The Basic Forces and Messenger Particles — Question 9

Question 9

A scalar coupling proportional to mass favors heavier fermions

In a simplified Higgs-decay model, the partial width is $\Gamma_f=C N_c m_f^2(1-4m_f^2/M^2)^{3/2}$. Let M=100 GeV and compare fermions A and B with masses 1 and 10 GeV and equal color multiplicity.

1. Find the width ratio if phase-space factors are neglected.

2. Include the supplied phase-space factors and calculate the corrected ratio.

Question 9 - Solution

A scalar coupling proportional to mass favors heavier fermions

(a) Method and calculation

$$\Gamma_B/\Gamma_A\simeq(m_B/m_A)^2.$$$$\boxed{\Gamma_B/\Gamma_A\simeq100}.$$

(b) Interpretation and further calculation

$$\frac{\Gamma_B}{\Gamma_A}=100\left(\frac{1-.04}{1-.0004}\right)^{3/2}.$$$$\boxed{\Gamma_B/\Gamma_A=94.1169}.$$

Physical check

The Higgs is a spin-zero excitation, distinct from spin-one gauge mediators. The toy formula omits radiative corrections and fixes all other factors.