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---
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title: RGEs
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permalink: /RGEs/
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---
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[Category:Calculations](/Category:Calculations "wikilink")
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Calculating the RGEs with SARAH
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-------------------------------
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SARAH calculates the renormalization group equations (RGEs) for the parameters of the (super)potential, the soft-breaking terms and the gauge couplings at one and two loop level. This is done by using the generic formulas of supersymmetry or a general quantum field theory . These expressions were extended by the results for several Abelian gauge groups and Dirac mass terms for gauginos . In addition, the gauge dependence in the running of the VEVs is included . Read also see [the detials about the calculation](/Generic_RGE_calculation "wikilink").
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SARAH writes the analytical expressions for all RGEs in separated files, and provides in addition a file to run the RGEs numerical with Mathematica, see [RGE Running with Mathematica](/RGE_Running_with_Mathematica "wikilink").
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The calculation is started via
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CalcRGEs[Options]
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### Options
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The different options are
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1. `TwoLoop`, Value: `True` or `False`, Default: `True`
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If also the two loop RGEs should be calculated.
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2. `ReadLists`, Value: `True` or `False`, Default: `False`
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If the RGEs have already be calculated, the results are saved in the output directory. The RGEs can be read from these files instead of doing the complete calculation again.
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3. `VariableGenerations`, Value: List of particles, Default: `{}`
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Some theories contain heavy superfields which should be integrated out above the SUSY scale. Therefore, it is possible to calculate the RGEs assuming the number of generations of specific superfields as free variable to make the dependence on these fields obvious. The new variable is named `NumberGenertions[X]`, where `X` is the name of the superfield.
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4. `NoMatrixMultiplication`, Values: `True` or `False`, Default: `False`
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Normally, the *β*-functions are simplified by writing the sums over generation indices as matrix multiplication. This can be switched off using this option.
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5. `IgnoreAt2Loop`, Values: a list of parameters, Default: `{}`
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The calculation of 2-loop RGEs for models with many new interactions can be very time-consuming. However, often one is only interested in the dominant effects of the new contributions at the 1-loop level. Therefore, <span>IgnoreAt2Loop -> $LIST</span> can be used to neglect parameters at the two-loop level The entries of $LIST can be superpotential or soft SUSY-breaking parameters as well as gauge couplings.
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6. `WriteFunctionsToRun`, `True` or `False`, Default: `True`
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Defines, if a file should be written to evaluate the RGEs numerically in <span>Mathematica</span>
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### Output arrays
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The *β*-functions for SUSY will be stored in the following arrays:
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1. `Gij`: Anomalous dimensions of all chiral superfields
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2. `BetaWijkl`: Quartic superpotential parameters
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3. `BetaYijk`: Trilinear superpotential parameters
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4. `BetaMuij`: Bilinear superpotential parameters
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5. `BetaLi`: Linear superpotential parameters
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6. `BetaQijkl`: Quartic soft-breaking parameters
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7. `BetaTijk`: Trilinear soft-breaking parameters
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8. `BetaBij`: Bilinear soft-breaking parameters
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9. `BetaSLi`: Linear soft-breaking parameters
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10. `Betam2ij`: Scalar squared masses
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11. `BetaMi`: Majorana Gaugino masses
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12. `BetaGauge`: Gauge couplings
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13. `BetaVEVs`: VEVs
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14. `BetaDGi`: Dirac gaugino mass terms
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and for non-SUSY models in
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- `Gammaij` or `GijS`: Anomalous dimensions of scalars
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- `GijF`: Anomalous dimensions of fermions
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- `GammaijHat`: gauge dependent part of running VEVs
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- `BetaGauge`: Gauge couplings
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- `BetaLijkl`: Quartic scalar couplings
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- `BetaYijk`: Interactions between two fermions and one scalar
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- `BetaTijk`: Cubic scalar interactions
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- `BetaMuij`: Bilinear fermion term
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- `BetaBij`: Bilinear scalar term
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- `BetaVEVs`: Vacuum expectation values
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These arrays are also saved in the directory
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../\SARAH/Output/"ModelName"/RGE
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### Conventions for the output
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All RGEs are saved in three-dimension arrays with the following conventions:
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{Parameter,1-Loop,2-Loop}
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- The first entry contains the name of the parameter
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- The second entry contains the one-loop *β*-function up to a factor 1/(16*π*<sup>2</sup>)
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- The third entry contains the two loop *β*-function up to a factor (1/(16*π*<sup>2</sup>))<sup>2</sup>
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For anomalous dimension the first entry is a bit different and the output reads
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{{Field1,Field2},1-Loop,2-Loop}
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Thus, the names of the two external fields are given which must not be identical in the presence of off-diagonal, anomalous dimensions.
