feynlag.pheno.loop¶
Loop-induced Higgs decays: $hto gg,gammagamma,Zgamma$ (Tier 3).
These channels vanish identically at tree level — the Higgs is colour- and charge-neutral, so they proceed only through a top-quark triangle and (for $gammagamma$/$Zgamma$) a $W$ loop. A tree-level library cannot derive them from the Lagrangian.
This module takes the effective-vertex route the roadmap recommends
(§16.3): it imports the standard closed-form one-loop form factors and assembles
the widths from them. This is a deliberate, documented exception to the
derive-from-the-Lagrangian ethos — the same judgment as the FeynRules-style CKM
insertion in feynlag.flavor — because the honest one-loop computation
(Passarino–Veltman reduction, UV renormalisation) is a project the size of the
whole library, while the physical answer is a textbook closed form.
Every formula here is verified numerically against the PDG values (see
tests/test_loop.py) and cited: the $gg$/$gammagamma$ form factors and the
$Zgamma$ closed form follow Djouadi’s review [Djouadi08] and Carena et al.
[CGLW13]; the NLO-QCD $gg$ $K$-factor follows Spira et al. [Spira95].
Functions
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Spin-$tfrac12$ (fermion-loop) form factor, $A_{1/2}(tau)=2[tau+(tau-1)f(tau)]/tau^2$. |
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Fermion form factor for $Zgamma$: $A_{1/2}^{Zgamma}=I_1(x,y)-I_2(x,y)$, with $x=4m^2/m_h^2$, $y=4m^2/m_Z^2$. |
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Spin-1 ($W$-loop) form factor, $A_1(tau)=-[2tau^2+3tau+3(2tau-1)f(tau)]/tau^2$. |
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$W$ form factor for $Zgamma$: $A_1^{Zgamma}=4(3-t_W^2)I_2+[(1+2/x)t_W^2-(5+2/x)]I_1$, with $t_W^2=s_W^2/c_W^2$. |
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The scalar one-loop function $f(tau)$. |
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The second one-loop function $g(tau)$ (needed for $Zgamma$). |
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$Gamma(htogammagamma)=frac{alpha^2 m_h^3}{256pi^3 v^2} big|A_1(tau_W)+sum_f N_c Q_f^2 A_{1/2}(tau_f)big|^2$. |
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$Gamma(hto gg)=frac{alpha_s^2 m_h^3}{72pi^3 v^2} big|tfrac34 A_{1/2}(tau_t)big|^2$ (leading order). |
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$Gamma(hto Zgamma)=frac{alpha^2 m_h^3}{512pi^3} left(1-frac{m_Z^2}{m_h^2}right)^3|mathcal A|^2$, with |