One-Loop Threshold Corrections in Scalar Sectors

As of version 4.14.0 SARAH is able to compute threshold corrections to scalar and yukawa couplings in effective theories by integrating out all heavy degrees of freedom from a full theory, see the related SARAH paper. There is also a file Example_Matching.nb, located in SARAH's root directory, which contains a comprehensive example of the MSSM->SM as well as MSSM->THDM matching.

The matching of effective couplings consists of the following ingedients:

diagonalisation of mass matrices at the matching scale Q_{\rm match} \gg v_{\rm SM}
diagrammatic calculation of effective couplings
- assume vanishing external momenta p² → 0

Thus, effective low-energy couplings can be expressed in terms of high-scale parameters.

SARAH provides two different possibilities to perform the matching between two arbitrary scalar sectors:

An analytical calculations within Mathematica
A fully numerical calculation using only the SPheno interface

The numerical approach clearly has the advantage to be able to diagonalise the mass spectrum in more complicated situations while the fully analytical approach always needs to neglect all dimensionful parameters that are below the TeV scale, e.g. v_{\rm SM}\to 0, which effectively neglects contributions of higher dimensional operators important near the TeV scale. Thus, the numerical approach should be prefered if BSM scales near or even below \mathcal{O}({\rm 1 TeV}) are considered. Also falvour violating effects are better studied with the numerical approach. However, the analytical approach is well suited to study the influence of certain fields, diagram topologies and couplings on the analytical level.

Feature	Numerical	Analytical approach
Sophisticated selection of specific contributions	no	yes
(Off)-diagonal WFR contributions	yes	yes
Effective scalar operators	yes	yes
Effective yukawa operators above and below the matching scale	yes	no
Mass matrix diagonalisation	numerical	analytical
Matching of EFT towers	yes	no
Export analytical result of threshold corrections	no	yes

Fully Analytical Approach

Thus, the computed matching conditions can be used to generate SPheno output which calculates e.g. Higgs boson masses in a very precise way (right firgure).

While the #Interactive Mode allows only for the computation of dedicatet effective couplings, the #Batch Mode provides a fully automatized creation of such a SPheno version based on one single steering file.

Initialization

For the actual computation, the model of the full theory (containing the heavy degrees of freedom e.g. the MSSM), has to be loaded

<<SARAH`
Start["MSSM"]

Then, certain simplifications have to be provided in order to allow for an analytical diagonalization of all mass matrices. VEVs in that are of the order of the electroweak scale have to be neglected, i.e. all fields in the low-energy EFT are assumed to be massless. To do so, put the simplification VEV->epsUV into the parametrisation list described below. epsUV marks dimensionful parameters that are to be neglected during the matching. Do not simply set the VEVs to zero as this causes problems in e.g. the calculation of tadpole equations (division by zero).

To get informations about how to initialise the matching routines type ?InitMatching:

InitMatching[options] initializes the matching routines used for the calculation of effective scalar couplings.

The function can be used in two modi controlled by the option InputFile-> False/filename.m
where filename.m is an input file located in the directory of the currently loaded model.

If an input file is given the matching is performed as described in the file.
If InputFile->False, the following options can be used:

 - Simplifications -> {vu->Sin[\[Beta]] epsUV, ...} : list of simplifications used for the analytic diagonalization of mass matrices as well as all other calculations.

 - Approximations -> {\[Beta]>0,\[Beta]<Pi/2} : assumptions used to simplify expressions

 - SolveTadpoles -> {mHu2,mHd2} : parameters to solve tadpoles for

After the initialization (no input file given) use the functions ?EFTcoupNLO and ?EFTcoupLO to calculate effective couplings.
See also Options[InitMatching].

Interactive Mode

This section assumes that InitMatching was invoked without the InputFile option but with sufficient Simplifications and Approximations such that the analytical diagonalisation was successfull. To check this one can have a look at ?M where the result of the diagonal mass matrices is stored.

As stated above, the interactive mode is not able to generate fully automatized SPheno code. However, certain features are dedicated to this mode to simplify the study of the analytical structer of the different contributions:

restrict the computation on certail diagram topologies (boxes, triangles, etc.)
explicitly exclude diagrams containing certain internal fields (e.g. exclude internal hh fields)
restrict the computation on diagrams with certain internal field types ( e.g. only diagrams with four internal scalars etc.)
control the computation of
- MS-DR conversion factors
- gauge coupling thresholds
- (off-)diagonal WFR contributions
turn on debugging to see which fields contribute through which diagrams in some way

this makes it easy to get an impression on how e.g. certain fields contribute and motivates choices for numerical studies including all contributions.

