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---
title: SPheno threshold corrections
permalink: /SPheno_threshold_corrections/
---
# SPheno threshold corrections
[Category:SPheno](/Category:SPheno "wikilink")
Threshold corrections in Supersymmetric models
----------------------------------------------
......@@ -12,67 +9,47 @@ In general, the running SM parameters depend on the SUSY masses. The reason are
### Calculation of gauge couplings
The first step is the calculation of $\\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)$, $\\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z)$ via
The first step is the calculation of $`\alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)`$, $`\alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z)`$ via
$\\begin{aligned}
\\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z) &= \\frac{\\alpha^{(5),\\overline{\\text{MS}}}(M_Z)}{1 - \\Delta\\alpha^{\\text{SM}}(M_Z) - \\Delta\\alpha^{\\text{NP}}(M_Z)} ,\\\\
\\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z) &= \\frac{\\alpha_S^{(5),\\overline{\\text{MS}}}(M_Z)}{1 - \\Delta\\alpha_S^{\\text{SM}}(M_Z) - \\Delta\\alpha_S^{\\text{NP}}(M_Z)} \\end{aligned}$
$` \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\alpha^{(5),\overline{\text{MS}}}(M_Z)}{1 - \Delta\alpha^{\text{SM}}(M_Z) - \Delta\alpha^{\text{NP}}(M_Z)} ,\\ \alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\alpha_S^{(5),\overline{\text{MS}}}(M_Z)}{1 - \Delta\alpha_S^{\text{SM}}(M_Z) - \Delta\alpha_S^{\text{NP}}(M_Z)} `$
Here, $\\alpha_S^{(5),\\overline{\\text{MS}}}$ and $\\alpha^{(5),\\overline{\\text{MS}}}$ are taken as input and receive corrections from the top loops as well as form new physics (NP):
Here, $`\alpha_S^{(5),\overline{\text{MS}}}`$ and $`\alpha^{(5),\overline{\text{MS}}}`$ are taken as input and receive corrections from the top loops as well as form new physics (NP):
$\\begin{aligned}
& \\Delta\\alpha^{\\text{SM}}(\\mu) = \\frac{\\alpha}{2\\pi} \\left(\\frac{1}{3}- \\frac{16}{9} \\log{\\frac{m_t}{\\mu}} \\right), \\hspace{1cm}
\\Delta\\alpha^{\\text{NP}}(\\mu) =\\frac{\\alpha}{2\\pi} \\left( -\\sum_i c_i \\log{\\frac{m_i}{\\mu}} \\right), &\\\\
& \\Delta\\alpha_S^{\\text{SM}}(\\mu) = \\frac{\\alpha_\\text{s}}{2\\pi} \\left( -\\frac{2}{3} \\log{\\frac{m_t}{\\mu}} \\right), \\hspace{1cm}
\\Delta\\alpha_S^{\\text{NP}}(\\mu) = \\frac{\\alpha_S}{2\\pi}\\left( \\frac{1}{2}-\\sum_i c_i \\log{\\frac{m_i}{\\mu}} \\right) .&\\end{aligned}$
$` \Delta\alpha^{\text{SM}}(\mu) = \frac{\alpha}{2\pi} \left(\frac{1}{3}- \frac{16}{9} \log{\frac{m_t}{\mu}} \right), \hspace{1cm} \Delta\alpha^{\text{NP}}(\mu) =\frac{\alpha}{2\pi} \left( -\sum_i c_i \log{\frac{m_i}{\mu}} \right), \\ \Delta\alpha_S^{\text{SM}}(\mu) = \frac{\alpha_\text{s}}{2\pi} \left( -\frac{2}{3} \log{\frac{m_t}{\mu}} \right), \hspace{1cm} \Delta\alpha_S^{\text{NP}}(\mu) = \frac{\alpha_S}{2\pi}\left( \frac{1}{2}-\sum_i c_i \log{\frac{m_i}{\mu}} \right) . `$
The sum runs over all particles *i* which are not present in the SM and which are either charged or coloured. The coefficients *c*<sub>*i*</sub> depends on the charge respectively colour representation, the generic type of the particle (scalar, fermion, vector), and the degrees of freedom of the particle (real/complex boson respectively Majorana/Dirac fermion).
