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# One-Loop Self-Energies and Tadpoles
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Loop Corrections
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----------------
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### One-loop corrections
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## One-loop corrections
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![fig](/Images/1-loop.png)
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... | ... | @@ -31,7 +26,7 @@ $`\mathscr{M}_{\phi^+_a \tilde{u} \tilde{d}^*} = 3 \times \sum_{i=1}^6 \sum_{j=1 |
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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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### Conventions
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The results will contain the Passarino Veltman integrals listed [here](/Loop_functions "wikilink"). The involved couplings are abbreviated by
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... | ... | @@ -40,7 +35,7 @@ The results will contain the Passarino Veltman integrals listed [here](/Loop_fun |
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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](/Loop_Masses "wikilink"). 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.
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##### Results
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### Results
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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
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... | ... | @@ -58,7 +53,7 @@ The information about the loop correction are also saved in the directory |
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../SARAH/Output/"ModelName"/$EIGENSTATES/Loop
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##### One Loop Tadpoles
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### One Loop Tadpoles
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The complete results as sums of the different contributions are saved in the two dimensional array
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... | ... | @@ -69,7 +64,7 @@ A list of the different contributions, including symmetry and charge factors, is |
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Tadpoles1LoopList[$EIGENSTATES];
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##### One Loop Self Energies
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### One Loop Self Energies
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The results are saved in the following two dimensional array
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... | ... | @@ -85,9 +80,10 @@ Also a list with the different contributions does exist: |
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SelfEnergy1LoopList[$EIGENSTATES]
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##### Examples
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### Examples
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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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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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... | ... | @@ -129,8 +125,8 @@ Also a list with the different contributions does exist: |
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in `SelfEnergy1LoopListSum[EWSB]`.
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Generic expressions
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-------------------
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### Generic expressions
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In all calculations, specific coefficient are involved:
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... | ... | @@ -140,7 +136,7 @@ In all calculations, specific coefficient are involved: |
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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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#### One-loop tadpoles
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1. Fermion loop (generic name in SARAH : `FFS`):
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$`T = 8 c_S c_C m_F \Gamma A_0(m_F^2)`$
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... | ... | @@ -151,7 +147,7 @@ We use in the following $\Gamma$ for non-chiral interactions and $`\Gamma_L`$/$` |
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3. Vector boson loop (generic name in SARAH : `SVV`):
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$`T = 6 c_S c_C \Gamma A_0(m_V^2)`$
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### One-loop self-energies
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#### One-loop self-energies
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##### Corrections to fermion
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... | ... | @@ -200,26 +196,28 @@ We use in the following $\Gamma$ for non-chiral interactions and $`\Gamma_L`$/$` |
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2. Scalar loop (generic name in SARAH : `SSV`):
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*Π*<sup>*T*</sup>(*p*<sup>2</sup>)= − 4*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>|*Γ*|<sup>2</sup>*B*<sub>22</sub>(*p*<sup>2</sup>, *m*<sub>*S*<sub>1</sub></sub><sup>2</sup>, *m*<sub>*S*<sub>2</sub></sub><sup>2</sup>)
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$`\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)`$
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3. Vector boson loop (generic name in SARAH : `VVV`):
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$`\Pi^T(p^2) = |\Gamma|^2 c_s c_C c_R ( -(4 p^2 + m_{V,1}^2+ m_{V,2}^2) B_0 (p^2, m_{V,1}^2,m_{V,2}^2)-8 B_{22}(p^2, m_{S,1}^2,m_{S,2}^2))`$
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3. Vector boson loop (generic name in SARAH : `VVV`):
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*Π*<sup>*T*</sup>(*p*<sup>2</sup>)=|*Γ*|<sup>2</sup>*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>(−(4*p*<sup>2</sup>+*m*<sub>*V*<sub>1</sub></sub><sup>2</sup>+*m*<sub>*V*<sub>2</sub></sub><sup>2</sup>)*B*<sub>0</sub>(*p*<sup>2</sup>,*m*<sub>*V*<sub>1</sub></sub><sup>2</sup>,*m*<sub>*V*<sub>1</sub></sub><sup>2</sup>)−8*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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4. Vector-Scalar-Loop (generic name in SARAH : `SVV`):
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$`\Pi^T(p^2) = |\Gamma|^2 c_S c_C c_R B_0(p^2, m_V^2, m_S^2)`$
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4. Vector-Scalar-Loop (generic name in SARAH : `SVV`):
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*Π*<sup>*T*</sup>(*p*<sup>2</sup>)=|*Γ*|<sup>2</sup>*c*<sub>*S*</sub>*c*<sub>*C*</sub>*c*<sub>*R*</sub>*B*<sub>0</sub>(*p*<sup>2</sup>, *m*<sub>*V*</sub><sup>2</sup>, *m*<sub>*S*</sub><sup>2</sup>)
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We need here only the diagrams involving three point interactions because the 4-point interactions are related to them due to gauge invariance.
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#### Output
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### Output
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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 `Fortran`code for SPheno. This enable SPheno to calculate the loop-corrected masses for all particles as discussed below.
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CalcLoopCorrections[Eigenstates,Options];
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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 (<span>OnlyWith -> <span>Particle1,Particle2,...</span></span>). Only corrections involving these fields as internal particles are included.
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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.
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See also
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--------
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- [Loop functions](/Loop_functions "wikilink")
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- [Calculation of the mass spectrum with SPheno](/SPheno_mass_calculation "wikilink") |
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- [Loop functions](/Loop_functions)
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- [Calculation of the mass spectrum with SPheno](/SPheno_mass_calculation) |