Physics 215B QFT Winter 2018 Assignment 9

Transcription

1 University of California at San Diego Department of Physics Prof. John McGreevy Physics 215B QFT Winter 2018 Assignment 9 Due 12:30pm Wednesday, March 21, Emergence of the Dirac equation. Consider a chain of free fermions with H = t n c nc n+1 + h.c. Show that the low-energy excitations at a generic value of the filling are described the the massless Dirac lagrangian in 1+1 dimensions. Find an explicit choice of d gamma matrices which matches the answer from the lattice model. Show that the right-movers are right-handed γ 5 γ 0 γ 1 = 1 and the left-movers are left-handed. 2. Anomaly cancellation in the Standard Model. If we try to gauge a chiral symmetry (such as hypercharge in the Standard Model (SM)), it is important that it is actually a symmetry, i.e. is not anomalous. In D = 3 + 1, a possible anomaly is associated with a choice of three currents, out of which to make a triangle diagram. We ll call a G 1 G 2 G 3 anomaly the diagram with insertions of currents for G 1, G 2 and G 3. Generalizing a little, we showed that the divergence of the current for G 1 is µ j Aµ 1 = 1 32π 2 ɛµνρσ Fµν 2B Fρσ 3C ( 1) f tr R(f) {T1 A, T2 B }T3 C. The sum is over each Weyl fermion, R(f) is its representation under the combined group G 1 G 2 G 3, and T1 A are a basis of generators of the Lie algebra of G 1 etc. in the representation of the field f. By ( 1) f I mean ± for left- and right-handed fermions respectively. We consider the possibilities in turn. (a) Convince yourself that the divergence of the U(1) Y hypercharge current gets a contribution of the form ( µ J µ Y = Yl 3 ) Y 3 g 2 r 32π 2 ɛµνρσ B µν B ρσ left right f from the triangle with three insertions of the current itself (here B is the hypercharge gauge field strength). The sum on the RHS is over all leftand right-handed Weyl spinors weighted by the cube of their hypercharge. Check that this sum evaluates to zero in the SM. 1

2 (b) Show that any anomaly of the form SU(N)U(1) 2 or SU(N)G 1 G 2 is zero. (c) (Easy) Convince yourself that there is no SU(3) 3 anomaly for QCD. (d) Check that there is never an SU(2) 3 anomaly. (Hint: the generators satisfy {τ a, τ b } = 2δ ab.) (e) Show that the SU(3) 2 U(1) Y anomaly demands that 2Y Q Y u Y d = 0. Check that this is true in the SM. (f) Show that a necessary condition for hypercharge to not have an anomaly with the Electroweak gauge bosons on the RHS is Y L + 3Y Q = 0, where Y L and Y Q are the hypercharges of the left-handed leptons and quarks. Check that this works out in the SM. (g) There is another kind of anomaly called a gravitational anomaly. This is a violation of current conservation in response to coupling to curved space. An example is of the form µ j µ Y = atrr R where R is a two-form related to the curvature of spacetime (analogous to the field strength F ). The coefficient a is proportional to left try l right try r. Check that this too vanishes for hypercharge in the Standard Model. These conditions, plus the assumption that the right-handed neutrino is neutral, actually determine all the hypercharge assignments. 3. Grassmann brain-warmers. (a) A useful device is the integral representation of the grassmann delta function. Show that d ψ 1 e ψ 1 (ψ 1 ψ 2 ) = δ(ψ 1 ψ 2 ) in the sense that dψ 1 δ(ψ 1 ψ 2 )f(ψ 1 ) = f(ψ 2 ) for any grassmann function f. (Notice that since the grassmann delta function is not even, it matters on which side of the δ we put the function: dψ 1 f(ψ 1 )δ(ψ 1 ψ 2 ) = f( ψ 2 ) f(ψ 2 ).) (b) Recall the resolution of the identity on a single qbit in terms of fermion coherent states 1 = d ψdψ e ψψ ψ ψ. (1) Show that 1 2 = 1. (The previous part may be useful.) 2

3 (c) In lecture I claimed that a representation of the trace of a bosonic operator was tra = d ψdψ e ψψ ψ A ψ, and the minus sign in the bra had important consequences. (Here ψ c = ψ ( ψ) ). Check that using this expression you get the correct answer for tr(a + bc c) where a, b are ordinary numbers. (d) Prove the identity (1) by expanding the coherent states in the number basis. 4. Fermionic coherent state exercise. [Bonus] Consider a collection of fermionic modes c i with quadratic hamiltonian H = ij h ijc i c j, with h = h. (a) Compute tre βh by changing basis to the eigenstates of h ij (the singleparticle hamiltonian) and performing the trace in that basis: tr... = ɛ (b) Compute tre βh by coherent state path integral. Compare! (c) [super bonus problem] Consider the case where h ij What can you say about the thermodynamics? is a random matrix. 5. Right-handed neutrinos. [from Iain Stewart, and hep-ph/ ] Consider adding a right-handed singlet (under all gauge groups) neutrino N R to the Standard Model. It may have a majorana mass M; and it may have a coupling g ν to leptons, so that all the dimension 4 operators are L N = N R i/ N R M 2 N c RN R M 2 N R N c R + ( g ν NR H T i L j ɛ ij + h.c. ) n ɛ=c ɛc ɛ=0,1... where N c R = C ( NR ) T is the the charge conjugate field, C = iγ2 γ 0 (in the Dirac representation), H is the Higgs doublet, L is the left-handed lepton doublet, containing ν L and e L. Take the mass M to be large compared to the electroweak scale. Integrate out the right-handed neutrinos at tree level. [Hint: you may find it useful to work in terms of the Majorana field which satisfies N = N c.] N N R + N c R 3

