... | ... | @@ -7,30 +7,29 @@ Loop Corrections |
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### One-loop corrections
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[fig:](/Image:1-loop.png "wikilink") \[fig:1loopDiagrams\]
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![fig](/Images/1-loop.png)
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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:
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- The self-energies*Π* of scalars and scalar mass matrices
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- The self-energies*Σ*<sup>*L*</sup>,*Σ*<sup>*R*</sup>,*Σ*<sup>*S*</sup> for fermions and fermion mass matrices
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- The transversal self-energy*Π*<sup>*T*</sup> of massive vector bosons
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- The self-energies $`\Pi`$ of scalars and scalar mass matrices
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- The self-energies $`\Sigma_L`$, $`\Sigma_R`$, $`\Sigma_S`$ for fermions and fermion mass matrices
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- The transversal self-energy $`\Pi^T`$ of massive vector bosons
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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
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ℳ = Symmetry × Colour × Couplings × Loop-Function
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$`\mathscr{M} = `$ Symmetry $`\times`$ Colour $`\times`$ Couplings $`\times`$ Loop-Function
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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 in E.2 of Ref. https://arxiv.org/abs/0806.0538.
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Here ’Symmetry’ and ’Colour’ are real factors. The loop-functions are expressed by standard Passarino-Veltman integrals*A*<sub>0</sub> and*B*<sub>0</sub> and some related functions
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*B*<sub>1</sub>
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,*B*<sub>22</sub>,*F*<sub>0</sub>,*G*<sub>0</sub>,*H*<sub>0</sub>,*B̄*<sub>22</sub> as defined in Ref. .
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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
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ℳ<sub>*S**S**S*</sub> = *c*<sub>*S*</sub> × *c*<sub>*F*</sub> × |*c*|<sup>2</sup>*B*<sub>0</sub>(*p*<sup>2</sup>, *m*<sub>*S*<sub>1</sub></sub><sup>2</sup>, *m*<sub>*S*<sub>2</sub></sub><sup>2</sup>)
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$`\mathscr{M}_{SSS}=c_S\times c_F \times |c|^2 B_0\left(p^2, m_{s1}^2,m_{s2}^2\right) `$
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In the case of an external charged Higgs*ϕ*<sup>+</sup> = ((*H*<sub>*d*</sub><sup>−</sup>)<sup>\*</sup>, *H*<sub>*u*</sub><sup>+</sup>) together with down- and up-squarks in the loop the correction to the charged Higgs mass matrix becomes
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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
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$`\mathscr{M}_{\phi^+_a \tilde{u} \tilde{d}^\*} = 3 \times \sum_{i=1}^6 \sum_{j=1}^6 |c(\phi^+_a \tilde{u}_i \tilde{d}^\*_j)|^2 B_0(p^2,m_{\tilde u_i}^2,m_{\tilde d_j}^2)`$
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$`\mathscr{M}_{\phi^+_a \tilde{u} \tilde{d}^*} = 3 \times \sum_{i=1}^6 \sum_{j=1}^6 |c(\phi^+_a \tilde{u}_i \tilde{d}^*_j)|^2 B_0(p^2,m_{\tilde u_i}^2,m_{\tilde d_j}^2)`$
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*c*(*ϕ*<sub>*a*</sub><sup>+</sup>*ũ*<sub>*i*</sub>*d̃*<sub>*j*</sub><sup>\*</sup>) 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*<sup>2</sup> dependence is kept.
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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.
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##### Conventions
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... | ... | @@ -57,7 +56,7 @@ and includes the following information |
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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 <span>rMS</span>. <span>rMS = 0</span> gives to the results in$`\overline{\text{DR}}`$ scheme and <span>rMS = 1</span> corresponds to$`\overline{\text{MS}}`$ scheme.
