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# Loop Masses
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The information about the [one-](/One-Loop_Self-Energies_and_Tadpoles "wikilink") and [two-loop](/Two-Loop_Self-Energies_and_Tadpoles "wikilink") corrections to the one- and two-point functions can be used to calculate the loop corrected mass spectrum. The renormalized mass matrices (or masses) are related to the tree-level mass matrices (or masses) and the self-energies as follows.
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The information about the [one-](/One-Loop_Self-Energies_and_Tadpoles) and [two-loop](/Two-Loop_Self-Energies_and_Tadpoles) corrections to the one- and two-point functions can be used to calculate the loop corrected mass spectrum. The renormalized mass matrices (or masses) are related to the tree-level mass matrices (or masses) and the self-energies as follows.
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### Loop corrected masses
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#### Real scalars
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For a real scalar*ϕ*, the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop corrected mass matrix squared*m*<sub>*ϕ*</sub><sup>2, (*L*)</sup> is related to the tree-level mass matrix squared*m*<sub>*ϕ*</sub><sup>2, (*T*)</sup> and the self-energies via
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For a real scalar $`\Phi`$, the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop corrected mass matrix squared $`m_\Phi^{2,(L)}`$ is related to the tree-level mass matrix squared $`m_\Phi^{2,(T)}`$ and the self-energies via
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*m*<sub>*ϕ*</sub><sup>2, (*L*)</sup>(*p*<sup>2</sup>)=*m*<sub>*ϕ*</sub><sup>2, (*T*)</sup> − ℜ(*Π*<sub>*ϕ*</sub><sup>(1*L*)</sup>(*p*<sup>2</sup>)) − ℜ(*Π*<sub>*ϕ*</sub><sup>(2*L*)</sup>(0))
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$`m_\Phi^{2,(L)}(p^2)= m_\Phi^{2,(T)} - \mathcal{R}\left(\Pi_\Phi^{(1L)}(p^2)\right) -\mathcal{R}\left(\Pi_\Phi^{(2L)}(0)\right) `$
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The one-shell condition for the eigenvalue $`M_{\Phi_i}^2(p^2)`$ of the loop corrected mass matrix $`m_\Phi^{2,(L)}`$ reads
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$`Det\left[p_i^2 \mathcal{1} - M_{\Phi_i}^2(p^2) \right]=0`$
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The one-shell condition for the eigenvalue*M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>(*p*<sup>2</sup>) of the loop corrected mass matrix*m*<sub>*ϕ*</sub><sup>2, (*L*)</sup>(*p*<sup>2</sup>) reads
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*D**e**t*\[*p*<sub>*i*</sub><sup>2</sup>**1**−*M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>(*p*<sup>2</sup>)\] = 0,
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A stable solution of eq. (\[eq:propagator\]) for each eigenvalue*M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>(*p*<sup>2</sup> = *M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>) is usually just found via an iterative procedure. In this approach one has to be careful how*m*<sub>*ϕ*</sub><sup>2, (*T*)</sup> is defined: this is the tree-level mass matrix where the parameters are taken at the minimum of the effective potential evaluated at the same loop-level at which the self-energies are known. The physical masses are associated with the eigenvalues*M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>(*p*<sup>2</sup> = *M*<sub>*ϕ*<sub>*i*</sub></sub><sup>2</sup>). In general, for each eigenvalue the rotation matrix is slightly different because of the*p*<sup>2</sup> dependence of the self-energies. The convention by SARAH and SPheno is that the rotation matrix of the lightest eigenvalue is used in all further calculations and the output.
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A stable solution of this equation for each eigenvalue $`M_{\Phi_i}^2\left( p^2 = M_{\Phi_i}^2\right)`$ is usually just found via an iterative procedure. In this approach one has to be careful how $`m_\Phi^{2,(T)}`$ is defined: this is the tree-level mass matrix where the parameters are taken at the minimum of the effective potential evaluated at the same loop-level at which the self-energies are known. The physical masses are associated with the eigenvalues $`M_{\Phi_i}^2\left( p^2 = M_{\Phi_i}^2\right)`$. In general, for each eigenvalue the rotation matrix is slightly different because of the $`p^2`$ dependence of the self-energies. The convention by SARAH and SPheno is that the rotation matrix of the lightest eigenvalue is used in all further calculations and the output.