4 Show that the leading term in the expansion in 1/M is a dimension-5 operator made of Standard Model fields. Explain the consequences of this operator for neutrino physics, assuming a vacuum expectation value for the Higgs field. Place a bound on M assuming that the observed neutrinos have masses m ν < 0.5 ev. 6. Gross-Neveu model. [Bonus] Now that we ve learned about fermionic path integrals, consider the partition function for an N-vector of fermionic spinor fields in D dimensions: Z = [dψd ψ]e is[ψ], S[ ψ] ( = d D x ψa i/ ψ a g ( ψa ψ a) ) 2. N (a) At the free fixed point, what is the dimension of the coupling g as a function of the number of spacetime dimensions D? Show that it is classically marginal in D = 2, so that this action is (classically) scale invariant. (b) We will show that this model in D = 2 exhibits dimensional transmutation in the form of a dynamically generated mass gap. Here are the steps: first use the Hubbard-Stratonovich trick to replace ψ 4 by σψ 2 + σ 2 in the action, where σ is a scalar field. Then integrate out the ψ fields. Find the saddle point equation for σ; argue that the saddle point dominates the integral for large N. Find a translation invariant saddle point. Plug the saddle point configuration of σ back into the action for ψ and describe the resulting dynamics. 7. When is the QCD interaction attractive? [Bonus] Write the amplitude for tree-level scattering of a quark and antiquark of different flavors (say u and d) in the t-channel (in Feynman ξ = 1 gauge). Compare to the expression for e µ scattering in QED. First fix the initial colors of the quarks to be different say the incoming u is red and the incoming d is anti-green. Show that the potential is repulsive. Now fix the initial colors to be opposite say the incoming u is red and the incoming d is anti-red so that they may form a color singlet. Show that the potential is attractive. Alternatively or in addition, describe these results in a more gauge invariant way, by characterizing the potential in the color-singlet and color-octet channels. You can do this problem either by choosing a specific basis for the generators of SU(3) in the fundamental (a common one is called the Gell-Mann matrices), or using more abstract group theory methods. 4

5 8. Simplicial homology and the toric code. [Bonus] In lecture we discussed the (de Rham) cohomology of the exterior derivative d acting on vector spaces (over R) of differential forms on some smooth manifold X. The dimensions b p (X, R) of the cohomology groups are topological properties of X. This same data is manifested in many other ways; in this problem we study another one, along with an important and familiar physical realization of it. You could say that this problem is the punchline to the previous problem about the toric code. (a) The toric code we ve discussed so far has qbits on the links l 1 ( ) of a graph. But the definition of the Hamiltonian involves more information than just the links of the graph: we have to know which vertices v lie at the boundaries of each link l, and we have to know which links are boundaries of which faces. The Hamiltonian has two kinds of terms: a plaquette operator B p = l p X l associated with each 2-cell (plaquette) p 2 ( ). and star operators, A s = l 1 (s) Z l, associated with each 0-cell (site) s 0 ( ). Here I ve introduced some notation that will be useful, please be patient: k denotes a collection of k-dimensional polyhedra which I ll call k-simplices or more accurately k-cells k-dimensional objects making up the space. (It is important that each of these objects is topologically a k-ball.) This information constitutes (part of) a simplicial complex, which says how these parts are glued together: d d (2) where is the (signed) boundary operator. For example, the boundary of a link is l = s 1 s 0, the difference of the vertices at its ends. The boundary of a face p = l p l is the (oriented) sum of the edges bounding it. By 1 (s) I mean the set of links which contain the site s in their boundary (with sign). Think of this collection of objects as a triangulation (or more generally some chopping-up) of a smooth manifold X. Convince yourself that the sequence of maps (12) is a complex in the sense that 2 = 0. (b) [not actually a question] This means that the simplicial complex defines a set of homology groups, which are topological invariants of X, in the following way. (It is homology and not cohomology because decreases the degree k). To define these groups, we should introduce one more gadget, which is a collection of vector spaces over some ring R (for the ordinary toric code, R = Z 2 ) Ω p (, R), p = 0...d dim(x) 5

