Table of Contents
- 1 Model information
- 2 Input parameters
- 3 Boundary conditions
- 4 Convergence tester
- 5 Renormalization group equations (RGEs)
- 6 Pole masses
- 7 Lightest supersymmetric particle (LSP)
- 8 User-defined replacement rules
- 9 Observables
- 10 Output blocks
- 11 Model-specific switches
- 12 References
Symbol: FSModelName
Default value: unset
Description:
Name of the model class within the generated code. If FSModelName
is set to the string "@CLASSNAME@", it will be replaced by the
createmodel script to the name of the FlexibleSUSY model given
during the ./createmodel --name=<model_name> command.
Symbol: FSEigenstates
Default value: SARAH`EWSB
Description:
The name of the particle eigenstates in SARAH. Usually,
``SARAH``EWSB`` corresponds to the mass eigenstates after breaking of
the electroweak symmetry.
Symbol: FSDefaultSARAHModel
Default value: unset
Description:
Name of the SARAH model to be used. A sub-model can be specified
after a /.
Example: In the constrained CP-conserving MSSM (CMSSM) the SARAH
model MSSM is used as:
FSDefaultSARAHModel = MSSM;
Example: In the constrained CP-violating MSSM (CMSSMCPV) the SARAH
model MSSM together with the sub-model CPV is used:
FSDefaultSARAHModel = MSSM/CPV;
Symbol: FSBVPSolvers
Default value: { TwoScaleSolver }
Description:
A list of algorithms to use for solving the boundary value problem.
One or both of TwoScaleSolver or SemiAnalyticSolver may be
specified in the list.
Symbol: MINPAR
Default value: {}
Description:
In the MINPAR variable a list of input parameters for the spectrum
generator can be given, which is read of the MINPAR block of the
SLHA input file.
MINPAR is supposed to contain a list. The list elements are
two-component lists, where the first in element is an integer number
representing the index inside the MINPAR block. The second element
is the input parameter. The input parameter must be either a symbol
or a sign of the form Sign[p], where p is the name of a model
parameter.
Example: In the CMSSM the MINPAR block has the form:
MINPAR = {
{1, m0},
{2, m12},
{3, TanBeta},
{4, Sign[\[Mu]]},
{5, Azero}
};
In this case the input parameters can be given in the SLHA input file as:
Block MINPAR # Input parameters
1 125 # m0
2 500 # m12
3 10 # TanBeta
4 1 # SignMu
5 0 # Azero
Note
Unspecified parameters are assumed to be zero.
Symbol: EXTPAR
Default value: {}
Description:
The EXTPAR variable is a list of input parameters for the spectrum
generator, which is read of the EXTPAR block of the SLHA input file.
The list assigned to the EXTPAR variable must have the same form as
the MINPAR variable.
Example: In the NUTNMSSM the EXTPAR block has the form:
EXTPAR = {
{61, LambdaInput},
{62, KappaInput},
{63, ALambdaInput},
{64, AKappaInput},
{65, MuEff}
};
In this case the input parameters can be given in the SLHA input file as:
Block EXTPAR # Input parameters 61 0.650 # LambdaInput 62 0.164 # KappaInput 63 763.8 # ALambdaInput 64 1268.2 # AKappaInput 65 265.2 # MuEff
Note
Unspecified parameters are assumed to be zero.
Symbol: IMMINPAR
Default value: {}
Description:
The IMMINPAR variable is a list of input parameters for the spectrum
generator, which is read of the IMMINPAR block of the SLHA input
file. The list assigned to the IMMINPAR variable must have the same
form as the MINPAR variable.
Example: In the CP-violating MSSM (CMSSMCPV) the IMMINPAR block
has the form:
IMMINPAR = {
{2, Imm12},
{5, ImAzero}
};
In this case the input parameters can be given in the SLHA input file as:
Block IMMINPAR
2 10 # Imm12
5 10 # ImAzero
Note
Unspecified parameters are assumed to be zero.
Symbol: IMEXTPAR
Default value: {}
Description:
The IMEXTPAR variable is a list of input parameters for the spectrum
generator, which is read of the IMEXTPAR block of the SLHA input
file. The list assigned to the IMEXTPAR variable must have the same
form as the MINPAR variable.
Example: In the CP-violating MSSM (MSSMCPV) the IMEXTPAR block
has the form:
IMEXTPAR = {
{1, ImM1Input},
{2, ImM2Input},
{3, ImM3Input},
{23, ImMuInput}
};
In this case the input parameters can be given in the SLHA input file as:
Block IMEXTPAR
1 100 # Im(M1(MSUSY))
2 100 # Im(M2(MSUSY))
3 100 # Im(M3(MSUSY))
23 100 # Im(Mu(MSUSY))
Note
Unspecified parameters are assumed to be zero.
Symbol: FSAuxiliaryParameterInfo
Default value: {}
Description:
In the FSAuxiliaryParameterInfo variable additional input or extra
parameters can be defined, and extra information provided can be
provided about existing input parameters. FSAuxiliaryParameterInfo
is expected to be a list, whose element are two-component lists. The
first element of this list is a symbol representing the parameter.
The second element is a list of properties for that parameter,
specified as replacement rules. The supported properties are
InputParameter: A value ofTrueorFalseindicating if the parameter is an input parameter.LesHouches: The name of the SLHA block from which the parameter should be read, if it is an input parameter.MassDimension: A number specifying the mass dimension of the parameter.ParameterDimensions: A list specifying the vector- or matrix-type of the input parameter. A list of the form{N,M}withNandMbeing integer numbers defines a NxM matrix. A list of the form{N}, withN> 1 defines a vector withNrows. A list of the form{1}or{}defines a scalar.
Example: In the MSSM the FSAuxiliaryParameterInfo variable has
the form:
FSAuxiliaryParameterInfo = {
{Aeij, { LesHouches -> AeijIN,
ParameterDimensions -> {3,3},
InputParameter -> True
} },
{Adij, { LesHouches -> AdijIN,
ParameterDimensions -> {3,3},
InputParameter -> True
} },
{Auij, { LesHouches -> AuijIN,
ParameterDimensions -> {3,3},
InputParameter -> True
} }
};
Here, three 3x3 matrix-valued parameters are specified: Aeij,
Adij and Auij. They are defined as input parameters. These
matrices are read from the blocks AeijIN, AdijIN and AuijIN,
respectively.
