(a) Using the antisymmetric tensor product rule for representations used earlier for…

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

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

Problem

(a) Using the antisymmetric tensor product rule for

SU(N)

representations used earlier for isospin and flavour multiplets, verify that the antisymmetric product

[5\otimes5]_A

of two fundamental representations of

SU(5)

has dimension ten, matching the

10

of one generation. (b) The dimensional estimate for the muon decay rate is

\Gamma_\mu \sim G_F^2 m_\mu^5

. Replacing

G_F

by the unified analogue

\alpha_G/M_X^2

and

m_\mu

by the proton mass

m_p = 0.938\,\mathrm{GeV}

, estimate the proton lifetime for a unification scale

M_X = 1\times10^{16}\,\mathrm{GeV}

and

\alpha_G = 1/42

. (c) The pairwise crossing scales of the three Standard Model gauge couplings, run at one loop, lie at

1.0\times10^{13}

2.4\times10^{14}

and

1.0\times10^{17}\,\mathrm{GeV}

, and a seesaw mechanism for neutrino masses typically needs a heavy right-handed neutrino mass in the range

10^{14}

10^{15}\,\mathrm{GeV}

. State whether

M_X = 1\times10^{16}\,\mathrm{GeV}

falls inside the span of the three crossing scales, and whether the lifetime found in part (b) is compatible with the Super-Kamiokande bound

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