i^3 \log \frac{M_{\phi _H}^2}{Q^2} \nonumber \\&\quad + \sum _{\psi ^H} \frac{1}{16 \pi ^2} \frac{C^i(\psi ^H)}{3} I_2^i(\psi ^H) g_i^3 \log \frac{M_{\psi _H}^2}{Q^2}, \end{aligned}$$(15)
 2.\(\overline{\text {MS}}\)–\(\overline{\text {DR}}\) conversion: required if an \(\overline{\text {MS}}\) and a \(\overline{\text {DR}}\) renormalised quantity are to be matched. This is e.g. the case if nonSUSY models are matched onto SUSY ones. There are two different contributions which affect the quartic couplings:
 The finite shifts of the gauge couplings for an SU(N) group are [50]$$\begin{aligned} \delta g_i = \frac{1}{16 \pi ^2} g_i^3 \frac{N}{6}. \end{aligned}$$(16)
 Quartic vertices receive an additional shift from \(\overline{\text {MS}}\)–\(\overline{\text {DR}}\) conversion from the diagrams shown in Fig. 2. The amplitude difference of this diagram between the two schemes iswhere \(c_1\) and \(c_2\) are the two involved vertices between two scalars and two vector bosons.$$\begin{aligned} {\mathcal {M}}= c_1 c_2 , \end{aligned}$$(17)

2.4 Above and below the matching scale: threshold corrections to fermionic couplings
3 Implementation in SARAH and SPheno
In the last section all necessary ingredients for a matching of two arbitrary renormalisable scalar sectors at the oneloop level were introduced. In this section we describe the implementation as well as the usage in the computer programs SARAH and SPheno.
3.1 General information about SARAH and SPheno

The input parameters of the model

The choice for the renormalisation scale

The boundary conditions at the electroweak scale, at the renormalisation scale and at the GUT scale

Optional: a condition to dynamically determine the GUT scale, e.g. \(g_1(m_{\mathrm{GUT}})=g_2(m_{\mathrm{GUT}})\)