These input parameters can be given in the SLHA input file as:
Block AeijIN
1 1 100
1 2 100
1 3 100
2 1 100
2 2 100
2 3 100
3 1 100
3 2 100
3 3 100
Block AdijIN
1 1 200
1 2 200
1 3 200
2 1 200
2 2 200
2 3 200
3 1 200
3 2 200
3 3 200
Block AuijIN
1 1 300
1 2 300
1 3 300
2 1 300
2 2 300
2 3 300
3 1 300
3 2 300
3 3 300
Note
Unspecified parameters are assumed to be zero.
Symbol: RealParameters
Default value: { All }
Description:
RealParameters is a list, which contains the names of all model
parameters, which should be treated as real parameters. By default,
RealParameters is set to { All }, meaning that by default all
paramerters are treated to be real. If RealParameters is set to the
empty list {}, FlexibleSUSY takes the information which paramerters
are real and which are complex from the SARAH model file.
Example: In the complex Standard Model (cSM), the parameters mu2
and \[Lambda] should be defined to be real:
RealParameters = { mu2, \[Lambda] };
Note: The gauge couplings and VEVs are always assumed to be real in SARAH.
Example: In the CP-violating MSSM (CMSSMCPV), the B[\[Mu]]
parameter should be defined to be real:
RealParameters = { B[\[Mu]] };
In FlexibleSUSY, spectrum generators with maximum 3 boundary conditions can be generated. These boundary conditions are named "high-scale", "susy-scale" and "low-scale" boundary condition and are described in the following.
However, it is possible to disable the high-scale boundary condition. In order to do so, set:
OnlyLowEnergyFlexibleSUSY = True; (* disable high-scale BC, default: False *)
Symbol: LowScale
Default value: unset
Description:
The scale of the low-scale boundary condition, at which the model is matched to the Standard Model.
Note
LowScale is ignored if FlexibleEFTHiggs == True
Example: In the CMSSM the low-energy scale should be set to the Z or top pole mass. This choice is achieved by the following expression:
LowScale = LowEnergyConstant[MZ];
Symbol: LowScaleFirstGuess
Default value: unset
Description:
First guess of the low-energy scale.
Note
LowScaleFirstGuess is ignored if FlexibleEFTHiggs == True
Example: In the CMSSM the first guess for the low-energy scale should be set to the Z or top pole mass:
LowScaleFirstGuess = LowEnergyConstant[MZ];
Symbol: LowScaleInput
Default value: {}
Description:
With the LowScaleInput variable boundary conditions at the
low-energy scale can be specified. LowScaleInput is a list. Please
refer to ref input_format for details about the list format.
At the low-energy scale, FlexibleSUSY automatically determines the three gauge couplings from the SLHA input parameters \alpha_{em}, M_Z and G_F or M_W.
Note
LowScaleInput is ignored if FlexibleEFTHiggs == True
Example: In the CMSSM LowScaleInput is given as follows:
LowScaleInput = {
{Yu, Automatic},
{Yd, Automatic},
{Ye, Automatic},
{vd, 2 MZDRbar / Sqrt[GUTNormalization[g1]^2 g1^2 + g2^2] Cos[ArcTan[TanBeta]]},
{vu, 2 MZDRbar / Sqrt[GUTNormalization[g1]^2 g1^2 + g2^2] Sin[ArcTan[TanBeta]]}
};
The method to determine the weak mixing angle can be chosen by setting
the variable FSWeakMixingAngleInput to either Automatic,
FSFermiConstant or FSMassW. FSWeakMixingAngleInput is set to
Automatic by default.
Value of FSWeakMixingAngleInput |
Parameters from which weak mixing angle is determined |
|---|---|
FSFermiConstant |
G_F and M_Z |
FSMassW |
M_W and M_Z |
Automatic (default) (recommended) |
chose most precise method automatically |
Example: Automatically chose most precise method to determine the weak mixing angle:
FSWeakMixingAngleInput = Automatic; (* recommended *)
Note
If FSWeakMixingAngleInput = FSMassW; is chosen,
FlexibleSUSY looks for the definition of the weak mixing
angle in the symbol SARAH`Weinberg. If
SARAH`Weinberg is not defined, FlexibleSUSY uses the
expression assigned to FSWeakMixingAngleExpr, which is
by default set to
ArcSin[Sqrt[1-Mass[SARAH`VectorW]^2/Mass[SARAH`VectorZ]^2]].
Symbol: SUSYScale
Default value: unset
Description:
The scale of the susy-scale boundary condition, which is defined to be between the low-scale and the high-scale. This is the scale at which the electroweak symmetry breaking conditions are imposed by default, see ref input_format.
Example: In the CMSSM the SUSY scale should be set to the geometric average of the two stop masses. This choice is achieved by the following expression:
SUSYScale = Sqrt[Product[M[Su[i]]^(Abs[ZU[i,3]]^2 + Abs[ZU[i,6]]^2), {i,6}]];
Symbol: SUSYScaleFirstGuess
Default value: unset
Description:
First guess of the SUSY scale.
Example: In the CMSSM a reasonable first guess for the SUSY scale can be given by the following combination of the mSUGRA parameters:
SUSYScaleFirstGuess = Sqrt[m0^2 + 4 m12^2];
Symbol: SUSYScaleInput
Default value: {}
Description:
With the SUSYScaleInput variable boundary conditions at the SUSY
scale can be specified. SUSYScaleInput is a list. Please refer to
ref input_format for details about the list format.
Example: In the NUTNMSSM SUSYScaleInput is given as follows:
SUSYScaleInput = {
{\[Lambda], LambdaInput},
{\[Kappa], KappaInput},
{vS, Sqrt[2] MuEff / LambdaInput}
};
Symbol: HighScale
Default value: unset
Description:
This is the scale of the high-scale boundary condition.
Example: In the CMSSM the high-energy scale, M_X, is given by the equality of the gauge couplings g_1(M_X) and g_2(M_X):
HighScale = g1 == g2;
Symbol: HighScaleFirstGuess
Default value: unset
Description:
First guess of the high-energy scale.
