UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Thursday, 4 June, 2015 1:30 pm to 4:30 pm PAPER 47 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 Let ? be the unitary operator corresponding to charge conjugation. Take as given that positive and negative frequency spinors, us (p) and vs (p) respectively, are related by vs (p) = CusT (p) and us (p) = C?sT (p) where C is a 4 x 4 spin-matrix and T denotes the transpose. Show that, for a Dirac field ?(x), ?(x)Ô-1 = nCTT(x). Given that C is a unitary matrix with the property that CruTC-1 = - 74 show that [CTC-1, y4] = 0 and argue that this implies CT = +C. For the remainder of the problem, you may take as given CT = - C. Show that if (x) satisfies the Dirac equation, so does /c(x) = CT. Show that CvL (x)C-1 has the same chirality as VL. In just a few paragraphs, explain with a simple example why both C and CPviolation are necessary conditions for baryogenesis to occur in the early universe. [Here C stands for charge conjugation and CPstands for the product of charge conjugation with parity.] 2 Consider SU (2) gauge theory coupled to a 2-component complex scalar field ø 1 4 L =-= Fa Fa, V +(DHb)+(Du) -10 v 2 ? 0 2 02 2 with > > 0 and v2 > 0. The covariant derivative is Duo = duo + igAaTap, with the generators related to the Pauli o-matrices Ta = ga/2 and the field strength tensor Fav = Ou Ag - Oy Al - geabc Ab Ac. Identify the relevant degrees-of-freedom and determine their masses (ignoring quan- tum corrections). Comment on the number of massless fields based on symmetry princi- ples. What are the interaction terms involving the scalar field? Explicitly determine any terms in £ which give rise to gauge-boson self-interactions and sketch the vertices. Part III, Paper 47
UNIVERSITY OF CAMBRIDGE 3 3 Attempt to answer all of the following parts. At the end of the question (continued on the next page) a number of expressions are given which you may use without proof. (a) For each of the following elastic scattering processes, draw all of the relevant tree-level, Standard Model Feynman diagrams: (i) Vue > Vue (ii) Vee > Vee (iii) Dee- > Dee- (b) Treating the neutrinos as strictly massless, and working in the limit where the electron mass can be neglected at leading order, use the effective weak Lagrangian GF = C = - C X h veya(1 - 75)e eyQ(1 - 75)ve V2 (E{e, 1} i + vera (1 - 15) ve era (CV - CAY5)e + ... (the omitted terms