A list of particles for which the two and three body decays should be calculated.
3.2 Available options to perform the matching
The matching of two scalar sectors can be motivated by a precise investigation of very different properties of the theories to be matched. The largest contributions to threshold corrections often have their origin in one common sector of the heavy spectrum. It can be of particular interest to track this origin down in order to learn more about which parts of a given UVtheory are essential for the predictions in an EFT framework. For this purpose an analytical evaluation of threshold corrections is preferred. The analytical solutions can also easily be ported to other computer programs which is a key feature of many existing SARAH routines.
As already discussed, the matching of an EFT onto a UVcomplete model does not only influence many lowenergy observables but also enters the RGE running and other predictions above the matching scale. Considering the whole picture of the matching procedure and its numerical influence in all sectors of the theories to be matched, a numerical calculation of threshold corrections is preferred because it can easily be embedded into existing routines of the generated SPheno code.
 1.
An analytical calculation within Mathematica
 2.
A fully numerical calculation using only the SPheno interface.
It is important to stress that both options are not based on the same routines, but have been implemented independently. Thus, they offer the possibility to double check the obtained results. A schematic comparison of the two approaches is given in Fig. 4. A summary of the description given here is also available at the SARAH wiki page.^{3}
For the analytical calculation it is necessary that all mass matrices in the model can be diagonalised analytically. Thus, it is usually necessary to work with a set of assumptions which simplify the most general mass matrices in a given model. In theories with spontaneous symmetry breaking, a high degree of different mixing patterns is introduced through the presence of VEVs. It has already been argued that, if these VEVs are responsible for the generation of masses in the EFT, i.e. if the lowenergy Lagrangian is invariant under the symmetries broken by these VEVs, a common assumption is to neglect all small VEVs. In addition, flavour violating effects are usually negligible. The only exception are scenarios in which large contributions to flavour violation occur in the new physics sector. This could for instance happen in the MSSM with large offdiagonal trilinear softterms which can have a big effect on the Higgs boson mass [67]. Thus, if any of these assumptions is not justified, it is necessary to switch to the purely numerical calculation.
Although the focus of the Mathematica interface is on the derivation of analytical expressions for the matching conditions, additional routines have been implemented to make these results easily usable in numerical calculations. This has the advantage that the obtained code for numerical evaluations can be much faster than the fully numerical interface because many simplifications can be performed on the analytical level. In addition, the obtained results can be exported into LaTeX files which makes a evaluation of the expressions in a human readable format possible. On the other hand, the fully numerical implementation has several advantages: (i) the RGE running above the matching scale can be performed, (ii) shifts to fermionic couplings can be included, (iii) several EFTs appearing in models with more than one matching scale can be automatically linked (iv) flavour violating effects can be included.
 1.
Model files: in principle, one can set up specific model files for the UV theory where for instance EW VEVs are dropped from the very beginning. However, this complicates further studies of the UV theory. Thus, we are going to work in the following with the default model files delivered with SARAH. For instance, we use the MSSM implementation which includes EW VEVs and apply the simplifying assumption to neglect these VEVs during the matching procedure. However, the considered EFT may require the development of further model files. For instance, various splitSUSY models that contain only the fermionic degrees of freedom of their corresponding SUSY models already have been implemented in the new SARAH version.
 2.Normalisation of couplings: in many models studied in literature, the coefficients in front of the scalar couplings are often chosen differently from Eq. (1). For instance, a common convention for the SM Lagrangian reads Thus, after replacing \(H^0 \rightarrow \frac{1}{\sqrt{2}} \left( h + i G + v\right) \) the vertex in Eq. (1) between four Higgs fields h isTherefore, the correct matching condition to calculate \(\lambda _{SM}\) becomes$$\begin{aligned} {\tilde{\kappa }}_{hhhh}=  3 \lambda _{SM}. \end{aligned}$$(21)where \(\kappa _{i}\) denotes treelevel vertices in the UV theory while \({\bar{\delta }}\) are the corresponding oneloop shifts. The relative normalizations between operators in the considered UV and the effective theory, such as for example the factor \(\nicefrac {1}{3}\) in Eq. (22), have to be provided by the user.$$\begin{aligned}&\lambda _{SM} \nonumber \\&= \frac{1}{3} \left( \kappa _{hhhh} + {\bar{\delta }} \kappa _{hhhh} + \sum _{s,t,u}\, \sum _{x \in \{\phi ^H\}} \frac{ \left( \kappa _{hhx} \right) ^2}{M_x^2} \right) ,\nonumber \\ \end{aligned}$$(22)
 3.Superposition of fields: when matching a scalar sector involving multiple (light) scalar fields with identical quantum numbers, often linear combinations of external fields contribute to the matching of different parameters. For instance, consider the couplings \(\lambda _4\) and \(\lambda _5\) in a THDM:where the two SU(2) doublets \(H_1\) and \(H_2\) have the same hypercharge. We find that any vertex involving \(\lambda _4\) receives also contributions from \(\lambda _5\) and vice versa. For instance consider the couplings$$\begin{aligned}&{\mathcal {L}}_{THDM} \nonumber \\&= \dots  \lambda _4 H_1^\dagger H_2^2  \frac{1}{2}\left( \lambda _5 (H_1^\dagger H_2)^2 + \text {h.c.} \right) , \end{aligned}$$(23)$$\begin{aligned} \kappa _{h_1 h_2 H_1^+ H_2^}&= \frac{1}{2} \left( \lambda _4 + \lambda _5 \right) , \end{aligned}$$(24)after splitting the two doublets into their charged (\(H^\pm _{1,2}\)), CPeven (\(h_{1,2}\)) and CPodd (\(A_{1,2}\)) components (note that the gauge eigenstates introduced here also correspond to the mass eigenstates as we assume vanishing VEVs). For simplicity, we assume real parameters . Thus, to obtain the matching conditions for \(\lambda _4\) and \(\lambda _5\) separately, it is necessary to calculate the superpositions$$\begin{aligned} \kappa _{h_1 A_2 H_1^+ H_2^}&= \frac{1}{2} i \left( \lambda _4  \lambda _5 \right) , \end{aligned}$$(25)$$\begin{aligned} \lambda _4&= \left( {\mathcal {M}}(h_1 h_2 H_1^+ H_2^)+i {\mathcal {M}}(h_1 A_2 H_1^+ H_2^)\right) , \end{aligned}$$(26)These conditions are user input as well.$$\begin{aligned} \lambda _5&= \left( {\mathcal {M}}(h_1 h_2 H_1^+ H_2^)i {\mathcal {M}}(h_1 A_2 H_1^+ H_2^)\right) . \end{aligned}$$(27)
3.3 Analytical approach
 1.
one can calculate individual effective couplings in an interactive mode or
 2.
use a batch mode to calculate several matching conditions at once and to optionally obtain LaTeX , Fortran and SPheno outputs.
 Open image in new window