Example: In the CMSSM a reasonable initial guess for the high-energy scale is:
HighScaleFirstGuess = 2.0 10^16;
Symbol: HighScaleMinimum
Default value: unset
Description:
Minimum value of the high-energy scale during the iteration.
Example: In the E6SSM the high-energy scale can vary a lot between the iteration steps. For this reason, it makes sense to use a minimum high-energy scale in intermediate steps as:
HighScaleMinimum = 1.0 10^4;
Symbol: HighScaleMaximum
Default value: unset
Description:
Maximum value of the high-energy scale during the iteration.
Example: In the E6SSM the high-energy scale can vary a lot between the iteration steps. For this reason, it makes sense to use a maximum high-energy scale in intermediate steps as:
HighScaleMaximum = 5.0 10^17;
Symbol: HighScaleInput
Default value: {}
Description:
With the HighScaleInput variable boundary conditions at the
high-energy scale can be specified. HighScaleInput is a list.
Please refer to ref input_format for details about the list format.
Example: In the CMSSM HighScaleInput is set to the mSUGRA boundary
conditions:
HighScaleInput = {
{T[Ye], Azero Ye},
{T[Yd], Azero Yd},
{T[Yu], Azero Yu},
{mHd2, m0^2},
{mHu2, m0^2},
{mq2, UNITMATRIX[3] m0^2},
{ml2, UNITMATRIX[3] m0^2},
{md2, UNITMATRIX[3] m0^2},
{mu2, UNITMATRIX[3] m0^2},
{me2, UNITMATRIX[3] m0^2},
{MassB, m12},
{MassWB,m12},
{MassG, m12}
};
Symbol: InitialGuessAtLowScale
Default value: {}
Description:
With the InitialGuessAtLowScale variable initial values for the
model MS-bar/DR-bar parameters can be given at the low-energy scale
LowScale.
Note
InitialGuessAtLowScale is ignored if FlexibleEFTHiggs == True
Example: In the CMSSM InitialGuessAtLowScale is given as follows:
InitialGuessAtLowScale = {
{vd, LowEnergyConstant[vev] Cos[ArcTan[TanBeta]]},
{vu, LowEnergyConstant[vev] Sin[ArcTan[TanBeta]]},
{Yu, Automatic},
{Yd, Automatic},
{Ye, Automatic}
};
Symbol: InitialGuessAtSUSYScale
Default value: {}
Description:
Note
InitialGuessAtSUSYScale is only used if FlexibleEFTHiggs == True
With the InitialGuessAtSUSYScale variable initial values for the
model MS-bar/DR-bar parameters can be given at the SUSY scale
SUSYScale.
Example: In the MSSMEFTHiggs InitialGuessAtSUSYScale is given as follows:
InitialGuessAtSUSYScale = {
{Yu, Automatic},
{Yd, Automatic},
{Ye, Automatic}
{MassB, Ms},
{MassWB, Ms},
{MassG, Ms},
{mq2, UNITMATRIX[3] Ms^2},
{mu2, UNITMATRIX[3] Ms^2},
{md2, UNITMATRIX[3] Ms^2},
{ml2, UNITMATRIX[3] Ms^2},
{me2, UNITMATRIX[3] Ms^2},
{\[Mu], Ms},
{B[\[Mu]], Sqr[Ms]/(TanBeta + 1/TanBeta)},
{T[Yu], Ms/TanBeta Yu},
{T[Yd], Ms TanBeta Yd},
{T[Ye], Ms TanBeta Ye},
{T[Yu][3,3], (Ms/TanBeta + Xtt Ms) Yu[3,3]}
};
Symbol: InitialGuessAtHighScale
Default value: {}
Description:
With the InitialGuessAtHighScale variable initial values for the
model MS-bar/DR-bar parameters can be given at the high-energy scale
HighScale.
Example: In the CMSSM InitialGuessAtHighScale is given as
follows:
InitialGuessAtHighScale = {
{\[Mu] , 1.0},
{B[\[Mu]], 0.0}
};
Symbol: EWSBOutputParameters
Default value: {}
Description:
In the EWSBOutputParameters variable the model parameters must be
specified, which are fixed by the electroweak symmetry breaking (EWSB)
conditions, \partial V_\text{Higgs}/\partial v_i = 0. The
length of the EWSBOutputParameters list must be equal to the number
of EWSB conditions.
Example: In the CMSSM EWSBOutputParameters is given as follows:
EWSBOutputParameters = { B[\[Mu]], \[Mu] };
The elements of the EWSBOutputParameters must be _real_ parameters.
In a model with complex parameters, as in the CMSSMCPV for example,
EWSBOutputParameters is set to be:
EWSBOutputParameters = { Re[B[\[Mu]]], Im[B[\[Mu]]], \[Mu] };
Symbol: EWSBInitialGuess
Default value: {}
Description:
In the EWSBInitialGuess variable initial guesses for some or all
of the EWSB output parameters can be specified.
Example: In the VCMSSM EWSBInitialGuess is defined as:
EWSBInitialGuess = {
{TanBeta, vu / vd},
{MuSq, \[Mu]^2}
};
Symbol: EWSBSubstitutions
Default value: {}
Description:
In the EWSBSubstitutions variable, substitutions for model
parameters in terms of other parameters can be given.
EWSBSubstitutions should be a list of two-component lists, in which
the first element is the parameter to be substituted for, and the
second element is the expression to be substituted in its place.
Example: In the VCMSSM EWSBSubstitutions is defined as:
EWSBSubstitutions = {
{vd, vMSSM Cos[ArcTan[TanBeta]]},
{vu, vMSSM Sin[ArcTan[TanBeta]]},
{\[Mu], Sign[\[Mu]] Sqrt[MuSq]}
};
Symbol: FSSolveEWSBTreeLevelFor
Default value: {}
Description:
In the FSSolveEWSBTreeLevelFor variable the model parameters can be
specified, which are fixed by the tree-level electroweak symmetry
breaking (EWSB) conditions when the running (tree-level) masses are
calculated. The length of the FSSolveEWSBTreeLevelFor list must be
either zero (default) or equal to the number of EWSB conditions. If
FSSolveEWSBTreeLevelFor is the empty list, then the temporary EWSB
output parameters are chosen automatically as follows:
- In SUSY models, by default the soft-breaking squared Higgs mass parameters are fixed by the tree-level EWSB equation temporarily when the running (tree-level) masses are calculated.
