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---
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title: Loop Masses
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permalink: /Loop_Masses/
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---
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[Category:Calculations](/Category:Calculations "wikilink") 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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### 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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*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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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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#### 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*<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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#### 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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#### 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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$\\begin{aligned}
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m_\\chi^{(1L)} (p^2) &=& m_\\chi^{(T)} - \\frac{1}{2} \\bigg\[ \\Sigma^\\chi_S(p^2) + \\Sigma^{\\chi,T}_S(p^2)
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+ \\left(\\Sigma^{\\chi,T}_L(p^2)+ \\Sigma^\\chi_R(p^2)\\right) m_\\chi^{(T)}
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\\nonumber \\\\
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&& \\hspace{16mm}
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+ m_{\\chi}^{(T)} \\left(\\Sigma^{\\chi,T}_R(p^2) + \\Sigma^\\chi_L(p^2) \\right)
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\\bigg\] \\end{aligned}$
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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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#### Dirac fermions
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For a Dirac fermion*Ψ* one obtains the one-loop corrected mass matrix via
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$\\begin{aligned}
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\\label{eq:DiracLoop}
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m_\\Psi^{(1L)}(p^2) = m_\\Psi^{(T)} - \\Sigma^+_S(p^2) - \\Sigma^+_R(p^2) m_\\Psi^{(T)} - m_\\Psi^{(T)} \\Sigma^+_L(p^2) .\\end{aligned}$
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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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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") |