Phys 624 - Quantum Field Theory
University of Maryland, College Park
Fall 2016, Professor: Ted Jacobson
Homework
HW0 - due at beginning of class Thursday, Sept. 08.
HW12 - due at beginning of class Thursday, Dec. 15

1. S16.4 (Landau pole with standard model fermions)  (Note: To keep this simple, define the renormalized charges at mtop2, and run up the energy scale starting from mtop, using the |p2| >> mtop2 approximation in the vacuum polarization graphs. The fine structure constant at mtop is eR2(mtop)/4π ~ 1/127.

2. 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) Derive the numerical factor.

3. Linear sigma model

4. gauge invariance and the little group for massless spin-1 particles
HW11 - due at beginning of class Tuesday, Nov. 22

1. Chiral symmetry: In class we showed that conservation of the Dirac current (10.110) follows from the invariance of the action under constant phase changes of the spinor field, when the spinor field satisfies its equation of motion. This is an example of Noether's theorem, which is discussed in section 3.3. I explained it a bit differently, as follows: Under a a phase shift of the spinor field, (10.99), with a non constant  phase, the variation of the action is not generally zero, but comes only from the term involving the derivative of the phase contracted with some current vector. However, if the fields are varied away from a solution to the equation of motion, then the variation of the action must vanish for any variation, including this one. Integrating by parts, this implies that the current is conserved.

Show that:
(a) the Dirac action is invariant under the axial phase transformation ψ→e-iαγ5ψ\psi\rightarrow e^{-i\alpha\gamma^5}\psi with constant alpha, if and only if the mass vanishes;
(b) this symmetry in the massless case implies conservation of the axial vector current, ψ¯γμγ5ψ\bar\psi\gamma^\mu\gamma^5\psi (by an argument analogous analogous to the one I gave above);
(c) conservation of the vector and axial vector currents implies that the L and R currents are separately conserved;
(d) when the mass is zero, the action is invariant under separate constant phase transformations of the L and R spinors, and show directly from this that the L and currents are separately conserved.

2. Read Chapter 17 (The anomalous magnetic moment)

3. 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.)
HW10 - due at beginning of class Thursday, Nov. 10

 1. S 13.6 (a,b,c only) (parity violation)

2. Bhabha scattering
HW9 - due at beginning of class Thursday, Nov. 3

 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. To check yourself, the result, according to my sources, is
[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^(but I didn't check it myself, so I can't swear it's correct). 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).]

2. (spontaneous symmetry breaking and Higgs mechanism) (i) Read http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism. (ii) verify equations (9) and (12), and (iii) post any questions you have about this at Piazza.
HW8 - due at beginning of class Thursday, Oct. 27

 1.    (boost to the eigenspinors) In class we will show directly that the spinors u(p) and v(p) in (11.25) satisfy the Dirac equation with positive and negative energy, respectively. These solutions can also be obtained from (11.17) by application of a Lorentz boost. (a) What hyperbolic boost angle produces energy E and momentum p starting with zero momentum? (b) Apply the appropriate boosts to the L & R chirality components of (11.17) and show that you recover (11.25). [Hint: exp[w·σ\boldsymbol\sigma]=cosh(w) + sinh(w) (w/w)·σ\boldsymbol\sigma, where w = √w·w, as noted in class.]

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 at beginning of class Thursday, Oct. 20.

1. Nonrelativistic limit of Dirac equation

 2.  S10.1(h) (Dimensional/physical analysis)  Do ONLY part (h) of this problem, and find the Rydberg instead of the inverse Rydberg. You might remember some or all of the expressions; and you could of course use full blown dimensional analysis, after identifying on physical grounds which one of the four quantities e, m, c, ℏ  \hbar should not appear, but the point of this exercise is to practice a simplified method: in units with ℏ=c=1\hbar = c = 1, write each quantity as the appropriate power of the the electron mass times the appropriate power of the fine structure constant, and determine the latter just by using physics to infer what power of the electric charge should appear in the quantity. [If you are looking at the 4th printing or earlier, note that "m" should be ℏ\hbar/mecm_e c in that part, and "time-dependent" should be time-independent".]

