MATHEMATICAL TRIPOS Part III Tuesday, 6 June, 017 9:00 am to 1:00 pm PAPER 305 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 (a) The matrix B is defined so that { B 1 γ µ γ µ µ = 0 B = γ µ µ = 1,,3. Show that B 1 γ 5 B = γ 5 where γ 5 = iγ 0 γ 1 γ γ 3. (b) Thetime-reversal operator ˆT is anti-linear, ˆT ( α φ +β ψ ) = α ˆT φ +β ˆT ψ. By making an appropriate assumption (which you should state) about how momentum eigenstates transform under ˆT and expanding in terms of such eigenstates, show that ˆTφ ˆTψ = φ ψ for arbitrary scalar states φ and ψ, i.e. ˆT is an anti-unitary operator. (c) Recall the expansion for a Dirac field ψ(x), ψ(x) = [ b s (p)u s (p)e ip x +d s (p)v s (p)e ip x]. p,s Show that under time reversal, ˆTψ(x)ˆT 1 = Bψ(x T ), where x µ T = ( x0, x). [The intrinsic phase η T = 1 in this question and you may assume that ˆTb s (p)ˆt 1 = ( 1) 1 s b s (p T ), ˆTd s (p)ˆt 1 = ( 1) 1 s d s (p T ), ( 1) 1 s u s (p T ) = γ 5 Cu s (p), ( 1) 1 s v s (p T ) = γ 5 Cv s (p), where p µ T = (p0, p) and as part of your derivation you should find a relation between B and C.] (d)for theremainderofthisquestionyou mayassumethat ˆT ψ(x)ˆt 1 = ψ(x T )B 1. Consider a U(1) gauge theory with gauge field A µ (x). Given that the term in the Lagrangian, ga µ ψγ µ ψ, where g is a real constant, leads to an interaction which is time-reversal invariant, determine how A µ (x) transforms under ˆT. How does ia ψσ µν γ 5 ψf µν transform under ˆT, where a is a real constant, σ µν = i [γµ,γ ν ] and F µν = µ A ν ν A µ? Could such an interaction arise in the Standard Model? (e) Explain briefly why C violation and CP violation are required for baryogenesis. What are the other two conditions required?
3 (a) Considertheelectroweak partofthestandardmodelwithgaugegroupsu() L U(1) Y. The covariant derivative acting on the complex scalar doublet φ is D µ φ = ( µ +igw a µτ a + i g B µ ) φ, where W a µ are the three SU() L gauge bosons and B µ is the U(1) Y gauge boson. Write down the part of the electroweak Lagrangian involving only gauge and scalar fields. Describe how the gauge symmetry is spontaneously broken to U(1) EM via the Higgs mechanism. In particular, you should relate the gauge fields after symmetry breaking (W µ ±, Z µ, A µ ) to the original gauge fields, define the Weinberg angle θ W, show that one gauge boson remains massless and find tree-level expressions for the masses of the other three gauge bosons and the Higgs boson. [Hint: After giving a brief justification you may consider the expansion φ(x) = 1 ( 0 v +H(x) where v is a real constant and H(x) is a real scalar field.] Draw Feynman diagrams for all the tree-level interactions involving both Z and H fields. (b) The interaction term in the Lagrangian which governs the decay of the Higgs boson,h,totwoz bosonsisl HZZ = m Z v Z µ Z µ H,wherev isthehiggsvacuumexpectation value. Without neglecting any masses, derive an expression for the decay rate Γ(H ZZ) supposing that m H > m Z. [Hint: The following expressions may be used without proof: 0 Z µ Z(k,s) = ǫ µ (k,s), Z spins s ), ǫ µ (k,s)ǫ ν (k,s) = g µν + k µk ν M Z, and the decay rate for A(p) B(k 1 )+C(k ) is, Γ(A BC) = 1 m A d 3 k 1 (π) 3 k 0 1 d 3 k (π) 3 k 0 (π) δ () (p k 1 k ) spins M, where m A is the mass of particle A.] [TURN OVER
3 (a) State whether or not each of the following processes is allowed at tree level in the Standard Model: (i) u d c s (ii) u c d s (iii) b ue + ν e (iv) c dµ + ν µ. For those which are allowed draw all possible tree-level Feynman diagrams. (b) Suppose that the chiral structure of the weak charged-current interaction has not been determined and so the part of the effective Lagrangian relevant for u d c s is L eff = G F [ dγ α (P +Qγ 5 )u ][ cγ α (P +Qγ 5 )s ], where P and Q are real constants and G F = g. Neglecting quark masses, CKM matrix 8MW factors and the effects of the strong interaction (e.g. hadronization), show that ( u(p1 ) d(p ) c(k 1 ) s(k ) ) = F(s) [ H 1 (1+cos θ)+h cosθ ], where θ is the angle between p 1 and k 1, F(s) is some function of s = (p 1 + p ) which you should find, and the constants H 1 and H should be expressed in terms of P and Q. [Hints: Work in the centre of momentum frame. The following expressions may be used without proof: Tr(γ µ 1...γ µn ) = 0 for n odd, Tr(γ µ γ ν γ ρ γ σ ) = (g µν g ρσ g µρ g νσ +g µσ g νρ ), Tr(γ 5 γ µ γ ν γ ρ γ σ ) = iǫ µνρσ, ǫ αβσρ ǫ αβλτ = (δ σ λ δρ τ δσ τ δρ λ ), and the differential cross section for A(p A )+B(p B ) C(p C )+D(p D ) is, ( 1 1 d 3 )( p C d 3 ) p D = v A v B p 0 A p0 B (π) 3 p 0 C (π) 3 p 0 (π) δ () (p A +p B p C p D ) 1 M, D where A, B, C and D are spin-1/ fermions.] (c) What happens to your expression at high energy (large s)? Comment on this. (d) Find an expression for the asymmetry A(θ) = (θ) (θ +π). (θ +π) (θ)+ Why might the experimental measurement of this asymmetry be a useful way to constrain the chiral structure of the weak charged-current interaction? spins
5 To leading order, the scale dependence of a renormalized coupling g(µ) is given by µ dg dµ = β 0 16π g3, where µ is the renormalization scale and β 0 is a real constant. (a) Briefly explain the consequences of β 0 being positive (as in QCD) or negative. (b) For α s = g /π and assuming that β 0 > 0, derive a expression for α s (µ) in terms of an energy scale Λ where α s diverges. (c) In QCD, suppose that Λ = Λ for µ < m b (where n f = ) and Λ = Λ 5 for m b < µ < m t (where n f = 5). Here m b and m t are respectively the bottom and top quark masses. By requiring that α s (µ) is continuous at µ = m b, show that, ( ) p/r mb Λ 5 = Λ, Λ where p and r are positive integers which you should find. [You may assume that β 0 = 11 3 N 3 n f in an SU(N) gauge theory with n f flavours of quarks.] (d) The cross section for e + (p 1 )e (p ) hadrons beyond leading order may be written as σ = πα 3q 3 Q f K( α s (µ ),q /µ ), f where Q f is the charge of quark flavour f, q = p 1 +p and at O(α s) K ( α s (µ ),q /µ ) = 1+ α s(µ ) π + α s(µ ) π One way to choose the arbitrary scale µ is to require that dk dlnµ = 0. [ 1.99 0.11n f β ] 0 ln( q /µ ). By using this prescription with the leading order scale dependence of α s (µ), find an expression for µ in terms of n f and q. END OF PAPER