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Remarks
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-------
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### GUT normalization
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The gauge couplings of *U*(1) gauge groups are often normalized at the GUT scale with respect to a specific GUT group. Therefore, it is possible to define for each gauge coupling the GUT-normalization by the corresponding entry in the parameters file. See [parameters.m](/parameters.m "wikilink") for more information.
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### Index contraction
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Generally, the results contain sums over the generation indices of the particles in the loop. SARAH always tries to write them as matrix multiplications, in order to shorten the expressions. Therefore, new symbols are introduced:
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1. `MatMul[A,B,C,...][i,j]`: (*A**B**C*…)<sub>*i*, *j*</sub>. Matrix multiplication, also used for vector-matrix and vector-vector multiplication.
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2. `trace[A,B,C,...]`: Tr(*A**B**C*…). Trace of a matrix or of a product of matrices.
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3. `Adj[M]`: *M*<sup>†</sup>. Adjoint of a matrix
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4. `Tp[M]`: *M*<sup>*T*</sup>. Transposed of a matrix
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Some remarks about the output:
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1. To differ between generation and other indices during the calculation, `Kronecker[i,j]` is used for generation indices instead of `Delta[i,j]`.
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2. The results for the scalar masses are simplified by using abbreviations for often appearing traces, see also Ref. . The definition of the traces are saved in the array `TraceAbbr`.
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3. If the model contains parameters with three indices, matrix multiplication is automatically switched off and the results are given as sum over the involved indices. In addition, these expressions are simplified by replacing a parameter with three indices by a sum of parameters with two indices. The *β* function in this form a saved in <span>NAME <> 3I</span> with <span>NAME</span> stands for the standard array containing the RGEs.
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Examples
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--------
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1. <span>***β*-function of Yukawa coupling**</span> The Yukawa couplings of the MSSM are saved in `BetaYijk`. The first entry consists of
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BetaYijk[ [1,1]]: Ye[i1,i2] ,
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i.e. this entry contains the *β*-functions for the electron Yukawa coupling. The corresponding one-loop *β*-function is
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BetaYijk[ [1,2]]:
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(-9*g1^2*Ye[i1,i2])/5-3*g2^2*Ye[i1,i2]+3*trace[Yd,Adj[Yd]]*Ye[i1,i2]+
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trace[Ye,Adj[Ye]]*Ye[i1, i2]+3*MatMul[Ye,Adj[Ye],Ye][i1, i2]
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The two-loop *β*-function is saved in `BetaYijk[ [1,3]]` but we skip it here because of its length.
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2. **Anomalous dimensions:**
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- the anomalous dimensions for leptons in the SM is saved in `GijF` read
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{{e[{i1}], e[{i2}]}, (3*g1^2*Xi*Kronecker[i1, i2])/5 +
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MatMul[Ye, Adj[Ye]][i1, i2], (-846*g1^4*Kronecker[i1, i2] -
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5*(3*(11*g1^2 - 85*g2^2 + 60*trace[Yd, Adj[Yd]] + 20*trace[Ye, Adj[Ye]])*
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MatMul[Ye, Adj[Ye]][i1, i2] + 10*MatMul[Ye, Adj[Ye], Ye, Adj[Ye]][i1,
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i2]))/200}
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- In models with Vector-like states usually anomalous dimensions show up which mix fields. For instance
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{{SuR[{gen1, col1}], St1[{col2}]}, 2*MatMul[conj[Yu], Yt][i1],
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((-2*g1^2)/5 + 6*g2^2 - 8*ScalarProd[Yt, conj[Yt]] - 6*trace[Yu, Adj[Yu]])*
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MatMul[conj[Yu], Yt][i1] -
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2*(MatMul[conj[Yu], Tp[Yd], conj[Yd], Yt][i1] +
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MatMul[conj[Yu], Tp[Yu], conj[Yu], Yt][i1])}
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Here, `SuR`are the up-squarks from the MSSM and `St1` are new superfields. Note, even if the names for scalars are shown by conventions, the anomalous dimension apply for the superfields!
