 Solution

Take an on-shell W boson of fixed mass $M_W$, zero transverse recoil, negligible charged-lepton mass, and unrestricted angular acceptance. In its rest frame the charged lepton has momentum magnitude $p_0=M_W/2$. A longitudinal Lorentz boost to the laboratory preserves transverse momentum, soBoth angular roots contribute to the same $p_T$. Set $x=p_T/p_0$ and $a=\sqrt{1-x^2}$. The change-of-variable Jacobian determinant has $|dc/d{p}_T|=x/(p_{0}a)$, and the two-root angular weight is $\frac{3}{8}[(1-a{)}^2+(1+a{)}^2]=\frac{3}{4}(2-x^2)$. ConsequentlyThis is the ideal W-decay Jacobian peak. The divergence at the endpoint is integrable; it is a change-of-variable effect, not an infinite decay probability. Reversing the polar axis does not change this transverse momentum density, because the sum of the two roots removes the term odd in $c$.