One-Loop Matching

Use ?EFTcoupNLO

EFTcoupNLO[{field1[index1],field2[index2]},...,options] calculates the effective coupling between field1, field2, ... with the generation indices index1, index2, ... at the one-loop order in the limit of vanishing external momenta.

The most important options and there default values are:

- ExcludeFields -> {}  : fields contained in this list do not enter the calculation of the effective coupling i.e. diagrams with internal propagators of these fields are excluded. Example: {Chi, Cha}.

- InternalPatterns -> {} : compute only diagrams with certain internal field-type patterns e.g. two internal scalars/fermions {SS,FF} etc. For an empty list all patterns are computed. Example: {S,SS,SSS,SSSS} computes corrections from heavy scalars only.

- Topologies -> {} : restrict the calculation to a subset of topologies. For an empty list, all topologies are considered. Example: Topologies -> {D} computes box diagrams only. Also a group of topologies e.g.  {B} can be given which includes all topologies that contain a two-point function.

- ExcludeTopologies -> {OffdiagonalWFRs} : excludes topologies from the calculation. The filtering of ExcludeTopologies is also applied on the topology groups given in the Topology list.

- Debug -> False : If set to True, all amplitudes are multiplied with a term debug[<topology>][<fieldinsertions>] that identifies the topology and the internal fields.
- ShiftMSDR -> Automatic : includes the MD-DR conversion in the calculation. If set to 1, the conversion is computed even in a non-SUSY model, if set to 1, the conversion is computed exclusively.

- GaugeThresholds -> True : wheter to include the computation of gauge coupling thresholds which may enter the tree-level effective coupling.

See also Options[EFTcoupNLO].

Example: EFTcoupNLO[{hh[1],hh[1],hh[1],hh[1]}, Topologies->{C}] computes all triangle diagrams.

Al list of all options is:

Topologies -> $LIST
- Default: {}
- Description: list of topologies to include into the calculation. If empty, all topologies are used. Topologies are denoted as in Appendix A of ARXIVLINK.
- Example: {B[4][1],B[4][2][1], B[4][2][2]} or equivalently {B[4]}.
ExcludeTopologies -> $LIST
- Default: {OffdiagonalWFRs}
- Description: list of topologies to be excluded from the calculation. The filtering of ExcludeTopologies is also applied on the topology groups given in the Topologies option, e.g. if {B[4]} is given in the topologies list but B[4][2][2] in the ExcludeTopologies list, then only B[4][1] and B[4][2][1] are computed.
- Example: {OffdiagonalWFRs, DiagonalWFRS} to exclude all contributions on external legs.
ExcludeFields -> $LIST
- Default: {}
- Description: list of fields to be excluded when appearing as internal fields.
- Example: {Cha,Chi} e.g. to exclude electroweakinos within a split SUSY scenario.
InternalPatterns -> $LIST
- Default: {}
- Description: Description: compute only diagrams with certain internal field-type patterns. For an empty list all patterns are computed.
- Example: {S,SS,SSS,SSSS} computes corrections from heavy scalars only while FF computes diagrams that contain exactly two internal fermions.
GaugeThresholds->$BOOL
- Default: True
- Description: whether to include the contributions from gauge coupling thresholds to the amplitude.
ShiftMSDR-> 0/1/2/Automatic
- Default: Automatic
- Description: whether to include the MS − DR conversion factors. 0: no, 1: inclusive, 2: exclusive, Automatic : decide between 1 and 0 depending on the type of considered model (SUSY or non-SUSY).
Debug -> $BOOL
- Default: False
- Description: multiplies each amplitude with a debug variable marking its topology and field insertions
- Example: debug[C[4][1]][hh[2], hh[2],hh[2]] may be multiplied with the expression of the triangle diagram with three heavy internal Higgs bosons.
SimplifyResults -> $BOOL
- Default: True
- Description: whether to simplify the results using the given assumptions
LoopReplace -> $FUNCTION
- Default: AnalyticLoopFunctions
- Description: the amplitudes contain loop functions in the FormCalc notation (e.g. a B₀(0, m₁², m₂²) function is denoted by B0i[bb0,0,m1^2,m2^2] ). The function AnalyticLoopFunctions replaces them with the IR-save loop functions defined in Appendix B of ARXIVLINK. However, for a better readability one may set this to the Identity function.
- Example: Identity

Tree-Level Matching

Use EFTcoupLO. Note that only the options ExcludeFields, Debug and SimplifyResults are available.

