Intercollegiate post-graduate course in High Energy Physics. Paper 1: The Standard Model

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1 Brunel University Queen Mary, University of London Royal Holloway, University of London University College London Intercollegiate post-graduate course in High Energy Physics Paper 1: The Standard Model Monday, 10 February 014 Time allowed for Examination: 3 hours Answer ALL questions Books and notes may be consulted 1

2 The Standard Model and beyond part 1. Elastic electron-proton scattering. (a) Draw the Feynman diagram for the lowest order (electromagnetic) process contributing to electron-proton scattering. [3] The matrix element squared for the lowest order electromagnetic electron-proton scattering (under the assumption of the proton being a structureless, point-like Dirac particle) with given four-momenta, can be written ( e M = q e (p i ) + p (P i ) e (p f ) + p (P f ), ) [ 1 4 Tr pf / + m m γµ p i / + m m γν ] [ Pf / + M Tr M γ P/ i + M µ M where q = p f p i = P i P f, with m denoting the electron's mass, and M that of the proton. (b) Evaluate the Dirac traces here to give M = e 4 m M q 4 [ (pf.p f ) (p i.p i ) + (p f.p i ) (p i.p f ) M (p f.p i ) m (P f.p i ) + m M ]. γ ν ], [8] (c) Assuming four-vectors p i = (E, p), p f = (E, p ), P i = (M, 0), P f = (E f, P f ), show that, in the limit m E, energy-momentum conservation implies E E M = q M. [5] (d) Hence show that the cross section, dσ dω = m 4π E /E 1 + (E/M) sin θ/ M,

3 where M = 16π α EE m q 4 [1 + q 4EE ( ) 1 + E E + m M EE ( )] E E M can be simplied, in the limit E m but E M, to dσ dω = α cos θ 4E sin 4 θ, where θ is the angle between the outgoing and incoming electron. [4] CONTINUED 3

4 . Collinear factorisation of matrix elements. Throughout this question one should take the on-shell quark and gluon masses to be zero: p = g = 0. Consider an n + 1 particle process, with amplitude M (n+1), in which a quark and a gluon are produced; the momenta of the quark and the gluon are denoted p and g respectively, likewise their colour indices are i and a. In the limit that the quark and gluon momenta are collinear (p.g = E p E g (1 cos θ pg ) 0) the amplitude is dominated by diagrams involving propagators of the form 1/ (p + g), i.e. it is dominated by graphs in which the gluon is radiated by the quark leg coming out of the n particle process. This is depicted in gure 1, where the right-hand side represents the sum of all graphs involving a quark of momentum P = p + g and colour j branching to the collinear quark-gluon pair. lim p.g 0 n- n-1 M (n+1) 1 p, i g, a n- n-1 (n) M P, j p, i j 1 g, a Figure 1: Collinear limit (p + g) 0 for an arbitrary n + 1 particle process involving the production of a quark and gluon. Neglecting terms that are nite as p.g 0, using standard Feynman rules, in the collinear limit, the amplitude may then be written M (n+1) = ɛ (g) µ u (p) ( ig s T a ijγ µ ) i (p/ + g/) (p + g) M(n) j, where M (n) j denotes all contributions to the n + 1 particle amplitude, except the q (p + g, j) q (p, i) + g (g, a) branching, g s is the strong coupling constant and Tij a is a Gell-Mann matrix. (a) Derive the complex conjugate amplitude: M (n+1) = gs p.g T a j i M(n) j γ 0 (p/ + g/) γ ν u (p) ɛ (g) ν. [5] 4

5 (b) Summing over gluon polarizations and colour indices (a and i) gives ( ) pol,col M(n+1) M (n+1) = g sc F η µν + gµ n ν +n µ g ν δ (p.g) n.g jj u (p)γ µ (p/ + g/) M (n) j M (n) j γ 0 (p/ + g/) γ ν u (p), where η µν here denotes the usual metric tensor, and n is an unphysical gauge vector arising in the sum over gluon polarizations: n is a light-like four-vector, n = 0, which is arbitrary except for the constraints n.p 0 and n.g 0. Perform a further sum over the external quark spins in this expression to give the full spinpolarization- and colour-summed squared amplitude as ( ) M (n+1) M (n+1) = g sc F η µν + gµ n ν +n µ g ν δ (p.g) n.g jj [ Tr M (n) j ] γ 0 (p/ + g/) γ ν p/γ µ (p/ + g/) M (n) j. [5] (c) After straightforward Dirac algebra this matrix element simplies further to yield [ ] M (n+1) M (n+1) = g sc F δ jj Tr M (n) (p.g)(n.g) j γ 0 (n. (p + g) (p/ + g/) + n.pp/ p.gn/) M (n) j. Keeping only the dominant O (1/p.g) terms, one can replace in the trace and the 1/n.g part of the denominator p = zp, g = (1 z) P, where z is the momentum fraction of the daughter quark with respect to the parent with momentum P (i.e. z is just a scalar number). Using these momentum relations write M (n+1) M (n+1) in the form M (n+1) M (n+1) = g s P p.g qq (z) Tr [ M (n) j ] γ 0 P/ δ jj M (n) j, recording explicitly the form you obtain for the function P qq (z). [5] (d) Using the completeness relation for a fermion with (light-like) momentum P, uj (P ) u j (P ) = P/ δ jj, derive the factorized form of the spin- and coloursummed squared matrix element for the n + 1 particle process, in terms of the n particle one: M (n+1) M (n+1) = g s P P qq (z) M (n) M (n) 5

