PHYSICS 663 QUANTUM FIELD THEORY II PROBLEM SET 3 Due in Class: Monday, October 22, 2018 Problem 1) /25 Points Problem 2) /25 Po

PHYSICS 663 QUANTUM FIELD THEORY II PROBLEM SET 3 Due in class: Monday, October 22, 2018

Problem 1) /25 points

Problem 2) /25 points

Problem 3) /25 points

Problem 4) /25 points

Problem 5) /25 points

TOTAL: /125 points

1 1. Consider the Lagrangian for the scalar Yukawa model with fermions 1 1 1 λ L = − ∂ σ∂µσ − M 2σ2 − ψ¯(γµ ∂ + m)ψ − gψψσ¯ − σ4 (0.1) 2 µ 2 i µ 4! describing an hermitian scalar ﬁeld σ and a Dirac ﬁeld ψ interacting via a Yukawa interaction with real coupling g as well the scalar self interaction with real coupling λ.

a) Compute the contribution of the fermion loop to the 1-loop renormalized scalar eﬀective potential in the massless fermion limit (m = 0).

b) Using the computation of part (a), re-derive the fermion 1-loop contribution to the beta function for λ, the σ4 coupling.

2. a) Consider√ the electron-positron annihilation of total center of mass frame energy s into a charged scalar particle φ− with charge Qse and mass mφ and + − its antiparticle φ+ of charge −Qse and mass mφ: e e → φ+φ−. Here s is 2 µ µ the Mandelstam variable s = −(p+ + p−) with p+ and p− being the energy 4-momentum of the positron and electron respectively. To leading order in the electromagnetic ﬁne structure constant and neglecting the mass of the electron and positron, but keeping the scalar mass mφ, write the T matrix element in Rξ gauge and show that it is independent of ξ.

Compute the electron and positron spin averaged angular diﬀerential cross section in the center of mass frame for measuring the charge Qse scalar as a function of the scattering angle between it and the incident electron. Compute the leading order total cross section for this process.

b) Consider the√ process electron-positron annihilation of total center of mass frame energy s into a muon-antimuon pair: e+e− → µ+µ−. The muon has the same electric charge, e, as the electron and its antiparticle, the antimuon, has the same charge as the positron. The interaction of the muon and antimuon with the electromagnetic field is identical to that of the electron and positron with the electromagnetic field. To leading order in the electromagnetic fine structure constant and neglecting the mass of the electron and positron, but keeping the muon and antimuon mass mµ, express the T matrix element in ξ gauge and show that it is independent of ξ.

Compute the electron and positron spin averaged angular diﬀerential cross section in the center of mass frame for measuring the muon as a function of the scattering angle between the incident electron and the outgoing muon. Here neither the muon nor antimuon spin is detected. Compute the leading order total cross section for this process.

2 c) Compute the ratio of these leading order total cross sections

+ − σe e →φ+φ− Rs = (0.2) σe+e−→µ+µ−

What is this ratio in the limit when mµ → 0 and mφ → 0?

− + Let f (f ) denotes a fermion (anti-fermion) of charge Qf e (−Qf e) and mass mf which couple to the photon analogously to the electron and muon coupling. Compute the ratio of leading order total cross sections

σe+e−→f +f − Rf = (0.3) σe+e−→µ+µ−

in the me = 0 limit. What is this ratio in the limit when mµ → 0 and mf → 0?

d) This time, consider√ the process muon-antimuon annihilation of total center of mass frame energy s into a electron-positron pair: µ+µ− → e+e−. Compute the muon and anti-muon spin averaged angular diﬀerential cross section in the center of mass frame for measuring the electron as a function of the scattering angle between the incident muon and the outgoing electron. Here neither the electron nor positron spin is detected. Once again, you can neglect the mass of the electron, but maintain the muon mass.

Compute the leading order total cross section for this process.

Compute the ratio of total cross sections

σ + − + − e e →µ µ (0.4) σµ+µ−→e+e− as a function of the Mandelstam variable s and the muon mass.