Default: Open image in new window

Description: list of specific parametrisations of selected model parameters

Example: Open image in new window

 Open image in new window

Default: Open image in new window

Description: list of assumptions for parameters in the model in order to simplify the expressions

Example: Open image in new window

 Open image in new window

Default: Open image in new window

Description: list of parameters which are obtained by the tadpole equations

Example: Open image in new window

 Open image in new window

Default: Open image in new window

Description: if set to Open image in new window , the calculation of vertices is skipped, but results stored in a previous session are used. This can be a significant performance boost.

 Open image in new window

Default: Open image in new window

Description: can be used to define an input file containing all necessary information

If the interactive mode is demanded, the option Open image in new window has to be omitted while values for Open image in new window , Open image in new window and Open image in new window should be provided to allow for an analytical diagonalization of all mass matrices. The usage of the batch mode requires only the option Open image in new window and serves a high reproducibility of the obtained results by providing only one single input file.
The provided assumptions and parametrisations are used to calculate analytical expressions for all masses and rotation matrices. If this is not possible, because Mathematica cannot diagonalize the mass matrices analytically (using the buildin functions Open image in new window and Open image in new window ), one can either use the purely numerical interface explained in Sect. 3.4 or choose appropriate simplifying assumptions.
3.3.1 Interactive mode: calculating individual matching conditions
Initializing the matching routines using the Open image in new window function with the options described in the previous paragraph while not specifying the option Open image in new window enables the interactive mode. The necessary vertices of the highscale theory are calculated or loaded from a previous session and the masses/rotation matrices are derived. However, no further calculations are performed at this point.
Example initialization: consider a highscale MSSM scenario where all SUSY particles have a degenerate mass Open image in new window while only the SM Higgs remains light. A possible parametrisation may look like
Note that the symbols Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window are not defined in the MSSM model file. Thus, additional information about these symbols must be provided using the Open image in new window option, otherwise they are assumed to be arbitrary complex numbers. The initialization is invoked by

The symbol Open image in new window is used to indicate dimensionful parameters Open image in new window which are to be neglected in the UV theory. One should always use this parameter instead of the simpler rule Open image in new window to avoid problems caused by a division by 0.

It is recommended to express all matching conditions in terms of the running parameters of the effective theory, see Sect. 2.3. Therefore, we express the MSSM gauge and Yukawa couplings by the SM ones using the suffix Open image in new window which marks the running parameters (instead of Open image in new window we e.g. specify it to be Open image in new window ). For these parameters, only the treelevel matching conditions are required. The oneloop matching conditions for the gauge couplings, discussed in Sect. 2.3, are automatically derived.

Open image in new window is the SARAH internal symbol for the Kronecker delta \(\delta _{ab}{} \). We use it here to include only contributions from third generation Yukawa couplings, and to force diagonal soft masses for the sfermions.

