# Regularization Schemes and Higher Order Corrections

William B. Kilgore  Physics Department, Brookhaven National Laboratory, Upton,
New York 11973, USA.
<EMAIL_ADDRESS>
###### Abstract

I apply commonly used regularization schemes to a multiloop calculation to
examine the properties of the schemes at higher orders. I find complete
consistency between the conventional dimensional regularization scheme and
dimensional reduction, but I find that the four-dimensional helicity scheme
produces incorrect results at next-to-next-to-leading order and singular
results at next-to-next-to-next-to-leading order. It is not, therefore, a
unitary regularization scheme.

##  I Introduction

Dimensional regularization ’t Hooft and Veltman (1972) is an elegant and
efficient means of handling the divergences that arise in perturbation theory
beyond the tree level. Among its many favorable qualities it respects gauge
and Lorentz invariance and allows one to handle both ultraviolet and infrared
divergences in the same manner. The application of dimensional regularization
to different kinds of problems has led to the development of a variety of
regularization schemes, which share the dimensional regularization of momentum
integrals, but differ in their handling of external (or observed) states and
of spin degrees of freedom.

The original formulation of dimensional regularization ’t Hooft and Veltman
(1972), known as the ’t Hooft-Veltman (HV) scheme, specifies that observed
states are to be treated as four-dimensional, while internal states are to be
treated as $D_{m}=4-2\,{\varepsilon}$ dimensional. That is, both their momenta
and spin degrees of freedom were to be continued from four to $D_{m}$
dimensions. It turns out that one has the freedom to choose the value of the
trace of the Dirac unit matrix to take its canonical value of four, so
fermions continue to have two spin degrees of freedom, even though their
momenta are continued to $D_{m}$ dimensions. Internal gauge bosons, however,
have $D_{m}-2$ spin degrees of freedom (internal massive gauge bosons have
$D_{m}-1$ degrees of freedom).

A slight variation on the HV scheme has come to be called conventional
dimensional regularization (CDR) Collins (1984). In this variation, all
particles and momenta are taken to be $D_{m}$ dimensional. This often turns
out to be computationally more convenient, since one set of rules governs all
interactions. This is particularly so when computing higher order corrections
to theories subject to infrared sensitivities, like QCD. In the HV scheme, if
two external states have infrared sensitive overlaps, they must be treated as
internal, or $D_{m}$ dimensional states. In the CDR scheme, all states are
already treated as $D_{m}$ dimensional, so there is no possibility of failing
to properly account for infrared overlaps.

A third variation, called dimensional reduction (DRED) Siegel (1979), was
devised for application to supersymmetric theories. In supersymmetry, it is
essential that the number of bosonic degrees of freedom is exactly equal to
the number of fermionic degrees of freedom. This requirement is violated in
the HV and CDR schemes. In the DRED scheme, the continuation to $D_{m}$
dimensions is taken as a compactification from four dimensions. Thus, while
space-time is taken to be four-dimensional and particles have the standard
number of degrees of freedom, momenta span a $D_{m}$ dimensional vector space
and momentum integrals are regularized dimensionally.

A fourth variation, the four-dimensional helicity (FDH) scheme Bern and
Kosower (1992); Bern et al. (2002), was developed primarily for use in
constructing one-loop amplitudes from unitarity cuts. The most efficient
building blocks for such calculations are tree-level helicity amplitudes,
which necessarily have two spin degrees of freedom for both fermions and gauge
bosons. The FDH scheme resembles the DRED scheme in that it regularizes
momentum integrals dimensionally while maintaining the spin degrees of freedom
of a four-dimensional theory (and therefore appears to be a valid
supersymmetric regularization scheme Bern et al. (2002)), but there are
crucial differences, which I will discuss in detail.

The fact that the HV scheme respects the unitarity of the $S$-matrix was
proven at its introduction ’t Hooft and Veltman (1972). The arguments which
establish the validity of the HV scheme carry over to the CDR scheme and
establish that it too is a valid regularization scheme. After some initial
confusion over the proper renormalization procedure van Damme and ’t Hooft
(1985); Capper et al. (1980); Jack et al. (1994a) for the DRED scheme, it was
established that it too is a proper, unitary regularization scheme Jack et al.
(1994a) and that it is indeed equivalent to the CDR scheme Jack et al.
(1994b). The FDH scheme has never been subjected to such stringent
examination. It has been used successfully in a number of landmark next-to-
leading order (NLO) calculations, but it has never been established whether it
is a proper, unitary regularization scheme, or merely a set of shortcuts that
allow expert users to obtain correct results.

In this paper, I will perform a well-known multiloop calculation in the
various regularization schemes. I will show that while the HV and CDR scheme
calculations yield the correct result and the DRED scheme calculation, while
far more complicated is completely equivalent, the FDH scheme calculation
yields incorrect results which inevitably violate unitarity at sufficiently
high order. A detailed comparison of the various calculations identifies the
source of the unitarity violations in the FDH scheme.

The plan of this paper is as follows: in section two, I will describe the test
calculation to be performed and present the result to be obtained. In sections
three, four and five, I will describe in detail the calculation to next-to-
next-to-leading order (NNLO) as it is performed in the CDR, DRED and FDH
schemes, respectively. In section six, I present partial results at N3LO which
solidify the conclusion that the CDR and DRED schemes are equivalent and
correct, but that the FDH scheme violates unitarity. In section seven, I will
discuss my results and draw my conclusions.

##  II The Test Environment

To test the regularization schemes, I will calculate two quantities: the
massless nonsinglet contributions to

  1. 1.

the hadronic decay width of a fictitious neutral vector boson $V$, of mass
$M_{V}$;

  2. 2.

the single photon approximation to the total hadronic annihilation cross
section for an electron – positron pair.

I will perform these calculations by means of the optical theorem, taking the
imaginary part of the forward scattering amplitudes. In both cases, this means
taking the imaginary part of the vacuum polarization tensor sandwiched between
external states. Since the optical theorem is a direct consequence of the
unitarity of the $S$-matrix, any unitary regularization scheme must give the
same result, once one expands in terms of a standard coupling. To avoid
complications involving prescriptions for handling $\gamma_{5}$ and the Levi-
Civita tensor, I will take $V$ to have only vectorlike couplings. In this way,
the vacuum polarization tensor for the $V$ boson will be identical to that of
the off shell photon, up to coupling constants and so the QCD expansion of the
two results will differ only by constant numerical factors.

Each regularization scheme will start from the same four-dimensional
Lagrangian,

$\begin{split}{\cal
L}=&-\frac{1}{2}A^{a}_{\mu}\left(\partial^{\mu}\partial^{\nu}(1-\xi^{-1})-g^{\mu\nu}\Box\right)A^{a}_{\nu}-g\,f^{abc}(\partial^{\mu}\,A^{a\,\nu})A^{b}_{\mu}\,A^{c}_{\nu}-\frac{g^{2}}{4}f^{abc}\,f^{ade}\,A^{b\,\mu}\,A^{c\,\nu}\,A^{d}_{\mu}\,A^{e}_{\nu}\\\
&+i\sum_{f}\,\overline{\psi}_{f}^{i}\left(\delta_{ij}\not{\partial}-i\,g\,t^{a}_{ij}\not{A}^{a}-i\,g_{V}\,Q_{f}\not{V}\right)\,\psi_{f}^{j}-\overline{c}^{a}\Box\,c^{a}+g\,f^{abc}\left(\partial_{\mu}\,\overline{c}^{a}\right)\,A^{b\,\mu}\,c^{c}\,,\end{split}$
(1)

where $A^{a\,\mu}$ is the QCD gauge field, $V^{\mu}$ is the massive vector
boson, $\psi_{f}$ is the quark field of flavor $f$, $\overline{c}^{a}$ and
$c^{a}$ are the Faddeev-Popov ghost fields, $g$ is the QCD coupling, $g_{V}$
is the $V$ gauge coupling and $Q_{f}$ represents the charge of the quark
flavor $f$ under the $V$ symmetry. I will not be computing nontrivial
corrections in $g_{V}$, so there is no need to specify the $V$-self
interaction parts of the Lagrangian.

Figure 1: Sample diagrams of one-, two- and three-loop contributions to the
vacuum polarization of $V$.

The result to N3LO is well known Chetyrkin et al. (1979); Dine and Sapirstein
(1979); Celmaster and Gonsalves (1980); Gorishnii et al. (1988, 1991),

$\begin{split}\Gamma^{V}_{had}=&\Gamma^{V}_{0,\,{\rm had}}\,{\cal
F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2}=M_{V}^{2})\hskip
70.0pt\Gamma^{V}_{0,\,{\rm
had}}=\frac{\alpha_{V}\,M_{V}}{3}\,N_{c}\sum_{f}\,Q_{f}^{2}\\\
\sigma^{e^{+}\,e^{-}\to\ {\rm had}}(Q^{2})=&\sigma_{0}^{e^{+}\,e^{-}\to\ {\rm
had}}(Q^{2})\,{\cal F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2})\hskip
40.0pt\sigma_{0}^{e^{+}\,e^{-}\to\ {\rm
had}}(Q^{2})=\frac{4\,\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\sum_{f}\,Q_{f}^{2}\\\
\end{split}$ (2)

and

$\begin{split}{\cal
F}({\alpha_{s}^{\overline{\rm{MS}}}},Q^{2})=\hskip-12.0pt&\hskip
12.0pt\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left({\beta_{1}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}+{\beta_{0}^{\overline{\rm{MS}}}}^{\,2}\,\ln^{2}\frac{\mu^{2}}{Q^{2}}\right)\right]\right.\\\
&+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left[\left(-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right)\right.\\\
&\qquad\qquad\times\left.\left(1+2{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right)\right]\\\
&+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\left[-C_{F}^{3}\frac{69}{128}+C_{F}^{2}\,C_{A}\left(-\frac{127}{64}-\frac{143}{16}\zeta_{3}+\frac{55}{4}\,\zeta_{5}\right)\right.\\\
&\qquad\qquad+C_{F}\,C_{A}^{2}\left(\frac{90445}{3456}-\frac{2737}{144}\,\zeta_{3}-\frac{55}{24}\,\zeta_{5}\right)\\\
&\qquad\qquad+C_{F}^{2}\,N_{f}\left(-\frac{29}{128}+\frac{19}{8}\,\zeta_{3}-\frac{5}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\,N_{f}\left(-\frac{485}{54}+\frac{56}{9}\,\zeta_{3}+\frac{5}{12}\,\zeta_{5}\right)\\\
&\left.\left.\qquad\qquad+C_{F}\,N_{f}^{2}\left(\frac{151}{216}-\frac{19}{36}\,\zeta_{3}\right)-\frac{1}{4}\,\pi^{2}\,C_{F}\,{\beta_{0}^{\overline{\rm{MS}}}}^{2}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{4}\right)\right\\}\,.\end{split}$
(3)

To obtain the hadronic decay width at LO, NLO and NNLO, I need to compute the
QCD corrections to the vacuum polarization of the $V$ (photon) at $1$, $2$ and
$3$ loops, respectively. Sample diagrams are shown in Fig. (1).

###  II.1 Methods

In each scheme, I will need to compute the vacuum polarization of $V$ and the
necessary coupling renormalization constants. As a cross-check on the
reliability of my calculational framework, I reproduce known results on the
QCD $\beta$-functions and mass anomalous dimensions to three-loop order, as
well as the three-loop QCD contributions to the $\beta$-function of $V$ (where
needed).

In all calculations, I generate the contributing diagrams using QGRAF Nogueira
(1993). The symbolic algebra program FORM Vermaseren (2000) is used to
implement the Feynman rules and perform algebraic manipulations to reduce the
result to a set of Feynman integrals to be performed and their coefficients.
The set of Feynman integrals are then reduced to master integrals using the
program REDUZE Studerus (2010). Using the method of Ref. Davydychev et al.
(1998), the vertex corrections can be expressed in terms of the same
propagator integrals used to compute the vacuum polarization and wave function
renormalizations. The complete set of master integrals at one, two and three
loops are shown in Fig. (2).

a) b)
c)

Figure 2: Master integrals for the evaluation of vacuum polarization at a) one
loop, b) two loops and c) three loops.

