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.
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.
The dirivation you can find in this paper: https://books.google.se/books?id=BipjDwAAQBAJ&pg=PA79&lpg=PA79&dq=polar+angle+of+the+electron+in+W+center&source=bl&ots=lONtgjxHP3&sig=ACfU3U3yqZFAwqCJJjlUa1vhlNhexsU8Rg&hl=zh-CN&sa=X&ved=2ahUKEwiO5Z-QycnkAhWMpYsKHZK0CYEQ6AEwAHoECAgQAQ#v=onepage&q=polar%20angle%20of%20the%20electron%20in%20W%20center&f=false