### Calculation of the Weinberg angle
The next step is the calculation of the running Weinberg angle $\\sin\\Theta^{{{\\overline{\\mathrm{DR}}}}}$ and electroweak VEV *v*. For that the one-loop corrections *δ**M*<sub>*Z*</sub><sup>2</sup> and *δ**M*<sub>*W*</sub><sup>2</sup> to the *Z*- and *W*-mass are needed. And an iterative procedure is applied with $\\Theta^{{{\\overline{\\mathrm{DR}}}}}_W = \\Theta_W^{\\text{SM}}$ in the first iteration together with:
The next step is the calculation of the running Weinberg angle $`\sin\Theta^{{{\overline{\mathrm{DR}}}}}`$ and electroweak VEV *v*. For that the one-loop corrections *δ**M*<sub>*Z*</sub><sup>2</sup> and *δ**M*<sub>*W*</sub><sup>2</sup> to the *Z*- and *W*-mass are needed. And an iterative procedure is applied with $`\Theta^{{{\overline{\mathrm{DR}}}}}_W = \Theta_W^{\text{SM}}`$ in the first iteration together with:
$\\begin{aligned}
v^2 =& (M_Z^2 + \\delta M_Z^2) \\frac{(1- \\sin^2\\Theta^{{{\\overline{\\mathrm{DR}}}}}_W)\\sin^2\\Theta^{{\\overline{\\mathrm{DR}}}}_W}{\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}} \\\\
\\sin^2\\Theta^{{{\\overline{\\mathrm{DR}}}}}_W =& \\frac{1}{2} - \\sqrt{\\frac{1}{4} - \\frac{\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}}{\\sqrt{2} M_Z^2 G_F (1-\\delta_r)}}\\end{aligned}$
$` v^2 = (M_Z^2 + \delta M_Z^2) \frac{(1- \sin^2\Theta^{{{\overline{\mathrm{DR}}}}}_W)\sin^2\Theta^{{\overline{\mathrm{DR}}}}_W}{\pi \alpha^{{{\overline{\mathrm{DR}}}}}} \\ \sin^2\Theta^{{{\overline{\mathrm{DR}}}}}_W = \frac{1}{2} - \sqrt{\frac{1}{4} - \frac{\pi \alpha^{{{\overline{\mathrm{DR}}}}}}{\sqrt{2} M_Z^2 G_F (1-\delta_r)}}`$
Here, *G*<sub>*F*</sub> is the Fermi constant and *δ*<sub>*r*</sub> is defined by
$\\delta_r = \\rho \\frac{\\delta M_W^2}{M_W^2} - \\frac{\\delta M_Z^2}{M_Z^2} + \\delta_{VB}$
$`\delta_r = \rho \frac{\delta M_W^2}{M_W^2} - \frac{\delta M_Z^2}{M_Z^2} + \delta_{VB}`$
where *δ*<sub>*V**B*</sub> are the corrections to the muon decay *μ* → *e**ν*<sub>*i*</sub>*ν̄*<sub>*j*</sub> which are calculated at one-loop as well. The *ρ* parameter is calculated also at full one-loop and the known two-loop SM corrections are added.