Most of the master integrals are trivial iterated-bubble diagrams and the
others were evaluated long ago Chetyrkin et al. (1980); Kazakov (1984). As an
additional cross-check, the integral reduction and evaluation is also
performed using the program MINCERGorishnii et al. (1989); Larin et al.
(1991).

###  II.2 Notation

The various schemes that I will consider span a variety of vector spaces, each
with their own metric tensor. To establish some level of consistency, I will
denote the metric tensor of classical four-dimensional space-time as
$\eta^{\mu\nu}$; the metric tensor of the $D_{m}$ dimensional vector space in
which momentum integrals are regularized will be denoted as
$\hat{g}^{\mu\nu}$; and the metric tensor of the largest vector space will be
denoted $g^{\mu\nu}$. Where it does not vanish, the complement of
$\hat{g}^{\mu\nu}$ will be denoted as
$\delta^{\mu\nu}=g^{\mu\nu}-\hat{g}^{\mu\nu}$. Similarly, the Dirac matrices
$\gamma^{\mu}$, will be denoted $\gamma_{(4)}^{\mu}$ when they are strictly
four-dimensional, $\hat{\gamma}^{\mu}$ when they span the $D_{m}$ dimensional
space and $\bar{\gamma}^{\mu}$ in the space spanned by $\delta^{\mu\nu}$.

I will now present the details of the calculation in the CDR, DRED and FDH
schemes.

##  III Conventional Dimensional Regularization

In the CDR scheme, the calculation is quite straightforward. The Lagrangian
and Feynman rules are just the same as for a four-dimensional calculation,
except that the Dirac matrices $\gamma^{\mu}$ and the metric tensor
$g^{\mu\nu}$ have been extended to span a $D_{m}$ dimensional vector space.
That is,

$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\,g^{\mu\nu}\,,\qquad
g^{\mu\nu}\,g_{\mu\nu}=D_{m}\,,\qquad\gamma^{\mu}\,\gamma_{\mu}=D_{m}\,,\qquad
g^{\mu\nu}\equiv\hat{g}^{\mu\nu}\,.$ (4)

The Dirac trace, $\mathop{\rm Tr\left[{1}\right]}\nolimits=4$, retains its
standard normalization.

Although $D_{m}$ is given the representation $D_{m}=4-2\,{\varepsilon}$, the
sign of ${\varepsilon}$ is not determined. If it is taken to be positive, so
that $D_{m}<4$, then the Feynman integrals that one encounters are convergent
under the rules of ultraviolet power counting. On the other hand, infrared
power counting would prefer ${\varepsilon}<0\Rightarrow D_{m}>4$. In practice,
the sign of ${\varepsilon}$ does not matter and it can be used to regularize
both infrared and ultraviolet divergences. Regardless of the sign of
${\varepsilon}$, it is important that the vector space in which momenta take
values is larger than the standard $3+1$ dimensional space-time. This means
that the standard four-dimensional metric tensor $\eta^{\mu\nu}$ spans a
smaller space than the $D_{m}$ dimensional metric tensor, and the four-
dimensional Dirac matrices $\gamma^{0,1,2,3}$ form a subset of the full
$\gamma^{\mu}$,

$g^{\mu\nu}\,g_{\mu}^{\rho}=g^{\nu\rho}\,,\qquad\qquad
g^{\mu\nu}\,\eta_{\mu}^{\rho}=\eta^{\nu\rho}\,,\qquad\qquad\eta^{\mu\nu}\,\eta_{\mu}^{\rho}=\eta^{\nu\rho}\,.$
(5)

These considerations are of particular importance when considering chiral
objects involving $\gamma_{5}$ and the Levi-Civita tensor, but will play a
role in our discussion below.

Because the Dirac trace is unchanged, fermions still have exactly two degrees
of freedom in the CDR scheme. Gauge bosons, however, acquire extra spin
degrees of freedom in the $D_{m}$ dimensional vector space. The spin sum over
polarization vectors in a physical (axial) gauge takes the form

$-g_{\mu\nu}\,\sum_{\lambda}\epsilon^{*\,\mu}(k,\lambda)\,\epsilon^{\nu}(k,\lambda)=g_{\mu\nu}\,\left(g^{\mu\nu}-\frac{k^{\mu}\,n^{\nu}+n^{\mu}\,k^{\nu}}{k\cdot
n}\right)=D_{m}-2=2-2\,{\varepsilon}\,,$ (6)

where $n$ is the axial gauge reference vector. For massive vector bosons, the
spin sum becomes

$-g_{\mu\nu}\,\sum_{\lambda}\epsilon^{*\,\mu}(k,\lambda)\,\epsilon^{\nu}(k,\lambda)=g_{\mu\nu}\,\left(g^{\mu\nu}-\frac{k^{\mu}\,k^{\nu}}{M^{2}}\right)=D_{m}-1=3-2\,{\varepsilon}\,,$
(7)

###  III.1 Renormalization

The renormalization constants in the CDR scheme are defined as

$\begin{split}\Gamma^{(B)}_{AAA}&=Z_{1}\Gamma_{AAA}\,,\qquad\psi^{(B)\,i}_{f}=Z^{\frac{1}{2}}_{2}\,\psi^{i}_{f}\,,\qquad
A^{(B)\,a}_{\mu}=Z^{\frac{1}{2}}_{3}\,A^{a}_{\mu}\\\
\Gamma^{(B)}_{c\overline{c}A}&=\widetilde{Z}_{1}\Gamma_{q\overline{q}A}\,,\qquad\
c^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,c^{a}\,,\qquad\ \
\overline{c}^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,\overline{c}^{a}\,,\\\
\Gamma^{(B)}_{q\overline{q}A}&=Z_{1\,F}\Gamma_{q\overline{q}A}\,,\qquad\xi^{(B)}=\xi\,Z_{3}\,,\end{split}$
(8)

where $\Gamma_{abc}$ represents the vertex function involving fields $a$, $b$
and $c$.

Although we treat the quark fields as massless, we can compute the mass
anomalous dimension by introducing a fictitious scalar particle $\phi$ and
computing the $\beta$-function of its Yukawa coupling to the quarks. The
equivalence is clear from the standard model, where the Higgs Yukawa coupling
and the fermion mass are proportional at leading electroweak order and must
behave the same under QCD renormalization. For this purpose, I introduce one
more renormalization constant,
$\Gamma^{(B)}_{q\overline{q}\phi}=Z_{1\,\phi}\Gamma_{q\overline{q}\phi}$. One
can introduce a wave function renormalization for $\phi$, $Z_{3\,\phi}$, but
it will not contribute because $Z_{3\,\phi}=1+{\cal O}(\alpha_{\phi})$. Note
also that I do not need to compute the QCD corrections to the $\beta$-function
for $\alpha_{V}$, which will start at order $\alpha_{V}^{2}$ because of the
Ward Identity.

In the $\overline{\rm MS}$ scheme, the couplings renormalize as

$\begin{split}{\alpha_{s}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}\,{\alpha_{s}^{\overline{\rm{MS}}}}\,,\qquad
Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}=\frac{Z_{1}^{2}}{Z_{3}^{3}}=\frac{Z_{1\,F}^{2}}{Z_{2}^{2}\,Z_{3}}=\frac{\widetilde{Z}_{1}^{2}}{\widetilde{Z}_{3}^{2}\,Z_{3}}\\\
{\alpha_{\phi}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}\,{\alpha_{\phi}^{\overline{\rm{MS}}}}\,,\qquad
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}=\frac{Z_{1\,\phi}^{2}}{Z_{2}^{2}\,Z_{3\,\phi}}\end{split}$
(9)

The structure of the renormalization constants
$Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}$ and
$Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}$ is determined entirely by their
lowest order ($1/{\varepsilon}$) poles, which in turn define the
$\beta$-functions.

$\begin{split}{\beta^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}\left(1+\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{s}^{\overline{\rm{MS}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{MS}}}}}{\pi}-\sum_{n=0}^{\infty}\,{\beta_{n}^{\overline{\rm{MS}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+2}\\\
{\beta_{\phi}^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}+\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{s}^{\overline{\rm{MS}}}}}\,{\beta^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})\right)\left(1+\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{MS}}}}}}{\partial{\alpha_{\phi}^{\overline{\rm{MS}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{\phi}^{\overline{\rm{MS}}}}}{\pi}\left({\varepsilon}+\sum_{n=0}^{\infty}\,{\beta_{\phi\,,n}^{\overline{\rm{MS}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+1}\right)\\\
\end{split}$ (10)

The mass anomalous dimension,

${\gamma^{\overline{\rm{MS}}}}({\alpha_{s}^{\overline{\rm{MS}}}})=\frac{\mu^{2}}{m^{{\overline{\rm
MS}}}}\frac{d}{d\mu^{2}}m^{{\overline{\rm
MS}}}=\sum_{n=0}^{\infty}-{\gamma_{n}^{\overline{\rm{MS}}}}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{n+1}$
(11)

is defined in terms of $m$, rather than $m^{2}$, with the result that
${\gamma_{n}^{\overline{\rm{MS}}}}=\frac{1}{2}{\beta_{\phi\,,n}^{\overline{\rm{MS}}}}$.
The results for ${\beta_{n}^{\overline{\rm{MS}}}}$ and
${\gamma_{n}^{\overline{\rm{MS}}}}$ through three loops are given in Appendix
A.

###  III.2 Vacuum polarization in the CDR scheme

The imaginary part of the unrenormalized vacuum polarization tensor in the CDR
scheme is

$\begin{split}\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{CDR}}\right]&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{55}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{1711}{48}-\frac{15}{4}\,\zeta_{2}-19\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\,\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)\right.\\\
&\hskip-30.0pt-\frac{3}{32}\,C_{F}^{2}+C_{F}\,C_{A}\left(\frac{487}{48}-\frac{33}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{11}{6}+\frac{3}{2}\,\zeta_{3}\right)\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{143}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{50339}{576}-\frac{231}{32}\,\zeta_{2}-\frac{109}{2}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-30.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{4417}{288}+\frac{21}{16}\,\zeta_{2}+\frac{19}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(12)

Upon renormalizing the QCD coupling according to Eq. (9), setting
${\alpha_{V}^{B}}\to{\alpha_{V}}\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}$,
and dropping terms of order $({\varepsilon})$, I obtain

$\begin{split}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(13)

In this way of performing the calculation, all of the QCD states that appear
are internal states, so the HV scheme gives exactly the same result.

###  III.3 Total Decay rate and annihilation cross section in the CDR scheme

The decay rate and the annihilation cross section are determined by computing
the imaginary part of the forward scattering amplitude. For the decay rate,
this means attaching the polarization vector ${\varepsilon}^{\mu}(Q,\lambda)$
and its conjugate ${\varepsilon}^{\nu}(Q,\lambda)^{*}$ ($Q^{2}=M_{V}^{2}$) and
averaging over the spins,

$\Gamma^{{CDR}}_{V\to\ {\rm hadrons}}=\frac{1}{M_{V}}\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]\,{\varepsilon}^{\nu}(Q,\lambda)^{*}\,,$
(14)

where

$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{N_{\rm
spins}}\left(-g^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}\right)\,.$ (15)

Notice that because the imaginary part of the vacuum polarization tensor is
finite, it does not matter whether the spin sum is taken in
$D_{m}=4-2\,{\varepsilon}$ dimensions as in the CDR scheme or in four
dimensions as in the HV scheme as the difference is of order ${\varepsilon}$.
The result is

$\begin{split}\Gamma^{{CDR}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(16)

in agreement with Eqs. (LABEL:eqn:knownresult-3).

For the annihilation cross section $\sigma_{e^{+}\,e^{-}\to\ {\rm hadrons}}$,
one attaches fermion bilinears to each end of the vacuum polarization tensor
and averages over the spins.