- In non-SUSY models, by default the parameters given in
EWSBOutputParametersare fixed by the tree-level EWSB equation temporarily when the running (tree-level) masses are calculated.
Symbol: MatchingScaleInput
Default value: {}
Description:
Note
MatchingScaleInput is only used if FlexibleEFTHiggs == True
In the MatchingScaleInput variable, relations between the parameters
of the full model and the Standard Model (the EFT) at the SUSYScale
can be specified.
An important application is the relation between the vacuum
expectation values (VEVs) in a SUSY model and v in the Standard
Model: In FlexibleEFTHiggs the running Yukawa couplings of the full
model are determined from a pole mass matching of the Standard Model
fermions (which need to be present in both models). For this
determination the running VEVs of the full model must be known and
non-zero. MatchingScaleInput allows the user for example to fix the
running VEVs of the full model as a function of the running SM-like
VEV v in the full model.
Example: In the MSSM the vacuum expectation values v_u and v_d are related to the MSSM SM-like VEV v = \sqrt{v_u^2 + v_d^2} as
v_u &= v \sin\beta , \\ v_d &= v \cos\beta .
To fix v_u and v_d in the MSSM in this way,
MatchingScaleInput can be used:
MatchingScaleInput = {
{vu, VEV Sin[ArcTan[TanBeta]]},
{vd, VEV Cos[ArcTan[TanBeta]]}
};
where TanBeta is an input parameter. The symbol VEV is a
FlexibleSUSY constant which is assigned the value
\text{VEV} = \frac{2 m_Z}{\sqrt{g_Y^2 + g_2^2}} ,
where m_Z is the running Z boson mass in the full model, detetermined by requiring the equality of the Z boson pole masses of the full model and the Standard Model. g_Y and g_2 are the running gauge couplings of U(1)_Y and SU(2)_L in the full model, respectively. These two gauge couplings are calculated using the 1-loop threshold correction for \alpha_{\text{em}} and the running weak mixing angle, \cos\theta_W = m_W / m_Z. m_W is the running W boson mass in the full model, detetermined by requiring the equality of the W boson pole masses of the full model and the Standard Model.
The variables LowScaleInput, SUSYScaleInput and HighScaleInput
are lists which specify the boundary conditions for the running model
parameters at the corresponding scale. The boundary conditions can be
expressed as follows.
A running model parameter can be assigned at the corresponding scale
to a fixed numerical value or a value which is the result of the
evaluation of an expression. Such an assignment is made by a
two-component list, {p, v}, where the first list element must be the
model parameter (p in this case) and the second list element is a
numerical value or an expression.
Example: An example is the mSUGRA boundary condition in the CMSSM at the GUT scale:
HighScaleInput = {
{T[Ye], Azero*Ye},
{T[Yd], Azero*Yd},
{T[Yu], Azero*Yu},
{mHd2, m0^2},
{mHu2, m0^2},
{mq2, UNITMATRIX[3] m0^2},
{ml2, UNITMATRIX[3] m0^2},
{md2, UNITMATRIX[3] m0^2},
{mu2, UNITMATRIX[3] m0^2},
{me2, UNITMATRIX[3] m0^2},
{MassB, m12},
{MassWB,m12},
{MassG, m12}
};
The model parameters in the expression in the second list element are
running parameters at the corresponding scale. I.e. the setting
{T[Ye], Azero*Ye} means T_{y_e}(Q) := A_0 y_e(Q), where
Q is the scale.
For matrix- or vector-valued assignments, the following convenient symbols can be used in the second list element:
UNITMATRIX[rows] (* quadratic unit matrix with ``rows' rows *) UNITMATRIXCOMPLEX[rows] (* complex quadratic unit matrix with ``rows' rows *) ZEROMATRIX[rows,cols] (* zero matrix with ``rows' rows and ``cols' columns *) ZEROMATRIXCOMPLEX[rows,cols] (* complex zero matrix with ``rows' rows and ``cols' columns *) ZEROVECTOR[rows] (* zero vector with ``rows' rows *) ZEROVECTORCOMPLEX[rows] (* complex zero vector with ``rows' rows *)
On the r.h.s. of the assignment it is possible to refer to a model
parameter, which is read from an SLHA input block. These model
parameter input blocks are named after the model parameter output
blocks concatenated with an additionan "IN" (see the SLHA-2 standard,
arXiv:0801.0045, Section 4.1.3). To refer to such an input model
parameter on the r.h.s. of an assignment one can either add an entry
in FSAuxiliaryParameterInfo or use the LHInput[p] symbol,
where p is the name of the model parameter.
Example:
SUSYScaleFirstGuess = Sqrt[Sqrt[LHInput[mq2[3,3]] * LHInput[mu2[3,3]]]];
SUSYScaleInput = {
{mq2, 2 g2^2 LHInput[mq2]}
};
It is also possible to access the \beta functions on the
r.h.s. of an assignment using the BETA head: BETA[p]
represents the \beta function of the parameter p using the
loop level given in the SLHA input. BETA[l,p] represents the
l-loop \beta function of the parameter p.
Example:
HighScaleInput = {
{\[Lambda], BETA[g1] + BETA[g2] + BETA[1,Yu][3,3]}
};
Since FlexibleSUSY 1.4.0, the user can perform a temporary parameter
definition to be used in the boundary conditions using the
FSTemporary[] head.
If a parameter p set in a boundary conditions in the form
FSTemporary[p,<expr>], the following happens: Immediately after the RG
running the value of the parameter is saved locally. Afterwards, the
parameter is assigned to <expr>. Now, all further boundary
conditions are imposed and calculations are performed (calculation of
running masses, solution of the EWSB conditions, etc.). Finally, the
parameter p is restored to the locally saved value.
