Phys 624 - Quantum Field Theory
University of Maryland, College Park
Fall 2017, Professor: Ted Jacobson
Homework
HW0 - due at beginning of class Thursday, Sept. 07.
HW13 - due by Thursday,  Dec. 14, 12pm, in envelope @ PSC 3151.
 HW12 - due by Friday,  Dec. 1, 5pm, in envelope @ PSC 3151.

1. Read Chapter 17 (The anomalous magnetic moment)

2. Exotic contributions to g-2   (For part (a), you may evaluate the final integral to lowest order in me/mh. For part (c), you may assume the axion mass is much smaller than the electron mass, as that is anyway implied by other bounds. This problem was written before the Higgs was discovered. For part (b), instead of deriving a bound, just use the observed Higgs mass and compute the g-2 contribution, and compare to the precision of the current experimental measurements.)
HW11 - due by Tuesday,  Nov. 21, at the beginning of class. (You're welcome to turn it in before then if you like.)

1. Lorentz invariance of  Weyl, Dirac and Majorana Lagrangians

2. Seesaw mechanism

3. Bhabha scattering
 HW10 - due by Friday,  Nov. 10, 5pm, in envelope @ PSC 3151.

 1. S 13.3 (particle decays to e+e−)

[Note: To compute the total decay rate, you should sum over the spins of the final particles and, for the cases of vector and axial vector decaying particle, average over the initial polarizations. What will enter is spherically symmetrical after the spin and polarization sums, which makes the phase space integration straightforward. You may invoke previous results for the phase space integral. To check yourself, the result for the matrix element is (the following displays correctly on Firefox and Safari but not on my Chrome browser)

 [13]∑si,s+,s-|M|2=2mϕ2{gS2(1-4x2),gP2,23gV2(1+2x2),23gA2(1-4x2)}[\frac{1}{3}]\sum_{s_i,s_+,s_-}|\{\mathcal M}|^2=2m_\phi^2\{g_S^2(1-4x^2),\;\; g_P^2,\; \;\frac{2}{3}g_V^2(1+2x^2),\;\; \frac{2}{3}g_A^2(1-4x^2)\} >The factor [1/3] applies in the vector and axial vector cases. For the form of M, compare with what you see in (13.22) for the QED v
2. S 13.6 (a,b,c only) (parity violation)

In part (a), the point is that the neutrinos don't couple to the photon, but they do couple to the Z vector boson, which is a massive, neutral spin-1 field. The coupling constant for both the photon and the Z boson is dimensionless. In part (b), "Assume the Z couples to the electron in the same way as it couples to neutrinos." means to assume that the coupling constants gV and gA are the same for the couplings to electrons and neutrinos. Also, useful Dirac matrix identities are in appendix A.4. And note that ϵμναβϵμνγδ=−4δ[αγδβ]δ, where the square bracket denotes the antisymmetric part, e.g. [ab] = (ab - ba)/2. For part (c), look at section 11.1, where the chirality projection operators are introduced.

HW9 - due by Friday,  Nov. 3, 5pm, in envelope @ PSC 3151.
1. Axion lifetime: An axion is a hypothetical pseudoscalar particle that might couple to the electromagnetic field via the interaction in Eqn. (4) of this particle data group article.  (a) Use dimensional analysis and physical reasoning to argue that the rate for the axion to decay to two photons is given by Eqn. (6) in the same article, up to a numerical factor. (b) Work out the Feynman rule for the axion-photon-photon vertex. (In momentum space, a derivative becomes a factor ±ip. Feynman rules for derivative couplings are discussed in 7.4.2. See section 9.2 of Schwartz for another situation where the interaction term in the Lagrangian has derivatives.) (c) Compute the axion decay rate to lowest order in perturbation theory and show that you get the numerical factor in Eqn. (6). Be careful about symmetry factors and identical particles (photons are bosons). (d) Given the form of the coupling in Eqn. (4), explain why the axion must be a pseudoscalar (odd under parity), and why the presence of a nonzero vacuum expectation value for the  axion would "spontaneously" violate P, T and CP symmetry, but not C , PT, or CPT. (Sections 11.4-6 might be helpful, but note much of that concerns fermions, which are not at issue here. Hint: How does the four-index alternating symbol transform under P and T?) 2. Linear sigma model : HW7, problem 3, was an introduction to spontaneously broken symmetry, first global, then gauge symmetry and the Higgs mechanism. The present problem generalizes and looks more deeply at the case of a broken global symmetry. The scalar field has N components,  so you should think of it as a vector in an N-dimensional space. The potential is spherically symmetric in that space. Part (a) is the symmetric case, with the minimum of the potential at zero field. In Part (b), the quadratic term in the potential is negative, so the minimum is located on a (hyper)sphere of some radius. The ground state picks some field expectation value at this radius, and that spontaneously breaks the symmetry. Small radial fluctuations around the minimum have a quadratic potential, hence a nonzero mass, while small fluctuations tangent to the sphere have no change of potential, so are massless. These are the "Goldstone bosons", and there are N-1 of them. In (c) you'll find that the scattering amplitude for four Goldstone bosons vanishes if one of them has zero momentum. This is because an infinitesimal shift of one Goldstone field will not change the Lagrangian unless the field has a nonzero derivative. Finally, in part (d) you break the symmetry of the potential itself, by adding a tilt term. This now selects a unique minimum, gives the fomer Goldstone bosons a mass, and produces a nonzero scattering amplitude even when the momentum vanishes.
HW8 - due by Friday,  Oct. 27, 5pm, in envelope @ PSC 3151.

