In order to investigate the properties of elementary particles, physicists per- form scattering experiments. The most important quantity of a scattering process is the cross section. There exist two important tasks for quantum field theory:
(a) hνin
hνout
e−
e− -
3
s
(b) pin
pout
e− pel
- ϑ
3
s
Fig. 2.9. Compton effect
(i) the computation of cross sectionsσ(scattering states), and
(ii) the computation of the rest energies of elementary particles (bound states).
It turns out that the task (i) is much easier to handle than (ii), since we can use the methods of perturbation theory pictured by Feynman diagrams. The ultimate, extremely ambitious goal is the creation of a theory which predicts the existence and properties (e.g., the masses and the magnetic moments) of all fundamental particles and forces in nature.
Compton scattering.In each scattering process, physicists measure the crucial cross sectionσ.In 1929, for the cross section of the Compton scattering of light at crystals, Klein and Nishina computed the formula
σ=
S2
f(ϑ)dΩ with
f(ϑ) :=
β2(1 + cos2ϑ) +β(1−β)2
ãλ2eα2
8π2 (2.39)
whereβ :=λin/λout.Here, we use the following notation:
• me mass of the electron,−echarge of the electron,
• λin(resp.λout) wave length of the incoming (resp. outgoing) photon,
• hPlanck’s quantum of action,
• Compton wave length of the electron λe:= h
mec = 10−12m,
• dimensionless fine structure constant in quantum electrodynamics α:= e2
4πε0c = 1 137.04 whereε0 is the electric field constant of a vacuum.
-- - jin
y s
S2R
n jout
ϑ
Fig. 2.10.Cross section for the Compton scattering of light
We integrate over the unit sphere S2 with the scattering angle ϑ and the surface element
dΩ= cosϑ dϑdϕ
where the geographic latitudeϑand the geographic longitudeϕvary in the interval [−π/2, π/2] and [−π, π], respectively. The differential
dσ:=f(ϑ)dΩ and the integralσ =
S2dσ are called the differential cross section and the total cross section, respectively.
The famous Klein–Nishina formula (2.39) shows that Compton scatter- ing is a second-order effect with respect to the fine structure constant α.
Explicitly,
σ= λ2eα2 2π
4 3 −8γ
3 +104γ2 15 +. . .
, γ:= λe λin.
If the energy of the incoming photons is sufficiently low,γ1, then we get the classical formula
σ= 2λ2eα2
3π = 0.665ã10−28m2
which was obtained by Joseph John Thomson at the end of the 19th century.23 Observe that this classical approximation formula does not depend on the scattering angleϑ.Physicists measure cross sections in barns. By definition, 1 barn = 10−28m2.
Physical interpretation of the cross section.Let us now discuss the physical meaning of the cross section σ. Consider the situation pictured in Figure 2.10. We choose a sufficiently large sphereS2R of radiusR about the scattering center. Let n and ∆S be the outer normal unit vector and the surface element of the sphereS2R, respectively. The incoming photon stream can be described by the energy current density vector
23This can be found in the standard textbook on electrodynamics by Jackson (1975).
jin=invin
wherein andvindenote the energy density and the velocity vector, respec- tively. In a typical experiment, the incoming photon stream is homogeneous.
Therefore, we assume that the vectorjin is constant. By the scattering pro- cess, we obtain the outgoing energy current density vector field
jout=outvout
which depends on the position vectorx, but not on time. Now to the point.
The decisive quantity
E= (t1−t0)
S2R
joutndS (2.40)
is equal to the amount of outgoing energy that flows through the sphereS2R
during the time interval [t0, t1].This amount of energy can be measured by experiment. Naturally enough,E is proportional to ||jin|| (incoming energy flow). The coefficient of proportionalityσdefined by
E =σ(t1−t0)||jin|| (2.41) has the physical dimension of area (m2). Therefore,σis called the total cross section of the scattering process. We want to show that there exists a function f such that
σ=
S2R
f(ϕ, ϑ)dΩ. (2.42)
In fact, from (2.40) and (2.41) we get σ=
S2R
joutn
||jin|| dS.
Naturally enough, the outgoing energyEdoes not depend on the choice of the radiusRif the radius is sufficiently large. Because of the equalitydS=R2dΩ, we assume that the productR2jout does not depend onR, and hence
(joutn)(P)
||jin|| =f(ϕ, ϑ), P∈S2R
where ϕand ϑ are the geographic longitude and the geographic latitude of the pointP, respectively. This implies the desired formula (2.42).
Concerning Rutherford’s experiment on the scattering of α-particles at protons (Fig. 2.8 on page 114), observe that it does not make any sense to consider the total cross section in this case, since the integral
S2Rf dΩ
is divergent. Therefore, we need a localized version∆σ of the cross section called the differential cross section. The idea is to consider a regular subset S of the sphereS2R that surrounds the pointP ∈ S2R. We now measure the scattered energy flow that passes through the partS of the sphere S2R.The quantity
E(S) := (t1−t0)
SjoutndS
is equal to the amount of outgoing energy that flows through the partS of the sphereS2R during the time interval [t0, t1].Similarly as above, we define the cross sectionσ(S) with respect toS by the relation
E(S) =σ(S)(t1−t0)||jin||. Hence σ(S) =
Sf(ϕ, ϑ)dΩ. Contracting the setS to the pointP ∈S2R, we define
dσ
dΩ(P) := lim
S→P
σ(S)
area (S) =f(ϕ, ϑ).
Mnemonically, it is convenient to write σ(S) =
S
dσ where dσ:=f(ϕ, ϑ)dΩ.
Compton effect and quantum electrodynamics.The Klein–Nishina formula (2.39) for the cross section of the Compton effect can be obtained from quantum electrodynamics. This highlight of quantum electrodynamics will be thoroughly studied in Volume II. Quantum electrodynamics represents a quantum field theory which describes the interactions between electrons, positrons, and photons. This is a perturbation theory with respect to the fine structure constant
α= 1
137.04.
The smallness of the fine structure constant is responsible for the great success of perturbation theory in quantum electrodynamics. We will show in Volume II that the Klein–Nishina formula follows from using second-order Feynman diagrams along with time-consuming computations based on Dirac matrices.
Observe that the Klein–Nishina formula does not depend on the polarization of the photons. In fact, this formula averages over the polarizations of the incoming photons and sums over the polarizations of the outgoing photons.
General cross sections in elementary particle physics. The def- inition of total cross section σ and differential cross sectiondσ introduced in (2.41ff) above applies to all types of scattering processes in physics. In particle accelerators, one defines cross sections with respect to the particle number. In this case, the incoming particle stream is described by the particle number current density vector,
jin=invin
where in and vin denote the particle density and the velocity vector, re- spectively. In definition (2.41) we then have to replace the energyE by the particle number N.