Chapter 1 Interactions And Field
1.2 The Electromagnetic Interaction
Interactions and fields
Particle Physics
The Electromagnetic Interaction
Fig 1
The electromagnetic interaction involves all particles with an electric charge and the uncharged photon, which is the carrier of the electromagnetic field. The electromagnetic force is responsible for the bound system of atoms and for the binding of atoms together to form molecules. The strength of the electromagnetic force decreases as the square of the distance between interacting particles, and its range is, in principle, unlimited. The wave nature of electromagnetic radiation is characterized by oscillating electric and magnetic fields. Dayalbagh Educational Institute
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Interactions and fields Particle Physics In a classical view of the exchange of energy between two atoms, an excited atom radiates electromagnetic waves, which are absorbed by another atom. In a quantum description of electromagnetic radiation, photons are associated with the electromagnetic field. The exchange of energy between two atoms, in quantum physics, is interpreted as the emission of a photon by the excited atom and the absorption of the photon by the second atom. We extend the model of photons connected with a moving electromagnetic field by saying that a stationary field has a particle-like nature. We consider the interaction of a stationary electric field produced by charge q with another charge q', in terms of the interaction of q' with the photons that make up the stationary electric field of q. With this description , we replace the Coulomb action through a distance between two electrons as a force resulting from the transfer of momentum when there is an exchange of photons between electrons. Because of its electric field, each electron is a potential emitter or absorber of photons. The probability of emission or absorption of a photons is taken proportional to the energy stored in the electric field. In this model of electromagnetic interaction, photons are “tossed� back and forth between electrons. Dayalbagh Educational Institute
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Fig 2
The recoil of the electrons upon ejection and reception of photons is equivalent to a repulsive force.
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Basic EM Processes In Fig 12(a) an electron (e-) emits a photon (Îł). This process does not simultaneously conserve energy and momentum. You can see this by looking at the process in a reference frame in which the electron is initially at rest.
Fig 12(a)
In this frame the initial energy is the rest-mass m0c
2
of the electron. To eject a photon,the electron must end up with less energy than its rest mass. But the electron still exists hence the process is never observed, which makes it virtual. Further Interpretation of this Feynman diagram
Fig i(a): A free electron
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Interactions and fields Particle Physics This Feynman diagram, represents the basic electromagnetic process e- → e- γ and the amplitude for this process is proportional to the charge on the electron or more precisely to √ α where α, the finestructure constant, is given by
This is a small number and indicates electromagnetic interactions are relatively weak.
that
In Feynman diagrams it is possible to change the direction of the arrows provided that the particle is replaced by its antiparticle.
Fig i(b): Photon emission
Thus figure i(c) represents the reverse process e- γ → e-, corresponding to the absorption of a photon by the incoming electron.
(The photon and its antiparticle are indistinguishable.)
Fig i(c): Photon absorption Dayalbagh Educational Institute
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Interactions and fields Particle Physics Similarly, by reversing the arrow on the incoming electron and changing it to the antiparticle, the positron, we arrive at figure i(d), which represents electron-positron pair production.
Fig: i(d) Pair production
(e) e + e- annihilation
Finally, if all the arrows in figure i(d) are reverset we get figure i(e) which corresponds to electron positron annihilation. The coupling is the same in each diagram. The mathematical treatment the interaction is described by a Lagrangian L and if each of the diagrams in figure represents one term in the Lagrangian. On the understanding that diagrams i(c), i(d) and i(e) follow i(b) only the latter is included in the Lagrangian.
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In Fig 12(b) a photon is emitted by one electron and absorbed by another electron. The overall process represented by this reaction is the scattering of one electron by another electron through the electromagnetic interaction.
Fig 12 (b)
Further Interpretation of this Feynman diagram The Feynman diagram shown in figure ii(a) which represents one contribution to the elastic scattering of one electron by another via virtual photon exchange. The coupling at the second vertex is again √ Îą.
Fig (a): A first- order process Dayalbagh Educational Institute
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Interactions and fields Particle Physics The amplitude for this process is essentially the product of three factors: (i) the amplitude that the incoming electron emits a photon at vertex 1, (ii) an amplitude that the photon propagates from the space-time point 1 to the space time point vertex 2, the photon ‘propagator’, and (iii) the amplitude that the photon is absorbed by the second incoming electron at vertex 2. The amplitude for the overall process is thus proportional to α and is described as a first-order process. Other more complicated diagrams can contribute to the process e-e- → e-e-; for instance by combining figure i(b) with figure i(d) one gets the second-order diagram, with an amplitude proportion to α2, shown in figure ii(b)
Fig i(b): Photon emission
(d) Pair production
Each vertex contributes a factor √ α to the amplitude. Dayalbagh Educational Institute
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Interactions and fields Particle Physics Proceeding in this way one can construct diagrams proportional to α3 , α4 and so on, and indeed the total amplitude for the process e-e- → e-e- is an infinite sum of diagrams of increasing order, and hence decreasing amplitude, because α «1. Perturbation theory can be used to calculate the total amplitude, and hence the cross-section for the process e-e- → e-e-, to any degree of accuracy required.