Example in U1xMSSM3G: Temporarily rotate the gauge couplings to the
triangular basis:
g1T = (g1*gX - g1X*gX1)/Sqrt[gX^2 + gX1^2];
gXT = Sqrt[gX^2 + gX1^2];
g1XT = (g1X*gX + g1*gX1)/Sqrt[gX^2 + gX1^2];
SUSYScaleInput = {
{FSTemporary[g1], g1T},
{FSTemporary[gX], gXT},
{FSTemporary[g1X], g1XT},
{FSTemporary[gX1], 0},
{xS, vSInput},
{x2, Sqrt[4*MZpInput^2 - gX^2*(vu^2 + vd^2)]/(2*gX*Sqrt[1 + TanBetaX^2])},
{x1, (TanBetaX*Sqrt[4*MZpInput^2 - gX^2*(vu^2 + vd^2)])/(2*gX*Sqrt[1 + TanBetaX^2])},
{L[lw], 0},
FSSolveEWSBFor[{mHd2, mHu2, mC12, lw, mS2}]
};
In this example the gauge couplings, defined in the triangular basis,
are used in every calculation performed at the SUSY scale. This
includes the calculation of x1 and x2 as well as solving the EWSB
conditions.
The scale, at which the electroweak symmetry breaking (EWSB)
conditions are imposed can be specified by adding
FSSolveEWSBFor[parameters] to the corresponding boundary condition.
The argument parameters must be the list of model parameters which
are fixed by the electroweak symmetry breaking conditions.
Example: Impose the EWSB conditions at the low-energy scale:
LowScaleInput = {
FSSolveEWSBFor[EWSBOutputParameters]
};
If FSSolveEWSBFor[EWSBOutputParameters] is not given in any boundary
condition, then it is added to SUSYScaleInput. This implies, that
by default, the EWSB conditions are imposed at the scale SUSYScale.
In low-energy models (models where OnlyLowEnergyFlexibleSUSY ===
True) parameters, which are _not_ set in any boundary condition are
automatically input at the SUSYScale. The values of these
parameters are automatically read from the corresponding SLHA input
blocks.
To disable the automatic input of unspecified parameters, set:
AutomaticInputAtMSUSY = False; (* default: True *)
It is possible to fix model parameters at a scale by performing an iteration. Two kinds of iterations are supported:
Model parameters can be fixed by requiring that a function is minimal.
The parameters to be fixed and the function to be minimized must be
specified by the symbol FSMinimize[parameters, f], where
parameters is the list of parameters to be fixed and f is the
scalar function to be minimized.
Example:
SUSYScaleInput = {
FSMinimize[{vd,vu}, (LowEnergyConstant[MZ] - Pole[M[VZ]])^2 / STANDARDDEVIATION[MZ]^2
+ (LowEnergyConstant[MH] - Pole[M[hh[1]]])^2 / STANDARDDEVIATION[MH]^2]
};
Model parameters can be fixed by requiring that a function is zero.
The parameters to be fixed and the function whose zero should be found
must be specified by the symbol FSFindRoot[parameters, f], where
parameters is the list of parameters to be fixed and f is the
vector-valued function to be zero.
Example:
SUSYScaleInput = {
FSFindRoot[{vd,vu}, {LowEnergyConstant[MZ] - Pole[M[VZ]], LowEnergyConstant[MH] - Pole[M[hh[1]]]}]
};
FlexibleSUSY solves the given boundary value problem (BVP) by running to each scale and imposing the corresponding boundary conditions until a convergent solution has been found.
The convergence criterion can be customized using the
FSConvergenceCheck variable. The default is:
FSConvergenceCheck = Automatic; (* default *)
If FSConvergenceCheck is set to Automatic, then the following
convergence criteria are used:
- In SUSY models the BVP solver stops if the maximum number of
iterations has been reached (
FlexibleSUSY[1], see SLHA input file or the maximum relative difference of the DR-bar masses of the SUSY particles at the SUSY scale between two successive iterations is less than the precision goal (FlexibleSUSY[0], see SLHA input file). - In non-SUSY models the BVP solver stops if the maximum number of
iterations has been reached (
FlexibleSUSY[1], see SLHA input file or the maximum relative difference of all MS-bar masses of the model at the SUSY scale between two successive iterations is less than the precision goal (FlexibleSUSY[0], see SLHA input file).
To create a custom convergence tester, the FSConvergenceCheck
variable must be set to a list containing the running masses and/or
running parameters to be compared between two successive iterations.
The BVP solver stops if the maximum number of iterations has been
reached (FlexibleSUSY[1]) or the maximum relative difference of all
running masses and/or parameters given in the FSConvergenceCheck
list at the SUSY scale between two successive iterations is less than
the precision goal (FlexibleSUSY[0]).
Example: In the following MSSM example the running masses of all
massive particles as well as the running parameters g1, g2, g3, Yu,
Yd[3,3], Ye, B[\[Mu]], \[Mu] are tested for convergence.
FSConvergenceCheck = {
M[hh], M[Ah], M[Hpm],
M[Su], M[Sd], M[Se],
M[Chi], M[Cha], M[Glu],
M[Fu], M[Fd], M[Fe],
M[VZ], M[VWm],
g1, g2, g3, Yu, Yd[3,3], Ye, B[\[Mu]], \[Mu]
};
Note
For matrix- or vector-valued parameters every component is used in the convergence test, if the matrix/vector indices are omitted.
The loop order of the RGEs to be used can be selected in the model
file using the FSRGELoopOrder variable: By setting FSRGELoopOrder =
0; no RGEs will be generated by SARAH. By setting FSRGELoopOrder =
1; only one-loop RGEs will be generated by SARAH. By setting
FSRGELoopOrder = 2; the two-loop RGEs will be generated by SARAH
(this is the default).
Example:
FSRGELoopOrder = 2; (* generate two-loop RGEs using SARAH *)
In order to tune the spectrum generator for speed, the precision of
the pole mass calculation can be selected for each particle. There
are three different pole mass calculation algorithms available:
LowPrecision, MediumPrecision and HighPrecision. Please
refer to Section 6.5 of Ref. [1406.2319] for details.
By default, the pole masses of all particles are calculated with
MediumPrecision, except for the CP-even, CP-odd and charged Higgs
bosons, which are calculated with HighPrecision in order to include
some momentum-dependent 2-loop corrections.
Example:
DefaultPoleMassPrecision = MediumPrecision;
HighPoleMassPrecision = {hh, Ah, Hpm};
MediumPoleMassPrecision = {};
LowPoleMassPrecision = {};
FlexibleSUSY can generate the helper function get_lsp(), which
returns the mass of the lightest supersymmetric particle (LSP) as well
as the particle type. The particles which are candidates for being an
LSP must be specified in the PotentialLSPParticles variable.
