Feynman Parameters-Peskin&Schroeder 6.44

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In summary, Feynman Parameters--Peskin&Schroeder 6.44 describes the difficulties in deriving equation 6.44 from Peskin and Schroeder. Peskin and Schroeder give the algebra for the equation, but it appears that you need to use the Dirac equation to get another term. After staring at the equation for a while, you find that the terms cancel and that q2 is not 0 in the Δ term.
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Feynman Parameters--Peskin&Schroeder 6.44

I'm having some trouble deriving equation 6.44 on p.191 of Peskin and Schroeder (my book is the fifth printing, December 1997). The algebra comes close but I think that you need to do some arguing to actually get his answer--maybe this requires some Dirac equation reasoning that I'm not seeing. Here's the problem:

We're working on a Feynman integral for electrons and photons, so everything is sandwiched between u(p,s) electron state-functions. The precise step I'm on is converting the denominator of the integral into the integral ∫dx dy dz δ(1-x-y-z) 2/D3. We are given that:
D = k2+2k(yq-zp)+yq2+zp2-(x+y)m2+iε

With the Dirac delta in the integral, we can always assume 1=x+y+z. He makes the definition l=k+yq-zp.

Then he says
After a bit of algebra we find that D simplifies to:
D=l2-Δ+iε where Δ = -xyq2+(1-z)2m2

So I tried doing the algebra, and immediately it seems we need to use the Dirac equation to get another m2 term.

Starting with the given D, I got
D=l2-[z2p2]+2zpyq-y2q2+yq2+[zp2]-(1-z)m2+iε

So after staring at that a while, I decided there's no way to get anything else into the coefficient of m2 without applying the Dirac equation. So I made the substitution p2 = m2 in the bracketed terms. [This is the only time I applied the Dirac equation. Is it valid?]

This gets us closer--it changes the coefficient of m2 into what we want. A few steps later I found:

D=l2+2zpyq+yzq2+yxq2-(1-z)2m2+iε

=l2-Δ+iε+yzq2+2zpyq

=l2-Δ+iε+yz[(p+q)2-p2]
Aside from that one application of the Dirac equation, I didn't use anything but algebra. Why are those pesky terms there? Should we be saying they cancel, somehow, or that q=0? (In Ryder's QFT, he mentions something like this on p.344, but I don't really get it.) If that's the case, why would q not drop out of the Δ term? Did I misapply the Dirac equation?

Thanks for any and all help!
 
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  • #2
that term is zero.just see the feynman diagram of vertex correction where
p'=p-k+k',now he defines k'=k+q so p'=p+q,now p' and p are outgoing and ingoing states so use the onshell condition p'2=p2=m2
p'2=p2+q2+2pq.so q2+2pq=0 which is needed.
 
  • #3
Ah! Thanks very much andrien. That's really helpful. That answers both of my questions: why the terms cancel and why q2 is not 0 in the Δ term.

What had me really confused was this snippet from Ryder:
In the denominator of the integrand, putting p2=p'2=m2, (p-p')2=q2=0 ...
(Lewis H. Ryder, Quantum Field Theory, Second edition, p.344)

I guess based on what you say, that quote from Ryder is wrong?
 
  • #4
In PS they're letting q^2 be off-shell, as they mention that in a scattering process q^2 < 0.

So they don't use the on-shell condition q^2 = 0 for photons.

In Ryder, are they just computing a diagram where the photon is real? I don't have a copy in front of me, but if they are then they use that condition.
 
  • #5
The diagram is the same as the one in this post: https://www.physicsforums.com/showthread.php?t=690885 . That quote comes from the "Anomalous magnetic moment of the electron" section of Ryder, so I'm assuming they're calculating the same diagram and using the same method.

About the notation for the diagrams: In both Ryder and Peskin&Schroeder, q is the incoming photon and p is the incoming electron. (There is small difference in some of the other definitions but I don't think they matter). The integration variable (the free variable inside the e-e-photon vertex) is k.

If we are allowed to apply the Dirac equation to the incoming/outgoing electrons, then is it possible that Ryder is saying we can apply the Klein Gordon equation to the incoming photon? But then it is on-shell, right, so they must be doing something different from P&S? Does that mean there's a similar but slightly different way to evaluate that diagram? Or is it just a typo in Ryder?
 
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  • #6
what actually Ryder has done is not wrong but it may confuse you because he has used some simplification rather earlier.When you write the vertex part then after some manipulation you try to express it with two form factors(there would be four if electromagnetism does not conserve parity) which are multiplied by terms γμ and σμv respectively.Now the physical interpretation of those factors is given by the amount of charge(F1(0)) and amount of magnetic moment [1+F2(0)](ryder uses λ type thing for F's).The interpretation of the form factor really comes while taking the limit q2→0 which is what ryder has done but rather early.This F2(0) really gives α/2∏ correction which is schwinger term.P&S rather do it after expressing everything with q2 and then take the limit q2→0 in last for getting schwinger term.I hope it will clear you confusion.
 
  • #7
Hmm that is interesting. Why would P/S not do the same thing (usually removing unnecessary variables is an obvious way to make things easier)?

Anyway, thanks for all your help!
 

FAQ: Feynman Parameters-Peskin&Schroeder 6.44

What are Feynman Parameters?

Feynman Parameters are a mathematical tool used in quantum field theory to simplify the integration of certain types of Feynman diagrams. They are particularly useful in calculations involving loop diagrams.

What is Peskin and Schroeder 6.44?

Peskin and Schroeder 6.44 refers to a specific equation in the textbook "An Introduction to Quantum Field Theory" by Michael Peskin and Daniel Schroeder. It involves the use of Feynman Parameters in calculating loop integrals.

How do Feynman Parameters work?

Feynman Parameters work by introducing a parameter (usually denoted by "x") that allows for the simplification of a loop integral. This parameter can then be integrated over and the resulting formula can be applied to various Feynman diagrams.

Why are Feynman Parameters important in quantum field theory?

Feynman Parameters are important in quantum field theory because they allow for the calculation of loop integrals, which are necessary for predicting and understanding the behavior of subatomic particles. They also help to simplify complex calculations and make them more manageable.

In what types of calculations are Feynman Parameters commonly used?

Feynman Parameters are commonly used in calculations involving loop diagrams, such as in particle interactions and decay processes. They are also frequently used in calculations involving quantum electrodynamics (QED) and quantum chromodynamics (QCD).

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