NTNU Trondheim, Institutt for fysikk

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1 NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel Allowed tools: mathematical tables Some formulas can be found on p Concepts. Please answer concise! a.) Explain what the ϑ vacuum of QCD is. (2 pts) b.) Give an explanation of the Goldstone theorem and how it can be avoided. (2 pts) c.) Assume that a local conservation law of the type µ T µ σ = 0 for a tensor T holds in a classical field theory. Explain why this relation can be broken on the quantum level. (2 pts) d.) Give 2 examples for symmetries broken on the quantum level. (2 pts) e.) The running coupling of the Higgs potential satisfies at one-loop dλ dt = 1 { 12λ 2 +12λy 2 16π 2 t 12yt λ( 3g 2 +g 2) + 3 [ 2g 4 +(g 2 +g 2 ) 2]} 16 with t = ln(µ 2 /µ 2 0). Explain schematically why this implies a lower and an upper bound for the Higgs mass m h as function of the scale µ. (6 pts) f.) Explain why (why not) the term F Fµν µν can be neglected in the generating functional of QED; in non-abelian theories like QCD. Assume in both cases a pure gauge theory, i.e. the absence of fermions. (6 pts) a.) A linear superposition of vacua n with fixed winding number, ϑ = n= e inϑ n ; its a common eigenstate of the YM Hamiltonian H and the operator U (ν) for large gauge transformations connecting vacua with different winding number ν = n 1 n 2. b.) The spontaneous breaking of a (global) continuous symmetry leads to the appearance of massless Goldstone bosons. If the symmetry is gauged, these Goldstone bosons are converted into the longitudinal degrees of freedom of massive gauge bosons. c.) The integration measure Dφ i of a collection of field φ i can change non-trivially under the classical symmetry; the equation µ T µ σ = 0 translated into an operator equation is ambiguous; any possible regularization procedure breaks the symmetry. d.) Chiral anomaly, breaking of scale invariance. f. The coupling λ must be positive for the vacuum state to be stable. A lower bound on m h = 2λv means small values of λ, and thus we neglect all terms of O(λ), dλ dt 1 [ 12y 4 16π 2 t + 3 ( 2g 4 +(g 2 +g 2 ) 2)] 3y4 t 16 4π 2 page 1 of 2 pages

2 Higgs couplings are proportional to the particle masses, and thus the bracket can be approximated by the large top Yukawa coupling, which drives m h (µ) for large enough µ to zero. This happens the earlier, the smaller the starting value of m h. An upper bound for m H corresponds to the case of large λ. Therefore we need to keep now only the self-coupling of the Higgs in the beta-function, leading to a Landau pole at β(λ) = dλ dt = 3λ2 4π 2, (1) { } 4π 2 µ L = µ 0 exp. (2) 3λ 0 We need λ to be small enough to do perturbation theory. Including a cutoff, Λ, µ L > Λ, and asking for λ <, we obtain the inequality ( ) 3λ(µ 0 ) Λ 2 4π 2 ln < 1. (3) Using m 2 H = 2λ(v)v2 and v = µ 0, this translates into an upper bound for m H as function of the cutoff scale. f.) i) A non-trivial vacuum structure as the ϑ vacuum has only physical consequences if the corresponding classical tunneling solutions have a finite action. Imposing τ d4 x FF < for an Euclidean Yang-Mills theory gives F O(τ 3 ) and A O(τ 2 ). Then in the abelian case, the total derivative goes as K O(A A) O(τ 5 ). Thus the surface term in d 4 FF = dω µ K µ 0 (4) Ω Ω vanishes and FF does not influence physical quantities in an abelian theory In an non-abelian case, we can choose a pure gauge field at spatial infinity, µ 2 0 A µ = i g ( µu)u 1, (5) which results in F µν = 0 for τ and ensures that the action is finite. On the other hand, a pure gauge transformation U which becomes constant for τ, i.e. depends in this limit only on the angles, gives A O(τ 1 ) and thus K O(AAA) O(τ 3 ). As a result, the surface integral may become non-zero in the limit τ. ii) The mappings S 3 S 3 relevant for the non-abelian case are characterised by winding number ν corresponding to non-trivial gauge field configurations. By contrast, the mappings U(1) S 3 S 1 S 3 can be only contracted to the identity. 2. Effective action. a). The effective action Γ[φ c ] is the generating functional for 1PI Green functions Γ (n), Γ[φ c ] = 1 d 4 x 1 d 4 x n Γ (n) (x 1,...,x n )φ c (x 1 ) φ c (x n ). n! n page 2 of 2 pages

