# When can we ignore the delta function in th Feynman rules?

• Vereinsamt
In summary, the amplitude for the process e^-e^+ -> mu^-mu^+ can be written using Feynman rules as -iM = [\bar{v}(p_2)(-ie\gamma^\mu)u(p_1)] \frac{-ig_{\mu\nu}}{q^2}[\bar{u}(k_1)(-ie\gamma^\nu)v(k_2)], but the delta function integration is not included in this expression as it is part of the S matrix.

#### Vereinsamt

in peskin-schroeder and http://www.hep.phy.cam.ac.uk/batley/particles/handout_04.pdf" [Broken] the amplitude for $$e^-e^+\rightarrow \mu^- \mu^+$$ is written using feynman rules as follows
$$-iM=[\bar{v}(p_2)(-ie\gamma^\mu )u(p_1)] \frac{-ig_{\mu\nu}}{q^2}[\bar{u}(k_1)(-ie\gamma^\nu )v(k_2)]$$

but what about the delta function integeration? is it already done here?

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I may well be wrong here, but that looks like the M matrix part of the S matrix, and it is the S matrix that has the delta function in it, so you shouldn't be expecting a delta function.

again i DID only just do this, so i may be wrong.

The delta function in the Feynman rules is used to represent the conservation of four-momentum at each vertex in the diagram. In the specific example you provided, the delta function integration is already accounted for in the expression for the amplitude. This is because the delta function in this case is represented by the factor of -ig_{\mu\nu}/q^2, which is the propagator for the photon. This factor ensures that the four-momentum is conserved at that vertex. Therefore, there is no need to explicitly include the delta function integration in this case. However, there may be cases where the delta function integration needs to be included separately, such as when there are multiple particles involved in the interaction. It is important to carefully follow the Feynman rules and pay attention to the conservation of four-momentum at each vertex to determine when the delta function integration needs to be included.

## 1. What is the delta function in Feynman rules?

The delta function, also known as the Dirac delta function, is a mathematical function used in quantum field theory to represent the probability amplitude of a particle interacting at a specific point in space-time.

## 2. When can we ignore the delta function in Feynman rules?

The delta function can be ignored in Feynman rules when it represents a point where the interaction between particles does not occur or has a negligible effect on the overall calculation. This is because the delta function is often used to simplify calculations and can be treated as a constant.

## 3. How does ignoring the delta function affect the Feynman diagram?

Ignoring the delta function in the Feynman diagram means that the interaction at that specific point is not included in the calculation. This may lead to a simpler diagram with fewer vertices and propagators, but it also means that the resulting calculation may not be as accurate.

## 4. Can we always ignore the delta function in Feynman rules?

No, we cannot always ignore the delta function in Feynman rules. It depends on the specific interaction being studied and the level of accuracy required in the calculation. In some cases, the delta function represents an important interaction that cannot be ignored.

## 5. How do we determine when to ignore the delta function in Feynman rules?

The decision to ignore the delta function in Feynman rules is based on the specific problem being studied and the desired level of accuracy. This is a judgement call that requires knowledge and understanding of the physical system being studied and the mathematical techniques being used.

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