UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Friday, 6 June, 2014 1:30 pm to 4:30 pm PAPER 45 THE STANDARD MODEL Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 Define the 4 x 4 matrix B to be one which transforms the Dirac matrices as follows 0 µ ByH* B-1 = ? . Given 75 = - iy0 ~1,2,3, show that By5* B-1 = ~ 5. Consider a Dirac field / which can be written as an integral over plane wave solutions ?(x) = X h 6º (p)us (p)e-ip-a + dst (p)vs (p)eip.r i p,s Show that the time-reversal transformation maps v and v as føleres: p = P R (2 ) . Î(x)Î-1= Bv(CT) Î?(x)1-1 = 7(IT)B-1 where pT = (p0,-p) and IT = (-x0, x). (For simplicity let us assume intrinsic phases NT = 1 throughout this problem. ) You may use without proof (-1)7-su-5*(PT) =- 75Cus(p) (-1)2-8 0-s*(PT) = - 75Cvs(p). You should have obtained a relation between B and C. Use this and the defining property of B stated at the start of this problem to show that CHIC-1 =- 1H where y"T is the transpose of y". Consider the decay of a neutron n to a proton p: n -> pe ve, treating the neutron and proton (as well as the electron and anti-neutrino) as Dirac fields. Given that the effective interaction mediating this decay is CI = - V2 GF It lept, u Jun + h.c., with Je = DerH(1 -25) e and Jpn = py"(gv + 9AY5)n, derive a condition which ga/gv must satisfy if this interaction is to be invariant under time-reversal. Part III, Paper 45
UNIVERSITY OF CAMBRIDGE 3 2 The electroweak theory of the Standard Model consists of an SU(2)LXU(1)y gauge theory which undergoes spontaneous symmetry breaking via the Higgs mechanism. Let w& (a=1,2,3) be the SU(2)L gauge bosons with coupling g and Bu be the U(1)y gauge boson with coupling g'. The scalar field ø transforms as a doublet under SU(2)L, has hypercharge +, and has a Lagrangian of the form Co=(Dud)+(D"?)-?2|?|2->|?|4 (>>0). Explicitly write down the terms arising from covariant differentiation of o, i.e. from Duo. Show that mass terms for 3 out of 4 gauge bosons are generated if the scalar field acquires a nonzero vacuum expectation value. Be clear about the relation between the original fields (Wa, Bu) and the ones after symmetry breaking, (WH, z0, Au). Show how the electroweak theory includes the electron e and electron neutrino Ve. In particular, write down gauge-invariant terms in the electroweak Lagrangian which contain the coupling of these fermions to the gauge bosons and others which contain the fermion-scalar interactions. How does the latter lead to a nonzero electron mass? Part III,