3 Show that Γ (2) is equal to minus the inverse propagator or inverse 2-point function. (8 pts) b). How is the effective potential V eff connected to the effective action Γ[φ c ]? For what is the effective potential V eff useful? (4 pts) a.) We show first that δγ[φc] δφ c(y) = J(y) by computing the functional derivative w.r.t. φ c, Then we write δγ[φ c ] δφ c (y) = d 4 x δw δφ c (y) δ(x 1 x 2 ) = δφ(x 1) δφ(x 2 ) = d 4 x δφ c (y) φ c(x) J(y). d 4 x δφ(x 1) δφ(x 2 ) using the chain rule. Next we insert φ(x) = δw/ and J(x) = δγ/δφ(x) to obtain δ(x 1 x 2 ) = d 4 x δ 2 W δj(x 1 ) δ 2 Γ δφ(x)δφ(x 2 ). Setting J = φ = 0, it follows d 4 xig(x,x 1 )Γ (2) (x,x 2 ) = δ(x 1 x 2 ) (6) or Γ (2) (x 1,x 2 ) = ig 1 (x,x 1 ). b.) The effective potential V eff is the lowest order term of the gradient expansion of the effective actionγ[φ c ], i.e.itequalstheeffectiveactionγ[φ c ](dividedbythespace-timevolume)forconstant fields φ c. The minima of the effective potential V eff gives the vev of the fields, including loop corrections. 3. Pressure of λφ 4 theory. The pressure P of a scalar gas with λφ 4 interaction is [ P = π2 T λ 16π ( 3ln µ 4πT + 31 )( ) 2 ] λ 35 +C +O(λ 3 ), 16π 2 where the parameter µ is the renormalisation scale and C is a constant. Derive from this expression the beta function β(µ) = b 1 λ 2 +O(λ 3 ). (8 pts) The pressure as a physical quantity should not depend on the renormalisation scale µ. Thus the explicit µ dependence in P has to be cancelled by the µ dependent running of the coupling λ. Thus β = µdλ/dµ can be determined from 0 = µ dp dµ = π2 T 4 [ π 2 µ dλ + 3 ( ) λ 2 dµ 8 16π }{{} 2 +O(λ )] 3. β=3λ 2 /16π 2 page 3 of 2 pages

4 4. Georgi-Glashow model. The Georgi-Glashow model contains a SO(3) gauge field A a µ supplemented by a triplet of real Higgs scalars Φ = {φ 1,φ 2,φ 3 }, L = 1 4 F2 +(D µ Φ) 2 V(Φ). a.) Write down the explicit form of Fµν a and the covariant derivative (D µ φ) a as function of the gauge fields. (8 pts) b.) Break spontaneously the gauge symmetry choosing a uniform vev in the potential ( ) V(φ) = λ 2 φ 2 i η 2. 4 i What is the remaining gauge symmetry? What are the masses of the fields? (6 pts) c.) What type of topologically non-trivial solution contains this model? (2 pts) d.) Consider the model as an effective theory with its parameters defined at the scale Λ. Describe schematically how the new L = L( k sλ) looks like after integrating out modes with (Euclidean momenta) s Λ k Λ and 0 < s < 1. (8 pts) e.) Commentonwhathappensinthequantumcase, ifyouaddaleft-chiralfermiondoublet L = (u,d) t to the model. (2 pts) a.) Applying the general D µ = µ + iga a µt a to a specific fields amounts to choose the right matrix representation of the gauge group. In contrast to the SM where a complex Higgs doublet transforms according the fundamental representation, it is most convenient to put the the three real Higgses in Georgi-Glashow model into the adjoint representation with (T a A ) bc = if abc. Moreover, the structure constants of the rotation group SO(3) are f abc = ε abc and thus The field-strength follows similarly as D µ φ a = µ φ a gε abc A b µφ c. F a µν = µ A a ν ν A a µ gε abc A b µa c ν. b.) Choosing the vev e.g. in the 3-direction, φ a = (0,0,v), leaves a residual SO(2) U(1) symmetry unbroken, corresponding to rotations around the 3-axis in isospin space. Thus one can view the Georgii-Glashow model as a toy model for the electroweak sector of the SM, where the gaugefielda 3 µ playstheroleofthephoton, thetwomassivestatesa ± = (A 1 ±ia 2 )/ 2correspond to W ±, and the Z boson is missing. Choosing the unitary gauge, and plugging φ = (0,0,h+v) t into D µ φ a D µ φ a gives L = 1 2 D µφ a D µ φ a = 1 2 (gv)2 (A 1 µa µ1 +A 2 µa µ2 )+... page 4 of 2 pages

5 Thus the masses of the gauge bosons are m 1,2 = gv and m 3 = 0. c.) A 3 d-dimensional topological defect contains in its core the unbroken vacuum φ 1 =... = φ d = 0. In the Georgi-Glashow model, the three constraints φ i (x 1,x 2,x 3 ) = 0 define three twodimensional surface in R 3, and their intersection defines a point-like defect which the external field of a monopole. d). The values of the operators contained in L would be changed, relevant ones (ρ λ and m 2 ) increase, marginal ones(g) change logarithmically. An infinite tower of new irrelevant interactions appears, L = Λ 2 φ b φ b D µ φ a D µ φ a +..., suppressed by the high-energy cutoff. e). Adding a single left-chiral fermion doublet would make the theory non-renormalisable because of the chiral anomaly. Some formulas Z[J] = Dφ exp{i d 4 x(l(x)+j(x)φ(x))} = e iw[j] (7) φ c (x) = δw[j] D µ = µ +iga µ with A µ = A a µt a [T a,t b ] = if abc T c (T a A) bc = if abc F µν = 1 ig [D µ,d ν ] page 5 of 2 pages