I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example.
I have an integral that looks approximately as
$I = \int^1_0 dx \int_{-\infty}^{\infty} \frac{d^dk}{(2\pi)^d}\frac{(k+p)\cdot\gamma}{((k+px)^2+m^2x^2)^2}$
where $d = 4-2\epsilon$, which is used often in "dimensional regularization" in physics and $\gamma$ is the Dirac gamma matrices also used in physics.
I approach this integral in two different methods:
1) First, I shift $k = k'-px$ and assume $d^4k = d^4k'$ since integration is from $-\infty $ to $\infty$, I get, $I = \int^1_0 dx \int_{-\infty}^{\infty} \frac{d^dk}{(2\pi)^d}\frac{(k+p(1-x))\cdot\gamma}{(k^2+m^2x^2)^2}.$ (By the way, $p^2 = (p\cdot\gamma) (p\cdot\gamma)$).
Now say I integrate to get $I = \int^1_0 dx f(x,p\cdot\gamma)$, then take deriavtive with respect to $p\cdot\gamma$:
$\frac{d}{d p\cdot\gamma} I = \int_0^1 dx\frac{\partial}{\partial p\cdot\gamma}(f(x,p\cdot\gamma)) $.
2) This time, I take the derivative w.r.t $p\cdot\gamma$ first to get:
$\frac{d}{d p\cdot\gamma} I=\int^1_0 dx \int_{-\infty}^{\infty} \frac{d^dk}{(2\pi)^d}\frac{\partial}{\partial p\cdot\gamma}\frac{(k+p)\cdot\gamma}{((k+px)^2+m^2x^2)^2}$
$=\int^1_0 dx \int_{-\infty}^{\infty} \frac{d^dk}{(2\pi)^d}(\frac{1}{((k+px)^2+m^2x^2)^2}+\frac{((k+p)\cdot\gamma)(2x(k+px)\cdot\gamma)}{((k+px)^2+m^2x^2)^3})$
Now, I shift $k=k'-px$ again, and I get a different answer.
Why are they different from each other, and if I want to get $\frac{d}{d p\cdot\gamma} I$, which one should I use? I would assume that second method is correct, if there is difference in answer, but all the textbooks have answers that match with my first method; which seems bizarre.
The reason we run into problems is we have not really regularized our loop integral. We are dealing with an ill-defined quantity, and it is natural that we run into contradictions.
The way to go is to regularize first, which in essence defines our object of interest, and then manipulate it. We will then see that some formal operations we use are justified, and some are not.
If we use momentum cutoff regularization, in general we can not shift anymore, since the integration domain is no longer infinite. Similarly, to perform analytic continuation in spacetime dimension, we usually employ spherical coordinates, hence lose manifest translation invariance. Of course, the shift invariance will come back upon lifting regularization if we are dealing with a convergent integral to begin with. Not always true otherwise.
Also, our expression is a matrix, the size of which depends on spacetime dimension. It is hard to make sense of a matrix with a noninteger number of rows and columns. It would be a good idea to work with the final amplitude, which is scalar.