Feynman Diagrams and Forces Between Particles
PARTICLES

Photo: Courtesy of The Archives,California Institute of Technology

The Basic Idea
If the electromagnetic field is defined in terms of the force on a charged particle, then it is tempting to say that the field itself consists of photons which cause a force on a charged particle by being absorbed by it or simply colliding with it - as in the Photo-electric effect.

The electric repulsion between two electrons could then be understood as follows:

One electron emits a photon and recoils; the second electron absorbs the photon and acquires its momentum. This simple process can be pictured with a Feynman diagram:

Clearly the recoil of the first electron and the impact of the second electron with the photon drive the electrons away from each other. So much for repulsive forces. How can attraction be represented in this way ?

 ELECTROMAGNETIC SCATTERING Here two charged particles approach each other and scatter by exchanging a virtual photon. Animation reproduced by kind permission of (c) J. Eric Slone www.FeynmanOnline.com.

The uncertainty principle makes this possible . The attraction between an electron and a positron may be described as follows: the electron emits a photon with momentum directed away from the positron and thus recoils towards the positron. This entails a degree of definiteness in the momentum of the photon. There must be a corresponding uncertainty in its position - it could be on the other side of the positron so that it can hit it and knock it towards the electron .In the diagram below the wavy line represents the SAME photon.

Heisenberg's uncertainty principle is a quantitative statement of these ideas .It asserts that it is impossible to specify both the position snd the momentum of a particle.If the position is determined in the finite interval Dx and the momentum in the intervalDp then these intervals obey the inequality

 Dx.Dp > h / 4p

so that they cannot be both reduced to zero.

The uncertainty principle can also be regarded as an expression of the conflict between wavelike and particle-like properties. If we use de Broglie's relation

 p = h /l

and express momentum in terms of wavelength then the uncertainty principle becomes

Dx.Dk > 1 / 4p

where k = l-1-is the wave number.We thus have the impossibility of describing simultaneously particle and wavelike properties.

Thus it is possible for a pair of particles to exchange a photon whose momentum is in either direction along the line joining the particles. Which direction depends on whether the particles have the same or opposite electric charges.

Electromagnetic Events

The basic events in Electromagnetic interactions are the emission and absorption of photons by charged particles. These events are represented by a single vertex in a Feynman diagram. The strength of the force is given by the intensity of the field which is proportional to the number of photons; hence the greater the force on a particle the more photons it will emit or absorb. We could say that :

'The charge on a particle is proportional to the
probability that it will emit or absorb a photon.'

The basic electromagnetic events are thus :

• 1) Mutual repulsion of an electron/electron or positron/positron pair.
• 2) Mutual attraction of electron and a positron.
• 3) Annihilation of an electron and positron to form a photon.
• 4) Creation of an electron positron pair from a photon.
The Feynman diagrams for the last two are shown below:

Other Fundamental Forces

A description of this type applies to each of the fundamental forces - strong,weak, electromagnetic and gravitational. Each has its own field quantum - like the photon. These 'Force Particles' are termed Vector Bosons.

.

The weak force is associated with three bosons, called the W+ , W- and Zo . The W particles carry the electric charge and when they are emitted or absorbed by a particle they change its identity. The exchange of W particles is responsible for processes such as beta decay :
.

 n ----> p + e- + n Beta Decay Equation

As the strength of a force is proportional to the probability of emitting a boson, the weakness of the weak force implies that this decay is a rare event and that the neutron is fairly stable.

Quarks and Leptons

Since the proton and neutron are not elementary particles, the process of beta decay must be analysed at the quark level. As the quark composition of a neutron is ( udd ) and a proton is ( uud ) then a neutron transforms into a proton if a down quark becomes an up quark,the Basic Event being the vertex

d -> u + W-
.Emission of a W- or a W+ thus changes the ' flavour' of the quark.Vertices involving the Zo particle leaves a quark flavour unchanged - it is similar in some ways to the photon. The Zo processes do exert a force on the neutrino - this was first demonstrated experimentally in 1974. The vector bosons themselves were observed directly in 1983. The following diagrams show the basic weak events at the quark level:

This means that the following diagram could also be used for showing beta decay at the quark level :

The range of a force may be understood in terms of exchange of field quanta too.The electromagnetic force, which has infinite range is due to the exchange of photons which travel at light speed and thus have zero rest mass in accordance with Special Relativity. Bosons with non-zero rest mass , m , travelling at less than light speed would not be able to move as far as a photon would before being absorbed by another particle. A simple formula for their range is given by :

 R = h / 2pcm YUKAWA FORMULA

- this is an inverse relation between distance and mass - characteristic of quantum phenomena.

Some Questions To Think About Or Try

1) Explain briefly how both attractive and repulsive forces may be explained by the idea of particle exchange.

2) What is a ' Field Quantum ' ? Give some examples.

3) Draw a Feynman diagram for Proton decay according to the scheme:

 p ----> n + e+ + n

4) In a neutron star gravitational collapse causes valence electrons to combine with protons. Write a decay equation for this process and draw a Feynman diagram. What would be the mass of 1 cm3 of closely packed neutrons ? (take the radius of a hydrogen nucleus to be 4 x 10-15 m.

5) (a) If the Uncertainty Principle is stated as DpDx > = h / 2p ,calculate the uncertainty in an electron's momentum if it is to be confined to a region the size of the nucleus.

(b) If particle energy is given by the relativistic invariant

 E = ( m02c4 + p2c2 )1/2

produce a reasoned argument that the Kinetic energy of such a particle is approximately given by K = pc. Evaluate this and show that the KE of such electrons must exceed 20 MeV. Beta decay electrons never have more than a fraction of this energy.Comment.

6) If the radius of a hydrogen atom is 5.3 x 10 -11 m, produce a quantitative argument to show that electrons can appear as free particles inside the confines of an atomic radius.(The KE of an electron in the ground state of Hydrogen is 13.6eV)

7) Calculate approximate rest masses for Intermediate Vector Bosons (W,Z particles) and Gluons, using the Yukawa formula.

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