In order to simplify the analytical calculation, we assume that all parameters are real. This is translated by Open image in new window . The object Open image in new window is the SARAH internal command for complex conjugation.
As expected, the spectrum at the matching scale contains one massless CPeven Higgs boson which corresponds to the SMlike Higgs boson. Also all SMlike fermions remain massless while the heavy fields are degenerate in the mass parameter Open image in new window .
The rotation matrices are stored in the array Open image in new window and read in our example
Let us now continue with the description of the analytical interface. After the successful initialization and calculation of all mass and rotation matrices, one can compute the leading order (LO) and nextto leading order (NLO) corrections to an amplitude with the external fields given in the list Open image in new window
 Open image in new window

Default: Open image in new window

Description: list of topologies to include into the calculation. If empty, all topologies are used. Topologies are denoted as in “Appendix A”.

Example: Open image in new window or equivalently Open image in new window .

 Open image in new window

Default: Open image in new window

Description: list of topologies to be excluded from the calculation. The filtering of Open image in new window is also applied on the topology groups given in the Open image in new window option, e.g. if Open image in new window is given in the Open image in new window list but Open image in new window in the Open image in new window list, then only Open image in new window and Open image in new window are computed.

Example: Open image in new window to exclude all contributions on external legs.

 Open image in new window

Default: Open image in new window

Description: list of fields to be excluded when appearing as internal fields.

Example: Open image in new window e.g. to exclude electroweakinos within a split SUSY scenario.

 Open image in new window

Default: Open image in new window

Description: compute only diagrams with certain internal fieldtype patterns. For an empty list all patterns are computed.

Example: Open image in new window computes corrections from heavy scalars only while Open image in new window computes diagrams that contain exactly two internal fermions.

 Open image in new window

Default: Open image in new window

Description: whether to include the contributions from oneloop gauge coupling thresholds to the treelevel amplitude or not

 Open image in new window

Default: Open image in new window

Description: whether to include the \(\overline{\text {MS}}\overline{\text {DR}} \) conversion factors. 0: no, 1: inclusive, 2: exclusive, Open image in new window : decide between 1 and 0 depending on the type of considered model (SUSY or nonSUSY). Open image in new window means that only the conversion factor is calculated while Open image in new window gives the full result plus conversion factor (default for SUSY models).

 Open image in new window

Default: Open image in new window

Description: multiplies each amplitude with a debug variable marking its topology and field insertion

Example: the term Open image in new window Open image in new window may be multiplied with the expression of the amplitude of the triangle diagram ( Open image in new window , see Eq. (A.6)) with three heavy internal Higgs bosons ( Open image in new window ).

 Open image in new window

Default: Open image in new window

Description: whether to simplify the results using the given assumptions or not.

 Open image in new window

Default: Open image in new window

Description: the amplitudes contain loop functions in the FormCalc notation (e.g. a \(B_0(0,m_1^2,m_2^2)\) function is denoted by Open image in new window ). The function Open image in new window replaces them with the IRsave loop functions defined in “Appendix B”. However, for a better readability one may set this to the Open image in new window function.

Example: Open image in new window

Example calculation: proceeding with the highscale MSSM example i.e. the MSSM \(\rightarrow \) SM matching we can use the introduced functions to calculate the expressions for the effective quartic coupling \(\lambda _{SM}{} \) of the SM Higgs boson at the matching scale. The treelevel matching condition is calculated as follows
the output reads
where we have introduced the stop mixing parameter \(X_t=A_t  \mu \tan ^{1}\beta \), yields
3.3.2 Batch mode
The complexity of the calculation requires a high degree of reproducibility of the results. For this purpose it is possible to write input files that contain all necessary information for the matching to a given EFT model. This includes all information already discussed in the interactive mode. In addition, the correspondence between effective couplings in the lowenergy model and amplitudes in the UV model, as it was demonstrated for the THDM matching, have to be defined.
The batch mode is invoked during the initialisation by specifying the input file Open image in new window located in the directory of the loaded SARAH model
The mandatory content of the input file is

Open image in new window : defines a name for the current setup. This also determines the name of the output directory in which the results are saved into as well as the file name of the SPheno binary.

Open image in new window : the parametrisation in the UV. This is equivalent to option Open image in new window when running Open image in new window without an input file.

