Particle Physics WS2015/16
Prof. A. Schöning
Sheet 9, to be returned 22nd Dec 2016
Group:
up to three names:
Question:
1
2
3
4
Total
Points:
10
10
10
10
40
Score:
Question 1 (10 points): Bhabha scattering
The unpolarized differential cross section for Bhabha scattering (e+ e− → e+ e− ) can be calculated
in leading order QED.
(a) Which diagrams contribute to the process? Write down the matrix elements for each diagram
(not the spin averaged ones!).
The resulting matrix element is: |M|2 /2e4 =
s2 +u2
t2
+
2u2
ts
+
u2 +t2
s2
(b) What is the origin of each of the three terms?
(c) Re-write all three terms in dependence of the centre-of-mass scattering angle θ∗ , neglect the
electron mass.
(d) What is the value/behaviour of the contributions for θ∗ = 0, π/2, π? Sketch (in one plot and to
scale) the absolute value of the individual contributions as a function of θ∗ .
Question 2 (10 points): Mott Cross Section
The differential cross section for electron-nucleon scattering with the nucleon being at rest (labsystem) and no nucleon spin (Jnucl = 0) is given by the Mott formula:
4πα2 E 0
θ
dσ
cos2
=
2
dQ
Q4 E
2
.
(1)
Here E and E 0 are the energy of the electron before and after the scattering process, respectively,
and θ is the scattering angle.
(a) Calculate the scattered electron energy E 0 in the lab system as a function of the scattering angle
θ, with the nucleon being at rest and assuming the nucleon mass to be M .
dσ
(b) Express equation (1) in terms of the differential cross dΩ
, with Ω being the solid angle of
the scattered electron. Express the negative four-momentum square Q2 = −q 2 by the term
containing the scattering angle and insert above result for E 0 .
(c) Discuss the result and compare it to the Rutherford scattering formula. What are the differences
and what are their physical origins?
(d) Why does the differential cross section vanish for back-scattered electrons θ → π?
Question 3 (10 points): Nuclear form-factors
Form-factors F ~q 2 of nuclei are determined in electron scattering experiments via the ratio of
the measured cross-section and the theoretical prediction for point-like charge distribution (Mott
cross-section):
2
dσ
dσ =
F ~q 2 dΩExp
dΩMott
The measured cross-section of the electron scattering off the 16 O nuclei
p normalised to the Mott crosssection dσExp / dσMott as a function of the momentum transfer q = |~q 2 | is shown in Figure 1a.
(a) Write down the electric charge distribution for the point-like particle. Following the nuclear
form-factor definition write down the corresponding form-factor.
(b) Write down the electric charge distribution inside the 16 O nucleus, assuming a spherically homogeneous charge distribution with a radial extent R. Following the nuclear form-factor definition
write down the corresponding nuclear form-factor (in the integral form).
(c) Use computer numerical tools to get the analytic form of the form-factor in (b) . Plot F ~q 2
and read off the first 3 minima as a function of |qR|.
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Particle Physics WS2015/16
Prof. A. Schöning
Sheet 9, to be returned 22nd Dec 2016
(d) Using Figure 1a and plot from (c) estimate the R of
|q| = 1.5 fm.
16
Question 3 continued. . .
O. Take that the first minimum is at
How do the positions of the minima and the overall scale of the experimental differential crosssection change qualitatively
(e) if the electrons are scattered off a heavier isotope?
(f) if the electrons are scattered off a nucleus of the same mass but with higher charge Z?
(g) if the energy of the incoming electrons (Ee ) is reduced by a factor of 2?
Question 4 (10 points): Form Factors
Elastic e− p → e− p scattering via the exchange of a single photon can be described by the Rosenbluth
formula
E3 G2E + τ G2M
α2
dσ
2 θ
2 θ
2
=
cos
+ 2τ GM sin
.
dΩ
1+τ
2
2
4E12 sin4 ( θ2 ) E1
Here E1 and E3 are the energy of the incoming and outgoing electron, respectively; τ = Q2 /4m2p
and GE (Q2 ) (GM (Q2 )) is the electric (magnetic) form factor of the proton.
dσ
, which is the Mott scattering cross section
(a) Re-express the Rosenbluth formula in terms of dΩ
0
of
dσ
α2
E3
θ
=
cos2 .
2
dΩ 0
2
4E1 sin4 ( θ2 ) E1
(b) The ratio of
dσ
dΩ
dσ
/
dΩ 0
is linear in tan2 (θ/2). Express the gradient and the intersect as functions of GE (Q2 ) and
GM (Q2 ).
(c) Obtain values for GE (0.292 GeV2 ) and GM (0.292 GeV2 ) from Figure 1b.
(a) Normalised differential cross-section dσExp / dσMott (b) Low energy e− p → e− p elastic scatfor the electron-16 O scattering as a function of |q|.
tering data.
Figure 1
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