I calculated an expression for a decay rate of a decay of Higgs boson into two $W$ bosons. The expression is as follows

$$ \Gamma = \frac{g^2\sqrt{1-4\frac{M_W^2}{M_H^2}}}{64\pi \hbar} \frac{M_H^3}{M_W^2}\left [ 1-4\frac{M_W^2}{M_H^2}+12\frac{M_W^4}{M_H^4}\right ].$$

What I want to get is a number. I can use known masses for higgs, $M_H = 125$ $GeV/c^2$, and W boson, $M_W = 80$ $Gev/c^2$. What I dont understand is, do I need to divide by the $m_e$ and multiply by $\hbar$ to have the expression be dimensionless as i have the factor $\frac{M_H^3}{\hbar M_W^2}$. And with using these masses, I get the square root to be imaginary number. I am confused by this and dont know how to proceed.

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    $\begingroup$ $m_e$?? where did that come from? In natural units, $\hbar=1$ and the width is in GeVs. But, do you understand you are below threshold for decay? $\endgroup$ – Cosmas Zachos Jun 17 at 18:46
  • $\begingroup$ The decay rate has units of either energy (as the "decay width") or time$^{-1}$. You're not going to get a pure number out. $\endgroup$ – probably_someone Jun 17 at 19:12
  • $\begingroup$ @CosmasZachos , ok I guess i havent expected it to be in units of energy. So i understand that part now. But if I am below treshold for decay, how do i get the width expressed with a number with appropriate units? It should be possible as it is part of a problem question. $\endgroup$ – Riz Do Jun 17 at 19:23
  • $\begingroup$ I suspect this is between you and your instructor. A pure imaginary width is what you do expect to get for impossible real decays. $\endgroup$ – Cosmas Zachos Jun 17 at 19:49

You get a pure imaginary number because the decay from rest of a Higgs into two W bosons is not allowed kinematically, for the masses you used.

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