OneLoop SelfEnergies and Tadpoles
Oneloop corrections
SARAH calculates the analytical expressions for the oneloop corrections to the tadpoles and the oneloop selfenergies for all particles. For states which are a mixture of several gauge eigenstates, the selfenergy matrices are calculated. For doing that, SARAH is working with gauge eigenstates as external particles but uses mass eigenstates in the loop. The calculations are performed in {{\overline{\mathrm{DR}}}}
scheme using ’t Hooft gauge. In the case of nonSUSY models SARAH switches to {{\overline{\mathrm{MS}}}}
scheme. This approach is a generalization of the procedure applied in Ref. to the MSSM. In this context, the following results are obtained:
 The selfenergies
\Pi
of scalars and scalar mass matrices  The selfenergies
\Sigma_L
,\Sigma_R
,\Sigma_S
for fermions and fermion mass matrices  The transversal selfenergy
\Pi^T
of massive vector bosons
The approach to calculate the loop corrections is as follows: all possible generic diagrams at the oneloop level shown in Fig. [fig:1loopDiagrams] are included in SARAH. Each generic amplitude is parametrized by
\mathscr{M} =
Symmetry \times
Colour \times
Couplings \times
LoopFunction
Here ’Symmetry’ and ’Colour’ are real factors. The loopfunctions are expressed by standard PassarinoVeltman integrals A_0
and B_0
and some related functions B_1
, B_{22}
, F_0
, G_0
, H_0
, \bar{B}_{22}
as defined on this page and in E.2 of Ref. https://arxiv.org/abs/0806.0538.
As first step to get the loop corrections, SARAH generates all possible Feynman diagrams with all field combinations possible in the considered model. The second step is to match these diagrams to the generic expressions. All calculations are done without any assumption and always the most general case is taken. For instance, the generic expression for a purely scalar contribution to the scalar selfenergy reads
\mathscr{M}_{SSS}=c_S\times c_F \times c^2 B_0\left(p^2, m_{s1}^2,m_{s2}^2\right)
In the case of an external charged Higgs \Phi^+=((H_d^)^*, H_u^+))
together with down and upsquarks in the loop the correction to the charged Higgs mass matrix becomes
\mathscr{M}_{\phi^+_a \tilde{u} \tilde{d}^*} = 3 \times \sum_{i=1}^6 \sum_{j=1}^6 c(\phi^+_a \tilde{u}_i \tilde{d}^*_j)^2 B_0(p^2,m_{\tilde u_i}^2,m_{\tilde d_j}^2)
where c(\phi^+_a \tilde{u}_i \tilde{d}^*_j)
is the charged Higgssdownsup vertex where the rotation matrix of the charged Higgs are replaced by the identity matrix to get the projection on the gauge eigenstates. One can see that all possible combinations of internal generations are included, i.e. also effects like flavour mixing are completely covered. Also the entire $p^2$ dependence is kept.
Conventions
The results will contain the Passarino Veltman integrals listed here. The involved couplings are abbreviated by

Cp[p1,p2,p3]
andCp[p1,p2,p3,p4]
for nonchiral, three and four point interactions involving the particlesp1
p4
. 
Cp[p1,p2,p3][PL]
andCp[p1,p2,p3][PR]
for chiral, threepoint interactions involving the fieldsp1
p3
.
The self energies can be used for calculating the radiative corrections to masses and mass matrices, respectively. We have summarized the needed formulas for this purpose here. For calculating the loop corrections to a mass matrix, it is convenient to use unrotated, external fields, while the fields in the loop are rotated. Therefore, SARAH adds to the symbols of the external particle in the interaction an U
for ’unrotated’, e.g. Sd
→ USd
. The mixing matrix associated to this field in the vertex has to be replaced by the identity matrix when calculating the correction to the mass matrix.
Results
The results for the loop corrections are saved in two different ways. First as list containing the different loop contribution for each particle. Every entry reads
{Particles, Vertices, Type, Charge Factor, Symmetry Factor}
and includes the following information

