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The proton decay rate in the standard $SU(5)$ grand unified theory is given by $$ \Gamma \sim \left(\frac{g^2}{M_x^2}\right)^2 m_p^5 =\frac{g^4}{M_x^4}m_p^5 $$ Naively we could push up the bound for the decay rate $\Gamma$ arbitrarily high by increasing the mass of the $X$ boson, $M_x$. However, it seems that we set $M_x$ to be the scale of

(1). $M_x \sim M_{gut}$ which is the energy scale when all three gauge couplings meet

or

(2). $M_x$ set to be the scale GUT's Higgs scale where the $SU(5)$ is broken down to the standard model group $SU(3) \times SU(2) \times U(1)$.

My questions are that:

  1. Are scales in point (1) and (2) are the same? Are they somehow related? It seems to me (1) and (2) can be different.

  2. Why can we arbitrarily push up the scale of $M_x$ to the scale where all three gauge couplings meet? (as long as it is below the Planck scale?)

  3. If possible, can we sketch other ways of varying the proton decay rate in these theories? (How exactly can SUSY help?) The current bound in experiment seems to suggest the half-life is about $10^{34}$ years..

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It isn’t necessary that the GUT gauge bosons obtain masses that are the same as the GUT breaking scale. If these are different there will be threshold effects that are calculable functions of the ratio of these scales and group theoretic factors related to the reps the gauge bosons transform in.

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