Let us work in dimension $D = 4-2\epsilon$.

In 4-dimension, we can write $\text{Tr}[A B]$, where $A$ and $B$ are string of gamma matrices, as

$\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$, where $\Gamma^m = \{1,\gamma_5,\gamma^\mu,\gamma_5\gamma^\mu,\sigma_{\mu\nu}\}$ are complete set of gamma matrices spanning the dirac space in 4-dim.

As it is well-known, generalizing this to non-integer $D$ dimension causes difficulties since $\gamma_5$ (defined as $\gamma_5= i\gamma^0\gamma^1\gamma^2\gamma^3$ in 4-dim.) cannot be well-defined.

One often does not need to work with the explicit form of $\gamma_5$, but uses the two relations to evaluate the trace:



However, in $D$ dimension, the two relations cannot be satisfied simultaneously; and people use different $\gamma_5$-schemes to treat $\gamma_5$ in $D$-dimension.

However, when you evaluate both of the traces $\text{Tr}[A B]$ and $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$ in $D$-dimension by using different $\gamma_5$-schemes they do not necessarily agree.

As an example,

take $A = (\gamma\cdot p_1)\gamma^\alpha(\gamma\cdot p_2)$ and $B =\gamma^\beta(\gamma\cdot p_1)(\gamma\cdot p_2)$.

Then evaluation of $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$ requires $\gamma_5$ scheme choice.

Then $\text{Tr}[A B] = -4~(D-2)~(2~(p_1\cdot p_2)^2 - p_1^2~ p_2^2)$

$\left(\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]\right)_{\text{t'Hooft-Veltman}} = -4~(D-2)~\left((D-2)~(p_1\cdot p_2)^2 - (D-3)p_1^2~ p_2^2\right)$

$\left(\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]\right)_{\text{NDR}} = -4D~(p_1\cdot p_2)^2 + 8p_1^2~ p_2^2$

where 'NDR' is naive dimensional regularization scheme and 't'Hooft-Veltman' is t'Hooft-Veltman-scheme.

All three results agree when $D$ is taken to be 4, but do not agree in $\epsilon$ terms. Is there a way to ensure agreement down to $\epsilon$ piece?


The answer to your question is laid out here:

I have a question though: given that you sound privy to the essence of the issue, do you happen to be the author of one of the above papers?

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  • $\begingroup$ Hi, I am not an author of one of these papers, and from what I can tell I do not think these papers address the question I am asking. $\endgroup$ – Quantization Nov 27 '18 at 21:08
  • $\begingroup$ The papers tried to look at this DR related issue via a different angle: a regularization staying with the fixed dimension. Since you are familiar with the topic, do you think these papers lead to blind alleys? $\endgroup$ – MadMax Nov 28 '18 at 14:52

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