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The information about the loop correction are also saved in the directory
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../\SARAH/Output/"ModelName"/$EIGENSTATES/Loop
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../SARAH/Output/"ModelName"/$EIGENSTATES/Loop
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##### One Loop Tadpoles
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... | ... | @@ -78,9 +77,9 @@ The results are saved in the following two dimensional array |
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The first column gives the name of the particle, the entry in the second column depends on the type of the field
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1. Scalars: one-loop self energy*Π*(*p*<sup>2</sup>)
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2. Fermions: one-loop self energies for the different polarizations (</math>\Sigma^L(p^2)</math>,</math>\Sigma^R(p^2)</math>,*Σ*<sup>*S*</sup>(*p*<sup>2</sup>))
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3. Vector bosons: one-loop, transversal self energy*Π*<sup>*T*</sup>(*p*<sup>2</sup>)
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1. Scalars: one-loop self energy $`\Pi(p^2)`$
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2. Fermions: one-loop self energies for the different polarizations ($`\Sigma^L(p^2),\, \Sigma^R(p^2),\, \Sigma^S(p^2)`$)
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3. Vector bosons: one-loop, transversal self energy $`\Pi^T(p^2)`$
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Also a list with the different contributions does exist:
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... | ... | @@ -88,20 +87,20 @@ Also a list with the different contributions does exist: |
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##### Examples
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1. <span>**One-loop tadpoles**</span> The correction of the tadpoles due to a chargino loop is saved in
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1. **One-loop tadpoles** The correction of the tadpoles due to a chargino loop is saved in
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Tadpoles1LoopList[EWSB][[/1|1]];
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and reads
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{bar[Cha],Cp[Uhh[{gO1}],bar[Cha[{gI1}]],Cha[{gI1}]],FFS,1,1/2}
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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](/Loop_functions "wikilink")).
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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](/Loop_functions "wikilink")).
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The corresponding term in `Tadpoles1LoopSum[EWSB]` is
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4*sum[gI1,1,2, A0[Mass[bar[Cha[{gI1}]]]^2]*
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Cp[phid,bar[Cha[{gI1}]],Cha[{gI1}]]*Mass[Cha[{gI1}]]]
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2. <span>**One-loop self-energies**</span>
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2. **One-loop self-energies**
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1. 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
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{Ah,Cp[conj[USd[{gO1}]],USd[{gO2}],Ah[{gI1}],Ah[{gI1}]],SSSS,1,1/2}
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... | ... | @@ -135,70 +134,72 @@ Generic expressions |
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In all calculations, specific coefficient are involved:
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- *c*<sub>*S*</sub> 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.
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- *c*<sub>*C*</sub> 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.
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- *c*<sub>*R*</sub> is 2 for real fields and Majorana fermions in the loop and 1 otherwise.
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- $`c_S`$ 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.
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- $`c_S`$ 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.
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- $`c_R`$ is 2 for real fields and Majorana fermions in the loop and 1 otherwise.
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We use in the following*Γ* for non-chiral interactions and*Γ*<sub>*L*</sub>/</math>\Gamma_R</math> 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.
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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.
