# RGEs

## Calculating the RGEs with SARAH

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 [1][2] or a general quantum field theory [3][4][5][6]. These expressions were extended by the results for several Abelian gauge groups [7][8] and Dirac mass terms for gauginos [9]. In addition, the gauge dependence in the running of the VEVs is included [10][11]. Read also see the detials about the calculation.

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.

The calculation is started via

CalcRGEs[Options]

### Options

The different options are

1. TwoLoop, Value: True or False, Default: True
If also the two loop RGEs should be calculated.
2. ReadLists, Value: True or False, Default: False
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.
3. VariableGenerations, Value: List of particles, Default: {}
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.
4. NoMatrixMultiplication, Values: True or False, Default: False
Normally, the $\beta$-functions are simplified by writing the sums over generation indices as matrix multiplication. This can be switched off using this option.
5. IgnoreAt2Loop, Values: a list of parameters, Default: {}
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, IgnoreAt2Loop -> $LIST can be used to neglect parameters at the two-loop level The entries of$LIST can be superpotential or soft SUSY-breaking parameters as well as gauge couplings.
6. WriteFunctionsToRun, True or False, Default: True
Defines, if a file should be written to evaluate the RGEs numerically in Mathematica

### Output arrays

The $\beta$-functions for SUSY will be stored in the following arrays:

1. Gij: Anomalous dimensions of all chiral superfields
2. BetaWijkl: Quartic superpotential parameters
3. BetaYijk: Trilinear superpotential parameters
4. BetaMuij: Bilinear superpotential parameters
5. BetaLi: Linear superpotential parameters
6. BetaQijkl: Quartic soft-breaking parameters
7. BetaTijk: Trilinear soft-breaking parameters
8. BetaBij: Bilinear soft-breaking parameters
9. BetaSLi: Linear soft-breaking parameters
10. Betam2ij: Scalar squared masses
11. BetaMi: Majorana Gaugino masses
12. BetaGauge: Gauge couplings
13. BetaVEVs: VEVs
14. BetaDGi: Dirac gaugino mass terms

and for non-SUSY models in

• Gammaij or GijS: Anomalous dimensions of scalars
• GijF: Anomalous dimensions of fermions
• GammaijHat: gauge dependent part of running VEVs
• BetaGauge: Gauge couplings
• BetaLijkl: Quartic scalar couplings
• BetaYijk: Interactions between two fermions and one scalar
• BetaTijk: Cubic scalar interactions
• BetaMuij: Bilinear fermion term
• BetaBij: Bilinear scalar term
• BetaVEVs: Vacuum expectation values

These arrays are also saved in the directory

../\SARAH/Output/"ModelName"/RGE

### Conventions for the output

All RGEs are saved in three-dimension arrays with the following conventions:

{Parameter,1-Loop,2-Loop}

• The first entry contains the name of the parameter
• The second entry contains the one-loop $\beta$-function up to a factor $1/(16 \pi^2)$
• The third entry contains the two loop $\beta$-function up to a factor $(1/(16 \pi^2))^2$

For anomalous dimension the first entry is a bit different and the output reads

{{Field1,Field2},1-Loop,2-Loop}


Thus, the names of the two external fields are given which must not be identical in the presence of off-diagonal, anomalous dimensions.

## Remarks

### GUT normalization

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 for more information.

### Index contraction

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:

1. MatMul[A,B,C,...][i,j]: $(A B C \dots)_{i,j}$. Matrix multiplication, also used for vector-matrix and vector-vector multiplication.
2. trace[A,B,C,...]: $\mbox{Tr}(A B C \dots)$. Trace of a matrix or of a product of matrices.
3. Adj[M]: $M^\dagger$. Adjoint of a matrix
4. Tp[M]: $M^T$. Transposed of a matrix

1. To differ between generation and other indices during the calculation, Kronecker[i,j] is used for generation indices instead of Delta[i,j].
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.
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 $\beta$ function in this form a saved in NAME <> 3I with NAME stands for the standard array containing the RGEs.

## Examples

1. $\beta$-function of Yukawa coupling The Yukawa couplings of the MSSM are saved in BetaYijk. The first entry consists of

BetaYijk[ [1,1]]:  Ye[i1,i2] ,

i.e. this entry contains the $\beta$-functions for the electron Yukawa coupling. The corresponding one-loop $\beta$-function is

BetaYijk[ [1,2]]:
trace[Ye,Adj[Ye]]*Ye[i1, i2]+3*MatMul[Ye,Adj[Ye],Ye][i1, i2]

The two-loop $\beta$-function is saved in BetaYijk[ [1,3]] but we skip it here because of its length.