Topologies

The topologies are described in appendix A in LINK TO PAPER. To get a list of all available topologies and groups of topologies see the list TopoNotation.

Fortran Output

If a standard RGE running, matching and mass calculation using SPheno is demanded, the Batch mode described in the next section should be used. However, for the usage of pieces of the calculations obtained in the interactive mode in other programs or custom SPheno versions one can either export the Mathematica expressions or use the function GenerateSelfDefinedFuntion[“MyRoutineName”, ] to generate a Fortran routine. By default, all symbols are assumed to be complex. Ensure to set conj[parameter]=parameter for all real parameters as well as to replace Mathematica symbols (e.g. greek letters, mathematical typesetting etc.) with undefined ones that are compatible with yout Fortran code.

Batch Mode

For a fully functional high-scale SPheno version the Batch mode is recommended as it can create

a high-scale SPheno version which uses the pre-computed matching conditions as boundary condition at the high scale and
LaTeX output for the evaluation of the matching conditions, mass matrices, ... in a human readable way.

To start the batch mode run InitMatching[InputFile-> “File name located in the directory of the loaded model.”] where the .m file may contain the following information

(* ----------------------------------------- *)
(* Informations about the Matching           *)
(* ----------------------------------------- *)

$NameUV="HighScaleMSSMlowMA"; (* Name of the  output directory and the SPheno binary *)

$ParametrisationUV = { (* similar to the Parametrisation-option of InitMatching *)
   vd -> epsUV,
   vu -> epsUV,
   Yu[a_,b_] :> Delta[3,b] Delta[a,3] Yu[a,b],
   Yd[a_,b_] :> Delta[3,b] Delta[a,3] Yd[a,b],
   Ye[a_,b_] :> Delta[3,b] Delta[a,3] Ye[a,b],
   T[Yu][a__] :> Delta[a] Azero Yu[a],
   T[Yd][a__] :> Delta[a] Azero Yd[a],
   T[Ye][a__] :> Delta[a] Azero Ye[a],
   mq2[a__] :> Delta[a] m0^2,
   mu2[a__] :> Delta[a] m0^2,
   md2[a__] :> Delta[a] m0^2,
   me2[a__] :> Delta[a] m0^2,
   ml2[a__] :> Delta[a] m0^2,
   MassB->m12,
   MassWB->m12,
   MassG->m12,
   B[\[Mu]] -> epsUV^2,
   \[Mu]->MuSUSY,
   conj[x_] -> x
};

$AssumptionsMatching={ (* similar to the Assumptions-option of InitMatching *)
 TanBeta>0,
 m0>0,
 m12>0,
 MuSUSY>0
}

$ExcludeFieldsMatching={};  (* similar to the ExcludeFields-option of EFTcoupNLO *)

$SolveTadpolesUV = {mHd2,mHu2}; (* similar to the SolveTadpoles-option of InitMatching *)

$MatchingConditions = { (* conditions to be applied at the matching scale *)
   Lambda1 -> -1/6 hh[1].hh[1].hh[1].hh[1],
   Lambda2 -> -1/6 hh[2].hh[2].hh[2].hh[2],
   Lambda3 -> -hh[1].hh[1].Hpm[2].conj[Hpm[2]],
   Lambda4 -> hh[1].hh[2].Hpm[2].conj[Hpm[1]] + I*hh[1].Ah[2].Hpm[1].conj[Hpm[2]],
   Lambda5 -> hh[1].hh[2].Hpm[2].conj[Hpm[1]] - I*hh[1].Ah[2].Hpm[1].conj[Hpm[2]],
   Lambda6 -> -hh[1].hh[2].Hpm[1].conj[Hpm[1]],
   Lambda7 -> -hh[1].hh[2].Hpm[2].conj[Hpm[2]]
   };

(* ----------------------------------------- *)
(* LaTeX Output                              *)
(* ----------------------------------------- *)