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### One-loop tadpoles
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1. Fermion loop (generic name in SARAH : `FFS`):
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</math>T = 8 c_S c_C m_F \\Gamma A_0(m_F^2)</math>
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$`T = 8 c_S c_C m_F \Gamma A_0(m_F^2)`$
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2. Scalar loop (generic name in SARAH : `SSS`):
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</math>T = - 2 c_S c_C \\Gamma A_0(m_S^2)</math>
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$`T = - 2 c_S c_C \Gamma A_0(m_S^2)`$
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3. Vector boson loop (generic name in SARAH : `SVV`):
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</math>T = 6 c_S c_C \\Gamma A_0(m_V^2)</math>
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$`T = 6 c_S c_C \Gamma A_0(m_V^2)`$
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### One-loop self-energies
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##### Corrections to fermion
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1. Fermion-scalar loop (generic name in SARAH : `FFS`):
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$\\begin{aligned}
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\\Sigma^S(p^2) &=& m_F c_S c_C c_R \\Gamma^1_R \\Gamma^{2,\*}_L B_0(p^2,m_F^2,m_S^2)
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\\\\
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\\Sigma^R(p^2) &=& - c_S c_C c_R \\frac{1}{2} \\Gamma^1_R \\Gamma^{2,\*}_R
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B_1(p^2,m_F^2,m_S^2) \\\\
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\\Sigma^L(p^2) &=& - c_S c_C c_R \\frac{1}{2} \\Gamma^1_L \\Gamma^{2,\*}_L
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B_1(p^2,m_F^2,m_S^2) \\end{aligned}$
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2. Fermion-vector boson loop (generic name in SARAH : `FFV`):
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$\\begin{aligned}
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\\Sigma^S(p^2) &=& - 4 c_S c_C c_R m_F \\Gamma^1_L \\Gamma^{2,\*}_R
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1. Fermion-scalar loop (generic name in SARAH : `FFS`):
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$`
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\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) \\
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\Sigma^R(p^2) = - c_S c_C c_R \frac{1}{2} \Gamma^1_R \Gamma^{2,*}_R
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B_1(p^2,m_F^2,m_S^2) \\
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\Sigma^L(p^2) = - c_S c_C c_R \frac{1}{2} \Gamma^1_L \Gamma^{2,*}_L
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B_1(p^2,m_F^2,m_S^2)
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`$
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2. Fermion-vector boson loop (generic name in SARAH : `FFV`):
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$`
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\Sigma^S(p^2) = - 4 c_S c_C c_R m_F \Gamma^1_L \Gamma^{2,*}_R
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B_0(p^2,m_F^2,m_S^2) \\\\
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\\Sigma^R(p^2) &=& - c_S c_C c_R \\Gamma^1_L \\Gamma^{2,\*}_L B_1(p^2,m_F^2,m_S^2)
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\\\\
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\\Sigma^L(p^2) &=& - c_S c_C c_R \\Gamma^1_R \\Gamma^{2,\*}_R B_1(p^2,m_F^2,m_S^2) \\end{aligned}$
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\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)\\
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\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)
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`$
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##### Corrections to scalar
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1. Fermion loop (generic name in SARAH : `FFS`):
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*Π*(*p*<sup>2</sup>)=*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>((*Γ*<sub>*L*</sub><sup>1</sup>*Γ*<sub>*L*</sub><sup>2, \*</sup>+*Γ*<sub>*R*</sub><sup>1</sup>*Γ*<sub>*R*</sub><sup>2, \*</sup>)*G*<sub>0</sub>(*p*<sup>2</sup>,*m*<sub>*F*</sub><sup>2</sup>,*m*<sub>*S*</sub><sup>2</sup>)+(*Γ*<sub>*L*</sub><sup>1</sup>*Γ*<sub>*R*</sub><sup>2, \*</sup>+*Γ*<sub>*R*</sub><sup>1</sup>*Γ*<sub>*L*</sub><sup>2, \*</sup>)*B*<sub>0</sub>(*p*<sup>2</sup>,*m*<sub>*F*</sub><sup>2</sup>,*m*<sub>*S*</sub><sup>2</sup>))