Open image in new window : a list of simplifications which are only applied at the matching scale.

Open image in new window : a list of assumptions at the matching scale equivalent to the Open image in new window option when running Open image in new window without input file.

Open image in new window : the equivalent to the option Open image in new window of Open image in new window .

Open image in new window : a list of matching conditions which relates a parameter in the EFT to amplitudes in the highscale model containing light external fields only, similar to Eqs. (26) and (27).
Here, we skipped most of the lines for Open image in new window because they are similar to the definition of Open image in new window in the last subsection. For simplicity, we set here all trilinear sfermion couplings as well as the matching scale Open image in new window equal to Open image in new window .
$ SARAH_Directory/Output/ $ Model/EWSB/Matching/ $ NameUV
Thus, one can work with the results within other Mathematica sessions as well.

Open image in new window : if set to Open image in new window , all information obtained during the matching is exported into a LaTeX file ready to be compiled by standard LaTeX compilers.

Open image in new window : a list containing replacement rules that define the correspondence between LaTeX and Mathematica expressions which are for instance used in the defined parametrisation. This will improve the readability of the LaTeX document significantly.
 1.
Export the Mathematica expressions into Fortran code and write a corresponding SPheno.m file
 2.
Run the EFT model using this SPheno.m
The export into SPheno routines is enabled with the first line. This option is sufficient to obtain Fortran routines for all matching conditions at the oneloop level. All other information must be given to automatically generate a suitable Open image in new window for the EFT model. Most variables have a 1:1 correspondence to the standard variables (without the Open image in new window prefix) used in Open image in new window files discussed in “Appendix C”. The new option is Open image in new window which defines at which scale the matching should be performed.
 The file Open image in new window Open image in new window , located in the output directory of the MSSM model, which contains the matching conditions in Fortran format where most of the terms in the sum have been omitted as they are not important for the discussion.
 a Mathematica file named Open image in new window Open image in new window which is located in the model directory of the SM. This file may look like One can see that this file contains the information given to Open image in new window (line 1–12). In addition, the information about the matching and the corresponding Fortran routines (using parameter without the Open image in new window prefix) have been automatically added by SARAH (line 14–16 and 21–26).
 1.
Copy the SARAH output to a new subdirectory of your SPheno installation^{4}
 2.Copy the code to a new SPheno subdirectory
 3.Compile the code
 4.Run SPheno
3.3.3 Matching at two scales
3.4 Numerical approach
 1.
MatchingToModel is used to define the UV model(s), i.e. the model directory in SARAH.
 2.
IncludeParticlesInThresholds defines the list of particles which are included in the loop calculations.
 3.
AssumptionsMatchingScale is used to define simplifying assumptions at the matching scale. A common choice is to neglect the contributions from EW VEVs or other small parameters.
 4.
BoundaryMatchingScaleUp defines the boundary conditions to relate the parameters of the UV theory to the running parameters of the EFT when the RGEs run from low to high scales.
 5.
BoundaryMatchingScaleDown defines the boundary conditions to relate the parameters of the UV theory to the running parameters of the EFT when the RGEs run from high to low scales.
 6.
ParametersToSolveTadpoleMatchingScale defines the parameters that are fixed by the tadpole equations in the full theory.
Several examples for the usage of these options are given below.
3.4.1 One matching scale without RGE running above
Note that for simple highscale theories, without additional light fields, the Open image in new window could also be stored in the SM model directory as the two models are technically the same. The newly introduced models are described in Sect. 3.5.
The definitions are very similar to the analytical approach: the symbol Open image in new window again has been used to neglect specific parameters at the matching scale. An important difference is that we have not singled out the contributions from only third generation Yukawas because this would not give any performance improvement for the numerical calculation. Note, that it is also not necessary to define the matching for the Yukawas when running down. Moreover, we have used the option to define the scale where the RGE running should stop as function of an input parameter ( Open image in new window ).^{5} Thus, SPheno will run the RGEs only to that scale and evaluate the SUSY boundary conditions.
 1.Run MakeSPheno of SARAH with the new input file
 2.Copy the files and compile SPheno
 201