Particles
: The particles in the loop. 
Vertices
: The needed Vertex for the correction is given. 
Charge Factor
: If several gauge charges of one particle are allowed in the loop, this factor will be unequal to one. In the case of the MSSM, only the a factor of 3 can appear because of the different colors. 
Symmetry Factor
: If the particles in the loop indistinguishable, the weight of the contribution is only half of the case of distinguishable particles. If two different charge flows are possible in the loop, the weight of the diagram is doubled, e.g. loop with charged Higgs andWboson. The absolute value of the factor depends on the type of the diagram.
The results differ in general between the\overline{\text{MS}}
and\overline{\text{DR}}
renormalization scheme by a constant term which is reflected in the variable rMS. rMS = 0 gives to the results in\overline{\text{DR}}
scheme and rMS = 1 corresponds to\overline{\text{MS}}
scheme.
The information about the loop correction are also saved in the directory
../SARAH/Output/"ModelName"/$EIGENSTATES/Loop
One Loop Tadpoles
The complete results as sums of the different contributions are saved in the two dimensional array
Tadpoles1LoopSums[$EIGENSTATES]
The first column gives the name of the corresponding VEV, the second entry the oneloop correction. A list of the different contributions, including symmetry and charge factors, is
Tadpoles1LoopList[$EIGENSTATES];
One Loop Self Energies
The results are saved in the following two dimensional array
SelfEnergy1LoopSum[$EIGENSTATES]
The first column gives the name of the particle, the entry in the second column depends on the type of the field
 Scalars: oneloop self energy
\Pi(p^2)
 Fermions: oneloop self energies for the different polarizations (
\Sigma^L(p^2),\, \Sigma^R(p^2),\, \Sigma^S(p^2)
)  Vector bosons: oneloop, transversal self energy
\Pi^T(p^2)
Also a list with the different contributions does exist:
SelfEnergy1LoopList[$EIGENSTATES]
Examples

Oneloop tadpoles: The correction of the tadpoles due to a chargino loop is saved in
Tadpoles1LoopList[EWSB][[/11]];
and reads
{bar[Cha],Cp[Uhh[{gO1}],bar[Cha[{gI1}]],Cha[{gI1}]],FFS,1,1/2}
The meaning of the different entries is: (i) a chargino (
Cha
) is in the loop, (ii) the vertex with an external, unrotated Higgs (Uhh
) with generation indexgO1
and two charginos with indexgI1
is needed, (iii) the generic type of the diagram isFFS
, (iv) the charge factor is 1, (v) the diagram is weighted by a factor\frac{1}{2}
with respect to the generic expression (see here). The corresponding term inTadpoles1LoopSum[EWSB]
is4*sum[gI1,1,2, A0[Mass[bar[Cha[{gI1}]]]^2]* Cp[phid,bar[Cha[{gI1}]],Cha[{gI1}]]*Mass[Cha[{gI1}]]]

Oneloop selfenergies

The correction to the down squark matrix due to a four point interaction with a pseudo scalar Higgs is saved in
SelfEnergy1LoopList[EWSB][ [1,12]]
and reads {Ah,Cp[conj[USd[{gO1}]],USd[{gO2}],Ah[{gI1}],Ah[{gI1}]],SSSS,1,1/2}This has the same meaning as the term
sum[gI1,1,2,A0[Mass[Ah[{gI1}]]^2]* Cp[conj[USd[{gO1}]],USd[{gO2}],Ah[{gI1}],Ah[{gI1}]]]/2
in
SelfEnergy1LoopSum[EWSB]
. 
Corrections to the Z boson are saved in
SelfEnergy1LoopList[EWSB][ [15]]
. An arbitrary entry looks like {bar[Fd], Fd, Cp[VZ, bar[Fd[{gI1}]], Fd[{gI2}]], FFV, 3, 1/2}and corresponds to
(3*sum[gI1, 1, 3, sum[gI2, 1, 3, H0[p^2, Mass[bar[Fd[{gI1}]]]^2, Mass[Fd[{gI2}]]^2]* (conj[Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PL]]* Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PL] + conj[Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PR]]* Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PR]) + 2*B0[p^2,Mass[bar[Fd[{gI1}]]]^2,Mass[Fd[{gI2}]]^2]* Mass[bar[Fd[{gI1}]]]*Mass[Fd[{gI2}]]* Re[Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PL]* Cp[VZ,bar[Fd[{gI1}]],Fd[{gI2}]][PR])]])/2
in
SelfEnergy1LoopListSum[EWSB]
.

Generic expressions
In all calculations, specific coefficient are involved:

c_S
is the symmetry factor: if the particles in the loop are indistinguishable, the weight of the contribution is only half of the weight in the case of distinguishable particles. If two different charge flows are possible in the loop, the weight of the diagram is doubled. 
c_S
is a charge factor: for corrections due to vector bosons in the adjoint representation this is the Casimir of the corresponding group. For corrections due to matter fields this can be, for instance, a color factor for quarks/squarks. For corrections of vector bosons in the adjoint representation this is normally the Dynkin index of the gauge group. 
c_R
is 2 for real fields and Majorana fermions in the loop and 1 otherwise.
We use in the following $\Gamma$ for nonchiral interactions and \Gamma_L
/\Gamma_R
for chiral interactions. If two vertices are involved, the interaction of the incoming particle has an upper index 1 and for the outgoing field an upper index 2 is used.
Oneloop tadpoles

Fermion loop (generic name in SARAH :
FFS
):T = 8 c_S c_C m_F \Gamma A_0(m_F^2)

Scalar loop (generic name in SARAH :
SSS
):T =  2 c_S c_C \Gamma A_0(m_S^2)