With the obtained information, the running gauge couplings at *M*<sub>*Z*</sub> are given by
$\\begin{aligned}
& g_1^{{{\\overline{\\mathrm{DR}}}}}(M_Z) =
\\frac{\\sqrt{4 \\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}}{\\cos\\theta_{W}^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}, \\hspace{0.5cm}
g_2^{{{\\overline{\\mathrm{DR}}}}}(M_Z) =
\\frac{\\sqrt{4\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}}{\\sin\\theta_{W}^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}, \\hspace{0.5cm}
g_3^{{{\\overline{\\mathrm{DR}}}}}(M_Z) = \\sqrt{4 \\pi \\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z)} &\\end{aligned}$
$` g_1^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\sqrt{4 \pi \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)}}{\cos\theta_{W}^{{{\overline{\mathrm{DR}}}}}(M_Z)}, \hspace{0.5cm} g_2^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\sqrt{4\pi \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)}}{\sin\theta_{W}^{{{\overline{\mathrm{DR}}}}}(M_Z)}, \hspace{0.5cm} g_3^{{{\overline{\mathrm{DR}}}}}(M_Z) = \sqrt{4 \pi \alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z)} `$
### Calculation of the Yukawa couplings
The running Yukawa couplings are also calculated in an iterative way. The starting point are the running fermion masses in $\\overline{\\mathrm{DR}}$ obtained from the pole masses given as input:
The running Yukawa couplings are also calculated in an iterative way. The starting point are the running fermion masses in $`\overline{\mathrm{DR}}`$ obtained from the pole masses given as input:
$\\begin{aligned}
m_{l,\\mu,\\tau}^{{{\\overline{\\mathrm{DR}}}},\\text{SM}} =& m_{l,\\mu,\\tau} \\left(1 - \\frac{3}{128 \\pi^2}(g^{{{\\overline{\\mathrm{DR}}}},2}_1+g^{{{\\overline{\\mathrm{DR}}}},2}_2)\\right) \\\\
m_{d,s,b}^{{{\\overline{\\mathrm{DR}}}},\\text{SM}} =& m_{d,s,b} \\left(1- \\frac{\\alpha^{{\\overline{\\mathrm{DR}}}}_S}{3\\pi}-\\frac{23 \\alpha_S^{{{\\overline{\\mathrm{DR}}}},2}}{72 \\pi^2} + \\frac{3}{128 \\pi^2} g^{{{\\overline{\\mathrm{DR}}}},2}_2 - \\frac{13}{1152 \\pi^2} g^{{{\\overline{\\mathrm{DR}}}},2}_1\\right)\\\\
m_{u,c}^{{{\\overline{\\mathrm{DR}}}},\\text{SM}} =& m_{u,c} \\left(1- \\frac{\\alpha^{{\\overline{\\mathrm{DR}}}}_S}{3\\pi}-\\frac{23 \\alpha^{{{\\overline{\\mathrm{DR}}}},2}_S}{72 \\pi^2} + \\frac{3}{128 \\pi^2} g^{{{\\overline{\\mathrm{DR}}}},2}_2 - \\frac{7}{1152 \\pi^2} g^{{{\\overline{\\mathrm{DR}}}},2}_1\\right)\\\\
m^{{{\\overline{\\mathrm{DR}}}},\\text{SM}}_t =& m_t \\left\[1 + \\frac{1}{16\\pi^2} \\left(\\Delta m_t^{(1),qcd} +\\Delta m_t^{(2),qcd} + \\Delta m_t^{(1),ew}\\right)\\right\]\\end{aligned}$
$` m_{l,\mu,\tau}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{l,\mu,\tau} \left(1 - \frac{3}{128 \pi^2}(g^{{{\overline{\mathrm{DR}}}},2}_1+g^{{{\overline{\mathrm{DR}}}},2}_2)\right) \\ m_{d,s,b}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{d,s,b} \left(1- \frac{\alpha^{{\overline{\mathrm{DR}}}}_S}{3\pi}-\frac{23 \alpha_S^{{{\overline{\mathrm{DR}}}},2}}{72 \pi^2} + \frac{3}{128 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_2 - \frac{13}{1152 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_1\right)\\ m_{u,c}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{u,c} \left(1- \frac{\alpha^{{\overline{\mathrm{DR}}}}_S}{3\pi}-\frac{23 \alpha^{{{\overline{\mathrm{DR}}}},2}_S}{72 \pi^2} + \frac{3}{128 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_2 - \frac{7}{1152 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_1\right)\\ m^{{{\overline{\mathrm{DR}}}},\text{SM}}_t = m_t \left[1 + \frac{1}{16\pi^2} \left(\Delta m_t^{(1),qcd} +\Delta m_t^{(2),qcd} + \Delta m_t^{(1),ew}\right)\right]`$