$\sigma^{{CDR}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=\frac{2}{Q^{2}}\frac{e^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\gamma^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\gamma^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\,.$
(17)

Because this is a forward scattering amplitude, the spinor bilinears can be
combined into a trace,

$\frac{1}{2}\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\gamma^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\gamma^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\gamma^{\mu}\not{p}_{e^{-}}\,\gamma^{\nu}}\right]}\nolimits=\left(-Q^{2}\,g^{\mu\,\nu}+Q^{\mu}\,Q^{\nu}\right)\,,$
(18)

where the last identification results from the fact that
$Q^{\mu}=p_{e^{-}}^{\mu}+p_{e^{+}}^{\mu}$,
$p_{e^{-}}\cdot\,Q=p_{e^{+}}\cdot\,Q=Q^{2}/2$. The result is

$\begin{split}\sigma^{{CDR}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(19)

again in agreement with Eqs. (LABEL:eqn:knownresult-3).

Thus, I have established that I can reproduce the known results in the CDR
scheme through three-loop order, which is a strong check on my computational
framework.

##  IV Dimensional Reduction

In dimensional reduction, one starts from standard four-dimensional space-time
and compactifies to a smaller vector space of dimension
$D_{m}=4-2\,{\varepsilon}<4$ in which momenta take values. The particles in
the spectrum, however, retain the spin degrees of freedom of four dimensions.
That is, both fermions and gauge bosons still have two degrees of freedom.
This is by design, of course, since it is required by supersymmetry. All Dirac
algebra can be treated as four-dimensional. However, now the four-dimensional
metric tensor $\eta^{\mu\nu}$ spans a larger space than the $D_{m}$
dimensional metric $\hat{g}^{\mu\nu}$ that might arise from tensor momentum
integrals,

$\hat{g}^{\mu\nu}\,\eta_{\mu}^{\rho}=\hat{g}^{\nu\rho}\,.$ (20)

There is also a very serious consequence of the fact that the $D_{m}$
dimensional vector space is smaller than four-dimensional space-time. The Ward
Identity only applies to the $D_{m}$ dimensional vector space. This means that
the $2\,{\varepsilon}$ spin degrees of freedom that are not protected by the
Ward Identity must renormalize differently than the $2-2\,{\varepsilon}$
degrees of freedom that are protected. In supersymmetric theories, the
supersymmetry provides the missing Ward Identity which demands that the
$2\,{\varepsilon}$ spin degrees of freedom be treated as gauge bosons. In
nonsupersymmetric theories, however, they must be considered to be distinct
particles, with distinct couplings and renormalization properties. It is
common to refer to these extra degrees of freedom as “${\varepsilon}$-scalars”
or as “evanescent” degrees of freedom.

Once the evanescent degrees of freedom (which I will label
$A_{e}^{a\,\tilde{\mu}}$, to distinguish them from the gluons, $A^{a\,\mu}$)
are recognized as independent particles, it is apparent that their couplings
are also independent, not only of the QCD coupling, but of one another. That
is, the coupling $g_{e}$ of the evanescent gluons to the quarks is not only
distinct from $g$, the coupling of QCD, but is also distinct from
$\lambda_{i}$, the quartic couplings of the evanescent gluons to themselves.
(The quartic gauge coupling of QCD splits into three independent quartic
couplings of the evanescent gluons.) Note that the massive vector boson
$V^{\mu}$ also has evanescent degrees of freedom, $V_{e}^{\tilde{\mu}}$, which
couple to quarks with strength $g_{Ve}$.

Thus, the Lagrangian in the DRED scheme becomes:

$\begin{split}{\cal
L}=&-\frac{1}{2}A^{a}_{\mu}\left(\partial^{\mu}\partial^{\nu}(1-\xi^{-1})-\hat{g}^{\mu\nu}\Box\right)A^{a}_{\nu}-g\,f^{abc}(\partial^{\mu}\,A^{a\,\nu})A^{b}_{\mu}\,A^{c}_{\nu}-\frac{g^{2}}{4}f^{abc}\,f^{ade}\,A^{b\,\mu}\,A^{c\,\nu}\,A^{d}_{\mu}\,A^{e}_{\nu}\\\
&+i\sum_{f}\,\overline{\psi}_{f}^{i}\left(\delta_{ij}\not{\partial}-i\,g\,t^{a}_{ij}\not{A}^{a}-i\,g_{V}\,Q_{f}\not{V}\right)\,\psi_{f}^{j}-\overline{c}^{a}\Box\,c^{a}+g\,f^{abc}\left(\partial_{\mu}\,\overline{c}^{a}\right)\,A^{b\,\mu}\,c^{c}\\\
&+\frac{1}{2}A_{e\,\tilde{\mu}}^{a}\Box\
A_{e}^{a\,\tilde{\mu}}-g\,f^{abc}(\partial^{\mu}\,A_{e}^{a\,\tilde{\nu}})A^{b}_{\mu}\,A^{c}_{e\,\tilde{\nu}}+\frac{g^{2}}{2}f^{abc}\,f^{adf}\,A^{b\,\mu}\,A_{e}^{c\,\tilde{\nu}}\,A^{d}_{\mu}\,A_{e\,\tilde{\nu}}^{f}-\frac{1}{4}\sum_{i}\lambda_{i}\,H_{i}^{bcdf}\,A_{e}^{b\,\tilde{\mu}}\,A_{e}^{c\,\tilde{\nu}}\,A_{e\,\tilde{\mu}}^{d}\,A_{e\,\tilde{\nu}}^{f}\\\
&+\sum_{f}\,\overline{\psi}_{f}^{i}\left(g_{e}\,t^{a}_{ij}\not{A}_{e}^{a}+g_{Ve}\,Q_{f}\not{V}_{e}\right)\,\psi_{f}^{j}\,.\end{split}$
(21)

As mentioned above, the quartic coupling of the evanescent gluons splits into
three terms, which mix under renormalization. One can choose the tensors
$H_{i}^{bcde}$ to be Harlander et al. (2006a)

$\begin{split}H_{1}^{bcde}=&\frac{1}{2}\left(f^{abc}\,f^{ade}+f^{abe}\,f^{adc}\right)\\\
H_{2}^{bcde}=&\delta^{bc}\delta^{de}+\delta^{bd}\delta^{ce}+\delta^{be}\delta^{cd}\\\
H_{3}^{bcde}=&\frac{1}{2}\left(\delta^{bc}\delta^{de}+\delta^{be}\delta^{cd}\right)-\delta^{bd}\delta^{ce}\,,\end{split}$
(22)

Although the quartic couplings enter the $\beta$-functions and anomalous
dimension at three loops and are essential to the renormalization program,
they do not explicitly contribute to the calculation at hand.

Now that the correct spectrum has been identified, one must carefully consider
the renormalization program. The naïve application of the principle of minimal
subtraction leads to the violation of unitarity van Damme and ’t Hooft (1985).
Because the contributions of evanescent states and couplings to scattering
amplitudes are weighted by a factor ${\varepsilon}$, the leading one-loop
contribution is finite and therefore not subtracted. As one proceeds to higher
orders, there is a mismatch among the counterterms such that the
renormalization program fails to remove all of the ultraviolet singularities.

A successful renormalization program for the DRED scheme Jack et al. (1994a,
b) applies the principle of minimal subtraction to the evanescent Green
functions (that is, Green functions with external evanescent states)
themselves. At each order, the renormalization scheme renders the evanescent
Green functions finite. Since evanescent Green functions enter into the
scattering amplitudes of physical particles at order ${\varepsilon}$ and they
are rendered finite by renormalization, they never contribute to physical
scattering amplitudes.

The evanescent coupling still contributes to Green functions with only
physical external states, but the contribution is rendered finite by the
prescribed renormalization program Jack et al. (1994a, b); Harlander et al.
(2006b, a). Because the evanescent coupling, $\alpha_{e}$ renormalizes
differently than the gauge coupling $\alpha_{s}$, the two cannot be
identified, even at the end of the calculation. One can choose a
renormalization point where the two coincide, but they evolve differently
under renormalization group transformations and their values will diverge as
one moves away from the renormalization point.

Still, the evanescent coupling is essentially a fictitious quantity and one
finds that if one computes a physical quantity in the DRED scheme and then
converts the running couplings of the DRED scheme to those of a scheme such as
CDR that has no evanescent couplings, the factors of $\alpha_{e}$ drop out
Harlander et al. (2006b, a).

###  IV.1 Renormalization

The renormalization constants in the DRED scheme are defined as

$\begin{split}\Gamma^{(B)}_{AAA}&=Z_{1}\Gamma_{AAA}\,,\qquad\psi^{(B)\,i}_{f}=Z^{\frac{1}{2}}_{2}\,\psi^{i}_{f}\,,\qquad
A^{(B)\,a}_{\mu}=Z^{\frac{1}{2}}_{3}\,A^{a}_{\mu}\\\
\Gamma^{(B)}_{c\overline{c}A}&=\widetilde{Z}_{1}\Gamma_{q\overline{q}A}\,,\qquad\
c^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,c^{a}\,,\qquad\ \
\overline{c}^{(B)\,a}=\widetilde{Z}^{\frac{1}{2}}_{3}\,\overline{c}^{a}\,,\\\
\Gamma^{(B)}_{q\overline{q}A}&=Z_{1\,F}\Gamma_{q\overline{q}A}\,,\qquad\xi^{(B)}=\xi\,Z_{3}\,,\\\
\Gamma^{(B)}_{q\overline{q}e}&=Z_{1\,e}\Gamma_{q\overline{q}e}\,,\qquad
A^{(B)\,a}_{e\,\mu}=Z^{\frac{1}{2}}_{3\,e}\,A^{a}_{e\,\mu}\,,\qquad\Gamma^{(B)\,i}_{eeee}=Z^{i}_{1\,eeee}\,\Gamma^{i}_{eeee}\,,\\\
\Gamma^{(B)}_{q\overline{q}V_{e}}&=Z_{1\,Ve}\Gamma_{q\overline{q}V_{e}}\,,\qquad
V^{(B)}_{e\,\mu}=Z^{\frac{1}{2}}_{3\,Ve}\,V_{e\,\mu}\,.\end{split}$ (23)

In addition, I again introduce the fictitious scalar that allows me to compute
the mass anomalous dimension for massless quarks. Note that while the Ward
Identity protects $\alpha_{V}$ from leading QCD corrections, it does not
protect $\alpha_{Ve}$. That is why I need to introduce renormalization
constants for the vertex and wave-function and why I need to compute the
$\beta$-function of $\alpha_{Ve}$.

In the ${\overline{\rm DR}}$ scheme (modified minimal subtraction in the DRED
scheme), the couplings renormalize as

$\begin{split}{\alpha_{s}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\alpha_{s}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}=\frac{Z_{1}^{2}}{Z_{3}^{3}}=\frac{Z_{1\,F}^{2}}{Z_{2}^{2}\,Z_{3}}=\frac{\widetilde{Z}_{1}^{2}}{\widetilde{Z}_{3}^{2}\,Z_{3}}\,,\\\
{\alpha_{e}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\alpha_{e}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,e}^{2}}{Z_{2}^{2}\,Z_{3\,e}}\,,\\\
{\alpha_{Ve}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}\,{\alpha_{Ve}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,Ve}^{2}}{Z_{2}^{2}\,Z_{3\,Ve}}\,,\\\
\
{\alpha_{\phi}^{B}}&=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}\,{\alpha_{\phi}^{\overline{\rm{DR}}}}\,,\qquad
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}=\frac{Z_{1\,\phi}^{2}}{Z_{2}^{2}\,Z_{3\,\phi}}\,.\end{split}$
(24)

and the $\beta$-functions are given by

$\begin{split}{\beta^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\left(1+\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{s}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\\\
{\beta_{e}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\left(1+\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{e}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-{\varepsilon}\frac{{\alpha_{e}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{e,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\\\
{\beta_{Ve}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\\\ &\hskip
100.0pt\times\left(1+\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{Ve}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{Ve}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{Ve}^{\overline{\rm{DR}}}}}{\pi}\left({\varepsilon}+\sum_{i,j,k,l,m}\,{\beta_{Ve,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\right)\\\
{\beta_{\phi}^{\overline{\rm{DR}}}}=\mu^{2}\frac{d}{d\,\mu^{2}}\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}&=-\left({\varepsilon}\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{s}^{\overline{\rm{DR}}}}}\,{\beta^{\overline{\rm{DR}}}}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{e}^{\overline{\rm{DR}}}}}\,{\beta_{e}^{\overline{\rm{DR}}}}+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\eta_{i}^{{\overline{\rm
DR}}}}}\,{\beta_{\eta_{i}}^{\overline{\rm{DR}}}}\right)\\\ &\hskip
100.0pt\times\left(1+\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}\frac{\partial
Z_{{\alpha_{\phi}^{\overline{\rm{DR}}}}}}{\partial{\alpha_{\phi}^{\overline{\rm{DR}}}}}\right)^{-1}\\\
&=-\frac{{\alpha_{\phi}^{\overline{\rm{DR}}}}}{\pi}\left({\varepsilon}+\sum_{i,j,k,l,m}\,{\beta_{\phi,\,ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\right)\\\ \end{split}$ (25)

Through three-loop order, the $\eta_{i}$ do not contribute to the QCD
$\beta$-function, ${\beta^{\overline{\rm{DR}}}}$, nor to the vacuum
polarization of $V$ (or $V_{e}$). To three-loop order, I find agreement with
known results Harlander et al. (2006b, a) and derive new results for the
$\beta$-function of $\alpha_{Ve}$. The coefficients of the $\beta$-functions
and anomalous dimensions are given in Appendix B.

