Past exam of the particle Astrophysics course of the University of Cambridge / 2013 / Paper 1 / 2 / a / Solution

A grand unified theory embeds the strong and electroweak gauge groups in a larger group, for example $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y\subset SU(5)$. quarks and leptons can belong to common representations, while heavy extra gauge or scalar bosons connect them. The resulting interactions can violate baryon number, unlike ordinary perturbative Standard Model reactions. Running gauge couplings and proton-decay searches test such ideas; unification is a theoretical possibility, not an established observation.^[ref]

baryogenesis must explain why a nearly matter-antimatter-symmetric early state evolves into a surviving baryon asymmetry. In the usual local CPT-invariant framework, the Sakharov conditions require baryon-number violation, C and CP violation, and departure from thermal equilibrium. baryon conservation would forbid creating a net excess; exact conjugation symmetries would pair baryon-producing processes with equal antibaryon production; equilibrium with no imposed asymmetry would erase a source term through detailed balance.

For heavy particles $X$ and $\bar{X}$, let decay channels carry different baryon numbers. CP-violating interference can make their branching fractions differ, even though CPT symmetry requires equal total widths. Defining the mean baryon number produced per decay as $ϵ$, an out-of-equilibrium abundance can generate $n_B/s$ of order $ϵ$ times its initial number per entropy, reduced by inefficient production and washout. A rough condition is that decays become effective after inverse production has ceased, with the relevant reaction rate compared with the Hubble parameter. Complex couplings alone do not guarantee a nonzero rate asymmetry: a relative absorptive phase is also needed.

Survival matters as much as production. Electroweak sphalerons violate $B+L$ while conserving baryon-minus-lepton number $B-L$, so a minimal GUT-generated asymmetry with $B-L=0$ can be washed out if sphalerons and the other relevant reactions equilibrate. Generating nonzero $B-L$, as in suitable heavy-Majorana-neutrino leptogenesis, can avoid this problem. A viable GUT baryogenesis model must generate an asymmetry and demonstrate that subsequent reactions do not erase it. The Sakharov conditions are necessary ingredients under these assumptions, not a numerical prediction of the observed abundance.