PY751, Assignment 3, due in class Monday, 10/23/2017

Question 1: e−e− → e−e− scattering in QED

Compute the differential scattering cross section dσ/d cos θ for scattering of unpolarized

electrons in the center of mass frame. Neglect the electron mass. The matrix element

receives contributions from 2 Feynman diagrams which differ by interchange of the two

fermions in the final state (or equivalently the initial state). This interchange of fermions

implies a relative minus sign between the two diagrams which you must add by hand. The

differential cross section has singularities for 2 values of θ. Explain the physical origin of

these singularities.

Question 2: Axion-electron scattering: ae− → ae−

The axion is a hypothetical spinless (pseudo)scalar particle with a negligibly small mass.

Axions occur in many theories of beyond the standard model physics. Their couplings to

matter can be small enough for the axions to have evaded detection. For this problem,

assume that axions have no mass and couple only to electrons. The scattering process

which you explore in this problem is similar to Compton scattering (Chapter 5.5 in Peskin

& Schroeder).

1. Draw the 2 distinct Feynman diagrams corresponding to the process ae− → ae−.

Label all momenta in the 2 diagrams and note that the momenta in the 2 internal

propagators in the diagrams are different.

2. Assume that the coupling of the axion to (massive) electrons is “vectorial” ψγµψ∂µv/f

where v is the vectorial axion field, ψ is the electron field, and f � me is a constant

with dimension of mass. This coupling in the Lagrangian leads to the Feynman rule

−ikµγµ/f for an electron-electron-axion vertex where k is the axion momentum. The

sign convention is for the axion momentum to flow into the vertex. Show explicitly that

the Lorentz invariant matrix element M for vectorial axion-electron scattering vanishes

after summing both diagrams. (FYI: In general, the vanishing of this amplitude follows

from a Ward Identity. It can be used to demonstrate that the polarization of a spin 1

particle which is parallel to its four-momentum does not contribute to matrix elements

of a conserved current. In this example, the vectorial axion couples like the parallel

polarization of a spin 1 particle.) Hint: External particles are on shell. For the axion

this means that k2 = 0 and the spinors satisfy (pµγµ −m)up = 0 = up(pµγ

µ −m) and

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(pµγµ +m)vp = 0 = vp(pµγ

µ +m).

3. Consider now an axial coupling of the axion ψγµγ5ψ∂µa/f with Feynman rule

−ikµγµγ5/f . This time the invariant matrix element does not vanish. Simplify the

matrix element as much as you can before squaring, summing and averaging over spins.

4. In the “Lab frame” the electron is initially at rest. The incoming axion has frequency

ω. Determine the frequency ω′ of the outgoing axion as a function of ω and the

scattering angle θ.

5. Simplify the 2d phase space in the lab frame, obtaining an expression of the form∫d(cos θ)G(ω, cos θ)

where G is a function which you need to determine. It is convenient to write the

expression in terms of ω and ω′ instead of ω and cos θ.

6. Use your invariant matrix element and phase space to compute the differential cross

section dσ/d cos θ.

Plot the differential cross section as a function of cos θ for the special case of ω = me =

0.5 MeV, f = 1 GeV with Mathematica. Label the axes with appropriate units.

7. In the limit ω � me we have ω′/ω → 1. Calculate the total cross section in this limit.

8. Calculate the differential annihilation cross section for unpolarized e+e− → aa in the

center of mass frame. Hint: Crossing symmetry.

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Question 3: Pdfs and deep inelastic neutrino scattering

From the course web page, you can download a Mathematica file which allows you to

plot the parton distribution functions. Take a look at the example file and plot some of the

parton distribution functions. By numerically integrating the appropriate combinations of

pdfs verify the sum rules which we discussed in class. (This is to familiarize yourself with

the pdfs before using them for the problem.)

1. Write down the sum rule which follows from the fact that the charge of the proton is

+1. Verify that the pdf’s in the Mathematica file satisfy this consistency check. Note

that you may have to introduce a cutoff on your integral to get reasonable answers.

Why is that?

2. Derive the differential cross sections

dσ̂

dQ2dx(νLdL → µ−LuL) ,

dσ̂

dQ2dx(νLuR → µ−LdR) .

Here the L/R subscripts refer to helicities of the fermions. For your calculation you

may assume that all fermions are massless so that fermion helicities imply definite

chiralities. The calculation is very similar to the QED calculation and you should feel

free to recycle the QED results. The massive W boson propagator is −igµν/(p2−m2W )

plus terms which do not contribute. (These terms are proportional to pµpν which

vanishes once contracted with the fermion currents because of the Ward Identity). For

the W boson coupling to fermions use igγµPL/√

2 where g is the W gauge coupling

and PL = (1− γ5)/2 is a chiral projection operator.

3. Using your partonic cross sections derive the deep inelastic neutrino-proton scattering

cross sectiondσ

dQ2(νLp

+ → µ−LX)

and plot dσ/dQ2 for the range 0 < Q2 < s for√s = 100 GeV. Use αweak ≡ g2/4π =

1/29.5 and use picobarn/GeV 2 as units to plot the differential cross section.

Note that the cross section assumes that the incoming beam of neutrinos is purely

left-handed, can you explain why it is reasonable to assume that neutrino beams in

experiments are polarized like this?

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