Some grand-unified models push the unified gauge boson mass up towards the scale near…

PHYSICS · Subjective · Fermi University Physics · Particle Physics & Quantum Fields · Beyond the Standard Model · Beyond the Standard Model

Symmetry and Flavour, Ch Grand unification: running couplings, SU(5), proton decay and the hierarchy problem

Problem

Some grand-unified models push the unified gauge boson mass up towards the scale near which the three couplings would meet if the supersymmetric coefficients held, of order

M_X \approx 3\times10^{16}\,\mathrm{GeV}

. Using the dimensional estimate

\tau_p \approx \frac{M_X^4}{\alpha_G^2 m_p^5},

with unified coupling

\alpha_G = 1/42

and proton mass

m_p = 0.938\,\mathrm{GeV}

, estimate the proton lifetime for this

M_X

and state whether it is compatible with the Super-Kamiokande limit

\tau(p\to e^+\pi^0) > 2.4\times10^{34}\,\mathrm{yr}

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