with
$\\begin{aligned}
\\Delta m_t^{(1),qcd} &= -\\frac{16 \\pi \\alpha_S^{{\\overline{\\mathrm{DR}}}}}{3} \\left(5 + 3 \\log\\frac{M_Z^2}{m_t^2} \\right) \\\\
\\Delta m_t^{(2),qcd} &= -\\frac{64 \\pi^2 \\alpha_S^{{{\\overline{\\mathrm{DR}}}},2} }{3} \\left(\\frac{1}{24}+\\frac{2011}{384\\pi^2}+\\frac{\\ln2}{12}-\\frac{\\zeta(3)}{8\\pi^2}+\\frac{123}{32\\pi^2} \\log\\frac{M_Z^2}{m_t^2}+\\frac{33}{32\\pi^2} \\left(\\log\\frac{M_Z^2}{m_t^2}\\right)^2 \\right)\\\\
\\Delta m_t^{(1),ew} &= - \\frac{4}{9} g_2^{{{\\overline{\\mathrm{DR}}}},2} \\sin^2 \\Theta^{{\\overline{\\mathrm{DR}}}}_W \\left(5 + 3 \\log\\frac{M_Z^2}{m_t^2}\\right)\\end{aligned}$
$` \Delta m_t^{(1),qcd} = -\frac{16 \pi \alpha_S^{{\overline{\mathrm{DR}}}}}{3} \left(5 + 3 \log\frac{M_Z^2}{m_t^2} \right) \\ \Delta m_t^{(2),qcd} = -\frac{64 \pi^2 \alpha_S^{{{\overline{\mathrm{DR}}}},2} }{3} \left(\frac{1}{24}+\frac{2011}{384\pi^2}+\frac{\ln2}{12}-\frac{\zeta(3)}{8\pi^2}+\frac{123}{32\pi^2} \log\frac{M_Z^2}{m_t^2}+\frac{33}{32\pi^2} \left(\log\frac{M_Z^2}{m_t^2}\right)^2 \right)\\ \Delta m_t^{(1),ew} = - \frac{4}{9} g_2^{{{\overline{\mathrm{DR}}}},2} \sin^2 \Theta^{{\overline{\mathrm{DR}}}}_W \left(5 + 3 \log\frac{M_Z^2}{m_t^2}\right)`$
The two-loop parts are taken from Refs. . The masses are matched to the eigenvalues of the loop-corrected fermion mass matrices calculated as
*m*<sub>*f*</sub><sup>(1*L*)</sup>(*p*<sub>*i*</sub><sup>2</sup>)=*m*<sub>*f*</sub><sup>(*T*)</sup> − *Σ̃*<sub>*S*</sub><sup>+</sup>(*p*<sub>*i*</sub><sup>2</sup>)−*Σ̃*<sub>*R*</sub><sup>+</sup>(*p*<sub>*i*</sub><sup>2</sup>)*m*<sub>*f*</sub><sup>(*T*)</sup> − *m*<sub>*f*</sub><sup>(*T*)</sup>*Σ̃*<sub>*L*</sub><sup>+</sup>(*p*<sub>*i*</sub><sup>2</sup>)
Here, the pure QCD and QED corrections are dropped in the self-energies *Σ̃*. Inverting this relation to get the running tree-level mass matrix *m*<sub>*f*</sub><sup>(*T*)</sup> leads to $Y_d^{{{\\overline{\\mathrm{DR}}}}}$, $Y_u^{{{\\overline{\\mathrm{DR}}}}}$, $Y_e^{{{\\overline{\\mathrm{DR}}}}}$. Since the self-energies depend also one the Yukawa matrices, this calculation has to be iterated until a stable point is reached. Optionally, also the constraint that the CKM matrix is reproduced can be included in the matching.
Here, the pure QCD and QED corrections are dropped in the self-energies *Σ̃*. Inverting this relation to get the running tree-level mass matrix *m*<sub>*f*</sub><sup>(*T*)</sup> leads to $`Y_d^{{{\overline{\mathrm{DR}}}}}`$, $`Y_u^{{{\overline{\mathrm{DR}}}}}`$, $`Y_e^{{{\overline{\mathrm{DR}}}}}`$. Since the self-energies depend also one the Yukawa matrices, this calculation has to be iterated until a stable point is reached. Optionally, also the constraint that the CKM matrix is reproduced can be included in the matching.
Threshold corrections in Non-Supersymmetric models
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