Turn on/off all oneloop contributions to the matching. By default, they are turned on. This might be helpful to check the size and importance of the oneloop corrections.
 202

Turn on/off the contributions from the offdiagonal wavefunction renormalisation. By default, they are turned off. See Sect. 2.2 for more details.
3.4.2 Running above the matching scale
Thus, SPheno stops the running once the condition \(g_1(Q)=g_2(Q)\) is fulfilled.
In addition, the matching conditions for the Yukawas are changed to
The need for the normalization onto the treelevel rotation matrix elements Open image in new window is described in the next section. In that way, we can include the oneloop shifts to all Yukawa couplings which are necessary to have a consistent RGE running with twoloop SUSY RGEs between the matching and GUT scale, see also the discussion in Sect. 2.4. Note, we did not consider any generation indices for the involved fermions, i.e. the result of ShiftCoupNLO is a \(3\times 3\) matrix. If one wants to safe program runtime it is possible to consider the oneloop shifts to the top Yukawa couplings only.
Moreover, the shifts for the gauge couplings are applied automatically.
3.4.3 Several matching scales
With the above settings one can now implement an arbitrary number of matching scales. However, as we have noted already in Sect. 3.3.3, the polemass matching to the SM is automatically included in the SPheno output. Thus, if a second matching scale, which is not too far away from the EW scale, is needed, one can simply rely on that. However, if more than two matching scales are needed, or if the matching to the SM should take place at such a high scale where the polemass matching might suffer from numerical problems,^{6} one can now start to build up towers of EFTs by defining more matching scales in SPheno.m. For instance, the full input to define the tower
SM \(\rightarrow \) THDM \(\rightarrow \) THDM + electroweakinos \(\rightarrow \) MSSM
Here, \(g_{1,2}^{u,d}{} \) are the splitSUSY couplings between the Higgs boson and a HiggsinoGaugino pair, see e.g. Ref. [10]. We include these corrections by considering the oneloop amplitude between the Higgs boson and a pair of neutralinos. In this example we have also used another feature: we have not explicitly defined the generation indices of the involved neutralinos. The reason for this is: even if the neutralino mass matrix contains only zero’s under the given approximations (\(\mu ,M_i \ll M_{\mathrm{SUSY}}{} \)), it is not clear how the mass eigenstates are ordered in the numerical run. Therefore, we have used the name of the gauge eigenstates. By doing that, SPheno checks during the numerical evaluation which of the mass eigenstates has the biggest contribution of the given gauge eigenstate. Of course, if the rotation matrix for the neutralinos is not equivalent to the unit matrix, i.e. if some mixing appears for instance because of effects of nonvanishing \(\mu \), one needs to define
Thus, the rotation to mass eigenstates, which should take place just at the weak scale, is divided out.
3.4.4 Summary
3.5 Included models and input files in SARAH
The names of the new models that make use of the numerical approach are listed in Table 1. Also for the analytical approach several input files are now included in SARAH. Those are summarised in Table 2. Based on these examples and by the explanations in this section, it is now straightforward for the users to implement their own scenarios.
4 Examples, selfconsistency checks and comparisons with other codes
The following section describes realistic examples of practical applications of the presented framework. We consider different highscale SUSY scenarios which were already studied intensively in literature. In particular comparisons between predictions for the SM Higgs boson mass derived with our generic setup against dedicated tools and calculations are made. In this context, we demonstrate also the perfect agreement between the two available options to use SARAH/SPheno for numerical studies. Finally, we also show that one can easily obtain precise results for other highscale extensions for which no other tool existed so far.
4.1 Lowenergy limits of the MSSM
The names of the new models which are part of SARAH 4.14.0. The hierarchy in the last column refers to Fig. 7. For the splitNMSSM also a light singlet is present, i.e. the hierarchy is similar to \((c)\), but not identical
Model name  EFT  UV model(s)  Hierarchy 