Example: In the MSSM the lightest supersymmetric particles might be:
PotentialLSPParticles = { Chi, Sv, Su, Sd, Se, Cha, Glu };
User-defined replacement rules can be applied to the beta functions,
self-energies/ tadpoles and vertices. The rules are specified by the
FSBetaFunctionRules, FSSelfEnergyRules and FSVertexRules
variables, respectively.
Example: Set the gauge couplings g1 and g2 to zero in all 1-loop,
2-loop and 3-loop beta functions:
FSBetaFunctionRules = {
{g1 -> 0, g2 -> 0}, (* applied to 1L beta functions *)
{g1 -> 0, g2 -> 0}, (* applied to 2L beta functions *)
{g1 -> 0, g2 -> 0} (* applied to 3L beta functions *)
};
Example: Set the mass of the Z boson and the corresponding ghost field to zero in the 1-loop self-energies/ tadpoles:
FSSelfEnergyRules = {
{ (Mass|Mass2)[VZ|gZ] -> 0 } (* applied to 1L self-energies/tadpoles *)
};
Example: Set the gauge couplings g1 and g2 to zero in all
vertices:
FSVertexRules = {
g1 -> 0,
g2 -> 0
};
FlexibleSUSY can calculate various observables. To enable the calculation of a specific observable, the corresponding symbol must be added to an extra SLHA output block, see Output blocks . In the following the supported observables are listed.
In the context of [1602.05581], FlexibleSUSY has been extended to calculate the effective couplings of CP-even and CP-odd Higgs bosons to two photons or two gluons up to NNNLO. The following table lists the Mathematica symbols to enable the calculation of these effective couplings.
| Coupling | Symbol |
|---|---|
| CP-even Higgs to two photons, h\rightarrow\gamma\gamma | FlexibleSUSYObservable`CpHiggsPhotonPhoton |
| CP-odd Higgs to two photons, A\rightarrow\gamma\gamma | FlexibleSUSYObservable`CpPseudoScalarPhotonPhoton |
| CP-even Higgs to two gluons, h\rightarrow gg | FlexibleSUSYObservable`CpHiggsGluonGluon |
| CP-odd Higgs to two gluons, A\rightarrow gg | FlexibleSUSYObservable`CpPseudoScalarGluonGluon |
Example:
ExtraSLHAOutputBlocks = {
{EFFHIGGSCOUPLINGS, NoScale,
{{1, FlexibleSUSYObservable``CpHiggsPhotonPhoton},
{2, FlexibleSUSYObservable``CpHiggsGluonGluon},
{3, FlexibleSUSYObservable``CpPseudoScalarPhotonPhoton},
{4, FlexibleSUSYObservable``CpPseudoScalarGluonGluon} } }
};
Since version 2.0, FlexibleSUSY can calculate the BSM contributions to the anomalous magnetic moment of the muon, a_\mu^{\text{BSM}} at the 1-loop level, including the leading 2-loop QED logarithmic corrections. The following table lists the Mathematica symbols to enable the calculation of a_\mu^{\text{BSM}}.
| Observable | Symbol |
|---|---|
| a_\mu^{\text{BSM}} | FlexibleSUSYObservable`aMuon |
| \Delta a_\mu^{\text{BSM}} | FlexibleSUSYObservable`aMuonUncertainty |
\Delta a_\mu^{\text{BSM}} is obtained by varying the renormalization scale by a factor 2. It therefore represents a a lower a bound of the theoretical uncertainty.
Example:
ExtraSLHAOutputBlocks = {
{FlexibleSUSYLowEnergy,
{{0, FlexibleSUSYObservable``aMuon},
{1, FlexibleSUSYObservable``aMuonUncertainty} } }
};
FlexibleSUSY contains an interface to GM2Calc, which can be used to calculate the MSSM contributions to the anomalous magnetic moment of the muon, a_\mu^{\text{MSSM}}. GM2Calc calculates a_\mu^{\text{MSSM}} at the 1-loop level, takes all known 2-loop contributions into account and performs a resummation of \tan\beta-enhanced contributions.
Note
GM2Calc version 1.*.* is restricted to CP-conserving MSSM without sfermion flavour violation. Thus, the GM2Calc interface in FlexibleSUSY can only be used for MSSM models with CP and sfermion flavour conservation.
| Observable | Symbol |
|---|---|
| a_\mu^{\text{MSSM}} | FlexibleSUSYObservable`aMuonGM2Calc |
| \Delta a_\mu^{\text{MSSM}} | FlexibleSUSYObservable`aMuonGM2CalcUncertainty |
Example:
ExtraSLHAOutputBlocks = {
{FlexibleSUSYLowEnergy,
{{2, FlexibleSUSYObservable``aMuonGM2Calc},
{3, FlexibleSUSYObservable``aMuonGM2CalcUncertainty} } }
};
Since version 2.0 FlexibleSUSY can calculate the BSM contributions to the electric dipole moments (EDM) of fermions at the 1-loop level in models with complex parameters. The following table lists the Mathematica symbols to enable the calculation of the EDM d_f^{\text{BSM}} of the fermion f.
| Observable | Symbol |
|---|---|
| d_f^{\text{BSM}} | FlexibleSUSYObservable`EDM[f] |
Example: To calculate the EDMs of the electron, muon and tau lepton in the CP-violating MSSM, add the following to the FlexibleSUSY model file:
ExtraSLHAOutputBlocks = {
{FlexibleSUSYLowEnergy,
{{23, FlexibleSUSYObservable``EDM[Fe[1]]},
{24, FlexibleSUSYObservable``EDM[Fe[2]]},
{25, FlexibleSUSYObservable``EDM[Fe[3]]} } }
};
The user can define additional SLHA output blocks. These additional
blocks must be defined in the FlexibleSUSY model file using the
ExtraSLHAOutputBlocks variable. The ExtraSLHAOutputBlocks
variable is a nested list of the following form:
ExtraSLHAOutputBlocks = {
{<blockname>, [<scale>,]
{{<index>, <expression>},
{<index>, <expression>},
{<index>, <expression>}}
},
...
};
<blockname> is the name of the output block.
Optionally, the renormalization scale can be added to the block head.
