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---
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title: Vertices
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permalink: /Vertices/
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---
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# Vertices
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[Category:Calculations](/Category:Calculations "wikilink")
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Calculation of vertices
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-----------------------
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... | ... | @@ -11,15 +8,11 @@ Calculation of vertices |
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Vertices are not automatically calculated during the initialization of a model like this is done for mass matrices and tadpole equations. However, the calculation can be started very easily. In general, SARAH is optimized for the extraction of three- and four-point interactions with renormalizable operators. That means, usually only the following generic interactions are taken into account in the calculations: interactions of two fermions or two ghosts with one scalar or vector bosons (`FFS`, `FFV`, `GGS`, `GGV` ), interactions of three or four scalars or vector bosons (`SSS`, `SSSS`, `VVV`, `VVVV` ), as well as interactions of two scalars with one or two vector bosons (`SSV`, `SSVV` ) or two vector bosons with one scalar (`SVV` ).
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In this context, vertices not involving fermions are calculated by
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$\\begin{aligned}
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V(\\eta_a,\\eta_b,\\eta_c) \\equiv & i \\frac{\\partial^3 \\mathfrak{L}}{\\partial \\eta_a \\partial \\eta_b \\partial \\eta_c} = C \\Gamma \\\\
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V(\\eta_a,\\eta_b,\\eta_c, \\eta_d) \\equiv= & i \\frac{\\partial^4 \\mathfrak{L}}{\\partial \\eta_a \\partial \\eta_b \\partial \\eta_c \\partial \\eta_d} = C \\Gamma\\end{aligned}$
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$` V(\eta_a,\eta_b,\eta_c) \equiv i \frac{\partial^3 \mathfrak{L}}{\partial \eta_a \partial \eta_b \partial \eta_c} = C \Gamma \\ V(\eta_a,\eta_b,\eta_c, \eta_d) \equiv= i \frac{\partial^4 \mathfrak{L}}{\partial \eta_a \partial \eta_b \partial \eta_c \partial \eta_d} = C \Gamma`$
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Here,*η*<sub>*i*</sub> are either scalars, vector bosons, or ghosts. The results are expressed by a coefficient*C* which is a Lorentz invariant and a Lorentz factor*Γ* which involves*γ*<sub>*μ*</sub>,*p*<sub>*μ*</sub>, or*g*<sup>*μ**ν*</sup>. Vertices for Dirac fermions are first expressed in terms of Weyl fermions. The two vertices are then calculated separately. Taking two Dirac fermions*F*<sub>*a*</sub> = (*Ψ*<sub>*a*</sub><sup>1</sup>, *Ψ*<sub>*a*</sub><sup>2\*</sup>),*F*<sub>*b*</sub> = (*Ψ*<sub>*b*</sub><sup>1</sup>, *Ψ*<sub>*b*</sub><sup>2\*</sup>) and distinguishing the two cases for fermion–vector and fermion–scalar couplings, the vertices are calculated and expressed by
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$\\begin{aligned}
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V(\\bar F_a,F_b,V_c) &= \\{V(\\Psi_a^{1 \*},\\Psi_b^1,V_c), V(\\Psi_a^2,\\Psi_b^{2\*},V_c)\\} \\equiv \\{C^L \\gamma_\\mu P_L, C^R \\gamma_\\mu P_R\\} \\\\
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V(\\bar F_a,F_b,S_c) &= \\{V(\\Psi_a^2,\\Psi_b^1,S_c), V(\\Psi_a^{1\*},\\Psi_b^{2\*},V_c)\\} \\equiv \\{C^L P_L, C^R P_R\\}\\end{aligned}$
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$` V(\bar F_a,F_b,V_c) = \{V(\Psi_a^{1 \*},\Psi_b^1,V_c), V(\Psi_a^2,\Psi_b^{2\*},V_c)\} \equiv \{C^L \gamma_\mu P_L, C^R \gamma_\mu P_R\} \\ V(\bar F_a,F_b,S_c) = \{V(\Psi_a^2,\Psi_b^1,S_c), V(\Psi_a^{1\*},\Psi_b^{2\*},V_c)\} \equiv \{C^L P_L, C^R P_R\}`$
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Here, the polarization operators*P*<sub>*L*, *R*</sub> are used.
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The user can either calculate specific vertices for a particular set of external states or call functions that SARAH derives all existing interactions from the Lagrangian. The first option might be useful to check the exact structure of single vertices, while the second one is needed to get all vertices to write model files for other tools.
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1. `gamma[lor]`: Gamma matrix*γ*<sub>*μ*</sub>
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2. `g[lor1,lor2]`: Metric tensor*g*<sub>*μ**ν*</sub>
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3. `Mom[particle,lor]`: Momentum*p*<sub>*P*</sub><sup>*μ*</sup> of particle*P*
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4. `PL`, `PR`: Polarization operators$P_L = \\frac{1 - \\gamma_5}{2}$,$P_R = \\frac{1+\\gamma_5}{2}$
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4. `PL`, `PR`: Polarization operators$`P_L = \frac{1 - \gamma_5}{2}`$,$`P_R = \frac{1+\gamma_5}{2}`$
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5. `1`: If the vertex is a Lorentz scalar.
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6. `LorentzProduct[_,_]:` A non commutative product of terms transforming under the Lorentz group
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