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---
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title: One-Loop Self-Energies and Tadpoles
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permalink: /One-Loop_Self-Energies_and_Tadpoles/
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---
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# One-Loop Self-Energies and Tadpoles
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[Category:Calculations](/Category:Calculations "wikilink")
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Loop Corrections
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----------------
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... | ... | @@ -12,7 +9,7 @@ Loop Corrections |
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[fig:](/Image:1-loop.png "wikilink") \[fig:1loopDiagrams\]
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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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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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... | ... | @@ -31,7 +28,7 @@ As first step to get the loop corrections, SARAH generates all possible Feynman |
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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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$\\mathscr{M}_{\\phi^+_a \\tilde{u} \\tilde{d}^\*} = 3 \\times \\sum_{i=1}^6 \\sum_{j=1}^6 |c(\\phi^+_a \\tilde{u}_i \\tilde{d}^\*_j)|^2 B_0(p^2,m_{\\tilde u_i}^2,m_{\\tilde d_j}^2)$
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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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... | ... | @@ -57,7 +54,7 @@ and includes the following information |
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3. `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.
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4. `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 and*W*-boson. The absolute value of the factor depends on the type of the diagram.
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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 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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... | ... | @@ -82,7 +79,7 @@ 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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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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Also a list with the different contributions does exist:
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... | ... | @@ -142,7 +139,7 @@ In all calculations, specific coefficient are involved: |
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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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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*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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### One-loop tadpoles
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... | ... | |