$EFTcouplingsToTeX=True;
$AdditionalTeXsymbols={
 {Lambda1, "\\lambda_1"},
 {Lambda2, "\\lambda_2"},
 {Lambda3, "\\lambda_3"},
 {Lambda4, "\\lambda_4"},
 {Lambda5, "\\lambda_5"},
 {TanBeta, "t_{\\beta}"},
 {m0, "m_0"},
 {m12, "M_{1/2}"},
 {MA, "M_A"},
 {MuSUSY, "\\mu"},
 {Azero, "A_0"}
};


(* ----------------------------------------- *)
(* SPheno Output                             *)
(* ----------------------------------------- *)

$ExportToSPheno=True;
$SPhenoEFTmodel="THDM-IInoZ2";
$SPhenoMINPAR={
 {1, m0},
 {2, mGauginos},
 {3, TanBeta},
 {4, MuSUSY},
 {5, Azero},
 {6, MA}};

$SPhenoBoundaryHighScale={};
$SPhenoBoundaryRenScale={
 {M12, -MA^2 TanBeta/(1+TanBeta^2)}
 };
$SPhenoTadpoles={M112,M222};
$SPhenoMatchingScale=m0;
$SPhenoRenScale=MA^2;
$SPhenoMatchingEWSB=Default[THDMII];

The option concerning the matching which is new compared to the interactive mode is $MatchingConditions. It defines the matching condition which is applied at the matching scale $SPhenoMatchingScale. The first entry of the replacement list is a coupling in the low-energy theory (in this case of the THDM) while the second one is the corresponding amplitude in the UV-model. The prefactors are due to different normalizations of the fields in the two SARAH models.

The LaTeX options should be self-explanatory. The list AdditionalTeXsymbols should contain all additional symbols which are not defined in the UV-model or haven't got a LaTeX definition yet.

The generated Fortran and LaTeX routines are saved into the output directory of the UV model: $SARAH_Directory/Output/Modelname/EWSB/Matching/$NameUV. While the TeX files can be compiled with any standard LaTex compiler, the Fortran routines contain only the effective couplings but are included during the SPheno generation.

The variables prepended with $SPheno will be used to generate a new SPheno.m and correspond to the usual variables used there. The new SPheno.m will then include the Fortran routines in the $MatchingConditions applied at the scale $SPhenoMatchingScale. The location of the SPheno.m is the directory of the model $SPhenoEFTmodel. It can be used in a seperate Mathematica kernel to make SPheno as usual.

Numerical Approach

The key difference to the analytical approach is the ability to also compute threshold corrections to Yukawa couplings, necessary for the running above the matching scale.

Example

(*---------------------------------------------------*)
(* information for matching to MSSM *)
(*---------------------------------------------------*)

MatchingToModel= {"MSSM"};
MatchingScale = {m0};

IncludeParticlesInThresholds={
 {hh,Ah,Hpm,Su,Sd,Se,Sv,Chi,Cha}
};

AssumptionsMatchingScale={
 {
  {vd,epsUV*Cos[ArcTan[TanBeta]]},
  {vu,epsUV*Sin[ArcTan[TanBeta]]}
 }
};


BoundaryMatchingScaleUp={
{
  {Yu, Sqrt[1+TanBeta^2]/TanBeta*Yu},
  {Yd, Sqrt[1+TanBeta^2]*Yd},
  {Ye, Sqrt[1+TanBeta^2]*Ye}
 }
};

BoundaryMatchingScaleDown={
{
  {\[Lambda], -1/3 EFTcoupNLO[hh[1].hh[1].hh[1].hh[1]]}
 }
};

ParametersToSolveTadpoleMatchingScale={
 {mHd2,mHu2}
};

Implemented Models and Hierarchies

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	∼(c)
SplitSUSY/MSSMlowMA	THDM+EWkinos	MSSM	(d)
SplitSUSY/MSSM_2scale	SM	MSSM -> SM+EWkinos	(e)
SplitSUSY/MSSM_3scale	SM	MSSM -> THDM+EWkinos -> THDM	(f)

File Name	EFT	UV mode	hierarchy
MSSM/Matching_HighScaleSUSY.m	SM	MSSM	(a)
NMSSM/Matching_HighScaleSUSY.m	SM	NMSSM	(a)
MSSM/Matching_SplitSUSY.m	SM+EWkinos	NMSSM	∼(c)
MSSM/Matching_THDM.m	THDM	MSSM	(b)
SMSSM/Matching_SplitSUSY.m	SM+singlet+EWkinos	SMSSM	(c)

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