6 basis vectors for which are p-simplices: Ω p (, R) = span R {σ p } that is, we associate a(n orthonormal) basis vector to each p-simplex (which I ve just called σ), and these vector spaces are made by taking linear combinations of these spaces, with coefficients in R. Such a linear combination of p-simplices is called a p-chain. It s important that we can add (and subtract) p-chains, C + C Ω p. A p-chain with a negative coefficient can be regarded as having the opposite orientation. We ll see below how better to interpret the coefficients. The boundary operation on p induces one on Ω p. A chain C satisfying C = 0 is called a cycle, and is said to be closed. So the pth homology is the group of equivalence classes of p-cycles, modulo boundaries of p + 1 cycles: H p (X, R) ker ( : Ω p Ω p 1 ) Ω p Im ( : Ω p+1 Ω p ) Ω p This makes sense because 2 = 0 the image of : Ω p+1 Ω p is a subset of ker ( : Ω p Ω p 1 ). It s a theorem that the dimensions of these groups are the same for different (faithful-enough) discretizations of X. Furthermore, their dimensions (as vector spaces over R) b p (X) contain (much of 1 ) the same information as the Betti numbers defined by de Rham cohomology. For more information and proofs, see the great book by Bott and Tu, Differential forms in algebraic topology. (c) A state of the toric code on a cell-complex can be written (for the hamiltonian described above, this is in the basis where Z l is diagonal) as an element of Ω 1 (X, Z 2 ), Ψ = Ψ(C) C C where C is an assignment of an element of Z 2 in X (the eigenvalue of Z l ). For the case of Z 2 coefficients, 1 = 1 mod 2 and we don t care about the orientations of the cells. Show that the conditions for a state Ψ(C) to be a groundstate of the toric code (A s Ψ = Ψ s and B p Ψ = Ψ p) are exactly those defining an element of H 1 (X, Z 2 ). (d) Consider putting a spin variable on the p-simplices of. More generally, let s put an N-dimensional hilbert space H N span{ n, n = 1..N} on each 1 I don t want to talk about torsion homology. 6

7 p-simplex, on which act the operators N X n n ω n 0 ω 0... = 0 0 ω 2... n= , N Z n n + 1 = n= where ω N = 1 is an nth root of unity. If you haven t already, check that they satisfy the clock-shift algebra: ZX = ωxz. For N = 2 these are Pauli matrices and ω = 1. Consider the Hamiltonian H = J p 1 A s J p+1 s p 1 with A s µ p+1 B µ g p σ 1 (s) p Z σ B µ σ µ X σ. σ p Z σ This is a lattice version of p-form Z N gauge theory, at a particular, special point in its phase diagram. Show that 0 = [A s, A s ] = [B µ, B µ ] = [A s, B µ ], s, s, µ, µ so that for g p = 0 this is solvable. (e) Show that the groundstates of H p (with g p = 0) are in one-to-one correspondence with elements of H p (, Z N ). 9. Continuum U(1) version of canonical loop operators. [Bonus] In the toric code on T 2, there are four degenerate groundstates which are not distinguishable by local operators. They are made from a given groundstate (say the one where all the electric flux lines are contractible) by the action of the algebra of loop operators, defined (in the basis where the plaquette operator is l p X l) by W (C) = X l, V (Č) = Z l l C l Č where C is a path through the lattice, and Č is a path through the dual lattice (the lattice whose sites are the faces of the original lattice). 7

8 The groundstate degeneracy on the torus arises from the fact that the groundstates must represent the algebra W x Vˇy = Vˇy W x, where α is a loop wrapping the α-direction, as in the figure at right. (This relation follows because these two curves share a single link, on which the operators anticommute.) Here we would like to understand the analogs of these operators in (the deconfined phase of) U(1) gauge theory (first in D = and then in D = 3 + 1). The analogy is: X l e i l A, Z l e i l dl E /g 2. The analog of W (C) is the Wilson loop W (C) = e i C A, the phase produced by adiabatically moving a charge along the path C. The analog of V (Ĉ) depends on the number of dimensions. In D = 2+1, j µ ɛ µνρ F νρ /g 2 is the vortex current. Consider V (Ĉ) = ei Ĉ dxµ ɛ µνρf νρ. Check that the exponent is dimensionless. Interpret V (Ĉ) as the phase produced by adiabatically moving a vortex along the curve Ĉ, We would like to show that W x V y Wx 1 Vy 1 is a nontrivial phase, just like in the toric code. We ll do it using the path integral. Convince yourself that T ( Wx V y W 1 x V 1 y ) = W (Cx )V (C y ) where C x and C y are indicated in the right figure. Time goes vertically. In the left figure, the curves are closed by the periodic identifications of the spatial torus. = Write a path integral expression for the RHS. Do the gaussian integral. Show that the answer is e ial(cx,cy) where l(c, C ) is the linking number of the two curves, and a is a constant. Find a. 8

9 Reproduce this result using canonical methods, [A i (x, t), E j (y, t)] = ig 2 δ 2 (x y) and the Baker-Campbell-Hausdorff formula. In D = 3 + 1, the analog of j µ is a two-form: j µν ɛ µνρσ F ρσ /g 2, the vortex string current. So we may consider surface operators V Σ e i Σ j which are the phase produced by moving a vortex string along this worldsheet trajectory. Using canonical methods or path integrals, study the commutation of W x with V yz e i yz j where the integral is over the yz plane at x = 0. Bonus: using the path integral, consider the effects of charged matter on this calculation. 9