Decay of the Higgs boson
Consider the decay of the Higgs boson into a fermion pair \(f\bar{f}\).
The quantum state of the fermions has the form \[|f\bar{f}\rangle = \int \frac{d\boldsymbol{p}_1}{(2\pi)^3\sqrt{2E_1}}\frac{d\boldsymbol{p}_2}{(2\pi)^3\sqrt{2E_2}}\widetilde{\psi}(\boldsymbol{p}_1,s_1;\boldsymbol{p}_2,s_2)|f(\boldsymbol{p}_1,s_1),\bar{f}(\boldsymbol{p}_2,s_2)\rangle,\] where \(\widetilde{\psi}\) is the two-particle wave function of the fermions in the momentum representation and \(s_i\) are the spin 4-vectors. In the rest frame of a fermion \(s=(0,\boldsymbol{\xi})\).
Find \(\widetilde{\psi}\) using the matrix element \[\langle f(\boldsymbol{p}_1,s_1),\bar{f}(\boldsymbol{p}_2,s_2)|S-1|H\rangle.\] ✓ Now compute the wave function in the coordinate representation: \[\psi(\boldsymbol{x}_1,\boldsymbol{x}_2) = \int \frac{d\boldsymbol{p}_1}{(2\pi)^3}\frac{d\boldsymbol{p}_2}{(2\pi)^3}\widetilde{\psi}(\boldsymbol{p}_1,s_1;\boldsymbol{p}_2,s_2)e^{+i\left(\boldsymbol{p}_1\cdot \boldsymbol{x}_1+\boldsymbol{p}_2\cdot\boldsymbol{x}_2\right)}.\] Show that \(\psi(\boldsymbol{x}_1,\boldsymbol{x}_2)\) depends on the difference of the coordinates \(\boldsymbol{r}=\boldsymbol{x}_1-\boldsymbol{x}_2\) and compute the wave function \(\psi(\boldsymbol{r})\).
Expand the wave function in spherical harmonics and study how it depends on \(\boldsymbol{\xi}_1\cdot \hat{\boldsymbol{r}}\), \(\boldsymbol{\xi}_2\cdot \hat{\boldsymbol{r}}\), \(\boldsymbol{\xi}_1\cdot\boldsymbol{\xi}_2\). Interpret the result.