Vector boson loop (generic name in SARAH :
SVV
):T = 6 c_S c_C \Gamma A_0(m_V^2)
Oneloop selfenergies
Corrections to fermion

Fermionscalar loop (generic name in SARAH :
FFS
):
\Sigma^S(p^2) = m_F c_S c_C c_R \Gamma^1_R \Gamma^{2,*}_L B_0(p^2,m_F^2,m_S^2) \\ \Sigma^R(p^2) =  c_S c_C c_R \frac{1}{2} \Gamma^1_R \Gamma^{2,*}_R B_1(p^2,m_F^2,m_S^2) \\ \Sigma^L(p^2) =  c_S c_C c_R \frac{1}{2} \Gamma^1_L \Gamma^{2,*}_L B_1(p^2,m_F^2,m_S^2)

Fermionvector boson loop (generic name in SARAH :
FFV
):
\Sigma^S(p^2) =  4 c_S c_C c_R m_F \Gamma^1_L \Gamma^{2,*}_R B_0(p^2,m_F^2,m_S^2) \\\\ \Sigma^R(p^2) =  c_S c_C c_R \Gamma^1_L \Gamma^{2,*}_L B_1(p^2,m_F^2,m_S^2)\\ \Sigma^L(p^2) =  c_S c_C c_R \Gamma^1_R \Gamma^{2,*}_R B_1(p^2,m_F^2,m_S^2)
Corrections to scalar

Fermion loop (generic name in SARAH :
FFS
):
\Pi(p^2) = c_S c_C c_R \left( (\Gamma^1_L \Gamma^{2,*}_L + \Gamma^1_R \Gamma^{2,*}_R ) G_0(p^2,m_F^2, m_S^2) + (\Gamma^1_L \Gamma^{2,*}_R + \Gamma^1_R \Gamma^{2,*}_L ) B_0(p^2,m_F^2, m_S^2) \right)

Scalar loop (two 3point interactions, generic name in SARAH :
SSS
):
\Pi(p^2) = c_S c_C c_R \Gamma^1\Gamma^2 B_0 (p^2, m_F^2,m_S^2)

Scalar loop (4point interaction, generic name in SARAH :
SSSS
):
\Pi(p^2) =  c_S c_C \Gamma A_0(m_S^2)

Vector bosonscalar loop (generic name in SARAH :
SSV
):
\Pi(p^2) = c_S c_C c_R \Gamma^1 \Gamma_2 F_0(p^2, m_F^2, m_S^2)

Vector boson loop (two 3point interactions, generic name in SARAH :
SVV
):
\Pi(p^2) = c_S c_C c_R \frac{7}{2} \Gamma^1 \Gamma^{2,*} B_0(p^2,m_F^2,m_S^2)

Vector boson loop (4point interaction, generic name in SARAH :
SSVV
):
\Pi(p^2) = c_S c_C \gamma A_0(m^2_V)
Corrections to vector boson

Fermion loop (generic name in SARAH :
FFV
):
\Pi^T(p^2) = c_S c_C c_R (( \Gamma_L^1^2+\Gamma_R^1^2) H_0 (p^2, m_V^2, m_F^2) + 4 R_e (\Gamma_L^1 \Gamma_R^2) B_0(p^2, m_V^2,m_F^2))

Scalar loop (generic name in SARAH :
SSV
):
\Pi^T(p^2) = 4 c_s c_C c_R \Gamma^2 B_{22}(p^2,m_{S,1}^2,m_{S,2}^2)

Vector boson loop (generic name in SARAH :
VVV
):
\Pi^T(p^2) = \Gamma^2 c_s c_C c_R ( (4 p^2 + m_{V,1}^2+ m_{V,2}^2) B_0 (p^2, m_{V,1}^2,m_{V,2}^2)8 B_{22}(p^2, m_{S,1}^2,m_{S,2}^2))

VectorScalarLoop (generic name in SARAH :
SVV
):
\Pi^T(p^2) = \Gamma^2 c_S c_C c_R B_0(p^2, m_V^2, m_S^2)
We need here only the diagrams involving three point interactions because the 4point interactions are related to them due to gauge invariance.
Output
The oneloop expressions are saved in the SARAH internal Mathematica format and can be included in the LaTeX output. In addition, all selfenergies and onetadpoles are exported into Fortran
code for SPheno. This enable SPheno to calculate the loopcorrected masses for all particles as discussed below.
CalcLoopCorrections[Eigenstates,Options];
As usual, Eigenstates
can be for instance in the case of the MSSM either GaugeES
for the gauge eigenstates or EWSB
for the eigenstates after EWSB. If the vertices for the given set of eigenstates were not calculated before, this is done before the calculation of the loop contributions begins. As option a list with fields can be given (OnlyWith > {Particle1,Particle2,...}
). Only corrections involving these fields as internal particles are included.