The "proof" that is being looked for is, I suppose, along the following lines: the appearance of $\gamma_5$ means that you need something which changes sign under parity, and the 4 indices means you need a 4-index tensor. To get antisymmetry, you need the $\epsilon$ tensor, all of whose contractions with itself or with the metric vanish, so that in all dimensions greater than 4 you can't make anything. In 3d you can't make anything either, and in 2d, you can only make $\epsilon_{\mu\nu} \delta{\rho\sigma}$ and other index combinations on the bottom, but it doesn't work because the result has the same symmetry structure in $\mu\nu\rho\sigma$, and the $\epsilon$ and $\delta$ parts have mixed symmetry, so they will vanish in the appropriate combination with the right symmetry between all 4 indices. You can show this in a straightforward way.
But this question is really nonsense as posed, although it is probably derived from grad-school homework, because there is no unambiguous continuation of $\gamma_5$ to other dimensions. One tradition is to consider $\gamma_5$ as the same product of $i\gamma_0\gamma_1\gamma_2\gamma_3$ as in 4d, so that it is not a higher dimensional rotationally invariant construction. In this case, the statement is false, you get a non-rotationally invariant answer for the contraction in dimensions 5 and above.
But a better answer is that all this is really about voodoo. The process of dimensional regularization can only be understood and internalized after you understand that it is derived from an earlier method of propagator modification called analytic regularization. Analytic regularization simply modifies the $k^2 $ at the bottom of Feynman propagators to $k^{2-\epsilon}$ in each field, and takes the limit as $\epsilon$ goes to zero. It makes sense non-perturbatively, as it is converting the Schwinger representation of random walks to Schwinger representations of Levy flights, and in Stochastic quantization it just replaces the Laplacians with fractional Laplacians appropriate to describing the propagation of Levy flights. It is very intuitive, and rather straightforward, but it gives a mess when you sit down to do actual calculations.
The reason it gives a mess is because the propagator modification modifies the perturbation integrals in a nonuniform way, with a different regularizing power of k at large k depending on the particle type in the loops, and the number of particle in the loop, and so on. This doesn't make any difference, ultimately the only thing that matters to get a finite analytically continuable answer is that you replace $d^4k$ with $d^4 k /k^\epsilon$ for some $\epsilon$. This is what dimensional regularization is all about. Once you make the loop integrals, you just introduce a little bit of extra falloff at large k, and make it smooth, and if you introduce it in the exponent, it's an analytic function of $\epsilon$, and you can throw away the pole parts with appropriate subtraction.
This process of decorating integrations with denominators is reinterpreted as continuing the dimension as a useful intuition trick for Veltman and 'tHooft. This way, they can make an integral table for various rotationally invariant k integrals integrated over a measure with an extra completely unimportant nondiverging constant factor C(a) multiplying the integration measure:
$$C(a) d^4 k \over k^a$$
where $C(a) = S(4-a)/S(4)$, where
$$S(d) = {\pi^{d/2}\over\Gamma(d/2+1)}$$
is the volume of a d dimensional sphere.
With this convention, you use Feynman parameters to make a diagram rotationally invariant, and then the integrations get a small denominator, and any rotationally integral gets a value which is a smooth interpolation of the values it takes in any dimension less or greater than 4.
But you don't need to continue any of the numerator decorations. You can do the numerator operations as if in 4d. The point is that you don't need to make sensible interpolations between the dimensions for any of the other quantities in the numerator, and focusing on this is an endless source of confusion for students. Best to just leave the numerator in 4d.
Q&A (4869)
