Loop integral computation

AI Thread Summary
The discussion centers on computing a complex loop integral using two methods: rewriting the numerator and Feynman parametrization. The first method simplifies the integral, yielding a result consistent with expected outcomes, while the second method introduces an unwanted term involving the commutator of gamma matrices. The discrepancy suggests an issue with the Feynman parametrization approach, potentially related to infrared (IR) or ultraviolet (UV) divergences. The participant seeks clarification on where the calculation may have gone wrong, referencing Srednicki's textbook for guidance. The conversation highlights the challenges of loop integral computations in quantum field theory.
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Homework Statement
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I am trying to compute the following loop integral:
$$
\require{cancel}
\displaystyle
\begin{align}
I= \int \frac{d^4k}{(2\pi)^4}\bar u(p')\frac{[k^2k^\mu - (k\cdot p)\cancel{k}\gamma^\mu -(k\cdot p')\gamma^\mu\cancel{k}]}{[k^2-M^2+i\epsilon][(k-p)^2-m^2+i\epsilon][(k-p')^2-m^2+i\epsilon]}u(p),
\end{align}
$$
with 2 different methods.
First Method: Rewrite the numerator and cancel the denominator. The numerator is
$$
\begin{align}
N^\mu &= k^2k^\mu - (k\cdot p)\cancel{k}\gamma^\mu -(k\cdot p')\gamma^\mu\cancel{k}
\nonumber \\
&= k^2k^\mu - \frac{1}{2}(k^2-[(k-p)^2-m^2])k_\nu \gamma^\nu\gamma^\mu -\frac{1}{2}(k^2-[(k-p')^2-m^2])k_\nu\gamma^\mu\gamma^\nu
\nonumber \\
&= \frac{1}{2}[(k-p)^2-m^2]k_\nu \gamma^\nu\gamma^\mu +\frac{1}{2}[(k-p')^2-m^2]k_\nu\gamma^\mu\gamma^\nu.
\end{align}
$$
Thus,
$$
\begin{align}
\require{cancel}
\displaystyle
I =& \frac{1}{2}\int\frac{d^4k}{(2\pi)^4}\bar u(p') \frac{1}{[k^2-M^2+i\epsilon]} \left( \frac{k_\nu}{[(k-p')^2-m^2+i\epsilon]}\gamma^\nu\gamma^\mu + \frac{k_\nu}{[(k-p)^2-m^2+i\epsilon]}\gamma^\mu\gamma^\nu \right) u(p)
\nonumber \\
&= \frac{1}{2}A\bar u(p')(p'_\nu\gamma^\nu\gamma^\mu + p_\nu\gamma^\mu\gamma^\nu) u(p)
\nonumber \\
&= Am \bar u(p') \gamma^\mu u(p),
\end{align}
$$
where ##A## is a scalar.
Second Method: Feynman Parametrization. The denominator becomes:
$$
\require{cancel}
\displaystyle
\begin{align}
[k^2-M^2+i\epsilon]^{-1}[(k-p)^2-m^2+i\epsilon]^{-1}[(k-p')^2-m^2+i\epsilon]^{-1}=
\int_0^1dx\int_0^{1-x}dy \frac{2}{[l^2-\Delta^2+i\epsilon]^3},
\end{align}
$$
where ##l^\mu=(k-xp-yp')^\mu##, ##\Delta^2=(x+y)^2m^2+(1-x-y)M^2-xyq^2##, and, ##q^\mu=(p'-p)^\mu##.
Next, we perform the shift in the numerator:
$$
\begin{align}
N^\mu = k^2k^\mu - (k\cdot p)\cancel{k}\gamma^\mu -(k\cdot p')\gamma^\mu\cancel{k} = A_1 \bar u(p') \gamma^\mu u(p) + A_2 \bar u(p') \frac{i\sigma^{\mu\nu}q_\nu}{2m} u(p),
\end{align}
$$
where ##A_1## and ##A_2## are scalars. This method clearly gives a different result.
The first method gives the correct result (the result should not contain a ##i\sigma^{\mu\nu}q_\nu/2m## term), so the problem must be related with the Feynman Parametrization somehow.
The denominator ##[(k-p)^2-m^2+i\epsilon]=[k^2-2(k\cdot p)+i\epsilon]## (and the other one with ##p'##) could be problematic when ##k\to 0##, however there are other terms in the numerator (which I did not included here) that give the correct result with the same denominator and using Feynman Parametrization, i.e., using the second method. On the other hand these same terms (which give the correct result) are UV finite but the one in question is not. Is the problem in the IR or UV or both?
I believe I didn't make any mistakes on the passages, so what am I missing?
 
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I believe this calculation is done explictly in Srednicki's textbook or its solution manual. Enjoy!
 
mad mathematician said:
I believe this calculation is done explictly in Srednicki's textbook or its solution manual. Enjoy!
where exactly?
 
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