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1. Fermion loop (generic name in SARAH : `FFS`):
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$` \Pi(p^2) = c_S c_C c_R \left( (\Gamma^1_L \Gamma^{2,*}_L + \Gamma^1_R \Gamma^{2,*}_R ) G_0(p^2,m_F^2, m_S^2) + (\Gamma^1_L \Gamma^{2,*}_R + \Gamma^1_R \Gamma^{2,*}_L ) B_0(p^2,m_F^2, m_S^2) \right) `$
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2. Scalar loop (two 3-point interactions, generic name in SARAH : `SSS`):
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*Π*(*p*<sup>2</sup>)=*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>*Γ*<sup>1</sup>*Γ*<sup>2, \*</sup>*B*<sub>0</sub>(*p*<sup>2</sup>, *m*<sub>*F*</sub><sup>2</sup>, *m*<sub>*S*</sub><sup>2</sup>)
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2. Scalar loop (two 3-point interactions, generic name in SARAH : `SSS`):
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$`\Pi(p^2) = c_S c_C c_R \Gamma^1\Gamma^2 B_0 (p^2, m_F^2,m_S^2)`$
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3. Scalar loop (4-point interaction, generic name in SARAH : `SSSS`):
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*Π*(*p*<sup>2</sup>)= − *c*<sub>*S*</sub>*c*<sub>*C*</sub>*Γ**A*<sub>0</sub>(*m*<sub>*S*</sub><sup>2</sup>)
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3. Scalar loop (4-point interaction, generic name in SARAH : `SSSS`):
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$`\Pi(p^2) = - c_S c_C \Gamma A_0(m_S^2) `$
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4. Vector boson-scalar loop (generic name in SARAH : `SSV`):
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*Π*(*p*<sup>2</sup>)=*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>*Γ*<sup>1</sup>*Γ*<sup>2, \*</sup>*F*<sub>0</sub>(*p*<sup>2</sup>, *m*<sub>*F*</sub><sup>2</sup>, *m*<sub>*S*</sub><sup>2</sup>)
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4. Vector boson-scalar loop (generic name in SARAH : `SSV`):
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$`\Pi(p^2) = c_S c_C c_R \Gamma^1 \Gamma_2 F_0(p^2, m_F^2, m_S^2)`$
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5. Vector boson loop (two 3-point interactions, generic name in SARAH : `SVV`):
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$`\Pi(p^2) = c_S c_C c_R \frac{7}{2} \Gamma^1 \Gamma^{2,*} B_0(p^2,m_F^2,m_S^2)`$
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5. Vector boson loop (two 3-point interactions, generic name in SARAH : `SVV`):
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$\\Pi(p^2) = c_S c_C c_R \\frac{7}{2} \\Gamma^1 \\Gamma^{2,\*} B_0(p^2,m_F^2,m_S^2)$
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6. Vector boson loop (4-point interaction, generic name in SARAH : `SSVV`):
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$`\Pi(p^2) = c_S c_C \gamma A_0(m^2_V)`$
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6. Vector boson loop (4-point interaction, generic name in SARAH : `SSVV`):
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*Π*(*p*<sup>2</sup>)=*c*<sub>*S*</sub>*c*<sub>*C*</sub>*Γ**A*<sub>0</sub>(*m*<sub>*V*</sub><sup>2</sup>)
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##### Corrections to vector boson
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1. Fermion loop (generic name in SARAH : `FFV`):
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*Π*<sup>*T*</sup>(*p*<sup>2</sup>)=*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>((|*Γ*<sub>*L*</sub><sup>1</sup>|<sup>2</sup>+|*Γ*<sub>*R*</sub><sup>1</sup>|<sup>2</sup>)*H*<sub>0</sub>(*p*<sup>2</sup>,*m*<sub>*V*</sub><sup>2</sup>,*m*<sub>*F*</sub><sup>2</sup>)+4*R**e*(*Γ*<sub>*L*</sub><sup>1</sup>*Γ*<sub>*R*</sub><sup>2</sup>)*B*<sub>0</sub>(*p*<sup>2</sup>,*m*<sub>*V*</sub><sup>2</sup>,*m*<sub>*F*</sub><sup>2</sup>))
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1. Fermion loop (generic name in SARAH : `FFV`):
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$`\Pi^T(p^2) = c_S c_C c_R (( |\Gamma_L^1|^2+|\Gamma_R^1|^2) H_0 (p^2, m_V^2, m_F^2) + 4 R_e (\Gamma_L^1 \Gamma_R^2) B_0(p^2, m_V^2,m_F^2))`$
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2. Scalar loop (generic name in SARAH : `SSV`):
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2. Scalar loop (generic name in SARAH : `SSV`):
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*Π*<sup>*T*</sup>(*p*<sup>2</sup>)= − 4*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>|*Γ*|<sup>2</sup>*B*<sub>22</sub>(*p*<sup>2</sup>, *m*<sub>*S*<sub>1</sub></sub><sup>2</sup>, *m*<sub>*S*<sub>2</sub></sub><sup>2</sup>)
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3. Vector boson loop (generic name in SARAH : `VVV`):
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