Energy conservation in e-e+ annihilation

In summary, the conversation revolves around understanding the conservation of energy in e^-e^+ annihilation, specifically in solving Exercise 3.5(b) of Halzen and Martin's book. The focus is on the time dependent term in the scattering amplitude and how it relates to the energy conservation in the process. The transition amplitude is found to be proportional to a delta function, but with the wrong argument. The mistake is attributed to a misunderstanding of the "rule" of forming the matrix element.
  • #1
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Hi,

I'm trying to convince myself (mathematically) that energy is conserved at the vertex in [itex]e^-e^+[/itex] annihilation, while solving Exercise 3.5(b) of Halzen and Martin's book (page 83).

I am looking at the time dependent term in the scattering amplitude [itex]T_{fi}[/itex] alone, to recover the delta function term which asserts the energy conservation. I know that I can look at the process in forward time or backward time, with the roles of the e+ and e- reversed.

Suppose the energies of the incoming electron and positron are E and E' respectively, whereas the energy of the outgoing photon is [itex]\omega[/itex] ([itex]\hbar = 1[/itex]).

From what I understand, the transition amplitude is proportional to

[tex]\int dt\,(e^{-i(-E')t})^{*}e^{-i\omega t}e^{-iEt} = 2\pi\delta(E+E'+\omega)[/tex]

which gives the wrong answer of course, since the argument of the delta function should be [itex]E+E'-\omega[/itex].

I believe I am making a mistake in interpreting their "rule":

The rule is to form the matrix element,

[tex]\int d^{4}x\,\phi^{*}_{outgoing}V\phi_{ingoing}[/tex]

where ingoing and outgoing always refer to the arrows on the particle (electron) lines.

What does this mean? I'm a bit confused with the convention.

Thanks in advance.
 
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  • #3
The signs in your exponents are wrong. It should be
[tex]e^{i\omega}e^{-i (E+E')}[/tex], corresponding to E+E' in the initial state and omega in the final state.
 
  • #4
clem said:
The signs in your exponents are wrong. It should be
[tex]e^{i\omega}e^{-i (E+E')}[/tex], corresponding to E+E' in the initial state and omega in the final state.

Thanks clem. I was earlier thinking that the photon contribution will be through the potential term which is sandwiched between the final and initial states, which (I thought) referred to particles alone.
 

1. What is energy conservation in e-e+ annihilation?

Energy conservation in e-e+ annihilation refers to the principle that the total energy before and after the collision between an electron and a positron must remain the same. This means that the total energy of the particles before the collision, which is in the form of kinetic energy and mass energy, must be equal to the total energy of the particles after the collision, which is also in the form of kinetic energy and mass energy.

2. How is energy conserved in e-e+ annihilation?

In e-e+ annihilation, the energy is conserved through the creation of new particles. When an electron and a positron collide, they can produce a variety of different particles, such as photons, muons, and neutrinos. These new particles carry the excess energy from the collision, ensuring that the total energy remains constant.

3. Why is energy conservation important in e-e+ annihilation?

Energy conservation is important in e-e+ annihilation because it is a fundamental principle of physics. It helps us understand and predict the outcomes of particle collisions and ensures that our understanding of energy is consistent and accurate. Additionally, e-e+ annihilation is often used in particle accelerators to produce high-energy particles, and energy conservation is crucial for these experiments to be successful.

4. What are some applications of e-e+ annihilation and energy conservation?

E-e+ annihilation has several important applications in physics research. One application is in the study of the fundamental building blocks of matter and the forces that govern their interactions. Another application is in medical imaging, where e-e+ annihilation is used to produce positron-emitting isotopes for PET scans. In both cases, energy conservation is essential for accurately interpreting the results of these experiments.

5. Is energy conservation always observed in e-e+ annihilation?

Yes, energy conservation is always observed in e-e+ annihilation. This is because the laws of physics, including the conservation of energy, are universal and apply to all physical processes. While the exact outcomes of e-e+ annihilation may vary, the total energy must always remain the same before and after the collision, as dictated by the principle of energy conservation.

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