Adjusting the rate of proton decay in the standard $\rm SU(5)$ grand unified theory

Question

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..

Answer 1

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.

Answer 2

1. (1) and (2) are thought to be the same*. When there is SU(5) symmetry, there is only one coupling constant for SU(5), not three. When the SU(5) symmetry is broken by taking a vacuum expectation value (vev) for a Higgs field (such as a 24 representation), the symmetry becomes SU(3) x SU(2) x U(1) with three different coupling constants. When taking a vev of an SU(5) GUT Higgs, the vev can be plugged back into the action to verify what particles get mass, which would be related to the scale of symmetry breaking.

*Note that typical running of coupling constants for SU(3) x SU(2) x U(1) doesn't quite lead to a single coupling constant based on the computations we can perform, but we cannot always find exact solutions and what is found is very close to unification. Nevertheless, SUSY helps make this easier to have all three coupling constants unify precisely.

2. Since we don't know what scale the GUT scale is at experimentally, we can consider different values and compute the consequences. However, the running of the coupling constants suggests a value value near $10^{16}$ GeV. Perhaps it is claimed that SUSY can make this a little higher.

3. The proton can decay from X and Y bosons in SU(5). The electroweak Higgs is an SU(2) doublet, which is typically placed in a 5 Higgs of SU(5). This 5 Higgs leads to 3+2, giving a color triplet Higgs that can also lead to proton decay, giving a doublet-triplet problem. SUSY can help in various ways, there are a lot of proposals. Orbifolds allow for symmetry breaking without a Higgs field, which is one direction to explore. From there, various membrane dynamics may change things as well. However, more simply, string-inspired models, or really, Spin(10) GUT motivates the flipped SU(5) x U(1) GUT from Barr in 1982. By 1987, the flipped SU(5) GUT model was "revitalized" in the SUSY GUT community by considering a 10 Higgs instead of a 24 Higgs, which leads to a "missing partner mechanism" to solve the doublet-triplet splitting problem. While SU(5) x U(1) in non-supersymmetric GUT is often not considered not truly unified, the notion of U(5) = SU(5) x U(1) is quite natural in string theory, I believe because one can think of combining five branes with U(1) symmetry and then connect the branes with strings to get the symmetry enhanced to U(5). Also, the fact that the 10 Higgs can be used only with flipped SU(5) x U(1) GUT and not SU(5) is another nice advantage because it is not too large and also it is more natural to find the 10 Higgs from higher gauge groups, such as exceptional Lie algebras.