3. This problem addresses the 1-loop ﬁeld anomalous dimensions and beta functions in the O(2) linear sigma model with fermions.

a) Construct the most general Lorentz and parity invariant Lagrangian com- posed of all possible operators with mass dimension less than or equal to four in d = 4 space-time dimensions constructed out of a bare hermitian scalar field, σ0(x), a bare hermitian pseudoscalar field, π0(x), and a bare Dirac field, ψ0(x) ¯ and its conjugate, ψ0 which is also invariant under the O(2) transformations

σ0(x) → cos θσ0(x) + sin θπ0(x)

3 π0(x) → − sin θσ0(x) + cos θπ0(x) θ iγ5 ψ(x) → e 2 ψ0(x) (0.5) where θ is a real space-time independent constant which parametrizes the trans- formation.

b) Renormalize the model by rescaling the fields and coupling parameters and shifting the bare scalar squared mass parameter. Introduce an arbitrary mass scale µ and include in the rescaling of the couplings appropriate with powers of µ such that the renormalized parameters carry the same mass dimension in d dimensions that the associated bare parameters carry in d = 4 dimensions. Rewrite the Lagrangian you constructed in part (a) in terms of the renormalized fields and parameters in the form L + LCT where L has the same form as the Lagrangian of part (a) except that the fields and couplings are renormalized with the renormalized couplings multiplied by the power of µ appearing in the rescaling. Identify the counter-term Lagrangian LCT .

c) Using dimensional regularization and minimal subtraction, compute the 1- loop anomalous dimensions of the σ, π and ψ ﬁelds.

d) Using dimensional regularization and minimal subtraction, compute the 1- loop β functions for the Yukawa coupling g and the spin-zero self coupling λ.

You can lift appropriate results from the calculation done in class and in the notes for the scalar Yukawa model. In particular, you do not need to evaluate any momentum space integrals which were not already encountered.

4. For weak interaction tree level processes at energy scales much less than the mass of the W vector boson, it is a good approximation to use the local 4-Fermi interaction.

One such process is the decay of the muon. The muon is a spin 1/2 particle of mass mµ '105 MeV whose interactions have the exact same structure as to those of the electron. Since the mass of the muon is much less than the that of the W boson which is MW ' 80 GeV, we can well approximate the leading order weak interaction controlled muon decay by the local 4-Fermi Lagrangian

GF ¯ µ ¯ L4−F ermi = √ [ψe(x)γ (1 − γ5)ψν (x)][ψν (x)γµ(1 − γ5)ψµ(x)] + (h.c.) 2 e µ (0.6)

Here ψe, ψνe , ψµ, ψνµ are the spin 1/2 electron, electron-neutrino, muon, and muon-neutrino ﬁelds respectively while GF is a constant (the Fermi constant)

4 which carries mass dimension negative two. Hence this model is non-renormalizable but nonetheless forms a very good approximate eﬀective theory to extract the leading order results for weak interaction processes such as muon decay which occur at energies much less than the W mass.

2 − − Using the 4-Fermi interaction, compute the O(GF ) decay rate, Γ(µ → e νµν¯e), of the rest frame unpolarized muon to an electron, a muon-neutrino and an anti electron-neutrino as a function of GF and the muon mass, mµ, in the limit where the electron, the muon-neutrino and the anti electron-neutrino are all taken as massless and where none of the ﬁnal state fermion spins are detected.

Using the experimentally measured muon lifetime 1 τµ = − − Γ(µ → e ν¯eνµ) ' 2.2 × 10−6 sec (0.7)

along with the value of the muon mass, extract the numerical value of GF . 5. Consider the photon pair production process e+e− → γγ where the electron, − + µ µ e , and the positron, e , carry 4-momentum p1 and p2 respectively and the 2 µ µ photons carry 4-momentum q1 , q2

a) Draw the Feynman graphs which contribute to the T matrix element for this process to O(e2)

b) Using the Feynman rules, write the T matrix element corresponding to the graphs of part (a) working in Coulomb gauge

c) Show that the result of part (b) is gauge invariant.

d) In the e+e− center of mass frame, calculate the e+, e− spin averaged angular diﬀerential cross for detecting one of the photons while neither photon polar- ization is detected.

HINT: It could prove convenient to relate the various traces encountered here to those evaluated in class and in the notes for Compton scattering.

e) Compute the e+, e− spin averaged total cross section for e+e− → γγ with the ﬁnal photon polarizations undetected. Find the leading behavior in both the low energy limit, |~p| << m, and the high energy limit, |~p| >> m.