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3. <span>***β*-function of soft-breaking masses and abbreviations for traces**</span> The soft-breaking mass of the selectron is the first entry of `Betam2ij`
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Betam2ij[ [1,1]]: me2[i1,i2]
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and the one-loop *β*-function is saved in `Betam2ij[ [1,2]]`:
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(-24*g1^2*MassB*conj[MassB]+10*g1^2*Tr1[1])*Kronecker[i1,i2]/5 +
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4*mHd2*MatMul[Ye,Adj[Ye]][i1,i2]+4*MatMul[T[Ye],Adj[T[Ye]]][i1,i2] +
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2*MatMul[me2,Ye,Adj[Ye]][i1,i2]+4*MatMul[Ye, ml2, Adj[Ye]][i1,i2] +
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2*MatMul[Ye,Adj[Ye],me2][i1,i2]
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The definition of the element `Tr1[1]` is saved in `TraceAbbr[ [1,1]]`:
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{Tr1[1], -mHd2 + mHu2 + trace[md2] + trace[me2] - trace[ml2] +
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trace[mq2] - 2*trace[mu2]}
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4. <span>**Number of generations as variable**</span>: With
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CalcRGEs[VariableGenerations -> {q}]
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the number of generations of the left-quark superfield is handled as variable. Therefore, the one-loop *β*-function of the hypercharge couplings reads
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(63*g1^3)/10 + (g1^3*NumberGenerations[q])/10
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5. <span>**No matrix multiplication**</span> Using matrix multiplication can be switched off by
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CalcRGEs[NoMatrixMultiplication -> True]
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The one-loop *β*-function for the electron Yukawa coupling is now written as
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sum[j2,1,3,sum[j1,1,3,conj[Yd[j2,j1]]*Yu[i1,j1]]*Yd[j2,i2]] +
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2*sum[j2,1,3,sum[j1,1,3,conj[Yu[j1,j2]]*Yu[j1,i2]]*Yu[i1,j2]] +
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sum[j2,1,3,sum[j1,1,3,conj[Yu[j2,j1]]*Yu[i1,j1]]*Yu[j2,i2]] +
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(3*sum[j2,1,3,sum[j1,1,3,conj[Yu[j1,j2]]*Yu[j1,j2]]]*Yu[i1,i2])/2 +
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(3*sum[j2,1,3,sum[j1,1,3,conj[Yu[j2,j1]]*Yu[j2,j1]]]*Yu[i1,i2])/2 -
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(13*g1^2*Yu[i1,i2])/15-3*g2^2*Yu[i1,i2]-(16*g3^2*Yu[i1,i2])/3
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6. <span>**Ignoring parameters at two-loop**</span> Using
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CalcRGEs[IgnoreAt2Loop -> {T[L1],T[L2],L1,L2}]
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in the MSSM with trilinear *R*pV would ignore the *λ* and *λ*′ coupling as well as their soft-breaking equivalents in the calculation of the 2-loop RGEs.
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Output
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------
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The RGEs calculated by SARAH are outputted in different formats:
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- they are written in the internal SARAH format in the output directory
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- they are included in the [LaTeX](/LaTeX "wikilink") output in a much more readable format
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- they are exported into [,a format which can be used together with `NDSolve` of Mathematica to solve the RGEs numerically within Mathematica](/RGE_Running_with_Mathematica "wikilink")
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- they are exported into `Fortran`code which is used by [SPheno](/SPheno "wikilink")
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See also
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--------
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- [RGE Running with Mathematica](/RGE_Running_with_Mathematica "wikilink")
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- [Generic RGE calculation](/Generic_RGE_calculation "wikilink")
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References
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---------- |
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\ No newline at end of file |