HighScaleSUSY/MSSM  SM  MSSM  (a) 
HighScaleSUSY/NMSSM  SM  NMSSM  (a) 
HighScaleSUSY/MSSMlowMA  THDM  MSSM  (b) 
SplitSUSY/MSSM  SM+EWkinos  MSSM  (c) 
SplitSUSY/NMSSM  SM+singlet+EWkinos  SMSSM  \(\sim \)(c) 
SplitSUSY/MSSMlowMA  THDM+EWkinos  MSSM  (d) 
SplitSUSY/MSSM_2scale  SM  MSSM \(\rightarrow \) SM+EWkinos  (e) 
SplitSUSY/MSSM_3scale  SM  MSSM \(\rightarrow \) THDM+EWkinos \(\rightarrow \) THDM  (f) 
Input files for the analytical approach which are now delivered with SARAH. The hierarchy in the last column refers to Fig. 7. For the splitNMSSM also a light singlet is present, i.e. the hierarchy is similar to \((c)\), but not identical
File name  EFT  UV model  Hierarchy 

MSSM/Matching_HighScaleSUSY.m  SM  MSSM  (a) 
NMSSM/Matching_HighScaleSUSY.m  SM  NMSSM  (a) 
MSSM/Matching_SplitSUSY.m  SM+EWkinos  NMSSM  \(\sim \)(c) 
MSSM/Matching_THDM.m  THDM  MSSM  (b) 
SMSSM/Matching_SplitSUSY.m  SM+singlet+EWkinos  SMSSM  (c) 
4.1.1 SplitSUSY: MSSM \(\rightarrow \) SM & electroweakinos & gluinos
We have compared the analytical expressions of Ref. [10] for the oneloop thresholds with the results of SARAH and found perfect agreement. Thus, we can immediately go to the discussion of the comparison of the numerical results of SPheno and SusyHD. Even if the expressions for the thresholds agree, there are many other ingredients which enter the Higgs mass prediction. Most importantly, the determination of the top Yukawa coupling which affects all comparisons shown here. Also higherorder corrections for highscale SUSY scenarios are implemented to some extent in other codes which are not (yet) available in our generic setup. The corresponding model in SARAH which we have set up for this scenario is
SplitSUSY/MSSM
We show in Fig. 8 the calculated Higgs boson mass by SusyHD^{7} and SPheno as function of \(M_{\mathrm{SUSY}}{} \) for two different choices of \(M_{\chi }\). First of all, one can see that the overall agreement is very good between all calculations: for the two calculations implemented in SARAH/SPheno we find agreement up to the numerical precision, while the biggest difference between SPheno and SusyHD is well below one GeV for all considered values of \(M_{\mathrm{SUSY}}{} \).
4.1.2 Highscale SUSY: MSSM \(\rightarrow \) SM
HighScaleSUSY/MSSM
Moreover, we find that the polemass matching becomes also numerically unstable – at least in SPheno– once \(M_{\mathrm{SUSY}} \gg v\) is used because the loop functions used for the polemass calculations are not optimised for these cases: we see in Fig. 9 that the polemass matching breaks down at \(M_{\mathrm{SUSY}} \simeq 5 \cdot 10^5\) GeV. Nevertheless, we find that the agreement between the polemass matching and the direct matching procedure presented here is very good for SUSY scales up to 100 TeV. One finds also that the fixedorder calculation agrees perfectly with the polemass matching for \(M_{\mathrm{SUSY}}{} \) below 1 TeV. Of course, for larger SUSY scales, the discrepancy between the fixedorder calculation and all EFT calculations grows very rapidly.
4.1.3 Highscale SUSY with intermediate \(M_A: MSSM \rightarrow \) THDM
The dominant threshold corrections to \(\lambda _1\)–\(\lambda _7\) involving third generation Yukawa couplings are available in literature [29]. We have double checked the analytical expressions derived by SARAH and found full agreement.
Since we have demonstrated the importance of performing the matching to the THDM properly for the case of a light second Higgs doublet, it is clear that codes were developed to include these effects. The first tool in this direction was MhEFT which uses a purely EFT ansatz [36]. In a recent update of FeynHiggs a hybrid ansatz combining the fixedorder calculation with higherorder terms was implemented [40].The overall agreement between both codes turned out to be good once a careful translation between the parameters in both renormalisation schemes was done. Since MhEFT is much closer to the ansatz of SARAH/SPheno we are going to compare our results with this tool.^{9} For this purpose, we have set up the model