NoScale (default) specifies that the block head should have no
scale. CurrentScale specifies that the scale written in the block
head should be the current scale of the model. Otherwise, <scale>
can be numeric value.
The fields inside the block are specified in form of a list of 2-component lists, where the first entry is an integer number representing the field index. The second entry is an expression to be evaluated and whose numeric result is written to the field value.
Example: In the MSSM mode file the following additional output blocks are defined:
ExtraSLHAOutputBlocks = {
{FlexibleSUSYOutput, NoScale,
{{0, Hold[HighScale]},
{1, Hold[SUSYScale]},
{2, Hold[LowScale]} } },
{FlexibleSUSYLowEnergy,
{{21, FlexibleSUSYObservable``aMuon} } },
{EFFHIGGSCOUPLINGS, NoScale,
{{1, FlexibleSUSYObservable``CpHiggsPhotonPhoton},
{2, FlexibleSUSYObservable``CpHiggsGluonGluon},
{3, FlexibleSUSYObservable``CpPseudoScalarPhotonPhoton},
{4, FlexibleSUSYObservable``CpPseudoScalarGluonGluon} } },
{ALPHA, NoScale,
{{ArcSin[Pole[ZH[2,2]]]}}},
{HMIX , {{1, \[Mu]},
{2, vu / vd},
{3, Sqrt[vu^2 + vd^2]},
{4, M[Ah[2]]^2},
{101, B[\[Mu]]},
{102, vd},
{103, vu} } },
{Au, {{1, 1, T[Yu][1,1] / Yu[1,1]},
{2, 2, T[Yu][2,2] / Yu[2,2]},
{3, 3, T[Yu][3,3] / Yu[3,3]} } },
{Ad, {{1, 1, T[Yd][1,1] / Yd[1,1]},
{2, 2, T[Yd][2,2] / Yd[2,2]},
{3, 3, T[Yd][3,3] / Yd[3,3]} } },
{Ae, {{1, 1, T[Ye][1,1] / Ye[1,1]},
{2, 2, T[Ye][2,2] / Ye[2,2]},
{3, 3, T[Ye][3,3] / Ye[3,3]} } },
{MSOFT, {{1, MassB},
{2, MassWB},
{3, MassG},
{21, mHd2},
{22, mHu2},
{31, SignedAbsSqrt[ml2[1,1]]},
{32, SignedAbsSqrt[ml2[2,2]]},
{33, SignedAbsSqrt[ml2[3,3]]},
{34, SignedAbsSqrt[me2[1,1]]},
{35, SignedAbsSqrt[me2[2,2]]},
{36, SignedAbsSqrt[me2[3,3]]},
{41, SignedAbsSqrt[mq2[1,1]]},
{42, SignedAbsSqrt[mq2[2,2]]},
{43, SignedAbsSqrt[mq2[3,3]]},
{44, SignedAbsSqrt[mu2[1,1]]},
{45, SignedAbsSqrt[mu2[2,2]]},
{46, SignedAbsSqrt[mu2[3,3]]},
{47, SignedAbsSqrt[md2[1,1]]},
{48, SignedAbsSqrt[md2[2,2]]},
{49, SignedAbsSqrt[md2[3,3]]} } }
};
In the MSSM the dominant two-loop Higgs pole mass corrections [arxiv:hep-ph/0105096, arxiv:hep-ph/0112177, arxiv:hep-ph/0212132, arxiv:hep-ph/0206101, arxiv:hep-ph/0305127] can be used by setting in the model file
UseHiggs2LoopMSSM = True; (* use 2-loop Higgs corrections *)
The known 3-loop Higgs pole mass corrections of the order O(\alpha_t\alpha_s^2 + \alpha_b\alpha_s^2) [arxiv:hep-ph/0803.0672, arxiv:hep-ph/1005.5709, arxiv:1409.2297, arxiv:1708.05720] can be used by setting in the model file
UseHiggs3LoopMSSM = True; (* use 3-loop Higgs corrections *)
Note
The Himalaya library must be linked to FlexibleSUSY in order to enable the 3-loop contributions:
./configure \
--with-models=MSSMNoFVatMGUTHimalaya \
--enable-himalaya \
--with-himalaya-incdir=${HIMALAYA_DIR}/source/include \
--with-himalaya-libdir=${HIMALAYA_DIR}/build
MSSMNoFVatMGUTHimalaya is a pre-defined FlexibleSUSY model which
includes the 3-loop contributions to the light CP-even Higgs mass from
Himalaya. ${HIMALAYA_DIR} is the path to the Himalaya directory.