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#### Complex scalars
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For a complex scalar*η* the one-loop corrected mass matrix squared is related to the tree-level mass and the one-loop self-energy via
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$`m_\eta^{2,(L)}(p^2)= m_\eta^{2,(T)} - \Pi_\eta^{(1L)}(p^2)`$
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*m*<sub>*η*</sub><sup>2, (1*L*)</sup>(*p*<sup>2</sup>)=*m*<sub>*η*</sub><sup>(*T*)</sup> − *Π*<sub>*η*</sub><sup>(1*L*)</sup>(*p*<sub>*i*</sub><sup>2</sup>),
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The same on-shell condition, eq. (\[eq:propagator\]), as for real scalars is used.
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The same on-shell condition as for real scalars is used (see previous section).
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#### Vector bosons
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For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosons*V* are given by
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*m*<sub>*V*</sub><sup>2, (1*L*)</sup> = *m*<sub>*V*</sub><sup>2, (*T*)</sup> − ℜ(*Π*<sub>*V*</sub><sup>*T*, (1*L*)</sup>(*p*<sup>2</sup>))
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$`m_V^{2,(L)}(p^2)= m_V^{2,(T)} - \Pi_V^{T,(1L)}(p^2)`$
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#### Majorana fermions
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The one-loop mass matrix of a Majorana fermion*χ* is related to the tree-level mass matrix*m*<sub>*χ*</sub><sup>(*T*)</sup> and the different parts of the self-energies by
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The one-loop mass matrix of a Majorana fermion*χ* is related to the tree-level mass matrix $`m_\chi^{(T)}`$ and the different parts of the self-energies by
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$` m_\chi^{(1L)} (p^2) = m_\chi^{(T)} - \frac{1}{2} \bigg\[ \Sigma^\chi_S(p^2) + \Sigma^{\chi,T}_S(p^2) + \left(\Sigma^{\chi,T}_L(p^2)+ \Sigma^\chi_R(p^2)\right) m_\chi^{(T)} \nonumber \\ \hspace{16mm} + m_{\chi}^{(T)} \left(\Sigma^{\chi,T}_R(p^2) + \Sigma^\chi_L(p^2) \right) \bigg\] `$
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$` m_\chi^{(1L)} (p^2) = m_\chi^{(T)} - \frac{1}{2} \bigg[ \Sigma^\chi_S(p^2) + \Sigma^{\chi,T}_S(p^2) + \left(\Sigma^{\chi,T}_L(p^2)+ \Sigma^\chi_R(p^2)\right) m_\chi^{(T)} + m_{\chi}^{(T)} \left(\Sigma^{\chi,T}_R(p^2) + \Sigma^\chi_L(p^2) \right) \bigg]`$
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Note,(*T*) is used to assign tree-level values while*T* denotes a transposition. Eq. (\[eq:propagator\]) can also be used for fermions by taking the eigenvalues of*m*<sub>*χ*</sub><sup>2, (1*L*)</sup> = *m*<sub>*χ*</sub><sup>(1*L*)\*</sup>*m*<sub>*χ*</sub><sup>(1*L*)</sup>.
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Note, $`(T)`$ is used to assign tree-level values while $`T`$ denotes a transposition. This equation can also be used for fermions by taking the eigenvalues of $` m_\chi^{1,(1L)} (p^2)= \left(m_\chi^{(1L)} (p^2)\right)^* m_\chi^{(1L)} (p^2)`$.
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#### Dirac fermions
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For a Dirac fermion*Ψ* one obtains the one-loop corrected mass matrix via
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For a Dirac fermion $`\Psi`$ one obtains the one-loop corrected mass matrix via
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$` m_\Psi^{(1L)}(p^2) = m_\Psi^{(T)} - \Sigma^+_S(p^2) - \Sigma^+_R(p^2) m_\Psi^{(T)} - m_\Psi^{(T)} \Sigma^+_L(p^2) .`$
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$` m_\Psi^{(1L)}(p^2) = m_\Psi^{(T)} - \Sigma^+_S(p^2) - \Sigma^+_R(p^2) m_\Psi^{(T)} - m_\Psi^{(T)} \Sigma^+_L(p^2) .`$
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Here, the eigenvalues of(*m*<sub>*Ψ*</sub><sup>(1*L*)</sup>)<sup>†</sup>*m*<sub>*Ψ*</sub><sup>(1*L*)</sup> are used in eq. (\[eq:propagator\]) to get the pole masses.
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Here, the eigenvalues of $`(m_\Psi)^* m_\Psi`$ are used in the iterative procedure to get the pole masses.
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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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- [Using SPheno for two-loop masses](/Using_SPheno_for_two-loop_masses "wikilink") |
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- [Loop functions](/Loop_functions)
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- [Calculation of the mass spectrum with SPheno](/SPheno_mass_calculation)
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- [Using SPheno for two-loop masses](/Using_SPheno_for_two-loop_masses) |