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---
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title: Tree Masses
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permalink: /Tree_Masses/
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---
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# Tree Masses
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[Category:Calculations](/Category:Calculations "wikilink")
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General calculation
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-------------------
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SARAH uses the definition of the rotations defined in the model file to calculate the mass matrices for particles which mix. In general, all field rotations as listed in '[Supported particle mixing](/Supported_particle_mixing "wikilink")' are possible. The mass matrices for scalars are calculated by
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$M^S_{ij} = \\frac{-\\partial^2 \\mathfrak{L}}{\\partial {\\phi}_i \\partial {\\phi}_j^\*}$
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$`M^S_{ij} = \frac{-\partial^2 \mathfrak{L}}{\partial {\phi}_i \partial {\phi}_j^\*}`$
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where *ϕ* can be either real or complex, i.e. the resulting *M*<sup>*S*</sup> corresponds to *M*<sub>*C*</sub> or *M*<sub>*R*</sub> as [as define here](/Supported_particle_mixing "wikilink"). In the mass matrices of states which include Goldstone bosons also the *R*<sub>*ξ*</sub> dependent terms are included.
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The mass matrices for fermions are calculated as
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$M^F_{ij} = \\frac{-\\partial^2 \\mathfrak{L}}{\\partial {\\psi}^x_i {\\psi}^y_j}$
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$`M^F_{ij} = \frac{-\partial^2 \mathfrak{L}}{\partial {\psi}^x_i {\psi}^y_j}`$
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with *x* = *y* = 0 for Majorana fermions, and *x* = 1, *y* = 2 for Dirac fermions.
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