By comparing ${\beta_{Ve,\,\,}^{\overline{\rm{DR}}}}$ and
${\gamma^{\overline{\rm{DR}}}}$ in Eqs. (61-62), we see that the term
“${\varepsilon}$-scalar” is a misnomer. If the evanescent part of $V$ were a
true scalar, its $\beta$-function would coincide (but for a factor of $2$)
with the mass anomalous dimension. The pure
${\alpha_{s}^{\overline{\rm{DR}}}}$ terms do coincide, because there is no
nonvanishing contraction of the Lorentz indices of the evanescent $V$ and
those of the gluons. Because there are contractions between the Lorentz
indices of the evanescent $V$ and those of the evanescent gluons, however,
terms involving ${\alpha_{e}^{\overline{\rm{DR}}}}$ do not agree.

Calculations in the DRED scheme naturally produce results in terms of
${\alpha_{s}^{\overline{\rm{DR}}}}$ while the standard result has been
expressed in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$. One can always
convert one renormalized coupling to another. The rule for converting
${\alpha_{s}^{\overline{\rm{DR}}}}\to{\alpha_{s}^{\overline{\rm{MS}}}}$,
derived in Refs. Kunszt et al. (1994); Harlander et al. (2006b), is

${\alpha_{s}^{\overline{\rm{DR}}}}={\alpha_{s}^{\overline{\rm{MS}}}}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{C_{A}}{12}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\frac{11}{72}C_{A}^{2}-{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}\frac{C_{F}\,N_{f}}{16}+\ldots\right]$
(26)

When the result is expressed in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$,
all ${\alpha_{e}^{\overline{\rm{DR}}}}$ terms drop out.

###  IV.2 Vacuum polarization in the DRED scheme

In the DRED scheme, there are two independent transverse vacuum polarization
tensors,

$\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{DRED}}\right]=\frac{-Q^{2}\,\hat{g}_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}\,\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]-Q^{2}\,\frac{\delta_{\mu\nu}}{2\,{\varepsilon}}\,\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]\,,$
(27)

where

$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{51}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{497}{16}-\frac{15}{4}\zeta_{2}-15\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[-{\varepsilon}\,\frac{3}{4}-{\varepsilon}^{2}\,\frac{29}{8}+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)-\frac{3}{32}C_{F}^{2}+\left(\frac{77}{8}-\frac{33}{4}\zeta_{3}\right)\,C_{F}\,C_{A}-\left(\frac{7}{4}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{141}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{15301}{192}-\frac{231}{32}\,\zeta_{2}-\frac{193}{4}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-20.0pt\left.\left.+C_{F}\,N_{f}\left(-\frac{1355}{96}+\frac{21}{16}\,\zeta_{2}+\frac{17}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{3}{4}C_{F}^{2}-\frac{3}{8}C_{F}\,C_{A}+\frac{3}{16}C_{F}\,N_{f}-{\varepsilon}\left(\frac{47}{8}C_{F}^{2}-\frac{11}{4}C_{F}\,C_{A}+\frac{7}{4}C_{F}\,N_{f}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[-\frac{9}{8}C_{F}^{2}-{\varepsilon}\left(\frac{141}{16}C_{F}^{2}+\frac{21}{16}C_{F}\,C_{A}\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\,,\end{split}$
(28)

and

$\begin{split}\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]&={\alpha_{Ve}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{{\varepsilon}+2\,{\varepsilon}^{2}+\left(4-\frac{3}{2}\zeta_{2}\right){\varepsilon}^{3}+{\cal
O}({\varepsilon}^{4})\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{2}+{\varepsilon}\frac{29}{4}+{\varepsilon}^{2}\,\left(\frac{227}{8}-\frac{15}{2}\zeta_{2}-6\,\zeta_{3}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[-1-4\,{\varepsilon}-{\varepsilon}^{2}\left(\frac{27}{2}-5\,\zeta_{2}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{9}{8}C_{F}^{2}+\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)+\frac{279}{32}C_{F}^{2}+\frac{199}{32}C_{F}\,C_{A}-\frac{17}{16}C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(\frac{3139}{64}-\frac{189}{16}\,\zeta_{2}-\frac{45}{4}\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{2473}{64}-\frac{231}{32}\,\zeta_{2}-\frac{75}{8}\,\zeta_{3}\right)\right.\\\
&\hskip-20.0pt\left.\left.+C_{F}\,N_{f}\left(-\frac{207}{32}+\frac{21}{16}\,\zeta_{2}+\frac{3}{2}\,\zeta_{3}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[-\frac{1}{{\varepsilon}}\frac{9}{4}C_{F}^{2}-\frac{129}{8}C_{F}^{2}-\frac{3}{8}C_{F}\,C_{A}\right.\\\
&\hskip-30.0pt\phantom{1}\left.-{\varepsilon}\left(\left(\frac{671}{8}-\frac{189}{8}\zeta_{2}-9\,\zeta_{3}\right)\,C_{F}^{2}+\frac{53}{16}C_{F}\,C_{A}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(C_{F}^{2}-\frac{1}{4}C_{F}\,C_{A}+\frac{1}{8}C_{F}\,N_{f}\right)+\frac{13}{2}C_{F}^{2}-\frac{3}{2}C_{F}\,C_{A}+\frac{15}{16}C_{F}\,N_{f}\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\varepsilon}\left(\left(31-\frac{21}{2}\zeta_{2}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}-\left(\frac{53}{8}-\frac{21}{8}\zeta_{2}-\frac{3}{8}\zeta_{3}\right)\,C_{F}\,C_{A}+\left(\frac{157}{32}-\frac{21}{16}\zeta_{2}\right)\,C_{F}\,N_{f}\right)+{\cal
O}({\varepsilon}^{2})\right]\\\ &\hskip-50.0pt\phantom{1}\left.+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\,,\\\
\end{split}$ (29)

where $\displaystyle{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)$
denotes terms for which the sum of the powers of
$\displaystyle\left(\frac{{\alpha_{s}^{B}}}{\pi}\right)$ and
$\displaystyle\left(\frac{{\alpha_{e}^{B}}}{\pi}\right)$ is at least three.

Upon renormalization according to Eq. (24) and expanding in terms of
${\alpha_{s}^{\overline{\rm{MS}}}}$ according to Eq. (26), I find that

$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{DRED}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}\frac{3}{4}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}{\beta_{20}^{\overline{\rm{DR}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{2}\left[-C_{F}^{2}\frac{3}{32}+C_{F}\,C_{A}\left(\frac{121}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{11}{16}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left(\left(\frac{{\alpha_{s}^{B}}}{\pi},\frac{{\alpha_{e}^{B}}}{\pi}\right)^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{DRED}}\right]&={\cal
O}({\varepsilon})\,.\end{split}$ (30)

###  IV.3 Total Decay rate and annihilation cross section in the DRED scheme

As in the CDR scheme, the decay rate and annihilation cross section are
determined from the imaginary part of the forward scattering amplitude.

$\Gamma^{{DRED}}_{V\to\ {\rm hadrons}}=\frac{1}{M_{V}}\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED}}\right]\,{\varepsilon}^{\nu}(Q,\lambda)^{*}\,,$
(31)

where

$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{3}\left(-\hat{g}^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}-\delta^{\mu\nu}\right)\,.$
(32)

The evanescent part of the spin average contracts only with the $\Pi_{B}(Q)$
term, which has been renormalized to be of order $({\varepsilon})$, so that
the result is

$\begin{split}\Gamma^{{DRED}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(33)

just like in the CDR calculation.

For the annihilation cross section $\sigma_{e^{+}\,e^{-}\to\ {\rm hadrons}}$,
one attaches fermion bilinears to each end of the vacuum polarization tensor
and averages over the spins.

$\begin{split}\sigma^{{DRED}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}&=\frac{2}{Q^{2}}\frac{e^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\hat{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\hat{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\\\
&+\frac{2}{Q^{2}}\frac{e_{\ell\,e}^{2}}{4}\sum_{\lambda\,\lambda^{{}^{\prime}}}\frac{{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\bar{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}}{Q^{2}}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{DRED},\,{\alpha_{V}}\to\alpha}\right]\frac{{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\bar{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}}{Q^{2}}\,,\end{split}$
(34)

where $e_{\ell\,e}$ represents the coupling of the evanescent photon to the
electron. Combining the spinor bilinears into traces,

$\begin{split}\frac{1}{2}&\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\hat{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\hat{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\gamma^{\mu}\not{p}_{e^{-}}\,\gamma^{\nu}}\right]}\nolimits=\left(-Q^{2}\,\hat{g}^{\mu\,\nu}+Q^{\mu}\,Q^{\nu}\right)\\\
\frac{1}{2}&\sum_{\lambda\,\lambda^{{}^{\prime}}}{{\left\langle\overline{v}(p_{e^{+}},\lambda)\left|\bar{\gamma}^{\mu}\right|u(p_{e^{-}},\lambda^{{}^{\prime}})\right\rangle}}{{\left\langle\overline{u}(p_{e^{-}},\lambda^{{}^{\prime}})\left|\bar{\gamma}^{\nu}\right|v(p_{e^{+}},\lambda)\right\rangle}}=\frac{1}{2}\mathop{\rm
Tr\left[{\not{p}_{e^{+}}\,\bar{\gamma}^{\mu}\not{p}_{e^{-}}\,\bar{\gamma}^{\nu}}\right]}\nolimits=\left(-Q^{2}\,\delta^{\mu\,\nu}\right)\,\end{split}$
(35)

The final result is

$\begin{split}\sigma^{{DRED}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{3}{32}+C_{F}\,C_{A}\,\left(\frac{123}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{11}{16}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(36)

again in agreement with Eqs. (LABEL:eqn:knownresult-3). As promised, under the
DRED scheme renormalization program, evanescent Green functions are rendered
finite by renormalization and contribute to scattering amplitudes at order
$({\varepsilon})$. Also as promised, the results are completely equivalent to
those of the CDR scheme.

##  V The Four-Dimensional Helicity Scheme

In the four-dimensional helicity scheme, one defines an enlarged vector space
of dimensionality $D_{m}=4-2\,{\varepsilon}$, in which loop momenta take
values, as in the CDR scheme. In addition, one defines a still larger vector
space, of dimensionality $D_{s}=4$, in which internal spin degrees of freedom
take values. The precise rules for the FDH scheme are given in Ref. Bern et
al. (2002). They are:

  1. 1.

As in ordinary dimensional regularization, all momentum integrals are
integrated over $D_{m}$ dimensional momenta. Metric tensors resulting from
tensor integrals are $D_{m}$ dimensional.

  2. 2.