3.  S11.1 (Dirac matrix identities) 

4.  S11.9 (right handed neutrino, Majorana mass & see-saw mechanism) The 4th and 5th printings differ on this problem. If you already did the problem using the 4th printing version, just indicate that on your hw. Otherwise, please use the 5th printing version, linked here. [4th printing version: Change part (d) to: "If m=100GeV, what value of M would yield a light neutrino mass 0.1 eV, and what would be the heavy neutrino mass?  For 11.9(c), by "satisfies the Klein-Gordon equation" he means with a mass-matrix that can be diagonalized. By taking linear combinations of the fields you can decouple the system into components, each of which satisfies the Klein-Gordon equation with a different mass.] Hint : To show that it satisfies the KG eqn you should use the nice identity σ(μσ¯ν)=ημν\sigma^{(\mu}\bar\sigma^{\nu)}=\eta^{\mu\nu} (if you can't read this math, try another browser - Safari and Firefox work for me).]
HW6 - due at beginning of class Thursday, Oct. 13.

1. S3.6 (Yukawa potential) Skip part (f). See Sec. 3.4 (Coulomb's law) for comparison.

2. (a) Verify that the massive spin-1 field operator, Eqn (8.64), satisfies the field equation coming from the Proca Lagrangian (8.23). (b) Evaluate the vacuum correlation function ⟨0|Aμ(x)Aν(y)|0⟩\langle 0|A_\mu(x)A_\nu(y)|0\rangle. This will involve doing the sum over the three polarizations. Express the result in a Lorentz covariant way, in terms of η_μν, p_μ (with p_0 = √p2+m2),  and the mass m. Leave the result as an integral over the momentum.

3. (Energy positivity for the Proca and Maxwell vector fields) Find the Hamiltonian density for the Proca Lagrangian (8.23), and show that the energy density is positive, apart from a total divergence term. To do this: (i) Find the momentum π^i conjugate to A_i,  and the momentum π^0 conjugate to A_0. The latter should vanish identically. (ii) Express the Hamiltonian density in terms of A_i, π^i, and A_0. (For the spatial derivatives term involving A_i use the shorter notation F_ij F_ij.) (iii) Find the equation of motion for A_0, solve it, and substitute back into the Hamiltonian density to eliminate A_0. At this point, the result should be positive, except for a total divergence. (iv) Now consider the massless case. Show that A_0 now acts just as a Lagrange multiplier, and find the constraint that its equation of motion imposes. This is the Gauss law constraint of electromagnetism. (v) Show that if you substitute this Gauss law constraint back into the Hamiltonian density, again the Hamiltonian density is positive, up to a total divergence.
HW5 - due at beginning of class Thursday, Oct. 06.

1. S5.3 (muon decay phase space) Before doing this problem read section S5.1.1. For part (a), it is probably wise to use Lorentz invariance to simplify the integral over the phase space by working in the CM frame for two of the final particles. For part (b) the stated value for the muon mass is not the most accurate. Go to the Particle Data Group website, http://pdg.lbl.gov/, to look up the current best value. (And, by the way, note that the electron neutrino that emerges from the muon decay is an antineutrino.)

2. 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. This is how it works in the Standard Model. For part (a), also draw the relevant Feynman diagram. 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.

3. Problem S7.9 (unstable particles and widths) Before doing this problem, read sections S7.4.1 and S24.1.0,2,4. For part (a), assume scalar particles. Correct the statement in (d) to read "Loops of psi will have imaginary parts if and only if √p2 is greater than twice the mass of psi.."
HW4 - due at beginning of class Thursday, Sept. 29.

The linked file consists of three problems:
1. Proof of Wick's theorem.
2. Feynman rules for ϕ3\phi^3 theory, and practice with symmetry factors.
3. Mass as an interaction.\phi^3
HW3 - due at beginning of class Thursday, Sept. 22.
HW2 - due at beginning of class Thursday, Sept. 15.

1. The LHC is currently producing 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 γγ→e+e-\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 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) Find the conjugate momenta and then the Hamiltonian and mode expansions in terms of raising and lowering operators, using the real and imaginary parts of the field. (b) Next 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.)  (c) Show that the current (1.129) is divergenceless (∂μjμ=0\partial_\mu j^\mu=0), and show that this implies that the charge Q is conserved. (d) Verify (1.130-131).
HW1 - due at beginning of class Thursday, Sept. 08.