1.  S11.1 (Dirac matrix identities) 

2.  S11.4 (Gordon identity) [By "on-shell spinors" he means, for example, that p-slash u(p) = m u(p). Hint: Use this property to introduce p-slash and q-slash, and then fiddle with the gamma matrices.] [The first and second printings have an error in this problem. If you have one of those printings, ignore the last sentence ,which starts after the words "dipole moment".]

3.  (Fermi oscillator) Suppose b and b* satisfy the algebraic relations {b,b}=0={b*,b*}, and {b,b*}=1, where b* is the Hermitian conjugate of b and { , } represents the anticommutator. Show that the spectrum of b*b consists of just two values, 0 and 1, corresponding to eigenstates |0> and |1>, and that b|0>=0, b*|0>=|1>, b|1>=|0>.
HW7 - due by Friday,  Oct. 20, 5pm, in envelope @ PSC 3151.
HW6 - due by Friday,  Oct. 13, 5pm, in envelope @ PSC 3151.

1. Consider two real scalar fields, A and B, with particle masses mA and mB < 1/2 mA, coupled by an interaction g AB2. Calculate the decay rate of an A particle to a pair of B particles, at lowest order in  perturbation theory. Check that your result has dimension of inverse time. What is the rate in the limit mB << 1/2 mA?

2. S5.3 (muon decay phase space)
Before doing this problem read section S5.1.1. Although not stated explicitly, the given squared matrix element includes the sum over final spins and average over initial spin, so you need only consider the momentum phase space. Note that the electron neutrino that emerges from muon decay is actually an antineutrino.  Suggestion:  For part (a), first carry out the 2-body phase space integral for each possible value of the electron antineutrino  momentum.  Correction: For part (b) use a more accurate value for the muon mass, 105.66 MeV. Your result will be around 0.5% smaller than the measured lifetime. What could be the primary reason for that discrepancy? (The relative size of the correction from the neglected electron mass is of order me2/ mµ2, which is too small to account for the discrepancy.)

3. Problem S7.8 (mu and tau decay)
Assume the W couples universally, i.e. couples to electron, muon, and tauon with the same coupling constant g, converting them  into the corresponding neutrino. This is how it works in the Standard Model. For part (a), also draw the relevant Feynman diagram. For part (c), interpret the question this way: If the calculation were done with a vector W boson and spinor electron and neutrinos, the coupling constant would have been dimensionless. Use dimensional analysis to determine what power of the muon mass would appear in the decay rate in that case. For part (d), set g = 0.3, which corresponds to g2/4π = e2/4π = 1/137. In part (e), Explain why the prediction of the tauon mass depends only on the muon mass and the muon and tauon lifetimes. Also, note that in (e) and (f), the subscript on one of the neutrinos should be tau, not mu. 

4. Problem S7.9 (unstable particles and widths) Before doing this problem, read section 2.9 of Timo's notes, and sections S7.4.1,  S24 pp. 452-455 and S24.1.4 of Schwartz. (The latter refers to some things we haven't covered, but should still be more or less understandable.) For part (a), assume scalar particles. In part (c), he is referring to a Feynman propagator. Correct the statement in (d) to read "Loops made of two psi propagators will have imaginary parts if and only if the external √p2 is greater than twice the mass of psi.." In part(e) the answer to the second question is explained in 24.1. This gives the bird's eye view of what's going on, from a non-perturbative viewpoint. What he's getting at in the first question is, I think, the perturbative version of this. That is, in view of (c),  the imaginary part of the loop can only come from the loop propagators being on-shell. To see this you have to keep track of all the factors of i, including the one in the residue theorem, and use (c) when evaluating the imaginary part of the propagator contributing to the residue at the pole.
HW5 - due by Friday,  Oct. 6, 5pm, in envelope @ PSC 3151.
HW4 - due by Friday,  Sept. 29, 5pm, in envelope @ PSC 3151.
HW3 - due by Friday,  Sept. 22, 5pm, in envelope @ PSC 3151.
HW2 - due by Friday,  Sept. 15, 5pm, in envelope @ PSC 3151.

1. The LHC  produces head-on collisions of protons with energy 6.5 TeV each. If instead one of the protons were at rest, what energy would the other proton need to have in order to produce a collision with the same energy in the zero-momentum frame? (Tip: Express the energy in the ZM frame as a Lorentz invariant scalar constructed from the two 4-momenta, and use this to solve for the energy.)

2. For a given source distance, there is an effective maximum energy above which the universe is opaque to extragalactic gamma rays due to the pair creation process\gamma\gamma\rightarrow e^+e^-, where the second photon comes from the "background light". To get a sense for where this cutoff energy lies, compute the threshold energy for a gamma ray to collide with a 1 micron wavelength photon to make an electron-positron pair. Compare your result with the graph in Fig. 3 of http://arxiv.org/abs/0807.2822.

3. Read Section 1.9 of Weigand, on the complex scalar field. (a) The theory of the complex field can be thought of as the theory of a pair of real fields, the real and imaginary parts. Work out this description of the quantization. That is, write the Lagrangian in terms of the real and imaginary parts of the field, find the conjugate momenta of these parts, and then the Hamiltonian and mode expansions in terms of raising and lowering operators. (b) Next, adopt Weigand's definition of the complex momentum in the first equation of (1.20), and express that in terms of the momenta conjugate to the real and imaginary parts of the field, and then use your results from part (a) to verify (1.121-125). (I don't approve of the words written around (1.120), which assert that the field and it's hermitian conjugate are independent fields. They are not. But what you show in this problem is that his conclusions are nevertheless correct.)  (c) Show that the current (1.129) is divergenceless (\partial_\mu j^\mu=0), and show that this implies that the charge Q is conserved, i.e. that any change of total charge in a volume V is accounted for by a flux of current through the boundary of V. (d) Verify (1.130-131).
HW1 - due at beginning of class Thursday, Sept. 07.