All “observed” external states are taken to be four-dimensional, as are their
momenta and polarization vectors. This facilitates the use of helicity states
for observed particles.

  3. 3.

All “unobserved” or internal states are treated as $D_{s}$ dimensional, and
the $D_{s}$ dimensional vector space is taken to be larger than the $D_{m}$
dimensional vector space. Unobserved states include virtual states inside of
loops, virtual states inside of trees as well as external states that have
infrared sensitive overlaps with other external states.

  4. 4.

Both the $D_{s}$ and $D_{m}$ dimensional vector spaces are larger than the
standard four-dimensional space-time, so that contraction of four-dimensional
objects with $D_{m}$ or $D_{s}$ dimensional objects yields only four-
dimensional components.

To keep track of the many vector spaces and their overlapping domains, I give
the result of the contractions of the various metric tensors with one another,

$\begin{split}g^{\mu\nu}\,g_{\mu\nu}&=D_{s}\,,\qquad\
\hat{g}^{\mu\nu}\,\hat{g}_{\mu\nu}=D_{m}\,,\qquad\eta^{\mu\nu}\,\eta_{\mu\nu}=4\,,\qquad\delta^{\mu\nu}\,\delta_{\mu\nu}=D_{x}=D_{s}-D_{m}\\\
g^{\mu\nu}\hat{g}^{\rho}_{\nu}&=\hat{g}^{\mu\rho}\,,\qquad
g^{\mu\nu}\eta^{\rho}_{\nu}=\eta^{\mu\rho}\,,\qquad\hat{g}^{\mu\nu}\eta^{\rho}_{\nu}=\eta^{\mu\rho}\,,\\\
g^{\mu\nu}\delta^{\rho}_{\nu}&=\delta^{\mu\rho}\,,\qquad\hat{g}^{\mu\nu}\delta^{\rho}_{\nu}=0\,,\qquad\quad\
\eta^{\mu\nu}\delta^{\rho}_{\nu}=0\,.\\\ \end{split}$ (37)

Like the HV scheme, the FDH scheme treats observed states as four-dimensional.
In inclusive calculations, however, where there are infrared overlaps among
external states, the external states are taken to be $D_{s}$ dimensional in
the infrared regions.

As in the DRED scheme, spin degrees of freedom take values in a vector space
that is larger than that in which momenta take values. It would seem,
therefore, that the same remarks regarding the Ward Identity and the
conclusion that the $D_{x}=D_{s}-D_{m}$ dimensional components of the gauge
fields and their couplings must be considered as distinct from the $D_{m}$
dimensional gauge fields and couplings would apply. That is not, however, how
the FDH scheme is used. All field components in the $D_{s}$ dimensional space
are treated as gauge fields and no distinction is made between the couplings.
It is common, however, to define an interpolating scheme, the “$\delta_{R}$”
scheme, in which $D_{s}=4-2\,{\varepsilon}\,\delta_{R}$. The parameter
$\delta_{R}$ interpolates between the HV scheme ($\delta_{R}=1$) and the FDH
scheme ($\delta_{R}=0$). Using this scheme gives one a handle on the impact of
the evanescent degrees of freedom on the result, but not on the impact of a
distinct evanescent coupling.

It is claimed Bern et al. (2002) that the essential difference between the FDH
and DRED schemes is that in the former $D_{m}>4$, while in the latter
$D_{m}<4$. It must be this difference, then, that allows for the very
different handling of the evanescent couplings and degrees of freedom. We
shall see what impact this choice has in the calculation and discussion below.

###  V.1 Renormalization

I will not give detailed results for the renormalization parameters of the FDH
scheme. There is no point in doing so because, as I will show, the rules of
the FDH scheme enumerated in the previous section are not consistent with a
successful renormalization program. The first sign that there is a problem
with the renormalization program comes in the computation of the one-loop
renormalization constants. In particular, the gluon vacuum polarization tensor
splits into two independent components,
$\Pi_{A}^{\mu\nu}=\Pi_{A}(Q^{2})\,\left((-Q^{2}\hat{g}^{\mu\nu}+Q^{\mu}\,Q^{\nu}\right)$
and $\Pi_{B}^{\mu\nu}=\Pi_{B}(Q^{2})\,\delta^{\mu\nu}$, both of which are
singular. This is a clear warning that what the FDH scheme calls the gluon is
in fact two distinct sets of degrees of freedom. If I ignore $\Pi_{B}$ and
just renormalize $\Pi_{A}$, I find the usual result that

${\beta_{0}^{\overline{\rm{FDH}}}}=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,.$
(38)

Note that I also get this result if I take the spin average (trace) of the
full vacuum polarization tensor. Because $\Pi_{B}$ is weighted by a factor of
$2\,{\varepsilon}$, its contribution to the spin average is not singular.
Because the leading order term in the quantities being calculated is of order
one, and the NLO term of order $\alpha_{s}$, this result for the one-loop
$\beta$-function is all that is needed to compute the renormalized cross
section at NNLO. Furthermore, the many NLO results that have been obtained
using the FDH scheme have all renormalized using the above result for
${\beta_{0}^{\overline{\rm{FDH}}}}$.

When I try to proceed to the two-loop beta function, I find that both
$\Pi_{A}$ and $\Pi_{B}$ contribute singular terms to the spin-averaged vacuum
polarization, while if I again ignore $\Pi_{B}$ and renormalize $\Pi_{A}$, I
obtain the usual value for $\beta_{1}$,

${\beta_{1}^{\overline{\rm{FDH}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,.$
(39)

This seems to be the choice made in Ref. Bern et al. (2002) as they quote only
the result for terms proportional to $Q^{\mu}Q^{\nu}$, which would be part of
my $\Pi_{A}$. Since the standard lore has been that
${\alpha_{s}^{\overline{\rm{FDH}}}}$ and ${\alpha_{s}^{\overline{\rm{DR}}}}$
coincide, at least through second order corrections, this seems to be the most
reasonable choice. Furthermore, it means that the conversion to
${\alpha_{s}^{\overline{\rm{MS}}}}$ will be Kunszt et al. (1994); Bern et al.
(2002)

${\alpha_{s}^{\overline{\rm{FDH}}}}={\alpha_{s}^{\overline{\rm{MS}}}}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{C_{A}}{12}+\ldots\right]$
(40)

As it turns out, it does not matter what choice one makes as even the one-loop
result for ${\beta_{0}^{\overline{\rm{FDH}}}}$, which seems safe if only
because it is familiar, leads to the violation of unitarity.

###  V.2 Vacuum polarization in the FDH scheme

Leaving aside the question of renormalization beyond one-loop, I will proceed
with the calculation of the $V$-boson vacuum polarization. In performing
calculations in the FDH scheme, it becomes apparent that the results are
identical, term-by-term. to the calculation in the DRED scheme, except that
the evanescent gluons are identified as gluons and the coupling $\alpha_{e}$
is set to $\alpha_{s}$. Therefore I find that

$\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{FDH}}\right]=\frac{-Q^{2}\,\hat{g}_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}\,\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]-Q^{2}\,\frac{\delta_{\mu\nu}}{2\,{\varepsilon}}\,\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]\,,$
(41)

where

$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt1+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{3}{4}+{\varepsilon}\left(\frac{45}{8}-6\,\zeta_{3}\right)+{\varepsilon}^{2}\,\left(\frac{439}{16}-\frac{15}{4}\zeta_{2}-15\,\zeta_{3}-9\,\zeta_{4}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(\frac{11}{16}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right)-\frac{15}{32}C_{F}^{2}+\left(\frac{37}{4}-\frac{33}{4}\zeta_{3}\right)\,C_{F}\,C_{A}-\left(\frac{25}{16}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{235}{32}-\frac{111}{8}\,\zeta_{3}+\frac{45}{2}\,\zeta_{5}\right)+C_{F}\,C_{A}\left(\frac{14521}{192}-\frac{231}{32}\,\zeta_{2}-\frac{193}{4}\,\zeta_{3}-\frac{99}{8}\,\zeta_{4}-\frac{15}{4}\,\zeta_{5}\right)\right.\\\
&\hskip-20.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{1187}{96}+\frac{21}{16}\,\zeta_{2}+\frac{17}{2}\,\zeta_{3}+\frac{9}{4}\,\zeta_{4}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(42)

and

$\begin{split}\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{\varepsilon}\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\hskip-50.0pt{\varepsilon}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{{\varepsilon}}C_{F}\,\left[\frac{1}{2}+{\varepsilon}\frac{13}{4}+{\varepsilon}^{2}\,\left(\frac{119}{8}-\frac{5}{2}\zeta_{2}-6\,\zeta_{3}\right)+{\cal
O}({\varepsilon}^{3})\right]\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{2}\,\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{2\,{\varepsilon}}\left[\frac{1}{{\varepsilon}}\left(-\frac{1}{8}C_{F}^{2}+\frac{7}{16}C_{F}\,C_{A}\right)-\frac{29}{32}C_{F}^{2}+\frac{139}{32}C_{F}\,C_{A}-\frac{1}{8}C_{F}\,N_{f}\right.\\\
&\hskip-30.0pt+{\varepsilon}\left(C_{F}^{2}\left(-\frac{245}{64}+\frac{21}{16}\,\zeta_{2}-3\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{1837}{64}-\frac{147}{32}\,\zeta_{2}-9\,\zeta_{3}\right)\right.\\\
&\hskip-20.0pt\left.\left.\left.+C_{F}\,N_{f}\left(-\frac{25}{16}+\frac{3}{2}\,\zeta_{3}\right)\right)+{\cal
O}({\varepsilon}^{2})\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\right)\right\\}\,.\\\
\end{split}$ (43)

Upon renormalizing such that

${\left(\frac{\alpha_{s}^{B}}{\pi}\right)}\to{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{-{\varepsilon}}\left(1-\frac{{\beta_{0}^{\overline{\rm{FDH}}}}}{{\varepsilon}}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\right)\,,\qquad\qquad{\alpha_{V}^{B}}\to{\alpha_{V}}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{-{\varepsilon}}\,,$
(44)

I find that

$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{FDH}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{3}{4}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left[-C_{F}^{2}\frac{15}{32}+C_{F}\,C_{A}\left(\frac{131}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\left(-\frac{5}{8}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{FDH}}\right]&={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{1}{2}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left[\frac{1}{{\varepsilon}}\left(-C_{F}^{2}\frac{1}{8}-C_{F}\,C_{A}\frac{1}{48}+C_{F}\,N_{f}\frac{1}{12}\right)\left(1+3{\varepsilon}\ln\frac{\mu^{2}}{Q^{2}}\right)\right.\\\
&\left.\left.-C_{F}^{2}\frac{29}{32}+C_{F}\,C_{A}\frac{131}{96}+C_{F}\,N_{f}\frac{5}{12}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\right)\right\\}\\\
&\hskip-45.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left\\{{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\frac{1}{2}C_{F}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}{\beta_{0}^{\overline{\rm{FDH}}}}\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-50.0pt\phantom{1}+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\left[\frac{1}{{\varepsilon}}\left(-C_{F}^{2}\frac{1}{8}-C_{F}\,C_{A}\frac{1}{48}+C_{F}\,N_{f}\frac{1}{12}\right)\left(1+3{\varepsilon}\ln\frac{\mu^{2}}{Q^{2}}\right)\right.\\\
&\left.\left.-C_{F}^{2}\frac{29}{32}+C_{F}\,C_{A}\frac{45}{32}+C_{F}\,N_{f}\frac{5}{12}\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(45)

###  V.3 Total Decay rate and annihilation cross section in the FDH scheme

The results of the vacuum polarization calculation look to be disastrous as
$\Pi_{B}$ is singular at order $\alpha_{s}^{2}$. However, the rules of the FDH
scheme, enumerated above, specify that external states are taken to be four-
dimensional. This means that the spin average of the vector polarizations is

$\frac{1}{N_{\rm
spins}}\sum_{\lambda}{\varepsilon}^{\mu}(Q,\lambda)\,{\varepsilon}^{\nu}(Q,\lambda)^{*}=\frac{1}{3}\left(-\eta^{\mu\nu}+\frac{Q^{\mu}\,Q^{\nu}}{M_{V}^{2}}\right)\,,$
(46)

which annihilates $\left.\Pi_{B}^{\mu\nu}\right|_{FDH}$. For the annihilation
rate, the rules are a bit ambiguous, as they could be read to mean that the
lepton spinors are four-dimensional but the vertex ($\gamma^{\mu}$) connecting
them to the loop part of the amplitude is $D_{s}$ dimensional. This would
bring $\left.\Pi_{B}^{\mu\nu}\right|_{FDH}$ into the calculation and lead to a
singular result at order $\alpha_{s}^{2}$. However, Rule $4$ could also be
read to mean that the vertex sandwiched between four-dimensional states is
also reduced to being four-dimensional.

