# The Higgs decay and the threshold

I am using this paper by Guido Altarelli as a reference for the various decay widths of the Higgs. Altarelli yields the well-known formula for the WW and ZZ channels $$\Gamma(VV)=\delta_V\beta_V\frac{G_FM_h^3}{16\pi\sqrt{2}}\left(1-4x+12x^2\right)$$ being $V=W,Z$, $G_F$ the Fermi's constant, $M_h$ the Higgs mass and $x=M_V^2/M_h^2$. The factors are $\delta_V=1(2)$ for Z(W) and $\beta_V=\sqrt{1-4x}$. This formula makes sense only over the threshold $M_h\ge 2M_V$ and, in this case, these are the dominant decay channels otherwise the main decay is $H\rightarrow b{\bar b}$. Now, all this seems at odds with what is seen currently at LHC as $M_h=125.09\ GeV$ well below the threshold and the decay into beauty quarks does not seem so dominant. Could you explain what is going on?

• For $V=Z_0$ $M_V=90GeV$, then $x=.5184$, so $\beta_V= \sqrt{1-4x}$ gets imaginary. The formula is invalid for a 125GeV-Higgs. I think the formula was used to make the figure 55b for variable H-mass far beyond 125GeV in the range where the formula is valid. BTW in the text is written: "For decay into a real pair of V’s, withV=W,Z one obtains in Born approximation" followed by the formula. How the curve was calculated below the VV-threshold, as the virtual particle $M_V$-mass can adapt another values as the on-shell mass it might be still used below the VV-creation threshold, but I don't know. – Frederic Thomas Mar 22 '18 at 17:52