Fig ii(b): A second- order process
Figure 12(c) represents the interaction of an electron and proton (p) by the emission and absorption of a virtual photon.
Fig 12 (c) Dayalbagh Educational Institute
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Interactions and fields Particle Physics While the overall reaction conserves both energy and momentum, these quantities are not conserved during the short time the photon exists. It is the unobservability of the virtual photon that allows the nonconservation of energy and momentum during this time interval. If the uncertainty in energy ΔE and uncertainty in momentum Δpx are consistent with Heisenberg's principle of indeterminacy, the violation of the conservation laws is in agreement with quantum theory. Recall that the uncertainty in momentum is related to the uncertainty in position Δx through
and the uncertainty in energy to the uncertainty in the Δt by
If the uncertainty in energy ΔE and uncertainty in momentum Δpx are consistent with Heisenberg's principle of indeterminacy, the violation of the conservation laws is in agreement with quantum theory.
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Interactions and fields Particle Physics Where ħ is Planck's constant divided by 2π. If the process takes place in a short enough space and time interval the" borrowed " energy ΔE and momentum Δpx do no violate quantum theory. The quantity Δt, Fig 12(b) is the time interval between the emission and absorption of the virtual photons. Since photon energies ΔE range from zero to infinity depending on their frequency, the time interval between emission and absorption Δt ≥ ħ/ΔE varies from very long intervals to very short intervals.
Fig 12(d) is the worldline diagram for annihilation between electron and a positron(e+), the antiparticle of the electron. In Fig 12(d) the electron emits a photon, is scattered as a virtual electron, and then positron-electron annihilation occurs.
Fig 12 (d)
The overall process conserves energy and momentum, and both photons are observed.
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Charge is conserved in this process because the net charge of the electron-positron system before the interaction (+1e – 1e), or zero, and the final charge of the system is zero since photons carry no charge. The intersection of the world-lines of two particles is called a vertex. In fig 12(a) there is one vertex; in Figures 12(b) and (c) there are two vertices. Using such vertices, more complicated processes can be diagrammed. With the use of quantum electrodynamics, a very sophisticated theory that unites quantum theory and classical electrodynamic, it can be shown that the relative probability of a final state occurring through a virtual process depends on the number of vertices between the initial and final state. For example, the probability of the emission of a virtual photon by an electron depends on the energy associated with the electron's electric field. Since the energy of the field is proportional to the square of the electric field (E= kee/r2), the probability of the emission of a virtual photon is proportional to e2, where e is the charge for the electron (Fig 12(a). The probability of the absorption of a virtual photon by an electron is proportional to e2 and the probability of emission and absorption is then probability to e4 (Fig 12(b)). Dayalbagh Educational Institute
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If we take as one the proportional of a strong interaction occurring (see below), the probability of an electromagnetic interaction taking place is 10 -2 as great. On this scale, the probability of emission and absorption is 10-4. The "appearance" of the electron in more complicated virtual process occurs with smaller probabilities.
Quantum Electrodynamics (QED) QED is the quantum theory of electromagnetic interactions. Classical electromagnetism: Force between charged particle arise from the electric field
Fig 3
act instantaneously at a distance Charge of electron
e.g. electron-proton scattering ep → ep propagated by the exchange of photons
charge of proton Dayalbagh Educational Institute
photon propagator
→
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Interactions and fields Quantum Picture:
Particle Physics
Force between charged particle described by exchange of photons. Strength of interaction is related to charge of particles interacting. Feynman rules: Vertex term: each photon-charged particle interaction gives a factor of fermion charge, Q. Propagator term: each photon gives a factor of
where q is the photon four-momentum. Matrix element is proportional to product of vertex and propagator terms.
