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---
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title: SPheno threshold corrections
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permalink: /SPheno_threshold_corrections/
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---
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# SPheno threshold corrections
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[Category:SPheno](/Category:SPheno "wikilink")
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Threshold corrections in Supersymmetric models
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----------------------------------------------
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... | ... | @@ -12,67 +9,47 @@ In general, the running SM parameters depend on the SUSY masses. The reason are |
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### Calculation of gauge couplings
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The first step is the calculation of $\\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)$, $\\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z)$ via
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The first step is the calculation of $`\alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)`$, $`\alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z)`$ via
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$\\begin{aligned}
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\\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z) &= \\frac{\\alpha^{(5),\\overline{\\text{MS}}}(M_Z)}{1 - \\Delta\\alpha^{\\text{SM}}(M_Z) - \\Delta\\alpha^{\\text{NP}}(M_Z)} ,\\\\
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\\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z) &= \\frac{\\alpha_S^{(5),\\overline{\\text{MS}}}(M_Z)}{1 - \\Delta\\alpha_S^{\\text{SM}}(M_Z) - \\Delta\\alpha_S^{\\text{NP}}(M_Z)} \\end{aligned}$
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$` \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\alpha^{(5),\overline{\text{MS}}}(M_Z)}{1 - \Delta\alpha^{\text{SM}}(M_Z) - \Delta\alpha^{\text{NP}}(M_Z)} ,\\ \alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\alpha_S^{(5),\overline{\text{MS}}}(M_Z)}{1 - \Delta\alpha_S^{\text{SM}}(M_Z) - \Delta\alpha_S^{\text{NP}}(M_Z)} `$
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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):
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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):
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$\\begin{aligned}
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& \\Delta\\alpha^{\\text{SM}}(\\mu) = \\frac{\\alpha}{2\\pi} \\left(\\frac{1}{3}- \\frac{16}{9} \\log{\\frac{m_t}{\\mu}} \\right), \\hspace{1cm}
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\\Delta\\alpha^{\\text{NP}}(\\mu) =\\frac{\\alpha}{2\\pi} \\left( -\\sum_i c_i \\log{\\frac{m_i}{\\mu}} \\right), &\\\\
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& \\Delta\\alpha_S^{\\text{SM}}(\\mu) = \\frac{\\alpha_\\text{s}}{2\\pi} \\left( -\\frac{2}{3} \\log{\\frac{m_t}{\\mu}} \\right), \\hspace{1cm}
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\\Delta\\alpha_S^{\\text{NP}}(\\mu) = \\frac{\\alpha_S}{2\\pi}\\left( \\frac{1}{2}-\\sum_i c_i \\log{\\frac{m_i}{\\mu}} \\right) .&\\end{aligned}$
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$` \Delta\alpha^{\text{SM}}(\mu) = \frac{\alpha}{2\pi} \left(\frac{1}{3}- \frac{16}{9} \log{\frac{m_t}{\mu}} \right), \hspace{1cm} \Delta\alpha^{\text{NP}}(\mu) =\frac{\alpha}{2\pi} \left( -\sum_i c_i \log{\frac{m_i}{\mu}} \right), \\ \Delta\alpha_S^{\text{SM}}(\mu) = \frac{\alpha_\text{s}}{2\pi} \left( -\frac{2}{3} \log{\frac{m_t}{\mu}} \right), \hspace{1cm} \Delta\alpha_S^{\text{NP}}(\mu) = \frac{\alpha_S}{2\pi}\left( \frac{1}{2}-\sum_i c_i \log{\frac{m_i}{\mu}} \right) . `$
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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).
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### Calculation of the Weinberg angle
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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:
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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:
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$\\begin{aligned}
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v^2 =& (M_Z^2 + \\delta M_Z^2) \\frac{(1- \\sin^2\\Theta^{{{\\overline{\\mathrm{DR}}}}}_W)\\sin^2\\Theta^{{\\overline{\\mathrm{DR}}}}_W}{\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}} \\\\
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\\sin^2\\Theta^{{{\\overline{\\mathrm{DR}}}}}_W =& \\frac{1}{2} - \\sqrt{\\frac{1}{4} - \\frac{\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}}{\\sqrt{2} M_Z^2 G_F (1-\\delta_r)}}\\end{aligned}$
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$` v^2 = (M_Z^2 + \delta M_Z^2) \frac{(1- \sin^2\Theta^{{{\overline{\mathrm{DR}}}}}_W)\sin^2\Theta^{{\overline{\mathrm{DR}}}}_W}{\pi \alpha^{{{\overline{\mathrm{DR}}}}}} \\ \sin^2\Theta^{{{\overline{\mathrm{DR}}}}}_W = \frac{1}{2} - \sqrt{\frac{1}{4} - \frac{\pi \alpha^{{{\overline{\mathrm{DR}}}}}}{\sqrt{2} M_Z^2 G_F (1-\delta_r)}}`$
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Here, *G*<sub>*F*</sub> is the Fermi constant and *δ*<sub>*r*</sub> is defined by
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$\\delta_r = \\rho \\frac{\\delta M_W^2}{M_W^2} - \\frac{\\delta M_Z^2}{M_Z^2} + \\delta_{VB}$
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$`\delta_r = \rho \frac{\delta M_W^2}{M_W^2} - \frac{\delta M_Z^2}{M_Z^2} + \delta_{VB}`$
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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.
