Loop Masses

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The information about the one- and two-loop 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.

Loop corrected masses

Real scalars

For a real scalar[math]\phi[/math], the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop corrected mass matrix squared[math]m_\phi^{2,(L)}[/math] is related to the tree-level mass matrix squared[math]m_{\phi}^{2,(T)}[/math] and the self-energies via

[math]\label{eq:RealScalarLoop} m_\phi^{2,(L)}(p^2) = m^{2,(T)}_\phi - \Re(\Pi^{(1L)}_{\phi}(p^2)) - \Re(\Pi^{(2L)}_{\phi}(0)) [/math]

The one-shell condition for the eigenvalue[math]M^2_{\phi_i}(p^2)[/math] of the loop corrected mass matrix[math]m_\phi^{2,(L)}(p^2)[/math] reads

[math]\mathrm{Det}\left[ p^2_i \mathbf{1} - M^2_{\phi_i}(p^2) \right] = 0, \label{eq:propagator}[/math]

A stable solution of eq. ([eq:propagator]) for each eigenvalue[math]M^2_{\phi_i}(p^2=M^2_{\phi_i})[/math] is usually just found via an iterative procedure. In this approach one has to be careful how[math]m_\phi^{2,(T)}[/math] 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[math]M^2_{\phi_i}(p^2=M^2_{\phi_i})[/math]. In general, for each eigenvalue the rotation matrix is slightly different because of the[math]p^2[/math] 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.

Complex scalars

For a complex scalar[math]\eta[/math] the one-loop corrected mass matrix squared is related to the tree-level mass and the one-loop self-energy via

[math]m_\eta^{2,(1L)}(p^2) = m_\eta^{(T)} - \Pi^{(1L)}_{\eta}(p^2_i) ,[/math]

The same on-shell condition, eq. ([eq:propagator]), as for real scalars is used.

Vector bosons

For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosons[math]V[/math] are given by

[math]m^{2,(1L)}_{V} = m^{2,(T)}_{V} - \Re(\Pi^{T,(1L)}_{V}(p^2))[/math]

Majorana fermions

The one-loop mass matrix of a Majorana fermion[math]\chi[/math] is related to the tree-level mass matrix[math]m_\chi^{(T)}[/math] and the different parts of the self-energies by

[math]\begin{aligned} 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] \end{aligned}[/math]

Note,[math](T)[/math] is used to assign tree-level values while[math]T[/math] denotes a transposition. Eq. ([eq:propagator]) can also be used for fermions by taking the eigenvalues of[math]m_\chi^{2,(1L)}= m_\chi^{(1L)*} m_\chi^{(1L)}[/math].

Dirac fermions

For a Dirac fermion[math]\Psi[/math] one obtains the one-loop corrected mass matrix via

[math]\begin{aligned} \label{eq:DiracLoop} 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}[/math]

Here, the eigenvalues of[math](m_\Psi^{(1L)})^\dagger m_\Psi^{(1L)}[/math] are used in eq. ([eq:propagator]) to get the pole masses.

See also