HighScaleSUSY/MSSMlowMA
in SARAH. We show in Fig. 11 the results of MhEFT and SPheno when varying \(\tan \beta \) or \(M_A\) for a fixed SUSY scale of 100 TeV. The agreement between both codes is always good. The maximal difference for comparable calculations is about 0.5 GeV and can be even smaller for \(M_A\) below 500 GeV and arbitrary values of \(\tan \beta \). The differences are due to the threeloop RGEs which are included in MhEFT in the running between \(m_t\) and \(M_A\) while SPheno uses always twoloop RGEs. This explains the flattening of the difference as the top quark Yukawa coupling runs fastest near the weak scale. One can also see that the impact of the additional twoloop corrections implemented in MhEFT is very moderate. Thus the main source of the difference is the determination of the running top Yukawa coupling. In contrast, the additional oneloop corrections due to gauginos, which were presented very recently also in Ref. [40], can be easily included in SPheno using the numerical matching interface. For the considered choice of parameters these have numerically a bigger effect than the twoloop corrections and cause a shift of 1–1.5 GeV.
4.2 Highscale NMSSM
HighScaleSUSY/NMSSM
Of course, one could now start to consider also other lowenergy limits of the NMSSM. However, this is beyond the scope of this paper here and interesting applications are given elsewhere [71].
5 Summary
We have presented an extension of the Mathematica package SARAH which derives the oneloop matching conditions for effective scalar couplings based on a UV theory. Two different approaches exists, which are based on either an analytical or fully numerical calculation. The full agreement between both calculations and analytical results available in literature has been pointed out. Furthermore, good agreement with specialised codes to study Split or Highscale SUSY like SusyHD or MhEFT was shown. Since our approach is completely general, it can be used to study UV completions of a large variety of BSM models with and without an extended Higgs sector.
Footnotes
 1.
SARAH is available at hepforge: sarah.hepforge.org.
 2.
stauby.de/sarah_wiki/.
 3.
 4.
SPheno can be downloaded from spheno.hepforge.org.
 5.
The naming of this keyword, which was originally introduced for other purposes, might be misleading because the chosen scale need not be connected to any GUT theory.
 6.
We elaborate a bit on that issue in Sect. 4.1.2.
 7.
During this comparison we found a bug in the twoloop RGEs of \(\lambda _{SM}{} \) for splitSUSY as implemented in SusyHD. The contribution \(\frac{21}{2} {\tilde{g}}_{2d}^2 {\tilde{g}}_{2u}({\tilde{g}}_{1d}^2+{\tilde{g}}_{1u}^2)\) misses one power of \({\tilde{g}}_{2u}\). We fixed that and in all following results the patched version of SusyHD is used.
 8.
Strictly speaking, one obtains a THDM typeIII when integrating out all SUSY fields in the MSSM because the ‘wrong’ Yukawa couplings \(\sim H_d^* {\bar{q}} u\) are loopinduced. However, this becomes mainly important for flavour violating observables and has no visible impact on our discussion of the Higgs boson mass prediction here.
 9.
For simplicity, we modified MhEFT to take \(A_t\) as input instead of \(X_t^{\overline{\text {MS}}}\).
Notes
Acknowledgements
We thank Mark Goodsell for many fruitful discussions about the matching of scalar couplings and other related topics as well as Pietro Slavich for proof reading the manuscript. FS is supported by the ERC Recognition Award ERCRA0008 of the Helmholtz Association. MG acknowledges financal support by the GRK 1694 “Elementary Particle Physics at Highest Energy and highest Precision”.
Supplementary material
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