Assuming this interpretation, I find that

$\begin{split}\Gamma^{{FDH}}_{V\to\ {\rm
hadrons}}=&\frac{{\alpha_{V}}\,M_{V}}{3}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,,\end{split}$
(47)

and

$\begin{split}\sigma^{{FDH}}_{e^{+}\,e^{-}\to\ {\rm
hadrons}}=&\frac{4\pi\,\alpha^{2}}{3\,Q^{2}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,\left\\{1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,C_{F}\,\frac{3}{4}\left[1+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}\,{\beta_{0}^{\overline{\rm{MS}}}}\,\ln\frac{\mu^{2}}{Q^{2}}\right]\right.\\\
&\hskip-45.0pt\left.+{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{2}\,\left[-C_{F}^{2}\,\frac{15}{32}+C_{F}\,C_{A}\,\left(\frac{133}{32}-\frac{11}{4}\zeta_{3}\right)+C_{F}\,N_{f}\,\left(-\frac{5}{8}+\frac{1}{2}\,\zeta_{3}\right)\right]+{\cal
O}\left({\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\right)\right\\}\,.\end{split}$
(48)

The results agree with one another, are correct through NLO and are finite
through NNLO. Unfortunately, the NNLO terms are not correct! Because the
discrepancy is finite, there remains the possibility that the conversion from
${\alpha_{s}^{\overline{\rm{FDH}}}}$ to ${\alpha_{s}^{\overline{\rm{MS}}}}$
given in Eq. (40) is incorrect, although this would contradict previous
results Kunszt et al. (1994); Bern et al. (2002). If this were the case, then
one would expect that the N3LO result would also be finite but incorrect. If,
instead, the finite discrepancy at NNLO is the result of a failure of the
renormalization program, the N3LO result should be singular.

##  VI Partial results at N3LO

Although first computed some time ago, the vacuum polarization at four loops
Gorishnii et al. (1988, 1991) remains a formidable calculation. It is only
necessary, however, to look at a small part of the calculation: the terms
proportional to the square of the number of fermion flavors, $N_{f}^{2}$. This
is fortunate for a couple of reasons: 1) there are only three four-loop
diagrams to be computed, see Fig. (3), (plus three more in the DRED scheme,
where the gluons are replaced by evanescent gluons); and 2) the contributions
from renormalization in the CDR and FDH schemes come only from the leading
term in the QCD $\beta$-function ($\beta_{0}$ and $\beta_{0}^{2}$). Thus, my
result will not depend on how the higher order terms of the $\beta$-function
are chosen in the FDH scheme.

Figure 3: Four-loop diagrams that contribute to the $N_{f}^{2}$ term at N3LO.

###  VI.1 The CDR scheme

In the CDR scheme, there are only three four-loop diagrams that need to be
calculated. The first two are simply iterated-bubble diagrams and are
essentially trivial. The third is slightly nontrivial, so I again use my
QGRAF-FORM-REDUZE suite of programs to address the problem. All of the four-
loop master integrals can be found in Ref. Baikov and Chetyrkin (2010). I find
the result of the four-loop calculation to be

$\begin{split}\Im\left[\left.\Pi^{(B)}_{\mu\nu}(Q)\right|_{{CDR}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}\\\
&\qquad\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,C_{F}\,N_{f}^{2}\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{121}{288}-\frac{1}{3}\zeta_{3}\right)+\frac{2777}{576}-\frac{3}{8}\zeta_{2}-\frac{19}{6}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\end{split}$
(49)

Renormalizing, I find

$\begin{split}\Im\left[\left.\Pi_{\mu\nu}(Q)\right|_{{CDR}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&=\frac{-Q^{2}\,g_{\mu\nu}+Q_{\mu}Q_{\nu}}{3}{\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\,C_{F}\,N_{f}^{2}\\\
&\times\left[\frac{151}{216}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{11}{48}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\end{split}$
(50)

Using this term to compute the $\alpha_{s}^{3}\,N_{f}^{2}$ contribution to the
decay rate and annihilation cross section as in Eqs. (14,17), I find the
result expected from Eqs. (LABEL:eqn:knownresult-3).

###  VI.2 The DRED scheme

In the DRED scheme, there are three extra four-loop diagrams to compute,
obtained by replacing gluon propagators with evanescent gluon propagators. I
find

$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\,\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\phantom{+}{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{13}{32}-\frac{1}{3}\zeta_{3}\right)+\frac{7847}{1728}-\frac{3}{8}\zeta_{2}-\frac{53}{18}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\\\
&\left.+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{{\varepsilon}}\frac{3}{64}-\frac{83}{128}\right]\right\\}\\\
\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{Ve}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\,\left\\{\vphantom{{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}}\right.\\\
&\phantom{+}{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{72\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\frac{73}{432}+\frac{3595}{2592}-\frac{1}{4}\zeta_{2}-\frac{1}{3}\zeta_{3}\right]\\\
&\left.+{\left(\frac{\alpha_{e}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{48\,{\varepsilon}^{2}}-\frac{1}{{\varepsilon}}\frac{11}{48}-\frac{155}{96}+\frac{3}{8}\zeta_{2}\right]\right\\}\end{split}$
(51)

Upon renormalizing according to Eq. (24) and converting the coupling to
${\alpha_{s}^{\overline{\rm{MS}}}}$, I obtain

$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-55.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{MS}}}}{\pi}\right)}^{3}\,\left[\frac{151}{216}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{11}{48}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{DRED}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\cal
O}({\varepsilon})\,.\end{split}$ (52)

As for the CDR scheme, this leads to the expected result for the decay rate
and annihilation cross section.

###  VI.3 The FDH scheme

In the FDH scheme, however, I find that

$\begin{split}\Im\left[\left.\Pi^{(B)}_{A}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\\\
&\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[\frac{1}{48\,{\varepsilon}^{2}}+\frac{1}{{\varepsilon}}\left(\frac{23}{64}-\frac{1}{3}\zeta_{3}\right)+\frac{13453}{3456}-\frac{3}{8}\zeta_{2}-\frac{53}{18}\zeta_{3}-\frac{1}{2}\zeta_{4}\right]\,,\\\
\Im\left[\left.\Pi^{(B)}_{B}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&={\alpha_{V}^{B}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\left(\frac{4\,\pi}{Q^{2}\,e^{\gamma_{E}}}\right)^{4\,{\varepsilon}}C_{F}\,N_{f}^{2}\\\
&\times{\left(\frac{\alpha_{s}^{B}}{\pi}\right)}^{3}\,\left[-\frac{1}{144\,{\varepsilon}^{2}}-\frac{1}{{\varepsilon}}\frac{13}{216}-\frac{295}{1296}+\frac{1}{8}\zeta_{2}-\frac{1}{3}\zeta_{3}\right]\,.\end{split}$
(53)

I renormalize according to

${\alpha_{s}^{B}}=\left(\frac{\mu^{2}\,e^{\gamma_{E}}}{4\,\pi}\right)^{\varepsilon}\,{\alpha_{s}^{\overline{\rm{FDH}}}}\,\left[1-{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}\frac{{\beta_{0}^{\overline{\rm{FDH}}}}}{{\varepsilon}}+{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{2}\left(\frac{{\beta_{0}^{\overline{\rm{FDH}}}}^{2}}{{\varepsilon}^{2}}-\frac{1}{2}\frac{{\beta_{1}^{\overline{\rm{FDH}}}}}{{\varepsilon}}\right)\right]\,,$
(54)

keeping only terms proportional to
${\alpha_{s}^{\overline{\rm{FDH}}}}^{3}\,N_{f}^{2}$. Such terms can only come
from the ${\beta_{0}^{\overline{\rm{FDH}}}}$ and
${\beta_{0}^{\overline{\rm{FDH}}}}^{2}$ terms, so any uncertainty about
${\beta_{1}^{\overline{\rm{FDH}}}}$ has no effect here. The renormalized
result is

$\begin{split}\Im\left[\left.\Pi_{A}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-75.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\,\left[-\frac{1}{192\,{\varepsilon}}+\frac{1843}{3456}-\frac{1}{24}\zeta_{2}-\frac{19}{36}\zeta_{3}+\left(\frac{3}{16}-\frac{1}{6}\zeta_{3}\right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{1}{48}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,,\\\
\Im\left[\left.\Pi_{B}(Q)\right|_{{FDH}}\right]_{\alpha_{s}^{3}\,N_{f}^{2}}&\\\
&\hskip-75.0pt={\alpha_{V}}\,N_{c}\,\sum_{f}\,Q_{f}^{2}\,C_{F}\,N_{f}^{2}{\left(\frac{\alpha_{s}^{\overline{\rm{FDH}}}}{\pi}\right)}^{3}\,\left[\frac{1}{144\,{\varepsilon}^{2}}-\frac{5}{432\,{\varepsilon}}-\frac{869}{2592}+\frac{1}{18}\zeta_{2}-\frac{5}{27}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{1}{36}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\right]\,.\end{split}$
(55)

The demand that external states be four-dimensional removes the $\Pi_{B}$
term, but there is also a pole in $\Pi_{A}$ and no finite renormalization to
put the result in terms of ${\alpha_{s}^{\overline{\rm{MS}}}}$ can remove it.
I must therefore conclude that the FDH scheme is not consistent with
unitarity.

##  VII Discussion

In this paper, I have performed a high-order calculation in each of three
regularization schemes: the conventional dimensional regularization (CDR)
scheme; the dimensional reduction (DRED) scheme; and the four-dimensional
helicity (FDH) scheme. Of these, the CDR scheme is by far the most widely
used, and was, in fact, used to compute the original results that I use as my
test basis. The FDH scheme has primarily been used to produce one-loop
helicity amplitudes, although it has been used in a few cases in two-loop
calculations and also as a supersymmetric regulator. The primary purpose of
this paper was to put the FDH scheme to a stringent test and determine its
reliability in a high-order calculation. The DRED scheme is primarily used as
a supersymmetric regulator and is quite cumbersome for nonsupersymmetric
calculations. It is, however, closely related to the FDH scheme and has been
demonstrated Jack et al. (1994a, b); Harlander et al. (2006b, a) to be
equivalent to the CDR scheme through four loops. A close comparison of the
details of the calculations in the FDH and DRED schemes helps to identify
where and when things go wrong with the former.

In the cases of the CDR and DRED schemes, I have reproduced the known result
for the hadronic decay width of a massive vector boson (or equivalently, the
$e^{+}e^{-}$ annihilation rate to hadrons) through NNLO, and a few terms at
N3LO. This represents computing the QCD corrections to the vacuum polarization
of the photon ($V$ boson) through three loops, with partial results at four
loops. In addition, I have reproduced the renormalization parameters of QCD
($\beta$-function(s), mass anomalous dimension) through three-loop order. This
establishes that I have theoretical control over all of the needed
calculations through three-loop order. In order to obtain the partial N3LO
result in the DRED scheme, I also needed the three-loop QCD corrections to the
$\beta$-function of the evanescent photon ($V$ boson).

The calculation of the $V$ boson decay rate provides another instance of the
equivalence the CDR and DRED schemes at the four-loop level Harlander et al.
(2006a). The ability to obtain the correct result using the DRED scheme
required a delicate balance of the many extra couplings and their
renormalization effects upon one another. Indeed, given the complexity needed
to make the DRED scheme work, it seems that there should be little surprise
that the FDH scheme, with its greater simplicity, should fail.