2. Anomalous dimensions:

• the anomalous dimensions for leptons in the SM is saved in GijF read
{{e[{i1}], e[{i2}]}, (3*g1^2*Xi*Kronecker[i1, i2])/5 +
MatMul[Ye, Adj[Ye]][i1, i2], (-846*g1^4*Kronecker[i1, i2] -
i2]))/200}

• In models with Vector-like states usually anomalous dimensions show up which mix fields. For instance
{{SuR[{gen1, col1}], St1[{col2}]}, 2*MatMul[conj[Yu], Yt][i1],
((-2*g1^2)/5 + 6*g2^2 - 8*ScalarProd[Yt, conj[Yt]] - 6*trace[Yu, Adj[Yu]])*
MatMul[conj[Yu], Yt][i1] -
2*(MatMul[conj[Yu], Tp[Yd], conj[Yd], Yt][i1] +
MatMul[conj[Yu], Tp[Yu], conj[Yu], Yt][i1])}

Here, SuRare 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!
3. $\beta$-function of soft-breaking masses and abbreviations for traces The soft-breaking mass of the selectron is the first entry of Betam2ij

 Betam2ij[ [1,1]]:           me2[i1,i2]

and the one-loop $\beta$-function is saved in Betam2ij[ [1,2]]:

(-24*g1^2*MassB*conj[MassB]+10*g1^2*Tr1[1])*Kronecker[i1,i2]/5 +
2*MatMul[Ye,Adj[Ye],me2][i1,i2]

The definition of the element Tr1[1] is saved in TraceAbbr[ [1,1]]:

{Tr1[1], -mHd2 + mHu2 + trace[md2] + trace[me2] - trace[ml2] +
trace[mq2] - 2*trace[mu2]}
4. Number of generations as variable: With

CalcRGEs[VariableGenerations -> {q}]

the number of generations of the left-quark superfield is handled as variable. Therefore, the one-loop $\beta$-function of the hypercharge couplings reads

 (63*g1^3)/10 + (g1^3*NumberGenerations[q])/10
5. No matrix multiplication Using matrix multiplication can be switched off by

CalcRGEs[NoMatrixMultiplication -> True]

The one-loop $\beta$-function for the electron Yukawa coupling is now written as

  sum[j2,1,3,sum[j1,1,3,conj[Yd[j2,j1]]*Yu[i1,j1]]*Yd[j2,i2]] +
2*sum[j2,1,3,sum[j1,1,3,conj[Yu[j1,j2]]*Yu[j1,i2]]*Yu[i1,j2]] +
sum[j2,1,3,sum[j1,1,3,conj[Yu[j2,j1]]*Yu[i1,j1]]*Yu[j2,i2]] +
(3*sum[j2,1,3,sum[j1,1,3,conj[Yu[j1,j2]]*Yu[j1,j2]]]*Yu[i1,i2])/2 +
(3*sum[j2,1,3,sum[j1,1,3,conj[Yu[j2,j1]]*Yu[j2,j1]]]*Yu[i1,i2])/2 -
(13*g1^2*Yu[i1,i2])/15-3*g2^2*Yu[i1,i2]-(16*g3^2*Yu[i1,i2])/3
6. Ignoring parameters at two-loop Using

CalcRGEs[IgnoreAt2Loop -> {T[L1],T[L2],L1,L2}]

in the MSSM with trilinear $R$pV would ignore the $\lambda$ and $\lambda'$ coupling as well as their soft-breaking equivalents in the calculation of the 2-loop RGEs.

## Output

The RGEs calculated by SARAH are outputted in different formats:

## References

1. S. P. Martin and M. T. Vaughn, “Two loop renormalization group equations for soft supersymmetry breaking couplings,” Phys. Rev. D 50, 2282 (1994) Erratum: [Phys. Rev. D 78, 039903 (2008)] doi:10.1103/PhysRevD.50.2282, 10.1103/PhysRevD.78.039903 [hep-ph/9311340].
2. Y. Yamada, “Two loop renormalization group equations for soft SUSY breaking scalar interactions: Supergraph method,” Phys. Rev. D 50 (1994) 3537 doi:10.1103/PhysRevD.50.3537 [hep-ph/9401241].
3. M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization,” Nucl. Phys. B 222 (1983) 83. doi:10.1016/0550-3213(83)90610-7
4. M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings,” Nucl. Phys. B 236 (1984) 221.
5. M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings,” Nucl. Phys. B 249 (1985) 70.
6. M. x. Luo, H. w. Wang and Y. Xiao, “Two loop renormalization group equations in general gauge field theories,” Phys. Rev. D 67 (2003) 065019 doi:10.1103/PhysRevD.67.065019 [hep-ph/0211440].
7. R. M. Fonseca, M. Malinsky, W. Porod and F. Staub, “Running soft parameters in SUSY models with multiple U(1) gauge factors,” Nucl. Phys. B 854 (2012) 28 doi:10.1016/j.nuclphysb.2011.08.017 [arXiv:1107.2670 [hep-ph]].
8. R. M. Fonseca, M. Malinský and F. Staub, “Renormalization group equations and matching in a general quantum field theory with kinetic mixing,” Phys. Lett. B 726 (2013) 882 doi:10.1016/j.physletb.2013.09.042 [arXiv:1308.1674 [hep-ph]].
9. M. D. Goodsell, “Two-loop RGEs with Dirac gaugino masses,” JHEP 1301 (2013) 066 doi:10.1007/JHEP01(2013)066 [arXiv:1206.6697 [hep-ph]].
10. M. Sperling, D. Stöckinger and A. Voigt, “Renormalization of vacuum expectation values in spontaneously broken gauge theories,” JHEP 1307 (2013) 132 doi:10.1007/JHEP07(2013)132 [arXiv:1305.1548 [hep-ph]].
11. M. Sperling, D. Stöckinger and A. Voigt, “Renormalization of vacuum expectation values in spontaneously broken gauge theories: Two-loop results,” JHEP 1401 (2014) 068 doi:10.1007/JHEP01(2014)068 [arXiv:1310.7629 [hep-ph]].