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%% subsection 6.1 The `t Hooft-Veltman Scheme [slac-pub-7144-0-0-6-1 in slac-pub-7144-0-0-6: ^slac-pub-7144-0-0-6 >slac-pub-7144-0-0-6-2]
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\subsection{\usemenu{slac-pub-7144::context::slac-pub-7144-0-0-6-1}{The `t Hooft-Veltman Scheme}}\label{subsection::slac-pub-7144-0-0-6-1}
In the `t Hooft-Veltman scheme \cite{13}, the $d$ dimensions are
split into 4 and $d-4$; the corresponding structures
are distinguished by no superscript, by a tilde and by a hat,
respectively. There are Lorentz indices in $d$, 4
and $d-4$ dimensions and the corresponding metric
tensors $g_{\mu \nu}$, $\tilde{g}_{\mu \nu}$ and $\hat{g}_{\mu \nu}$.
While all the $\g$-matrices are taken in $d$ dimensions, their indices
are split in $4$ and $d-4$ components, according to the rules
\bea
\label{HVrules}
g_{\mu \nu} &=& \tilde{g}_{\mu \nu} + \hat{g}_{\mu \nu} \nonumber \\
\tilde{g}_{\mu \nu} \tilde{g}^{\mu \nu} &=& 4 \quad , \quad
\hat{g}_{\mu \nu} \hat{g}^{\mu \nu} = d-4 \quad , \nonumber \\
\tilde{g}_{\mu \nu} \hat{g}^{\mu \nu} &=& 0 \quad .
\eea
The $\g$-matrices in 4 dimensions ($\tilde{\g}^\mu$) and $(d-4)$
dimensions
($\hat{\g}^\mu$) are defined by $\tilde{g}^{\mu \nu} \g_\nu$
and $\hat{g}^{\mu \nu} \g_\nu$, respectively. Assuming the usual
anticommutation relations of the $d$-dimensional Dirac matrices
in terms of the $d$-dimensional metric tensor $g_{\mu \nu}$,
one gets the
following rules for $\tilde{\g}^\mu$ and $\hat{\g}^\mu$.
\be
\label{anticomm}
\left\{ \tilde{\g}^\mu, \tilde{\g}^\nu \right\} = 2 \tilde{g}^{\mu \nu}
\quad , \quad
\left\{ \hat{\g}^\mu, \hat{\g}^\nu \right\} = 2 \hat{g}^{\mu \nu}
\quad , \quad
\left\{ \tilde{\g}^\mu, \hat{\g}^\nu \right\} = 0 \quad .
\ee
The commutation relations with $\g_5$ are postulated to be
\be
\left\{ \tilde{\g}^\mu, \g_5 \right\} = 0
\quad , \quad
\left[ \hat{\g}^\mu, \g_5 \right] = 0
\quad ,
\ee
which is equivalent to defining $\g_5$ by the product
$i \tilde{\g_0} \tilde{\g_1} \tilde{\g_2} \tilde{\g_3} $.
This is the only way known to treat $\g_5$ without running into
algebraic inconsistencies
\cite{31}.
Finally, we mention that the chiral vertices in $d$-dimensions
can be defined in different ways, all having the same formal limit
when $d \to 4$. For left- and right- handed currents we follow
the common practice and
use
\be
\label{currents}
\tilde{\g}^\mu L = R \g^\mu L \quad \mbox{and} \quad
\tilde{\g}^\mu R = L \g^\mu R \quad .
\ee
There are several possibilities
to define the operators $O_7$ and $O_8$ in $d$ dimensions
(with identical 4 dimensional limit);
for example the term $\sigma^{\mu \nu}$
in eq.
(\docLink{slac-pub-7144-0-0-1.tcx}[operators]{1.2}) can be defined to be
\be
\label{def1}
\sigma^{\mu \nu} = \frac{i}{2} [\g^\mu,\g^\nu] \quad ,
\ee
where $\g^\mu$ and $\g^\nu$ are the "$d$-dimensional"
matrices, or, alternatively,
\be
\label{def2}
\sigma^{\mu \nu} = \frac{i}{2} [\tilde{\g}^\mu,\tilde{\g}^\nu] \quad .
\ee
A difference in the
definition will result in a difference of the finite terms
in the one-loop matrix elements of these operators.
As the calculation of the one-loop matrix elements of
$O_7$ and $O_8$
is relatively
easy, once an exact definition of the operators has been
specified, we give in this appendix only the result for the two-loop
contribution of the operator $O_2$.