One-Loop Self-Energies and Tadpoles

One-loop corrections

SARAH calculates the analytical expressions for the one-loop corrections to the tadpoles and the one-loop self-energies for all particles. For states which are a mixture of several gauge eigenstates, the self-energy 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 non-SUSY 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 self-energies \Pi of scalars and scalar mass matrices
The self-energies \Sigma_L, \Sigma_R, \Sigma_S for fermions and fermion mass matrices
The transversal self-energy \Pi^T of massive vector bosons

The approach to calculate the loop corrections is as follows: all possible generic diagrams at the one-loop level shown in Fig. [fig:1loopDiagrams] are included in SARAH. Each generic amplitude is parametrized by

\mathscr{M} = Symmetry \times Colour \times Couplings \times Loop-Function

Here ’Symmetry’ and ’Colour’ are real factors. The loop-functions are expressed by standard Passarino-Veltman 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 self-energy 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 up-squarks 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 Higgs-sdown-sup 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] and Cp[p1,p2,p3,p4] for non-chiral, three and four point interactions involving the particles p1 - p4.
Cp[p1,p2,p3][PL] and Cp[p1,p2,p3][PR] for chiral, three-point interactions involving the fields p1 - 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 andW-boson. 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 one-loop 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: one-loop self energy \Pi(p^2)
Fermions: one-loop self energies for the different polarizations (\Sigma^L(p^2),\, \Sigma^R(p^2),\, \Sigma^S(p^2))
Vector bosons: one-loop, transversal self energy \Pi^T(p^2)

Also a list with the different contributions does exist:

SelfEnergy1LoopList[$EIGENSTATES]

Examples

One-loop tadpoles: The correction of the tadpoles due to a chargino loop is saved in
```
Tadpoles1LoopList[EWSB][[/1|1]];
```
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 index gO1 and two charginos with index gI1 is needed, (iii) the generic type of the diagram is FFS, (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 in Tadpoles1LoopSum[EWSB] is
```
4*sum[gI1,1,2, A0[Mass[bar[Cha[{gI1}]]]^2]*
   Cp[phid,bar[Cha[{gI1}]],Cha[{gI1}]]*Mass[Cha[{gI1}]]]
```

One-loop self-energies

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 non-chiral 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.

One-loop 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)

One-loop self-energies

Corrections to fermion

Fermion-scalar 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)
Fermion-vector 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 3-point 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 (4-point interaction, generic name in SARAH : SSSS):
\Pi(p^2) = - c_S c_C \Gamma A_0(m_S^2)
Vector boson-scalar 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 3-point 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 (4-point 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))
Vector-Scalar-Loop (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 4-point interactions are related to them due to gauge invariance.

Output

The one-loop expressions are saved in the SARAH internal Mathematica format and can be included in the LaTeX output. In addition, all self-energies and one-tadpoles are exported into Fortrancode for SPheno. This enable SPheno to calculate the loop-corrected 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.

One Loop_Self Energies_and_Tadpoles