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## Scattering Processes of Elementary Particles, an Introduction to Feynman Diagrams and to the Feynman Calculus

Walter Pfeifer
Switzerland
20 pages, 9 line figures

2011

### Preface

Feynman’s diagrams are handy aids to deal with quantum electrodynamic phenomena as scattering, decay and pair annihilation/production.
To each event one or more diagrams can be assigned which are the starting points for formulating the so-called invariant matrix element. This quantity is the main part of the cross section of the mentioned particle process.
The procedure with Feynman diagrams is recipe like and a help in dealing with particle phenomena.
In this publication we restrict ourselves to scattering processes. We will compare the results with the calculations in reference [1] (Pfeifer, W., 2009) and consider the same agreements as in [1] (see the preface in [1]).

### Contents

Preface 3
Contents 4
1 Cross section formulas 5
1.1 The differential cross section of electrons scattered off electrons 5
1.2 The cross section for the scattering of photons off free electrons 7
2 Feynman diagrams in quantum electrodynamics 9
3 The Feynman calculus 12
3.1 The Feynman rules 12
3.2 Electron-electron scattering 13
3.3 Photon-electron scattering 15
3.4 Other scattering processes 16
Epilogue 18
References 19
Index 20

### 1 Cross section formulas

In this chapter we write down the formulas for scattering cross sections which have been developed in reference [1], sections 6.1 and 6.2. The quantities and the symbols are described in detail. The formulas will be used to check the expressions obtained by the Feynman calculus given in chapter 3.

### 1.1 The differential cross section of electrons scattered off electrons

From reference [1], equation (6.1.40), we copy the following expression for the scattering cross section given in the centre of mass system

(1.1.1)

This expression gives the transition rate  from state  to the state  per unit incoming flux with an outgoing electron moving within a solid angle   . The cross section has the dimension [area].

Equation 1.1.1 contains the following quantities

: differential solid angle of the scattered particle,

m, e          : mass and charge of the electron,

E, E’        : relativistic energy of incoming/outgoing electrons,

: velocity of the incoming electron,

c             : velocity of light,

: three-dimensional momentum of the scattered electron,

: modified Planck constant ,

: invariant matrix element ; it is also named “invariant amplitude” (Lorenz

invariant) and contains the dynamics of the process. It determines the

transition probability from the initial state of both primary particles to the

final state,

: relativistic momenta  of the incoming/outgoing electron,

: relativistic momenta of the target electron before/after the reaction,

: ordinal numbers of the reacting electron specifying the direction of

spin,

: ordinal numbers of the target electron.

In equation (1.1.1) the normalizing factor /  has been set to 1 according to reference [1], equation (6.1.81).

The part  in equation (1.1.1) is named “phase space ”.

Due to reference [1], equation (6.1.19), the invariant matrix element reads

(1.1.2)

This formula contains the following additional quantities

: amplitude of the fermion state r,

Due to reference [1], equation (3.2.19), it is a dimensionless four

spinor, in fact a column spinor,

: line spinor containing the conjugate complex elements of .

: a matrix of the Dirac equation . These matrices are given in [1],

equation (3.1.2),

: element of the metric tensor    with

(1.1.3)

: Fourier transformed propagator  of electron-electron scattering,

(see [1], equation (5.6.9)),  (1.1.4)

: real infinitesimal quantity. In the following it will be neglected,

: electric field constant , . Using the

fine structure constant ( in the system of units SI it reads

) we can write in eq. (1.1.4)  (1.1.5)

(1.1.6)

With the aid of the relations (1.1.2) up to (1.1.6) we obtain

(1.1.7)

### 1.2 The cross section for the scattering of photons off free electrons

According to reference [1], equation (6.2.20), the differential cross section for photon scattering reads

(1.2.1)

The following quantities appear

: differential solid angle of scattered photons,

: energy of the incoming/outgoing photon. The relation

(1.2.2)

holds (cf [1], equation (4.4.3))

: relativistic momentum of the incoming/outgoing photon,

: directional indices of the orthonormalized unit vector ,

: scattering angle.

The invariant matrix element is defined analogically as in section 1.1.

In reference [1], equation (6.2.2), it is given as follows (the factor may be moved in the formula)

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