Perhaps, it is worth considering how it is that the FDH scheme has been used
successfully in so many calculations. Its most common use has been in the
construction of one-loop scattering amplitudes via unitarity cuts, using four-
dimensional helicity amplitudes as the primary building blocks. Thus, it is
natural that it restricts observed (external) states to be four-dimensional.
Because the FDH scheme defines that $D_{s}>D_{m}>4$, this restriction excludes
evanescent fields from appearing as external states. This is very important
because, as one can see from comparing Eqs. (30) and (55), terms involving
external evanescent states are the most dangerous. Even though it does not
renormalize evanescent states and couplings properly the FDH is able to get
the nonevanescent part of the vacuum polarization tensor correct at NLO, while
the evanescent part is ready to contribute a finite error at NLO. Because the
DRED scheme defines $4>D_{m}$, the evanescent states are parts of the
classical four-dimensional states. It would not seem natural to exclude them
from appearing as external states. Instead, they are handled through the
renormalization program so that their effects are removed from physical
scattering amplitudes. In the FDH scheme, the evanescent states are instead
additions to the four-dimensional states (as are the extra degrees of freedom
that come from regularizing momentum integrals) and there is no barrier to
excluding them as observed states.

In an FDH scheme calculation, a tree-level term is strictly four-dimensional
and is free from evanescent contributions. (Depending on interpretation, this
may be a stronger condition than is given in the rules of Ref Bern et al.
(2002), but it is the actual condition imposed if one defines the tree-level
amplitude as being a four-dimensional helicity amplitude.) Because evanescent
terms are absent at tree-level, they cannot generate ultraviolet poles at one
loop. Even if one were to renormalize them properly, as in the DRED scheme,
there would be nowhere to make the counter-term insertion! In fact, the one-
loop contributions are not even finite, as the counting over the number of
states ($2{\varepsilon}$) makes the result of order ${\varepsilon}$. This is
clearly illustrated in Eq. (28). Neither $\alpha_{s}$, nor $\alpha_{e}$ appear
at LO. Therefore, the contributions at NLO are finite for $\alpha_{s}$ and of
order ${\varepsilon}$ (because of the counting over the number of states) for
$\alpha_{e}$. In more complicated QCD calculations, $\alpha_{s}$ will appear
at LO and will therefore contribute an ultraviolet pole at one-loop, which
will be removed by renormalization. $\alpha_{e}$, however, will still make its
first appearance at NLO and that contribution will be of order
${\varepsilon}$. Thus, one can expect that the FDH scheme, used as above,
should be reliable for computing NLO corrections through finite order
(${\varepsilon}^{0}$). The error from improperly identifying evanescent
quantities should be of order ${\varepsilon}$. At NNLO and beyond however, the
failure to properly identify and renormalize the evanescent parameters leads
to incorrect results and the violation of unitarity.

So, as suggested Bern et al. (2002), one of the FDH scheme’s most important
assets is that it defines $D_{s}>D_{m}>4$. This feature is also the scheme’s
undoing, though not of necessity. Because the effects of external evanescent
states can be removed (or indeed never seen) by imposing a four-dimensionality
restriction, and because the effects of internal evanescent states therefore
contribute at order ${\varepsilon}$ at one loop, it appears that one can
simply ignore the distinction between gauge and evanescent terms. In contrast,
because the DRED scheme must deal with external evanescent terms from the
beginning, its advocates were forced to develop a successful renormalization
program Jack et al. (1994a, b). Extensive testing Jack et al. (1994a, b);
Harlander et al. (2006b, a) has shown that this program works to at least the
fourth order and that it handles the effects of both internal and external
evanescent contributions. As I remarked earlier, calculations in the DRED and
FDH schemes are term-by-term identical, except for the identification of the
couplings and propagating states. Thus, one could make the FDH scheme a
unitary regularization scheme for nonsupersymmetric calculations by
recognizing the distinction between gauge and evanescent terms and adopting
the DRED scheme’s renormalization program. This would, of course, do away with
any notion of the FDH scheme being simple, but it would at least be correct.
The FDH scheme would still be distinguished from the DRED scheme by the fact
that $D_{s}>D_{m}>4$, which facilitates helicity amplitude calculations and,
in chiral theories, improves its situation with regard to $\gamma_{5}$ and the
Levi-Civita tensor Siegel (1980); Stockinger (2005). Furthermore, with a valid
renormalization program, the requirement of four-dimensional observed states
could be made optional. This would lead to two linked, slightly different,
schemes, just like the HV and CDR schemes. This suggestion has already been
made by Signer and Stöckinger Signer and Stockinger (2009) who in fact define
their version of the DRED scheme to have precisely the $D_{s}>D_{m}>4$
hierarchy of the FDH scheme.

Thus, in conclusion, the CDR and DRED schemes are correct and equivalent ways
of performing QCD calculations through N3LO. The FDH scheme, however, has been
shown to be incorrect and to violate unitarity beyond NLO when applied to
nonsupersymmetric theories. It must therefore be viewed as a shortcut for
performing NLO calculations and should only be used for such calculations with
great caution.

#### Acknowledgments:

This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.

##  Appendix A Renormalization parameters for the CDR scheme

To three-loop order, I find the coefficients of the $\beta$-function to be

$\begin{split}{\beta_{0}^{\overline{\rm{MS}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,,\qquad\qquad{\beta_{1}^{\overline{\rm{MS}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,,\\\
{\beta_{2}^{\overline{\rm{MS}}}}&=\frac{2857}{3456}C_{A}^{3}-\frac{1415}{3456}C_{A}^{2}\,N_{f}-\frac{205}{1152}C_{A}\,C_{F}\,N_{f}+\frac{1}{64}C_{F}^{2}\,N_{f}+\frac{79}{3456}C_{A}\,N_{f}^{2}+\frac{11}{576}C_{F}\,N_{f}^{2}\,,\end{split}$
(56)

while the coefficients of the mass anomalous dimension are

$\begin{split}{\gamma_{0}^{\overline{\rm{MS}}}}&=\frac{3}{4}C_{F}\,,\hskip
80.0pt{\gamma_{1}^{\overline{\rm{MS}}}}=\frac{3}{32}C_{F}^{2}+\frac{97}{96}C_{F}\,C_{A}-\frac{5}{48}C_{F}\,N_{f}\,,\\\
{\gamma_{2}^{\overline{\rm{MS}}}}&=\frac{129}{128}C_{F}^{3}-\frac{129}{256}C_{F}^{2}\,C_{A}+\frac{11413}{6912}C_{F}\,C_{A}^{2}-\left(\frac{23}{64}-\frac{3}{8}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{139}{864}+\frac{3}{8}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\frac{35}{1728}C_{F}\,N_{f}^{2}\,,\end{split}$
(57)

in agreement with known results Tarasov et al. (1980); Larin and Vermaseren
(1993); Chetyrkin (1997); Vermaseren et al. (1997).

##  Appendix B Renormalization parameters for the DRED scheme

The coefficients of the QCD $\beta$-function,
${\beta^{\overline{\rm{DR}}}}({\alpha_{s}^{\overline{\rm{DR}}}})$ through
three loops are:

$\begin{split}{\beta_{20}^{\overline{\rm{DR}}}}&=\frac{11}{12}C_{A}-\frac{1}{6}N_{f}\,,\hskip
60.0pt{\beta_{30}^{\overline{\rm{DR}}}}=\frac{17}{24}C_{A}^{2}-\frac{5}{24}C_{A}\,N_{f}-\frac{1}{8}C_{F}\,N_{f}\,,\\\
{\beta_{40}^{\overline{\rm{DR}}}}&=\frac{3115}{3456}C_{A}^{3}-\frac{1439}{3456}C_{A}^{2}\,N_{f}-\frac{193}{1152}C_{A}\,C_{F}\,N_{f}+\frac{1}{64}C_{F}^{2}\,N_{f}+\frac{79}{3456}C_{A}\,N_{f}^{2}+\frac{11}{576}C_{F}\,N_{f}^{2}\,,\\\
{\beta_{31}^{\overline{\rm{DR}}}}&=-\frac{1}{16}C_{F}\,N_{f}\left(\frac{3}{2}C_{F}\right)\,,\qquad\quad{\beta_{22}^{\overline{\rm{DR}}}}=-\frac{1}{16}C_{F}\,N_{f}\left(\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}\right)\,,\end{split}$
(58)

where the notation is that

${\beta^{\overline{\rm{DR}}}}({\alpha_{s}^{\overline{\rm{DR}}}})=-{\varepsilon}\frac{{\alpha_{s}^{\overline{\rm{DR}}}}}{\pi}-\sum_{i,j,k,l,m}\,{\beta_{ijklm}^{\overline{\rm{DR}}}}\,{\left(\frac{\alpha_{s}^{\overline{\rm{DR}}}}{\pi}\right)}^{i}\,{\left(\frac{\alpha_{e}^{\overline{\rm{DR}}}}{\pi}\right)}^{j}\,{\left(\frac{\eta_{1}^{{\overline{\rm
DR}}}}{\pi}\right)}^{k}\,{\left(\frac{\eta_{2}^{{\overline{\rm
DR}}}}{\pi}\right)}^{l}\,{\left(\frac{\eta_{3}^{{\overline{\rm
DR}}}}{\pi}\right)}^{m}\,.$ (59)

The last three indices of ${\beta_{ijklm}^{\overline{\rm{DR}}}}$ are omitted
when they are all equal to $0$.

The $\beta$-function of evanescent QCD coupling,
${\beta_{e,\,\,}^{\overline{\rm{DR}}}}({\alpha_{e}^{\overline{\rm{DR}}}})$ is