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With the obtained information, the running gauge couplings at *M*<sub>*Z*</sub> are given by
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$\\begin{aligned}
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& g_1^{{{\\overline{\\mathrm{DR}}}}}(M_Z) =
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\\frac{\\sqrt{4 \\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}}{\\cos\\theta_{W}^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}, \\hspace{0.5cm}
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g_2^{{{\\overline{\\mathrm{DR}}}}}(M_Z) =
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\\frac{\\sqrt{4\\pi \\alpha^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}}{\\sin\\theta_{W}^{{{\\overline{\\mathrm{DR}}}}}(M_Z)}, \\hspace{0.5cm}
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g_3^{{{\\overline{\\mathrm{DR}}}}}(M_Z) = \\sqrt{4 \\pi \\alpha_S^{{{\\overline{\\mathrm{DR}}}}}(M_Z)} &\\end{aligned}$
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$` g_1^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\sqrt{4 \pi \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)}}{\cos\theta_{W}^{{{\overline{\mathrm{DR}}}}}(M_Z)}, \hspace{0.5cm} g_2^{{{\overline{\mathrm{DR}}}}}(M_Z) = \frac{\sqrt{4\pi \alpha^{{{\overline{\mathrm{DR}}}}}(M_Z)}}{\sin\theta_{W}^{{{\overline{\mathrm{DR}}}}}(M_Z)}, \hspace{0.5cm} g_3^{{{\overline{\mathrm{DR}}}}}(M_Z) = \sqrt{4 \pi \alpha_S^{{{\overline{\mathrm{DR}}}}}(M_Z)} `$
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### Calculation of the Yukawa couplings
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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:
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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:
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$\\begin{aligned}
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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) \\\\
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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)\\\\
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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)\\\\
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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}$
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$` m_{l,\mu,\tau}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{l,\mu,\tau} \left(1 - \frac{3}{128 \pi^2}(g^{{{\overline{\mathrm{DR}}}},2}_1+g^{{{\overline{\mathrm{DR}}}},2}_2)\right) \\ m_{d,s,b}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{d,s,b} \left(1- \frac{\alpha^{{\overline{\mathrm{DR}}}}_S}{3\pi}-\frac{23 \alpha_S^{{{\overline{\mathrm{DR}}}},2}}{72 \pi^2} + \frac{3}{128 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_2 - \frac{13}{1152 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_1\right)\\ m_{u,c}^{{{\overline{\mathrm{DR}}}},\text{SM}} = m_{u,c} \left(1- \frac{\alpha^{{\overline{\mathrm{DR}}}}_S}{3\pi}-\frac{23 \alpha^{{{\overline{\mathrm{DR}}}},2}_S}{72 \pi^2} + \frac{3}{128 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_2 - \frac{7}{1152 \pi^2} g^{{{\overline{\mathrm{DR}}}},2}_1\right)\\ m^{{{\overline{\mathrm{DR}}}},\text{SM}}_t = m_t \left[1 + \frac{1}{16\pi^2} \left(\Delta m_t^{(1),qcd} +\Delta m_t^{(2),qcd} + \Delta m_t^{(1),ew}\right)\right]`$
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with
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$\\begin{aligned}
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\\Delta m_t^{(1),qcd} &= -\\frac{16 \\pi \\alpha_S^{{\\overline{\\mathrm{DR}}}}}{3} \\left(5 + 3 \\log\\frac{M_Z^2}{m_t^2} \\right) \\\\
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\\Delta m_t^{(2),qcd} &= -\\frac{64 \\pi^2 \\alpha_S^{{{\\overline{\\mathrm{DR}}}},2} }{3} \\left(\\frac{1}{24}+\\frac{2011}{384\\pi^2}+\\frac{\\ln2}{12}-\\frac{\\zeta(3)}{8\\pi^2}+\\frac{123}{32\\pi^2} \\log\\frac{M_Z^2}{m_t^2}+\\frac{33}{32\\pi^2} \\left(\\log\\frac{M_Z^2}{m_t^2}\\right)^2 \\right)\\\\
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\\Delta m_t^{(1),ew} &= - \\frac{4}{9} g_2^{{{\\overline{\\mathrm{DR}}}},2} \\sin^2 \\Theta^{{\\overline{\\mathrm{DR}}}}_W \\left(5 + 3 \\log\\frac{M_Z^2}{m_t^2}\\right)\\end{aligned}$
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$` \Delta m_t^{(1),qcd} = -\frac{16 \pi \alpha_S^{{\overline{\mathrm{DR}}}}}{3} \left(5 + 3 \log\frac{M_Z^2}{m_t^2} \right) \\ \Delta m_t^{(2),qcd} = -\frac{64 \pi^2 \alpha_S^{{{\overline{\mathrm{DR}}}},2} }{3} \left(\frac{1}{24}+\frac{2011}{384\pi^2}+\frac{\ln2}{12}-\frac{\zeta(3)}{8\pi^2}+\frac{123}{32\pi^2} \log\frac{M_Z^2}{m_t^2}+\frac{33}{32\pi^2} \left(\log\frac{M_Z^2}{m_t^2}\right)^2 \right)\\ \Delta m_t^{(1),ew} = - \frac{4}{9} g_2^{{{\overline{\mathrm{DR}}}},2} \sin^2 \Theta^{{\overline{\mathrm{DR}}}}_W \left(5 + 3 \log\frac{M_Z^2}{m_t^2}\right)`$
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The two-loop parts are taken from Refs. . The masses are matched to the eigenvalues of the loop-corrected fermion mass matrices calculated as
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*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>)
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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.
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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.
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Threshold corrections in Non-Supersymmetric models
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--------------------------------------------------
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