6 where M (n) is the amplitude for the n particle process, related to M (n) by M (n) j = u j (P ) M (n) j, and (hence) M (n) j = M (n) j γ 0 u j (P ). Comment on whether this result is interesting. [5] CONTINUED 6

7 3. Spontaneous breaking of global symmetry in a complex scalar eld theory. The Lagrangian density for a complex scalar (φ) eld theory is given by L = ( µ φ) ( µ φ ) m φ φ λ (φ φ). (a) Show that the ground/vacuum state eld conguration φ 0 satises i) φ 0 = 0 for m m > 0 and ii) φ 0 = for λ m < 0. [6] (b) Comment briey on the dierence in the vacuum state obtained for m < 0 with respect to that found for m > 0. [] (c) Assuming m < 0 and taking as the vacuum state for φ m φ 0 = φ 0, φ 0 = λ, determine the Lagrangian density in terms of two real scalar elds, φ 1 and φ, reparameterizing φ as φ = φ (φ 1 + iφ ). [8] (d) Comment on the nature of the various terms in the Lagrangian that results in terms of φ 1 and φ, in particular, comment on the masses of φ 1 and φ and whether or not these are what you might have expected them to be, based on what you know of spontaneous symmetry breaking. [4] CONTINUED 7

8 4. Abelian gauge invariance for a complex scalar eld theory. The Lagrangian density for a complex scalar (φ) eld theory is given by L = ( µ φ) ( µ φ ) V (φ, φ ) V (φ, φ ) = m φ φ λ (φ φ). (a) Determine how the potential V (φ, φ ) changes under a local U (1) symmetry transformations φ Uφ, U = e iqλ, Λ = Λ (x). [3] (b) Determine how the derivative term µ φ changes under the same U (1) transformation. [4] (c) Dening the covariant derivative as D µ = µ + iqa µ, with A µ transforming as A µ A µ = UA µ U + i q ( µ U) U = A µ µ Λ, determine the result of the same U (1) transformation applied to D µ φ. [6] (d) Hence show that L gauged = (D µ φ) ( D µ φ ) V (φ, φ ) 1 4 F µνf µν, where F µν = µ A ν ν A ν. is U (1) gauge invariant. [7] CONTINUED 8

9 5. Euler-Lagrange eld equations and symmetry transformations. (a) Consider a Lagrangian density L ( φ (i), µ φ (i)) and the corresponding action, S = d 4 x L ( φ (i), µ φ (i)), where (i) labels various elds; i = 1,..., N. Consider small variations of the elds φ (i) (x, t) φ (i) (x, t) + δφ (i) (x, t), the variations δφ (i) all being zero at space-time innity (the boundary of the action integral). Applying the variational principle, in particular, by imposing the action be extremised with respect to the eld variations (δs = 0) derive the Euler- Lagrange dierential equations obeyed by the elds φ (i). [10] (b) Suppose that L, the Lagrangian density itself, is invariant under some symmetry transformation group. Under an innitesimal transformation associated with the `ath' generator of this symmetry group we denote the change in the elds and their derivatives naturally as φ (i) φ (i) + δ a φ (i), ν φ (i) ν φ (i) + ν δ a φ (i), where the subscript a on δ a is simply there to clarify that the innitesimal change δ is to be associated with a `rotation' by the `ath' generator of the group only. Assuming that the elds φ (i) and their derivatives ν φ (i) satisfy the Euler-Lagrange eld equations, compute the change in the Lagrangian density δl and show that invariance of the Lagrangian implies the conservation of four-vector currents: ν J ν a = 0 where J ν a = i L ( ν φ (i) ) δ aφ (i). [10] CONTINUED 9

10 . Goldstone propagator. p p p m + iϵ 3. Ghost propagator. 6. Computation of the width for Higgs boson decay to i fermion-antifermion pairs. = p p ξm + iϵ 4. W bosonsince propagator. the Higgs boson is a scalar particle, rather than associating a polarization vector or spinor to its presence as an external particle ig in µnu amplitudes (as would be the case if = it were a vector boson or a fermion), p instead p one msimply + iϵ associates a trivial factor `1' to each external Higgs boson in the amplitude Vertices 1. Higgs-fermion-fermion-vertex The Feynman rule [PS95b]. for a Higgs For quarks boson coupling one should to a multiply fermion iswith depicted an additional in Fig.. δ AB (A, B color indices) to ensure color-charge conservation: h 0 f f = i p ξm + iϵ = i m f v = ie sinθ w (9) (10) (11) m f m W (1) Figure : The vertex Feynman rule for a Higgs boson coupling to a fermion; e is the electric charge and θ w the Weinberg angle, while m f and m W are, respectively, the mass of the fermion and the W boson. (a) Denoting the fermion momentum by p and the anti-fermion momentum by k, write down the amplitude for a Higgs boson decaying into a fermion anti-fermion pair. [5] (b) Compute the amplitude squared for a Higgs boson decaying into a fermion pair, summed over nal-state fermion spins and colours, averaged over incoming polarizations, eliminating all momenta in terms of m h (the Higgs boson mass) and m f. Do not neglect the fermion mass. [8] (c) Using the expression for the two-body Lorentz invariant phase space dlips = 1 4π p 4m h dω, where p is the three-momentum of either decay product in the Higgs boson rest frame, and dω is the solid angle, compute the width for a Higgs boson decaying into a fermion anti-fermion pair, again, eliminating all momenta in terms of m h and m f. Do not neglect the fermion mass. [7] END OF PAPER 10