In the paper https://projecteuclid.org/euclid.cmp/1103922050, the equation 5.4 seems to be lacking a minus: $$\left(\dfrac{m_B}{m_P}\right)^8\dfrac{1}{m_B}\sim 10^{122}yrs$$ seems to be OK only if $n=-8$ (anyway, I am also doubtful about how he computes the 8 factor), since if we plug $\hbar\sim 10^{-34}J\cdot s$, $c^2=10^{17}m^2/s^2$, $m_B\sim 1GeV=10^{-27}kg$ and $m_P\sim 10^{19}GeV$, then $$(10^{-19})^{-8}\cdot \dfrac{10^{-34}}{10^{-10}}\sim 10^{128}s\sim 10^{122}yrs$$ Am I right? Is the argumenf of the paper also valid about for the fermion case? Should it be $$\tau=\left(\dfrac{m_f}{m_P}\right)^n\dfrac{1}{m_f}$$ and how to get the thumb rule for guessing the $n$ for fermions?
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Maybe, I think I got partial answer to my confusion... The RATE of proton decay should be $$\Gamma=\left(\dfrac{m_B}{m_P}\right)^8m_B\sim 10^{-128}s^{-1}\sim (10^{122}yr)^{-1}$$ However, I do not understand yet how to pick up the $8th$ power (i.e. the power counting is a mystery yet from the paper Feynman rules, but I presume it has to do with the Born rule and the coefficient of the Feynman graphs...). Moreover, I am not sure of how to get a similar result for fermions. Why should the coefficient for fermions be different from baryons? After all, if we allow for B and/or L symmetries, maybe B-L symmetry, how to understand all this?
answered Dec 29, 2020 at 15:30