# Derivation of Jacobian peak in particle physics

I am reading some papers on particle phynomenology.

The mathematical derivation of Jacobian peak in some paper is wrong. However, it gives the right prediction.

If you have a W boson decaying in to an electron and a neutrion in the CM frame, then the electron transverse momentum and its 3-momentum must have the following relation as the figure indicated:

The angle $$\theta^ \star$$ has the following relation:

A basic QFT calculation gives the differential cross section in CM frame as:

Now if we differentiate the cross section with the transverse monemtum, namely, using the chain rule:

Many paper says the result is:

However, if one really do the calculation, one should get exactly a minus sign in the front of the right hand side of the above eq, since there is a minus sign in front of the $$4P_T^2/(m_w)^2$$ in the expression for $$\cos\theta^\star$$.

I cannot figure out why.

• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Sep 11 at 22:27