$\begin{split}{\beta_{e,\,0\,2}^{\overline{\rm{DR}}}}&=\frac{1}{2}C_{A}-C_{F}-\frac{1}{4}N_{f}\,,\qquad{\beta_{e,\,1\,1}^{\overline{\rm{DR}}}}=\frac{3}{2}C_{F}\,,\\\\[5.0pt]
{\beta_{e,\,0\,3}^{\overline{\rm{DR}}}}&=\frac{3}{8}C_{A}^{2}-\frac{5}{4}C_{A}\,C_{F}+C_{F}^{2}-\frac{3}{16}C_{A}\,N_{f}+\frac{3}{8}C_{F}\,N_{f}\,,\qquad{\beta_{e,\,1\,2}^{\overline{\rm{DR}}}}=-\frac{3}{8}C_{A}^{2}+\frac{5}{2}C_{A}\,C_{F}-\frac{11}{4}C_{F}^{2}-\frac{5}{16}C_{F}\,N_{f}\,,\\\
{\beta_{e,\,2\,1}^{\overline{\rm{DR}}}}&=-\frac{7}{64}C_{A}^{2}+\frac{55}{48}C_{A}\,C_{F}+\frac{3}{16}C_{F}^{2}+\frac{1}{16}C_{A}\,N_{f}-\frac{5}{24}C_{F}\,N_{f}\\\\[5.0pt]
{\beta_{e,\,0\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{8}\qquad{\beta_{e,\,0\,2010}^{\overline{\rm{DR}}}}=\frac{5}{4}\qquad{\beta_{e,\,0\,2001}^{\overline{\rm{DR}}}}=\frac{3}{4}\\\
{\beta_{e,\,0\,1200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{27}{64}\qquad{\beta_{e,\,0\,1020}^{\overline{\rm{DR}}}}=-\frac{15}{4}\qquad{\beta_{e,\,0\,1002}^{\overline{\rm{DR}}}}=\frac{21}{32}\qquad{\beta_{e,\,0\,1101}^{\overline{\rm{DR}}}}=-\frac{9}{16}\\\
{\beta_{e,\,0\,4}^{\overline{\rm{DR}}}}&=-\left(\frac{7}{4}+\frac{9}{4}\zeta_{3}\right)\,C_{F}^{3}+\left(\frac{17}{8}+\frac{15}{2}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}-\left(\frac{3}{4}+\frac{69}{16}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\left(\frac{1}{16}+\frac{9}{16}\zeta_{3}\right)\,C_{A}^{3}\\\
&+\left(\frac{13}{32}-\frac{33}{16}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{1}{32}+\frac{51}{32}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\left(\frac{21}{128}+\frac{9}{32}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}-\left(\frac{1}{128}C_{F}-\frac{7}{256}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,1\,3}^{\overline{\rm{DR}}}}&=\left(\frac{13}{2}-3\,\zeta_{3}\right)\,C_{F}^{3}-\left(10-6\,\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{133}{32}-\frac{15}{4}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}-\left(\frac{25}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{A}^{3}\\\
&+\left(\frac{13}{16}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\frac{9}{8}\left(1-\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}+\left(\frac{7}{32}-\frac{3}{8}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}+\frac{3}{64}\,C_{A}\,N_{f}^{2}\\\
{\beta_{e,\,2\,2}^{\overline{\rm{DR}}}}&=-\left(\frac{139}{64}-\frac{27}{4}\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{793}{128}+18\,\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{1587}{256}+\frac{207}{16}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}-\left(\frac{427}{512}+\frac{45}{16}\zeta_{3}\right)\,C_{A}^{3}\\\
&-\left(\frac{569}{256}-\frac{99}{16}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{31}{16}-\frac{171}{32}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\left(\frac{871}{1024}-\frac{45}{32}\zeta_{3}\right)\,C_{A}^{2}\,N_{f}+\left(\frac{1}{16}C_{F}-\frac{1}{256}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,3\,1}^{\overline{\rm{DR}}}}&=\frac{129}{64}C_{F}^{3}-\frac{457}{128}C_{F}^{2}\,C_{A}+\frac{11875}{3456}C_{F}\,C_{A}^{2}-\frac{3073}{4608}C_{A}^{3}\\\
&-\left(\frac{23}{32}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{157}{1728}+\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}+\frac{463}{2304}C_{A}^{2}\,N_{f}-\left(\frac{35}{864}C_{F}+\frac{5}{576}C_{A}\right)\,N_{f}^{2}\\\
{\beta_{e,\,0\,3100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{9}{64}+\frac{243}{128}N_{f}\qquad{\beta_{e,\,0\,3010}^{\overline{\rm{DR}}}}=\frac{5}{8}-\frac{45}{64}N_{f}\qquad{\beta_{e,\,0\,3001}^{\overline{\rm{DR}}}}=\frac{3}{32}-\frac{81}{64}N_{f}\\\
{\beta_{e,\,1\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{219}{16}\qquad{\beta_{e,\,1\,2010}^{\overline{\rm{DR}}}}=\frac{145}{48}\qquad{\beta_{e,\,1\,2001}^{\overline{\rm{DR}}}}=\frac{73}{8}\\\
{\beta_{e,\,2\,1100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{1125}{1024}\qquad{\beta_{e,\,2\,1010}^{\overline{\rm{DR}}}}=\frac{105}{128}\qquad{\beta_{e,\,2\,1001}^{\overline{\rm{DR}}}}=\frac{615}{512}\\\
{\beta_{e,\,0\,2200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{1413}{512}-\frac{729}{1024}N_{f}\qquad{\beta_{e,\,0\,2020}^{\overline{\rm{DR}}}}=-\frac{115}{32}+\frac{135}{64}N_{f}\qquad{\beta_{e,\,0\,2002}^{\overline{\rm{DR}}}}=-\frac{161}{256}-\frac{567}{512}N_{f}\\\
{\beta_{e,\,0\,2110}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{75}{8}\qquad{\beta_{e,\,0\,2101}^{\overline{\rm{DR}}}}=-\frac{471}{128}+\frac{243}{256}N_{f}\qquad{\beta_{e,\,0\,2011}^{\overline{\rm{DR}}}}=-\frac{85}{8}\\\
{\beta_{e,\,0\,1300}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{1701}{1024}\qquad{\beta_{e,\,0\,1210}^{\overline{\rm{DR}}}}=-\frac{405}{128}\qquad{\beta_{e,\,0\,1201}^{\overline{\rm{DR}}}}=\frac{1701}{512}\\\
{\beta_{e,\,0\,1120}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{135}{32}\qquad{\beta_{e,\,0\,1111}^{\overline{\rm{DR}}}}=\frac{135}{16}\qquad{\beta_{e,\,0\,1102}^{\overline{\rm{DR}}}}=-\frac{81}{128}\\\
{\beta_{e,\,0\,1021}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{315}{32}\qquad{\beta_{e,\,0\,1012}^{\overline{\rm{DR}}}}=-\frac{315}{32}\qquad{\beta_{e,\,0\,1003}^{\overline{\rm{DR}}}}=\frac{63}{128}\qquad\end{split}$
(60)

The mass anomalous dimension in the DRED scheme is

$\begin{split}{\gamma_{10}^{\overline{\rm{DR}}}}&=\frac{3}{4}C_{F}\\\\[5.0pt]
{\gamma_{20}^{\overline{\rm{DR}}}}&=\frac{3}{32}C_{F}^{2}+\frac{91}{96}C_{A}\,C_{F}-\frac{5}{48}C_{F}\,N_{f}\qquad{\gamma_{11}^{\overline{\rm{DR}}}}=-\frac{3}{8}C_{F}^{2}\qquad{\gamma_{02}^{\overline{\rm{DR}}}}=\frac{1}{4}C_{F}^{2}-\frac{1}{8}C_{A}\,C_{F}+\frac{1}{16}C_{F}\,N_{f}\\\\[5.0pt]
{\gamma_{30}^{\overline{\rm{DR}}}}&=\frac{129}{128}C_{F}^{3}-\frac{133}{256}C_{F}^{2}\,C_{A}+\frac{10255}{6912}C_{F}\,C_{A}^{2}-\left(\frac{23}{64}-\frac{3}{8}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}-\left(\frac{281}{1728}+\frac{3}{8}\zeta_{3}\right)\,C_{A}\,C_{F}\,N_{f}-\frac{35}{1728}C_{F}\,N_{f}^{2}\\\
{\gamma_{21}^{\overline{\rm{DR}}}}&=-\frac{27}{64}C_{F}^{3}-\frac{21}{32}C_{F}^{2}\,C_{A}-\frac{15}{256}C_{F}\,C_{A}^{2}+\frac{9}{64}C_{F}^{2}\,N_{f}\\\
{\gamma_{12}^{\overline{\rm{DR}}}}&=\frac{9}{8}C_{F}^{3}-\frac{21}{32}C_{F}^{2}\,C_{A}+\frac{3}{64}C_{F}\,C_{A}^{2}+\frac{3}{128}C_{F}\,C_{A}\,N_{f}+\frac{3}{16}C_{F}^{2}\,N_{f}\\\
{\gamma_{03}^{\overline{\rm{DR}}}}&=-\frac{3}{8}C_{F}^{3}+\frac{3}{8}C_{F}^{2}\,C_{A}-\frac{3}{32}C_{F}\,C_{A}^{2}+\frac{1}{16}C_{F}\,C_{A}\,N_{f}-\frac{5}{32}C_{F}^{2}\,N_{f}-\frac{1}{128}C_{F}\,N_{f}^{2}\\\\[5.0pt]
{\gamma_{02100}^{\overline{\rm{DR}}}}\hskip-10.0pt&\hskip
10.0pt=\frac{3}{8}\qquad{\gamma_{02010}^{\overline{\rm{DR}}}}=-\frac{5}{12}\qquad{\gamma_{02001}^{\overline{\rm{DR}}}}=-\frac{1}{4}\qquad\\\
{\gamma_{01200}^{\overline{\rm{DR}}}}\hskip-10.0pt&\hskip
10.0pt=-\frac{9}{64}\qquad{\gamma_{01101}^{\overline{\rm{DR}}}}=\frac{3}{16}\qquad{\gamma_{01020}^{\overline{\rm{DR}}}}=\frac{5}{4}\qquad{\gamma_{01002}^{\overline{\rm{DR}}}}=-\frac{7}{32}\end{split}$
(61)

The above results for ${\beta^{\overline{\rm{DR}}}}$,
${\beta_{e,\,\,}^{\overline{\rm{DR}}}}$ and ${\gamma^{\overline{\rm{DR}}}}$
all agree with the results of Refs. Harlander et al. (2006b, a)

The QCD contributions to the $\beta$-function of the evanescent part of a non-
QCD gauge coupling is a new result. I find

$\begin{split}{\beta_{Ve,\,1\,0}^{\overline{\rm{DR}}}}&=\frac{3}{2}C_{F}\qquad{\beta_{Ve,\,0\,1}^{\overline{\rm{DR}}}}=-C_{F}\\\\[5.0pt]
{\beta_{Ve,\,2\,0}^{\overline{\rm{DR}}}}&=\frac{3}{16}C_{F}^{2}+\frac{91}{48}C_{F}\,C_{A}-\frac{5}{24}C_{F}\,N_{f}\qquad{\beta_{Ve,\,1\,1}^{\overline{\rm{DR}}}}=-\frac{11}{4}C_{F}^{2}-\frac{3}{4}C_{F}\,C_{A}\qquad{\beta_{Ve,\,0\,2}^{\overline{\rm{DR}}}}=C_{F}^{2}+\frac{3}{8}C_{F}\,N_{f}\\\\[5.0pt]
{\beta_{Ve,\,3\,0}^{\overline{\rm{DR}}}}&=\frac{129}{64}C_{F}^{3}-\frac{133}{128}C_{F}^{2}\,C_{A}-\left(\frac{23}{32}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\frac{10255}{3456}C_{F}\,C_{A}^{2}-\left(\frac{281}{864}+\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}-\frac{35}{864}C_{F}\,N_{f}^{2}\\\
{\beta_{Ve,\,2\,1}^{\overline{\rm{DR}}}}&=-\left(\frac{139}{64}-\frac{27}{4}\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{331}{64}+\frac{81}{8}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\frac{11}{16}C_{F}^{2}\,N_{f}-\left(\frac{195}{256}-\frac{27}{8}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\frac{5}{64}C_{F}\,C_{A}\,N_{f}\\\
{\beta_{Ve,\,1\,2}^{\overline{\rm{DR}}}}&=\left(\frac{13}{2}-3\,\zeta_{3}\right)\,C_{F}^{3}-\left(\frac{7}{8}-\frac{9}{2}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}+\left(\frac{63}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{F}^{2}\,N_{f}+\left(\frac{7}{16}-\frac{3}{2}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}\\\
&-\left(\frac{3}{64}-\frac{3}{4}\zeta_{3}\right)\,C_{F}\,C_{A}\,N_{f}\\\\[5.0pt]
{\beta_{Ve,\,0\,3}^{\overline{\rm{DR}}}}&=-\left(\frac{7}{4}+\frac{9}{4}\zeta_{3}\right)\,C_{F}^{3}+\left(\frac{1}{8}+\frac{27}{8}\zeta_{3}\right)\,C_{F}^{2}\,C_{A}-\frac{27}{32}C_{F}^{2}\,N_{f}+\left(\frac{1}{16}-\frac{9}{8}\zeta_{3}\right)\,C_{F}\,C_{A}^{2}+\frac{3}{64}C_{F}\,C_{A}\,N_{f}+\frac{3}{64}C_{F}\,N_{f}^{2}\\\
{\beta_{Ve,\,0\,2100}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=\frac{3}{8}\qquad{\beta_{Ve,\,0\,2010}^{\overline{\rm{DR}}}}=-\frac{25}{6}\qquad{\beta_{Ve,\,0\,2001}^{\overline{\rm{DR}}}}=-\frac{1}{4}\\\
{\beta_{Ve,\,0\,1200}^{\overline{\rm{DR}}}}\hskip-13.0pt&\hskip
13.0pt=-\frac{63}{64}\qquad{\beta_{Ve,\,0\,1101}^{\overline{\rm{DR}}}}=\frac{21}{16}\qquad{\beta_{Ve,\,0\,1020}^{\overline{\rm{DR}}}}=\frac{65}{4}\qquad{\beta_{Ve,\,0\,1002}^{\overline{\rm{DR}}}}=-\frac{49}{32}\end{split}$
(62)

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