MATHEMATICAL TRIPOS Part III Friday, 6 June, 014 1:0 pm to 4:0 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.
1 Define the 4 4 matrix B to be one which transforms the Dirac matrices as follows Bγ µ B 1 = Given γ 5 = iγ 0 γ 1 γ γ, show that Bγ 5 B 1 = γ 5. { γ µ µ = 0 γ µ µ 0. ConsideraDiracfieldψ whichcanbewrittenasanintegraloverplanewavesolutions ψ(x) = [b s (p)u s (p)e ip x +d s (p)v s (p)e ip x] p,s where p follows: d p (π) (p 0 ). Show that the time-reversal transformation maps ψ and ψ as ˆTψ(x)ˆT 1 = Bψ(x T ) ˆT ψ(x)ˆt 1 = ψ(x T )B 1 where p T = (p 0, p) and x T = ( x 0, x). (For simplicity let us assume intrinsic phases η T = 1 throughout this problem.) You may use without proof ( 1) 1 s u s (p T ) = γ 5 Cu s (p) ( 1) 1 s v s (p T ) = γ 5 Cv s (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 where γ µt is the transpose of γ µ. Cγ µt C 1 = γ µ Consider the decay of a neutron n to a proton p: n pe ν e, 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 L I = G F J lept,µ Jµ pn + h.c., with J µ lept = ν e γ µ (1 γ 5 )e and J µ pn = pγ µ (g V +g A γ 5 )n, derive a condition which g A /g V must satisfy if this interaction is to be invariant under time-reversal.
The electroweak theory of the Standard Model consists of an SU() L U(1) Y gauge theory which undergoes spontaneous symmetry breaking via the Higgs mechanism. Let Wµ a (a = 1,,) be the SU() L gauge bosons with coupling g and B µ be the U(1) Y gauge boson with coupling g. The scalar field φ transforms as a doublet under SU() L, has hypercharge 1, and has a Lagrangian of the form L φ = (D µ φ) (D µ φ) µ φ λ φ 4 (λ > 0). Explicitly write down the terms arising from covariant differentiation of φ, i.e. from D µ φ. Show that mass terms for 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 (W a µ,b µ ) and the ones after symmetry breaking, (W ± µ,z 0 µ,a µ ). Show how the electroweak theory includes the electron e and electron neutrino ν e. 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? [TURN OVER
4 Consider the weak semileptonic decay K 0 π + e ν e where K 0 and π + are pseudoscalar mesons. Assume this decay proceeds due to the interaction L eff W = G F J lept,µ Jµ had + h.c. with J µ lept = ν e γ µ (1 γ 5 )e and J µ had = V usūγ µ (1 γ 5 )s. [Ignore K 0 K 0 mixing.] Explain why the following equalities hold π + (k) ūγ µ (1 γ 5 )s K 0 (p) = π + (k) ūγ µ s K 0 (p) = (p+k) µ f + (q )+(p k) µ f (q ) where q p k and f + (q ) and f (q ) are scalar (dimensionless) functions of q. Treating the electron and anti-neutrino as massless, derive the invariant scattering amplitude M. In preparation to calculate the decay rate Γ( K 0 π + e ν e ), find and justify an expression for M which is proportional to f + (q ) (i.e. has no f (q ) term). Show that the decay rate Γ( K 0 π + e ν e ) for this process can be written as an integral over pion momentum d k [ A (p q) k 0 q m ] K f+ (q ) where A is a dimensionful constant which you should determine. By working in the K 0 rest frame, or otherwise, show that the decay rate can be expressed as Γ = B b where the kinematic variable λ is defined as a dq λ / f + (q ) λ = (q ) +m 4 K +m π q m K q m π m Km π. You should determine the dimensionful constant B and the limits of integration a and b. [You may use without proof the following: Γ = 1 d k d q 1 d q m K (π) k 0 (π) q1 0 (π) q 0 (π) 4 δ (4) (p k q 1 q ) M spins Trγ α γ β γ τ γ δ = 4(g αβ g τδ g ατ g βδ +g αδ g βτ ) Trγ α γ β γ τ γ δ γ 5 = 4iǫ αβτδ d q 1 d q q 1 q δ(4) (Q q 1 q )q 1µ q ν = π Q µq ν + π 6 g µνq d q 1 d q q 1 q δ(4) (Q q 1 q ) = π ]
4 5 Renormalized coupling constants g i (µ) (i = 1,,) generally depend on a renormalization scale µ. For small values of the couplings, the scale dependence can be written to one-loop order as µ d dµ g i(µ) = b i g i +O(g5 i ). What are the consequences of b i being positive vs. negative? For α i = g i /4π, derive an expression relating α i(m Z ) to α i (µ). (m Z is the mass of the Z boson.) Let α 1, α, and α be the coupling constants of the respective Standard Model gauge groups U(1) Y, SU() L, and SU() c. Suppose there exists a scale M GUT > m Z at which the following holds: Show that this implies 5 α 1(M GUT ) = α (M GUT ) = α (M GUT ). α 1 (m Z) = α 1 (m Z) + b b 5 b 1 b [ ] 5 α 1 1 (m Z) α 1 (m Z). Now let us work to another order, considering a single coupling α. Define a = α/4π. Given µ d dµ a = β 0a β 1 a, show that for a suitable choice of Λ. a 1 (µ) = β 0 log µ Λ + β 1 loglog µ (1/log β 0 Λ + O µ ) Λ END OF PAPER