To make use of the 2-loop and/or 3-loop corrections the effective
\mu parameter must be specified using the EffectiveMu
variable:
EffectiveMu = \[Mu];
Note
When the 3-loop corrections are used, the following switches will be set automatically for consistency:
SARAH`UseHiggs2LoopMSSM = True; UseMSSMYukawa2Loop = True; (* use 2-loop SQCD corrections to yt and yb *) UseMSSMAlphaS2Loop = True; (* use 2-loop SQCD corrections to alpha_s *) UseMSSM3LoopRGEs = True; (* use 3-loop RGEs *)
In the NMSSM the dominant two-loop Higgs pole mass corrections from Ref. [arXiv:0907.4682] plus the MSSM-like contributions from Refs. [hep-ph/0105096, hep-ph/0112177, hep-ph/0212132, hep-ph/0206101, hep-ph/0305127] can be used by setting in the model file:
UseHiggs2LoopNMSSM = True; (* use 2-loop Higgs corrections *)
In addition, the effective \mu parameter must be specified using
the EffectiveMu variable, Furthermore, the tree-level value of the
effective CP-odd MSSM-like Higgs must be specified in the
EffectiveMASqr variable:
EffectiveMu = \[Lambda] vS / Sqrt[2]; EffectiveMASqr = (T[\[Lambda]] vS / Sqrt[2] + 0.5 \[Lambda] \[Kappa] vS^2) (vu^2 + vd^2) / (vu vd);
In the Standard Model the two-loop Higgs pole mass corrections of the order O(\alpha_t\alpha_s + \alpha_b\alpha_s) [arxiv:1407.4336], O((\alpha_t + \alpha_b)^2) [arxiv:1205.6497] and O(\alpha_\tau^2) can be used by setting in the model file:
UseHiggs2LoopSM = True;
The Standard Model the three-loop Higgs pole mass corrections of the order O(\alpha_t\alpha_s^2 + \alpha_t^2\alpha_s + \alpha_t^3) [arxiv:1407.4336, Eq.(3.2)] can be used by setting in the model file:
UseHiggs3LoopSM = True;
Note
When the 3-loop corrections are used, the following switches will be set automatically for consistency:
UseHiggs2LoopSM = True; UseSMYukawa2Loop = True; (* use 2-loop non-QCD corrections to m_t *) UseSMAlphaS3Loop = True; (* use 2- and 3-loop QCD corrections to alpha_s *) UseYukawa3LoopQCD = True; (* use 2- and 3-loop QCD corrections to m_t *) UseSM3LoopRGEs = True; (* use 3-loop RGEs *)
The Standard Model the 4-loop Higgs pole mass corrections of the order O(\alpha_t\alpha_s^3) [arxiv:1508.00912, Eq.(5.5)] can be used by setting in the model file:
UseHiggs4LoopSM = True;
Note
When the 4-loop corrections are used, the following switches will be set automatically for consistency:
UseHiggs2LoopSM = True; UseHiggs3LoopSM = True; UseSMAlphaS4Loop = True; (* use 2-, 3- and 4-loop QCD corrections to alpha_s *) UseYukawa4LoopQCD = True; (* use 2-, 3- and 4-loop QCD corrections to m_t *) UseSM3LoopRGEs = True; (* use 3-loop RGEs *) UseSM4LoopRGEs = True; (* use 4-loop RGEs *) UseSM5LoopRGEs = True; (* use 4-loop RGEs *)
In the split-MSSM (SplitMSSM) the two-loop Higgs pole mass
corrections from [arxiv:1312.5220, Eq. (4.8)] of the order
O(\alpha_t \alpha_s^2) can be used by setting in the model
file:
UseHiggs3LoopSplit = True;
In the Standard Model the known three-loop RGEs from [arxiv:1303.4364, arXiv:1307.3536] can be used by setting in the model file:
UseSM3LoopRGEs = True; (* use three-loop SM RGEs *)
In the MSSM the known three-loop RGEs from [hep-ph:0308231], [http://www.liv.ac.uk/~dij/betas/allgennb.log] can be used by setting in the model file:
UseMSSM3LoopRGEs = True; (* use three-loop MSSM RGEs *)
In the Standard Model the known four-loop RGEs from [arxiv:1508.00912, arXiv:1604.00853, arxiv:1508.02680] can be used by setting in the model file:
UseSM4LoopRGEs = True; (* use four-loop SM RGEs *)
In the Standard Model the known five-loop QCD RGE from [arxiv:1606.08659] can be used by setting in the model file:
UseSM5LoopRGEs = True; (* use five-loop SM QCD RGE *)
The known SM 2-, 3- and 4-loop QCD threshold corrections of order O(\alpha_s^2 + \alpha_s^3 + \alpha_s^4) to the strong coupling constant are known by [hep-ph/0004189, hep-ph/0512060]. They can be taken into account by setting in the model file:
UseSMAlphaS4Loop = True; (* use 2-, 3- and 4-loop threshold for αs *)
The known SM 2-loop threshold corrections of order O(\alpha_t \alpha_s + \alpha_t^2) to the running top mass are known by [arXiv:1604.01134]. They can be taken into account by setting in the model file:
UseSMYukawa2Loop = True; (* use 2-loop thresholds for mt *)
Note
These corrections require FlexibleSUSY to be configured with TSIL support.
In the MSSM the known two-loop SQCD relation between the top pole mass and the DR-bar top mass from [arxiv:hep-ph/0210258,arxiv:hep-ph/0507139] as well as between the MS-bar bottom mass in the Standard Model and the DR-bar bottom mass in the MSSM [arxiv:0707.0650] can be used by setting in the model file:
UseMSSMYukawa2Loop = True; (* use two-loop threshold for yt and yb *)
The known MSSM two-loop corrections of order O(\alpha_s^2 + \alpha_s\alpha_t + \alpha_s\alpha_b) to the strong coupling constant are known by [hep-ph/0509048, arXiv:0810.5101, arXiv:1009.5455]. They can be taken into account by setting in the model file:
UseMSSMAlphaS2Loop = True; (* use two-loop threshold for alpha_s *)
In non-SUSY models the known 3-loop (Standard Model) QCD corrections O(\alpha_s^3) [arxiv:hep-ph/9911434, arxiv:hep-ph/9912391] can be used in the determination of the running \overline{MS} top Yukawa coupling y_t at the low-energy scale by setting:
UseYukawa3LoopQCD = Automatic;
or:
UseYukawa3LoopQCD = True;
Note, that these 3-loop corrections are only applied at run-time if
the threshold correction loop order (block FlexibleSUSY[7]) is set
to a value > 2.
In addition, the 3-loop (Standard Model) QCD corrections O(\alpha_s^3) [arxiv:hep-ph/0004189] to the running \overline{MS} strong coupling \alpha_s can be used at the low-energy scale by setting:
UseSMAlphaS3Loop = True;
Note, that these 3-loop corrections are only applied at run-time if
the threshold correction loop order (block FlexibleSUSY[7]) is set
to a value > 2.
In non-SUSY models the known 4-loop (Standard Model) QCD corrections O(\alpha_s^4) [1604.01134], [1502.01030], [1606.06754] can be used in the determination of the running \overline{MS} top Yukawa coupling y_t at the low-energy scale by setting:
UseYukawa4LoopQCD = Automatic;
or:
UseYukawa4LoopQCD = True;
| [hep-ph:0308231] | Phys.Lett. B579 (2004) 180-188 [arxiv:hep-ph/0308231] |
| [1406.2319] | CPC 190 (2015) 139-172 [arxiv:1406.2319] |
| [1602.05581] | Eur.Phys.J. C76 (2016) no.9, 516 [arxiv:1602.05581] |
| [1604.01134] | Phys.Rev. D93 (2016) no.9, 094017 [arXiv:1604.01134] |
| [1502.01030] | Phys.Rev.Lett. 114 (2015) 14, 142002 [arXiv:1502.01030] |
| [1606.06754] | Phys.Rev.D 94 (2016) 7, 074025 [arXiv:1606.06754] |