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Electromagnetic Vertex Basic electromagnetic process: Initial state charged fermion (e, μ, τ or quark + antiparticles) Absorption or emission of a photon Final state charged fermion Examples: e → e γ
; eγ→e
Mathematically, EM interactions are described by a term for the interaction vertex and a term for the photon propagator
Coupling strength Fig 4 QED Conservation Laws Momentum, energy and charge is conserved at each vertex Fermion flavour (e, μ, τ, u, d...) is conserved: e.g. u → uγ allowed, c → uγ forbidden
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Matrix element is proportional to the fermion charge: M ∝ e Alternatively use the fine structure constant, a
strength of the coupling at the vertex is ∝ √α
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Interactions and fields
Particle Physics
Virtual Particles The force between two charged particles is propagated by virtual photons. A particle is virtual when its four-momentum squared, does not equal its rest mass: Allowed due to Heisenberg Uncertainty Principle: can borrow energy to create particle if energy (ΔE=mc2) repaid within time (Δt), where ΔEΔt≈h Example: electronpositron scattering creating a muon pair: e+e- → μ+ μ-. Fig 5
Four momentum conservation: Momentum transferred by the photon is: Squaring,
In QED interactions mass of photon propagator is nonzero. Only intermediate photons may be virtual. Final state ones must be real! Dayalbagh Educational Institute
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Real Processes A real process demands energy conservation, hence is a combination of virtual processes.
Figure 7: Electron-electron scattering, single photon Exchange
Any real process receives contributions from all possible virtual processes.
Figure 8: Two-photon exchange contribution
Number of vertices in a diagram is called its order. Each vertex has an associated probability proportional to a coupling constant, usually denoted as â€œÎąâ€?. In discussed processes this constant is
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QED Coupling Constant Strength of interaction between electron and photon
However, α is not really a constant... An electron is never alone: it emits virtual photons, these can convert to electron positron pairs...
Fig 14
Any test charge Will feel the e+e- pairs: true charge of the electron is screened. At higher energy (shorter distances) the test charge can see the “bare” charge of the electron. α varies as a function of energy and distance
Fig 15 Table 1
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Interactions and fields
Exercise 1
Particle Physics
Draw the lowest order Feynman diagrams for our favorite process: e+ e- → μ+ μ-. Discuss the corresponding Matrix element, (e+ e- → μ+ μ-).
Solution 1 where:
A similar process can be used to create pairs of quarks, e+ e- → q q. Discuss the corresponding Matrix element for this process (e+ e- → q q). This time the second vertex has a charge of +2/3e or −1/3e. Let’s write that as: Qqe. Then we get:
where: q is the same as before.
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Interactions and fields Particle Physics What can you say about the ratio of the cross sections, Cross sections, σ ∝|
|2.
Therefore
Please note: this is not the whole answer to the problem! We’ll look more at this process in the coming weeks.
Exercise 2
Draw the lowest and second order Feynman diagrams for electron-muon scattering e- μ- → e- μ-.
Discuss the corresponding matrix element, cross section for the lowest order.
, and
Solution 2
The top vertex is proportional to the charge of the electron, e.
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The bottom vertex is proportional the charge of the muon, e. The propagator is proportional to 1/q2 ; where q is the four momentum transfer squared between the electron and the muon. The matrix element, at lowest order, is proportional to the product of these terms:
Cross section is proportional to the square of the matrix element: Estimate the contribution of the second order diagrams to the cross section.
The second order diagrams all have four vertices, each proportional to e. To calculate the matrix elements we would also have to worry about the propagator term, but we are only doing an estimate here. Therefore each contribution to the matrix 4 element, 1, is proportional to e , and the contribution to the cross section is e8 or Îą4. Dayalbagh Educational Institute
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All second order diagrams are suppressed by a factor α in the matrix element and a factor α2 in the cross section. We can write this as:
The amplitude coefficients ai are some factors relative to 0, they can be calculated, be were are not going to do it! But we can assume for ai ~ 1. The cross section σ1 will not change significantly compared to the first order estimate, σ0, as the correction from the next orders is proportional to α2 = 1/1372.
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Exercise 3 Draw distinct Feynman diagrams that contribute to the following processes in lowest order: a) γ + e- → e- + γ b) e+ + e- → e+ + e-
Solution 3 a) γ + e- → e- + γ
b) e+ + e- → e+ + e+
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Exercise 4 Draw the lowest order Feynman diagram representing Delbruck scattering: γ + γ → γ + γ. Comment: This process defines scattering of light by light, and it has no analog in classical electrodynamics.
Solution 4
Exercise 5 Express the following interaction in symbol form: a) elastic scattering of an electron to produce anti neutrino and a positron b) annihilation of an anti proton with a neutron to produce three pions
Solution 5 a) νe + e+ → νe + e+ b) p + n → п- + п+ + п+ (or) p + n → п- + п0 + п0
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Definitions and Keywords Quantum electrodynamics (QED): this is the fully quantized, relativistic theory of electrons and positrons in interaction with the electromagnetic field. Renormalization: this is a procedure applied to quantized field theories whereby the appearance of infinite re-suits for physical quantities that have to be finite is properly avoided by re-definition of the masses and coupling constants.
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